Higher Order Perturbative Calculations in Strong
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UNIVERSITY OF CYPRUS
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PHYSICS DEPARTMENT
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Higher Order
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Perturbative Calculations in
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Strong Interaction Physics with Improved Discretized Actions
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for Quarks and Gluons PhD DISSERTATION
MARTHA CONSTANTINOU
APRIL 2008
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UNIVERSITY OF CYPRUS
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PHYSICS DEPARTMENT
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PhD Dissertation
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of
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Martha Constantinou
Advisor
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Haralambos Panagopoulos
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Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Physics Department at the University of Cyprus
c 2008 by Martha Constantinou Copyright All rights reserved
April 2008
PhD Candidate: Martha Constantinou
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Title of Dissertation: Higher order perturbative calculations in Strong Interaction Physics with improved discretized actions for quarks and gluons
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Dissertation Committee:
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Prof. Haralambos Panagopoulos, Research Supervisor
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Prof. Constantia Alexandrou
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Prof. Giorgios Archontis, Committee Chairman Prof. Athanassios Nicolaides
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Prof. Ettore Vicari
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Defended on April 14, 2008
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Dedicated to Ventsi who has been an inspiration to me
Acknowledgments First of all, I would like to express my sincerest gratitude to my advisor, Professor Haris
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Panagopoulos, whose guidance made this Thesis possible. I am grateful for all interesting computation that he proposed to me and trusted me for their completion. I thank him
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for his infinite patience and his continuous encouragement throughout my PhD studies. While working with him, I surely benefited from his large experience and knowledge. His perfectionism was a motivation for improving myself as a researcher and developing my
autonomy. It was certainly a privilege to be his student. Secondly, I want to thank Apostolos Skouroupathis and Fotos Stylianou for our collab-
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oration in many of the computations presented here. Special thanks go to Apostolos for being not only a good collaborator but a true friend. He was always available for interesting conversations and to hear my thoughts and concerns about our projects. I should
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also thank my office mates Andreas Christou, Giannis Koutsou, Savvas Polydorides and Phanourios Tamamis for providing a pleasant and enjoyable working environment during my graduate studies. I do not have enough words to thank my parents for their unconditional love and everlasting support. I am grateful to them for being close to me, for emphasizing on the
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importance of education, and insisting at the same time that there are more in life than work. Most of all, I owe them a huge thank you for creating the perfect family environment that kept me balanced at many difficult times. So mom and dad, I take this chance to let
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you know how much I love you and that you have been row models for me. A very special acknowledgment goes to Ventsi Ivanoff, for bringing joy to my life and
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for his spiritual support. His unique way of dealing with difficulties, helped me be stronger when facing one. His productive criticism, even though I often complained about it, has gradually improved me. I should thank him for his patience and forbearance whilst spending hundreds of hours working on this Thesis. Ventsi, this work is dedicated to you, because I consider it as a success of the both of us. I also feel the need to thank those who contributed in different ways to the realization of this work: ⋆ Sofia Papanastasiou who encouraged me to join the Lattice QCD community. ⋆ Maria Christoforou for being a good friend with genuine interest on the progress of my
work. ⋆ Yiota Andreou with her positive way of thinking that was transferring to me as well. Our brief lunch breaks were extremely refreshing for me. ⋆ Chrysanthi Demetriou and Zoe Demetriou for the fun time we had during their graduate studies the Physics. ⋆ My younger sisters Sotia and Maria-Mikaella for their endless love and care, that really
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contributed to who I am. ⋆ Stavroulla Andreou who I accidentally met a few years ago, and since then she has been next to me, supporting me through rough times. Most of all, she is a good listener
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whenever I need it. ⋆ All those of you, who have been by my side during my graduate studies. Your support
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is definitely appreciated. Dear friends, you will always have a special place in my heart.
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Last but not least, I thank the Research Promotion Foundation of Cyprus for financial support over the last two and a half years.
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Abstract
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In this Thesis we address a number of perturbative calculations in Quantum Chromodynamics, formulated on the lattice. We employ a variety of improved fermion and gauge field actions, which are currently used in numerical simulations. The calculations that we present are the following:
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• The evaluation of the relation between the bare coupling constant g0 and the renormalized one in the MS scheme, gMS . This computation is performed to 2 loops in perturbation theory, employing the standard Wilson action for gluons and the overlap action for
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fermions. We also derive the 3-loop coefficient of the bare β-function (βL (g0 )) and provide the recipe for extracting the ratio of energy scales, ΛL /ΛMS . Moreover, we generalize our results for fermions in an arbitrary representation of the gauge group SU(N).
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• We extend a systematic improvement method of perturbation theory for gauge fields on the lattice, to encompass all possible gluon actions made of closed Wilson loops. The
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improvement procedure entails resummation of an infinite, gauge invariant class of Feynman diagrams. Two different applications are presented: The additive renormalization of fermion masses, and the multiplicative renormalization ZV (ZA ) of the vector (axial) cur-
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rent. In many cases where nonperturbative estimates of renormalization functions are also available for comparison, the agreement with improved perturbative results is significantly
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better as compared to results from bare perturbation theory. • We study the critical value of the hopping parameter, κc , up to 2 loops in perturbation theory. This quantity is a typical case of a vacuum expectation value resulting in an additive
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renormalization. The clover improved action is employed for fermions and the Symanzik improved action for gluons. In order to compare our results to nonperturbative evaluations of κc coming from Monte Carlo simulations, we employ our improved perturbation theory method for improved actions. • An ongoing project regards the improvement of the fermion propagator and quark operators, to second order in the lattice spacing a, in 1-loop perturbation theory. The
1
Abstract
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computations are performed using clover fermions and Symanzik improved gluons. Our calculation has been carried out in a general covariant gauge. The higher order terms allow us to specify the corrections which must be applied to the quark operators, in order for them to be O(a2 ) improved. Our results are applicable also to the case of twisted
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mass fermions, which are currently being studied intensely by a number of collaborations worldwide.
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Per lhyh Sta pla sia ti paroÔsa Diatrib pragmatopoioÔntai diataraktiko upologismo sthn
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Kbantik Qrwmodunamik , sto formalismì tou Plègmato . Efarmìzoume beltiwmène fermionikè kai gklouonikè drsei , oi opo e qrhsimopoioÔntai eurèw sti arijmhtikè prosomoi¸sei . Oi upologismo pou parousizontai e nai oi akìloujoi:
Exagwg th sqèsh metaxÔ th gumn (bare) stajer sÔzeuxh
nakanonikopoihmènh (renormalized) sto sq ma
an
•
MS, gMS .
kai th epa-
H diexagwg tou upologismoÔ
g netai sthn txh diìrjwsh dÔo brìgqwn, me th qr sh fermion wn
overlap
kai gklouon wn
Ta apotelèsmat ma odhgoÔn ston kajorismì tou suntelest tri¸n brìgqwn
th gumn
β
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Wilson.
g0
βL (g0 ), kai parèqoume th suntag gia thn eÔresh tou lìgou th ΛL /ΛMS. Epiplèon, genikeÔoume ta apotelèsmat ma gia fermiìnia
sunrthsh ,
energeiak kl maka ,
SU(N).
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se opoiad pote anaparstash th omda bajm do
• Epèktash mia susthmatik mèjodou belt wsh th jewr a diataraq¸n gia ped a baj-
m do sto plègma, oÔtw ¸ste na e nai efarmìsimh gia opoiad pote drsh pou apotele tai
Wilson. H diadikas a belt wsh sunepgetai thn jroish mia olìdiagramtwn Feynman, ta opo a e nai anallo wta upì metasqhmatismoÔ
klhrh kathgor a
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apì kleistoÔ brìgqou
bajm do . DÔo diaforetikè efarmogè parousizontai: H prosjetik epanakanonikopo hsh
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th kr simh fermionik mza
kai h pollaplasiastik epanakanonikopo hsh
ZV (ZA )
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tou dianusmatikoÔ (axonikoÔ) reÔmato . Se arketè peript¸sei ìpou up rqan ektim sei twn stajer¸n epanakanonikopo hsh apì arijmhtikè prosomoi¸sei , h sumfwn a tou me
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ta dik ma beltiwmèna apotelèsmata e nai emfan¸ kalÔterh se sqèsh me ta apotelèsmata th gumn jewr a .
• Melèth th kr simh tim th paramètrou hopping, κc , mèqri dÔo brìgqou sth jewr a
diataraq¸n. H en lìgw posìthta e nai tupik per ptwsh anamenìmenh tim kenoÔ pou odh-
ge se prosjetik epanakanonikopo hsh. Gia ta fermiìnia epilèqjhke h beltiwmènh drsh
clover,
en¸ gia ta gklouìnia h beltiwmènh drsh
telesmtwn ma gia to
Symanzik.
Gia lìgou sÔgkrish twn apo-
κc me ta ant stoiqa twn prosomoi¸sewn Monte Carlo, qrhsimopoioÔme
3
Per lhyh
4
th mèjodo belt wsh th jewr a diataraq¸n pou anaptÔxame gia beltiwmène drsei .
•
H teleuta a ergas a ma anafèretai se èna upologismì o opo o br sketai se exèli-
xh kai afor th belt wsh tou fermionikoÔ diadìth, kaj¸ kai fermionik¸n telest¸n mèqri 2η txh th stajer plègmato
a
kai se jewr a diataraq¸n enì brìgqou. Oi upologi-
smo diexgontai me qr sh twn drsewn
clover
kai
Symanzik
kai pragmatopoioÔntai se mia
genik sunallo wth bajm da. Oi uyhlìterh txh ìroi ma epitrèpoun na kajor soume ti
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diorj¸sei pou prèpei na efarmostoÔn stou fermionikoÔ telestè , oÔtw ¸ste na beltiw-
O(a2 ). Ta apotelèsmat ma e nai efarmìsima kai sthn per ptwsh twn fermion wn twisted mass, pou qrhsimopoioÔntai entatik apì arketè ereunhtikè omde pagkosm w .
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Contents 9
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List of figures List of tables
12 12 16
2 The Wilson formulation and O(a) improvements 2.1 Standard Wilson quarks and gluons . . . . . . . . . . . . . . . . . . . . . .
21 21
O(a) improved actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 The clover fermion action . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 The Symanzik improved gluon action . . . . . . . . . . . . . . . . .
26 26 29
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Introduction to the Standard Model . . . . . . . . . . . . . . . . . . . . . . Perturbative calculations using improved actions . . . . . . . . . . . . . . .
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1.1 1.2
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1 Introduction
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3 O(a) improved overlap action
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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Nielsen-Ninomiya No-Go theorem . . . . . . . . . . . . . . . . . . . .
3.3 3.4 3.5
Locality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Ginsparg-Wilson relation . . . . . . . . . . . . . . . . . . . . . . . . . Overlap action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.1 3.2
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4 Twisted mass action
32 33
4.1 4.2 4.3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The lattice twisted mass action for degenerate quarks . . . . . . . . . . . . Calculations with twisted mass QCD . . . . . . . . . . . . . . . . . . . . .
44 45 48
4.4 4.5
Twisted mass QCD for nondegenerate quarks . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50 51
5
CONTENTS
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5 The background field formalism 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55 55
Background fields in the continuum theory . . . . . . . . . . . . . . . . . . The lattice background field method . . . . . . . . . . . . . . . . . . . . .
56 58
5.4
Vertices of the overlap action in the background field method . . . . . . . .
61
6 The running coupling and the β-function 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Renormalization group equation and β-function . . . . . . . . . . . . . . .
65 65 67
The step scaling function . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
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6.3
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5.2 5.3
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7 QCD with overlap fermions: Running coupling and the 3-loop β-function 75 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 7.2 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Description of the calculation . . . . . . . . . . . . . . . . . . . . . . . . . Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83 87
7.5
Generalization to an arbitrary representation . . . . . . . . . . . . . . . . . 7.5.1 The strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Adjoint representation . . . . . . . . . . . . . . . . . . . . . . . . .
93 93 96
7.6
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7.3 7.4
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The method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 8.2.1 Dressing the propagator . . . . . . . . . . . . . . . . . . . . . . . . 107 8.2.2 Dressing vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
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8.2
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8 Improved perturbation theory for improved lattice actions 105 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.4 8.5
Dressing QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
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8.3
8.2.3 Numerical values of improved coefficients . . . . . . . . . . . . . . . 112 8.2.4 The improvement procedure in a nutshell . . . . . . . . . . . . . . . 119 Application: 1-loop renormalization of fermionic currents . . . . . . . . . . 120
9 Two-loop additive mass renormalization with clover fermions and Symanzik improved gluons 125 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 9.2 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . 127
CONTENTS
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9.3 9.4
Computation and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Improved perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . 135
9.5
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
10 O(a2 ) improvements 145 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
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10.2 Improvement to the fermion propagator . . . . . . . . . . . . . . . . . . . . 146 10.2.1 Basic divergent integrals . . . . . . . . . . . . . . . . . . . . . . . . 148 10.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
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10.3 Improved operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 11 Conclusions
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Appendix B: Numerical integration
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Appendix A: Notation 166 A.1 Continuum QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 A.2 Lattice QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 171
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B.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 B.2 The integrator routine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 B.3 A particular example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
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Bibliography
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Appendix C: The algorithm for improving the Symanzik coefficients
184 188
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List of Figures The dependence of αs on the energy scale µ. . . . . . . . . . . . . . . . . .
2.1
A plaquette on a 2-dimensional slice of the hypercubic lattice. . . . . . . .
22
2.2 2.3
Graphical representation of Qµν (Eq. (2.22)) appearing in the clover action. The 4- and 6-link loops contributing to the gauge action of Eq. (2.26). . . .
28 30
3.1
The poles of Eq. (3.29) and the integration region C. . . . . . . . . . . . .
40
4.1
A plot of the pion propagator against time separation for the quenched approximation on a 323 × 64 lattice with coupling β = 6.2 ([24]). . . . . . .
45
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1.1
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r0 fPS as a function of (r0 mPS ) for β = 3.9 and β = 4.05 . . . . . . . . . . Nucleon mass as a function of m2π for β = 3.9, 4.05 . . . . . . . . . . . . .
52 53
5.1
The multiple character of a vertex in the background field method. . . . .
61
6.1
The value of αs (MZ 0 ) as derived from various processes and the average of these measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
7.1 7.2 7.3
Fermion contributions to the 1-loop function ν (1) . . . . . . . . . . . . . . . Fermion contributions to the 2-loop function ν (2) . . . . . . . . . . . . . . . Plot of the total 1-loop coefficient k (1) versus the overlap parameter ρ. . . .
82 82 90
7.4 7.5
Plot of the total 2-loop coefficient c(1,−1) versus ρ. . . . . . . . . . . . . . . Plot of the total 2-loop coefficient c(1,1) versus ρ. . . . . . . . . . . . . . . .
91 91
7.6
7.7
The 3-loop coefficient bL2 (Eq. (7.40)), plotted against ρ, for N = 3 and Nf = 0, 2, 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The non-pointlike nature of an overlap vertex. . . . . . . . . . . . . . . . .
92 94
7.8
A particular example of a 2-loop fermionic diagram. . . . . . . . . . . . . .
94
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4.2 4.3
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LIST OF FIGURES 7.9
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The ρ dependence of the ratio ΛL /Λ MS in the adjoint representation for N = 3 and Nf = 0, 1/2, = 1. . . . . . . . . . . . . . . . . . . . . . . . . . .
97
A cactus diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Improved coefficient c˜0 for N=2 and N=3 (plaquette action). . . . . . . . . 117
8.3 8.4 8.5
Improved coefficients c˜0 and c˜1 (tree-level Symanzik improved action). . . . 117 Improved coefficients c˜0 and c˜1 (Iwasaki action). . . . . . . . . . . . . . . . 118 Coefficients c0 , c1 , c3 and their dressed counterparts c˜0 , c˜1 , c˜3 for different
8.6
values of β c0 = 6 c0 /g02 (TILW actions). . . . . . . . . . . . . . . . . . . . . 118 1-loop contribution to the amputated Green’s function (bV,A ). . . . . . . . 121
8.7 8.8
1-loop contribution to the quark self-energy (bΣ ). . . . . . . . . . . . . . . 121 dr Plots of ZV,A and ZV,A for the plaquette, Iwasaki and TILW actions. . . . . 123
9.1
1-loop diagrams contributing to dm(1−loop) . . . . . . . . . . . . . . . . . . . 128
9.2 9.3
2-loop diagrams contributing to dm(2−loop) . . . . . . . . . . . . . . . . . . . 129 Total value of dm to 2 loops, for N = 3, Nf = 0 and c2 = 0. . . . . . . . . 134
9.4 9.5 9.6
Total value of dm to 2 loops, for N = 3, Nf = 2 and c2 = 0. . . . . . . . . 134 Total value of dm to 2 loops, for N = 3, Nf = 3 and c2 = 0. . . . . . . . . 135 Improved and unimproved values of dm up to 2 loops, as a function of cSW ,
9.7
for the plaquette action (β = 5.29, N = 3, Nf = 2). . . . . . . . . . . . . . 137 Improved and unimproved values of dm up to 2 loops, as a function of cSW ,
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for the Iwasaki action (β = 1.95, N = 3, Nf = 2). . . . . . . . . . . . . . . 137 Improved and unimproved values of dm up to 2 loops, as a function of cSW , for the DBW2 action (N = 3, Nf = 2). We set β = 0.87 and β = 1.04. . . . 138
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8.1 8.2
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10.1 1-loop diagrams contributing to the improvement of the fermion propagator. 146 10.2 1-loop diagrams contributing to the improvement of the bilinear operators. 160
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10.3 1-loop diagrams contributing to the extended operators. . . . . . . . . . . . 161
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A.1 The interaction vertices of quarks and gluons. . . . . . . . . . . . . . . . . 167
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List of Tables Numerical results for k (1) , c(1,−1) , c(1,1) . . . . . . . . . . . . . . . . . . . . . 88 (0) (1) Per diagram breakdown of the 1-loop coefficients ki and ki for various ρ values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.3 7.4
Contribution to ci of diagrams 1, 4, 5, 3+14+15, 6+12, 8+18. . . . . . 100 (0,1) Contribution to ci of diagrams 1, 2+13+16, 3+14+15, 4, 5. . . . . . . . 101
7.5 7.6 7.7
Contribution to ci of diagrams 6+12, 7+11, 8+18, 9+17, 10. . . . . . . 101 (1,−1) Contribution to ci of diagrams 1, 4, 5. . . . . . . . . . . . . . . . . . . 102 (1,−1) Contribution to ci of diagrams 3+14+15, 6+12, 7+11, 8+18. . . . . . . 102
7.8 7.9
Contribution to ci of diagrams 1, 2+13+16, 3+14+15, 4, 5. . . . . . . . 103 (1,1) Contribution to ci of diagrams 6+12, 7+11, 8+18, 9+17, 10. . . . . . . 103
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7.1 7.2
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(1,1)
(2,−1)
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7.10 The overlap parameter dependence of the 2-loop coefficients ci . . . . . 104 (2,1) 7.11 Numerical results for the 2-loop coefficients ci for different values of the overlap parameter ρ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Input parameters β, c0 , c1 , c3 , c˜0 , c˜1 , c˜3 (c2 = 0). . . . . . . . . . . . . . . 119 dr Results for ZV,A , ZV,A (Eq. (8.39), (8.40)) using ρ=1.0, ρ=1.4 . . . . . . . . 122
9.1 9.2 9.3
Input parameters c0 , c1 , c3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Total 1-loop coefficients ε(1) , ε(2) , and ε(3) . . . . . . . . . . . . . . . . . . . 140 Total 2-loop contribution to dm of order O(N 2 , c02 ). . . . . . . . . . . . . . 140
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Total 2-loop contribution to dm of order O(N 0 , c02 ). . . . . . . . . . . . . . 141 2-loop coefficients of dm containing closed fermion loops. . . . . . . . . . . 141
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9.4 9.5 9.6 9.7 9.8
Total dm coefficient containing the parameter c2 (part 1). . . . . . . . . . . 141 Total dm coefficient containing the parameter c2 (part 2). . . . . . . . . . . 142 1-loop contribution to dm for the Iwasaki action. . . . . . . . . . . . . . . 142
9.9
2-loop results for dm coming from diagrams 3, 4, 6, for the Iwasaki action.
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142
LIST OF TABLES
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9.10 2-loop results for dm coming from diagrams 7-11, 14-18, 24, 26, for the Iwasaki action. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 9.11 2-loop results for dm coming from diagrams 12, 13, 19, 20, for the Iwasaki action. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
9.14 1- and 2-loop results, and nonperturbative estimates for κc .
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9.12 2-loop results for dm coming from diagrams 21-23, 25, 27, 28, for the Iwasaki action. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 9.13 Results for dmdr (1−loop) (Eq. (9.23)), with N = 3. . . . . . . . . . . . . . . . 144 . . . . . . . . 144
10.1 The ε(0,i) coefficients of Eq. (10.28) for different actions. . . . . . . . . . . . 158
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10.2 The ε(1,i) coefficients of Eq. (10.28) for different actions. . . . . . . . . . . . 158 10.3 The ε(2,i) coefficients of Eq. (10.28) for different actions (part1). . . . . . . 158 10.4 The ε(2,i) coefficients of Eq. (10.28) for different actions (part2). . . . . . . 159 (i,1)
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10.5 The dependence of ε˜1 (Eq. (10.29)) on the Symanzik parameters. . . . . 159 (i,1) 10.6 The dependence of ε˜2 (Eq. (10.30)) on the Symanzik parameters. . . . . 159
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Chapter 1
Introduction to the Standard Model
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1.1
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Introduction
The Standard Model (SM) is a Theory which successfully describes, in terms of Quantum
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Fields, three of the four fundamental interactions: The electromagnetic, the weak and the strong; it leaves out the gravitational force which is negligible in the atomic and subatomic level. It was developed through 1970-1973 by the contributions of different scientists and
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it is consistent with both quantum mechanics and special relativity. A lot earlier, in 1950’s and 1960’s, a huge number of new particles had been detected
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in accelerators and there was an urgent need for a theory to explain their appearance. In 1964 M. Gell-Mann and G. Zweig proposed an underlying structure of fundamental elements, the so-called quarks, that could group in different combinations and form all
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particles, known as hadrons. Today we know that quarks come in six flavors (up, down, charm, strange, top and bottom) and for each quark there is a corresponding antiquark. They also carry an additional unobserved quantum number, the color, in order to avoid
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conflict with Pauli’s exclusion principle. For each flavor the quark can have any of the three colors (red, blue, green) and quarks with different flavors have different masses. The quark
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model separates hadrons in two categories: a. The mesons, with integer spin, consisting of a quark-antiquark pair and b. the baryons, with half-integer spin, which are bound states of three quarks or three antiquarks. This model not only could explain the already found particles, but could also predict the existence of new ones. More than a decade later, S. Weinberg in collaboration with S. Glashow, and A. Salam independently created the electroweak theory, unifying the electromagnetic and weak forces. The theory involves six leptons (electron, muon, tau, with their massless and 12
1.1. Introduction to the Standard Model
13
chargeless neutrinos) and the force carriers: Photon for electromagnetism and the W ± , Z bosons for the weak interaction. It is important to say that the above models are far from a mere hypothesis; the existence of both quarks and neutrinos has been experimentally verified, and the models have already passed very stringent experimental tests.
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M. Gell-Mann, H. Fritzsch and H. Leutwyler introduced the color force as the interaction between the fundamental constituents of the strong force, which are the quarks and gluons (the interaction mediators). Both quarks and gluons have color as an additional degree of
freedom; indeed gluons are self-interacting. With the formulation of the theory of the color force, Quantum Chromodynamics (QCD for short), the Standard Model was theoretically
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completed. Another fundamental feature of the SM is the prediction of the existence of a Higgs particle responsible for the nonvanishing masses of W ± and Z. The existence of the Higgs mechanism has not been confirmed experimentally yet, but it is hoped that this will
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be achieved in the LHC facilities, currently starting their operation at CERN.
We dedicate the rest of the section to the strong force and discuss different aspects of its theoretical formulation. When the quark model of QCD was first proposed, a lot
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of doubts arised concerning its validity since no quark was seen in experiments until that time. In 1973 G. ’t Hooft, H. Politzer, D. Gross and F. Wilczek studied two of the main features of the strong interactions: Asymptotic freedom and confinement. That is, when nucleons are given a lot of energy, the strong interaction actually weakens and the quarks can hardly interact with each other (asymptotic freedom). On the contrary, when the
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quarks are pulled apart, they appear to be tightly bound to each other within the nucleons (confinement). Thus, it would take infinite energy to separate them, explaining why they have never been seen as isolated entities.
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Quantum Chromodynamics has all the necessary features to describe the strong interactions. To start with, it is a nonabelian gauge theory based on the group SU(3) (the number 3 indicates the number of colors carried by quarks). This group has eight gener-
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ators corresponding to the number of massless gluons. Moreover, the nonabelian nature of QCD explains asymptotic freedom and the gluon self-interaction; it also leads to quark confinement, as has been numerically shown. The beauty of QCD lies in the fact that only a few parameters are needed to explain the whole spectrum of strong interactions; these are the coupling constant of the force and the quark masses. Despite its simple mathematical form, it is extremely complicated to solve it, even to perform fundamental computations like the hadron masses. Among the features of QCD is the ‘running’ of the fundamental coupling constant αs
1.1. Introduction to the Standard Model
14
with the energy scale. The term ‘constant’ is actually a misnomer since αs depends on the energy exchange µ, in each process under consideration. At high energies, or equivalently at small separations of the quarks, the coupling constant is small and the theory can be studied perturbatively. On the other hand, at low energies (< ∼ 1GeV) the coupling becomes
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large and perturbation theory breaks down completely. This behavior almost resulted in abandoning QCD: It was considered totally incapable of addressing the strong interactions because the coupling was too strong for perturbation theory to be of any use. In the Lagrangian of QCD (at least in the absence of fermions, or in presence of massless fermions) there is no particular dimensionful quantity in terms of which one could express
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dimensionful observables, such as the bound state spectrum. A solution to that, is to specify the value of the energy scale µ at which αs ∼ 1. This is the energy (known as ΛQCD ) at which the perturbative method cannot be applicable as we come down from high
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energies. The behavior of the coupling with respect to the energy scale is shown in Fig. 1.1. Note that ΛQCD merely sets the scale against which other quantities are to be measured.
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Αs
ßQCD
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Figure 1.1: The dependence of αs on the energy scale µ.
The failure of the perturbative expansion in the low energy sector is a major problem, since at that region quarks and gluons bind together into composite states, the hadrons. To
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study the world of hadrons and prove confinement, it is necessary to search for alternative, nonperturbative tools. This is where Lattice QCD (LQCD) comes into play. Lattice QCD was first suggested in 1974 by K. Wilson [1] as the mathematical tool to calculate observables in the nonperturbative region of QCD. The numerical computations are performed by means of Monte Carlo simulations, with input the bare coupling and the bare quark masses without having to tune additional parameters. The theory is formulated
1.1. Introduction to the Standard Model
15
on a finite-sized hypercubic space-time lattice, characterized by the spacing a (the distance between neighboring lattice points). By definition, the lattice introduces a momentum cutoff proportional to a−1 , excluding in this way the high frequencies and making the theory finite. Space and time are further allocated a finite extent, so that the number
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of degrees of freedom is finite. However, this number is still large, making simulations computationally very demanding; it is necessary to use some approximations in order to deal with it. Simulations are performed for a variety of lattice spacings and the continuum limit is reached by extrapolating the results to a → 0. An essential constraint in numerical calculations is the range of the lattice spacing that can be used. Typical algorithms slow
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down proportionally to a5 and thus simulations use values for a limited within the interval 0.05fm ≤ a ≤ 0.01fm. Performing numerical simulations is crucial both for predicting the dependence of the hadronic quantities on the bare parameters and for checking that QCD
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is the correct theory of the strong interactions, by comparing with experiments. For an appropriate comparison, the bare quantities must be renormalized.
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Although the idea of the lattice formulation was introduced to cope with the low energies, it can also serve the region of energies well above the energy scale relevant to hadronic properties. At that region we employ perturbation theory for the computations, which is
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useful for investigating the behavior of lattice theories near the continuum limit. It is also the key in connecting simulation results to physical quantities. For this to be achieved, the Lagrangian parameters and the matrix elements must be renormalized. Traditionally,
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the renormalization factors are defined perturbatively, even in many studies of the low energy sector. Ideally, for a truly nonperturbative result, the renormalization factors need
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to be calculated using simulations. Unfortunately, the range of physical distances covered by a single lattice is limited and it is very difficult to calculate the large scale differences of nonperturbative renormalizations. Furthermore, the mixing between operators at the
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quantum level is often too small to be distinguishable with numerical techniques. The above problems make perturbation theory necessary for the correct connection between
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experiments and theory and to verify that a single Lagrangian describes the high energy sector and explains at the same time the spectrum of light hadrons. In this Thesis, we use Lattice QCD in perturbation theory to investigate main issues regarding the running coupling, the quark masses, improvement methods of perturbation theory, and improved measurable quantities (quark propagator, quark operators). All computations have been performed with improved actions, the importance of which will be explained next. Each of our computations is motivated in the following section.
1.2. Perturbative calculations using improved actions
1.2
16
Perturbative calculations using improved actions
We have previously discussed the discretization of the continuum action in the framework of Lattice QCD. Starting from the continuum action for a single fermion, its naive discretization gives rise to 2d = 16 fermions rather than one; the extra unphysical fermions are called fermion doublers. This doubling phenomenon is demonstrated by the fermion
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propagator, which has sixteen poles for zero fermion mass, within the integration interval (first Brillouin zone). The main problem is that the doublers destroy the chirality properties (left-right symmetry) of the continuum theory. This failure is explained in the
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Nielsen-Ninomiya theorem [2] which leaves us with the choice of either having the unwanted doubler fermions on the lattice or eliminating the doubler modes at the expense
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of breaking chiral symmetry. The first attempt to resolve the doubling problem was given by Wilson [1], who added an extra term in the action. Besides all the advantages of the Wilson action, there are a number of drawbacks. One of them is the fact that lattice artifacts in physical quantities are proportional to a, rather that a2 . This will prove to be a great disadvantage.
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In principle, the only restriction while discretizing the QCD Lagrangian, is the recovery of the continuum limit. Yet, it is worthwhile to construct improved actions for a better behavior at all lattice spacings. One motivation is the discretization errors appearing in the lattice formalism, which affect simulation results and have to be removed; these can be dramatically reduced if one uses improved actions. While these errors are to be removed
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in the continuum limit, it is advantageous to have them under control at nonzero lattice spacing for two reasons: a. For large errors, it is not clear how to perform a safe and reliable extrapolation to a → 0 and b. unimproved actions force us to work with very
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small lattice spacings which is computationally very demanding (particularly in full QCD where simulations are very expensive in terms of computer time). So, it would be beneficial
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to work with improved actions at larger values of a and still have results of the same quality. Moreover, improved actions are expected to preserve more symmetries of the continuum theory. Over the years, many improved actions have been constructed, both for fermion and gauge fields. The most frequently used discretizations of the continuum fermion action are the clover [3], the staggered [4], and the Ginsparg-Wilson fermions (overlap fermions [5, 6, 7] and domain wall [8, 9]). All of the actions mentioned above have discretization errors of order O(a2 ) and reach the continuum limit faster than Wilson fermions. In the
following chapters we will concentrate on the actions used for our calculations: Clover and
1.2. Perturbative calculations using improved actions
17
overlap fermions. Regarding the gluon part of the action, there is the Symanzik improved action [10], which includes all possible closed loops made of 4 and 6 links (see Fig. 2.3). This action has 4 parameters that can have different values. In Table 8.1 we tabulate the most commonly used sets of these coefficients. We are particularily interested in the relationship between the bare coupling and the
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cutoff, an essential ingredient in lattice calculations. Usual Monte Carlo simulations require the knowledge of this relationship far from the critical point, where corrections to asymptotic scaling could become relevant. By definition, the β-function is the quantity
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that dictates this relationship between running coupling and the lattice spacing. Due to asymptotic freedom of QCD, the β-function can be expanded in terms of the coupling con-
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stant, with perturbative evaluations for the coefficients. The 1- and 2-loop coefficients are universal (regularization and renormalization scheme independent), while the rest of them depend on the regulator. We calculate the 3-loop coefficient using the overlap fermions, one of the most frequently used actions in simulations at the present time. Knowledge of the coupling is of high importance because its precise determination would fix the value
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of a fundamental parameter in the Standard Model. As a by-product of the β-function calculation, one can derive the Λ parameter of QCD, a quantity needed to convert dimensionless quantities coming from numerical simulations into measurable predictions for physical observables. It is mathematically defined as a particular solution of the renormalization group equation. This parameter is dimensionful and cannot be directly calculated
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on the lattice. Instead, one can obtain the ratio between the lattice Λ parameter, ΛL, and the scale parameter in some continuum renormalization scheme such as MS, ΛL /ΛMS.
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Another question we address is the critical value of the hopping parameter, κc , for clover fermions and Symanzik gluons. The demand for strict locality and absence of fermion dou-
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blers, results in the explicit breaking of chiral symmetry. Thus, setting the bare quark mass equal to zero is not sufficient to ensure chiral symmetry in the quantum continuum limit, because quantum corrections introduce an additive renormalization to the fermionic mass. For the renormalized mass to be zero, the bare mass must be renormalized multiplicatively and additively. The hopping parameter is an adjustable quantity used in simulations, that is directly related to the fermion mass. When it is assigned its critical value, chiral symmetry is restored. The computation of the hopping parameter is a typical case of a vacuum expectation value resulting in an additive renormalization; as such, it is characterized by a power (linear) divergence in the lattice spacing, and its calculation lies at the limits of
1.2. Perturbative calculations using improved actions
18
applicability of perturbation theory. The nature of perturbation theory indicates that when performing higher order calculations, the quantities under study are deduced with increased accuracy. At the same time, the calculations become notoriously complicated and demanding (i.e. there is a notable increase, both in the number of Feynman diagrams involved, and in the number of terms
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in each diagram), as compared to lower order studies. It is important to develop a method for improving the perturbative expansion, so that one can get improved results from lower order calculations. The method we propose, is called cactus improvement and sums up
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a whole subclass of tadpole diagrams, to all orders in perturbation theory. The effect of resummation is to replace various parameters in the action (coupling constant, Symanzik
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coefficients, clover coefficient) by ‘dressed’ values; the latter are solutions to certain coupled integral equations, which are easy to solve numerically. Some positive features of this method are: a. It is gauge invariant, b. it can be systematically applied to improve (to
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all orders) results obtained at any given order in perturbation theory, c. it does indeed absorb in the dressed parameters the bulk of tadpole contributions. This Thesis contains work carried out over the past three years and it is laid out as
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follows. Chapters 2-6 provide some background information for the work covered in the rest of the Thesis. Particularly, in Chapter 2 we explore the Wilson action (both for quarks and gluons) as well as two of the most widely used improved actions: The clover fermion
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action and the Symanzik gluon action. Chapter 3 is devoted to another class of improved actions of great complexity, the overlap fermions. Besides some standard material, we also
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derive the required vertices that are needed for a subsequent Chapter. A brief overview of the twisted mass fermion action is given in Chapter 4, while Chapters 5-6 regard the background field method and the running coupling (with emphasis on the β-function),
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respectively. The main calculations of the Thesis start in Chapter 7 with the evaluation of the relation between the bare coupling constant g0 and the renormalized one in the modified minimal
subtraction scheme MS, gMS . For convenience, we have worked with the background field technique, which only requires evaluation of 2-point Green’s functions for the problem at hand. The computation was performed to 2 loops in perturbation theory, employing the standard Wilson action for gluons and the overlap action for fermions. Our results depend explicitly on the number of fermion flavors (Nf ) and colors (N) and are tabulated for different values of the overlap parameter ρ in its allowed range (0 < ρ < 2). We also derive
1.2. Perturbative calculations using improved actions
19
the 3-loop coefficient of the bare β-function (βL (g0 )) and provide the recipe for extraction of the ratio of energy scales, ΛL /ΛMS. Moreover, we generalize our results to fermions in an arbitrary representation of the gauge group SU(N). In Chapter 8 we extend a previous systematic improvement of perturbation theory
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for gauge fields on the lattice, to encompass all possible gluon actions made of closed Wilson loops. Two different applications are presented: The additive renormalization of fermion masses (in Chapter 9), and the multiplicative renormalization ZV (ZA ) of the vector (axial) current. In many cases where nonperturbative estimates of renormalization functions are also available for comparison, the agreement with improved perturbative
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results is significantly better as compared to results from bare perturbation theory. In Chapter 9 we study the critical value of the hopping parameter, κc , up to 2 loops in perturbation theory. We employ the clover improved action for fermions and the Symanzik
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improved gluon action with 4- and 6-link loops. Our results are polynomial in cSW (clover parameter) and cover a wide range of values for the Symanzik coefficients ci . The depen-
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dence on the number of colors N and the number of fermion flavors Nf is shown explicitly. In order to compare our results to nonperturbative evaluations of κc coming from Monte Carlo simulations, we employ our improved perturbation theory method for improved ac-
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tions. The work covered in Chapter 10 is an ongoing project on the improvement of the fermion propagator and several quark operators, to order a2 in 1-loop perturbation theory.
