Perturbative Improvement of Fermion Operators in Strong Interaction Physics

Home , Strong interaction

UNIVERSITY OF CYPRUS PHYSICS DEPARTMENT

M.Sc THESIS

FOTOS STYLIANOU

JUNE 2010 UNIVERSITY OF CYPRUS PHYSICS DEPARTMENT

Perturbative Improvement of Fermion Operators in Strong Interaction Physics

M.Sc Thesis Fotos Stylianou

Advisor Prof. Haralambos Panagopoulos

Submitted in partial fulﬁllment of the requirements for the degree of Master of Science in the Physics Department at the University of Cyprus

June 2010 Dedicated to my girlfriend Irene Acknowledgments

The completion of this project would not have been possible without the support of family members, friends and colleagues. In this respect, I grasp this opportunity to acknowledge their contribution and to express my sincerest regards. Initially, I would like to express my deepest gratitude to my supervisor Prof. Haris Panagopoulos, for his knowledge sharing. Throughout the course of this project, his continuous guidance contributed both to my professional and personal development. I deeply appreciate his persistence in my participation in the lattice conference in China, which has been a once in a lifetime experience for me. I also take this opportunity to thank Prof. Haris Panagopoulos, for showing trust to my abilities and choosing to include me in research programs. Special thanks go to my colleague Dr. Martha Constantinou, for her uncondi- tional support, her experience and helpful ideas through diﬃcult times. I would also like to thank my colleague Dr. Apostolos Skouroupathis, for the cooperation and his constructive calculations. For all the joyful lunch breaks as well as their meaningful advice, I would like to thank my fellow roommates Phanos, Savvas, Phillipos, Marios and Demetris. IdonothaveenoughwordstoexpressmygratitudetoIliasIlia,forallthefun time and the support through long night studying hours. A very special acknowledgement goes out to my parents, who are always proud and supportive whatever my career choices might be. Thank you father: for your resourceful mind, your endless strength andpersistence.Thankyoumother:foryour delicious cuisine, your patience and understanding through my unstable working hours. You are both an inspiration to me, in your own unique way. Last but not least, I wish to express my gratitude to my girlfriend Irene Tziakouri, for her admirable patience, support and love during the last two years. I look forward to our new future together.

Fotos Stylianou June 2010 Abstract

The content of this thesis lies under the broad scientific area of Quantum Field Theory (QFT). QFT provides the theoretical tool for describing the weak, electromagnetic, and strong interactions between the elementary particles of nature; it merges successfully the principles of Quantum Mechanics and Special Relativity. In theoretical physics, Quantum Chromodynamics (QCD) is the generally accepted QFT of the strong interactions between quarks and gluons that make up hadrons. In this thesis we make use of a space-time lattice regularization in order to perform a series of perturbative analytic calculations in the framework of QCD, to second order in the coupling constant g (“1-loop”). Our aim is to minimize the (a2) systematic errors, induced by the finiteness of lattice spacing a, which arise in O non-perturbative estimates. To fulfill this task we calculate in 1-loop perturbation theory the (a2)discretizationeffectsoftherenormalization constants for various O fermionic operators (i.e., quark field, localbilinears,“extended”bilinears,andfour- fermion operators). The derived (a2 g2)correctionscanbeexplicitly subtracted O from respective non-perturbative estimates, in order to obtain reliable simulation results. The above operators find significant application in the prediction of various physical properties of hadronic matter, such as: (a) masses-decay constants of hadrons (quark field and local bilinears operators),(b)structurefunctionsmeasuringspin content, momentum and charge distributions in hadrons (extended bilinears operators ), (c) matrix elements of ∆S =2mesontransitions(four-fermionoperators). The subtraction procedure turns out to be significant in controlling the lattice artifacts. We demonstrate the importance ofthisprocedureviagraphicalcompari- son of the subtracted and non-subtracted lattice data. The subtracted data provide amorestableandreliablecontinuumextrapolationa 0. Another possible appli- → cation of our results is in constructing improved versions of the fermion operators under study. The novel aspect of our calculations is that they are carried out to second order in the lattice spacing, (a2). Consequently, they have addressed a number of new O issues, most notably the appearance of loop integrands with “strong” Infrared Di- vergences (convergent only beyond 6 dimensions). Such integrands are not present

1 2 Abstract

in (a1) improvement calculations; there, infrared divergent terms are seen to have O the same structure as in the (a0) case, by virtue of parity under integration, and O they can thus be handled by well-known techniques. We explain how to correctly extract the full (a2)dependence;infact,ourmethodisgeneralizabletoanyorder O in a. In all our calculations we employ improved actions both for fermion and gluon ﬁelds. For fermions we use the family of (a1) improved Wilson/clover/twisted mass O actions; particular cases of these actions are currently being studied intensely by the ETMC and other collaborations. For gluons we employ a 3-parameter family of Symanzik (a2) improved gluon actions, comprising all cases which are in common O use, such as the Wilson, tree-level Symanzik, Iwasaki, DBW2, and L¨uscher-Weisz actions. In order to provide results with the widest possible applicability, we have considered the following parameters of the aforementioned actions as free parameters:

clover coeﬃcient cSW,numberofcolorsNc,couplingconstantg,gaugeﬁxingpa-

rameter λ,barequarkmassmf and twisted mass parameter µ.TheSymanzik

coeﬃcients, ci, appear in a nonpolynomial way in the calculations and, thus, we

tabulate the corresponding results for diﬀerent choices of ci. Περίληψη

Το περιεχόμενο αυτής της διατριβής Μάστερ εντάσσεται στα πλαίσια της Θεωρίας Κβαντικών Πεδίων (ΘΚΠ).Η ΘΚΠ αποτελεί το θεωρητικό εργαλείο για την περιγραφή των ασθενών,των ηλεκτρομαγνητικών,και των ισχυρών αλληλεπιδράσεων ανάμεσα στα στοιχειώδη σωματίδια που αποτελούν την ύλη.Βασίζεται στις έννοιες των πεδίων και συνενώνει με επιτυχία τις αρχές της Κβαντικής Μηχανικής και της Ειδικής Θεωρίας της Σχετικότητας.Η Κβαντική Χρωμοδυναμική (ΚΧΔ)είναι η γενικά αποδεκτή ΘΚΠ για την περιγραφή των ισχυρών αλληλεπιδράσεων ανάμεσα στα συστατικά των αδρονίων (κουάρκς και γκλουόνια). Σε αυτή τη διατριβή Μάστερ χρησιμοποιήσαμε το χωροχρονικό πλέγμα ως ομα- λοποιητή για να πραγματοποιήσουμε μια σειρά από διαταρακτικούς υπολογισμούς,στα πλαίσια της ΚΧΔ,μέχρι2 ης τάξης ως προς την σταθερά σύζευξης g.Στόχοςμας είναι να βελτιώσουμε τα αποτελέσματα των προσομοιώσεων,μειώνοντας τα συστημα- τικά σφάλματα (a2) που οφείλονται στο πεπερασμένο μέγεθος της σταθεράς πλέγ- O ματος a.Γιατηνεπίτευξηαυτούτουστόχουυπολογίσαμεσεθεωρίαδιαταραχώνενός βρόχου, (g2),τασφάλματαδιακριτοποίησης (a2) των σταθερών επανακανονικο- O O ποίησης διαφόρων φερμιονικών τελεστών,όπως:τελεστής φερμιονικού πεδίου,τοπι- κοί και εκτεταμένοι διγραμμικοί τελεστές,και τελεστές με τέσσερα φερμιονικά πεδία. Αφαιρώντας τους προαναφερόμενους διορθωτικούς όρους (g2a2) από αντίστοιχους O μη-διαταρακτικούς υπολογισμούς,καταλήγουμε σε πιο αξιόπιστα αποτελέσματα προσο- μοιώσεων. Οι εν λόγω τελεστές βρίσκουν εφαρμογή στη μελέτη της δομής και των φυσικών ιδιοτήτων της αδρονικής ύλης,όπως:(α)μάζες και σταθερές διάσπασης αδρονίων,(β) συναρτήσεις δομής που μετρούν την κατανομή φορτίου,σπιν,και ορμής στο εσωτερικό των αδρονίων,(γ)πινακοστοιχεία ∆S =2μεσονικών μεταβάσεων. Επιδεικνύουμε την σημαντικότητα της διαδικασίας αφαίρεσης των σφαλμάτων δια- κριτοποίησης,που προτείνουμε,με γραφική σύγκριση δεδομένων από προσομοιώσεις πριν και μετά την αφαίρεση.Τα δεδομένα μετά τη διαδικασία αφαίρεσης εξασφαλίζουν πιο σταθερά και αξιόπιστα αποτελέσματα στο όριο a 0.Μιαάλληδυνατήεφαρμο- → γή των αποτελεσμάτων μας είναι στην κατασκευή βελτιωμένων εκφράσεων για τους τελεστές που μελετήσαμε. Το καινοτόμο σημείο των υπολογισμών μας έγκειται στο ότι εκτείνονται μέχρι 2η

3 4 Περίληψη

τάξη ως προς τη σταθερά πλέγματος, (a2).Ωςεκτούτου,βρεθήκαμεαντιμέτωποιμε O πρωτότυπα προβλήματα,με πιο αξιοσημείωτο την εμφάνιση ολοκληρωμάτων βρόχου με «ισχυρούς»υπέρυθρους απειρισμούς (συγκλίνουν μόνο πέραν των 6διαστάσεων).Τα ολοκληρώματα αυτού του είδους δεν εμφανίζονται σε υπολογισμούς βελτίωσης μέχρι (a1).Σευπολογισμούς (a1) οι όροι με υπέρυθρο απειρισμό έχουν την ίδια δομή O O με αυτούς που εμφανίζονται σε υπολογισμούς (a0),και συνεπώς αντιμετωπίζονται O με γνωστές μεθόδους.Εξηγούμε με λεπτομέρεια τη σωστή διαδικασία εξαγωγής των όρων (a2).Αξίζεινασημειωθείότιαυτήηδιαδικασίαπουαναπτύξαμεείναιγενικεύ- O σιμη σε οποιαδήποτε τάξη ως προς την σταθερά πλέγματος a. Σε όλους τους υπολογισμούς μας χρησιμοποιούμε βελτιωμένες δράσεις τόσο για φερμιονικά όσο και για γκλουονικά πεδία.Για τα φερμιόνια χρησιμοποιούμε μια οικο- γένεια από (a1) βελτιωμένες δράσεις (Wilson/clover/twistedmass).Συγκεκριμένες O περιπτώσεις αυτών των δράσεων χρησιμοποιούνται από την ETMC και άλλες ερευ- νητικές ομάδες.Για τα γκλουόνια χρησιμοποιούμε μια οικογένεια 3-παραμέτρων,από βελτιωμένες (a2) δράσεις Symanzik,ηοποίαπεριλαμβάνειτιςδράσεις:Wilson, tree O − level Symanzik, Iwasaki, DBW2, και Luescher Weisz. − Τα αποτελέσματα μας καθίστανται ευρέως εφαρμόσιμα,αφού θεωρήσαμε όλες τις