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We employ improved actions for both fermions (clover) and gauge fields (Symanzik). The dependence on the gauge parameter is explicitly shown. The terms of order a2 can be
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used to specify the required corrections to the quark operators, in order to achieve O(a2 ) improvement. Our results are applicable also to the case of twisted mass fermions, which are currently being studied intensely by a number of collaborations worldwide.
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Finally in Chapter 11 we summarize and conclude. The Appendices contain supplemental material that has been left out of the main body of the Thesis in order to improve
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readability. The work of Chapters 8-9 was carried out in collaboration with Apostolos Skouroupathis and the project of Chapter 10 is in progress with Fotos Stylianou. Most of the results presented here have already been published in the following papers: • M. Constantinou, H. Panagopoulos, A. Skouroupathis, Improved Perturbation Theory for Improved Lattice Actions, Phys.Rev. D74 (2006) 074503, [hep-lat/0606001]
• M. Constantinou, H. Panagopoulos, QCD with overlap fermions: Running coupling
1.2. Perturbative calculations using improved actions
20
and the 3-loop beta-function, Phys. Rev. D76 (2007) 114504, [arXiv:0709.4368] • M. Constantinou, H. Panagopoulos, Gauge theories with overlap fermions in an arbi-
trary representation: Evaluation of the 3-loop beta-function, Phys. Rev. D77 (2008) 57503, [arXiv:0711.4665]
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• A. Skouroupathis, M. Constantinou, H. Panagopoulos, Two-loop additive mass renormalization with clover fermions and Symanzik improved gluons, Phys. Rev. D77 (2008) 014513, [arXiv:0801.3146]
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and conference proceedings:
• M. Constantinou, H. Panagopoulos, A. Skouroupathis, Improving perturbation theory
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with cactus diagrams, PoS LAT2006 (2006) 155, [hep-lat/0612003]
• A. Skouroupathis, M. Constantinou, H. Panagopoulos, 2-loop additive mass renormalization with clover fermions and Symanzik improved gluons, PoS LAT2006 (2006)
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162, [hep-lat/0611005]
• M. Constantinou, H. Panagopoulos, The three-loop beta-function of SU(N) lattice
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gauge theories with overlap fermions, PoS LAT2007 (2007) 247, [arXiv:0711.1826]
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Chapter 2
2.1
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The Wilson formulation and O(a) improvements Standard Wilson quarks and gluons
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In this chapter we provide the standard Wilson action for both fermions and gluons, as well
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as O(a) improvements on this action. For a brief introduction on the lattice formulation of QCD and our notation, the reader should see Appendix A. For completeness allow us to repeat here some of this notation that is necessary for this section. In the formulation
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of Lattice QCD, the fermion fields Ψ(x), Ψ(x) live on the lattice sites x and carry color (i, j, ... = 1, ..., N), flavor (f = 1, ..., Nf ) and Dirac indices (α, β, ... = 1, ..., 4). We recall
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that N, Nf are the number of fermion colors and fermion flavors, respectively. The variables U(x, µ) are defined on the links connecting two neighboring lattice sites. The index µ = 0, ..., 3 labels the direction of the link and µ ˆ is the unit vector in the µth direction.
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The link variables are not linear in the gauge fields Gaµ (x), but they are defined in a way that the continuum action is recovered when setting a → 0 Uµ (x, x + aˆ µ) ≡ Uµ (x) = eiag0 T
ˆ a Ga (x+ aµ ) µ 2
(2.1)
({T a } (a = 1, ..., N 2 − 1) are the SU(N) generator matrices). By convention, the argument
of Gaµ is defined in the midpoint of the link (without affecting the continuum limit or the simulations) and U is an N × N unitary matrix satisfying U(x, x − aˆ µ) ≡ U−µ (x) = e−iag0 T 21
ˆ a Ga (x− aµ ) µ 2
= U † (x − aˆ µ, x)
(2.2)
2.1. Standard Wilson quarks and gluons
22
The local gauge transformation Λ(x) (being in the same representation as U) is employed on the fermion and gauge fields through the relations Ψ(x) → Λ(x)Ψ(x)
Ψ(x) → Ψ(xΛ† (x))
Uµ (x) → Λ(x)Uµ (x)Λ† (x + aˆ µ)
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(2.3)
One requires that the lattice action be invariant under the gauge transformations and thus, its gluon part must be constructed by gauge invariant objects. The simplest choice
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is the trace of the 1 × 1 loop, called plaquette
Pµν ≡ Uµ (x)Uν (x + aˆ µ)Uµ† (x + aˆ ν )Uν† (x)
(2.4)
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This is the usual product of link variables along the perimeter of a plaquette originating at x in the positive µ − ν directions. As can be realized from Fig. 2.1, there are two different
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orientations for each plaquette, which are Hermitian conjugate to each other. Thus, taking the trace over color indices ensures gauge invariance, whilst the real part is taken to average over the loop and its complex conjugate. U~n†+ˆν, ~n+ˆµ+ˆν
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~n + νˆ
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U~n†, ~n+ˆν
~n + µ ˆ + νˆ
U~n+ˆµ, ~n+ˆµ+ˆν
~n + µ ˆ
~n U~n, ~n+ˆµ
ν
µ
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Figure 2.1: A plaquette on a 2-dimensional slice of the hypercubic lattice.
The formulation of the action in terms of the link variables, rather than the gauge fields directly, serves to uphold gauge invariance. This is a symmetry we are not willing to give up, because otherwise we would have more parameters to tune (the different couplings for the quark-gluon, 3-gluon and 4-gluon interactions, the gluon mass that would not be zero). There would also arise many more operators at any given order in a, leading to increase of
2.1. Standard Wilson quarks and gluons
23
the discretization errors. Finally, the existing proofs of perturbative renormalizability of QCD, defined on the lattice, rely on strict gauge invariance [11]. In fact, it is recommended to preserve as many symmetries of the theory as possible at all values of a. By construction, the discretization breaks continuous rotational, Lorentz and translational symmetry.
XX f
f
Ψ (x)(DW + mf0 )Ψf (x) +
x
1 2N X X (1 − ReTr[Pµν ]) 2 g0 µ0} {j+k=i}
d4 k3 1 1 1 × 4 (2π) ω(k1 ) + ω(k3) ω(k1) + ω(k2 ) ω(k2) + ω(k3 ) "
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=
Z
− Xj (k1 , k3 ) χ†0 (k3 ) Xk (k3 , k2 )
− Xj (k1 , k3 ) Xk† (k3 , k2 ) χ0 (k2 )
− χ0 (k1 ) Xj† (k1 , k3 ) Xk (k3 , k2 )
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V2i (k1 , k2 )
(3.35)
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# ω(k1) + ω(k2) + ω(k3) + χ0 (k1 )Xj† (k1 , k3 )χ0 (k3 )Xk† (k3 , k2 )χ0 (k2 ) (3.36) ω(k1 )ω(k2)ω(k3)
! 4 4 Y d k d k 1 1 3 4 V3i (k1 , k2 ) = × (2π)4 (2π)4 4 pǫS ω(kp1 ) + ω(kp2 ) 4 ! " X 1 X ω(kp1 )ω(kp2 )ω(kp3 ) Xj (k1 , k3 )Xk† (k3 , k4 )Xl (k4 , k2 ) − 6 pǫS {j>0,k>0,l>0}
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Z Z
4
{j+k+l=i}
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+ ω(k1 ) + ω(k3) + ω(k4 ) + ω(k2) × h χ0 (k1 )Xj† (k1 , k3)Xk (k3 , k4 )Xl† (k4 , k2 )χ0 (k2 )
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+ χ0 (k1 )Xj† (k1 , k3 )Xk (k3 , k4 )χ†0 (k4 )Xl (k4 , k2 )
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+ χ0 (k1 )Xj† (k1 , k3 )χ0 (k3 )Xk† (k3 , k4)Xl (k4 , k2 ) + Xj (k1 , k3 )χ†0 (k3 )Xk (k3 , k4 )Xl† (k4 , k2 )χ0 (k2 ) + Xj (k1 , k3 )χ†0 (k3 )Xk (k3 , k4 )χ†0 (k4 )Xl (k4 , k2 )
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i + Xj (k1 , k3 )Xk† (k3 , k4 )χ0 (k4 )Xl† (k4 , k2 )χ0 (k2 ) ! X ω(kp1 )ω(kp2 ) ω(kp1 )/2 + ω(kp3 )/3 − × ω(k 1 )ω(k3 )ω(k4 )ω(k2 ) pǫS 4
χ0 (k1 )Xj† (k1 , k3)χ0 (k3 )Xk† (k3 , k4 )χ0 (k4 )Xl† (k4 , k2 )χ0 (k2 )
#
(3.37)
3.5. Overlap action
43
! 4 4 4 Y d k d k 1 d k 1 4 5 3 V44 (k1 , k2 ) = × (2π)4 (2π)4 (2π)4 12 pǫS ω(kp1 ) + ω(kp2 ) 5 " ! X ω(kp1 )ω(kp2 )ω(kp3 )ω(kp4 ) ω(kp1 )/6 + ω(kp5 )/30 × Z Z Z
pǫS5
+ X1 (k1 , k3 )X1† (k3 , k4 )X1 (k4 , k5)χ†0 (k5 )X1 (k5 , k2 ) + X1 (k1 , k3 )X1† (k3 , k4 )χ0 (k4 )X1† (k4 , k5 )X1 (k5 , k2 ) + X1 (k1 , k3 )χ†0 (k3 )X1 (k3 , k4 )X1† (k4 , k5 )X1 (k5 , k2 )
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i + χ0 (k1 )X1† (k1 , k3)X1 (k3 , k4 )X1† (k4 , k5 )X1 (k5 , k2 ) ! 1 X ω(kp1 )ω(kp2 ) ω(kp1 ) + ω(kp3 ) × − 6 pǫS 5 h X1 (k1 , k3 )χ†0 (k3 )X1 (k3 , k4 )X1† (k4 , k5 )χ0 (k5 )X1† (k5 , k2 )χ0 (k2 )
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h (X1 (k1 , k3 )X1† (k3 , k4 )X1 (k4 , k5 )X1† (k5 , k2)χ0 (k2 )
+ X1 (k1 , k3 )χ†0 (k3 )X1 (k3 , k4 )χ†0 (k4 )X1 (k4 , k5 )χ†0 (k5 )X1 (k5 , k2 )
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+ X1 (k1 , k3 )χ†0 (k3 )X1 (k3 , k4 )χ†0 (k4 )X1 (k4 , k5 )X1† (k5 , k2 )χ0 (k2 )
+ X1 (k1 , k3 )X1† (k3 , k4 )χ0 (k4 )X1† (k4 , k5 )χ0 (k5 )X1† (k5 , k2 )χ0 (k2 )
on
+ χ0 (k1 )X1† (k1 , k3)χ0 (k3 )X1† (k3 , k4 )χ0 (k4 )X1† (k4 , k5 )X1 (k5 , k2 ) + χ0 (k1 )X1† (k1 , k3)χ0 (k3 )X1† (k3 , k4 )X1 (k4 , k5 )χ†0 (k5 )X1 (k5 , k2)
C
+ χ0 (k1 )X1† (k1 , k3)χ0 (k3 )X1† (k3 , k4 )X1 (k4 , k5 )X1† (k5 , k2 )χ0 (k2 ) + χ0 (k1 )X1† (k1 , k3)X1 (k3 , k4 )χ†0 (k4 )X1 (k4 , k5 )χ†0 (k5 )X1 (k5 , k2) + χ0 (k1 )X1† (k1 , k3)X1 (k3 , k4 )χ†0 (k4 )X1 (k4 , k5 )X1† (k5 , k2 )χ0 (k2 )
a
X
ω 2 (kp1 )ω(kp2 )ω(kp3 ) × ω(k1)ω(k2 )ω(k3)ω(k4)ω(k5 )
i
th
+
χ0 (k1 )X1† (k1 , k3)X1 (k3 , k4 )X1† (k4 , k5 )χ0 (k5 )X1† (k5 , k2 )χ0 (k2 )
+
pǫS5
! ω(kp2 )[ω(kp1 )/2 + ω(kp3 )/6] + ω(kp4 )[ω(kp1 )/3 + ω(kp2 ) + ω(kp5 )/3] ×
M ar
χ0 (k1 )X1† (k1 , k3 )χ0 (k3 )X1† (k3 , k4 )χ0 (k4 )X1† (k4 , k5 )χ0 (k5 )X1† (k5 , k2 )χ0 (k2 )
(Sn : permutation group of the numbers 1 through n)
#
(3.38)
ou
Chapter 4
Introduction
an
4.1
tin
Twisted mass action
In this chapter we present a new approach to lattice fermions, that is known under the
st
name of twisted mass QCD (tmQCD). As previously mentioned, the Wilson action breaks chiral symmetry, which can be restored with the introduction of appropriate counterterms (for instance, additive fermion mass renormalization). The result of the absence of chiral
on
symmetry for nonzero lattice spacing, is that the Wilson-Dirac operator is not protected against zero modes, unless the bare quark mass is positive. However, due to additive mass
C
renormalization, the masses of the light quarks correspond to negative bare masses. One of the consequences of the zero modes is the following: After integration over the fermion and anti-fermion fields in the functional integral, there is a small eigenvalue of the Wilson-
a
Dirac operator in the fermionic determinant and the fermion propagators appearing in the correlation functions. Thus, in the quenched approximation, where the fermionic determinant is ignored, this eigenvalue in the quark correlation is not cancelled out upon division.
th
The results are large fluctuations in particular measurables that compromise the ensemble average. The gauge field configurations at which this happens, are called exceptional. For
M ar
this to become more obvious, allow us to present a plot from Ref. [24] showing the ensemble average of the pion propagator over all gauge configurations (dots) besides an exceptional one (dashed line). One can see that the dashed line deviates dramatically from the average, thus, the inclusion of the exceptional configuration in the ensemble average would lead to large errors. On the other hand, its omission makes the Monte Carlo simulations invalid.
44
4.2. The lattice twisted mass action for degenerate quarks
45
2
10
1
10
0
ou
10
−1
−2
10
10
20
30
40 t
50
60
an
0
tin
10
st
Figure 4.1: A plot of the pion propagator against time separation for the quenched approximation on a 323 × 64 lattice with coupling β = 6.2 ([24]).
on
A solution to this problem is the addition of a mass term [25] to the standard Wilson action, that is given the role of protecting the Dirac operator against these configurations. The resulting action has the benefit that certain observables are automatically free of O(a)
C
lattice artifacts. Some additional advantages of this action are the efficient simulations and that the operator mixing appears to be the same as in the continuum. The twisted
4.2
th
a
mass action can be used to study quarks at small masses, where the Wilson action would fail. Also, the properties and the interactions of hadrons can be probed nonperturbatively from first principles.
The lattice twisted mass action for degenerate
M ar
quarks
The twisted mass lattice action for a doublet of Nf = 2 mass degenerate quarks, written in the so called twisted basis {χ, χ}, ¯ is {χ}
Stm = a4
X x
h i χ(x) ¯ DW + m0 + iµq γ 5 τ 3 χ(x)
(4.1)
4.2. The lattice twisted mass action for degenerate quarks
46
with m0 real and positive. DW is the Wilson-Dirac operator as given in Eq. (2.6). The last term with the twisted mass parameter µq protects the Dirac operator against zero modes for any finite µq , since the twisted Dirac operator has positive determinant det(DW + m0 + iµq γ 5 τ 3 ) = det(Q2 + µ2q )
(4.2)
ou
where Q = γ 5 (DW + m0 ) is the hermitian Wilson operator; hence, the twisted Dirac operator does not have any zero eigenvalues. The isospin generator τ 3 acts in flavor space and its
tin
appearance means that isospin is no longer conserved (i.e. the up and down quark have opposite signs of the twisted mass leading to flavor symmetry breaking). Moreover, the twist term breaks parity symmetry (due to γ 5 ). These symmetries are restored in the continuum limit. The action remains invariant under the flavor-dependent axial transformations 5 τ3
an
Ψ = eiωγ 2 χ 3 5τ ¯ = χ Ψ ¯ eiωγ 2
(4.3)
st
with the mass parameters mixed to each other as
on
m′ = µq sin(ω) + m cos(ω)
µ′q = µq cos(ω) − m sin(ω)
(4.4)
C
In the full twist case, ω = π/2, the flavor symmetry is restored at a rate O(a2 ). This case is useful, since there is automatic cancellation of O(a) effects in quantities like energies and ¯ where operator matrix elements. The action can be written in the physical basis {Ψ, Ψ},
a
the µq term has been eliminated
th
{Ψ}
Stm = a4
X x
h i ¯ Ψ(x) DW tm + M Ψ
(4.5)
M ar
DW tm is the twisted Wilson operator DW tm
3
− → ← − − − → 1X 5 3 ← {γµ ( ∇ µ + ∇ µ ) − a r e−iωγ τ ∇ µ ∇ µ } = 2 µ=0
and M is the polar mass M=
q
m20 + µ2q
(4.6)
(4.7)
4.2. The lattice twisted mass action for degenerate quarks
47
In the continuum limit, where the last term of Eq. (4.6) vanishes, tmQCD can be seen as a change of variables which leaves the physical content of the theory unchanged if the rotation angle ω satisfies tan(ω) =
µq m0
(4.8)
Thus, in the continuum limit the axial rotation of the fermion fields (Eq. (4.3)) relates
ou
tmQCD to the standard QCD. In terms of the fields {χ, χ} ¯ the axial (A) and vector (V ) currents along with the pseudoscalar (P ) and scalar (S) densities are given by 3
3
Vµa = χγ ¯ µ τ2 χ S 0 = χχ ¯
tin
Aaµ = χγ ¯ µ γ 5 τ2 χ 3 P a = χγ ¯ 5 τ2 χ
(4.9)
an
Using the transformations of Eq. (4.3) we derive the expressions of the above composite fields in the physical basis cos(ω)Aa + ε3ab sin(ω)V b µ µ a ¯ µ γ 5 τ Ψ= A′ µ ≡ Ψγ 3 2 Aµ cos(ω)V a + ε3ab sin(ω)Ab a µ µ a ¯ µτ Ψ = V ′ µ ≡ Ψγ V 3 2
(a = 1, 2),
on
st
a
µ
(4.10)
(a = 3), (a = 1, 2),
(4.11)
(a = 3),
C
The same can be applied to the densities P, S leading to the following equalities P a
a
a
a ¯ 5 τ Ψ= P ′ ≡ Ψγ cos(ω)P 3 + i sin(ω) 1 S 0 2 2
(a = 1, 2)
0 ¯ = cos(ω)S 0 + 2i sin(ω)P 3 S ′ ≡ ΨΨ
th
(4.12)
(a = 3) (4.13)
M ar
¯ satisfy the standard Ward Identities The rotated physical fermion fields {Ψ, Ψ} a
a
∂µ A′ µ = 2MP ′ ,
a
∂µ V ′ µ = 0
(4.14)
while the twisted fields {χ, χ} ¯ satisfy the partially conserved axial current (PCAC) and
4.3. Calculations with twisted mass QCD
48
partially conserved vector current (PCVC) relations ∂µ Aaµ = 2mq P a + iµq δ 3a S 0
(4.15)
∂µ Vµa = −2µq ε3ab P b
(4.16)
We are particularily interested in the action written in the twisted basis, because it
ou
is the one used in simulations. This is due to the fact that the renormalization of gauge invariant correlation functions is simpler for the twisted fields {χ, χ}. ¯ The expression for the twisted mass propagator is
◦
p◦ 2µ + M(p)2 + µ2q
◦
◦
pµ =
1 sin(apµ ), a
an
with p and M defined through
tin
G(p) =
−iγµ p µ + M(p) − iµq γ 5 τ 3
r 2 M(p) = m0 + aˆ p , 2 µ
pˆµ =
2 apµ sin( ) a 2
(4.17)
(4.18)
st
The tree-level expression can be extracted by taking the Taylor expansion for small values
on
of the lattice spacing a and keeping terms up to O(a), obtaining G0 (p) = p2 + m20 + µ2q + am0 rp2
(4.19)
C
The first observation is that for zero bare mass (or even for m0 = am ˜ 0 ), the theory is free of O(a) effects, but this picture changes once we take into account the interactions
a
between quarks. Moreover, the inclusion of the twisted mass parameter does not affect the O(a) improvement of the m0 = 0, am ˜ 0 cases. The absence of µq in the O(a) term can be explained from the origin of the O(a) correction terms. The am0 rp2 term in Eq. (4.19)
M ar
th
comes from M(p)2 and no µq is involved because it points in a different direction in flavor space than the other terms.
4.3
Calculations with twisted mass QCD
In the previous section we provided the axial transformations Eq. (4.3) that relate the continuum tmQCD to the standard QCD. However, these transformations are not an exact symmetry of the theory, thus the equivalence holds up to cutoff effects for renormalized correlation functions. The counterterms that enter the renormalization of the bare quark
4.3. Calculations with twisted mass QCD
49
masses are χχ, ¯
m0 χχ, ¯
iµq χγ ¯ 5τ 3χ
(4.20)
The lattice symmetries forbid the additive renormalization of the twisted mass µq , so it is renormalized only multiplicatively, while the bare mass m0 undergoes additive and
mR = Zm (g02 , aµ)(m0 − mcr )
µq,R = Zµ (g02 , aµ)µq
ou
multiplicative renormalization
(4.21)
tin
µ denotes the renormalization scale dependence of the renormalization constants Z. The critical mass mcr is the value that m0 must be given, so that the untwisted renormalized
an
mass mR vanishes (recovery of the chiral limit). The renormalization factors Z can be related to those of the composite fields, using the PCAC and PCVC relations (Eqs. (4.15), (4.16)) as normalization conditions. This connec-
st
tion is established by choosing renormalization schemes such that the PCAC and PCVC relations hold, with renormalized currents, densities and renormalized masses. In partic-
on
ular, the vector Ward Identity implies that the renormalization factor of the pseudoscalar density ZP and Zµ obey ZP Zµ = 1 (4.22)
C
Moreover, the renormalization constants of the untwisted mass and the scalar density may be shown to satisfy ZS 0 Zm = 1
(4.23)
th
a
Using the above equations, the twist angle ω can be defined from the renormalized mass parameters µq,R ZS 0 µq (4.24) = tan(ω) = µR ZP m0 − mcr
M ar
Let us now include gauge fields in our theory, so that the total action is denoted by Stotal = SG + SW tm , where SG is the purely gluon part and SW tm is the fermion Wilson action (Eq. (4.1)) with the twisted mass parameter µq . The partition function is given by Z=
Z
D[χ, ¯ χ]D[U] e−S
(4.25)
4.4. Twisted mass QCD for nondegenerate quarks
50
and the n-point correlation function is 1 < O >= Z
Z
D[χ, χ]D[U] ¯ e−S O
(4.26)
where O is a product of local gauge invariant composite fields. By renormalizing the theory as indicated above, and performing the continuum limit, the correlation functions in the
¯ >QCD =< O′ [χ, χ] < O′ [ψ, ψ] ¯ >tmQCD
ou
physical and in the twisted basis are equivalent (4.27)
an
tin
It is, of course, implied that < O′ >QCD has been computed in standard QCD with renormalized quark mass MR q MR = m2R + µ2q,R (4.28)
while < O >tmQCD regards the computation in tmQCD with renormalized masses mR , µq,R . Thus, a correlation function in QCD can be expressed as a linear combination of correlation
st
functions in tmQCD at a particular twist angle ω. A computation of an expectation value in the twisted basis starts by performing the ¯ that take us to the inverse axial transformations (Eq. (4.3)) to the physical fields {Ψ, Ψ}
on
{χ, χ} ¯ basis. One then computes the resulting correlation function using the twisted mass action (Eq. (4.1)) with a choice of quark masses. Finally, the continuum limit is taken and
C
the result is the desired continuum QCD correlation function with quark mass equal to MR of Eq. (4.28).
Twisted mass QCD for nondegenerate quarks
a
4.4
th
So far we have discussed the Nf = 2 case of degenerate light quarks, but the action can be generalized to include a further doublet of non-degenerate quarks [26, 27]. Such
M ar
a generalization arises from the need to describe the heavier quarks, charm and strange. Since we want to use this action in simulations of full tmQCD, we must maintain the reality and positivity of the quark determinant. Thus, in the action we add a flavor off-diagonal splitting
{χ}
Stm = a4
X x
h i χ(x) ¯ DW + m0 + iµq γ 5 τ 3 + ǫq τ 1 χ(x)
(4.29)
4.5. Discussion
51
where ǫq is the mass splitting parameter and we demand µq , ǫq > 0. The additional term retains theq properties of tmQCD at full twist and it keeps the quark determinant real and
positive if m20 + µ2q > ǫq . The transition to the physical basis is achieved with the following field transformations τ 1 1 ) √ (1 + iτ 2 ) χ 2 2 1 1 ¯ = χ¯ √ (1 − iτ 2 ) exp(−iωγ 5 τ ) Ψ 2 2 exp(−iωγ 5
{Ψ}
Stm = a4
X x
tin
The action in this basis is now
(4.30)
ou
Ψ =
h i ¯ Ψ(x) DW tm + M Ψ
(4.31)
(4.32)
an
p where M = m2q + µ2q is again the polar mass. The partial conservation equations can be obtained in the same manner as in the degenerate case (4.33)
∂µ Vµa = −2µq ε3ab P b + iǫq ε1ab S b
(4.34)
on
st
∂µ Aaµ = 2m0 P a + iµq δ 3a S 0 + ǫq δ a1 P 0
where
τa χ (4.35) 2 For the description of the heavy doublet charm and strange (c,s) we associate the physical S a = χ¯
C
P 0 = χγ ¯ 5 χ,
quark mass with the mass parameter M, that is mstrange = M − ǫq
(4.36)
th
a
mcharm = M + ǫq
M ar
and the fermion determinant is positive if M > ǫq .
4.5
Discussion
Twisted mass QCD is frequently used in simulations due to the nice properties of the theory. The twisted mass term introduces an infrared bound on the spectrum of the WilsonDirac operator and as a result the quenched and partially quenched approximations are well-defined. In the continuum case, an axial rotation of the fermion fields can eliminate
4.5. Discussion
52
the twisted mass term, while on the lattice the twisted mass action and the standard Wilson action cannot be related by a change of variables; consequently they have different discretization errors. Moreover, tmQCD at maximal twist is automatically O(a) improved. The European Twisted Mass Colaboration (ETMC) is currently performing large scale
ou
simulations for two flavors of light quarks with degenarate mass, using maximal twist LQCD. A review can be found in Ref. [28]. Among their work, some recent studies are the following:
tin
1. Charged pseudoscalar mass and decay constant [29, 30]: The charged pseudoscalar mass meson mass, denoted by amPS , is extracted from the time exponential decay of the pseudoscalar correlation function, covering a range of values 300MeV < ∼ mPS < ∼ 550MeV. Moreover, the charged pseudoscalar decay constant fPS is determined from 2µq |h0|P a|πi| , m2PS
a = 1, 2
(4.37)
an
fPS =
due to the exact lattice PCVC relation with no need to calculate any renormalization
st
constant. In order to compare results at different values of the lattice spacing, one measures the hadronic scale r0 /a (r0 : Sommer parameter), which is defined via the force between
on
static quarks at intermediate distance. The results for r0 fPS are plotted in Fig. 4.2 as a function of r0 mPS , for β = 3.9, 4.05.
C
r0 fPS
a
0.42
th
0.38
β = 4.05 β = 3.9
M ar
0.34
0.30 (r0 mPS )2 0.26 0.0
0.5
1.0
1.5
2.0
Figure 4.2: r0 fPS as a function of (r0 mPS )2 for β = 3.9 and β = 4.05
4.5. Discussion
53
2. Nucleon Mass and other baryon masses: Ref. [31] presents simulation results on the nucleon mass and the ∆ baryon masses. Their evaluation was performed at four quark masses (corresponding to a pion mass 300MeV < ∼ mπ < ∼ 690MeV) and different lattice sizes. The masses of the nucleon and the ∆’s are
C
on
st
an
tin
ou
extracted from 2-point correlators using standard interpolating fields. Fig. 4.3 shows the nucleon mass as a function of the squared pion mass. The simulation’s parameters are given in Ref. [31].
th
a
Figure 4.3: Nucleon mass as a function of m2π for β = 3.9 on a lattice of size 243 × 48 (filled triangles) and on a lattice of size 323 × 64 (open triangles). Results at β = 4.05 are denoted by stars. More details can be found in Ref. [31]. There are many other applications of tmQCD, among them the pion mass mπ and the pion decay constant Fπ , both obtained from the long distance behavior of the 2-point
M ar
function [32, 33]
(AR )10 (x)(PR )1 (y)a (M
R ,0)
= cos(ω) (AR )10 (x)(PR )1 (y) (M ,ω) D E R 2 1 ˜ + sin(ω) V0 (x)(PR ) (y) (MR ,ω)
(4.38)
Twisted mass QCD has been also employed for the determination of the chiral condensate from the local scalar density. An analogous computation with Wilson fermions has
4.5. Discussion
54
never been performed, due to the cubic divergence that appears in the chiral limit. This evaluation is based on the relation
(PR )3 (x) (M
R ,ω)
= cos(ω) (PR )3 (x) (M
R ,0)
−
i sin(ω) (SR )0 (x) (M ,0) R 2
(4.39)
It is worth mentioning that using tmQCD there is no general recipe for by-passing the
M ar
th
a
C
on
st
an
tin
ou
lattice renormalization problems of the Wilson fermions. One must study each computation individually and decide whether it is advantageous to use some variants of tmQCD.
ou
Chapter 5
Introduction
an
5.1
tin
The background field formalism
The background field method was first introduced by B. DeWit [34], and some years later,
st
J. Honerkamp [35] and G. ’t Hooft [36] independently discussed several issues of the same technique. The aim of this method is the simplification of quantum computations related to gauge and gravitational theories; the study of renormalization constants is a good example
on
demonstrating the effectiveness of the method. In the last decade, the background field technique found application in the Standard Model as well [37, 38].
C
A characteristic of the classical limit of gauge field theories is the gauge symmetry, which is broken by quantum corrections. A particular example is the effective action, which for a given classical action sums up all quantum corrections, and can only be de-
a
termined in perturbation theory. A way to overcome this problem is the introduction of a background field; the resulting effective action is then gauge invariant with respect to gauge transformations of the background field. Therefore, numerical studies become technically
th
easier. Considering the lattice regularization, a gauge theory is renormalizable to all orders in perturbation theory [39, 40]. In addition, the implementation of the background field
M ar
does not require any further counterterms besides those already needed in its absence, as it happens in the continuum case. Although the background fields lead to a gauge invariant theory, it is common to give up this symmetry at intermediate levels of a computation (by performing calculations in momentum space). This originates from the perturbative treatment of the interaction with the background field. One of the first applications of the particular method in lattice gauge theories is the 55
5.2. Background fields in the continuum theory
56
computation of the matching between different couplings; it has been proven to be the most efficient and economical way. Indeed, the method was utilized for the evaluation of the coefficients d1 (¯ µa) [42, 43, 44] and d2 (¯ µa) [45] of the equation relating the running coupling in the MS renormalization scheme (α MS (¯ µ)) to the bare lattice coupling in the pure SU(N) gauge theory (α0 ), given by (5.1)
ou
α MS (¯ µ) = α0 + d1 (¯ µa)α02 + d2 (¯ µa)α03 + ...
tin
(α0 = g02/4π, α MS = g 2MS /4π, a : lattice spacing, µ ¯ : scale parameter). Next, we discuss the continuum version of the background field technique, the sequential additions to the total action and the possible choices for the gauge transformations. Furthermore, we provide expressions for the functional integral and the effective action. We then analyze the technique on the lattice and show how the action is altered. We also
Background fields in the continuum theory
on
5.2
st
an
describe two different ways of expressing the gauge transformations and explain why only one is preferable. Finally, we quote examples of the X’s (Eqs. (3.21)) appearing in the overlap action in the background field formalism, as well as the 3-gluon vertex.
Having introduced the general principles of the background field technique, let us now
a
C
describe its mathematical formalism in the continuum theory. An SU(N) gauge potential Gµ = Gaµ T a ({T a } being the generators of the algebra) is described by the Yang-Mills action Z 1 S[G] = − 2 dD xTr[Fµν (x) Fµν (x)] (5.2) 2g0
M ar
th
where g0 is the bare gauge coupling constant and Fµν (x) the field strength tensor (Fµν = ∂µ Gν − ∂ν Gµ + [Gµ , Gν ]). The main idea of the background field method, is that the gauge field is decomposed into two parts, the quantum (Q) and the background (A) field Gµ = Aµ + g0 Qµ
(5.3)
Aµ is a smooth external source field, which is not required to satisfy the Yang-Mills equa-
tions, while the quantum field is the integration variable of the functional integral. An infinitesimal gauge transformation of Gµ with parameter Λ can be distributed in many ways over Aµ and Qµ , but the most convenient choices are
5.2. Background fields in the continuum theory
57
a. the ‘background transformation’ δAµ = Dµ Λ,
δQµ = [Qµ , λ]
(5.4)
b. and the ‘quantum transformation’ δQµ = Dµ Λ + [Qµ , λ]
(5.5)
ou
δAµ = 0,
One has to reexpress the gauge action S[G] in terms of Aµ , Qµ (Gµ is replaced by
Z
dD x Tr[Dµ Qµ (x)Dν Qν (x)]
(5.6)
an
1 Sgf [A, Q] = − ξ0
tin
Eq. (5.3)) and add the gauge fixing term which breaks the quantum gauge invariance. The latter is chosen in such a way that it preserves the background gauge invariance
SF P [A, Q, c¯, c] = −2
Z
st
In the above equation, ξ0 is the bare gauge parameter and Dµ = ∂µ + i Aµ is the covariant derivative. The ghost action is now dD x Tr[Dµ c¯(x) (Dµ + ig0 Qµ (x)) c(x)]
(5.7)
on
External quantities are coupled only to Qµ , so that Aµ is always invariant under gauge transformations. Of course, for vanishing background field the total action S[A + g0 Q] +
C
Sgf [A, Q]+SF P [A, Q, c¯, c], coincides with its standard form (with the usual covariant derivative). The partition function Z[J, η¯, η] (Jµ (x), η¯, η: classical source fields) changes in the presence of Aµ and is given by the formula Z
D[Q]D[¯ c]D[c]e−Stotal[A,Q,¯c,c]+(J,Q)+(¯η,c)+(¯c,η)
th
a
1 Z[A, J, η¯, η] = N
(5.8)
with the normalization factor N ensuring that Z[0, 0, 0, 0] = 1. Note that Aµ is not coupled
M ar
to the source J. The scalar product(A, B) between two fields A, B of the same type is (A, B) =
Z
dD xAaµ (x) Bµa (x)
(5.9)
The partition function is invariant under the gauge symmetries of Aµ . Further, Z can be expanded in powers of A, J, η¯, η, with the coefficients being expectation values of products
constructed by local operators, at vanishing sources. In what follows, Z[A, J, η¯, η] will be considered as a well defined formal power series of A, J, η¯, and η. The expectation value
5.3. The lattice background field method
58
of an operator in the presence of a background field is defined through 1 < O >B = NB
Z
D[Q]D[¯ c]D[c] O[A, Q, c¯, c] e−Stotal[A,Q,¯c,c]
(5.10)
(NB is chosen so that < 1 >B = 1). In order to write an expression for the effective action
W [A, J, η¯, η] = ln(Z[A, J, η¯, η])
ou
Γ[A, J, η¯, η], one must find the expansion of the generating functional for the connected diagrams W (5.11)
tin
Following the procedure described in Ref. [40] we derive the effective action ¯ C] = W [A, J, η¯, η] − (J, Q⋆ ) − (¯ ¯ η) Γ[A, Q⋆ , C, η , C) − (C, Q⋆ =
δW , δJ
c⋆ =
δW , δ η¯
δW δη
an
where
c¯⋆ = −
(5.12)
(5.13)
st
Γ corresponds to the background field effective action considered as a functional of A and evaluated at Q = 0. It can be obtained from the calculation of the 1-particle-irreducible, Green’s functions of the background field. Z and W are invariant under background
on
gauge transformations whereas all sources and ghost fields transform in the same way as Q does (see Eq. (5.4)). The vertex functions Γ(j,k,l) (j, k, l correspond to the number of background, quantum and ghost fields respectively, appearing in the vertex) can be
The lattice background field method
a
5.3
C
obtained by the series expansion of Eq. (5.12) in powers of its arguments.
th
On the lattice, the background field method can be approached in more than one ways. Many choices can be made for the action, and the difference between them is irrelevant
M ar
in the continuum limit (for lattice theories with all symmetries necessary for renormalizability). Consider a 4 − D lattice, whose sides are labelled by a four-vector x and each dimension is characterized by its unit vector µ ˆ (|ˆ µ|: one lattice spacing in direction µ). The
variable Uµ (x) relates the link connecting the points x and x + aˆ µ. The gauge fields are introduced via the link variables appearing in the action and a convenient decomposition of the links in the presence of a background field is Uµ (x) = eiag0 Qµ (x) · eiaAµ (x)
(5.14)
5.3. The lattice background field method
59
where Qµ (x) and Aµ (x) are noncommuting N × N hermitian matrices satisfying Qµ (x) = Qaµ (x)T a ,
Aµ (x) = Aaµ (x)T a ,
Tr[T a T b ] =
δ ab 2
(5.15)
ou
Since the dependence of the link variable on Q and A appears in different exponentials, the gauge transformations can be viewed in two alternative ways: a. The quantum field Qµ can be seen as a matter field with covariant transformation, while the background field transforms as a pure gauge field. This is equivalent to the continuum background transformations of the previous section
tin
an
⇒
QΛµ (x) = Λ(x)Qµ (x)Λ−1 (x) 1 AΛµ (x) = ln(Λ(x)eiaAµ Λ−1 (x + aˆ µ)) ih a ih i Λ iag0 Qµ (x) −1 iaAµ (x) −1 Uµ (x) = Λ(x)e Λ (x) Λ(x)e Λ (x + aˆ µ)
(5.16)
st
When the forward and backward lattice derivatives act on a matrix valued function f defined on lattice sites, one gets the relations
on
− →A D µ f (n) = eiaAµ (n) f (n + µ ˆ)e−iaAµ (n) − f (n) ← − −iaAµ (n−ˆ µ) DA f (n − µ ˆ)eiaAµ (n−µ) − f (n) µ f (n) = e
(5.17) (5.18)
C
These are covariant upon substituting Eqs. (5.16), that is
⇋
DA µ
Qν (n)
Λ
= Λ(n)
⇋ DA µ
Qν (n) Λ−1 (n)
(5.19)
th
a
b. The second interpretation of the gauge transformation is analogous to the continuum quantum transformation. This is derived by keeping the background field invariant with the transformation attributed to the gauge field
M ar
Λ Λ h i UµΛ (n) ≡ eiag0 Qµ (n) eiaAµ (n) = Λ(n)Uµ (n)Λ−1 (n + µ ˆ)e−iaAµ (n) eiaAµ (n)
(5.20)
The introduction of the background field is accompanied by a gauge fixing term in the action to absorb the gauge transformation of Eq. (5.20) and thus ensures a finite functional
5.3. The lattice background field method
60
integral. A proper choice is Sgf =
a4 X X Tr[Dµ− Qµ Dν− Qν ] ξ0 µ,ν x
(5.21)
The lattice derivatives appearing in Sgf are covariant, so that the gauge fixing term is
SF P = 2a
XX µ
x
† −1
Tr[Dµ cˆ(x) (M ) Qµ (x) · Dµ + ig0 Qµ (x) c(x)]
with M the matrix (more details can be found in Ref [41]) eiag0 Qµ − 1 iag0 Qµ
an
M(Qµ ) =
tin
4
ou
invariant under the background field transformations of Eqs. (5.16). The ghost action must also be modified and take the form
(5.22)
(5.23)
Note that the measure term is not affected by the presence of a background field and Ng02 X X Tr[Qµ (x)Qµ (x)] + O(g04 ) 12 µ x
(5.24)
on
Sm =
st
can be written as
The background field and coupling constant renormalization is determined by the 2point function of the background field; no renormalization for the quantum and ghost
C
field is needed. The reason for this is that these fields appear only within the loops of a diagram (external lines correspond to background fields) and their renormalization factor
M ar
th
a
would be cancelled with those of the propagators. The gauge fixing parameter also needs to be redefined, since the longitutinal part of the propagator must be renormalized. The renormalized quantities can be written with respect to the bare ones B0µ = ZB BRµ
1/2
(5.25)
g02 = Zg gR2
(5.26)
ξ 0 = Zξ ξ R
(5.27)
with BR , gR , ξR being the renormalized quantities. The fact that exact gauge invariance is preserved in the background field formalism, leads to the following relation between the renormalization constants ZB , Zg 1/2
Zg ZB = 1
(5.28)
5.4. Vertices of the overlap action in the background field method
5.4
61
Vertices of the overlap action in the background field method
The main consequence of the introduction of the background field is the appearance of different vertices in perturbative calculations. For each vertex with gauge fields, we must
ou
take into account all variants of quantum and background fields. Depending on the calculation performed, some of the variations of the vertices can be excluded. For instance, each vertex in a diagram cannot have more background fields than the external lines of the
tin
diagram. Thus, if the diagrams involved in a computation have only two external gluons and a certain number of loop gluons, the 3-gluon vertex cannot have all gluons as background fields. Let us demonstrate the variations of a fermion-antifermion-2gluons vertex
st
an
which has 3 contributions, as shown in Fig. 5.1.
on
Figure 5.1: The multiple character of a vertex in the background field method. Dashed lines represent gluon fields; those ending on a cross stand for background gluons. Solid lines represent fermions. The use of the background field technique implies that instead of the generic gluon fields
C
(in the case of the overlap action these appear in Xi ’s of Eqs. (3.21)), one must consider all different combinations of background (A) and quantum (Q) fields which originate in
th
a
the links of Eqs. (2.6), (5.14). As already illustrated, in the definition of the link variable, the quantum field appears on the left of the background field and the algebra generators of A, Q do not commute. Thus, while taking all possible permutations of {A, Q} we
M ar
eliminate the variants where the background fields are placed on the left of the quantum fields. Hence, for the β-function calculation Eqs. (3.21) are written as
X1 (p′ , p) = X1Q (p′ , p) + X1A (p′ , p)
X2 (p′ , p) = X2QQ (p′ , p) + X2QA (p′ , p) + X2AA (p′ , p) X3 (p′ , p) = X3QQQ(p′ , p) + X3QQA (p′ , p) + X3QAA (p′ , p) + X3AAA (p′ , p)
X4 (p′ , p) = X4QQQQ(p′ , p) + X4QQQA(p′ , p) + X4QQAA (p′ , p)+X4QAAA (p′ , p)+X4AAAA (p′ , p) (5.29) Eqs. (5.29) must be reinserted into Eqs. (3.35) - (3.38) to obtain the components of the
5.4. Vertices of the overlap action in the background field method
62
desired fermion-antifermion-gluon vertices (Eq. (3.34)). For the calculation of the third coefficient of the β-function, we must consider all vertices with a fermion-antifermion pair and up to 4 gluons. The involving 2-loop diagrams have 2 external gluons (A) and 2 internal (loop) gluons (Q). This allows us to to also exclude from Eqs. (5.29) the parts including
ou
more than two Q’s, or two A’s, that is X3QQQ , X3AAA , X4QQQQ, X4QQQA and X4AAAA . The expressions for Xi ’s including only one kind of gluons, can be directly read from Eqs. (3.21) (for background gluons we set g0 = 1). The next equations correspond to the remaining
M ar
th
a
C
on
st
an
tin
nontrivial vertices.
5.4. Vertices of the overlap action in the background field method
Z 4 4 d k1 d k2 ′ g0 = δ(p − p − k1 − k2 ) × 2 (2π)4 " p′ +p X V2,µ Qµ (k1 )Aµ (k2 ) + Aµ (k2 )Qµ (k1 ) 2 µ # p′ +p Qµ (k1 )Aµ (k2 ) − Aµ (k2 )Qµ (k1 ) +iaV1,µ 2
63
X2QA (p′ , p)
Z 4 4 4 g02 d k1 d k2 d k3 ′ = δ(p − p − k1 − k2 − k3 ) × 4 (2π)8 " p′ +p X −a2 V1,µ Qµ (k1 )Qµ (k2 )Aµ (k3 ) + Aµ (k3 )Qµ (k2 )Qµ (k1 ) 2 µ # p′ +p Qµ (k1 )Qµ (k2 )Aµ (k3 ) − Aµ (k3 )Qµ (k2 )Qµ (k1 ) +iaV2,µ 2
an
tin
X3QQA (p′ , p)
ou
(5.30)
st
Z 4 4 4 g0 d k1 d k2 d k3 ′ δ(p − p − k1 − k2 − k3 ) × = 4 (2π)8 " p′ +p X −a2 V1,µ Qµ (k1 )Aµ (k2 )Aµ (k3 ) + Aµ (k3 )Aµ (k2 )Qµ (k1 ) 2 µ # p′ +p +iaV2,µ Qµ (k1 )Aµ (k2 )Aµ (k3 ) − Aµ (k3 )Aµ (k2 )Qµ (k1 ) 2
(5.31)
(5.32)
C
on
X3QAA (p′ , p)
M ar
th
a
Z 4 4 4 g02 d k1 d k2 d k3 2d4 k4 ′ δ(p − p − k1 − k2 − k3 − k4 ) × = 8 (2π)12 " p′ +p X −a2 V2,µ Qµ (k1 )Qµ (k2 )Aµ (k3 )Aµ (k4 ) + Aµ (k4 )Aµ (k3 )Qµ (k2 )Qµ (k1 ) 2 µ # p′ +p −ia3 V1,µ Qµ (k1 )Qµ (k2 )Aµ (k3 )Aµ (k4 ) − Aµ (k4 )Aµ (k3 )Qµ (k2 )Qµ (k1 ) (5.33) 2
X4QQAA(p′ , p)
Upon substituting the expression for Xi ’s in the overlap vertices, the latter become
extremely lengthy and complicated. For instance, the vertex with Q-Q-A-Ψ-Ψ consists of 9,784 terms, while the vertex with Q-Q-A-A-Ψ-Ψ has 724,120 terms. As an example, we present below the vertex with Q-Q-A-Ψ-Ψ, written in the language of Xi ’s.
5.4. Vertices of the overlap action in the background field method
V13,QQA
64
" # 1 1 QQA QQA† X3 χ0 (k1 ) X3 + + = (k1 , k2 ) − (k1 , k2 ) χ0 (k2 ) ω(k1 ) + ω(k2 ) ω(k1 )ω(k2 ) Z 4 d k3 1 1 1 × + 4 (2π) ω(k1 ) + ω(k3 ) ω(k1 ) + ω(k2 ) ω(k2 ) + ω(k3 ) " h −X1Q (k1 , k3 ) χ†0 (k3 ) X2QA (k3 , k2 ) − X2QQ (k1 , k3 ) χ†0 (k3 ) X1A (k3 , k2 ) V23,QQA
V33,QQA
†
†
†
†
− X1Q (k1 , k3 ) X2QA (k3 , k2 ) χ0 (k2 ) − X2QQ (k1 , k3 ) X1A (k3 , k2 ) χ0 (k2 ) − X1A (k1 , k3 ) X2QQ (k3 , k2 ) χ0 (k2 ) − X2QA (k1 , k3 ) X1Q (k3 , k2 ) χ0 (k2 ) †
†
†
†
− χ0 (k1 ) X1A (k1 , k3 ) X2QQ (k3 , k2 ) − χ0 (k1 ) X2QA (k1 , k3 ) X1Q (k3 , k2 )
†
†
†
†
χ0 (k1 )X1Q (k1 , k3 )χ0 (k3 )X2QA (k3 , k2 )χ0 (k2 ) + χ0 (k1 )X1A (k1 , k3 )χ0 (k3 )X2QQ (k3 , k2 )χ0 (k2 ) +
† † χ0 (k1 )X2QQ (k1 , k3 )χ0 (k3 )X1A (k3 , k2 )χ0 (k2 )
Z Z
d4 k3 d4 k4 1 (2π)4 (2π)4 4
Y
1 ω(kp1 ) + ω(kp2 )
!"
+
† † χ0 (k1 )X2QA (k1 , k3 )χ0 (k3 )X1Q (k3 , k2 )χ0 (k2 )
! 1 X − ω(kp1 )ω(kp2 )ω(kp3 ) × 6
# i
st
h
ω(k1 ) + ω(k2 ) + ω(k3 ) × ω(k1 )ω(k2 )ω(k3 )
an
+
i
tin
− χ0 (k1 ) X1Q (k1 , k3 ) X2QA (k3 , k2 ) − χ0 (k1 ) X2QQ (k1 , k3 ) X1A (k3 , k2 )
ou
− X1A (k1 , k3 ) χ†0 (k3 ) X2QQ (k3 , k2 ) − X2QA (k1 , k3 ) χ†0 (k3 ) X1Q (k3 , k2 )
on
+ pǫS4 pǫS4 h i † † † X1Q (k1 , k3 )X1Q (k3 , k4 )X1A (k4 , k2 ) + X1Q (k1 , k3 )X1A (k3 , k4 )X1Q (k4 , k2 ) + X1A (k1 , k3 )X1Q (k3 , k4 )X1Q (k4 , k2 ) + ω(k1 ) + ω(k2 ) + ω(k3 ) + ω(k4 ) × h † † † χ0 (k1 )X1Q (k1 , k3 )X1Q (k3 , k4 )X1A (k4 , k2 )χ0 (k2 ) + χ0 (k1 )X1Q (k1 , k3 )X1Q (k3 , k4 )χ†0 (k4 )X1A (k4 , k2 ) †
†
†
C
+ χ0 (k1 )X1Q (k1 , k3 )χ0 (k3 )X1Q (k3 , k4 )X1A (k4 , k2 ) + X1Q (k1 , k3 )χ†0 (k3 )X1Q (k3 , k4 )X1A (k4 , k2 )χ0 (k2 ) †
†
+ X1Q (k1 , k3 )χ†0 (k3 )X1Q (k3 , k4 )χ†0 (k4 )X1A (k4 , k2 ) + X1Q (k1 , k3 )X1Q (k3 , k4 )χ0 (k4 )X1A (k4 , k2 )χ0 (k2 ) †
†
†
a
+ χ0 (k1 )X1Q (k1 , k3 )X1A (k3 , k4 )X1Q (k4 , k2 )χ0 (k2 ) + χ0 (k1 )X1Q (k1 , k3 )X1A (k3 , k4 )χ†0 (k4 )X1Q (k4 , k2 ) †
†
†
+ χ0 (k1 )X1Q (k1 , k3 )χ0 (k3 )X1A (k3 , k4 )X1Q (k4 , k2 ) + X1Q (k1 , k3 )χ†0 (k3 )X1A (k3 , k4 )X1Q (k4 , k2 )χ0 (k2 ) †
†
th
+ X1Q (k1 , k3 )χ†0 (k3 )X1A (k3 , k4 )χ†0 (k4 )X1Q (k4 , k2 ) + X1Q (k1 , k3 )X1A (k3 , k4 )χ0 (k4 )X1Q (k4 , k2 )χ0 (k2 ) †
†
†
+ χ0 (k1 )X1A (k1 , k3 )X1Q (k3 , k4 )X1Q (k4 , k2 )χ0 (k2 ) + χ0 (k1 )X1A (k1 , k3 )X1Q (k3 , k4 )χ†0 (k4 )X1Q (k4 , k2 ) †
†
†
M ar
†
†
+ χ0 (k1 )X1A (k1 , k3 )χ0 (k3 )X1Q (k3 , k4 )X1Q (k4 , k2 ) + X1A (k1 , k3 )χ†0 (k3 )X1Q (k3 , k4 )X1Q (k4 , k2 )χ0 (k2 ) i † † + X1A (k1 , k3 )χ†0 (k3 )X1Q (k3 , k4 )χ†0 (k4 )X1Q (k4 , k2 ) + X1A (k1 , k3 )X1Q (k3 , k4 )χ0 (k4 )X1Q (k4 , k2 )χ0 (k2 ) ! X ω(kp1 )ω(kp2 ) ω(kp1 )/2 + ω(kp3 )/3 × − ω(k1 )ω(k3 )ω(k4 )ω(k2 ) pǫS4 h † † † χ0 (k1 )X1Q (k1 , k3 )χ0 (k3 )X1Q (k3 , k4 )χ0 (k4 )X1A (k4 , k2 )χ0 (k2 ) †
+ χ0 (k1 )X1Q (k1 , k3 )χ0 (k3 )X1A (k3 , k4 )χ0 (k4 )X1Q (k4 , k2 )χ0 (k2 ) +
† † † χ0 (k1 )X1A (k1 , k3 )χ0 (k3 )X1Q (k3 , k4 )χ0 (k4 )X1Q (k4 , k2 )χ0 (k2 )
i
#
(5.34)
ou
Chapter 6
6.1
an
tin
The running coupling and the β-function Introduction
st
In the Standard Model’s Lagrangian, the interactions of the fundamental constituents
on
of matter are described by different coupling constants, characterizing the strength of the electromagnetic, weak or strong forces. QCD is the gauge field theory of the strong interaction between quarks and gluons which is fully defined by a few parameters, namely
C
the strong coupling constant αs and the quark masses; here we will focus on αs . A common attribute of all couplings is that they are not really constants, but they depend on the
a
energy transfer µ in the interaction process. This explains why we often refer to them as running couplings (they ‘run’ with the energy scale). QCD has specific features, asymptotic freedom and confinement, which determine the
th
behavior of quarks and gluons in particle reactions at high and at low energy scales. It predicts that αs decrease with increasing energy or momentum transfer, and vanishes at
M ar
asymptotically high energies. For procedures with energy transfer within the interval 10100GeV, the predictions of the theory can be worked out as an expansion in powers of the corresponding coupling constant; the so-called perturbative expansion. For the strong interaction, at energy scale µ ∼ 100GeV, the coupling constant has been found to be αs = 0.12. At this scale, the perturbation theory works very well. However, if the energy scale decreases from 100GeV, the αs increases and at µ ∼ 1GeV it is so large that the perturbative expansion is not reliable. Nevertheless, low energy experiments are important since we can observe the world of hadrons, which perturbation theory has completely failed 65
6.1. Introduction
66
to describe. A nonperturbative solution of the theory is required to deal with the large αs regime. Over the years, numerical techniques have been developed, the numerical simulations of QCD formulated on a discrete lattice of space-time points. This method allows the nonperturbative study of the theory for low-energy scales.
ou
At the low-energy regime, the lattice formulation in cooperation with numerical simulations are used to systematically extract predictions of QCD. An enormous number of publications appear for the nonperturbative estimation of the strong coupling constant.
Many of these studies have been performed by the ALPHA Collaboration [46]. One of the main goal of the group is the calculation of the strong coupling constant, using numerical
tin
simulations combined with perturbative renormalization. The emphasis of its long-term projects is given on precision and control of the systematic errors. The aim is not only to compute the running coupling over a wide range of low energies, but to also reach a
an
range of high energies in order to make comparisons with perturbation theory. Some of the nonperturbative studies on the strong coupling constant in the quenched approximation
st
of QCD, but also in the unquenched, can be found in Refs. [47, 48, 49, 51, 52, 53]. The coefficients of the perturbative expansion of αs and of β-function are known for various fermion and gluon actions [54, 55, 56, 57, 58, 59, 60, 45].
on
A number of experimental tests of the strong coupling behavior are summarized in Ref. [61]. A world summary of measurements of αs , leads to an unambiguous verification of the running of αs and of asymptotic freedom, in excellent agreement with the predictions
C
of QCD. It has become standard to evaluate αs choosing the energy scale to be the mass of the Z-boson, µ = MZ 0 . Averaging the set of measurements shown in Fig. 6.1, balanced
a
between different particle processes and the available energy range, one finds αs (MZ 0 ) = 0.1189 ± 0.0010. The error estimates of Fig. 6.1 include the theoretical uncertainties. It is interesting to see that the lattice contribution to αs (MZ 0 ) has relatively small error bars
M ar
th
and it is very close to the average value of all processes. Onwards we omit the index s of αs for simplicity.
67
st
an
tin
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6.2. Renormalization group equation and β-function
Renormalization group equation and β-function
C
6.2
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Figure 6.1: The value of αs (MZ 0 ) as derived from various processes and the average of these measurements [62]. This graph has been taken from Review of Particle Physics (2006).
The introduction of the lattice as a regulator, causes the appearance of a new parameter, the lattice spacing a, expressing the distance between neighboring lattice points. Naturally, for
a
lattice theories, the measurable quantities depend on a, but must recover their continuum value when setting a → 0. One of the most important quantities to study is the coupling
M ar
th
constant and its connection with a and other observables. Unavoidably, one has to define the bare and renormalized value for each quantity. When referring to the bare value, we have in mind the one involved in the Lagrangian, while the renormalized is the one compared to experimental measurements. In general, bare values depend on a. Let us study a system living on a 4-dimensional lattice. A fluctuation of the lattice spacing, changes the number of lattice points and links. The real system is not affected by the mathematical formalism of the lattice, so the bare parameters must be tuned with a, so that the observables are not affected. We now define a measurable quantity O with dimensions of mass (or inverse length)
6.2. Renormalization group equation and β-function
68
ˆ depending on the bare parameters of dO. On the lattice, it takes a dimensionless form O the theory. The existence of the continuum limit implies that O(g0 , a) =
ˆ O adO
(6.1)
requiring at the same time lim O(g0 (a), a) = Ophysical
(6.2)
ou
a→0
tin
ˆ = adO Ophysical . Knowledge of For small a’s, Eqs. (6.1) - (6.2) become equal, therefore O ˆ allows the determination of g0 with respect to a and Ophysical ; hence, the the function O coupling constant must be accompanied by a measurable quantity. For a small enough, there is a global g0 (a) applicable to all quantities. This is convenient, since we can choose
an
a particular quantity to define g0 (a), for instance the quark-antiquark static potential. We assume that the quark-antiquark pair is separated by a distance R measured in physical units, and its lattice version is given by
st
1 R V (R, g0 (a), a) = Vˆ ( , g0(a)) a a
(6.3)
on
where R/a is their separation distance in lattice units. This is our initial point: Despite the variation of the lattice spacing the observable V (R, g0 (a), a) must be invariant. This is ensured by d V (R, g0 , a) = 0 da " ! # ∂ ∂g0 ∂ ⇒ a − −a V (R, g0 , a) = 0 ∂a ∂a ∂g0
th
a
C
a
(6.4)
M ar
The above equation is known as the Renormalization group equation, from which the Callan-Symanzik β-function is defined in its lattice form ∂g0 βL (g0 ) = −a ∂a
(6.5)
g, µ ¯
where µ ¯ is the renormalization scale and g (g0 ) the renormalized (bare) coupling constant. The lattice β-function expresses how the g0 changes under a variation of a. The renormalized coupling constant g depends on the renormalization scheme (parameterized by a scale
6.2. Renormalization group equation and β-function
69
µ), and this dependence is given in the same manner, by the renormalized β-function β(g) ≡ µ
dg dµ
(6.6)
We will adopt the modified minimal subtraction MS scheme, the most commonly used scheme for the analysis of experimental data. MS involves momentum integrals in D =
ou
4 − 2ǫ dimensions and subtracts off the resulting 1/ǫ poles and also ln(4π) − γE (γE is the Euler-Mascheroni constant).
tin
We relate α0 to the renormalized coupling constant α MS as defined in the MS scheme at a scale µ ¯; at large momenta, these quantities are associated as follows α MS (¯ µ) = α0 + d1 (¯ µa)α02 + d2 (¯ µa)α03 + ...
(6.7)
an
(α0 = g02 /4π, α MS = g 2MS /4π). The running coupling is a QCD quantity with high impor-
st
tance and its precise determination would fix the value of a fundamental parameter in the Standard Model. Moreover, the observables of an experimental process depending on an overall energy scale µ and some kinematical variables y, can be computed in a perturbative
on
series which is usually written in terms of the MS coupling constant 2 O(µ, y) = αMS (µ) + c(y)αMS (µ) + ...
(6.8)
C
From now on, we will denote the renormalized coupling constant g MS simply by g. It is well known that in the asymptotic limit for QCD (g0 → 0), one can write the expansion of the β-function in powers of the coupling constants, that is
th
a
βL (g0 ) = −b0 g03 − b1 g05 − bL2 g07 − ...
β(g) = −b0 g 3 − b1 g 5 − b2 g07 − ...
(6.9) (6.10)
M ar
The coefficients b0 , b1 are universal constants (regularization independent) provided in Eqs. (6.11), and are computed from 1- and 2-loop diagrams. On the contrary, bLi (i ≥ 2)
depends on the regulator; it must be determined perturbatively. There are alternative methods to determine b0 , b1 , among them, the static quark potential approach [63] (in the quenched approximation of QCD), involving a phenomenological parametrization of the
6.2. Renormalization group equation and β-function
70
interquark potential. Their expressions for any color (N) and flavor (Nf ) number is 11 2 N − Nf 3 3 34 2 13 1 1 b1 = N − Nf N− (4π)4 3 3 N
1 b0 = (4π)2
ou
The coefficient b2 in the MS scheme is
(6.11)
(6.12)
tin
1 2857 3 1709N 2 187 56N 1 11 2 b2 = + Nf N + Nf − + + − (4π)6 54 54 36 4N 2 27 18N
Coefficients d1 , d2 of Eq. (6.7) can be extracted from b0 , b1 , b2 , bL2 using Eqs. (6.9), (6.10), (7.5) and (7.6).
an
For Nf < 11N/3 the expansion of the β-function begins with a negative term, in other words, in the asymptotic high energy regime, the coupling decreases logarithmically with increasing energy. It reflect the observation that at high energy, quarks behave like free
st
particles. The universality of b0 , b1 means that they are independent of the definition of the renormalized coupling constant, as long as the bare coupling is small (gR = g0 + O(g02 )).
on
This statement can be proven in the following way by considering two renormalization couplings gA and gB , which can be expanded in powers of each other. Since the only dimensionless parameters are gA , gB , then gA is a function only of gB and vice versa
C
gA (gB ) = gB + c gB2 + O(gB3 )
(6.14)
a
gB (gA ) = gA − c gA2 + O(gA3 )
(6.13)
M ar
th
The leading coefficient is set to unity because of the condition that both gA , gB are equal to g0 at leading order. The two β-functions can be related using the definition of Eq. (6.6) and Eq. (6.13) ∂gA dgA (6.13) ∂gB ∂gA (6.14) βA (gA ) = µ = µ = βB (gB ) (6.15) dµ ∂µ ∂gB ∂gB where
∂gA (6.13) (6.14) = 1 + 2c gB + O(gB2 ) = 1 + 2c gA + O(gA2 ) ∂gB
(6.16)
On the other hand, an analogous equation can be written for βB (gB ) in terms of gA (6.14)
2 B 3 4 B 2 B B 3 4 βB (gA ) = −bB 0 gB − b1 gB + O(gB ) = −b0 gA − (b1 − 2c b0 ) gA + O(gA )
(6.17)
6.2. Renormalization group equation and β-function
71
Substituting the necessary ingredients back to Eq. (6.15), one finds 2 B B 3 4 2 βA (gA ) = [−bB 0 gA − (b1 − 2c b0 ) gA + O(gA )] · [1 + 2c gA + O(gA )] 2 B 3 4 = −bB 0 gA − b1 gA + O(gA )
(6.18)
B A B proving that bA 0 = b0 and b1 = b1 , that is they do not depend on the regularization nor
∂ ∂ Λ=0 −µ + β(g) ∂µ ∂g
This is expressed by the exact relation e
Z exp −
0
g
dg
′
1 1 b1 + − 2 ′ ′ ′ 3 β(g ) b0 (g ) b0 g
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Λ = µ b0 g
2 2 −b1/(2b0 ) −1/(2b0 g 2 )
(6.19)
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invariant parameter Λ
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the renormalization scheme. The solution to Eqs. (6.6) contains an integration constant, the renormalization group
(6.20)
and can be expanded in powers of g ′ or equivalently g0 . This implies that the energy
relation
on
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dependence of α can be specified completely in terms of the Λ parameter measured in energy units. The basic lattice mass was introduce as a lattice parameter ΛL . By definition, it is connected with the lattice spacing and the bare coupling via the following asymptotic
1 aΛL = exp − 2b0 g02
2 (b0 g02 )−b1 /2b0 1 + qg02 + O g04
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where is q called the correction factor
a
q=
b21 − b0 bL2 2b30
(6.21)
(6.22)
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Knowledge of the first correction to the 2-loop approximation of ΛL is important in order
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to verify the asymptotic prediction, Eq. (6.21). For large µ, the asymptotic solution of the renormalization group equation reads h 4π 2b1 ln[ln(µ2 /Λ2 )] 1 − b0 ln(µ2 /Λ2 ) b20 b20 ln(µ2 /Λ2 ) 4b2 1 b2 b0 5 i + 4 2 12 2 (ln[ln(µ2 /Λ2 )] − )2 + 2 − 2 8b1 4 b0 ln (µ /Λ ) µ→∞
α2 (µ) ∼
(6.23) (6.24)
and the integration constant Λ can be regarded as a fundamental parameter of QCD; once
6.3. The step scaling function
72
it is known, the running coupling of the strong interactions is fixes at all scales. Eq. (6.24) illustrates the asymptotic freedom property (αs → 0 for µ → ∞) and shows that QCD become strongly coupled at µ ∼ Λ. Although Λ and b2 depend on the renormalization scheme, they can be easily transformed between different schemes through the 1- and 2-loop coefficient relating the strong couplings in those schemes
⇒
−dA 1 ΛA = e 8πb0 ΛB
⇒
A A 2 B A bA 2 = b2 − b1 d1 + b0 (d2 − (d1 ) )
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2 A 3 αA (µ) = αB (µ) + dA 1 (µ)αB (µ) + d2 (µ)αB (µ) + ...
(6.25)
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Note that both couplings are taken at the same energy scale and the coefficients in their perturbative relation are pure numbers.
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The significance of the Λ parameter is that it can be given as an input parameter for perturbative predictions of jet cross sections and compare to high energy experiments, in order to test the agreement between theory and experiment. QCD on the lattice is renor-
The step scaling function
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6.3
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malized through the hadron spectrum and such a calculation would reveal the connection between low and high energies.
a
As mentioned in the introduction, numerical simulations are necessary for calculations in the high energy region, in order to make comparisons with estimates for αs (or Λ param-
th
eter) coming from perturbation theory. It is difficult to study this region by numerical simulations, due to the large scale separation involved. Actually, it requires simulations
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with a cutoff a−1 much larger than the largest energy scale and a lattice with large size L4 (L larger than a pion’s Compton wavelength). That is, to avoid discretization and finite-size errors a, µ, L must obey a−1 >> µperturbative >> µhadronic >> L−1
meaning that numerical simulations have to be performed on a lattice with L/a >> 70. The method that has been developed involves an intermediate finite-volume renormalization
6.3. The step scaling function
73
scheme, in which the scale evolution of the coupling is computed recursively from low to very high energies. The connection between the perturbative region of QCD and the nonperturbative hadronic regime requires a nonperturbative definition of the coupling constant. Using
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this definition, one should be able to calculate the coupling on the lattice with a small error. It is also important to have small cutoff effects and a perturbative expansion which is relatively easy to calculate up to 2 loops. A popular definition is that given by the Schr¨odinger functional (SF), which is the propagation amplitude for going from some field configuration at the time x0 = 0 to another configuration at x0 = T ; an introduction to
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SF can be found in Ref. [64]. In this scheme, the coupling’s dependence on the energy scale µ is associated with the running of a coupling g¯2 (L) where L = µ−1 . Starting from the low energy scale (L → Lmax ),
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at which g¯2 (Lmax ) ≡ u0 is fixed at some value, we double the size of L.3 The coupling at the scale 2L is related to g¯2(L) through a unique function, the step scaling function
(6.26)
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g¯(2L) ≡ u1 = σ(¯ g (L))
σ(¯ g 2 (L)) is the discrete version of the β-function, describing how the coupling constant
on
alters under scale variations. Using this function, one can compute recursively the coupling at scales 2−n Lmax un = g¯2 (2−n L) (6.27)
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over several orders of magnitude in µ, while keeping the lattice size at a manageable level. The step scaling function is computed with Monte Carlo simulations. The starting
th
a
point is a simulation on a lattice with a certain number of lattice points in each direction, L/a, and the bare coupling constant must tuned to such that the renormalized coupling gˆ has the desired value u. The next step is a simulation at 2L/a using the same bare
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parameters. This provides an estimate for the coupling constant called Σ(un , a/L) which is an approximation of the step scaling function σ(un ). In fact, there appear lattice artifacts, that is σ(un ) = Σ(un , a/L) + O(a2 )
(6.28)
The above procedure is repeated for different resolutions L/a and σ(un ) can be obtained by extrapolating to the continuum limit. Note that the whole procedure of doubling the 3
In general, we can change the value of L by a factor s, that is L → sL. Here, we use the most common choice, s = 2.