παραμέτρους των δράσεων ως ελεύθερες:αριθμός χρωμάτων Nc,σταθεράσύζευξης

g,παράμετροςclover cSW,παράμετροςεπιλογήςβαθμίδαςλ,γυμνήμάζατωνκουάρκς

mf ,καιπαράμετροςtwisted mass µ.ΟιπαράμετροιSymanzik, ci,εμπλέκονταιμε μη πολυωνυμικό τρόπο στους υπολογισμούς και για αυτό το λόγο επιλέξαμε τις 10 δημοφιλέστερες τιμές που χρησιμοποιούνται σε προσομοιώσεις. Contents

List of ﬁgures 9

List of tables 12

1Introduction 13 1.1 IntroductiontoQCDandLatticeQCD ...... 13 1.2 IssuesinNon-PerturbativeStudies ...... 17 1.3 Using Lattice Perturbation Theory as a Tool for Improving Non- PerturbativeResults ...... 21

2 (a2) Corrections to the One-Loop Fermion Propagator and Bilin- O ear Operators 26 2.1 Introduction...... 26 2.2 Description of the Calculation ...... 28 2.3 Evaluation of a Primitively Divergent Integral ...... 33 2.4 Corrections to the Fermion Propagator ...... 38 2.5 Corrections to Fermion Bilinear Operators ...... 43 2.6 Renormalization Constants in the RI-MOM Scheme ...... 57 2.7 Subtraction of the (a2 g2) Discretization Eﬀects ...... 60 O 3 (a2) Corrections to the One-Loop Matrix Elements of Four-Fermion O ∆S =2Operators 65 3.1 Introduction...... 66 3.2 Amputated Green’s Functions of Four-Fermion ∆S =2Operators.. . 73 3.3 Mixing and Renormalization of and F on the Lattice. . . . . 84 OXY OXY 4RenormalizationConstantsforFermionFieldandUltra-LocalOp- erators in Twisted Mass QCD 99 4.1 Introduction...... 100 4.2 Formulation ...... 100 4.2.1 Lattice actions ...... 100 4.2.2 Deﬁnition of operators and Renormalization condition . . . . . 102

5 6CONTENTS

4.3 Corrections to the Fermion Propagator ...... 103 4.4 Corrections to Fermion Bilinear Operators ...... 108 4.5 Quark Field and Quark Bilinear Renormalization Constants in the

RI′-MOMScheme...... 110 4.6 Analytic expressions for one-derivative operators ...... 118 4.7 StrongIRdivergentintegrals...... 122 4.8 Conversion to the contimuum MS scheme ...... 127 4.8.1 Conversionfactors ...... 127 4.8.2 Evolution to a reference scale ...... 128 4.9 Non-perturbative calculation ...... 129 4.10 Non-perturbativeresults ...... 131 4.10.1 Pionmassdependence ...... 132 4.10.2 Volumedependence...... 132 4.10.3 Pion pole subtraction ...... 133

4.10.4 Results in RI′-MOMscheme ...... 134 4.10.5 MSscheme ...... 138 4.11Conclusions ...... 140

5RenormalizationConstantsforTwist-2OperatorsinTwistedMass QCD 143 5.1 Introduction...... 143 5.2 Formulation ...... 145 5.2.1 Lattice action ...... 145 5.3 PerturbativeProcedure...... 147 5.3.1 Renormalization of Twist-2 Operators ...... 148 5.4 Non-Perturbative Calculation ...... 152 5.4.1 Evaluation of Correlators ...... 152 5.4.2 Renormalization Condition ...... 154 5.5 Results...... 156

5.5.1 RI′-MOMCondition ...... 156

5.5.2 RI′-MOMataReferenceScale...... 162 5.5.3 Conversion to MS...... 165 5.6 Conclusions ...... 169

Appendix A: Notation 171 A.1 ContinuumQCD ...... 171 A.2 LatticeQCD ...... 173

Bibliography 175 List of Figures

1.1 Hadron spectrum in the continuum limit. mπ, mK and mΞ are used as inputs to ﬁx free parameters of QCD...... 17

2.1 Graphical representation of Qµν (Eq. (2.4)) appearing in the clover action...... 29 2.2 The 4- and 6-link loops contributing to the gauge action of Eq. (2.5). 30 2.3 One-loop diagrams contributing to the fermion propagator. Wavy (solid) lines represent gluons (fermions)...... 39 2.4 One-loop diagram contributing to the bilinear operators. A wavy (solid) line represents gluons (fermions). A cross denotes the Dirac matrices Γ...... 44 2 2 2 2 2.5 The RCs Zq(1/a; a p ) and ZΓ(1/a; a p ) at β =3.9,evaluatedatthe

reference scale µ0 =1/a,plottedagainsttheinitialrenormalization 2 scale a2p◦ sin2(ap ).Filledsquares(emptycircles)areresults ≡ ρ ρ obtained with (without) the subtraction of the (g2a2) discretization ! O eﬀects, computed in perturbation theory. The solid lines are linear ﬁts to the data...... 64

3.1 Graphical representation of the rescaled unitarity constraint (triangle) of Eq. (3.10)...... 68 3.2 Constraints on the ρ¯, η¯ plane...... 69 3.3 Box diagrams contributing to neutral Kaon oscillation in the SM. .. 70 3.4 1-loop diagrams contributing to the amputated Green’s function of the 4-fermion operator XY .Wavy(solid)linesrepresentgluons O (fermions)...... 75 3.5 RI MOM computation of the multiplicative renormalization factor − at β =3.90...... 90 Z11

4.1 Zq,ZV,ZA,ZT at β =3.9,asafunctionofthepionmass: mπ =

0.302 GeV (aµ0 =0.004), mπ =0.375 GeV (aµ0 =0.0064), mπ =

0.429 GeV (aµ0 =0.0085)andmπ =0.468 GeV (aµ0 =0.01). ....133

7 8LISTOFFIGURES

4.2 ZS,ZP at β =3.9 (left panel) and β =4.05 (right panel) for various masses. The upper plot shows the results before the pion pole subtraction (Eq. (4.118)), while the lower ﬁgure the results upon the appropriate subtraction (Eq. (4.119))...... 135

4.3 ZP/ZS at β =3.9 and β =4.05 as a function of the momentum. In each plot we demonstrate the eﬀect of the (a2)-terms subtraction O by plotting the pure non-perturbative results (black circles) and the subtracted ones (magenta diamonds). The Goldstone boson contamination was also removed from the plotted data...... 136

4.4 Non-perturbative results on Zq for β =3.9, β =4.05 and β =4.20. The upper plots are the results of a chiral extrapolation procedure,

while the lower one is presented at mπ =0.470 GeV. Black circles represent the non-perturbative data, and magenta diamonds the (a2) O subtracted ones...... 137

4.5 The scale-independent ZV and ZA for β =3.9, β =4.05 and β =4.20. The upper plots are the results of a chiral extrapolation procedure,

while the lower one is presented at mπ =0.470 GeV. Black circles represent the non-perturbative data, and magenta diamonds the (a2) O subtracted ones...... 138

4.6 Results on Zq for β =3.9, β =4.05 and β =4.20.Theupperplots are the results of a chiral extrapolation procedure, while the lower

one is presented at mπ =0.470 GeV...... 140

4.7 Chiral results on ZS and ZP for β =3.9 and β =4.05...... 141

4.8 Results on ZT for β =3.9, β =4.05 and β =4.20.Theupperplots are the results of a chiral extrapolation procedure, while the lower

one is presented at mπ =0.470 GeV...... 142

5.1 One-loop diagrams contributing to the computation of the twist-2 operators. A wavy (solid) line represents gluons (fermions). A cross denotes an insertion of the operator under study...... 149

0ν 1 5.2 ZDA for β =3.9 (a− =2.217 GeV) and mπ =0.430 GeV for method 1(opensymbols)andmethod2(ﬁlledsymbols).Theupperplot corresponds to non-perturbative results, where the index A, B represents the set of momenta with spatial components 2 π/L(3, 3, 3) and 2 π/L(2, 2, 2), respectively. The lower plot shows the non-perturbative results after subtracting the perturbative (g2 a2)-terms, where the O two methods give almost identical results...... 155

0ν 5.3 As in Fig. 5.2 but for ZDV...... 156 LIST OF FIGURES 9

5.4 ZDV1 at β =3.9,asafunctionofthepionmass: mπ =0.302 GeV

(aµ0 =0.004), mπ =0.375 GeV (aµ0 =0.0064)andmπ =0.429 GeV

(aµ0 =0.0085). The left plot regards the unsubtracted non-perturbative results and the right one corresponds to the subtracted data. ....158 µµ 00 aver 1 5.5 ZDV (squares), ZDV (circles), ZDV (crosses), for β =3.9 (a− =2.217 GeV),

mπ =0.430 GeV using method 1. The upper plot corresponds to the purely non-perturbative results, while the lower plot shows the non- perturbative results after subtracting the perturbative terms of (a2). 160 O 5.6 Renormalization scale dependence for the Z-factors at β =3.9 and

mπ =0.430 GeV ...... 161 5.7 Renormalization scale dependence for the Z-factors at β =4.05 and

mπ =0.465 GeV ...... 161 5.8 Renormalization scale dependence for the Z-factors at β =4.20 and

mπ =0.476 GeV ...... 162

5.9 Renormalization factors in the RI′ MOM scheme at renormalization − scale 1/a,forβ =3.9,µ0 =0.0085.Theblackcirclescorrespondto the unsubtracted results, while the magenta diamonds to the results with perturbatively subtracted one loop (a2) artifacts...... 163 O 5.10 Same as Fig 5.9, but for β =4.05 and µ0 =0.0080...... 164

5.11 Same as Fig 5.9, but for β =4.20 and µ0 =0.0065...... 164

5.12 Renormalization factors at β =3.9, µ0 =0.0085 in the MS-scheme at renormalization scale 2 GeV. Black circles correspond to the unsubtracted results, while magenta diamonds correspond to the results with perturbatively subtracted 1-loop (a2) artifacts. The lines show O extrapolations to a2p2 =0using the subtracted results within the range a2p2 =1 2.2...... 168 ∼ 5.13 Same as Fig 5.12, but for β =4.05 and µ0 =0.0080...... 168

5.14 Same as Fig 5.12, but for β =4.20 and µ0 =0.0065...... 169 A.15 The interaction vertices of quarks and gluons. The interaction vertices of quarks and gluons. Solid (wavy) lines represent fermions (gluons). 172 List of Tables