6.3. The step scaling function
74
lattice size is equivalent to keeping the L fixed and changing the lattice spacing. The couplings at L and 2L are related through the integral ln(2) =
Z
2L
L
dL′ =− L′
Z
g¯(2L)
g¯(L)
dg β(g)
(6.29)
(6.30)
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σ(un ) = un + s0 u2n + s1 u3n + s2 u4n + ...
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such that the perturbative expansion of σ(u) can be derived from the asymptotic expansion of the β-function (Eq. (6.9))
where s0 = 2b0 ln(2)
(6.31)
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s1 = s20 + 2b1 ln(2)) 5 s2 = s30 + s0 s1 + 2b2 ln(2) 2
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From the determination of the scaling function, the evolution of the coupling can be computed straightforwardly by solving the recursive equations u0 = g¯2 (Lmax ), b un = σ(un+1)forn = 0, 1, 2, ...
(6.32)
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As an example, let us take u0 = 3.480 and demand that σ(u) should be used only in the range of couplings covered by the data, the recursion after six steps gives
a
g¯ = 1.053(12),
L = 2−6 Lmax
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The error estimates comes from propagating the statistical errors to the fit polynomial and
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solving the recursion using this function.
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Chapter 7
7.1
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QCD with overlap fermions: Running coupling and the 3-loop β-function Introduction
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In later years, use of nonultralocal actions which preserve chiral symmetry on the lattice
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has become more viable for numerical simulations. The two actions which are being used most frequently are overlap fermions [5, 6, 7] based on the Wilson fermion action and domain-wall fermions [8, 9]. Overlap fermions are notoriously difficult to study, both
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numerically and analytically. Many recent promising investigations involving simulations with overlap fermions have appeared; see, e.g., Refs. [65, 66, 67, 68, 69, 70, 71]. Regarding
a
analytical computations, the only ones performed thus far have been either up to 1 loop, such as Refs. [58, 72, 73, 74, 75, 76, 77], or vacuum diagrams at higher loops [78, 79]. The calculation presented in this chapter is the first one involving nonvacuum diagrams beyond
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the 1-loop level. We compute the 2-loop renormalization Zg of the bare lattice coupling constant g0 in
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the presence of overlap fermions and Wilson gluons. For convenience, we have worked with the background field technique, which only requires evaluation of 2-point Green’s function for the problem at hand. We relate g0 to the renormalized coupling constant g MS as defined
in the MS scheme at a scale µ ¯; at large momenta, these quantities are related as follows α MS (¯ µ) = α0 + d1 (¯ µa)α02 + d2 (¯ µa)α03 + ...
(7.1)
(α0 = g02/4π, α MS = g 2MS /4π, a : lattice spacing). The 1-loop coefficient d1 (¯ µa) has been 75
7.2. Theoretical background
76
known for a long time; several evaluations of d2 (¯ µa) have also appeared in the past ∼10 years, either in the absence of fermions [45, 80], or using the Wilson [59] or clover [81, 82] fermionic actions. Knowledge of d2 (¯ µa), together with the 3-loop MS-renormalized βfunction [83], allows us to derive the 3-loop bare lattice β-function, which dictates the
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dependence of lattice spacing on g0 . In particular, it provides a correction to the standard 2-loop asymptotic scaling formula defining ΛL . Ongoing efforts to estimate the running coupling from the lattice [64, 48, 84, 85] have relied on a mixture of perturbative and nonperturbative investigations. As a particular example, relating α MS to αSF (SF: Schr¨odinger Functional scheme, as advocated by the ALPHA Collaboration), entails an intermediate
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passage through the bare coupling α0 ; the conversion from α MS to α0 is then carried out perturbatively. While the main application of SU(N) gauge theories on the lattice regards QCD, where
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fermions are in the fundamental representation of SU(3), there has recently been some interest in gauge theories with fermions in other representations and with N 6= 3. Such
st
theories are being studied in various contexts [86, 87, 88, 89, 90, 91, 92, 93], e.g., supersymmetry, phase transitions, and the ’AdS/QCD’ correspondence. Our results depend explicitly on the number of fermion flavors (Nf ) and colors (N).
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Since the dependence of Zg on the overlap parameter ρ cannot be extracted analytically, we tabulate our results for different values in the allowed range of ρ (0 < ρ < 2), focusing on values which are being used most frequently in simulations. We also provide expressions
Theoretical background
a
7.2
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for our results for fermions being in an arbitrary representation, which has led to a separate publication [94].
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The renormalized β-function describes the dependence of the renormalized coupling con-
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stant g on the scale inherent in the renormalization scheme (chosen to be the MS scheme). A more extended introduction on the running coupling and the β-function is given in Chapter 6. A bare β-function is also defined for the lattice regularization (βL (g0 )) dgMS β(gMS ) = µ ¯ , d¯ µ a,g0
dg0 βL (g0 ) = −a da gMS , µ¯
(7.2)
where a is the lattice spacing, µ ¯ the renormalization scale and gMS (g0 ) the renormalized (bare) coupling constant. In the asymptotic limit, one can write the expansion of Eq. (7.2)
7.2. Theoretical background
77
in powers of g0 βL (g0 ) = −b0 g03 − b1 g05 − bL2 g07 − ...
(7.3)
3 5 7 β(gMS ) = −b0 gMS − b1 gMS − b2 gMS + ...
(7.4)
with the coefficients b0 , b1 being universal constants and regularization independent (Eq. (6.11)).
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On the contrary, bLi (i ≥ 2) depends on the regulator; it must be determined perturbatively. Here we present the calculation of bL2 using the overlap fermionic action and Wilson gluons.
tin
Bare (βL (g0 )) and renormalized (β(g)) β-functions can be related using the renormalization function Zg , defined through g0 = Zg (g0 , a¯ µ)g , that is 4 −1 ∂ ln Zg2 2 β (g0 ) = 1 − g0 Zg β(g0 Zg−1 ) ∂g02
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L
Computing Zg2 to 2 loops
(7.6)
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Zg2 (g0 , a¯ µ) = 1 + g02 (2b0 ln(a¯ µ) + l0 ) + g04 (2b1 ln(a¯ µ) + l1 ) + O(g06)
(7.5)
and inserting it in Eq. (7.5), allows us to extract the 3-loop coefficient bL2 . The quantities
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b0 , b1 , b2 and l0 have been known in the literature for quite some time [83, 58]; b0 and b1 are the same as those of the bare β-function, Eq. (6.11), and b2 in the MS scheme is
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2857 3 56N 1 1709N 2 187 1 11 2 + Nf b2 = N + Nf − + + − (4π)6 54 54 36 4N 2 27 18N
(7.7)
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a
The constant l0 is related to the ratio of the Λ parameters associated with the particular lattice regularization and the MS renormalization scheme l0 = 2b0 ln (ΛL /Λ MS)
(7.8)
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For overlap fermions the exact form of l0 appears in Ref. [58] 1 5 (1) l0 = − 0.16995599N + Nf − − k (ρ) 8N 72π 2
(7.9)
where k (1) (ρ) is the convergent part of the 1-loop fermionic contribution (denoted by kf (ρ)
Zg could be denoted: ZgL, MS renormalization scheme (MS). 4
to indicate its dependence on the regulator (L: lattice) and on the
7.2. Theoretical background
78
in Ref. [58]), presented in Table 7.1. Eq. (7.5) is valid order by order in perturbation theory and expanding it in powers of g02 the first nontrivial relation is bL2 = b2 − b1 l0 + b0 l1
(7.10)
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Thus, the evaluation of bL2 requires only the determination of the 2-loop quantity l1 . A direct outcome of our calculation is the 2-loop corrected asymptotic scaling relation between
valid (see Eq. (5.28))
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ZA (g0 , a¯ µ)Zg2 (g0 , a¯ µ) = 1
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a and g0 (Eq. (6.21)) where all quantities in the correction term q, except bL2 , are known. The most convenient and economical way to proceed with calculating Zg (g0 , a¯ µ) is to use the background field technique described in Chapter 5, in which the following relation is
(7.11)
where ZA is the background field renormalization function
(7.12)
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Aµ (x) = ZA (g0 , a¯ µ)1/2 AµR (x)
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(Aµ (AµR ): bare (renormalized) background field). In the notation of Chapter 5, the quantum field is represented by Qµ (x). In this framework, instead of Zg (g0 , a¯ µ), it suffices to compute ZA (g0 , a¯ µ) with no need to evaluate any 3-point functions. For this purpose,
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we consider the background field one-particle irreducible (1PI) 2-point function, both in ab the continuum (dimensional regularization, MS subtraction): ΓAA R (p)µν and on the lattice: ab ΓAA L (p)µν . In the notation of Ref. [45], these 2-point functions can be expressed in terms of scalar functions νR (p), ν(p)
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a
ab ab ΓAA δµν p2 − pµ pν (1 − νR (p)) /g 2 R (p)µν = −δ (1)
(7.13)
(2)
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νR (p) = g 2 νR (p) + g 4νR (p) + ...
X µ
ab ab ΓAA p2 [1 − ν(p)] /g02 L (p)µµ = −δ 3b
(ˆ pµ = (2/a) sin(aqµ /2)).
ν(p) = g02 ν (1) (p) + g04ν (2) (p) + ...
(7.14)
7.2. Theoretical background
79
There follows ZA =
1 − νR (p, µ ¯ , g) 1 − ν(p, a, g0 )
(7.15)
The gauge parameter λ must also be renormalized (up to 1 loop), in order to compare lattice and continuum results (1)
ZQ = 1 + g02 zQ + ...
(7.16)
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λ = Z Q λ0 ,
(ZQ : renormalization function of the quantum field). Using the quantum field 1PI 2-point QQ ab ab function in the continuum (ΓQQ R (p)µν ) and on the lattice (ΓL (p)µν ) through
δµν p2 − pµ pν (1 − ωR (p)) + λpµ pν
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ab ab ΓQQ R (p)µν = −δ
(1)
µ
ab ab 2 ΓQQ b [3 (1 − ω(p)) + λ0 ] L (p)µµ = −δ p
(7.18)
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X
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ωR (p) = g 2 ωR (p) + O(g 4 )
(7.17)
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ω(p) = g02 ω (1) (p) + O(g04 ) (1)
one can obtain the coefficient zQ (1)
(1)
zQ = ω (1) (p, a, g0 ) − ωR (p, µ ¯ , g)
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(7.19)
form
a
In terms of the perturbative expansions Eqs. (7.13), (7.14), (7.17), (7.18), Zg2 takes the
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(1) i h (1) (2) (1) ∂ν Zg2 = 1 + g02 (νR − ν (1) ) + g04 (νR − ν (2) ) + λ0 g04 (ω (1) − ωR ) R ∂λ λ=λ0
(7.20)
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The fermion part of ω (1) coincides with that of ν (1) , and similarly for the fermion part of (1) (1) ωR and νR . Consequently, one may write (1)
(1)
(1)
(1)
ω (1) − ωR = [ω (1) − ωR ]Nf =0 + [ν (1) − νR ] − [ν (1) − νR ]Nf =0
(7.21)
Since the quantities of interest are gauge invariant, we choose to work in the bare Feynman gauge, λ0 = 1, for convenience. In order to compute ZA we need the expressions for
7.2. Theoretical background (1)
(2)
80
(1)
νR , νR , zQ , ν (1) , ν (2) . The MS renormalized functions necessary for this calculation to 2 loops are [45, 59] (1) νR (p, λ)
N 11 p2 205 Nf 2 p2 3 1 10 = − ln 2 + + + + ln 2 − 16π 2 3 µ ¯ 36 2λ 4λ2 16π 2 3 µ ¯ 9
(1) ωR (p, λ)
N = 16π 2
p2 577 N2 −8 ln 2 + = 1) = − 6ζ(3) + µ ¯ 18 (16π 2 )2
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13 p2 97 Nf 2 p2 1 1 1 10 ln 2 + + (7.23) − + + + ln 2 − 6 2λ µ ¯ 36 2λ 4λ2 16π 2 3 µ ¯ 9
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(2) νR (p, λ
(7.22)
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401 Nf p2 1 p2 55 N 3 ln 2 − + − ln 2 + − 4ζ(3) µ ¯ 36 N µ ¯ 12 (16π 2 )2
(7.24)
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For the lattice quantities, the gluonic contributions (Nf = 0) have been presented in
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previous works [45, 80] (for the Wilson gauge action)
5N 1 ln (a2 p2 ) − + 0.137286278291N 2 48π 8N
(7.25)
ν (1) (p, λ0 = 1) = −
1 11N ln (a2 p2 ) − + 0.217098494367N 2 48π 8N
(7.26)
ν (2) (p, λ0 = 1) = −
N 3 ln (a2 p2 ) + − 0.01654461954 + 0.0074438722N 2 (7.27) 4 32π 128N 2
a
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ω (1) (p, λ0 = 1) = −
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The 1- and 2-loop coefficients d1 (µa), d2 (µa) relating the bare and renormalized coupling constants (Eq. (7.1)) can be directly evaluated from the above quantities
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h i (1) (1) d1 (µa) = −4π νR (p) − ν (p) d2 (µa) =
λ=λ0
d21 (µa)
(1) i h (2) (1) ∂νR (2) (1) − 4π νR (p) − ν (p) + λ0 (ω − ωR ) ∂λ λ=λ0 2
(7.28)
(7.29)
The fermionic contributions are associated with the diagrams of Fig. 7.1 and Fig. 7.2. In
the present work, ν (2) is perturbatively calculated for the first time using overlap fermions
7.2. Theoretical background
81
and Wilson gluons. For completeness, we also compute the coefficient ν (1) and compare it with previous results. The 1-loop diagrams (Fig. 7.1) correspond to ν (1) , and the 2-loop diagrams (Fig. 7.2) lead to ν (2) . Dashed lines ending in a cross represent the background field, while those inside loops denote the quantum field. Solid lines correspond to fermion fields
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and a dot stands for the mass counterterm. Note that, for overlap fermions, the mass counterterm equals zero, by virtue of the exact chiral symmetry of the overlap action; consequently, diagrams 19 and 20 both vanish. Certain 2-loop diagrams have infrared di-
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th
a
C
on
st
an
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vergences and become convergent only when grouped together (6+12, 7+11, 8+18, 9+17).
7.2. Theoretical background
82
1
2
2
5
6
3
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1
4
8
11
12
14
15
16
18
19
20
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7
10
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th
13
a
C
9
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Figure 7.1: Fermion contributions to the 1-loop function ν (1) . Dashed lines ending on a cross represent background gluons. Solid lines represent fermions.
17
Figure 7.2: Fermion contributions to the 2-loop function ν (2) . Dashed lines represent gluonic fields; those ending on a cross stand for background gluons. Solid lines represent fermions. The filled circle is a 1-loop fermion mass counterterm.
7.3. Description of the calculation
7.3
83
Description of the calculation
In recent years, overlap fermions are being used ever more extensively in numerical simulations, both in the quenched approximation and beyond. This fact, along with the desirable properties of the overlap action, was our motivation to calculate the β-function with this type of fermions. The important advantage of the overlap action is that it preserves chiral
ou
symmetry while avoiding fermion doubling. It is also O(a) improved. The main drawback of this action is that it is necessarily nonultralocal; as a consequence, both numerical simulations and perturbative studies are extremely difficult and demanding (in terms of human,
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as well as computer time). Let us remind that the overlap action is given by Eq. (3.18), its overlap parameter ρ is restricted by the condition 0 < ρ < 2 and the coupling constant is
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included in the link variables, present in the definition of X. The perturbative expansion of X in powers of g0 leads to the vertices needed for our calculation, a procedure described in Chapter 3. The resulting vertices are extremely lengthy and the implementation of the background field technique enlarges the expressions even more.
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• Algebraic manipulations:
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For the algebra involving lattice quantities, we make use of our symbolic manipulation package in Mathematica, with the inclusion of the additional overlap vertices, written in
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the background field language. These are enumerated to 4 different vertices, giving a total of 743,968 terms when expanded. The first step to evaluate the diagrams is the contraction among vertices, a step performed automatically once the vertices and the ‘incidence matrix’
a
of the diagram are specified. The outcome of the contraction is a preliminary expression for the diagram under study; there follow simplifications of the color dependence, Dirac
th
matrices and tensor structures. We use symmetries of the theory (permutation symmetry and lattice rotational invariance), or any other additional symmetry that may appear in individual diagrams, to keep the size of the expression down to a minimum. A significant
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feature of the overlap vertices is their non-pointlike nature. This becomes obvious if we go back to Eq. (3.34) and realize that each vertex comprises different Vji ’s (the index i symbolizes the number of gluon in the vertex and j represents the number of integrals over
dummy momenta). The most convenient and computer RAM saving way to manage each diagram is to divide it into subdiagrams, coming from different parts (Vji ’s) of the vertices.
For example, let us illustrate how diagram 5 of Fig. 7.2 is drawn apart into subdiagrams.
7.3. Description of the calculation
84
The particular diagram is constituted of two identical vertices: Q-A-Ψ-Ψ. This vertex consists of the parts V12 , V22 and the resultant subdiagrams are four, coming from the contraction of {V12 , V12 }, {V12 , V22 }, {V22 , V12 }, or {V22 , V22 }. Of course, the second and third subdiagrams, have been worked out together, because they are effectively equivalent. Some
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subdiagrams, due to oversized expressions, need to be divided into smaller contributions according to the appearing order of the background and gluons fields (for example, the 2-loop diagram 1 the two background fields arise from the same vertex, and these could
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belong to the same Xi or not. This gives us a criterion to organize the diagram into smaller contributions that are manipulated separately). • Logarithmic Contributions:
diagram may in principle depend on q as follows 2
2
2
2
2
2
2 2
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The external momentum q appears in arguments of trigonometric functions and each
4 µ qµ q2
+ O(q 4 , q 4 ln a2 q 2 )
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α0 + α1 q + α2 q ln a q + α3 q (ln a q ) + α4
P
(7.30)
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where the coefficients αi are typically 2-loop integrals with no external momenta, which must be evaluated numerically. (a.) The most demanding calculation is that of α2 and α3 , due to the required procedure;
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these terms are isolated from the convergent ones. Certain diagrams (15-18) have superfi-
a
cially divergent terms, which are responsible for the double logarithms. Diagrams 2, 3, 8, 9, 10 and 13-18 have subdivergences (a few thousand terms) leading to single logarithms. All divergent terms are obtain by applying 2 different types of subtractions: First we replace
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th
the fermion propagator with the gluon propagator, via 1 1 1 1 = + − D0 (k) D0 (k) kˆ2 kˆ2
(7.31)
where k can be the loop momentum p or p + aq. D0 (k) is the inverse overlap propagator P (Eq. (3.32)) and kˆ2 ≡ 4 µ sin2 (kµ /2) is the inverse gluon propagator (in the Feynman gauge). Eq. (7.31) is necessary, since the divergent integrals that appear in the literature are given in terms of kˆ 2 . The terms in parenthesis have 2 more powers of momentum than the l.h.s and in most cases are directly added to the rest of the convergent part. For some terms the repetition of this subtraction is needed until the overlap propagator
7.3. Description of the calculation
85
is totally eliminated from all divergent terms. Then follows another kind of subtraction in order to extract the external momentum from the gluon propagators (which cannot be accomplished with naive Taylor expansion) 1 1 1 1 = + − (ˆ p )2 p )2 (p\ + aq)2 (p\ + aq)2 (ˆ
(7.32)
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Eq. (7.32) might need to be applied in the parenthesis term (double subtraction), in order
to gain enough powers of aq, while the first term on the r.h.s yields factorized 1-loop expressions, whose q dependence is easily extracted. For our calculation, some diagrams
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require up to triple subtraction. After performing both types of subtractions, all divergent contribution is expressible in terms of known integrals. (b.) For the convergent terms we employ naive Taylor expansion in aq up to O(q 2 ) leading
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to the evaluation of α0 , α1 . This extraction makes explicit the functional dependence of each diagram on q; the coefficients of terms proportional to q 2 are integrals over the two internal momentum 4-vectors. Although this procedure seems to be straight forward, it
• Numerical integration:
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involves many millions of terms in intermediate stages. This results a huge amount of human and CPU time, and it is too costly on computer RAM.
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The required numerical integrations are performed by optimized Fortran programs which are generated by our Mathematica ‘integrator’ routine. The algorithm along with
a
details on the improvements that have been made for the purposes of the present work, are discussed in Appendix B. Each integral is expressed as a sum over the discrete Brillouin zone of finite lattices, with varying size L, and evaluated for different values of the overlap
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parameter ρ. The average length of the expression for each diagram, after simplifications, is about 2-3 hundred thousand terms, so that diagrams must be split into parts (usually
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of 2000 terms) to be integrated. The numerical values of these parts must then be added together to avoid running into systematic errors or spurious divergences. • Extrapolation:
Finally, we extrapolate the results to L → ∞; this procedure introduces an inherent
7.3. Description of the calculation
86
systematic error, which we can estimate quite accurately. This error is estimated in the following way: 51 different extrapolations are automatically performed by our extrapolation routine, using a broad spectrum of functional forms of the type X
ei,j L−i lnLj
(7.33)
i,j
tin
ou
For the nth form, the deviation dn is calculated using alternative criteria for quality of fit. P −2 These deviations are used to assign weights d−2 n n dn to each extrapolation, producing a final value together with the error estimate. Regarding the extrapolation of the 1-loop results, these errors are of order of magnitude 10−10 − 10−12 (the expressions are relatively simple and we integrated them for lattices with up to 128 lattice points per direction). On the other side, 2-loop results are usually integrated for L ≤ 28, giving errors of order of
an
magnitude 10−6 − 10−8 . Apparently, the difference between the two cases is substantial and there has been a major effort to minimize the errors by implementing alternative ways of extrapolating; they will be listed below.
st
The majority of the diagrams, are composed of several subdiagrams, which in their turn are split into many Fortran files to be integrated. There is no fundamental reason to
on
claim the same color dependence for all subdiagrams; on the contrary, the color structure arises from the combination of the algebra generators {T a }. They could for instance have a prefactor of the type N, 1/N, (N 2 −1)/N or (N 2 −2)/N. Infrared divergent diagrams
C
must be summed up before carrying out the extrapolation. On the other hand, for infrared convergent diagrams there is no obligation to proceed in the same way. Having as a goal
a
the error minimization, we experiment with different strategies of extrapolating: 1. The numerical values of a certain diagram can be expressed in terms of N and 1/N. Right after, we may extrapolate them to L → ∞ and get the desired result of the diagram
M ar
th
for different ρ’s. 2. Alternatively, one can add results with the same color prefactor. This way, a diagram can has the form ai N + a2 /N + a3 (N 2 −1)/N + ... . We then extrapolate each one of the ai ’s and bring the results into the form a N + b/N. 3. Another choice, is to extrapolate the contribution of each file separately and then write
them as a N + b/N. One observes that methods 2 and 3 are equivalent for diagrams with a single color prefactor. For each diagram, we performed all three variations of extrapolating and for each choice of ρ we selected the signal with the lowest error estimate. Furthermore, particular
7.4. Results
87
sets of diagrams, {2, 13, 16} and {3, 4, 15}, are grouped together although each one of them is infrared convergent. Their extrapolation is performed for each set (as if they were one diagram), as well as separately, using the 3 methods explained above. Once again we compare the final results coming from the different method and choose the one with the smallest error.
Results
ou
7.4
(1)
We denote the contribution of the ith 1-loop Feynman diagram to ν (1) (q) as νi (q); sim(2)
(2)
"
(0)
aq b 2 νi (q)=Nf ki + a2 q 2
2 2o (1) (2) ln a q + O((aq)4 ) ki + ki (4π)2
#
(7.34)
st
X qµ4 # 2 2 2 2 2 ln a q ln a q µ (4) (3) (0) (1) (2) +ci +ci ci +a2 q 2 ci +ci +O((aq)4 ) (7.35) 2 2 2 )2 (4π) (4π) (q
on
aq b 2 νi (q)=Nf
n
an
(1)
"
tin
ilarly, contributions of 2-loop diagrams to ν (2) (q) are indicated by νi (q). The quantities (1) (2) νi (q), νi (q) depend on N, Nf , ρ and aq according to the following formulae (λ0 = 1)
C
P (j) (j) where qb2 = 4 µ sin2 (qµ /2). The index i runs over diagrams, and the coefficients ki , ci (j) depend on the overlap parameter ρ. Moreover, ci contains the color structure and can be h i (j) (j,−1) (j,1) written as ci = ci /N + ci N . Comparison with continuum results and usage of Ward Identities requires
i
X
(0)
ki
X
= 0,
(2)
ki
2 = , 3
(4) ci
=0
M ar
•
i
X
(0)
ci = 0
(gauge invariance)
i
th
•
X
a
•
X
(2)
ci =
i
1 1 (3N − ) 2 16π N
(Lorentz invariance)
i
(3)
• c15 =
1 , 3N
4 (3) c16 = N, 3
5 (3) c17 = − N, 3
(3)
c18 =
N2 − 1 3N
(7.36)
We have checked that all the above conditions are verified by our results. Inserting these conditions in Eqs. (7.34), (7.35), the expressions for the total fermionic contribution to
7.4. Results
88
ν (1) (q) and ν (2) (q), after addition of all diagrams, take the form
(1)
k 0.020377(7) 0.01581702(2) 0.0133504717(2) 0.0116910952(1) 0.0104621922(2) 0.0095058191(2) 0.00874441051(7) 0.00813753230(4) 0.00766516396(3) 0.00732057894(3) 0.00710750173(2) 0.00703970232(7) 0.0071425543(2) 0.0074569183(2) 0.0080467046(1) 0.0090134204(1) 0.010526080(2) 0.0128914(2)
# 1 1 ln a2 q 2 + (3N− ) +O((aq)2 ) (7.38) 16π 2 N (4π)2
c(1,−1) -0.0096(6) -0.0044(1) -0.00321(6) -0.00244(4) -0.00191(1) -0.001606(6) -0.001397(3) -0.001241(1) -0.001107(1) -0.000979(1) -0.000849(2) -0.000706(3) -0.000543(4) -0.000335(7) -0.00005(1) 0.00034(1) 0.00093(6) 0.0020(1)
ou
N
i
a
ρ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
i
(1,1) +Nci
c(1,1) 0.124(3) 0.0118(5) 0.0045(1) 0.0030(1) 0.0022(6) 0.00176(2) 0.00145(1) 0.00124(1) 0.001051(9) 0.000872(3) 0.000710(8) 0.00052(1) 0.00033(3) 0.00007(1) -0.0002(1) -0.0004(1) -0.0021(5) -0.02(3)
tin
+Nf
X c(1,−1)
(7.37)
an
Nf =0
i
"
#
st
(2) (2) ν (q) = ν (q)
+ Nf
2 ln a2 q 2 (1) ki + + O((aq)2 ) 2 3 (4π)
on
Nf =0
" X
C
ν (1) (q) = ν (1) (q)
th
Table 7.1: Numerical results for k (1) ≡
P
i
(1)
ki , c(1,−1) ≡
P
(1,−1) , i ci
c(1,1) ≡
P
(1,1) . i ci
M ar
P (1) In Table 7.1 we tabulate the total 1-loop contribution k (1) ≡ i ki for 18 values of the overlap parameter (0 < ρ < 2). Each diagram was integrated for lattice size L4 , L ≤ 128 and then extrapolated to L → ∞. In all Tables and Figures, the errors accompanying (1)
our results are entirely due to this extrapolation. The coefficients ki (ρ) do not depend on the number of colors N, nor on the choice of regularization for the pure gluonic part of the action (Symanzik, Iwasaki, etc.). 1-loop diagram 1 is entirely responsible for the (2)
logarithmic contribution and it is independent of ρ, k1 =2/3. The numbers of k (1) in Table 7.1 are in agreement with corresponding numbers from Ref. [58].
7.4. Results
89
The 2-loop calculation of ν (2) (q) was accomplished for the same values of ρ and for P (1,−1) (1,1) P (1,1) L ≤ 28. Table 7.1 also presents the coefficients c(1,−1) ≡ , c ≡ , i ci i ci
versus ρ. Due to the extremely large size of the vertices involved, it is almost impossible to extend the results to larger L. Typically, the integration of 2000 terms is completed in
(0)
ou
∼ 7 days on 1 CPU; the present calculation comprises approximately 3500×2000 terms. Thus, if only a single CPU were available, our work would have required ∼50 years. In certain cases with large systematic errors we extended the integration up to L = 46. The (1)
per diagram results for ki and ki are collected in Table 7.2. The per diagram 2-loop (0,1) (0,−1) contributions for ci , ci are given in Tables 7.3 - 7.5 for 0.2 ≤ ρ ≤ 1.8. The rest of (1,1)
(1,−1)
tin
the convergent coefficients, ci , ci , can be found in Tables 7.6 - 7.9. Although the total single logarithmic contributions are ρ independent, the corresponding per diagram (2,1) (2,−1) coefficients, ci , ci , depend onρ; their values are provided in Tables 7.10 - 7.11. In
an
general, the overlap action leads to coefficients which are very small for most values of ρ. As a consequence, systematic errors, which are by and large rather small, tend to be
st
significant fractions of the signal for ρ > 1.4. From Eqs. (7.6), (7.20) - (7.25), (7.27) we find the following expression for l1 in terms of N, Nf , k (1) , c(1,−1) , c(1,1) 3 + 0.018127763034 − 0.007910118514 N 2 128"N 2 # 55 N (1) c(1,−1) 1 N 481 + Nf − 4ζ(3) − − 2k − + Nc(1,1) (7.39) (16π 2 )2 N 12 (16π 2)2 36 8π N
C
on
l1 = −
a
We can write the final form of the 3-loop coefficient bL2 for the β-function (Eq. (7.10)), including gluonic and fermionic contributions, using Eqs. (6.11), (7.7), (7.9) and (7.39) 11 + 0.000364106020 N − 0.000092990690 N 3 2 2048π N " (4π 2 − 1)2 + Nf − 0.000046883436 − 000013419574 N 2 4(16π 2)3 N 2 ! Nf 23 8ζ(3) 37 N + − + + (16π 2 )3 9N 3N 6
M ar
th
bL2 = −
# (4 N 3 + N − 3 N 2 N ) (11 N − 2 Nf ) c(1,−1) f f − + c(1,1) N + k (1) (7.40) 48π 2 N (16π 2 )2 N
7.4. Results
90
A large variety of possible numerical checks has been performed, as mentioned above: a. The total contribution to the gluon mass adds to zero, as expected. b. The coefficients of the non-Lorentz invariant terms cancel. c. The terms with double logarithms correspond to the continuum counterparts. This has been checked diagram by diagram. d. Terms
ou
with single logarithms add up to their expected value, which is independent of ρ (although the expressions per diagram are ρ-dependent). In Fig. 7.3 we plot the 1-loop coefficient k (1) with respect to ρ. Note that the errors are too small to be visible at this scale. The 2-loop coefficients c(1,−1) and c(1,1) are plotted in Figs. 7.4 - 7.5, respectively, for different values of the overlap parameter. The extrapolation
an
tin
errors are visible for ρ ≤ 0.4 and ρ ≥ 1.7. Substituting k (1) , c(1,−1) and c(1,1) into Eq. (7.40), we find the numerical results for the 3-loop contribution, bL2 , of the β-function. These are plotted in Fig. 7.6, choosing N = 3 and Nf = 0, 2, 3.
0.026
st
0.024 0.022
on
0.020
0.016 0.014
a
0.012
C
k(1)
0.018
0.010
th
0.008
M ar
0.006 0.0
0.2
0.4
0.6
0.8
1.0 ρ
Figure 7.3: Plot of the total 1-loop coefficient k (1) ≡ ρ.
1.2
P
i
1.4
(1)
ki
1.6
1.8
2.0
versus the overlap parameter
7.4. Results
91
0.004
0.002
0.000
ou
c(1,-1)
-0.002
-0.004
tin
-0.006
-0.008
0.2
0.4
0.6
0.8
1.0 ρ
1.2
1.4
1.6
st
-0.012 0.0
an
-0.010
on
Figure 7.4: Plot of the total 2-loop coefficient c(1,−1) ≡ 0.005
0.003
(1,−1) i ci
2.0
versus ρ.
C
0.004
P
1.8
a
0.001
th
c(1,1)
0.002
M ar
0.000
-0.001 -0.002 -0.003
-0.004 0.0
0.2
0.4
0.6
0.8
1.0 ρ
1.2
1.4
Figure 7.5: Plot of the total 2-loop coefficient c(1,1) ≡
1.6
P
(1,1) i ci
1.8
2.0
versus ρ.
92
ou
7.4. Results
tin
0.002
0.001
an
0
-0.001
st
b2L
Nf=0 -0.002
-0.003 Nf=3
-0.005
0.4
0.6
a
0.2
C
-0.004
on
Nf=2
0.8
1
1.2
1.4
1.6
1.8
ρ
M ar
th
Figure 7.6: The 3-loop coefficient bL2 (Eq. (7.40)), plotted against ρ, for N = 3 and Nf = 0 (horizontal red line), Nf = 2 (green line) and Nf = 3 (blue line).