1.1 Properties of quarks...... 14

2.1 Input parameters c0, c1, c3...... 31 2.2 The coefficients ε(0,i) (Eq. (2.40)) for different actions...... 40 2.3 The coefficients ε(1,i) (Eq. (2.40)) for different actions...... 40 2.4 The coefficients ε(2,1) ε(2,3) (Eq. (2.40)) for different actions. .... 40 − 2.5 The coefficients ε(2,4) ε(2,6) (Eq. (2.40)) for different actions. .... 41 (0,i) − 2.6 The coefficients ε˜1 (Eq. (2.42)) for different actions...... 41 (0,i) 2.7 The coefficients ε˜2 (Eq. (2.43)) for different actions...... 42 ( 1,1) ( 1,2) ( 1,3) 2.8 The coefficients εm− ,εm− ,εm− (Eq. (2.44)) for different actions. 43 (0,i) 2.9 The coefficients εS (Eq. (2.49)) for different actions...... 45 (1,i) 2.10 The coefficients εS (Eq. (2.49)) for different actions...... 45 (2,i) 2.11 The coefficients εS (Eq. (2.49)) for different actions...... 45 (0,i) (2,i) 2.12 The coefficients εP and εP (Eq. (2.50)) for different actions. ... 46 (0,i) 2.13 The coefficients εV (Eq. (2.51)) for different actions...... 48 (1,i) 2.14 The coefficients εV (Eq. (2.51)) for different actions...... 48 (2,1) (2,3) 2.15 The coefficients εV εV (Eq. (2.51)) for different actions. .... 48 (2,4) − (2,6) 2.16 The coefficients εV εV (Eq. (2.51)) for different actions. .... 49 (2,7) − (2,9) 2.17 The coefficients εV εV (Eq. (2.51)) for different actions. .... 49 (0,1) − (1,i) 2.18 The coefficients εA and εA (Eq. (2.52)) for different actions. ... 51 (2,1) (2,3) 2.19 The coefficients εA εA (Eq. (2.52)) for different actions. .... 51 (2,4) − (2,6) 2.20 The coefficients εA εA (Eq. (2.52)) for different actions. .... 51 (2,7) − (2,9) 2.21 The coefficients εA εA (Eq. (2.52)) for different actions. .... 52 (0,i) − 2.22 The coefficients εT (Eq. (2.54)) for different actions...... 54 (1,i) 2.23 The coefficients εT (Eq. (2.54)) for different actions...... 54 2.24 The coefficients ε(2,1) ε(2,3) (Eq. (2.54)) for different actions. .... 54 T − T 2.25 The coefficients ε(2,4) ε(2,6) (Eq. (2.54)) for different actions. .... 55 T − T 2.26 The coefficients ε(2,7) ε(2,9) (Eq. (2.54)) for different actions. .... 55 T − T 2.27 The coefficients ε(2,1),ε(2,4),ε(2,7) (Eq. (2.55)) for different actions. .. 57 T ′ T ′ T ′ 2.28 The coefficients ε(2,8),ε(2,9) (Eq. (2.55)) for different actions...... 57 T ′ T ′

10 LIST OF TABLES 11

(i) (i) 2.29 Values of the coeﬃcients bΓ and cΓ entering the 1-loop expression (2.77) of the amputated projected Green functions (p). Results VΓ are presented for the case of the tree-level Symanzik improved gluon action in the Landau gauge and the tm fermionic action at maximal twist...... 62

3.1 Classification of four-fermion operators according to P and use- OXY ful products of their discrete symmetries C, S′ and S′′.Fortheoper- F ators ,wemustexchangetheentriesofthecolumnsCS′ CS′′ OXY ↔ and CPS′ CPS′′.Notethat ˜ ˜ = and ˜ = ˜ ...... 84 ↔ OT T OTT OT T OTT 3.2 The coefficients ϵd+(0,1) ϵd+(0,6) and ϵd+(2,1) ϵd+(2,2)...... 91 l,m − l,m l,m − l,m 3.3 The coefficients ϵd+(2,3) ϵd+(2,10)...... 92 l,m − l,m (0,1) (0,6) (2,1) (2,2) 3.4 The coefficients ϵd− ϵd− and ϵd− ϵd− ...... 93 l,m − l,m l,m − l,m (2,3) (2,10) 3.5 The coefficients ϵd− ϵd− ...... 94 l,m − l,m 3.6 The coefficients ϵδ+(0,1) ϵδ+(0,6) and ϵδ+(2,1) ϵδ+(2,2)...... 95 l,m − l,m l,m − l,m 3.7 The coefficients ϵδ+(2,3) ϵδ+(2,10)...... 96 l,m − l,m (0,1) (0,6) (2,1) (2,2) 3.8 The coefficients ϵδ− ϵδ− and ϵδ− ϵδ− ...... 97 l,m − l,m l,m − l,m (2,3) (2,10) 3.9 The coefficients ϵδ− ϵδ− ...... 98 l,m − l,m ( 1,1) ( 1,2) ( 1,3) 4.1 Numerical results for the coefficients εm− ,εm− ,εm− (Eq. (4.29)) for different actions. The systematic errors in parentheses come from the extrapolation over finite lattice size, L ...... 108 →∞ 4.2 Action parameters used in the simulations...... 132 4.3 Momenta list for the various ensembles at β =3.9, 4.05, 4.20. ....132

4.4 Renormalization constants at β =4.05,µ0 =0.006 using two lattice sizes: 323 64 for (4,4,4,4) and 243x48 for (3,3,3,3), and at a scale of × (ap)2 2...... 133 ∼ 4.5 Final results of the renormalization constants Zq,ZS,ZP,ZT in the

MS scheme, as well as for the scale-independent ZP/ZS,ZV and ZA. The above values have been obtained by performing a chiral extrapolation and then by extrapolating linearly in a2p2.Statisticalerrors are are shown in the first parenthesis. The error in the second parenthesis is the systematic error due to the continuum extrapolation, the difference between results using the fit range p2 15 32 (GeV)2 ∼ − and the range p2 17 24 (GeV)2.Anerrorsmallerthanthelast ∼ − digit given for the mean value is not quoted...... 139

5.1 Action parameters used in the simulations...... 157 5.2 Statistical sample at β =3.9, 4.05, 4.20 for various momenta...... 157 12 LIST OF TABLES

5.3 The renormalization constants at β =3.9 with µ0 =0.0040, 0.0085 for lattice size: 243 48...... 158 × 5.4 Renormalization constants at β =4.05,aµ0 =0.0080 for lattice size 323 64...... 159 × 5.5 Renormalization constants at β =4.20,µ0 =0.0065 for lattice size: 323 64...... 159 × 5.6 Renormalization constants at β =4.05,µ0 =0.0080 using method 2 and two lattice sizes: 323 64 for (4,4,4,4) and 243x48 for the rest of × the momenta...... 159

5.7 Renormalization constants ZDV and ZDA in the MS scheme. The above values have been obtained by extrapolating linearly in a2p2. The error in the second parenthesis is the systematic error due to the extrapolation, namely the largest difference between results using the fit range a2p2 =1.2 2.7 and one of the ranges a2p2 =1 2.7, ∼ ∼ a2p2 =1.2 2.2. Statistical errors are in most cases smaller and are ∼ shown in the first parenthesis...... 169 Chapter 1

Introduction

In this chapter we provide a general overview of Quantum Chromodynamics (QCD), the modern theory of the strong interaction. We consider the basic features of QCD: “asymptotic freedom”, “conﬁnement” and the “running” of the coupling constant. Moreover we introduce the “lattice”, the most commonly used non-perturbative regulator of the continuum QCD theory. We discuss the main issues in non-perturbative studies and we explain how lattice perturbation theory can be used in order to improve non-perturbative results.

1.1 Introduction to QCD and Lattice QCD

In 1964 M. Gell-Mann [1] and G. Zweig [2, 3] independently proposed that nucleons (protons and neutrons) were not fundamental constituents of matter, but instead they were composed of smaller particles, the so-called “quarks”. The quarks are spin-1/2 elementary particles, which means that they are fermions obeying Fermi-Dirac statistics. In the “quark model”, quarks and anti-quarks group in different combinations, in order to form the more composite particles known as hadrons (mesons and baryons). From indirect experimental evidence (until today) we know that quarks come in three generations each of which contains two types of quarks, conventionally referred to as “flavors” (up, down, charm, strange, top and bottom). Quarks also carry an additional unobserved quantum number known as “color charge”, which takes three values: red, blue and green. This intrinsic property was introduced by O. W. Greenberg [4], Y. Nambu and M. Y. Han [5] in order to explain how quarkscoexistinsidesomehadrons(e.g.∆++), with otherwise identical quantum states, without violating the Pauli exclusion principle. Some other properties of quarks such as bare mass (m), electric charge (Q)andisospin(I3:the quantum number characterizing an approximation symmetry among hadrons which differ only by the substitution of certain up quarks by down quarks) are shown in Table 1.1 [6]. For anti-quarks the electric charge and isospin are of opposite sign. Quarks are the only elementary particles in the Standard Model to experience all four

13 14 Chapter 1. Introduction

Generation Flavor Bare mass (m)Electriccharge(q)Isospin(I3)

+0.96 First Down 1/3 e− 5.04 1.54 MeV 1/2 − − + +0.75 First up 2/3 e 2.55 1.05 MeV +1/2 − +25 Second Strange 1/3 e− 105 35 MeV 0 − + +0.07 Second Charm 2/3 e 1.27 0.11 GeV 0 − +0.17 Third Bottom 1/3 e− 4.20 0.07 GeV 0 − + +1.1,+1.2 Third Top 2/3 e 171.3 1.1, 1.2 GeV 0 − −

Table 1.1: Properties of quarks. fundamental interactions of nature: strong, weak, electromagnetic and gravitational. The strong interaction (also called the strong nuclear force, or color force) between quarks is mediated by massless spin-1 particles called “gluons”. There are eight gluons and each one of them carries one color charge and one anti-color charge. Unlike photons, gluons are also able to interact with each other. In the framework of quantum field theory (QFT), the most natural candidate model to describe the strong interactions is quantum chromodynamics (QCD). It is a local gauge field theory based on the non-abelian SU(3) color group proposed by M. Gell-Mann, H. Fritzsch and H. Leutwyler [7]. The number “3” indicates the numbers of colors carried by quarks. Quarks are introduced as spinor fields with Nf flavors, each in the fundamental representation of the color group. The SU(3) group has eight generators corresponding to the number of gluons. The gluons are vector fields in the adjoint representation of color group. A brief introduction to the mathematical formulation of QCD can be found in Appendix A.1. QCD not only does providesahighlysuccessfuldescriptionofstrong interaction phenomena (i.e. it predicts the whole spectrum of strong interactions with only a few input parameters), it also leads to several remarkable predictions concerning “exotic hadrons”. QCD exhibits two peculiar properties intimately connected with its non-abelian nature:

“Asymptotic freedom”, which means that in high energy reactions (or, equivalently, for small quark separations) partons (quarks and gluons) can hardly interact with each other. As a consequence quarks move within hadrons mostly as free non- interacting particles. This feature was ﬁrst discovered in 1973 by G. ’t Hooft [8], H. Politzer [9], D. Gross and F. Wilczek [10, 11]. It was made possible for the ﬁrst time to carry out precise calculations of observables in strong interaction physics, using QFT techniques of perturbation theory.