7.5. Generalization to an arbitrary representation
7.5
93
Generalization to an arbitrary representation
7.5.1
The strategy
Here we provide the prescription that generalizes our results for ν (1) and ν (2) (Eqs. (7.37), (7.38)) to an arbitrary representation r, of dimensionality dr . For the calculation under
ou
study, only the fermion part of the action is affected, with the link variables assuming the form Ux, x+µ = exp(i g0 Aaµ (x) Tra ) (7.41)
X a
Tra Tra ≡ ˆ1 cr ,
tr(Tra Trb ) ≡ δ ab tr = δ ab
In the fundamental representation F , one has a
≡T ,
N2 − 1 cF = , 2N
dF = N,
tF =
1 2
(7.42)
(7.43)
st
TFa
d r cr N2 − 1
an
[Tra , Trb ] = i f abc Trc ,
tin
where Tra denote the generators in the representation r, and satisfy the relations
on
Studying the color structures for each diagram with a fermionic loop reveals the appropriate substitutions one should make, in order to recast the results in an arbitrary representation. The generalization prescription can be summarized in what follows.
th
a
C
For the 1-loop contribution in the fundamental representation (Eq. (7.37)), the color structure is 1 tr(T a T b ) = δ ab tF = δ ab (7.44) 2 Since diagrams with a closed fermion loop are always accompanied by a factor of Nf , the straightforward substitution Nf −→ Nf · (2 tr )
(7.45)
M ar
gives the desired results in an arbitrary representation. For the 2-loop fermion contribution to ν (2) (Eq. (7.38)), things get a bit more complicated, because there are different types of color structures. Fortunately, they all obey a general pattern, which will be given below. In all our diagrams, vertices with a fermionantifermion pair and with two or more gluons contain also contributions involving integrations over internal momenta, and they are best depicted diagrammatically as non-pointlike vertices, as shown in Fig. 7.7. There is no propagator (and thus no poles) associated to the bold fermion lines. It is important to note that the color structures corresponding to such
7.5. Generalization to an arbitrary representation
94
Figure 7.7: The non-pointlike nature of an overlap vertex. Dashed lines represent gluon fields; those ending on a cross stand for background gluons. Solid lines represent fermions.
ou
vertices are identical to those in ultralocal theories (with bold lines replaced by ordinary propagators). The diagram of Fig. 7.8 actually contains two subdiagrams, arising from the
an
tin
non-pointlike contributions of the vertex of Fig. 7.7. As an example, the diagram of Fig. 7.8 actually contains two subdiagrams, arising from the non-pointlike contributions of the vertex appearing in Fig. 7.7. Subdiagram A has a
AB
B
st
A
C
color dependence of the type
on
Figure 7.8: A particular example of a 2-loop fermionic diagram. Dashed (solid) lines represent gluon (fermion) fields.
tr(T a T c T c T b ) = cr tr δ ab
(7.46)
th
a
(a, b color indices of the external lines), while subdiagram B has tr(T a T c T b T c ) = tr δ ab (cr −
N ) 2
(7.47)
M ar
Therefore, the color structure of the diagram in Fig. 7.8 has the form δ
ab
N α cr tr + β tr (cr − ) 2
(7.48)
In the fundamental representation, this expression becomes N2 − 1 1 δ α + β (− ) 4N 4N ab
(7.49)
7.5. Generalization to an arbitrary representation
95
Thus, starting from our result for this diagram, which has the following color dependence: (α′ N + β ′/N) δ ab , the prescription for converting it to another representation is ′
′
1 −1 ′ ′ − 4(α + β )(− ) δ ab = 4α 4N 4N N ′ ′ ′ → 4α cr tr − 4(α + β )(cr − ) tr δ ab 2
′N
2
(7.50)
ou
(α N + β /N) δ
ab
tin
One may check that all diagrams follow the formula above. For the computation of bL2 , we need also the expressions for b0 , b1 , b2 in an arbitrary representation
st
an
11 1 4 (7.51) b0 = N − tr Nf (4π)2 3 3 1 34 2 20 b1 = N − tr Nf N + 4cr (7.52) (4π)4 3 3 2857 3 1415 2 79 1 11 2 205 2 2 N + 2 tr Nf cr − cr N − N +4 tr Nf cr + N (7.53) b2 = (4π)6 54 18 54 9 54 In order to calculate the ratio ΛL /Λ MS, the quantity l0 is necessary. For overlap fermions, it equals
on
1 5 (1) l0 = − 0.16995599N + 2 tr Nf − −k 8N 72π 2
(7.54)
(2)
(q) N =0 " f
th
(2)
a
C
Moreover, according to the prescriptions given in Eqs. (7.45), (7.50), the results for ν (1) and ν (2) become " # 2 2 2 ln a q + O((aq)2 ) ν (1) (q) = ν (1) (q) + 2 tr Nf k (1) + (7.55) Nf =0 3 (4π)2 ν
(q) = ν
M ar
+4 tr Nf
# 2 2 N N ln a q c(1,1) − c(1,−1) cr − + (cr +N) +O((aq)2 ) (7.56) 2 2 (4π)4
Finally, Eqs. (7.55), (7.56) lead to the 3-loop coefficient of the bare β-function, which in an arbitrary representation has the form
7.5. Generalization to an arbitrary representation
96
1 11 + 0.000364106020 N − 0.000092990690 N 3 2 128 (4π) N " 1 1 1 cr 2 c2r + tr Nf + + − 0.00011964262 − 0.00003220865 cr N 32 (4π)2 N 2 2 (4π)4 N (4π)6
bL2 = −
184 130 − 64 ζ(3) cr + + 32 ζ(3) N 3 3
ou
tr Nf − 0.00001086180 N + 3 (4π)6 2
c(1,−1) k (1) 2 − (c + N) t N + N (2 cr − N) (−4 tr Nf + 11 N) + r r f 32 π 4 24 π 2 # (1,1) c N (4 tr Nf − 11 N) (7.57) + 24 π 2
an
7.5.2
tin
+
Adjoint representation
st
As a particular application, let us focus on the adjoint representation, A. The latter is
on
encountered, e.g., in the standard supersymmetric extension of gauge theories in terms of vector superfields, where the gluinos are Majorana fermions in the adjoint representation,
C
thus similar in many respects to Nf = 1/2 species of Dirac fermions. The generators now take the form (TAa )bc ≡ i fbac
(7.58)
th
a
and the dimensionality of A is dA = N 2 − 1. Moreover, cA = N,
tA =
d A cA =N N2 − 1
(7.59)
M ar
Eqs. (7.55) and (7.56) for ν (1) and ν (2) now read # 2 2 ln a q 4 + Nf N 2 k (1) + + O((aq)2 ) (7.60) 3 (4π)2 Nf =0 " # 2 2 ln a q (2) νadj (q) = ν (2) (q) + Nf N 2 2(c(1,1) − c(1,−1) ) + + O((aq)2 ) (7.61) Nf =0 32π 4
(1) νadj (q) = ν (1) (q)
"
7.5. Generalization to an arbitrary representation
97
It is interesting to find the numerical values of ΛL /Λ MS in the adjoint representation (the corresponding results in the fundamental representation appear in Ref. [58]), which can be done using our 1-loop results for k (1) . This ratio is defined through Eq. (7.8), where l0 in this case is 1 5 (1) = − 0.16995599N + 2 N Nf − −k 8N 72π 2
(7.62)
ou
l0adj
Our results for ΛL/Λ MS are plotted in Fig. 7.9 for N = 3 and Nf = 0, 1/2, 1. One may compare Fig. 7.9 with an analogous figure pertaining to fermions in the fundamental
Nf = 1/2
st
0.008
on
(ΛL/ΛMS)adjoint
0.010
an
0.012
tin
representation (Ref. [58]); in that case, one obtains 0.02 ≤ ΛL /Λ MS ≤ 0.025, for Nf = 1.
0.006
Nf = 0 (scaled down by a factor of 10)
C
0.004
Nf = 1
a
0.002
th
0.000 0.0
M ar
0.2
0.4
0.6
0.8
1.0 ρ
1.2
1.4
1.6
1.8
2.0
Figure 7.9: The ρ dependence of the ratio ΛL /Λ MS in the adjoint representation for N = 3 and Nf = 0 (horizontal line scaled down by a factor of 10 from its value 0.034711), Nf = 1/2 and Nf = 1.
7.6. Discussion
7.6
98
Discussion
We have calculated the 2-loop coefficient of the coupling renormalization function Zg , for the Yang-Mills theory with gauge group SU(N) and Nf species of overlap fermions. We used the background field method to simplify the computation; in this method there is no need of evaluating any 3-point functions. This is the first 2-loop calculation using overlap
ou
fermions with external momenta, and it proved to be extremely demanding in human and CPU time; this is due to the fact that we had to manipulate very large expressions (millions of terms) in intermediate stages.
tin
We used our numerical results of Zg to determine the 3-loop coefficient bL2 of the bare lattice β-function; the latter dictates the asymptotic dependence between the bare coupling
an
constant g0 and the lattice spacing a, required to maintain the renormalized coupling at a given scale fixed. Knowledge of bL2 provides the correction term to the standard asymptotic scaling relation between a and g0 , via Eq. (6.21). The dependence of Zg and bL2 on N and Nf is shown explicitly in our expressions. On the other hand, dependence on the overlap parameter ρ cannot certainly be given in closed
on
st
form; instead, we present our results for a large set of values of ρ in its allowed range. The 3-loop correction is seen to be rather small: This indicates that the perturbative series is very well behaved in this case, despite the fact that it is only asymptotic in nature. Furthermore, around the values of ρ which are most often used in simulations (1 ≤ ρ ≤ 1.6), fermions bring about only slight corrections to the 3-loop β-function, even compared to
C
pure gluonic contributions, as can be seen from Fig. 7.6. The only source of numerical error in our results has its origin in an extrapolation to infinite lattice size. Compatibly with the severe CPU constraints, numerical 2-loop
a
integration had to be performed on lattices typically as large as 284 , or even up to 464 in cases where an improved extrapolation was called for. An intermediate range for ρ
M ar
th
(0.6 ≤ ρ ≤ 1.3) showed the most stable extrapolation error, and this may be a sign of their suitability for numerical simulations. The present study, being the first of its kind in calculating 2-loop diagrams with overlap vertices and external momentum dependence, had a number of obstacles to overcome. One first complication is the size of the algebraic form of the Feynman vertices; as an example, the vertex with 4 gluons and a fermion-antifermion pair contains ∼724,000 terms when
expanded. Upon contraction these vertices lead to huge expressions (many millions of terms); this places severe requirements both on the necessary computer RAM and on the efficiency of the computer algorithms which we must design to manipulate such expressions
7.6. Discussion
99
automatically. Numerical integration of Feynman diagrams over loop momentum variables is performed on a range of lattices, with finite size L, and subsequent extrapolation to L → ∞. As it turns out, larger L are required for an accurate extrapolation in the present case, compared to ultralocal actions. In addition, since the results depend nontrivially
ou
on the parameter ρ of the overlap action, numerical evaluation must be performed for a sufficiently wide set of values of ρ, with an almost proportionate increase in CPU time. Extreme values of ρ (ρ > ∼ 2) show unstable numerical behavior, which is attributable ∼ 0, ρ < to the spurious poles of the fermion propagator at these choices; this forces us to even larger L. A consequence of all complications noted above is an extended use of CPU time: Our
M ar
th
a
C
on
st
an
tin
numerical integration codes, which ran on a 32-node cluster of dual CPU Pentium IV processors, required a total of ∼50 years of CPU time.
Tables
(0)
(0,−1)
c{3+14+15}
(0,−1)
c4
0.0030297(6) 0.003287(3) 0.0035578(5) 0.0038238(4) 0.0040929(3) 0.00436473(6) 0.00464517(2) 0.00493494(1) 0.00523630(2) 0.00555174(5) 0.00588422(6) 0.00623695(4) 0.0066142(2) 0.0070210(5) 0.007464(2) 0.007950(1) 0.008493(1) 0.009111(1)
C
0.00260(3) 0.00222(2) 0.001645(6) 0.001253(2) 0.000930(1) 0.000669(1) 0.0004522(5) 0.0002740(3) 0.0001310(2) 0.0000194(4) -0.0000625(7) -0.000115(1) -0.000142(1) -0.000147(1) -0.0001289(9) -0.000099(2) -0.000070(1) -0.000060(8)
th
a
-0.00173(3) -0.00186(4) -0.00160(1) -0.001497(4) -0.001477(4) -0.0014686(5) -0.0014870(3) -0.0015297(3) -0.0015952(3) -0.0016846(5) -0.0017974(6) -0.001937(2) -0.002100(2) -0.002289(1) -0.002515(3) -0.002762(6) -0.00309(2) -0.00344(3)
M ar
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
(0,−1)
c1
on
Table 7.2: Per diagram breakdown of the 1-loop coefficients ki
ρ
(1)
k2 0.019344(6) 0.018038350(6) 0.01753314922(4) 0.017322163470(3) 0.017278696275(4) 0.0173605843730(1) 0.017556782951(6) 0.017872114285(1) 0.018322547098(2) 0.0189345964405(7) 0.019747192643(9) 0.020815997561(1) 0.02222124823(1) 0.024081810674(6) 0.02658161963(1) 0.03002396495(1) 0.0349588593(1) 0.0425532(1)
ou
(1)
k1 0.001034(3) -0.00222133(1) -0.0041826775(2) -0.00563106832(2) -0.0068165041(2) -0.0078547653(2) -0.00881237245(7) -0.00973458199(4) -0.01065738314(3) -0.01161401750(3) -0.01263969091(2) -0.01377629524(7) -0.0150786939(2) -0.0166248923(2) -0.01853491504(4) -0.0210105446(1) -0.024432779(2) -0.0296618(1)
tin
(0)
k2 -0.04028528(3) -0.04397848498(6) -0.047732030565(5) -0.05156042074(3) -0.055479104515(8) -0.05950493780(1) -0.06365663388(1) -0.06795525969(2) -0.07242482258(3) -0.07709299443(1) -0.081992032002(4) -0.08715997701(2) -0.09264225671(1) -0.098493871766(7) -0.104782467614(10) -0.11159278151(1) -0.11903332401(1) -0.1272468911(5)
an
(0)
k1 0.04028529(3) 0.043978484984(9) 0.04773203057(2) 0.05156042075(1) 0.05547910452(1) 0.059504937801(3) 0.063656633886(10) 0.06795525970(2) 0.07242482265(4) 0.07709299444(2) 0.08199203200(1) 0.08715997701(2) 0.09264225672(2) 0.09849387180(3) 0.10478246764(3) 0.111592781506(3) 0.11903332403(2) 0.127246891(2)
st
ρ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
100
(0,−1)
Table 7.3: Contribution to ci
(1)
and ki
for various ρ values.
(0,−1)
c{6+12}
(0,−1)
c{8+18}
-0.0025115(4) -0.0026708(4) -0.0028296(3) -0.00298808(5) -0.00315083(6) -0.0.003319(1) -0.0034919(1) -0.00367242(8) -0.0038611(1) -0.0040591(1) -0.00426755(5) -0.0044881(2) -0.0047220(2) -0.00497085(9) -0.0052374(4) -0.0055220(5) -0.0058287(5) -0.0061601(5)
-0.004250(9) -0.00434(2) -0.004557(2) -0.004696(3) -0.004814(1) -0.0049262(1) -0.0050348(1) -0.0051445(1) -0.00525858(4) -0.00538102(4) -0.00551473(6) -0.0056641(1) -0.0058342(1) -0.0060321(7) -0.0062660(7) -0.006555(3) -0.006902(2) -0.007360(4)
0.00289(1) 0.00331(2) 0.003778(2) 0.004128(2) 0.0044170(7) 0.0046792(3) 0.0049171(3) 0.0051380(3) 0.0053482(3) 0.0055528(4) 0.0057565(4) 0.0059643(2) 0.0061828(2) 0.0064197(4) 0.0066865(5) 0.006999(2) 0.007386(3) 0.007896(3)
c5
(0,−1)
of diagrams 1, 4, 5, 3+14+15, 6+12, 8+18.
Tables c{3+14+15}
(0,1)
0.00240(3) 0.00262(5) 0.00245(1) 0.002454(4) 0.002535(4) 0.0026300(6) 0.0027545(4) 0.0029060(3) 0.0030829(3) 0.0032871(5) 0.0035185(6) 0.003782(2) 0.004074(2) 0.004402(3) 0.004768(4) 0.005178(6) 0.00568(2) 0.00623(2)
-0.000714(2) -0.0008117(9) -0.000916(2) -0.0010195(6) -0.0011269(3) -0.0012371(3) -0.0013485(2) -0.0014616(2) -0.0015763(1) -0.0016922(1) -0.0018092(1) -0.0019274(1) -0.0020469(2) -0.0021684(4) -0.0022931(7) -0.0024212(2) -0.0025571(7) -0.002703(2)
-0.00270(3) -0.00233(3) -0.001780(6) -0.001404(3) -0.0010990(9) -0.0008551(7) -0.0006539(6) -0.0004929(3) -0.0003671(2) -0.0002733(4) -0.0002106(8) -0.000178(1) -0.000172(1) -0.000195(2) -0.000240(2) -0.000304(2) -0.000382(4) -0.000439(7)
(0,1)
(0,1)
(0,1)
-0.0030895(6) -0.003362(3) -0.0036544(6) -0.0039430(4) -0.0042365(3) -0.00453901(6) -0.00485238(2) -0.00517919(1) -0.00552207(3) -0.00588395(5) -0.00626832(6) -0.00667913(4) -0.0071215(2) -0.0076015(5) -0.008127(2) -0.008707(1) -0.009357(1) -0.010099(2)
c5
(0,1)
(0,1)
(0,1)
c{6+12}
c{7+11}
c{8+18}
c{9+17}
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
0.004250(9) 0.00434(2) 0.004557(2) 0.004696(3) 0.004814(1) 0.0049262(1) 0.0050348(1) 0.0051445(1) 0.00525858(4) 0.00538102(4) 0.00551473(6) 0.0056641(1) 0.0058342(1) 0.0060321(7) 0.0062660(7) 0.006555(3) 0.006902(2) 0.007360(4)
-0.0006320(2) -0.000682(1) -0.0007205(2) -0.0007459(3) -0.00076162(2) -0.00076893(9) -0.00076874(6) -0.0007622(1) -0.00075053(9) -0.0007350(1) -0.00071691(4) -0.00069776(9) -0.00067891(5) -0.0006628(2) -0.0006509(3) -0.0006449(5) -0.0006450(2) -0.000649(1)
-0.00289(1) -0.00331(2) -0.003778(2) -0.004128(2) -0.0044170(7) -0.046792(3) -0.0049171(3) -0.0051380(3) -0.0053482(3) -0.0055528(4) -0.0057565(4) -0.0059643(2) -0.0061828(2) -0.0064197(4) -0.0066865(5) -0.006999(2) -0.007386(3) -0.007896(3)
0.0009153(5) 0.001002(1) 0.0010774(2) 0.0011410(3) 0.0011955(1) 0.0012415(2) 0.0012800(2) 0.0013115(2) 0.0013371(2) 0.0013576(3) 0.0013738(1) 0.00138700(6) 0.00139837(8) 0.0014091(1) 0.0014208(2) 0.0014346(3) 0.0014488(2) 0.001462(1)
M ar
th
a
C
ρ
(0,1)
Table 7.5: Contribution to ci
0.0022883(3) 0.0024238(4) 0.0025593(2) 0.00269701(6) 0.00283979(4) 0.00298856(3) 0.00314442(4) 0.00330843(7) 0.0034817(1) 0.00366520(8) 0.00386023(4) 0.00406807(4) 0.00429008(4) 0.0045278(1) 0.0047832(2) 0.0050574(2) 0.0053534(2) 0.0056730(6)
of diagrams 1, 2+13+16, 3+14+15, 4, 5.
on
Table 7.4: Contribution to ci
(0,1)
c4
ou
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
(0,1)
tin
c{2+13+16}
an
(0,1)
c1
st
ρ
101
(0,1)
c10
0.0001472(3) 0.0001734(1) 0.0002008(2) 0.00022928(3) 0.00025940(4) 0.00029154(4) 0.00032602(3) 0.00036319(3) 0.00040346(3) 0.00044735(2) 0.00049550(3) 0.00054880(3) 0.00060838(2) 0.00067577(2) 0.00075316(8) 0.0008431(2) 0.0009496(2) 0.0010789(4)
of diagrams 6+12, 7+11, 8+18, 9+17, 10.
Tables
102 (1,−1)
c{3+14+15}
-0.0109(2) -0.00436(2) -0.00248(1) -0.001493(8) -0.000913(1) -0.0004935(9) -0.0001571(3) 0.0001413(4) 0.0004281(4) 0.000723(1) 0.001041(2) 0.001407(2) 0.001846(2) 0.002398(4) 0.003130(9) 0.004180(4) 0.00578(4) 0.0086(1)
-0.0494(3) -0.01523(4) -0.00672(2) -0.00322(1) -0.001529(7) -0.000637(5) -0.000141(1) 0.0001316(7) 0.000264(1) 0.0003028(7) 0.0002729(5) 0.000177(1) 0.000015(3) -0.000214(1) -0.000548(4) -0.001052(1) -0.00189(2) -0.00338(3)
(1,−1)
ou
-0.00184(6) -0.00128(4) -0.00112(2) -0.001082(7) -0.0010521(8) -0.0010420(6) -0.0010455(3) -0.0010611(1) -0.0010885(2) -0.0011286(1) -0.0011836(4) -0.0012561(6) -0.001355(3) -0.0014852(5) -0.001669(3) -0.001940(6) -0.00233(3) -0.00307(6)
st
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
(1,−1)
c4
tin
c1
an
ρ
(1,−1)
(1,−1)
ρ
th M ar
c{6+12}
(1,−1)
c{7+11}
c{8+18}
0.00870(6) 0.00436(1) 0.002966(3) 0.0022765(2) 0.0018592(1) 0.001581059(7) 0.00138402(2) 0.00123931(5) 0.00113125(4) 0.00105148(4) 0.00099516(4) 0.00096112(8) 0.0009511(3) 0.0009714(4) 0.001036(1) 0.0011762(4) 0.001474(2) 0.002209(2)
-0.00034964(1) -0.00039464(6) -0.0004396(1) -0.00048364(6) -0.00052645(6) -0.00056740(4) -0.00060633(4) -0.00064287(4) -0.00067671(4) -0.00070758(4) -0.00073529(4) -0.00075955(3) -0.00078014(4) -0.00079695(4) -0.00080976(2) -0.00081839(2) -0.00082252(6) -0.0008218(1)
0.0437(4) 0.01214(9) 0.00426(5) 0.00126(4) -0.00004(1) -0.000738(4) -0.001127(3) -0.0013538(7) -0.0014841(4) -0.0015584(2) -0.0016033(4) -0.0016329(4) -0.0016608(8) -0.001703(6) -0.001756(4) -0.001876(6) -0.00210(3) -0.00266(2)
C
0.00049(3) 0.000363(5) 0.0003226(6) 0.0003011(4) 0.0002924(6) 0.0002911(4) 0.0002954(3) 0.00030475(8) 0.00031883(9) 0.0003385(1) 0.00036414(9) 0.0003974(1) 0.0004399(2) 0.0004947(9) 0.0005673(5) 0.000667(1) 0.000812(4) 0.00107(2)
a
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
c5
of diagrams 1, 4, 5.
on
Table 7.6: Contribution to ci
(1,−1)
Table 7.7: Contribution to ci
(1,−1)
(1,−1)
of diagrams 3+14+15, 6+12, 7+11, 8+18.
Tables
c{3+14+15}
(1,1)
0.0103(2) 0.00393(2) 0.00208(1) 0.001105(8) 0.000518(1) 0.0000815(9) -0.0002805(3) -0.0006127(4) -0.0009418(4) -0.001289(1) -0.001672(2) -0.002118(2) -0.002656(2) -0.003335(4) -0.004230(9) -0.005497(3) -0.00740(4) -0.0106(1)
0.115(3) 0.0072(3) 0.00144(5) 0.00038(2) -0.000026(9) -0.000179(4) -0.000255(3) -0.000302(2) -0.000329(1) -0.0003415(9) -0.000345(2) -0.000336(4) -0.000310(8) -0.000280(7) -0.00018(1) -0.00007(3) -0.0005(5) -0.02(3)
-0.0494(3) -0.01523(4) -0.00672(2) -0.00322(1) -0.001529(7) -0.000637(5) -0.000141(1) 0.0001316(7) 0.000264(1) 0.0003028(7) 0.0002729(5) 0.000177(1) 0.000015(3) -0.000214(1) -0.000548(4) -0.001052(1) -0.00189(2) -0.00338(3)
(1,1)
(1,1)
c{6+12}
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
-0.00870(6) -0.00436(1) -0.002966(3) -0.0022765(2) -0.0018592(1) -0.001581059(7) -0.00138402(2) -0.00123931(5) -0.00113125(4) -0.00105148(4) -0.00099516(4) -0.00096112(8) -0.0009511(3) -0.0009714(4) -0.001036(1) -0.0011762(4) -0.001474(2) -0.002209(2)
(1,1)
c{7+11}
C
ρ
M ar
th
a
0.0003652(1) 0.00040934(7) 0.00045220(8) 0.000493101(9) 0.00053206(4) 0.00056860(5) 0.00060266(5) 0.00063401(5) 0.00066253(4) 0.00068809(3) 0.00071059(3) 0.00072999(3) 0.00074631(4) 0.00075949(3) 0.00076959(2) 0.00077654(2) 0.0007803(1) 0.0007800(1) (1,1)
Table 7.9: Contribution to ci
(1,1)
c5
0.00187(6) 0.00129(4) 0.00112(2) 0.001080(7) 0.001053(1) 0.0010465(6) 0.0010560(2) 0.0010791(1) 0.0011158(2) 0.00116676(8) 0.0012348(3) 0.0013231(6) 0.001442(3) 0.0015955(6) 0.001810(3) 0.002110(8) 0.00256(3) 0.00342(6)
-0.00021(3) -0.000213(3) -0.000155(4) -0.000127(4) -0.0001182(5) -0.0001121(4) -0.00010878(6) -0.00010816(9) -0.0001091(1) -0.0001125(1) -0.0001182(1) -0.0001269(1) -0.0001389(2) -0.0001569(2) -0.0001823(3) -0.0002212(3) -0.000286(3) -0.000403(7)
of diagrams 1, 2+13+16, 3+14+15, 4, 5.
on
Table 7.8: Contribution to ci
(1,1)
c4
ou
(1,1)
tin
c{2+13+16}
an
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
(1,1)
c1
st
ρ
103
(1,1)
c{8+18}
(1,1)
c{9+17}
-0.0437(4) 0.00049(7) -0.01214(9) 0.0003(4) -0.00426(5) -0.0000(1) -0.00126(4) 0.00033(6) 0.00004(1) 0.00047(6) 0.000738(4) 0.00047(2) 0.001127(3) 0.000462(6) 0.001354(7) 0.000467(1) 0.0014841(4) 0.000467(9) 0.0015584(2) 0.000459(3) 0.0016033(4) 0.000468(8) 0.0016329(4) 0.00046(1) 0.0016608(8) 0.00046(3) 0.001703(6) 0.000444(7) 0.001756(4) 0.00047(1) 0.001876(6) 0.00066(1) 0.00210(3) 0.00018(6) 0.00266(1) -0.0000(2)
(1,1)
c10
0.0002(1) 0.00013(4) 0.000123(3) 0.000065(2) 0.000042(2) 0.0000261(8) 0.00001216(4) -0.0000010(1) -0.00001360(1) -0.00002609(2) -0.0000389(1) -0.00005355(3) -0.0000712(2) -0.0000933(6) -0.0001250(1) -0.0001744(1) -0.000266(4) -0.00038(5)
of diagrams 6+12, 7+11, 8+18, 9+17, 10.
Tables
104
(2,−1)
c{8+18}
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
-0.013366(6) -0.006513(9) -0.004235(7) -0.0031009(8) -0.0024235(1) -0.00197417(6) -0.00165489(8) -0.00141675(4) -0.00123263(4) -0.001086232(8) -0.00096723(3) -0.00086874(3) -0.00078598(2) -0.000715570(9) -0.00065502(4) -0.00060247(1) -0.00055648(5) -0.0005169(5)
0.013333(2) 0.006473(4) 0.004195(2) 0.0030608(3) 0.0023834(1) 0.00193408(3) 0.00161479(2) 0.00137665(2) 0.00119252(1) 0.00104613(1) 0.00092713(2) 0.000828632(5) 0.000745877(6) 0.000675469(7) 0.000614920(6) 0.00056237(1) 0.00051638(2) 0.00047587(1)
st
an
tin
c{3+14++15}
ou
(2,−1)
ρ
(2,−1)
(2,1)
(2,1)
(2,1)
ρ
c{2+13++16}
c{3+14+15}
c{8+18}
c{9+17}
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
0.0007574(4) 0.00062213(4) 0.0005500758(2) 0.0005026441(2) 0.0004686248(3) 0.0004433894(4) 0.00042474477(5) 0.0004116462(2) 0.00040370895(4) 0.0004010140(2) 0.0004040517(2) 0.0004137557(3) 0.0004316242(3) 0.0004599683(2) 0.0005023841(1) 0.0005646959(2) 0.000657053(8) 0.00080(1)
C
(2,1)
on
Table 7.10: The overlap parameter dependence of the 2-loop coefficients ci
-0.013333(2) -0.006473(4) -0.004195(2) -0.0030608(3) -0.0023834(1) -0.00193408(3) -0.00161479(2) -0.00137665(2) -0.00119252(1) -0.00104613(1) -0.00092713(2) -0.000828632(5) -0.000745877(6) -0.000675469(7) -0.000614920(6) -0.00056237(1) -0.00051638(2) -0.00047587(1)
-0.0005641(2) -0.00042061(1) -0.000342511(4) -0.000289971(1) -0.0002510600(5) -0.0002207785(4) -0.0001966701(8) -0.000177455(1) -0.0001624980(7) -0.000151587(1) -0.000144841(1) -0.0001426940(4) -0.0001459505(5) -0.0001559042(3) -0.0001745785(4) -0.0002051875(3) -0.0002530826(2) -0.0003280(2)
M ar
th
a
0.0132578(3) 0.006399(1) 0.0041184(8) 0.0029817(3) 0.00230196(1) 0.00185038(3) 0.00152810(2) 0.00128886(2) 0.00110281(1) 0.000954557(1) 0.00083376(1) 0.00073352(2) 0.000649062(9) 0.000576995(2) 0.000514822(6) 0.000460677(3) 0.000413133(3) 0.0003710854(7)
(2,1)
Table 7.11: Numerical results for the 2-loop coefficients ci overlap parameter ρ.
. (2,1)
c10
-0.000049(2) -0.0000(2) -0.0000(3) -0.00001(2) -0.000016(4) -0.000019(2) -0.0000220(4) -0.0000260(4) -0.0000312(2) -0.0000375(4) -0.0000455(2) -0.0000557(3) -0.000069(2) -0.000086(3) -0.000107(3) -0.000137(2) -0.00018(2) -0.00023(5)
for different values of the
ou
Chapter 8
8.1
an
tin
Improved perturbation theory for improved lattice actions Introduction
st
Since the earliest studies of quantum field theories on a lattice, it was recognized that quan-
on
tities measured through numerical simulation are characterized by significant renormalization effects, which must be properly taken into account before meaningful comparisons to corresponding physical observables can be made.
C
As has been rigorously demonstrated [11], the renormalization procedure can be formally carried out in a systematic way to any given order in perturbation theory. However,
a
calculations are notoriously difficult, as compared to continuum regularization schemes; furthermore, the convergence rate of the resulting asymptotic series is often unsatisfactory. A number of approaches have been pursued in order to improve the behavior of per-
th
turbation theory, among them Refs. [98, 99]. These approaches share in common the aim to reorganize perturbative series in terms of an expansion coefficient which would be more
M ar
suitable than the bare coupling constant g0 ; the definition of such a “renormalized” coupling constant is not unique, but can depend on the observables under study and on an energy scale. It is expected that such a definition will reabsorb a large part of the tadpole contributions which are known to dominate lattice perturbation theory. Some years ago, a method was proposed to sum up a whole subclass of tadpole diagrams, dubbed “cactus” diagrams, to all orders in perturbation theory [97, 100]; this procedure has a number of desirable features: • it is gauge invariant 105
8.1. Introduction
106
• it can be systematically applied to improve (to all orders) results obtained at any given order in perturbation theory • it does indeed absorb the bulk of tadpole contributions into an intricate redefinition of the coupling constant
ou
In cases where nonperturbative estimates of renormalization coefficients are also available for comparison, the agreement with cactus improved perturbative results is significantly better as compared to results from bare perturbation theory.
tin
Here we extend the improved perturbation theory method of Refs. [97, 100], to encompass the large class of actions which are used nowadays in simulations of QCD, a work
an
published in Physical Review D [101]. This class includes Symanzik improved gluon actions with any arbitrary combination of closed Wilson loops, combined with any fermionic action. The effect of resummation is to replace various parameters in the action (coupling constant, Symanzik coefficients, clover coefficient) by ”dressed” values; the latter are solutions to certain coupled integral equations, which are easy to solve numerically.
on
st
In this chapter we explain the procedure of dressing the gluon propagators, the gluon and fermion vertices, as a result of the summation of cactus diagrams to all orders. We show how these dressed constituents are employed to improve 1-loop and 2-loop Feynman diagrams coming from bare perturbation theory. In the end of the chapter we apply our resummation procedure to improve the multiplicative renormalization ZV and ZA of the
C
vector and axial current respectively, using overlap fermions and Symanzik gluons. Another application is presented in the next chapter, the additive renormalization of the fermion masses up to 2 loops, employing clover fermions and Symanzik gluons. In cases where
th
a
nonperturbative estimates of renormalization functions are also available for comparison, the agreement with improved perturbative results is significantly better as compared to results from bare perturbation theory. Finally, the improvement of QED is briefly discussed
M ar
in Section 8.4. Clearly, all resummation procedures, whether in the continuum or on the lattice, bear a caveat: A one-sided resummation could ruin desirable partial cancellations which might exist among those diagrams which are resummed and others which are not; what is worse, the end result might depend on the gauge. As we shall see, no partial cancellations will be
ruined in our procedure, due to the distinct N-dependence of the resummed diagrams (N is the number of colors); furthermore, our results will be gauge independent.
8.2. The method
8.2
107
The method
In this section, following the outline of Ref. [97], we start illustrating our method by showing how the gluon propagator is dressed by the inclusion of cactus diagrams. We will then dress gluon and fermion vertices as well. Finally, we will explain how this procedure is applied to Feynman diagrams at a given order in bare perturbation theory, concentrating
8.2.1
ou
on the 1- and 2-loop case.
Dressing the propagator
tin
We consider, for the sake of definiteness, the Symanzik improved gluon action involving
an
Wilson loops with up to 6 links, as presented in Chapter 2 (Eq. (2.26)); we recall that there is a set of four coefficients ci satisfying c0 + 8c1 + 16c2 + 8c3 = 1. We apply the usual parameterization of links in terms of the continuum gauge fields Aµ (x)
Aµ (x) = Aaµ (x) T a ,
Tr (T a T b ) = 21 δ ab
(8.1)
st
Ux,µ = exp i g0 a Aµ (x + aˆ µ/2) ,
on
where a is the lattice spacing (set to 1 from now on), µ ˆ is the unit vector in direction µ and T a is the generator of the SU(N) algebra. Use of the Baker-Campbell-Hausdorff (BCH) formula, leads to the following form for Ui
C
(1) 2 (2) 3 (3) 4 Ui = exp i g0 Fi + i g0 Fi + i g0 Fi + O(g0 ) (1)
In the above equation Fi
(8.2)
is simply the sum of the gauge fields on the links of loop i. For
a
this to become more clear, let us consider the plaquette as an example, where one has (1)
(j)
= Aµ (x+ˆ µ/2) + Aν (x+ˆ µ+ˆ ν /2) − Aµ (x+ˆ ν +ˆ µ/2) − Aν (x+ˆ ν /2)
th
F0
(8.3)
M ar
Fi (j > 1) are j-th degree polynomials in the gauge fields, constructed from nested commutators. The resummation procedure involves diagrams made of vertices containing (1) only Fi . We may now define the cactus diagrams which dress the gluon propagator: These are gauge invariant tadpole diagrams which become disconnected if any one of their vertices is removed (see Fig. 8.1). The original motivation of this procedure is the well known observation of ‘tadpole dominance’ in lattice perturbation theory. Each vertex is constructed
8.2. The method
108
(1)
pure gluon parts of the action. This fact implies that, the longitudinal
solely from the Fi
tin
ou
parts of all propagators will always cancel. Therefore, the effect of dressing is the same in all covariant gauges; this will be mathematically proven in Eq. (8.11).
an
Figure 8.1: A cactus diagram.