“Conﬁnement”, which means that the force between quarks increases as they are 1.1. Introduction to QCD and Lattice QCD 15

pulled apart. As a consequence it would take an infinite amount of energy to separate the quarks from hadrons. Due to this phenomenon, quarks and gluons appear to be tightly bound to each other within hadrons, explaining why they have never been experimentally observed as isolated particles. For this reason, much of what is known about quarks has been drawn from observations of the hadrons themselves. The eigenstates of QCD are quarks and gluons, whose free states are not directly observable. Hadrons observable in experiment are not eigenstates of QCD. QCD should explain this remarkable feature. However, this property of the theory has not been analytically proven yet, but it is verified within Lattice QCD (LQCD) formulations (non-perturbative numerical methods) proposed first by K. Wilson in 1974 [12].

The above properties are manifest in the behaviour of the “running” coupling con- 2 2 stant. In QCD the coupling constant, αs=g /(16π ), is the parameter that determines the strength of the color interactions. Because this parameter depends on the energy scale, µ,ofthephysicalprocessunderconsideration, it is called the “running” coupling constant:

In reactions involving large exchange of energy, µ 1GeV ,thecouplingconstant ≫ is small, α 0, and perturbation theory becomes a reliable tool. s → On the contrary, at low energies scales, µ 1GeV , the coupling constant is of order ≤ α 1andtheperturbativemethodsbreakdown.InthisregionLQCDprovidesa s ≃ framework for investigation of non-perturbative phenomena from ﬁrst principles.

AbriefintroductiontothemathematicalformulationofLQCDcanbefoundinAp- pendix A.2. LQCD is a non-perturbative scheme in which continuum QCD is replaced by a discrete statistical mechanical system. In this scheme the Minkowski space-time is replaced by a four dimensional euclidean grid with lattice spacing a.Thisdiscrete space-time lattice reduces the degrees of freedom of the theory to a numerable inﬁnity

(xµ = anµ where nµ Z), and acts as an ultraviolet regularization. Renormalized physi- ∈ cal quantities are expected to have a well defined limit as a 0. This distance between → neighboring lattice points, introduces a momentum cutoffat π/a,excludinginthisway the ultraviolet infinities. The range of the momenta is thus restricted to the first Brillouin zone:

π π B = k :

If the lattice volume is finite, V =L1 L2 L3 L4, this provides also an infrared regularization; in this case the number of degrees of freedom is finite, restricting further the allowed momenta of the first Brillouin zone into a discrete set given by: 16 Chapter 1. Introduction

2π nµ Lµ (kn)µ = nµ =0, 1,...,Nµ= . (1.2) ± a Nµ 2a

So in principle one deals with sums also in momentum space, where it is possible to perform numerical simulations of QCD using Monte Carlo methods. By discretizing the continuum QCD one hastogiveup,ingeneral,chiralsymmetry for massless quarks, mq = 0, and Poincar´e invariance (i.e. Lorentz and translational invariance). As in the continuum theory, LQCD is invariant under the global symmetry iθ iθ ψ(x) e ψ(x)(ψ(x) ψ(x)e− )whereθ is a continuous parameter. This symmetry → → is related to baryon number conservation and leads to a conserved vector current. It is impossible to construct an ultralocal LQCD theory with massless quarks, which is

iθγ5 iθγ5 also invariant under the transformation ψ(x) e ψ(x)(ψ(x) ψ(x)e− ). Due to → → chiral symmetry breaking there is no conserved axial current on the lattice. This feature is summarized in the no-go theorem, established by Nielsen and Ninomiya many years ago [13, 14, 15]. On the lattice Lorentz symmetry is inevitably reduced to that of the hypercubic group. This is the symmetry group generated by discrete π/2rotationsonthesixlatticeplanes and by reflections on the four dimensional hypercube. The hypercubic group is a subgroup of the orthogonal group O(4) (i.e. the symmetry group of reflections and continuous rotations); which is the Lorentz group O(3, 1) (i.e. the symmetry group of spatial rotations, boosts, and space-time reflections) analytically continued to the euclidean space-time 1. There is also invariance under translations by an integer multiple of the lattice spacing. On a finite lattice one usually imposes periodicity, so that lattice momentum is conserved modulo 2π.LQCDmaintainsgaugeinvariance,localityandunitarityforanynonzero a,andpreservestheinternalsymmetries:parity( ), charge conjugation ( ) and time P C reversal ( ). T Numerical simulations of QCD allow us to compute physical quantities such as hadron masses and hadronic matrix elements, which are otherwise inaccessible to investigations which use continuum QCD calculations. Lattice simulations can also give important contributions to our understanding of the strong interactions. In example they show that a large part of the proton and neutron mass (both particles are much heavier than their quark and gluon constituents) comes from the strong interactions between gluons. Lattice QCD has already made contact with experiments. For example, results for hadron spectrum from one of the most extensive calculations have been reported in Ref. [16], where the mass of the proton and neutron determined theoretically gives an error of less than 2%. Furthermore, as can be seen from Fig. 1.1, the agreement between Lattice QCD and experiment is excellent also for other hadron masses.

1 This relation between the two groups mactually regards those parts which are continuously connected to the identity. 1.2. Issues in Non-Perturbative Studies 17

Useful text books and article reviews, which cover at length non-perturbative aspects, can be found in Refs. [6, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26].

2000

Ω ∗ ∗ Ξ Σ 1500 Ξ ∆ Σ Λ * N 1000 K ρ Mass [MeV]

K 500 Input LQCD π experiment

Figure 1.1: Hadron spectrum in the continuum limit. mπ, mK and mΞ are used as inputs to ﬁx free parameters of QCD.

1.2 Issues in Non-Perturbative Studies

The results of lattice simulations are, inevitably, affected by statistical and systematic errors. In order to minimize these errors it is important to understand their origin. Since simulations of LQCD are based on Monte Carlo integration of the euclidean path integral, only a finite number of configurations can be generated and thus statistical errors arise in measurements. On the other hand systematic errors are of various nature:

1. Finite volume effects: Since an infinite system is represented by a finite lattice, the results obtained are expected to suffer from volume effects. These finite size effects are exponentially ML suppressed, e− , if we work on a large enough lattice [27]. Typically if the box size, L, is larger than 7 fm, then quantum mechanical properties of hadrons (of size smaller than 1 fm) are unaltered. Another consequence of the finiteness of the L is the momentum resolution. For N = L/a =64and1/a =2GeV the smallest momentum difference becomes ∆k 196MeV .Ifweareinterestedina ≃± much finer resolution we have to increase L, but the larger the box size the more computer power is needed. 18 Chapter 1. Introduction

2. Quenched approximation: For reasons of computing power, the so-called quenched approximation is often used in Monte Carlo simulations to save a couple of orders of magnitude in computer time. In order to perform a statistical sampling of functional integral, using numerical simulations, we ﬁrst analytically integrate the fermion variables (since Grassmann numbers cannot be implemented into a numerical code). At the end of this procedure we end up with a partition function of the gauge link ﬁelds with a fermionic determinant. In full QCD simulations the fermionic determinant is taken into account, while in the quenched approximation the determinant is replaced by one. In physical terms, this means that there are no sea (virtual) quarks in the calculations: the internal quark loops are neglected, this is often summarized by

saying that Nf = 0. The quenched approximation leads to an unknown systematic error in lattice calculations [28, 29]. Nowadays this situation is gradually changing, thanks to improvements for both

computers and algorithms. Most of the simulations are performed in Nf =2dynam- ical ﬂavor full QCD, where the up and down quarks are considered as degenerated.

Some collaborations have performed simulations using Nf = 2 + 1 dynamical ﬂavor full QCD, where the strange quark is also included in the fermionic determinant. Including the eﬀects of the three light sea quarks has dramatically improved the agreement between experiment and lattice QCD results [16].

3. Chiral extrapolations in the light quark masses: Lattice QCD has always been plagued by the fact that, in the limit of realistically small quark masses, simulation costs increase drastically due to the large correlation lengths of the light states, and the large lattice volumes that one employs in order to avoid finite-size effects. In particular, the physical up and down quark masses are too light to simulate on current lattices. For 1/a =2GeV ,realistic simulations require N = L/a > 70 to avoid finite volume effects. Furthermore the amount of time for the iterative solution of linear systems is very high for light quark masses (one of the most demanding parts of the simulation). To get results

for quantities involving light quarks, one typically extrapolates in mu = md from

the range [ms/4, 2ms](usingseveralunphysicallylarge quark masses, free from vol-

ume eﬀects). In current simulations, SU(2) isospin breaking eﬀects (md >mu)and electromagnetic contributions (q =+2/3, q = 1/3) are neglected, since these u d − are a few MeV in nature and smaller than the numerical resolution. The iso-spin

symmetric mass is deﬁned bym ¯ =(mu + md)/2. To study iso-spin breaking eﬀects

properly, dynamical lattices with physical values for mu and md are required. At the present, simulations are being performed with a physical value of the strange quark mass, and light quark masses near their physical values, which correspond to 1.2. Issues in Non-Perturbative Studies 19

a pion mass less than 200MeV.

4. Operator mixing: The following complexities arise from the fact that at finite lattice spacing the symmetries of the lattice are reduced with respect to those of the continuum theory. The set of operators one has to consider on the lattice is usually larger than those of the continuum theory. Also the number of independent renormalization constants in a lattice calculation is, in general, larger than in the continuum. Operators which are multiplicatively renormalizable in the continuum may lose this property on the lattice. In addition, on the lattice the possibilities for mixings under renormalization are larger than in the continuum. Such additional mixings are pure lattice artifacts and have to be carefully treated in order to get physical results from lattice simulations. The operators can in general mix with operators of the same, higher, and lower dimensions. In the case of mixings with operators of lower dimensionality the corresponding lattice renormalization constants contain a power divergent coefficient proportional to 1/an. In lattice simulations such coefficients have to be known very accurately, otherwise the power divergences overwhelm the signal as a 0, representing a serious computational challenge. In → lattice perturbation theory mixings with operators of lower dimensionality cannot be reliably computed.