A diagrammatic equation for the dressed gluon propagator (thick line) in terms of the bare propagator (thin line) and 1-particle irreducible (1PI) vertices (solid circle) reads +
+···
+
st
=
(8.4)
on
Our goal is to solve Eq. (8.4), so that we write the dressed propagator in terms of the bare parameters od the action. The 1PI vertex obeys the following recursive equation
+
M ar
th
a
C
=
+ ...
+
+
+
+
+
+. ..
+ +. . .
+. ..
(8.5)
Let us mention that each part of the above diagrammatic equations is made of 4 terms; each one coming from the plaquette, rectangle, chair or parallelogram contribution of the action. The tadpole diagrams of Eq. (8.5) are a contraction of vertices that exclude all
8.2. The method
109
anticommutators; one needs the Taylor expansion of Ui only up to O(g0 ). In order to put Eq. (8.5) into a mathematical form and solve it, let us first write down the bare inverse propagator D −1 resulting from the Symanzik gluon action (Eq. (2.26)), and from the gauge fixing term (Eq. (2.13)) =
X ρ
kˆρ2 δµν − kˆµ kˆρ δρν
dµρ +
1 ˆ ˆ kµ kν 1−ξ
(8.6)
ou
−1 Dµν (k)
For unexplained notation, see Subsection 2.2.2. The inverse propagator can thus be put in
1 ˆ ˆ kµ kν 1−ξ
tin
the form −1 (1) (2) (3) Dµν (k) ≡ c0 G(0) µν (k) + c1 Gµν (k) + c2 Gµν (k) + c3 Gµν (k) +
(8.7)
X
an
The matrices G(i) (k) (Eq. (2.29) ) are symmetric and transverse, i.e. they satisfy ˆ G(i) µν (k) kν = 0
(1)
st
ν
(1)
Each of them originates from a corresponding term Tr Fi
Fi
(8.8)
of the gluon action. Con-
on
sequently, each of the diagrams on the r.h.s. of Eq. (8.5), being the result of a contraction (1) with only two powers of Fi left uncontracted, will necessarily be equal to a linear combi-
G1PI (k) ≡
C
nation of G(i) (k); this implies that the 1PI vertex G1PI (k) (the l.h.s. of Eq. (8.5)) can be written as = α0 G(0) (k) + α1 G(1) (k) + α2 G(2) (k) + α3 G(3) (k)
(8.9)
a
Each of the quantities αi will in general depend on N, g0 , c0 , c1 , c2 , c3 , but not on the
M ar
th
momentum. We must now turn Eq. (8.5) into a set of 4 recursive equations for αi . Eq. (8.4) leads to the following expression for the dressed propagator D dr (k) D dr ≡
= D + D G1PI D + D G1PI D G1PI D + · · · = D
⇒ (D dr )−1 = (ˆ1 − G1PI D) D −1 = D −1 − G1PI = c˜0 G(0) + c˜1 G(1) + c˜2 G(2) + c˜3 G(3) +
1 ˆ ˆ kµ kν , 1−ξ
! ˆ1 (8.10) ˆ1 − G1PI D c˜i ≡ ci − αi
(8.11)
We observe that dressing affects entirely the transverse part of the inverse propagator,
8.2. The method
110
replacing the bare coefficients ci with improved ones c˜i , and leaves the longitudinal part intact. The same property carries over directly to the propagator itself; the consequence of this will be that our method leads to the same results in all covariant gauges. In terms of the dressed propagator, Eq. (8.5) can be drawn as
+
+ ...
+
(8.12)
ou
=
To proceed, we must evaluate the generic tadpole diagram of Eq. (8.12) coming from an n-point vertex of the action, in which n − 2 lines have been pairwise contracted. This
8.2.2
an
tin
calculation is explained in Subsection 8.2.3, after deriving the expressions for the improved vertices (Subsection 8.2.2). This leads to the numerical values of the dressed coefficients c˜i .
Dressing vertices
st
• We will begin by considering the 3-gluon vertex, coming from the action, Eq. (2.26). This vertex results from a Taylor expansion of Ui to 3rd order in g0 . Expressing Ui as in (1)
(2)
C
on
Eq. (8.2), we see that only terms of the form Tr(Fi Fi ) will appear in this vertex, since (3) (1) 3 Tr(Fi ) and Tr (Fi ) will vanish. By analogy with Eq. (8.12), the dressed 3-gluon vertex equals
+
+
+ ... (8.13)
a
=
Consistently with the dressing of propagators, each (2l + 1)-point vertex in Eq. (8.13) is a
M ar
th
sum of 4 parts (one from each type of Wilson loop in the action), made up of (1) 2l−1
Tr (Fi )
Denoting the bare 3-gluon vertex by (0)
V 3 = c0 V 3
(1)
+ c1 V 3
(2)
Fi
(2)
+ c2 V 3
(8.14)
(3)
+ c3 V 3
it is relatively straightforward to see from Eq. (8.13) that the dressed vertex, V3dr , is given
8.2. The method
111
by V3dr
=
3 X
∞ X (i g0 )2l+1
ci
i=0
l=0
2 F (2l+2; N) βil 2 (2l+1)! (N −1)
!
(i)
(i g0 )−1 V3
(8.15)
The summations inside parentheses are a mere multiple of those in Eq. (8.29); consequently, the result for V3dr turns out very simple (1)
+ c˜1 V3
(2)
+ c˜2 V3
(3)
+ c˜3 V3
(8.16)
ou
(0)
V3dr = c˜0 V3
tin
• We turn now to the 3-point fermion-antifermion-gluon vertex [100]. In the cases of Wilson and overlap fermions, these vertices remain unaffected, since the fermion actions
=
+
an
do not contain any closed Wilson loops on which the BCH formula might be applied. The vertex from the clover action (Eq. (2.20)), on the other hand, is amenable to improvement; we write
+ ...
+
(8.17)
st
where fermions are denoted by a dotted line. Just as in Eq. (8.15), we find =
l=0
on
∞ X (i g0 )2l 2 F (2l+2; N) β0l · 2 (2l+1)! (N −1)
!
=
c˜0 · c0
(8.18)
C
• Vertices with more fields would seem a priori more difficult to handle. To illustrate
th
a
the complications that may arise, let us consider the 4-gluon vertex. The BCH expansion (1) 4 (2) 2 (1) (3) of Tr(Ui ) contributes to this vertex in the form Tr (Fi ) , Tr (Fi ) and Tr(Fi Fi ). Such terms may in principle dress differently from each other. In addition, the dressed (1) 4 vertex produced from Tr (Fi ) will not be a multiple of its bare counterpart; rather, it will be a linear combination of two color tensors (which are independent for N > 3)
M ar
Tr{T a T b T c T d + permutations} and (δ ab δ cd + δ ac δ bd + δ ad δ bc )
(8.19)
This issue has been resolved in Ref. [97], and it generalizes directly to the present case. Actually, such complications will not appear while dressing 1- and 2-loop diagrams in (1) 4 typical cases: Terms of the type Tr (Fi ) must simply be omitted in order to avoid double counting, since their contribution is already included in dressing diagrams with one (2) 2 (1) (3) less loop. Thus, one is left only with: Tr (Fi ) and Tr(Fi Fi ); for both of these terms
8.2. The method
112
it is straightforward to show, just as in Eqs. (8.15, 8.16), that their dressing amounts to replacing ci by c˜i . • The same considerations as above apply to all higher vertices from both the gluon and fermion actions as well.
Numerical values of improved coefficients
ou
8.2.3
Let us now evaluate a typical diagram on the r.h.s. of Eq. (8.12) in order to derive numerical values for c˜i . Before contraction the vertex reads
an
tin
4 (ig0 )n X X n (1) n o −S → ci tr Fi,x,µν n! g02 i=1 x,µν Z dq1 dqn (ig0 )n ... · ei(q1 +···+qn )x tr{T a1 T a2 . . . T an } = 2 4 n! g0 (2π) (2π)4 n XY c0 qˆiµ Aaνi (qi ) − qˆiν Aaµi (qi )
ai \ ai −˚ qiµ Aaνi (qi ) + (q\ iµ +qiν )Aν (qi ) + (qiµ −qiν )Aν (qi )
+2c1
x,µν i=1
n X Y ai −iqiν /2 ai iqiµ /2 ai 2 (q\ +q )A (q ) − q ˆ e A (q ) − q ˆ e A (q ) iµ iν i iρ i iρ i ρ µ ν
x,µνρ µ6=ν6=ρ6=µ
on
+c2
st
x,µν i=1 n XY
i=1
n Y i=1
n Y
ai −(q\ ˆiρ e−iqiµ /2 Aaνi (qi ) + qˆiρ e−iqiν /2 Aaµi (qi ) iµ −qiν )Aρ (qi ) − q
a
+
ai (q\ ˆiρ eiqiµ /2 Aaνi (qi ) − qˆiρ eiqiν /2 Aaµi (qi ) iµ −qiν )Aρ (qi ) + q
C
+
i=1
n X Y ai ai ai \ \ −(q\ +q )A (q ) + (q −q )A (q ) + (q +q )A (q ) iν iρ i iµ iρ i iµ iν i µ ν ρ
th +c3
x,µνρ µ6=ν6=ρ6=µ
M ar
i=1
+
n 1 Y
3
i=1
ai \ ai \ ai (q\ iν −qiρ )Aµ (qi )−(qiµ −qiρ )Aν (qi )+(qiµ −qiν )Aρ (qi )
!
(8.20)
(ˆ q µ = 2i sin(qµ /2), ˚ qµ = 2i sin(qµ )). Each of the diagrams under consideration is a sum of 4 terms, one term for each of
the Wilson loops Ui (i = 0, 1, 2, 3) in the action, from which its n-point vertex may have originated. There are (n − 2)/2 1-loop integrals in the diagram; each of them corresponds
8.2. The method
113 (1)
to the contraction of two powers of Fi
via a dressed propagator, and will contribute one
power of βi (˜ c0 , c˜1 , c˜2 , c˜3 ), where
β2 = β3 =
Z
Z
−π π −π π −π π −π
d4 q (2π)4 d4 q (2π)4 d4 q (2π)4 d4 q (2π)4
dr dr 2 qˆµ2 Dνν (q) − 2 qˆµ qˆν Dµν (q)
dr dr dr (4ˆ qν2 − qˆν4 ) Dµµ (q) + qˆµ2 (4−ˆ qν2 ) Dνν (q) − 2 qˆµ qˆν (4−ˆ qν2 ) Dµν (q) 2 dr 2 dr 2 qˆµ (8−ˆ qν ) Dρρ (q)/2 − qˆµ qˆρ (8−ˆ qν ) Dµρ (q)/2
ou
β1 =
Z
π
dr dr 3 qˆµ2 (4−ˆ qν2 ) Dρρ (q)/2 − 3 qˆµ qˆν (4−ˆ qρ2 ) Dµν (q)/2
(8.21)
tin
β0 =
Z
We remind the reader that the index i corresponds to plaquette (i=0), rectangle (i=1),
an
chair (i=2) and parallelogram (i=3). Distinct values are assumed for µ, ν, ρ; no summation implied. Once again, we note that βi are gauge independent, since the longitudinal part
on
F (2; N) = δa1 a2 Tr {T a1 T a2 }
st
cancels in the loop contraction. For the contraction of the SU(N) generators we first evaluate F (n; N), which is the sum over all complete pairwise contractions of Tr{T a1 T a2 . . . T an }
F (4; N) = (δa1 a2 δa3 a4 + δa1 a3 δa2 a4 + δa1 a4 δa2 a3 ) Tr {T a1 T a2 T a3 T a4 } X 1 δa1 a2 δa3 a4 . . . δan−1 an Tr T P (a1 ) T P (a2 ) . . . T P (an ) (8.22) F (n; N) = n/2 2 (n/2)! P ∈S
C
n
(F (2n + 1; N) ≡ 0; Sn is the permutation group of n objects). The generating function
th
a
G(z; N) for this quantity, defined via
G(z; N) ≡
∞ X zn n=0
n!
F (n; N)
(8.23)
M ar
has been computed explicitly in Ref. [97], using Gaussian integration over the space of traceless Hermitian matrices [102], with the result G(z; N) = ez
2 (N −1)/(4N )
L1N −1 (−z 2 /2)
(8.24)
(Lαβ (x) : Laguerre polynomials). By differentiating Eq. (8.23), one gets the desired expres-
8.2. The method
114
sion for F (n; N)
dn G(z; N) (8.25) z=0 dz n Since 2 out of n generators remain uncontracted in our case, color contraction does not F (n; N) =
lead to F (n; N), but rather to n F (n; N) 2(N 2 −1)
(8.26)
ou
Thus, upon contraction, an n-leg diagram in Eq. (8.12), with its vertex coming from the term Ui of the Lagrangian (i = 0, 1, 2, 3), will merely result in the following multiple of G(i)
tin
ci (i g0 )n n F (n; N) (n−2)/2 (i) 4 βi G 2 2 g0 n! 2(N −1)
(8.27)
3 X
αi G
(i)
=
i=0
∞ X
ci (i g0 )n n F (n; N) (n−2)/2 (i) 4 βi G 2 2 g n! 2(N −1) n=4,6,8,... 0
(8.28)
st
i=0
3 X
an
We are finally in a position to set Eq. (8.12) in a mathematical form
on
Unknown in Eq. (8.28) are the coefficients αi ; they appear on the l.h.s., as well as inside the integrals βi of the r.h.s, by virtue of Eqs. (8.21), (8.11). We recall that G(i) are functions of the external momentum k and if these are independent of each other, then Eq. (8.28) amounts to 4 equations for the 4 coefficients αi . Actually, G(2) is not independent of the rest 3 functions G(i) , so that we have 3 equations for 3 coefficients, but this causes no
C
complication. In any case, c2 is typically set to zero in simulations. The generalization of our procedure for improved gluon actions with arbitrary numbers and types of Wilson loops is now evident. It is crucial to check at this stage that all
th
a
combinatorial weights are correctly incorporated in Eq. (8.28); this is indeed the case. Splitting Eq. (8.28) into 4 separate equations, and making use of Eq. (8.25), we can recast the infinite summations in closed form ∞ X
1 (i g0 )n n F (n; N) (n−2)/2 4 βi 2 2 g n! 2(N −1) n=4,6,8,... 0 ! ∞ X (i g0 )n 2(i g0 ) n/2 −1/2 = 1+ F (n+1; N) βi βi 2 2 n! g0 (N −1) n=0 2 = 1− G′ (z; N) 1/2 2 z (N −1) z=(i g0 βi )
M ar
αi = ci
(8.29)
8.2. The method
115
2 ′ ci −αi 2 (N −1) = ⇒ G (z; N) 1/2 ci z z=(i g0 βi ) N−1 1 2 2 −βi g02 (N −1)/(4N ) 2 LN −1 (g0 βi /2) + 2 LN −2 (g0 βi /2) (8.30) = e N
ou
In solving Eqs. (8.30), each choice of values for (ci , g0 , N) leads to a unique set of values for c˜i ≡ ci − αi . The latter are no longer normalized in the sense of Eq. (2.27);
one may equivalently choose, however, to express the results of our procedure in terms of a normalized set of improved coefficients, c˜i /C˜0 and an improved coupling constant g˜02 = g02/C˜0 , where C˜0 = c˜0 + 8˜ c1 + 16˜ c2 + 8˜ c3 . In fact, it is convenient to treat bare and
ci , g02
γ˜i ≡
c˜i , g02
β˜i (˜ c0 , c˜1 , c˜2 , c˜3 ) ≡ g02 βi (˜ c0 , c˜1 , c˜2 , c˜3 ) = βi (˜ γ0 , γ˜1, γ˜2 , γ˜3 ) (8.31)
an
γi ≡
tin
improved coefficients on an equal footing, by defining rescaled quantities as follows
st
The dressed propagators in β˜i will now contain a rescaled gauge parameter (1−ξ) → g02 (1−ξ), which is irrelevant since the longitudinal part does not contribute. Insertion of Eqs. (8.31) into Eq. (8.30), leads to the coupled equations satisfied by the rescaled
on
quantities γ˜i 1 ˜ γ˜i = 2 γi e−βi (N −1)/(4N ) N −1
N−1 1 LN −1 (β˜i /2) + 2 L2N −2 (β˜i /2) N
(8.32)
C
For the gauge groups SU(2) and SU(3), the Laguerre polynomials have a simple form, making Eqs. (8.32) more explicit ˜
th
a
(N = 2) : γ˜i = γi e−βi /8
M ar
(N = 3) :
−β˜i /6
γ˜i = γi e
β˜i 1− 12
!
β˜i β˜2 1− + i 4 96
!
(8.33)
Given the highly nonlinear nature of Eqs. (8.32), it is not a priori clear that a solution for γ˜i always exists. The converse, of course, is trivial: Finding the bare values γi which lead to a given set of dressed values γ˜i is immediate, since the integrals β˜i only depend on
γ˜i , not γi. As it turns out, this is always the case, for all physically interesting values of ci , and for all values of g0 ranging from g0 = 0 up to a certain limit value, well inside the
8.2. The method
116
strong coupling region. Fortunately, numerical solutions of Eqs. (8.32), (8.33) can be found very easily. We use a fixed point procedure, applicable to equations of the type x = f (x) γ˜i = fi (˜ γi )
⇒
(m)
γ˜i = lim γ˜i m→∞
(0)
,
where : γ˜i
= γi ,
(m+1)
γ˜i
(m)
= fi (˜ γi
) (8.34)
ou
In order for the procedure to converge (attractive fixed point), it must be that: |∂fi /∂˜ γi | < 1 in a neighborhood of γ˜i . This has been verified in a number of extreme cases. The algo-
tin
rithm developed for the evaluation of γ˜i appears in Appendix C. In what follows, we present the values of the improved coefficients for several gluon actions of interest, as derived from the algorithm of Appendix C. • Plaquette action:
(N = 2) :
2 c˜0 = e−g0 /(16 c˜0)
2 c˜0 = e−g0 /(12 c˜0)
C
(N = 3) :
g2 1− 0 24 c˜0
on
we obtain c˜0 for N = 2 and N = 3
st
an
Let us start with the plaquette action for which the undressed coefficients are c0 =1, c1 =c2 =c3 =0 [97]. In this case, Eqs. (8.32) reduce to only one equation, for γ˜0 , while γ˜i =γi=0 (i=1, 2, 3). This equation is further simplified greatly since the integral β˜0 can now be evaluated in closed form, which is β˜0 = 1/(2˜ γ0 ). Employing the above into Eq. (8.33)
g2 g04 1− 0 + 8 c˜0 384 c˜20
(8.35)
(8.36)
a
In Fig. 8.2 we plot c˜0 as a function of g02 , for N = 2 and N = 3. The range of g0 values, for √ which solutions exist, extends from g02 = 0 (where c˜0 = 1) up to 16 e/3 ≃ 3.23 (N = 2)
M ar
th
and 1.558 (N = 3); this covers the whole region of physical interest.
8.2. The method
117
1.00
N =2
0.90
c˜0 0.80
0.60
0.2
0.4
0.6
0.8 g02
1.0
1.2
1.4
1.6
tin
0.50 0.0
ou
N =3
0.70
an
Figure 8.2: Improved coefficient c˜0 for N=2 and N=3 (plaquette action).
Additionally, in Table 8.1 there are listed the unimproved and improved coefficients for several actions and different choices of β.
on
st
• Tree-level Symanzik action: The tree-level Symanzik improved action [10] corresponds to c0 =5/3, c1 = − 1/12 and c2 =c3 =0. The dressed coefficients c˜0 , c˜1 are now found using the algorithm of Appendix C (no closed form can be provided) and are shown in Fig. 8.3 for N = 3. In order to plot both c˜0 and c˜1 together, c˜1 is multiplied be a factor of -20. A similar scaling appears in 1.8
a
1.6
C
the next figures, whenever is necessary.
c˜0
1.4
c0 = 1.667 c1 = −0.833
th
1.2 1.0
M ar
0.8
−20 · c˜1
0.6 0.4
0.2 0.0
0.5
1.0
1.5
2.0
2.5
g02
Figure 8.3: Improved coefficients c˜0 and c˜1 (tree-level Symanzik improved action).
8.2. The method
118
• Iwasaki action: The Iwasaki set of parameter values [103] is c0 =3.648, c1 = − 0.331 and c2 =c3 =0. In principle, c0 and c1 depend on g0 , but they are typically kept constant in simulations. The corresponding dressed values are plotted in Fig. 8.4 (N = 3). 4.0
ou
c0 = 3.648 c1 = −0.331
3.5
c˜0
3.0 2.5
−10 · c˜1
tin
2.0
1.0 0.0
0.5
1.0
an
1.5
1.5
2.0
g02
2.5
3.0
3.5
st
Figure 8.4: Improved coefficients c˜0 and c˜1 (Iwasaki action).
on
• Tadpole improved L¨ uscher-Weisz (TILW) actions: Another class of gluon actions based on Symanzik improvement are the TILW actions [104, 105]. In this case, the coefficients c0 , c1 , c3 are optimized for each value of β = 2N/g02
2.6 2.4
C
separately, while c2 = 0 for all cases. In Fig. 8.5 we show the values of ci and of their dressed counterparts c˜i in a typical range for β : 8.0 ≤ β c0 ≤ 8.6 and N = 3. c0 c˜0
a
2.2
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2.0 1.8 1.6
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1.4 1.2
−10 · c1 −100 · c3 −10 · c˜1 −100 · c˜3
1.0
7.8
7.9
8.0
8.1
8.2
8.3
β · c0
8.4
8.5
8.6
8.7
Figure 8.5: Coefficients c0 , c1 , c3 (red/blue/green dots, respectively) and their dressed counterparts (dots joined by a line), for different values of β c0 = 6 c0 /g02 (TILW actions).
8.2. The method
119
• DBW2 action: Finally, the DBW2 gluon action [106] corresponds to c2 =c3 =0, and β-dependent values for c0 , c1 . Some standard values for c0 and c1 (obtained starting from β c0 = 6.0 and 6.3), as well as c˜0 and c˜1 are shown in Table 8.1.
0 0 0 0 0 0 -0.0128098 -0.0134070 -0.0142442 -0.0147710 -0.0154645 -0.0163414 0 0 0 0 0
c˜1
c˜3
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0 0 0 -0.083333 -0.083333 -0.083333 -0.151791 -0.154846 -0.159128 -0.161827 -0.165353 -0.169805 -0.331 -0.331 -0.331 -1.4086 -0.53635
c˜0
0.6797 0.70507 0.74977 1.32778 1.33311 1.39048 1.81690 1.82811 1.84700 1.85801 1.87425 1.89626 2.59313 2.72733 2.88070 8.80696 3.39826
0 0 0 -0.05489 -0.05530 -0.05982 -0.09631 -0.09703 -0.09835 -0.09908 -0.10021 -0.10177 -0.17151 -0.18930 -0.21048 -0.7313 -0.22528
tin
1.0 1.0 1.0 1.6666667 1.6666667 1.6666667 2.3168064 2.3460240 2.3869776 2.4127840 2.4465400 2.4891712 3.648 3.648 3.648 12.2688 5.29078
c3
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5.0 5.3 6.0 5.0 5.07 6.0 8.60/c0 8.45/c0 8.30/c0 8.20/c0 8.10/c0 8.00/c0 1.95 2.2 2.6 0.6508 1.1636
c1
st
c0
on
β
0 0 0 0 0 0 -0.00831 -0.00860 -0.00902 -0.00927 -0.00961 -0.01005 0 0 0 0 0
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Action Plaquette Plaquette Plaquette Symanzik Symanzik Symanzik TILW TILW TILW TILW TILW TILW Iwasaki Iwasaki Iwasaki DBW2 DBW2
The improvement procedure in a nutshell
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8.2.4
a
Table 8.1: Input parameters β, c0 , c1 , c3 , c˜0 , c˜1 , c˜3 (c2 = 0).
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The steps involved in the resummation of cactus diagrams can now be described quite succinctly: • Substitute gluon propagators in Feynman diagrams by their dressed counterparts. The latter are obtained by the replacement ci → c˜i = g02 γ˜i , where γ˜i are the solutions of Eqs. (8.32).
• Perform the same replacement, ci → c˜i , on the 3-gluon vertex.
8.3. Application: 1-loop renormalization of fermionic currents
120
• Account for dressing of the 3-point vertex from the clover action by adjusting the clover coefficient cSW : cSW → cSW · (˜ c0 /c0 ) . • In dressing subleading-order diagrams, avoid double counting, i.e., subtract terms which were included in dressing leading-order diagrams. These are very easy to
ou
identify and subtract: Writing a general subleading-order result (aside from an overall prefactor) as: a/N 2 + b + c Nf /N (Nf : number of fermion flavors), the quantity to subtract will include all of a/N 2 (because terms with BCH commutators are higher
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order in N), and it will be a multiple of (2N 2 − 3); thus subtraction boils down to the substitution 2 2 a/N + b + c Nf /N → b + a + c Nf /N (8.37) 3
Application: 1-loop renormalization of fermionic
st
8.3
an
(see Refs. [97, 107, 108, 109] for different applications of this). The remaining subleading vertices dress exactly as the propagators and 3-point vertices above.
on
currents
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In the last section of this chapter, we turn to an application of cactus improvement: ¯ The 1-loop renormalization of the vector (Vµ (x) ≡ Ψ(x)γ µ Ψ(x)) and axial (Aµ (x) ≡ ¯ Ψ(x)γ5 γµ Ψ(x)) currents using the overlap action. We employ Symanzik improved gluons;
a
hence, our results (depending also on the overlap parameter ρ) are presented for various sets of Symanzik coefficients and specific values of ρ. As mentioned in the introduction, a different application of our improvement method is that of the additive mass renormaliza-
th
tion for clover fermions. Both the unimproved and improved calculations are presented in the next chapter.
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The renormalization constants ZV , ZA are defined through V S (p) ≡ ZVS V (a) ,
AS (p) ≡ ZAS A(a)
(8.38)
where S is the renormalization scheme, p is the external momentum for zero lattice spacing, V S (p), AS (p) are the continuum operators and V (a), A(a) the lattice operators. Bare 1loop results for ZV , ZA have been computed in the literature [72, 20, 77] using overlap action. Using this action, one can show that the renormalization constants ZV and ZA are
8.3. Application: 1-loop renormalization of fermionic currents
121
equal [72], due to chiral symmetry (preserved by overlap fermions). In the MS scheme the constants read ZV,A (a, p) = 1 − g02 z1V,A ≡ 1 − g02
CF (bV,A + bΣ ) , 16π 2
CF =
N2 − 1 2N
(8.39)
following the notation of Refs. [20, 77]. bV,A and bΣ are 1-loop results pertaining to the
an
tin
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amputated 2-point function of the current and to the fermion self-energy, respectively; since ZV = ZA , we can write bV = bA .
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Figure 8.6: 1-loop contribution to the amputated Green’s function (bV,A ).
a
C
on
There is only one diagram contributing to bV,A (Fig. 8.6), and 2 diagrams needed for the calculation of bΣ (Fig. 8.7).
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Figure 8.7: 1-loop contribution to the quark self-energy (bΣ ).
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Using cactus improvement, Eq. (8.39) becomes dr dr ZV,A (a, p) = 1 − g02 z1V,A
(8.40)
dr To compute z1V,A we dress the Symanzik coefficients and the propagators as described in
dr the previous section. In Table 8.2 the values of ZV,A and ZV,A are presented for different sets of the Symanzik coefficients, choosing ρ = 1.0, ρ = 1.4. Systematic errors are too
dr small to affect any of the digits appearing in the table. The dependence of ZV,A and ZV,A on the overlap parameter ρ is more clearly shown in Fig. 8.8, where we plot our results
8.3. Application: 1-loop renormalization of fermionic currents
122
for three actions: Plaquette, Iwasaki and TILW. Note that improvement is more apparent for the case of the plaquette action. Indeed, from Table 8.2 one can clearly see that the effect of dressing is smaller for improved gluon actions. This, of course, could have been expected, since these actions were constructed in a way as to reduce lattice artifacts, in the first place.
tin
ou
dr dr ZV,A (ρ=1.0) ZV,A (ρ=1.4) ZV,A (ρ=1.4) 1.35247 1.14707 1.19615 1.29231 1.13574 1.16207 1.28735 1.13386 1.15932 1.23484 1.11311 1.13019 1.31941 1.15259 1.17690 1.32764 1.15613 1.18146 1.33677 1.16012 1.18651 1.34298 1.16282 1.18995 1.34972 1.16579 1.19369 1.35705 1.16908 1.19774 1.44921 1.21724 1.24847 1.38773 1.19256 1.21440 1.31940 1.16293 1.17656 1.45362 1.27543 1.25057
an
ZV,A (ρ=1.0) 1.26427 1.24502 1.24164 1.20418 1.27581 1.28223 1.28946 1.29434 1.29973 1.30569 1.39343 1.34872 1.29507 1.49631
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β=6/g02 6.00 5.00 5.07 6.00 3.7120 3.6018 3.4772 3.3985 3.3107 3.2139 1.95 2.20 2.60 0.6508
on
Action Plaquette Symanzik Symanzik Symanzik TILW TILW TILW TILW TILW TILW Iwasaki Iwasaki Iwasaki DBW2
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dr Table 8.2: Results for ZV,A , ZV,A (Eq. (8.39), (8.40)) using ρ=1.0, ρ=1.4 .
a
A comparison between our improved ZV,A values and some nonperturbative estimates [114], shows that improvement moves in the right direction. Cactus dressing had already
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th
been tested using clover fermions [100], and it turns out to be as good as standard tadpole improvement [99], but still not very close to nonperturbative results.
8.4. Dressing QED
123
2.0 dr
ZV,A (Iwasaki, β=1.95) 1.9
ZV,A (Iwasaki, β=1.95)
1.8
ZV,A (Plaquette, β=6.0)
dr
dr
ZV,A (TILW, β⋅c0=8.45) 1.7
ZV,A (TILW, β⋅c0=8.45) ZV,A (Plaquette, β=6.0)
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1.6 1.5
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1.4 1.3
1.1 1.0 0.6
0.8
1.0
1.2
an
1.2
1.4
1.6
1.8
2.0
st
ρ
8.4
on
dr Figure 8.8: Plots of ZV,A and ZV,A for the plaquette, Iwasaki and TILW actions. Labels have been placed in the same top-to-bottom order as their corresponding curves.
Dressing QED
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Cactus improvement can be easily carried over to Lattice Quantum Electrodynamics. In
a
this case, link variables commute; hence the first order BCH formula is exact and dressing includes the full contribution of diagrams with cactus topology. Dressing the propagator now proceeds precisely as in Eq. (8.29). The only difference is
th
that the result of contracting n−2 out of n generators of SU(N): n F (n; N)/(N 2 −1), must n now be replaced by: (n − 3)!! (the number of ways to pair n−2 out of n objects); this 2 results in:
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αi = ci
∞ X
n=4,6,8,...
1 (i g0 )n n (n−2)/2 −βi g02 /2 (n − 3)!! 2 β = 1 − e i g02 n! 2
2 ⇒ c˜i ≡ ci − αi = ci e−βi g0 /2
(8.41)
(βi are the c˜-dependent integrals defined in Eqs. (8.21)). As before, the 4 coupled equations (8.41) for c˜i assume their simplest form in the case
8.5. Discussion
124
of the plaquette action (c0 = 1, c1 = c2 = c3 = 0); in this case, β0 = 1/(2˜ c0 ) and we obtain 2 c0 ) , c˜0 = e−g0 /(4˜
c˜1 = c˜2 = c˜3 = 0
(8.42)
Dressing vertices is simpler than in the nonabelian case. For a bare (2m)-point vertex, (0) (1) (2) (3) denoted as: V2m = c0 V2m + c1 V2m + c2 V2m + c3 V2m , instead of contracting group genera-
=
i=0 3 X
(i) V2m ci
∞ X (i g0 )2l+2m (2m)! 2l + 2m l βi (2l − 1)!! (2l+2m)! (i g0 )2m 2m l=0
(i) V2m ci
2 e−βi g0 /2 =
i=0
3 X i=0
(i)
V2m c˜i
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=
3 X
(8.43)
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dr V2m
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tors, one must simply count the number of distinct pairings of 2l objects out of (2l+2m): 2l + 2m (2l − 1)!!. The dressed vertex becomes 2m
8.5
Discussion
on
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The above relations allow us to summarize the dressing procedure for QED very briefly: • Replace ci by c˜i (as given in Eq. (8.41)) throughout • Omit all diagrams which contain any cactus subdiagram, to avoid double counting
In closing, we remark that resummation of cactus diagrams is readily applicable to any ob-
a
C
servable in lattice gauge theories. This procedure for improving bare perturbation theory is gauge invariant, and can be applied in a systematic fashion to improve (to all orders) results obtained at any given order in perturbation theory. Another positive feature which
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reveals the simplicity of our method is the fact that the resummation procedure is applied by replacing the action parameters (coupling constant, Symanzik coefficients, clover co-
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efficients) by dressed values. In cases where the results are given as a polynomial of the parameters mentioned above, the improvement can directly be applied in the bare results, with no need of additional calculations. This leads to both human and computer time saving. The computation of the dressed renormalization factors ZV,A and the comparison with available nonperturbative results, shows that the improvement moves in the right direction and it is as good as the Lepage-Mackenzie tadpole improvement.
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Chapter 9
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Two-loop additive mass renormalization with clover fermions
Introduction
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9.1
an
and Symanzik improved gluons
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Here, we calculate the additive renormalization of the fermion mass in Lattice QCD, using clover fermions and Symanzik improved gluons. The calculation is carried out up to 2 loops in perturbation theory and it is directly related to the determination of the critical
C
value of the hopping parameter, κc . The clover fermion action [3] (SW) successfully reduces lattice discretization effects and approaches the continuum limit faster. This justifies the extensive usage of this action in
th
a
Monte Carlo simulations in recent years. The coefficient cSW appearing in this action is a free parameter for the current work and our results will be given as a polynomial in cSW . Regarding gluon fields, we employ the Symanzik improved action [10], which also aims
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at minimizing finite lattice spacing effects. For the coefficients parameterizing the Symanzik action, we consider several choices of values which are frequently used in the literature. The lattice discretization of fermions introduces some well known difficulties; demanding strict locality and absence of doublers leads to breaking of chiral symmetry. In order to recover this symmetry in the continuum limit one must set the renormalized fermion mass
(mR ) equal to zero. To achieve this, the mass parameter m◦ appearing in the Lagrangian must approach a critical value mc , which is nonzero due to additive renormalization. The mass parameter m◦ is directly related to the hopping parameter κ used in simula125
9.1. Introduction
126
tions. Its critical value, κc , corresponds to chiral symmetry restoration κc =
1 2 mc a + 8 r
(9.1)
1 mB ≡ mo − mc = 2a
1 1 − κ κc
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where a is the lattice spacing and r is the Wilson parameter. Using Eq. (9.1), the nonrenormalized fermion mass is given by (9.2)
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Thus, in order to restore chiral symmetry one must consider in simulations the limit mo → mc . This fact points to the necessity of an evaluation of mc .
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The perturbative value of mc is also a necessary ingredient in higher-loop calculations of the multiplicative renormalization of operators (see, e.g., Ref. [115]). In mass independent schemes, such renormalizations are typically defined and calculated at zero renormalized mass, and this entails setting the value of the Lagrangian mass equal to mc . The quantity which we study is a typical case of a vacuum expectation value resulting
on
st
in an additive renormalization; as such, it is characterized by a power (linear) divergence in the lattice spacing, and its calculation lies at the limits of applicability of perturbation theory. Previous studies of the hopping parameter and its critical value have appeared in the literature for Wilson fermions - Wilson gluons [107] and for clover fermions - Wilson gluons [108, 116]. The procedure and notation in our work is the same as in the above
C
references. Our results for κc (and consequently for the critical fermion mass) depend on the number of colors (N) and on the number of fermion flavors (Nf ). Besides that, there is
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a
an explicit dependence on the clover parameter cSW which, as mentioned at the beginning, is kept as a free parameter. On the other hand, the dependence of the results on the choice of Symanzik coefficients cannot be given in closed form; instead, we present it in a
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list of Tables and Figures. In order to compare our results to nonperturbative evaluations of κc coming from Monte Carlo simulations, we employ an improved perturbation theory
method for improved actions. In Section 9.2 we formulate the problem and describe our calculation of the necessary Feynman diagrams. Section 9.3 is a presentation of our results. Finally, in Section 9.4 we apply to our 1- and 2-loop results an improvement method, proposed by our group [117, 118, 101]. This method resums a certain infinite class of subdiagrams, to all orders in perturbation theory, leading to an improved perturbative expansion. We end this
9.2. Formulation of the problem
127
section with a comparison of perturbative and nonperturbative results. Our findings are summarized in Section 9.5.
9.2
Formulation of the problem
We employ the Wilson formulation of the QCD action on the lattice, with Nf flavors of
ou
degenerate clover (SW) [3] fermions (Eq. (2.20)). Regarding gluons, we use the Symanzik improved gauge field action (Eq. (2.26)), involving Wilson loops with 4 and 6 links and four parameters, c0 , c1 , c2 , c3 . In the full action we have to include the ghost (Eq. 2.14)
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and the measure terms (Eq. 2.15). The bare fermion mass mB must be set to zero for chiral invariance in the classical
an
continuum limit. Terms proportional to r in the action, as well as the clover terms, break chiral invariance. They vanish in the classical continuum limit; at the quantum level, they induce nonvanishing, flavor-independent fermion mass corrections. Numerical simulation algorithms usually employ the hopping parameter,
1 2 mo a + 8 r
st
κ≡
(9.3)
on
as an adjustable input. Its critical value, at which chiral symmetry is restored, is thus 1/8r classically, but gets shifted by quantum effects. The renormalized mass can be calculated in textbook fashion from the fermion self–
C
energy. Denoting by ΣL (p, mo , g) the truncated, one particle irreducible fermion 2-point function, we have for the fermion propagator ◦ −1 i p/ + m(p) − ΣL (p, mo , g) 1 X 2r X 2 µ ◦ p/ = γµ sin(apµ ), m(p) = m◦ + sin (ap /2) a µ a µ
a
S(p) =
th
where :
(9.4)
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To restore the explicit breaking of chiral invariance, we require that the renormalized mass vanish
S
−1
(0) =0 m◦ → mc
=⇒
mc = ΣL (0, mc , g)
(9.5)
The above is a recursive equation for mc , which can be solved order by order in perturbation theory. We denote by dm the additive mass renormalization of m◦ : mB = m◦ − dm. In
9.2. Formulation of the problem
128
order to obtain a zero renormalized mass, we must require mB → 0, and thus m◦ → dm. Consequently, mc = dm = dm(1−loop) + dm(2−loop)
(9.6)
The tree level value of the critical mass is zero, mc = 0.
2
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1
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Two diagrams contribute to dm(1−loop) , shown in Fig. 9.1. In these diagrams, the fermion mass must be set to its tree level value, mo → 0.
an
Figure 9.1: 1-loop diagrams contributing to dm(1−loop) . Wavy (solid) lines represent gluons (fermions).
st
The quantity dm(2−loop) receives contributions from a total of 26 diagrams, shown in Fig. 9.2. Genuine 2-loop diagrams must again be evaluated at mo → 0; in addition,
on
one must include to this order the 1-loop diagram containing an O(g 2 ) mass counterterm (diagram 23). Certain sets of diagrams, corresponding to 1-loop renormalization of propagators, must be evaluated together in order to obtain an infrared convergent result: These
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th
a
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are diagrams 7+8+9+10+11, 12+13, 14+15+16+17+18, 19+20, 21+22+23.
9.2. Formulation of the problem
129
5
6
7
9
10
11
12
13
15
16
17
21
22
8
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4
14
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3
19
20
24
25
26
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23
28
a
27
on
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18
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Figure 9.2: 2-loop diagrams contributing to dm(2−loop) . Wavy (solid, dotted) lines represent gluons (fermions, ghosts). Crosses denote vertices stemming from the measure part of the action; a solid circle is a fermion mass counterterm. The calculation of a diagram is similar to the procedure described in Chapter 7. Here
are summarized the required steps: • The appropriate vertices and the ‘incidence’ matrix are specified for the diagram under study, so that the contraction is performed.
• The color dependence, Dirac matrices and tensor structures are simplified and sym-
9.3. Computation and results
130
metries of the theory (permutation symmetry, lattice rotational invariance) are used, to minimize the length of the expression. • For the particular study, the external momentum is set to zero by definition of the critical mass calculation (Eq. (9.5)) and there are no logarithmic contributions.
ou
• Each diagram is a sum of trigonometric products (sines, cosines) of the loop momenta and it is convenient to bring it into a compact and canonical form.
• We then numerically integrate over the internal momenta using our ‘integrator’ rou-
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tine, which generates a highly optimized Fortran code. The final expression for most of the diagrams consists of many thousand terms and they cannot be integrated in
an
a single Fortran program. In such cases, we split the expression into parts of approximately 2000 terms and integrate separately each one of them. Of course, these are added together right after the integration. The numerical results are given for lattices of varying finite size L; for 1-loop diagrams: 4 ≤ L ≤ 128, while for 2-loop diagrams: 4 = Z
Z
DΨ DΨ DA O e−S[Ψ,Ψ,A]
(A.6)
ou
Clearly, the action is dimensionless and thus the Lagrangian has dimensions 5 [length]−4 , or equivalently [mass]4 . From the mass term we note that the fermion fields have dimen-
A.2
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sion [length]−3/2 and using the pure gluon term one can see that the coupling constant is dimensionless; thus the gauge fields have dimension [length]−1 .
Lattice QCD
nµ ǫ Z ,
a : lattice spacing
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xµ → nµ a ,
an
In a hypercubic space-time lattice (ultraviolet regulator), the continuum Euclidean coordinate xµ is replaced by a variable having discrete values
This discretization introduces a momentum cutoff which is inverse to the lattice spacing,
on
since the momenta are restricted in the finite interval −π/a ≤ p ≤ π/a (first Brillouin zone). Thus, the integrals transform to finite sums d4 x → a4
C
Z
X n
and all quantities calculated in the lattice are finite. The first thing that needs to be done
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a
is to convert the fermion and gauge fields into the lattice language. The discretized quarks are now described by Grassmann variables Ψ(n) and are placed on the lattice sites n, while
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the gluons live in the links between two neighboring lattice points. This way, gluons carry the interaction among quarks and at the same time they interact with each other. The lattice gauge fields are represented by the variable Uµ (n) defined as Uµ (n) = ei a g0 Aµ (n)
(A.7)
In many cases, it is convenient to work with dimensionless quantities, and this can be achieved by absorbing the dimension through appropriate powers of the lattice spacing. 5
We work in units where ~ = c = 1
A.2. Lattice QCD
169
For instance, the quark field has to be multiplied by a−3/2 Ψ(n) → a−3/2 Ψ(n) In order to write a lattice version of the QCD action6 , we discretize the derivative using the naive differences
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(A.8)
(A.9)
tin
− → 1 ∇ µ Ψ(n) = [Uµ (n)Ψ(n + aˆ µ) − Ψ(n)] a ← − 1 ∇ µ Ψ(n) = [Ψ(n) − Uµ (n − aˆ µ)−1 Ψ(n − aˆ µ)] a
an
where µ ˆ is the unit vector in direction µ. One of the desired properties of the lattice action is the gauge invariance for the reasons mentioned in Chapter 2. The lattice gauge transformations take the form Ψ(n) → Λ(n)Ψ(n)
st
Ψ(n) → Ψ(n)Λ† (n)
Uµ (n) → Λ(n)Uµ (n)Λ† (n)
(A.10)
on
For this purpose, the pure gluonic part of the action must be constructed by gauge invariant
C
elements. The simplest one is a product of link variables along the perimeter of a plaquette originating at n in the positive µ − ν directions (see Fig. 2.1). The ‘naive’ lattice action can thus be written as S[Ψ, Ψ, U] = a4
XX n
a
f
f
Ψ (n)(D + mf0 )Ψf (n)
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where
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2N X X 1 + 2 (1 − ReTr[Uµ (n)Uν (n + aˆ µ)Uµ† (n + aˆ ν )Uν† (n)]) g0 µ 0, tlist -> {}, xlist -> {}}; optionlist = List[options]; gaugelocal = gauge /. optionlist /. defaultlist; tlistlocal = tlist /. optionlist /. defaultlist; xlistlocalorig = xlist /. optionlist /. defaultlist; xlistlocal = xlistlocalorig; tlistTrueFalse = Table[!FreeQ[integrandlocal,t[i]], {i,Length[tlistlocal]}]; xlistTrueFalse = Table[!FreeQ[integrandlocal,x[i]], {i,Length[xlistlocal]}]; w[a__] := WriteString[ToString[file], " ", a, "\n"]; wc[a__]:= WriteString[ToString[file], " &", a, "\n"]; Label[p1p2]; expr = List /@ (If[Head[integrandlocal]===Plus, List@@integrandlocal, List[integra ndlocal]]);
expr = Drop[#,1]& /@ expr; expr = expr /. (a_[b_,rho[n_]] :> a[b /. {p[1] :> ToExpression[StringJoin["i",ToString[n]]], p[2] :> ToExpression[StringJoin["j",ToString[n]]]}]); substlist = {s2sq[p[1]]->s2sq1, s2sq[p[2]]->s2sq2, s2sq[p[1]+p[2]]->s2sq12, sisq[p[1]]->sisq1, sisq[p[2]]->sisq2, sisq[p[1]+p[2]]->sisq12, s2qu[p[1]]->s2qu1, s2qu[p[2]]->s2qu2, s2qu[p[1]+p[2]]->s2qu12, c2sq[p[1]]->c2sq1, c2sq[p[2]]->c2sq2, c2sq[p[1]+p[2]]->c2sq12, cisq[p[1]]->cisq1, cisq[p[2]]->cisq2, cisq[p[1]+p[2]]->cisq12, omegaO[p[1]]->omegao1[jov[ir]], omegainvO[p[1]]->omegainvo1[jov[ir]], omegabO[p[1]]->omegabo1[jov[ir]], omegabinvO[p[1]]->omegabinvo1[jov[ir]], mO[p[1]]->mo1[jov[ir]], omegaO[p[2]]->omegao2[jov[ir]], omegainvO[p[2]]->omegainvo2[jov[ir]], omegabO[p[2]]->omegabo2[jov[ir]], omegabinvO[p[2]]->omegabinvo2[jov[ir]], mO[p[2]]->mo2[jov[ir]], omegaO[p[1]+p[2]]->omegao12[jov[ir]], omegainvO[p[1]+p[2]]->omegainvo12[jov[ir]], omegabO[p[1]+p[2]]->omegabo12[jov[ir]], omegabinvO[p[1]+p[2]]->omegabinvo12[jov[ir]], mO[p[1]+p[2]]->mo12[jov[ir]], omegaplusinvO[0,p[1]]->omegaplusinvo1[jov[ir]], omegaplusinvO[0,p[2]]->omegaplusinvo2[jov[ir]], omegaplusinvO[0,p[1]+p[2]]->omegaplusinvo12[jov[ir]], omegaplusinvO[p[1],p[2]]->omegaplusinvo1p2[jov[ir]], omegaplusinvO[p[1],p[1]+p[2]]->omegaplusinvo1p12[jov[ir]], omegaplusinvO[p[2],p[1]+p[2]]->omegaplusinvo2p12[jov[ir]], m[p[1]]->m1[jov[ir]], m[p[2]]->m2[jov[ir]], m[p[1]+p[2]]->m12[jov[ir]], s2[2 a__]^2 :> si2[a], c2[2 a__]^2 :> ci2[a], s2[a__]^2 :> s22[a], c2[a__]^2 :> c22[a], s2[2 a__] :> si[a], c2[2 a__] :> ci[a], s2[a__]^4 :> s24[a], c2[a__]^4 :> c24[a], fhat[p[1]]-> fhat1[jov[ir]], hat2[p[1]]->hat1, fhat[p[2]]-> fhat2[jov[ir]], hat2[p[2]]->hat2, fhat[p[1]+p[2]]-> fhat12[jov[ir]], hat2[p[1]+p[2]]->hat12}; expr = expr /. substlist; xlistlocal = xlistlocalorig /. substlist; xlistlocal = xlistlocal /. mO -> m[jov[ir]]; expr = expr /. {dprop[p[1],a__] :> dprop["i",a], dprop[p[2],a__] :> dprop["j",a], dprop[p[1]+p[2],a__] :> dprop["ij",a]}; expr = expr /. dprop[a_,rho[i_],rho[j_]] :> dpropn[Sequence @@ (ToExpression[StringJoin[a,ToString[#],"n"]]& /@ {i,j}
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variables = Select[Variables[integrandlocal],(FreeQ[#,p[2]] && FreeQ[#,t] && FreeQ [#,x])&]; variablelist = Table[Cases[variables,_[_,rho[j]]],{j,4}]; variablelist = Append[Drop[variablelist,-1], Join[variablelist[[-1]], Select[variables,(!FreeQ[#,p[1]] && (FreeQ[#,rho] || (!FreeQ[#,dprop ]) || (!FreeQ[#,prop])))&]]]; variables = Complement[Variables[integrandlocal],variables]; variablelist = Join[variablelist,Table[Cases[variables,_[_,rho[j]]],{j,4}]]; variablelist = Append[Drop[variablelist,-1], Join[variablelist[[-1]], Select[variables,(FreeQ[#,rho] || (!FreeQ[#,dprop]) || (!FreeQ[#,pro p]))&]]];
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proplist = proplist /. {fhat[p[1]]->fhat1[jov[ir]], fhat[p[2]]->fhat2[jov[ir]], hat2[p[1]]-> hat1, hat2[p[2]]-> hat2, fhat[p[1]+p[2]]->fhat12[jov[ir]], hat2[p[1]+p[2]]-> hat12}; proplist = FortranForm /@ proplist;
B.2. The integrator routine
integrator2::wrongArgument = "integrator2 called with inappropriate argument" integrator2[integrand_, ___]:= Message[integrator2::wrongArgument] /; !(Complement[Union[If[Head[#]===Symbol,#,Head[#]]& /@ Variables[integrand]], {s2,c2,s2sq,sisq,s2qu,c2sq,cisq,r,m,var,hat2,fhat,prop,dprop,cprop0,cprop1, cprop2,cprop3,mO,omegaO,omegainvO,omegaplusinvO,omegabO,omegabinvO,x,t}]==={})
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proplist = {Select[propagatorslocal*dummy1*dummy2, FreeQ[#,p[2]]&] /. dummy1 -> 1 /. dummy2 -> 1}; proplist = {proplist[[1]], propagatorslocal/proplist[[1]]};
), Sequence @@ (ToExpression[StringJoin[a,ToString[#],"h"]]& /@ Complement[Range[4],{i,j}] ), jiw[ir]] /; (!(i===j)); expr = expr /. dprop[a_,rho[i_],rho[i_]] :> dpropd[Sequence @@ (ToExpression[StringJoin[a,ToString[#],"h"]]& /@ {i}), Sequence @@ (ToExpression[StringJoin[a,ToString[#],"h"]]& /@ Complement[Range[4],{i}]), jiw[ir]]; expr = expr /. {prop[p[1],a__] :> prop["i",a], prop[p[2],a__] :> prop["j",a],
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Do[rulelist = Rule[#,1]& /@ (variablelist[[-j]]); expr = (Join[{(#[[1]]) /. rulelist}, {(#[[1]]) / ((#[[1]]) /. rulelist)}, Drop[#,1]])& /@ expr, {j,Length[variablelist]}]; varlist=Union[integrandlocal[[Sequence @@ #]]&/@ Position[integrandlocal,var[_]]]; varlength = Max[(#[[1]])& /@ varlist, 1]; factorlist = ((#[[1]])& /@ expr); factorlist = Table[# /. var[j]->1 /. var[_]->0,{j,varlength}]& /@ factorlist; factorlist = factorlist /. r -> r[jov[ir]] /. m -> m[jov[ir]] /. mO -> m[jov[ir]] /. cprop0 -> cprop0[jiw[ir]] /. cprop1 -> cprop1[jiw[ir]] /. cprop2 -> cprop2[jiw[ir]] /. cprop3 -> cprop3[jiw[ir]]; factorlist = Map[FortranForm,factorlist,{2}];
jiw[ir]] /; (!(i===j)); expr = expr /. prop[a_,rho[i_],rho[i_]] :> propd[Sequence @@ (ToExpression[StringJoin[a,ToString[#],"h"]]& /@ {i}), Sequence @@ (ToExpression[StringJoin[a,ToString[#],"h"]]& /@ Complement[Range[4],{i}]), jiw[ir]]; expr = Map[FortranForm,expr,{2}];
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expr1 = Union[Take[#,{5,8}]& /@ expr]; expr2 = Union[Take[#,{6,8}]& /@ expr]; expr3 = Union[Take[#,{7,8}]& /@ expr]; expr4 = Union[Take[#,{8,8}]& /@ expr]; index3 = Table[Position[expr4,Drop[expr3[[j]],1]][[1,1]],{j,Length[expr3]}]; index2 = Table[Position[expr3,Drop[expr2[[j]],1]][[1,1]],{j,Length[expr2]}]; index1 = Table[Position[expr2,Drop[expr1[[j]],1]][[1,1]],{j,Length[expr1]}]; index = Table[Position[expr1,Take[expr[[j]],{5, 8}]][[1,1]], {j,Length[expr]}]; If[flagp1p2 == 0, flagp1p2 = 1; integrandlocal = integrandlocal /. {p[1]->p[2], p[2]->p[1]}; propagatorslocal = propagatorslocal /. {p[1]->p[2], p[2]->p[1]}; tlistlocalS = tlistlocal; tlistlocal = tlistlocal /. {p[1]->p[2], p[2]->p[1]}; xlistlocalS = xlistlocal; xlistlocalorig = xlistlocalorig /. {p[1]->p[2], p[2]->p[1]}; exprS = expr; expr1S = expr1; expr2S = expr2; expr3S = expr3; expr4S = expr4; index1S = index1; index2S = index2; index3S = index3; indexS = index; proplistS = proplist; factorlistS = factorlist; Goto[p1p2]]; If[Length[expr4S] < Length[expr4], tlistlocal = tlistlocalS; xlistlocalorig = xlistlocalorig /. {p[1]->p[2], p[2]->p[1]}; xlistlocal = xlistlocalS; expr = exprS; expr1 = expr1S; expr2 = expr2S; expr3 = expr3S; expr4 = expr4S; index1 = index1S; index2 = index2S; index3 = index3S; index = indexS; proplist = proplistS; factorlist = factorlistS]; Print["Inner loop: ",Length[expr4]," iterations"]; If[Length[expr4]>200,Print[" *****"]];
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B.2. The integrator routine
Sequence @@ (ToExpression[StringJoin[a,ToString[#],"h"]]& /@ Complement[Range[4],{i,j}]
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w["dimension dpropn(-20:20,-20:20,0:20,0:20,50)"]; w["dimension propn(-20:20,-20:20,0:20,0:20,50)"]; w["dimension modn(-39:80), modh(-39:80)"]; w["dimension g(4,4), g2(4,4), ginv(4,4)"]; w["integer indx(4)"]; w["dimension omegainvo1(50), omegabinvo1(50), omegaplusinvo1(50)"]; w["dimension omegao1(50), mo1(50), omegabo1(50)"]; w["dimension omegainvo2(50), omegabinvo2(50), omegaplusinvo2(50)"]; w["dimension omegao2(50), mo2(50), omegabo2(50)"]; w["dimension omegainvo12(50),omegabinvo12(50),omegaplusinvo12(50)"]; w["dimension omegao12(50), mo12(50), omegabo12(50)"]; w["dimension omegaplusinvo1p2(50),omegaplusinvo1p12(50)"]; w["dimension omegaplusinvo2p12(50)"]; w["dimension result(",varlength,")"]; If[Length[tlistlocal]>0, w["dimension t(",Length[tlistlocal],")"]]; If[Length[xlistlocal]>0, w["dimension x(",Length[xlistlocal],")"]]; w[""]; w["pi = 4.0*atan(1.d0)"]; WriteString[ToString[file], "CCCC ", "Enter minimum and maximum length of lattice (even, < 41)", "\n"]; w["read(*,*) nmin, nmax"]; WriteString[ToString[file], "CCCC ", "Write up to 300 sextuplets of values for r, m, c0, c1, c2, c3, one se t per line", "\n"]; WriteString[ToString[file], "CCCC ", "NB: No more than 50 different values for overlap parameters; idem for Iwasaki", "\n"]; w["ir = 1"]; w["iov = 1"]; w["iiw = 1"]; w["iovflag = 1"]; w["iiwflag = 1"]; WriteString[ToString[file], " 1 read(*,*,end=2) rtemp, mtemp, \n"]; wc[" c0temp, c1temp, c2temp, c3temp"]; w["do iov2 = 1, iov - 1"]; w[" if((rtemp-r(iov2)).lt.1.d-9 .and. (mtemp-m(iov2)).lt.1.d-9) then"]; w[" jov(ir) = iov2"]; w[" iovflag = 0"]; w[" endif"]; w["enddo"]; w["if(iovflag.eq.1) then"]; w[" r(iov) = rtemp"]; w[" m(iov) = mtemp"]; w[" jov(ir) = iov"]; w[" iov = iov + 1"]; w["endif"]; w["iovflag = 1"]; w["do iiw2 = 1, iiw - 1"]; w[" if((c0temp-cprop0(iiw2)).lt.1.d-9 .and. "]; wc[" (c1temp-cprop1(iiw2)).lt.1.d-9 .and. "]; wc[" (c2temp-cprop2(iiw2)).lt.1.d-9 .and. "]; wc[" (c3temp-cprop3(iiw2)).lt.1.d-9) then"]; w[" jiw(ir) = iiw2"]; w[" iiwflag = 0"]; w[" endif"]; w["enddo"]; w["if(iiwflag.eq.1) then"]; w[" cprop0(iiw) = c0temp"]; w[" cprop1(iiw) = c1temp"]; w[" cprop2(iiw) = c2temp"]; w[" cprop3(iiw) = c3temp"]; w[" cc1(iiw) = cprop2(iiw) + cprop3(iiw)"]; w[" cc2(iiw) = cprop1(iiw) - cprop2(iiw) - cprop3(iiw)"]; w[" jiw(ir) = iiw"]; w[" iiw = iiw + 1"]; w["endif"];
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w["program main"]; w["implicit real*8 (a-h,o-z)"]; w["real*8 k2p1, k4p1, k2p2, k4p2, k2p12, k4p12"]; w["real*8 m, m1, m2, m12, mo1, mo2, mo12"]; w["real*8 rtemp, mtemp, c0temp, c1temp, c2temp, c3temp"]; w["parameter (l = ",Length[expr],")"]; w["dimension a(l,4,300), b(l,4,300)"]; w["dimension s2(-500:500), c2(-500:500), s22(-500:500), c22(-500:500)"]; w["dimension s24(-500:500), c24(-500:500)"]; w["dimension si(-500:500), ci(-500:500), si2(-500:500), ci2(-500:500)"]; w["dimension r(50), r24(50), r12(50), m(50)"]; w["dimension cprop0(50), cprop1(50), cprop2(50), cprop3(50)"]; w["dimension cc1(50), cc2(50)"]; w["dimension jov(300), jiw(300)"]; w["dimension fhat1(50), fhat2(50), fhat12(50)"]; w["dimension m1(50), m2(50), m12(50)"]; w["dimension dpropd(0:20,0:20,0:20,0:20,50)"]; w["dimension propd(0:20,0:20,0:20,0:20,50)"];
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w[" enddo"]; w[" enddo"]; w[" do mu = 1,4"]; w[" do nu = 1,4"]; w[" ginv(mu,nu) = 0.d0"]; w[" enddo"]; w[" ginv(mu,mu) = 1.d0"]; w[" enddo"]; w[" call ludcmp(g,4,4,indx,d)"]; w[" do nu = 1, 4"]; w[" call lubksb(g,4,4,indx,ginv(1,nu))"]; w[" enddo"]; w[" propn(i1,i2,i3,i4,ir) = ginv(1,2)- ", gaugelocal, " * g2(1,2)"]; w[" dpropn(i1,i2,i3,i4,ir) = ginv(1,2)"]; w[" if (i1.ge.0.and.i2.ge.0) then"]; w[" propd(i1,i2,i3,i4,ir) = ginv(1,1)- ", gaugelocal, " * g2(1,1)"]; w[" dpropd(i1,i2,i3,i4,ir) = ginv(1,1) - hat1"]; w[" endif"]; w["enddo"]; WriteString[ToString[file], " 1001 ", "continue", "\n"]; w["enddo"]; w["enddo"]; w["enddo"]; w["enddo"]]; w[""]; w["do i = 1,l"]; w["do ir = 1,nr"]; w["a(i,1,ir) = 0.d0"]; w["enddo"]; w["enddo"]; w["do i1 = nhalf,n"]; w[" i1n = modn(i1)"]; w[" i1h = modh(i1)"]; w[" do i = 1,l"]; w[" do ir = 1,nr"]; w[" a(i,2,ir) = 0.d0"]; w[" enddo"]; w[" enddo"]; w[" do i2 = nhalf,n"]; w[" i2n = modn(i2)"]; w[" i2h = modh(i2)"]; w[" do i = 1,l"]; w[" do ir = 1,nr"]; w[" a(i,3,ir) = 0.d0"]; w[" enddo"]; w[" enddo"]; w[" do i3 = nhalf,n"]; w[" i3n = modn(i3)"]; w[" i3h = modh(i3)"]; w[" do i = 1,l"]; w[" do ir = 1,nr"]; w[" a(i,4,ir) = 0.d0"]; w[" enddo"]; w[" enddo"]; w[" do i4 = nhalf,n"]; w[" i4n = modn(i4)"]; w[" i4h = modh(i4)"]; w[" s2sq1 = (s22(i1)+s22(i2)+s22(i3)+s22(i4))"]; w[" sisq1 = (si2(i1)+si2(i2)+si2(i3)+si2(i4))"]; w[" c2sq1 = (c22(i1)+c22(i2)+c22(i3)+c22(i4))"]; w[" cisq1 = (ci2(i1)+ci2(i2)+ci2(i3)+ci2(i4))"]; If[!FreeQ[{expr,xlistlocal},s2qu1], w[" s2qu1 = (s24(i1)+s24(i2)+s24(i3)+s24(i4))"]]; w[" if (s2sq1.gt.1e-9) then"]; If[!FreeQ[{expr,proplist,xlistlocal},hat1], w[" hat1 = 0.25d0/s2sq1"]];
B.2. The integrator routine
w["iiwflag = 1"]; w["ir = ir + 1"]; w["goto 1"]; WriteString[ToString[file], " 2 nr = ir - 1 \n"]; w["iov = iov - 1"]; w["iiw = iiw - 1"]; w["do ir = 1, iov"]; w[" r24(ir) = 4.d0*r(ir)**2"]; w[" r12(ir) = 2.d0*r(ir)"]; w["enddo"]; w[""]; w["do 1000 n = nmin, nmax, 2"]; w["nmod2 = mod(n,2)"]; w["nhalf = n/2"]; w["n4 = n*4"]; w["do i = -12*n,12*n"]; w[" s2(i) = sin(i*pi/n)"]; w[" c2(i) = cos(i*pi/n)"]; w[" s22(i) = s2(i)**2"]; w[" c22(i) = c2(i)**2"]; w[" s24(i) = s2(i)**4"]; w[" c24(i) = c2(i)**4"]; w[" si(i) = sin(2*i*pi/n)"]; w[" ci(i) = cos(2*i*pi/n)"]; w[" si2(i) = si(i)**2"]; w[" ci2(i) = ci(i)**2"]; w["enddo"]; w[""]; w["do i = 1-n, 2*n"]; w[" itemp = mod(i+n,2*n)-n"]; w[" if(itemp.gt.0) itemp = min(itemp, n-itemp)"]; w[" if(itemp.lt.0) itemp = max(itemp,-n-itemp)"]; w[" modn(i) = itemp"]; w[" modh(i) = min(mod(i+n,n), n - mod(i+n,n))"]; w["enddo"]; w[""]; If[!FreeQ[integrand,prop] || !FreeQ[integrand,dprop], w["do i1 = -nhalf, nhalf"]; w["do i2 = -nhalf, nhalf"]; w["do i3 = 0, nhalf"]; w["do i4 = 0, nhalf"]; w["if (i1.eq.0.and.i2.eq.0.and.i3+i4.eq.0) goto 1001"]; w["do ir = 1, iiw"]; w[" s2sq1 = (s22(i1)+s22(i2)+s22(i3)+s22(i4))"]; w[" hat1 = 0.25d0/s2sq1"]; w[" k2p1 = 4* (s22(i1)+s22(i2)+s22(i3)+s22(i4))"]; w[" k4p1 = 16* (s24(i1)+s24(i2)+s24(i3)+s24(i4))"]; w[" dummy = (1-cc1(ir)*k2p1)"]; w[" do mu = 1, 4"]; w[" if (mu.eq.1) imu = i1"]; w[" if (mu.eq.2) imu = i2"]; w[" if (mu.eq.3) imu = i3"]; w[" if (mu.eq.4) imu = i4"]; w[" do nu = 1, mu"]; w[" if (nu.eq.1) inu = i1"]; w[" if (nu.eq.2) inu = i2"]; w[" if (nu.eq.3) inu = i3"]; w[" if (nu.eq.4) inu = i4"]; w[" g(mu,nu) = (1-dummy) * (2*s2(imu))*(2*s2(inu))"]; wc[" + cc2(ir) *(2*s2(imu))**3*(2*s2(inu))"]; wc[" + cc2(ir) *(2*s2(imu))*(2*s2(inu))**3"]; w[" g2(mu,nu) = (2*s2(imu))*(2*s2(inu))/k2p1**2"]; w[" if (mu.eq.nu) g(mu,nu) = g(mu,nu) + dummy * k2p1"]; wc[" - cc2(ir) * k4p1 - cc2(ir) * k2p1 * (2*s2(imu))**2"]; w[" if (nu.lt.mu) g(nu,mu) = g(mu,nu)"]; w[" if (nu.lt.mu) g2(nu,mu) = g2(mu,nu)"];