5. Matching between lattice and continuum renormalization scheme: Matrix elements of operators computed on the lattice using non-perturbative methods, still require renormalization in order to lead to meaningful quantities (free of ultraviolet divergences). Thus to make contact with phenomenology, results in lattice regularization schemes need to be matched to the standard continuum schemes. On the one hand experimental data are analyzed using the MS (or MS) continuum renormalization scheme of dimensionally regularized perturbation theory. On

the other hand lattice results are obtained in RI MOM (or RI′)regularization − scheme; a more suitable scheme for non-perturbative investigations of renormalization functions. It is well known that physical quantities in one scheme can be simply related to the same quantity in other schemes by a conversion function. There- fore, in order to obtain reliable non-perturbative estimates of physical quantities one requires the conversion of various renormalization group functions (conversion RI MS functions C → ,scaledownfunctionsR (µ, µ0)). The renormalization group O O functions are calculated in the framework of perturbation theory. Because these functions are regularization independent they are usually calculated in dimensional regularization. The perturbative 1-loop renormalization group functions are not suﬃcient since data show that the (α ) corrections can be large 10% 50%. O s ∼ − In almost all cases the renormalization group functions are known up to higher loops [30, 31, 32, 33] (2-loops or 3-loops and even in same cases up to 4-loops), 20 Chapter 1. Introduction

which means that the reliability of the matching functions is high.

6. Finiteness of the lattice spacing a:

The physical results are extracted by taking the continuum limit a 0. The con- → tinuum limit is reached by extrapolating (to a 0) the lattice results, obtained → form a variety of lattice spacings, roughly within the interval 0.05fm ! a ! 0.1fm. At ﬁnite a, lattice results are aﬀected by discretization errors, with size depending on the degree to which the lattice action and operators have been improved. While these discretization errors are to be removed in the continuum limit, it is advantageous to have them under control at nonzero lattice spacing for two reasons:

a) At large discretization eﬀects, it is notclearhowtoperformasafeandreliable extrapolation to a 0, → b) Unimproved actions force us to work with very small lattice spacings, making simulations computationally very demanding. Since typical QCD algorithms slow down proportionally to a5 (or even faster), it is very expensive in terms of computational time to reduce the discretization eﬀects by simply decreasing the lattice spacing a.

In principle the only restriction while discretizing the QCD action, is the recovery of the continuum limit. The simplest discretized action that recovers the continuum action in the limit a 0, was first proposed by Wilson [12]. One of the main → drawbacks of Wilson’s action is that the lattice artifacts in physical quantities are proportional to the lattice spacing, (a1). Therefore, it is worthwhile to construct O improved actions for a better behavior at the first orders of lattice spacings. The technique which goes under the name of “improvement” aims at removing the systematic errors due to the finiteness of the lattice spacing, which are generally of order a with respect to the continuum limit:

d 1 p p′ = a p p′ + (a ) . (1.3) ⟨ |Olattice| ⟩lattice ⟨ |Ocontinuum| ⟩continuum O $ % A systematic improvement program to reducetheseerrorsorderbyorderinlattice spacing, for Wilson fermions/gluons, was first proposed by Symanzik [34, 35] and then further developed, and applied to on-shell matrix elements by Lüscher, Weisz, Sheikholeslami, Wohlert and many others [36, 37, 38, 39, 40], and also developed to off-shell matrix elements by Capitani, Göckeler, Horsley, Rakow and Schierholz [41, 42]. In this formulation, higher dimensional irrelevant operators with appropriate coefficients are added to the Wilson action in order to cancel in on-shell matrix elements, all terms that in the continuum limit are effectively of (a1). In this O way, (a1) improvement is achieved in on-shell matrix elements (such as hadron O 1.3. Using Lattice Perturbation Theory as a Tool for Improving Non-Perturbative Results 21

masses), since the discretization errors coming from the action are reduced from (a1)to (a2): O O

d 2 p p′ = a p p′ + (a ) . (1.4) ⟨ |Olattice| ⟩improved lattice ⟨ |Ocontinuum| ⟩continuum O $ % Thus the continuum limit can be reached much faster with rate proportional to a2. Furthermore, the modifications in the lattice action permit accurate simulations at lattice spacings as large as 0.4fm. In this way, the speed of simulations can ∼ be increased by a factor of 103 106, allowing us to perform calculations on a ∼ high-performance personal computer [43]. Over the years, many improved actions have been constructed, both for fermion and gluon fields. The most frequently used fermion lattice actions are: clover [39], staggered [44], Ginsparg-Wilson (overlap [45, 46, 47, 48] and domain wall [49, 50]), twisted-mass [51, 52] and Stout Link Non- perturbative Clover (SLiNC) [53]. All these actions have discretization errors of order (a2)andreachthecontinuumlimitfaster than Wilson fermions. A not O so widely used fermionic action with (a3) errors is the D234 action [54]. The O gluon part of the Wilson action is already (a1) improved (the lattice corrections O with respect to the continuum are of order (a2)). A pure gauge action which is O (a2) improved is the Symanzik improved action [34, 35]. This action cancels all O (a2) effects, so the first corrections that are left are at least of order (a3) (in O O fact (a4)). All the above machinery improves partially the lattice theory (only on- O shell improvement is achieved). Simply improving the action does not remove (a1) O errors from operator matrix elements (off-shell quantities). To improve completely the lattice theory (off-shell improvement) one has to improve all operators under study. This is achieved by adding, to them, higher dimensional irrelevant operators with appropriate coefficients [41, 42].

1.3 Using Lattice Perturbation Theory as a Tool for Improving Non-Perturbative Results

Although the idea of the lattice formulation was introduced to investigate phenomena of low energy scales (relevant to hadronic properties), it can also be used in the regime of high energy scales. At this region it is possible to employ lattice perturbation theory in order to obtain physical results. A lengthy review on lattice perturbation theory can be found in Ref. [55]. In general all perturbative results of the continuum QCD theory (obtained using Pauli-Villars or dimensional regularization) can also be reproduced using the lattice regularization. Instead of reproducing already known results, one combines perturbative lattice calculations with numerical simulations, in order to obtain reliable phenomenological numbers at low energy scales. 22 Chapter 1. Introduction

In general, perturbative calculations are much more complicated on the lattice than in the continuum, because one has to deal with trigonometric functions and with diagrams that are not present in the continuum. Despite all the complications it is advantageous to use lattice instead of continuum perturbation theory, since it is possible to compare perturbative with non-perturbative results, testing also in this way the range of validity of the perturbative analytic expressions. Even though perturbative calculations cannot reveal the full content of the lattice field theory, they can still allow one to obtain valuable information. The connection of the matrix elements, simulated within the lattice scheme, to the continuum physical theory is established by performing a lattice renormalization. To achieve this connection all bare quantities such as Lagrangian parameters, fields, and operators have to be renormalized. Traditionally, the renormalization constants are provided by means of lattice perturbation theory, even in many studies at low energy scale [56, 57, 58]. A stand alone non-perturbative determination of the renormalization constants can turn out to be very challenging. This is because it is difficult to obtain a plateau for the signal over a large enough range of energies, so that one can numerically extract the values of the renormalization constants. Moreover, when mixings between operators can occur, the amount of mixing might be too small to be distinguished with numerical techniques, but not too small to be ignored. In practice, renormalization constants are obtained as a result of the combined effort of numerical simulations and perturbative calculations. Since perturbative calculations are performed analytically, the mixing patternsoftheoperatorsbecomemoretransparent. Thus perturbative lattice renormalization can also be used as a hint and a guide to determine the renormalization constants non-perturbatively. Additionally, the only possibility for having some analytical control over the continuum limit (while the lattice regulator is being removed) is given by lattice perturbation theory. Thus perturbative calculations are essential in the study of the general approach to the continuum limit, which includes the recovery of the continuum symmetries (broken by the lattice regulator), and the corrections which are of order an. As we have already mentioned in the previous section, lattice artifacts always lead to systematic errors in lattice estimates, which one can reduce by means of an improvement. Therefore there is a great interest in improving lattice QCD calculations. One-loop perturbative (a1) improvements, for both off-shell and on-shell quantities, have been O extensively studied since the early days of lattice field theory [41, 42, 59]. On the other hand, (a2)perturbativecalculationsdonotexisttodate. O The aim of this thesis is to minimize the systematic errors, that arise in quark, local bilinear, extended bilinear, and four-fermion renormalization constants, by reducing the aforementioned (a2) lattice artifacts. More specifically we calculate the leading dis- O cretization effects to (g2a2)oftherenormalizationconstants, in one-loop perturbation O 1.3. Using Lattice Perturbation Theory as a Tool for Improving Non-Perturbative Results 23 theory. The (g2a2)correctionscanbeexplicitlysubtractedfromthenon-perturbative O RI MOM determination of renormalization constants. Our results have already been − used by the ETM collaboration. Our calculations extend, to a rather large family of fermion/gluon actions, results which were previously known to (a0)and (a1) (modulo ln a). However, the truly novel O O feature in our calculations is that they are performed to second order in the lattice spacing a ( (a2,a2 ln a)). This fact introduces a number of complications, which are not present O in lower order results. In a nutshell, the reason for these complications is as follows: The extraction of a further power of a from a Feynman diagram strengthens, by one unit, the superficial degree of infrared (IR) divergence of the corresponding integrand over loop momenta. Thus, a priori, in a (a1) calculation, loop integrals would be IR convergent O only in D>5dimensions;however,ascanbeeasilydeducedbyinspection,themost divergent parts of the integrands are odd functions of the loop momenta, and will thus vanish upon integration. What is left behind is a less divergent integrand which is IR convergent in D>4, just as in the case of (a0)calculations,andcanthusbetreated O by standard methods, such as those of Ref. [60]. For (a2)calculations,ontheother O hand, integrands are IR convergent only at D>6, and their most divergent parts no longer vanish upon integration; a naive application of the procedure of Ref. [60] will fail to produce all (a2) contributions. The procedure which we propose in this work for O handling the above difficulty is in fact applicable to any order in a. In brief, it recasts the integrands as a sum of two parts: The first part can be exactly evaluated as a function of a,whilethesecondpartisnaivelyTaylorexpandible, as a polynomial to the desired order in a. In all our calculations we employ improved actions both for fermion and gluon fields. For fermions we use the family of (a1) improved Wilson/clover/twisted mass actions; O particular cases of these actions are currently being studied intensely by the ETMC and other collaborations. For gluons we employ a 3-parameter family of Symanzik (a2) O improved gluon actions, comprising all cases which are in common use, such as Wilson, tree-level Symanzik, Iwasaki, DBW2, and Lüscher-Weisz. In order to make our results generic as much as possible, and thus usable to a wide range of Lattice QCD researchers, we have considered the following parameters of the aforementioned actions as free parameters: clover coefficient cSW,numberofcolorsNc, coupling constant g,gaugefixingparameterλ,barequarkmassm0 and twisted mass parameter µ.TheSymanzikcoefficients,ci, appear in a nonpolynomial way in the calculations and, thus, we tabulate these results for different choices of ci. It is worth mentioning that we can easily repeat the calculations for other lattice actions and for other choices of the Symanzik coefficients. This thesis contains work carried out over the past two years and it is laid out as follows. In chapter 2 we calculate the (g2a2)correctionstothefermionpropagatorand O 24 Chapter 1. Introduction local bilinear operators. For this very first calculation we use massless Wilson/clover fermions and Symanzik improved gluons. At the end of this chapter we briefly explain how the ETM collaboration uses our results in order to “correct” their non-perturbative data. In chapter 3 we calculate the (g2a2)correctionstotherenormalizationconstants O of four-fermion operators. We use the same actions as in chapter 2. We pay particular attention to ∆S = 2 operators, both Parity Conserving and Parity Violating (S stands for strangeness). In chapter 4and5weextendthecalculationsofchapter2withmassive twisted mass fermions instead of massless clover fermions. Moreover, we calculate the (g2a2) corrections to the matrix elements of extended bilinear operators. In addition, O non-perturbative results are obtained using the twisted mass Wilson fermion formulation, employing two degenerate dynamical quarks, and the tree-level Symanzik improved gluon action for Pion masses in the range of about 450 260MeV and at β =3.9, 4.05, 4.20. − Furthermore, we use the aforementioned (a2) corrections in order to improve the respec- O tive non-perturbative results. The Appendices contain supplemental material regarding the mathematical formalism of continuum/lattice QCD theory. The work of the following chapters has been carried out in collaboration with Dr. Martha Constantinou and Dr. Apostolos Skouroupathis. All of the results presented here have already been included in the following papers:

M. Constantinou, V. Lubicz, H. Panagopoulos, F. Stylianou, (a2) corrections to O the one-loop propagator and bilinears of clover fermions with Symanzik improved gluons,JHEP10 (2009) 064, [hep-lat/0907.0381]

M. Constantinou, P. Dimopoulos, R. Frezzotti, G. Herdoiza, K. Jansen, V. Lu- bicz, H. Panagopoulos, G.C. Rossi, S. Simula, F. Stylianou, A. Vladikas, Non- perturbative renormalization of quark bilinear operators with Nf=2 (tmQCD) Wil- son fermions and the tree- level improved gauge action,JHEP1008 (2010) 068, [hep-lat/1004.1115]

M. Constantinou, P. Dimopoulos, R. Frezzotti, V. Lubicz, H. Panagopoulos, A. Sk- ouroupathis, F. Stylianou, Perturbative renormalization factors and (a2) correc- O tions for lattice 4-fermion operators with improved fermion/gluon actions, Phys.Rev. D 83 (2011) 074503, [hep-lat/1011.6059]

M. Constantinou, V. Lubicz, H. Panagopoulos, A. Skouroupathis, F. Stylianou and

members of the ETM Collaboration, BK -parameter from Nf =2twistedmasslattice QCD, Phys.Rev. D 83 (2011) 014505, [hep-lat/1009.5606]

C. Alexandrou, M. Constantinou, T. Korzec, H. Panagopoulos, F. Stylianou, Renor- malization constants for 2-twist operators in twisted mass QCD, Phys.Rev. D 83 (2011) 014503, [hep-lat/1006.1920] 1.3. Using Lattice Perturbation Theory as a Tool for Improving Non-Perturbative Results 25

C. Alexandrou, M. Constantinou, T. Korzec, H. Panagopoulos, F. Stylianou, Renor- malization Constants for Fermion Field and Ultra-Local Operators in Twisted Mass QCD, in preparation and conference proceedings:

M. Constantinou, H. Panagopoulos, F. Stylianou, (a2) corrections to the prop- O agator and bilinears of Wilson/clover fermions,PoSLATTICE2008 (2008) 213, [hep-lat/0811.4499] Presented at the ”XXVI International Symposium on Lattice Field Theory”, July 14-19, 2008, Williamsburg, VA, USA

M. Constantinou, V. Lubicz, H. Panagopoulos, A. Skouroupathis, F. Stylianou, (a2) corrections to 1-loop matrix elements of 4-fermion operators with improved O fermion/gluon actions,PoSLATTICE2009 (2009) 205, [hep-lat/1001.1241] Presented at the ”XXVII International Symposium on Lattice Field Theory”, July 26-31 2009, Peking University, Beijing, China

M. Constantinou, H. Panagopoulos, F. Stylianou, Perturbative renormalization of GPDs to (a2),forvariousfermion/gluonactions,PoSLATTICE2009 (2009) O 260, [hep-lat/1001.1498] Presented at the ”XXVII International Symposium on Lattice Field Theory”, July 26-31 2009, Peking University, Beijing, China

C. Alexandrou, M. Constantinou, T. Korzec, H. Panagopoulos, F. Stylianou, Renor- malization constants for one-derivative fermion operators in twisted mass QCD,PoS LATTICE2010 (2010) 224, [hep-lat/1012.2981] Presented at the ”XXVIII International Symposium on Lattice Field Theory”, June 14-19 2010, Villasimius, Sardinia, Italy Chapter 2

(a2) Corrections to the One-Loop O Fermion Propagator and Bilinear Operators

2.1 Introduction

A major issue facing Lattice Gauge Theory, since its early days, has been the reduction of effects which are due to the finite lattice spacing a, in order to better approach the elusive continuum limit. A systematic framework to address this issue is Symanzik’s program [34, 35], in which the regularized action is improved through a judicious inclusion of irrelevant operators with increasing dimensionality. Thus far, most efforts have been directed towards (a1) improvement; this is automatic in some cases (i.e. requires no O tuning of parameters), by symmetry considerations alone. Such is the case, for example, of the twisted mass formulation of QCD [61, 62] at maximal twist, where certain observables are (a1) improved, as a consequence of symmetries of the fermion action; setting the O maximal twist requires the tuning of only a single parameter in the action, i.e. the critical quark mass, and no further improvement of the operators is required. In other cases, such as with the clover fermion action, (a) corrections must be also O implemented on individual operators; such corrections take the form of an additional, finite (non UV-divergent) renormalization or anadmixtureofappropriatehigherdimen- sional operators. Determining the values of the renormalization functions or mixing coefficients requires an evaluation of appropriate Green’s functions, as dictated by the choice of renormalization scheme; these Green’s functions can be evaluated perturbatively or non-perturbatively. As regards the perturbative evaluation of Green’s functions for the “ultralocal” fermion Γ ¯ bilinear operators Oa = ΨλaΓΨ (where Γdenotes all possible distinct products of Dirac matrices, and λa is a flavor symmetry generator) and the related fermion propagator, the

26 2.1. Introduction 27 following types of calculations have appeared thus far in the literature:

i) One-loop calculations to (a0, ln a) have been performed in the past several years for O awidevarietyofactions,rangingfromWilsonfermions/gluonstooverlapfermions and Symanzik gluons [42, 63, 64, 65, 66, 67, 68, 69].

ii) There exist one-loop computations of (a1)corrections,withanarbitraryfermion O mass [42, 64].

Γ iii) The ﬁrst two-loop calculations of Green’s functions for Oa were completed recently, to (a0), for Wilson/clover/twisted-mass fermions and Wilson gluons [70, 71]. O iv) A number of (a0)resultshavealsobeenobtainedbymeansofstochasticperturba- O tion theory [72, 73, 74].

One-loop computations of (a2)correctionsdidnotexisttodate;indeedtheypresent O some novel difficulties, as compared to (a1). Extending (a0)calculationsupto (a1) O O O does not bring in any novel types of singularities. For instance, terms which were convergent to (a0) may now develop at worst an infrared (IR) logarithmic singularity in 4 O dimensions and the way to treat such singularities is well known. Also, in most of the cases, e.g. for m =0,termswhichwerealreadyIRdivergentto (a0) will not con- O tribute to (a1), by parity of loop integration. On the contrary, many IR singularities O encountered at (a2)wouldpersistevenupto6dimensions,makingtheirextraction O more delicate. In addition to that, there appear Lorentz non-invariant contributions in (a2)terms,suchas p4 /p2 (where p is the external momentum). O µ µ In this chapter we present a one-loopperturbativecalculation,to (a2), of the quan- ! O tum corrections to the fermion propagator S(p)andtotheGreen’sfunctionsΛΓ(p)for ¯ complete basis of local fermion bilinear currents z Ψ(z)ΓΨ(z). We employ a 3-parameter family of Symanzik improved gluon actions, comprising! all cases which are in common use (Wilson, tree-level Symanzik, Iwasaki [75, 76], DBW2 [77], Lüscher-Weisz [36, 37]). All calculations have been performed for generic values of the gauge parameter. For fermions we make use of the Wilson/clover action. Since the propagator and Green’s functions are meant to be used in mass independent renormalization schemes, our results have been obtained at vanishing fermionic masses. Also, by virtue of working in a massless scheme, all of our results are applicable to other ultralocal fermion actions as well, such as the twisted mass or Osterwalder-Seiler action [78]. The case of massive fermions (including non-degenerate flavors and twisted mass terms) will appear in a forthcoming chapter. Nevertheless, even at vanishing masses, our final expressions are quite lengthy, since they exhibit a rather nontrivial dependence on the external momentum (p), and they are explicit functions of the number of colors (Nc), gauge parameter (λ), lattice spacing (a), clover coefficient ( cSW)andcouplingconstant(g); furthermore, most numerical coeffi- Chapter 2. (a2) Corrections to the One-Loop Fermion Propagator and O 28 Bilinear Operators cients in these expressions depend on the Symanzik parameters of the gluon action, and we have tabulated them for the actions we have selected. Our results can be used to construct (a2) improved definitions of the fermion bilin- O ears. In particular, they have been used in Ref. [79, 80] to improve the non-perturbative determinations, with the RI MOM method as proposed in Ref. [81], of renormalization − constants of the quark field and bilinear quark operators. In this chapter we provide an outline of our calculational procedure, and we describe in detail the evaluation of a prototype IR divergent integral. We also list a basis of the divergent integrals which appear in the calculation, evaluated to the required order in a. Furthermore we present the corrections to thefermionpropagator,fermionbilinearsand their renormalization constants. At the end of this chapter we discus the usage of the results in order to “correct” non-perturbative estimates. The techniques employed in this chapter are readily applicable to the study of perturbative corrections of other Greens’s functions, to any desired order in a.Examples are matrix elements of 4-fermion operators appearing in effective weak Hamiltonians, and higher dimension twist-2 fermion bilinears involved in generalized parton distributions. We will be addressing these issues in forthcoming chapters.