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w[" do j4 = 1,n"]; w[" j4n = modn(j4)"]; w[" j4h = modh(j4)"]; w[" ij4n = modn(i4+j4)"]; w[" ij4h = modh(i4+j4)"]; w[" s2sq12 = (s22(j1+i1)+s22(j2+i2)+s22(j3+i3)+s22(j4+i4))"]; w[" s2sq2 = (s22(j1)+s22(j2)+s22(j3)+s22(j4))"]; w[" sisq12 = (si2(j1+i1)+si2(j2+i2)+si2(j3+i3)+si2(j4+i4))"]; w[" sisq2 = (si2(j1)+si2(j2)+si2(j3)+si2(j4))"]; w[" c2sq12 = (c22(j1+i1)+c22(j2+i2)+c22(j3+i3)+c22(j4+i4))"]; w[" c2sq2 = (c22(j1)+c22(j2)+c22(j3)+c22(j4))"]; w[" cisq12 = (ci2(j1+i1)+ci2(j2+i2)+ci2(j3+i3)+ci2(j4+i4))"]; w[" cisq2 = (ci2(j1)+ci2(j2)+ci2(j3)+ci2(j4))"]; w[" s2qu12 = (s24(j1+i1)+s24(j2+i2)+s24(j3+i3)+s24(j4+i4))"]; w[" s2qu2 = (s24(j1)+s24(j2)+s24(j3)+s24(j4))"]; w[" if (s2sq12.gt.1e-9 .and. s2sq2.gt.1e-9) then"]; w[" hat2 = 0.25d0/s2sq2 "]; w[" hat12 = 0.25d0/s2sq12"]; Do[If[tlistTrueFalse[[i]] && (!FreeQ[tlistlocal[[i]],p[2]]), w[" t(",i,") = 0.d0"]; Do[wc[" + ", FortranForm[tlistlocal[[i]] /. s2[b_] :> s2[b /. {p[1] :> ToExpression[StringJoin["i",ToString[ k]]], p[2] :> ToExpression[StringJoin["j",ToString[ k]]]}] /. c2[b_] :> c2[b /. {p[1] :> ToExpression[StringJoin["i",ToString[ k]]], p[2] :> ToExpression[StringJoin["j",ToString[ k]]]}]]], {k,4}]], {i, Length[tlistlocal]}]; w[" do ir = 1,iov"]; If[!FreeQ[proplist,fhat2] || !FreeQ[expr,fhat2], w[" m2(ir) = m(ir) + r12(ir)*s2sq2"]; w[" fhat2(ir) =1.d0/(m2(ir)**2 + sisq2 )"]]; If[!FreeQ[proplist,fhat12] || !FreeQ[expr,fhat12], w[" m12(ir) = m(ir) + r12(ir)*s2sq12"]; w[" fhat12(ir)=1.d0/(m12(ir)**2 + sisq12)"]]; If[!FreeQ[{integrand,xlistlocalorig},mO] || !FreeQ[{integrand,xlistlocalorig},omeg aO] || !FreeQ[{integrand,xlistlocalorig},omegainvO] || !FreeQ[{integrand,xlistlocalori g},omegaplusinvO] || !FreeQ[{integrand,xlistlocalorig},omegabO] || !FreeQ[{integrand,xlistlocalorig} ,omegabinvO], w[" mo2(ir) = m(ir) - r12(ir)*s2sq2"]; w[" omegao2(ir) = sqrt(sisq2 + mo2(ir)**2)"]; w[" omegainvo2(ir) = 1.d0/omegao2(ir)"]; w[" omegaplusinvo2(ir) = 1.d0/(omegao2(ir)+ m(ir))"]; w[" omegabo2(ir) = (omegao2(ir)-mo2(ir))"]; w[" omegabinvo2(ir) = 1.d0/(omegao2(ir)-mo2(ir))"]; w[" mo12(ir) = m(ir) - r12(ir)*s2sq12"]; w[" omegao12(ir) = sqrt(sisq12 + mo12(ir)**2)"]; w[" omegainvo12(ir) = 1.d0/omegao12(ir)"]; w[" omegaplusinvo12(ir) = 1.d0/(omegao12(ir)+ m(ir))"]; w[" omegabo12(ir) = (omegao12(ir)-mo12(ir))"]; w[" omegabinvo12(ir) = 1.d0/(omegao12(ir)-mo12(ir))"]; w[" omegaplusinvo1p2(ir) = 1.d0/(omegao1(ir)+ omegao2(ir))"]; w[" omegaplusinvo1p12(ir) = 1.d0/(omegao1(ir)+ omegao12(ir))"]; w[" omegaplusinvo2p12(ir) = 1.d0/(omegao2(ir)+ omegao12(ir))"]]; w[" enddo"]; w[" do ir = 1, nr"]; Do[If[xlistTrueFalse[[i]], w[" x(",i,") = 0.d0"]; If[Head[xlistlocal[[i]]]===Plus, Do[
B.2. The integrator routine
w[" do ir = 1,iov"]; If[!FreeQ[{expr,proplist,xlistlocal},fhat1], w[" m1(ir) = m(ir) + r12(ir)*s2sq1"]; w[" fhat1(ir) = 1.d0/(m1(ir)**2 + sisq1)"]]; If[!FreeQ[{integrand,xlistlocalorig},mO] || !FreeQ[{integrand,xlistlocalorig},omeg aO] || !FreeQ[{integrand,xlistlocalorig},omegainvO] || !FreeQ[{integrand,xlistlocalori g},omegaplusinvO] || !FreeQ[{integrand,xlistlocalorig},omegabO] || !FreeQ[{integrand,xlistlocalorig} ,omegabinvO], w[" mo1(ir) = m(ir) - r12(ir)*s2sq1"]; w[" omegao1(ir) = sqrt(sisq1 + mo1(ir)**2)"]; w[" omegainvo1(ir) = 1.d0/omegao1(ir)"]; w[" omegaplusinvo1(ir) = 1.d0/(omegao1(ir)+ m(ir))"]; w[" omegabo1(ir) = (omegao1(ir)-mo1(ir))"]; w[" omegabinvo1(ir) = 1.d0/(omegao1(ir)-mo1(ir))"]]; w[" enddo"]; Do[If[tlistTrueFalse[[i]] && (FreeQ[tlistlocal[[i]],p[2]]), w[" t(",i,") = 0.d0"]; Do[wc[" + ", FortranForm[tlistlocal[[i]] /. s2[b_] :> s2[b /. {p[1] :> ToExpression[StringJoin["i",ToString[ k]]], p[2] :> ToExpression[StringJoin["j",ToString[ k]]]}] /. c2[b_] :> c2[b /. {p[1] :> ToExpression[StringJoin["i",ToString[ k]]], p[2] :> ToExpression[StringJoin["j",ToString[ k]]]}]]], {k,4}]], {i, Length[tlistlocal]}]; w[" do i = 1,",Length[expr1]]; w[" do ir = 1,nr"]; w[" b(i,1,ir) = 0.d0"]; w[" enddo"]; w[" enddo"]; w[" do j1 = 1,n"]; w[" j1n = modn(j1)"]; w[" j1h = modh(j1)"]; w[" ij1n = modn(i1+j1)"]; w[" ij1h = modh(i1+j1)"]; w[" do i = 1,",Length[expr2]]; w[" do ir = 1,nr"]; w[" b(i,2,ir) = 0.d0"]; w[" enddo"]; w[" enddo"]; w[" do j2 = 1,n"]; w[" j2n = modn(j2)"]; w[" j2h = modh(j2)"]; w[" ij2n = modn(i2+j2)"]; w[" ij2h = modh(i2+j2)"]; w[" do i = 1,",Length[expr3]]; w[" do ir = 1,nr"]; w[" b(i,3,ir) = 0.d0"]; w[" enddo"]; w[" enddo"]; w[" do j3 = 1,n"]; w[" j3n = modn(j3)"]; w[" j3h = modh(j3)"]; w[" ij3n = modn(i3+j3)"]; w[" ij3h = modh(i3+j3)"]; w[" do i = 1,",Length[expr4]]; w[" do ir = 1,nr"]; w[" b(i,4,ir) = 0.d0"]; w[" enddo"]; w[" enddo"];
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w[" do ir = 1,nr"]; w[" do i = 1,l"]; w[" a(i,4,ir) = a(i,4,ir) * 0.5d0"]; w[" enddo"]; w[" enddo"]; w[" endif"]; w[" do ir = 1,nr"]; Do[If[expr[[j,3]]===FortranForm[1], w["a(",j,",3,ir) = a(",j,",3,ir) + a(",j,",4,ir)"], w["a(",j,",3,ir) = a(",j,",3,ir) + a(",j,",4,ir)*"]; wc[" ",expr[[j,3]]]], {j,Length[expr]}]; w[" enddo"]; w[" enddo"]; w[" if (i2.eq.n .or. (i2.eq.nhalf .and. nmod2.eq.0)) then"]; w[" do ir = 1,nr"]; w[" do i = 1,l"]; w[" a(i,3,ir) = a(i,3,ir) * 0.5d0"]; w[" enddo"]; w[" enddo"]; w[" endif"]; w[" do ir = 1,nr"]; Do[If[expr[[j,2]]===FortranForm[1], w["a(",j,",2,ir) = a(",j,",2,ir) + a(",j,",3,ir)"], w["a(",j,",2,ir) = a(",j,",2,ir) + a(",j,",3,ir)*"]; wc[" ",expr[[j,2]]]], {j,Length[expr]}]; w[" enddo"]; w[" enddo"]; w[" if (i1.eq.n .or. (i1.eq.nhalf .and. nmod2.eq.0)) then"]; w[" do ir = 1,nr"]; w[" do i = 1,l"]; w[" a(i,2,ir) = a(i,2,ir) * 0.5d0"]; w[" enddo"]; w[" enddo"]; w[" endif"]; w[" do ir = 1,nr"]; Do[If[expr[[j,1]]===FortranForm[1], w["a(",j,",1,ir) = a(",j,",1,ir) + a(",j,",2,ir)"], w["a(",j,",1,ir) = a(",j,",1,ir) + a(",j,",2,ir)*"]; wc[" ",expr[[j,1]]]], {j,Length[expr]}]; w[" enddo"]; w["enddo"]; w["do ir = 1,nr"]; Do[w["result(",k,") = 0.d0"]; Do[If[!(factorlist[[j,k]] === FortranForm[0]), w["result(",k,") = result(",k,") + a(",j,",1,ir) * "]; wc["(",N[factorlist[[j,k]],16],")"]], {j,Length[expr]}]; w["result(",k,") = result(",k,") * 2**4 / float(n)**8"], {k,varlength}]; w[" write(*,*) n,r(jov(ir)), m(jov(ir)),"]; wc[" cprop0(jiw(ir)), cprop1(jiw(ir)),"]; wc[" cprop2(jiw(ir)), cprop3(jiw(ir)), result"]; w["enddo"]; w["call flush(6)"]; WriteString[ToString[file], " 1000 ", "continue", "\n"]; w["stop"]; w["end"]; ]
B.2. The integrator routine
If[Head[xlistlocal[[i,j]]]===Times && Length[xlistlocal[[i,j]]]>5, wc[" +", FortranForm[Take[xlistlocal[[i,j]],4]]]; If[Length[Drop[xlistlocal[[i,j]],4]]>4, wc[" *", FortranForm[Take[Drop[xlistlocal[[i,j]],4],3]]]; wc[" *", FortranForm[Drop[Drop[xlistlocal[[i,j]],4],3]]], wc[" *", FortranForm[Drop[xlistlocal[[i,j]],4]]]], wc[" +", FortranForm[xlistlocal[[i,j]]]]], {j,Length[xlistlocal[[i]]]}], wc[" +", FortranForm[xlistlocal[[i]]]]]], {i, Length[xlistlocal]}]; w[" prop = ", proplist[[2]]]; Do[If[proplist[[2]]===FortranForm[1], w["b(",j,",4,ir) = b(",j,",4,ir) + "]; If[Head[expr4[[j,1,1]]]===Times && Length[expr4[[j,1,1]]] > 4, wc[" ",Take[#,4]& /@ expr4[[j,1]],"*"]; temp = Drop[#,4]& /@ expr4[[j,1]]; If[Head[temp[[1]]]===Times && Length[temp[[1]]] > 4, wc[" ",Take[#,4]& /@ temp,"*"]; wc[" ",Drop[#,4]& /@ temp], wc[" ",temp]], wc[" ",expr4[[j,1]]]], If[expr4[[j,1]]===FortranForm[1], w["b(",j,",4,ir) = b(",j,",4,ir) + prop"], w["b(",j,",4,ir) = b(",j,",4,ir) + prop*"]; wc[" ",expr4[[j,1]]]]], {j,Length[expr4]}]; w[" enddo"]; w[" endif"]; w[" enddo"]; w[" do ir = 1,nr"]; Do[If[expr3[[j,1]]===FortranForm[1], w["b(",j,",3,ir) = b(",j,",3,ir) + b(",index3[[j]],",4,ir)"], w["b(",j,",3,ir) = b(",j,",3,ir) + b(",index3[[j]],",4,ir)*"]; wc[" ",expr3[[j,1]]]], {j,Length[expr3]}]; w[" enddo"]; w[" enddo"]; w[" do ir = 1,nr"]; Do[If[expr2[[j,1]]===FortranForm[1], w["b(",j,",2,ir) = b(",j,",2,ir) + b(",index2[[j]],",3,ir)"], w["b(",j,",2,ir) = b(",j,",2,ir) + b(",index2[[j]],",3,ir)*"]; wc[" ",expr2[[j,1]]]], {j,Length[expr2]}]; w[" enddo"]; w[" enddo"]; w[" do ir = 1,nr"]; Do[If[expr1[[j,1]]===FortranForm[1], w["b(",j,",1,ir) = b(",j,",1,ir) + b(",index1[[j]],",2,ir)"], w["b(",j,",1,ir) = b(",j,",1,ir) + b(",index1[[j]],",2,ir)*"]; wc[" ",expr1[[j,1]]]], {j,Length[expr1]}]; w[" enddo"]; w[" enddo"]; w[" do ir = 1,nr"]; w[" prop = ", proplist[[1]]]; w[" if (i4.eq.n .or. (i4.eq.nhalf .and. nmod2.eq.0))"]; wc[" prop = 0.5d0 * prop"]; Do[If[expr[[j,4]]===FortranForm[1], w["a(",j,",4,ir) = a(",j,",4,ir) + prop*b(",index[[j]],",1,ir)"], w["a(",j,",4,ir) = a(",j,",4,ir) + prop*b(",index[[j]],",1,ir)*"]; wc[" ",expr[[j,4]]]], {j,Length[expr]}]; w[" enddo"]; w[" endif"]; w[" enddo"]; w[" if (i3.eq.n .or. (i3.eq.nhalf .and. nmod2.eq.0)) then"];
B.3. A particular example
B.3
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A particular example
It is useful to take a typical term that appears in a certain diagram and present the Fortran Code, generated by the integrator. The term is
−6 c2(p1 + 2p2 , rho(1))2 hat2(p1 ) omegabinvO(p2 ) omegabinvO(p1 + p2 )
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omegainvO(p2 )2 omegainvO(p1 + p2 )2 omegaplusinvO(p2 , p1 + p2 )2
s2(2p2 , rho(2)) s2(2p2 , rho(3))2 s2(2p1 + 2p2 , rho(2)) s2(2p1 + 2p2 , rho(4))2
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All the above symbolic functions are defined in the previous section.
do 1000 n = nmin, nmax, 2 nmod2 = mod(n,2) nhalf = n/2 n4 = n*4 do i = -12*n,12*n s2(i) = sin(i*pi/n) c2(i) = cos(i*pi/n) s22(i) = s2(i)**2 c22(i) = c2(i)**2 s24(i) = s2(i)**4 c24(i) = c2(i)**4 si(i) = sin(2*i*pi/n) ci(i) = cos(2*i*pi/n) si2(i) = si(i)**2 ci2(i) = ci(i)**2 enddo
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enddo if(iiwflag.eq.1) then cprop0(iiw) = c0temp cprop1(iiw) = c1temp cprop2(iiw) = c2temp cprop3(iiw) = c3temp cc1(iiw) = cprop2(iiw) + cprop3(iiw) cc2(iiw) = cprop1(iiw) - cprop2(iiw) - cprop3(iiw) jiw(ir) = iiw iiw = iiw + 1 endif iiwflag = 1 ir = ir + 1 goto 1 nr = ir - 1 iov = iov - 1 iiw = iiw - 1 do ir = 1, iov r24(ir) = 4.d0*r(ir)**2 r12(ir) = 2.d0*r(ir) enddo
B.3. A particular example
program main implicit real*8 (a-h,o-z) real*8 k2p1, k4p1, k2p2, k4p2, k2p12, k4p12 real*8 m, m1, m2, m12, mo1, mo2, mo12 real*8 rtemp, mtemp, c0temp, c1temp, c2temp, c3temp parameter (l = 1) dimension a(l,4,300), b(l,4,300) dimension s2(-500:500), c2(-500:500), s22(-500:500), c22(-500:500) dimension s24(-500:500), c24(-500:500) dimension si(-500:500), ci(-500:500), si2(-500:500), ci2(-500:500) dimension r(50), r24(50), r12(50), m(50) dimension cprop0(50), cprop1(50), cprop2(50), cprop3(50) dimension cc1(50), cc2(50) dimension jov(300), jiw(300) dimension fhat1(50), fhat2(50), fhat12(50) dimension m1(50), m2(50), m12(50) dimension dpropd(0:20,0:20,0:20,0:20,50) dimension propd(0:20,0:20,0:20,0:20,50) dimension dpropn(-20:20,-20:20,0:20,0:20,50) dimension propn(-20:20,-20:20,0:20,0:20,50) dimension modn(-39:80), modh(-39:80) dimension g(4,4), g2(4,4), ginv(4,4) integer indx(4) dimension omegainvo1(50), omegabinvo1(50), omegaplusinvo1(50) dimension omegao1(50), mo1(50), omegabo1(50) dimension omegainvo2(50), omegabinvo2(50), omegaplusinvo2(50) dimension omegao2(50), mo2(50), omegabo2(50) dimension omegainvo12(50),omegabinvo12(50),omegaplusinvo12(50) dimension omegao12(50), mo12(50), omegabo12(50) dimension omegaplusinvo1p2(50),omegaplusinvo1p12(50) dimension omegaplusinvo2p12(50) dimension result(1)
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do i = 1-n, 2*n itemp = mod(i+n,2*n)-n if(itemp.gt.0) itemp = min(itemp, n-itemp) if(itemp.lt.0) itemp = max(itemp,-n-itemp) modn(i) = itemp modh(i) = min(mod(i+n,n), n - mod(i+n,n)) enddo
do i = 1,l do ir = 1,nr a(i,1,ir) = 0.d0 enddo enddo do i1 = nhalf,n i1n = modn(i1) i1h = modh(i1) do i = 1,l do ir = 1,nr a(i,2,ir) = 0.d0 enddo enddo do i2 = nhalf,n i2n = modn(i2) i2h = modh(i2) do i = 1,l do ir = 1,nr
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pi = 4.0*atan(1.d0) Enter minimum and maximum length of lattice (even, < 41) read(*,*) nmin, nmax CCCC Write up to 300 sextuplets of values for r, m, c0, c1, c2, c3, one set per line CCCC NB: No more than 50 different values for overlap parameters; idem for Iwasaki ir = 1 iov = 1 iiw = 1 iovflag = 1 iiwflag = 1 1 read(*,*,end=2) rtemp, mtemp, & c0temp, c1temp, c2temp, c3temp do iov2 = 1, iov - 1 if((rtemp-r(iov2)).lt.1.d-9 .and. (mtemp-m(iov2)).lt.1.d-9) then jov(ir) = iov2 iovflag = 0 endif enddo if(iovflag.eq.1) then r(iov) = rtemp m(iov) = mtemp jov(ir) = iov iov = iov + 1 endif iovflag = 1 do iiw2 = 1, iiw - 1 if((c0temp-cprop0(iiw2)).lt.1.d-9 .and. & (c1temp-cprop1(iiw2)).lt.1.d-9 .and. & (c2temp-cprop2(iiw2)).lt.1.d-9 .and. & (c3temp-cprop3(iiw2)).lt.1.d-9) then jiw(ir) = iiw2 iiwflag = 0 endif CCCC
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ij4h = modh(i4+j4) s2sq12 = (s22(j1+i1)+s22(j2+i2)+s22(j3+i3)+s22(j4+i4)) s2sq2 = (s22(j1)+s22(j2)+s22(j3)+s22(j4)) sisq12 = (si2(j1+i1)+si2(j2+i2)+si2(j3+i3)+si2(j4+i4)) sisq2 = (si2(j1)+si2(j2)+si2(j3)+si2(j4)) c2sq12 = (c22(j1+i1)+c22(j2+i2)+c22(j3+i3)+c22(j4+i4)) c2sq2 = (c22(j1)+c22(j2)+c22(j3)+c22(j4)) cisq12 = (ci2(j1+i1)+ci2(j2+i2)+ci2(j3+i3)+ci2(j4+i4)) cisq2 = (ci2(j1)+ci2(j2)+ci2(j3)+ci2(j4)) s2qu12 = (s24(j1+i1)+s24(j2+i2)+s24(j3+i3)+s24(j4+i4)) s2qu2 = (s24(j1)+s24(j2)+s24(j3)+s24(j4)) if (s2sq12.gt.1e-9 .and. s2sq2.gt.1e-9) then hat2 = 0.25d0/s2sq2 hat12 = 0.25d0/s2sq12 do ir = 1,iov mo2(ir) = m(ir) - r12(ir)*s2sq2 omegao2(ir) = sqrt(sisq2 + mo2(ir)**2) omegainvo2(ir) = 1.d0/omegao2(ir) omegaplusinvo2(ir) = 1.d0/(omegao2(ir)+ m(ir)) omegabo2(ir) = (omegao2(ir)-mo2(ir)) omegabinvo2(ir) = 1.d0/(omegao2(ir)-mo2(ir)) mo12(ir) = m(ir) - r12(ir)*s2sq12 omegao12(ir) = sqrt(sisq12 + mo12(ir)**2) omegainvo12(ir) = 1.d0/omegao12(ir) omegaplusinvo12(ir) = 1.d0/(omegao12(ir)+ m(ir)) omegabo12(ir) = (omegao12(ir)-mo12(ir)) omegabinvo12(ir) = 1.d0/(omegao12(ir)-mo12(ir)) omegaplusinvo1p2(ir) = 1.d0/(omegao1(ir)+ omegao2(ir)) omegaplusinvo1p12(ir) = 1.d0/(omegao1(ir)+ omegao12(ir)) omegaplusinvo2p12(ir) = 1.d0/(omegao2(ir)+ omegao12(ir)) enddo do ir = 1, nr prop = 1 b(1,4,ir) = b(1,4,ir) + & hat2*omegabinvo12(jov(ir))*omegainvo12(jov(ir))**2*omegaplusinvo1p12(jov(ir))*
B.3. A particular example
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s22(2*i4 + 2*j4) enddo endif enddo do ir = 1,nr b(1,3,ir) = b(1,3,ir) + b(1,4,ir) enddo enddo do ir = 1,nr b(1,2,ir) = b(1,2,ir) + b(1,3,ir)* & s2(2*i2 + 2*j2) enddo enddo do ir = 1,nr b(1,1,ir) = b(1,1,ir) + b(1,2,ir)* & c22(2*i1 + j1) enddo enddo do ir = 1,nr prop = 1 if (i4.eq.n .or. (i4.eq.nhalf .and. nmod2.eq.0)) & prop = 0.5d0 * prop a(1,4,ir) = a(1,4,ir) + prop*b(1,1,ir)* & omegabinvo1(jov(ir))*omegainvo1(jov(ir))**2 enddo endif enddo if (i3.eq.n .or. (i3.eq.nhalf .and. nmod2.eq.0)) then do ir = 1,nr do i = 1,l
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a(i,3,ir) = 0.d0 enddo enddo do i3 = nhalf,n i3n = modn(i3) i3h = modh(i3) do i = 1,l do ir = 1,nr a(i,4,ir) = 0.d0 enddo enddo do i4 = nhalf,n i4n = modn(i4) i4h = modh(i4) s2sq1 = (s22(i1)+s22(i2)+s22(i3)+s22(i4)) sisq1 = (si2(i1)+si2(i2)+si2(i3)+si2(i4)) c2sq1 = (c22(i1)+c22(i2)+c22(i3)+c22(i4)) cisq1 = (ci2(i1)+ci2(i2)+ci2(i3)+ci2(i4)) if (s2sq1.gt.1e-9) then do ir = 1,iov mo1(ir) = m(ir) - r12(ir)*s2sq1 omegao1(ir) = sqrt(sisq1 + mo1(ir)**2) omegainvo1(ir) = 1.d0/omegao1(ir) omegaplusinvo1(ir) = 1.d0/(omegao1(ir)+ m(ir)) omegabo1(ir) = (omegao1(ir)-mo1(ir)) omegabinvo1(ir) = 1.d0/(omegao1(ir)-mo1(ir)) enddo do i = 1,1 do ir = 1,nr b(i,1,ir) = 0.d0 enddo enddo do j1 = 1,n j1n = modn(j1) j1h = modh(j1) ij1n = modn(i1+j1) ij1h = modh(i1+j1) do i = 1,1 do ir = 1,nr b(i,2,ir) = 0.d0 enddo enddo do j2 = 1,n j2n = modn(j2) j2h = modh(j2) ij2n = modn(i2+j2) ij2h = modh(i2+j2) do i = 1,1 do ir = 1,nr b(i,3,ir) = 0.d0 enddo enddo do j3 = 1,n j3n = modn(j3) j3h = modh(j3) ij3n = modn(i3+j3) ij3h = modh(i3+j3) do i = 1,1 do ir = 1,nr b(i,4,ir) = 0.d0 enddo enddo do j4 = 1,n j4n = modn(j4) j4h = modh(j4) ij4n = modn(i4+j4)
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B.3. A particular example
a(i,4,ir) = a(i,4,ir) * 0.5d0 enddo enddo endif do ir = 1,nr a(1,3,ir) = a(1,3,ir) + a(1,4,ir)* & si2(i3) enddo enddo if (i2.eq.n .or. (i2.eq.nhalf .and. nmod2.eq.0)) then do ir = 1,nr do i = 1,l a(i,3,ir) = a(i,3,ir) * 0.5d0 enddo enddo endif do ir = 1,nr a(1,2,ir) = a(1,2,ir) + a(1,3,ir)* & si(i2) enddo enddo if (i1.eq.n .or. (i1.eq.nhalf .and. nmod2.eq.0)) then do ir = 1,nr do i = 1,l a(i,2,ir) = a(i,2,ir) * 0.5d0 enddo enddo endif do ir = 1,nr a(1,1,ir) = a(1,1,ir) + a(1,2,ir) enddo enddo do ir = 1,nr result(1) = 0.d0 result(1) = result(1) + a(1,1,ir) * &(-6.) result(1) = result(1) * 2**4 / float(n)**8 write(*,*) n,r(jov(ir)), m(jov(ir)), & cprop0(jiw(ir)), cprop1(jiw(ir)), & cprop2(jiw(ir)), cprop3(jiw(ir)), result enddo call flush(6) 1000 continue stop end
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Appendix C: The algorithm for improving the Symanzik coefficients
In Chapter 8 we discussed an improvement method of perturbation theory for the Symanzik gluon action, that has the effect of replacing the bare parameters of the action with im-
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proved ones. This dressing is based on defining improved Symanzik coefficients c˜i that depend on the bare ones, ci , on the coupling constant g0 and on the number of colors N. The dressed coefficients obey the recursive Eqs. (8.32) that can be solved using a fixed
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point procedure. In this Appendix we present the algorithm that we have developed and takes as an input a set of values for the parameters g0 , N, ci and evaluates c˜i .
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Before providing our algorithm, let us set up the necessary notation: • nc is the number of colors N
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• betacoupling is related to the coupling constant g0 through the equation 2 nc g02
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betacoupling =
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• c(0), c(1), c(2), c(3) denote the bare Symanzik coefficients • gtilde(i) are the γ˜i ’s of Eq. (8.32)
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• cimpr(i) are the dressed Symanzik parameters given by (Eq. (8.31)) cimpr(i) =
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2 nc gtilde(i) betacoupling
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do i = 0, 3 gtildeold(i) = gtilde(i) enddo gtildenorm = gtilde(0) + 8*gtilde(1) + 16*gtilde(2) + 8*gtilde(3) call beta(gtilde(0)/gtildenorm,gtilde(1)/gtildenorm, & gtilde(2)/gtildenorm,gtilde(3)/gtildenorm,btilde,n) do i = 0, 3 btilde(i) = btilde(i)/gtildenorm enddo do i = 0, 3 gtilde(i) = gamma(i) * exp(-btilde(i)*(nc-1)/(4.d0*nc)) * & (1.d0 - btilde(i)/12.d0 - (nc-2)*btilde(i)/6.d0 & + (nc-2)*btilde(i)**2/96.d0) enddo call flush(6) if((abs(gtilde(0)-gtildeold(0)).gt.1.d-7).or. (abs(gtilde(1)-gtildeold(1)).gt.1.d-7).or. (abs(gtilde(2)-gtildeold(2)).gt.1.d-7).or. (abs(gtilde(3)-gtildeold(3)).gt.1.d-7) ) goto 3
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goto 1 continue stop end
c %%%%%%%%%%%%%%%%%%%% beta subroutine %%%%%%%%%%%%%%%%%%%%
subroutine beta(z0,z1,z2,z3,b,n) implicit real*8 (a-h,o-z) implicit integer*4 (i-n) real*8 k2p1, k4p1 real*8 m, m1, mo1 parameter (l = 11) dimension a(l,4,300) dimension b(0:3) dimension s2(-1600:1600), c2(-1600:1600), s22(-1600:1600)
nmod2 = mod(n,2) nhalf = (n+nmod2)/2 n4 = n*4 do i = -12*n,12*n s2(i) = sin(i*pi/n) c2(i) = cos(i*pi/n) s22(i) = s2(i)**2 c22(i) = c2(i)**2 s24(i) = s2(i)**4 c24(i) = c2(i)**4 si(i) = sin(2*i*pi/n) ci(i) = cos(2*i*pi/n) si2(i) = si(i)**2 ci2(i) = ci(i)**2 enddo
do i = 1,l do ir = 1,nr a(i,1,ir) = 0.d0 enddo enddo do i1 = nhalf,n do i = 1,l do ir = 1,nr a(i,2,ir) = 0.d0 enddo enddo do i2 = nhalf,n do i = 1,l do ir = 1,nr a(i,3,ir) = 0.d0 enddo enddo do i3 = nhalf,n do i = 1,l do ir = 1,nr a(i,4,ir) = 0.d0 enddo enddo do i4 = nhalf,n s2sq1 = (s22(i1)+s22(i2)+s22(i3)+s22(i4))
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write(*,*) write(*,*) c, betacoupling write(*,*) cimp call flush(6)
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do i = 0, 3 cimp(i) = gtilde(i) * 2 * nc / betacoupling enddo
pi = 4.0*atan(1.d0) Write up to 300 sextuplets of values for r, m, c0, c1, c2, c3, one set per line r(1) = 0. !! Dummy value, will never need it m(1) = 0. !! Dummy value, will never need it cc1(1) = z2 + z3 cc2(1) = z1 - z2 - z3 nr = 1 do ir = 1, nr r24(ir) = 4.d0*r(ir)**2 r12(ir) = 2.d0*r(ir) enddo
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& & &
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read(*,*,end=2) c(0), c(1), c(2), c(3), betacoupling do i = 0, 3 gamma(i) = c(i) * betacoupling/2/nc gtilde(i) = gamma(i) enddo
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3
colors, we present the unimproved’ c0, c1, c2, c3, ’ constant beta, and’ coefficients cimp0, cimp1, cimp2, cimp3’
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1
’For ’, nc, ’ ’coefficients ’the coupling ’the improved
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write(*,*) write(*,*) write(*,*) write(*,*)
dimension c22(-1600:1600), s24(-1600:1600), c24(-1600:1600) dimension si(-1600:1600), ci(-1600:1600), si2(-1600:1600) dimension ci2(-1600:1600) dimension r(300), r24(300), r12(300), m(300) dimension cc1(300), cc2(300) dimension fhat1(300), m1(300) dimension dprop1(4,4), prop1(4,4), g(4,4), ginv(4,4) integer indx(4) dimension omegainvo1(300), omegabinvo1(300), omegaplusinvo1(300) dimension omegao1(300), omegabo1(300), mo1(300) dimension result(4)
C. The algorithm for improving the Symanzik coefficients
program main implicit real*8 (a-h,o-z) implicit integer*4 (i-n) dimension btilde(0:3), c(0:3), gamma(0:3), & gtilde(0:3), gtildeold(0:3), cimp(0:3) parameter(nc=3) n = 64
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st on C
a
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186
a(2,3,ir) = a(2,3,ir) + a(2,4,ir) a(3,3,ir) = a(3,3,ir) + a(3,4,ir) a(4,3,ir) = a(4,3,ir) + a(1,4,ir)* & c22(i3) a(5,3,ir) = a(5,3,ir) + a(1,4,ir)* & s22(i3) enddo enddo if (i2.eq.n .or. (i2.eq.nhalf .and. nmod2.eq.0)) then do ir = 1,nr do i = 1,l a(i,3,ir) = a(i,3,ir) * 0.5d0 enddo enddo endif do ir = 1,nr a(1,2,ir) = a(1,2,ir) + a(2,3,ir) a(2,2,ir) = a(2,2,ir) + a(2,3,ir)* & c22(i2) a(3,2,ir) = a(3,2,ir) + a(3,3,ir)* & c22(i2) a(4,2,ir) = a(4,2,ir) + a(1,3,ir)* & s2(i2) a(5,2,ir) = a(5,2,ir) + a(4,3,ir)* & s2(i2) a(6,2,ir) = a(6,2,ir) + a(5,3,ir)* & s2(i2) a(7,2,ir) = a(7,2,ir) + a(1,3,ir)* & c22(i2)*s2(i2) enddo enddo if (i1.eq.n .or. (i1.eq.nhalf .and. nmod2.eq.0)) then do ir = 1,nr do i = 1,l a(i,2,ir) = a(i,2,ir) * 0.5d0 enddo enddo endif do ir = 1,nr a(1,1,ir) = a(1,1,ir) + a(1,2,ir)* & s22(i1) a(2,1,ir) = a(2,1,ir) + a(4,2,ir)* & s2(i1) a(3,1,ir) = a(3,1,ir) + a(1,2,ir)* & c22(i1)*s22(i1) a(4,1,ir) = a(4,1,ir) + a(2,2,ir)* & s22(i1) a(5,1,ir) = a(5,1,ir) + a(7,2,ir)* & s2(i1) a(6,1,ir) = a(6,1,ir) + a(1,2,ir)* & s22(i1) a(7,1,ir) = a(7,1,ir) + a(3,2,ir)* & s22(i1) a(8,1,ir) = a(8,1,ir) + a(5,2,ir)* & s2(i1) a(9,1,ir) = a(9,1,ir) + a(6,2,ir)* & s2(i1) a(10,1,ir) = a(10,1,ir) + a(3,2,ir)* & s22(i1) a(11,1,ir) = a(11,1,ir) + a(5,2,ir)* & s2(i1) enddo enddo do ir = 1,nr result(1) = 0.d0 result(1) = result(1) + a(1,1,ir) *
C. The algorithm for improving the Symanzik coefficients
sisq1 = (si2(i1)+si2(i2)+si2(i3)+si2(i4)) s2qu1 = (s24(i1)+s24(i2)+s24(i3)+s24(i4)) k2p1 = 4* (s22(i1)+s22(i2)+s22(i3)+s22(i4)) k4p1 = 16* (s24(i1)+s24(i2)+s24(i3)+s24(i4)) if (s2sq1.gt.1e-9) then hat1 = 0.25d0/s2sq1 do ir = 1,nr dummy = (1-cc1(ir)*k2p1) do mu = 1, 4 if (mu.eq.1) imu = i1 if (mu.eq.2) imu = i2 if (mu.eq.3) imu = i3 if (mu.eq.4) imu = i4 do nu = 1, mu if (nu.eq.1) inu = i1 if (nu.eq.2) inu = i2 if (nu.eq.3) inu = i3 if (nu.eq.4) inu = i4 g(mu,nu) = (1-dummy) * (2*s2(imu))*(2*s2(inu)) & + cc2(ir) *(2*s2(imu))**3*(2*s2(inu)) & + cc2(ir) *(2*s2(imu))*(2*s2(inu))**3 if (mu.eq.nu) g(mu,nu) = g(mu,nu) + dummy * k2p1 & - cc2(ir) * k4p1 - cc2(ir) * k2p1 * (2*s2(imu))**2 if (nu.lt.mu) g(nu,mu) = g(mu,nu) enddo enddo do mu = 1,4 do nu = 1,4 ginv(mu,nu) = 0.d0 enddo ginv(mu,mu) = 1.d0 enddo call ludcmp(g,4,4,indx,d) do nu = 1, 4 call lubksb(g,4,4,indx,ginv(1,nu)) enddo do mu = 1, 4 do nu = 1, mu prop1(mu,nu) = ginv(mu,nu) dprop1(mu,nu) = ginv(mu,nu) if (mu.eq.nu) dprop1(mu,nu) = dprop1(mu,nu) - hat1 if (nu.lt.mu) prop1(nu,mu) = prop1(mu,nu) if (nu.lt.mu) dprop1(nu,mu) = dprop1(mu,nu) enddo enddo prop = 1 if (i4.eq.n .or. (i4.eq.nhalf .and. nmod2.eq.0)) & prop = 0.5d0 * prop a(1,4,ir) = a(1,4,ir) + prop* & prop1(1,2) a(2,4,ir) = a(2,4,ir) + prop* & prop1(2,2) a(3,4,ir) = a(3,4,ir) + prop* & prop1(3,3) enddo endif enddo if (i3.eq.n .or. (i3.eq.nhalf .and. nmod2.eq.0)) then do ir = 1,nr do i = 1,l a(i,4,ir) = a(i,4,ir) * 0.5d0 enddo enddo endif do ir = 1,nr a(1,3,ir) = a(1,3,ir) + a(1,4,ir)
+ a(2,1,ir) * * 2**4 / float(n)**4
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+ a(3,1,ir) * + a(4,1,ir) * + a(5,1,ir) * * 2**4 / float(n)**4
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+ a(6,1,ir) * + a(7,1,ir) * + a(8,1,ir) * + a(9,1,ir) * * 2**4 / float(n)**4 + a(10,1,ir) * + a(11,1,ir) *
187
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a
C
on
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* 2**4 / float(n)**4
C. The algorithm for improving the Symanzik coefficients
&(8.) result(1) = result(1) &(-8.) result(1) = result(1) result(2) = 0.d0 result(2) = result(2) &(16.) result(2) = result(2) &(16.) result(2) = result(2) &(-32.) result(2) = result(2) result(3) = 0.d0 result(3) = result(3) &(8.) result(3) = result(3) &(8.) result(3) = result(3) &(-16.) result(3) = result(3) &(-8.) result(3) = result(3) result(4) = 0.d0 result(4) = result(4) &(24.) result(4) = result(4) &(-24.) result(4) = result(4) b(0) = result(1) b(1) = result(2) b(2) = result(3) b(3) = result(4) enddo return end
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