2.2 Description of the Calculation

Our calculation makes use of the clover (SW) action for fermions. For Nf ﬂavor species this action reads, in standard notation:

a3 S = ψ¯ (x)(r γ ) U ψ (x + aµ)+ψ¯ (x + aµ)(r + γ ) U ψ (x) F − 2 f − µ x, x+aµ f f µ x+aµ,x f x,& f, µ ' ( 4r + a4 ( + m )ψ¯ (x)ψ (x) a f f f &x, f a5 rc ψ¯ (x)σ F (x)ψ (x), (2.1) − 4 SW f µν µν f x,& f, µ, ν where the Wilson parameter r is set to r =1,f is a flavor index, σµν =[γµ,γν]/2 and the clover coefficient cSW is kept as a free parameter throughout. The relation of the link Ux, x+aµ matrices with the gauge fields Aµ(x), the varibles which have a direct correspondence with the continuum, is given by:

B B ig0aT Aµ (x+aµ/2) Ux, x+aµ = e , (2.2) where B =1,...,N2 1andT B are the SU(N )matricesinthefundamentalrepresenta- c − c tion. All quantities are dimensionful, while all powers of the lattice spacing a are shown explicitly. The tensor Fµν (x) is a lattice representation of the gluon ﬁeld tensor, deﬁned 2.2. Description of the Calculation 29 through: 1 F (x) (Q (x) Q (x)), (2.3) µν ≡ 8a2 µν − νµ where Qµν is the sum of the plaquette loops:

Qµν (x)=Ux, x+aµ Ux+aµ,x+aµ+aν Ux+aµ+aν,x+aν Ux+aν,x

+ Ux, x+aν Ux+aν,x+aν aµUx+aν aµ,x aµ Ux aµ,x − − − − + Ux, x aµ Ux aµ,x aµ aν Ux aµ aν,x aν Ux aν,x − − − − − − − − + Ux, x aν Ux aν,x aν+aµUx aν+aµ,x+aµ Ux+aµ,x, (2.4) − − − − as shown in Fig. 2.1.

µ ν

Figure 2.1: Graphical representation of Qµν (Eq. (2.4)) appearing in the clover action.

We perform our calculation for mass independent renormalization schemes, so that mf =0.Thissimpliﬁesthealgebraicexpressions,butatthesametimerequiresspecial treatment when it comes to IR singularities. By taking mf =0,ourcalculationand results are identical also for the twisted mass action and the Osterwalder-Seiler action in the chiral limit (in the so called twisted mass basis). For gluons we employ the Symanzik improved action, involving Wilson loops with 4 and 6 links , as shown in Fig. 2.2 (1 1 plaquette,1 2 rectangle,1 2 chair,and1 1 1 × × × × × parallelogram wrapped around an elementary 3-d cube), which is given by the relation:

2 S = c Re Tr 1 U + c Re Tr 1 U G g2 0 { − plaq.} 1 { − rect.} 0 rect. ) plaq&. & + c Re Tr 1 U + c Re Tr 1 U . (2.5) 2 { − chair} 3 { − paral.} chair& paral&. *

The coeﬃcients ci can in principle be chosen arbitrarily, subject to the following normalization condition, which ensures the correct classical continuum limit of the action:

c0 +8c1 +16c2 +8c3 =1. (2.6) Chapter 2. (a2) Corrections to the One-Loop Fermion Propagator and O 30 Bilinear Operators

Figure 2.2: The 4- and 6-link loops contributing to the gauge action of Eq. (2.5).

Some popular choices of values for ci used in numerical simulations will be considered in this work, and are itemized in Table 2.1; they are normally tuned in a way as to ensure (a2) improvement in the pure gluon sector. Our one-loop Feynman diagrams do not O involve pure gluon vertices, and the gluon propagator depends only on three combinations of the Symanzik parameters:

C c +8c +16c +8c =1, 0 ≡ 0 1 2 3 C c + c , (2.7) 1 ≡ 2 3 C c c c . 2 ≡ 1 − 2 − 3

Therefore, with no loss of generality all these sets of values have c2 =0. For the algebraic operations involved in evaluating the Feynman diagrams relevant to this calculation, we make use of our symbolic package in Mathematica. Next, we brieﬂy describe the required steps:

Algebraic manipulations: • The first step in evaluating each diagram is the contraction among vertices, which is performed automatically once the algebraic expression of the vertices and the topology (“incidence matrix”) of the diagram are specified. The outcome of the contraction is a preliminary expression for the diagram; there follow simplifications of the color dependence, Dirac matrices and tensor structures. We also fully exploit symmetries of the theory (periodicity, reflection, conjugation, hypercubic, etc.) to limit the proliferation of the algebraic expressions. 2.2. Description of the Calculation 31

Action c0 c1 c3 Plaquette 1.0 0 0 Symanzik 5/3 -1/12 0 TILW, βc0 =8.60 2.3168064 -0.151791 -0.0128098 TILW, βc0 =8.45 2.3460240 -0.154846 -0.0134070 TILW, βc0 =8.30 2.3869776 -0.159128 -0.0142442 TILW, βc0 =8.20 2.4127840 -0.161827 -0.0147710 TILW, βc0 =8.10 2.4465400 -0.165353 -0.0154645 TILW, βc0 =8.00 2.4891712 -0.169805 -0.0163414 Iwasaki 3.648 -0.331 0 DBW2 12.2688 -1.4086 0

Table 2.1: Input parameters c0, c1, c3.

Dependence on external momentum: • The above simpliﬁcations are followed by the extraction of all forms of functional dependence on the external momentum p (logarithmically divergent, Lorentz non-invariant, polynomial terms) and the lattice spacing (terms of order a0,a1,a2, and possibly ln a). Even though one-loop computations are normally a straighforward procedure, extending to (a2) introduces several complications, especially when isolating logarithms and O Lorentz non-invariant terms. As a ﬁrst task we want to reduce the number of infrared divergent integrals to a minimal set. To do this, we use two kinds of subtractions among the propagators, using the simple equalities:

2 4 sin4(q /2) 4 sin2(q /2) 1 1 µ µ − µ µ 2 = 2 + 2 2 , (2.8) q˜ qˆ + ! q˜ qˆ,! - .

D(q)=D (q)+ D(q) D (q) plaq − plaq " # 1 1 = D (q)+D (q) D− (q) D− (q) D(q), (2.9) plaq plaq plaq − " # where q stands for k or k + ap,whilek is the loop momentum and p is the external momentum. The denominator of the fermion propagator,q ˜2, is deﬁned as:

r 2 q q˜2 = sin2(q )+ m + qˆ2 , qˆ2 =4 sin2( µ ). (2.10) µ f 2 2 µ µ & , - & For the present work, one sets m =0andr = 1, as used in Eq. (2.8); D(q) is the 4 4 f × 1 1 Symanzik gluon propagator; the expression for the matrix D− (q) D− (q) , which is plaq − (q4), is independent of the gauge parameter, λ,anditcanbeeasilyobtainedinclosed O / 0 Chapter 2. (a2) Corrections to the One-Loop Fermion Propagator and O 32 Bilinear Operators form. Moreover, we have:

δ qˆ qˆ Dµν (q)= µν (1 λ) µ ν . (2.11) plaq qˆ2 − − (ˆq2)2

Terms in curly brackets of Eqs. (2.8) and (2.9) are less IR divergent than their unsubtracted counterparts, by two powers in the momentum. These subtractions are performed iteratively until all primitively divergent integrals (initially depending on the fermion and the Symanzik propagator) are expressed in terms of the Wilson gluon propagator. Having reduced the number of distinct divergent integrals down to a minimum, the most laborious task is the computation of these integrals, which is performed in a noninteger number of dimensions D>4. Ultraviolet divergences are explicitly isolated à la Zimmermann and evaluated as in the continuum. The remainders are D-dimensional, parameter-free, zero external momentum lattice integrals which can be recast in terms of Bessel functions, and finally expressed as sumsofapolepartplusnumericalconstants. We analytically evaluate an extensive basis of superfcially divergent loop integrals, listed in the next section; a few of these were calculated in Ref. [82]. The integrals of Eqs. (2.34), (2.36), (3.47), are the most demanding ones in the list; they must be evaluated to two further orders in a,beyondtheorderatwhichanIRdivergenceinitiallysetsin.Asa consequence, their evaluation requires going to D>6 dimensions. Fortunately, they are a sufficient basis for all massless integrals which can appear in any (a2) one-loop calcu- O lation; that is, any such computation can be recast in terms of (2.34), (2.36), (3.47), plus other integrals which are more readily handled. A correct way to evaluate the integrals of Eqs. (2.34), (2.36), (3.47) has not been presented previously in the literature, despite their central role in (a2)calculations,andthishaspreventedone-loopcomputationsto O (a2) thus far. The calculation of such an integral is given in detail in the next section. O Terms which are IR convergent can be treated by Taylor expansion in ap to the desired order. Alternatively, the extraction of the ap dependence may be performed using iteratively subtractions of the form:

f(k + ap)=f(k)+ f(k + ap) f(k) . (2.12) − ) * This leads to exact relations such as the following ones:

1 1 sin(2kµ + apµ)sin(apµ) = µ 2 ˜ 2 2 k!+ ap k − ! k!+ ap k˜ 2 2 apµ apµ ˆ 2 µ sin(kµ + 2 )sin( 2 ) k + k"+ ap , (2.13) 2 , - − ! k!+ap k˜ 2

apµ apµ 1 1 4 sin(kµ + )sin( ) = µ 2 2 . (2.14) 2 ˆ 2 2 k"+ ap k − ! k"+ ap kˆ 2 2.3. Evaluation of a Primitively Divergent Integral 33

In these relations the exact ap dependence of the remainders is under full control; this type of subtraction is especially useful when applied to the Symanzik propagator.

Numerical integration: • The required numerical integrations of the algebraic expressions for the loop integrands (a total of 40,000 terms) are performed by highly optimized Fortran programs; these ∼ are generated by our Mathematica “integrator” routine. Each integral is expressed as asumoverthediscreteBrillouinzoneofﬁnitelattices,withvaryingsizeN = L/a (44 N 4 1284), and evaluated for all values of the Symanzik coeﬃcients listed in ≤ ≤ Table 2.1 (corresponding to the Plaquette, Symanzik, Iwasaki, TILW and DBW2 action).

Extrapolation: • The last part of the evaluation is the extrapolation of the numerical results to inﬁnite lattice size. This procedure entails a systematic error, which is reliably estimated, using asophisticatedinferencetechnique;forone-loop quantities we expect a fractional error 7 smaller than 10− .

2.3 Evaluation of a Primitively Divergent Integral

The most diﬃcult part of this calculation, that requires careful attention, is the extraction of the dependence on the external momentum p and the lattice spacing a from the divergent terms. The singularities are isolated using the procedure explained in this section. We also present the list of primitively divergent integrals that appeared in our algebraic expressions. Divergent integrals which appear in calculations up to (a1)maybeevaluatedusing O the standard procedure of Kawai et al. [60], in which one subtracts and adds to the original integrand its naive Taylor expansion, to the appropriate order with respect to a, in D 4+ dimensions: The subtracted integrand, being UV convergent, is calculated in the → continuum limit a 0, using the methods of Ref. [83], while the Taylor expansion terms → are recast in terms of Bessel functions and are evaluated in the limit ϵ 0(ϵ (4 D)/2). → ≡ − The following integrals have been evaluated using the above procedure:

π 4 ◦ ◦ d k kµ kν 2 2 4 2 = δµν 0.014966695116 0.001256484446 a p • π (2π) ˆ 2 " − 1− k k + ap ) a2 p2 ln(a2p2) 0.001027789631 a2 p2 + − µ 192π2 * ln(a2p2) + a2 p p 0.003970508789 + (a4 p4), (2.15) µ ν − 48π2 O ) * Chapter 2. (a2) Corrections to the One-Loop Fermion Propagator and O 34 Bilinear Operators

3 π 4 ◦ d k kµ 2 2 4 2 = apµ 0.006184131744 + 0.001102333439 a p • π (2π) ˆ 2 " − 1− k k + ap ) ln(a2p2) p2 0.000174224479 a2 p2 + a2 p2 − µ 64π2 µ − 2 2 3* + (a5p5), (2.16) O

π 4 ◦ ◦ ◦ 2 2 d k kµ kν kρ ln(a p ) 4 2 =(δνρ apµ)S 0.000728769948 + 2 • π (2π) ˆ 2 2 " − 192π 1− (k ) k + ap ) * p p p +0.001027789631δ ap a µ ν ρ + (a3 p3), (2.17) µνρ µ − 48π2 p2 O

π 4 ˆ 4 d k µ kµ 2 2 4 2 =0.004050096698 0.000107954163 a p • π (2π) ˆ 2 2 − 1− 16(k!) k"+ ap 4 2 µ pµ 4 4 + a 2 2 + (a p ), (2.18) 1024! π p O

π 4 ◦ ◦ ◦ ◦ d k kµ kν kρ kσ 4 2 =0.001589337971 (δµν δρσ)S 0.001675948042 δµνρσ • π (2π) ˆ 2 2 − 1− (k ) k"+ ap 2 2 2 2 + δµνρσ 0.000186391491 a p +0.000410290033 a pµ

/ ln(a2p2) 0 +(δ a2 p p ) 0.000227848225 µν ρ σ S − 384π2 2 3 ln(a2p2) +(δ δ ) a2 p2 0.000245852737 + µν ρσ S − 768π2 2 3 0.000372782983(δ a2 p p ) − µνρ µ σ S 0.000062130497(δ δ a2 p2 ) − µν ρσ µ S p p p p + a2 µ ν ρ σ + (a4 p4), (2.19) 64π2 p2 O

π 4 ◦ ˆ 4 d k kν µ kµ 2 2 4 2 = apν 0.000800034900 + 0.000069705553 a p • π (2π) ˆ 2 2 " − 1− 16(k )!k + ap ) 4 2 2 2 ρ pρ +0.000107082394 a pν a 2 2 − 1280! π p ln(a2p2) p2 a2 p2 + (a5 p5), (2.20) − 2560π2 2 − ν O 2 3* 2.3. Evaluation of a Primitively Divergent Integral 35

4 π 4 ◦ ◦ " d k kν kρ µ kµ+apµ 2 2 = δνρ 0.000400017450 0.000034852777 a p 4 2 2 • π (2π) ˆ 2 2 − 1− 16(k !) (k"+ap ) ) 4 2 2 2 µ pµ +0.000105349447 a pν + a 2 2 2560! π p ln(a2p2) p2 + a2 ( 3p2 ) 5120π2 2 − ν * p2 + p2 + a2 p p 0.000006643045 ν ρ ν ρ − 2560π2 p2 ) 4 2 2 pµ ln(a p ) + µ + + (a4 p4), (2.21) 5120π2 (p2)2 5120π2 O ! * where: q q qˆ = 2 sin( µ ), qˆ2 =4 sin2( µ ), k◦ = sin(k ). (2.22) µ 2 2 µ µ µ & No summation over the indices µ, ν,ρ,σ is implied, unless otherwise stated. In addition,

()S means sum over inequivalent permutations.

In contrast to the above, some of the integrals in the present work, given that they must be evaluated to (a2), have Taylor expansions which remain IR divergent all the O way up to D 6dimensions.ArelateddifficultyregardsKawai’sprocedure:Subtracting ≤ from the original integral its Taylor expansion in D-dimensions to the appropriate order, the UV-convergent subtracted expression at which one arrives can no longer be evaluated in the continuum limit by naively setting a 0, because there will be (a2)correc- → O tions which must not be neglected. These novel difficulties plague integrals of Eqs. (2.34), (2.36), (3.47). Using a combination of momentum shifts, integration by parts and trigonometric identities, one may express Eqs. (2.36) and (3.47) in terms of (2.34) and other less divergent integrals. Thus, it suffices to address the evaluation of Eq. (2.34):

π d4k 1 A1(ap) 4 2 . (2.23) ≡ π (2π) ˆ2 1− k k"+ ap

This is a prototype case of an integral which is IR divergent in D 6dimensions;in ≤ fact, all other integrals encountered in the present calculation may be expressed in terms of A1(ap)plusotherintegralswhichareIRconvergentatD>4(andarethusamenable to a more standard treatment).

First we split the original integrand I into two parts:

1 I 2 = I1 + I2, (2.24) ≡ kˆ2 k"+ ap where I2 is obtained from I by a series expansion, with respect to the arguments of all Chapter 2. (a2) Corrections to the One-Loop Fermion Propagator and O 36 Bilinear Operators trigonometric functions, to subleading order, while I is simply the remainder I I : 1 − 2 2 2 k4 ˆ2 k4 k2 kˆ2 k4 (k + ap)2 k"+ ap k 12 k − − I1 = − − + + 2 ,2 - 2 , 2 -2 k2 kˆ2 k"+ ap 12 (k2) kˆ2 k"+ ap 12 (k2) (k + ap)2 k"+ ap

2 4 2 4 2 (k + ap)2 (k+ap) k"+ ap (k + ap) (k + ap) k"+ ap + − 12 − + − , (2.25) 2 , 2 2 - k2 (k + ap)2 k"+ ap 12 k2 ((k + ap)2) k"+ ap

1 (k + ap)4 k4 I2 = + + , (2.26) k2 (k + ap)2 2 2 2 2 2 2 '12 k ((k + ap) ) 12 (k ) (k + ap) ( where we make use of the shorthand notation q4 q4.TheintegrandI is free of ≡ µ µ 2 trigonometric functions, while I is naively Taylor expandable to (a2); its integral equals: 1 ! O π 4 d k 2 2 4 4 4 I1 =0.004210419649(1) + a p 0.0002770631001(3) + (a ,a ln a). (2.27) π (2π) O 1− The errors appearing in Eq. (2.27) come from extrapolations to inﬁnite lattice size.

To evaluate the integral of I2 we split the hypercubic integration region into a hypersphere of arbitrary radius µ about the origin (µ π)plustherest: ≤ π π = + . (2.28) π k µ π − k µ 1− 1| |≤ 21− 1| |≤ 3 The integral outside the hypersphere is free of IR divergences and is thus Taylor expandable to any order, giving for µ =3.14155:

π 4 d k 3 2 2 5 4 − − 4 I2 =6.42919(3) 10 + a p 6.2034(1) 10 + (a ). (2.29) π − k µ (2π) × × O 21− 1| |≤ 3 Due to its peculiar domain, this integral has been evaluated by a Monte Carlo routine, rather than as a sum over lattice points. The errors in Eq. (2.29) are thus Monte Carlo errors.

We are now left with the integral of I2 over a sphere. The most infrared divergent part of I ,withIRdegreeofdivergence 4, and can be integrated exactly,giving: 2 − d4k 1 1 a2 p2 4 2 2 = 2 1 ln( 2 ) . (2.30) k µ (2π) k (k + ap) 16π − µ 1| |≤ 2 3 The remaining two terms comprising I have IR degree of divergence 2, thus their 2 − calculation to (a2) can be performed in D-dimensions, with D slightly greater than O 4. Let us illustrate the procedure with one of these terms: k4/((k2)2 (k + ap)2). By 2.3. Evaluation of a Primitively Divergent Integral 37 appropriate substitutions of:

1 1 2(k ap) a2 p2 = + − · − , (2.31) (k + ap)2 k2 k2 (k + ap)2 we split this term as follows:

k4 k4 k4 ( 2(k ap) a2 p2) 4 k4(k ap)2 = + − · − + · (2.32) 2 2 2 2 3 2 4 2 5 (k ) (k + ap) '(k ) (k ) (k ) ( k4 (4(k ap)a2 p2 +(a2 p2)2) 4 k4(k ap)2 ( 2(k ap) a2 p2) + · + · − · − . 2 4 2 2 5 2 2 (k ) (k + ap) (k ) (k + ap) 3 The part in square brackets is polynomial in a and can be integrated easily, using D- dimensional spherical coordinates. The remaining part is UV-convergent; thus the integration domain can now be recast in the form:

= . (2.33) k µ k < − µ k < 1| |≤ 1| | ∞ 1 ≤| | ∞ The integral over the whole space can be performed using the methods of Ref. [83], while the integral outside the sphere of radius µ is naively Taylor expandable to the desired order in a; since at zeroth-order Taylor expansion it is already of (a3) it may be safely O dropped. The same procedure is applied to the last term of I2.Addingthecontributions from all the steps described above, we check that the result is independent of µ.

The integrals, with strong IR divergences (convergent only beyond D>6), encountered in the present calculation are listed below:

π d4k 1 ln(a2p2) 4 2 =0.036678329075 2 • π (2π) ˆ 2 − 16π 1− k k"+ ap 4 2 2 2 µ pµ 4 4 +0.0000752406(3) a p + a 2 2 + (a p ), (2.34) 384!π p O

π 4 ◦ ◦ 2 2 d k kµ kν ln(a p ) 4 2 = δµν 0.004327913824 2 • π (2π) ˆ 2 2 " − 64π 1− (k ) k + ap ) +0.00025539124(8) a2 p2 0.000135654113 a2 p2 − µ

4 2 2 2 pµ ln(a p ) p + a2 µ + a2 p2 1536π2 p2 768π2 µ − 2 ! 2 3* 4 1 pµ + a2 p p 0.0003788538(2) + µ µ ν 32π2 a2 p2 − 768π2 (p2)2 ) ! (p2 + p2) ln(a2p2) µ ν + + (a4 p4). (2.35) − 384π2 p2 768π2 O * Chapter 2. (a2) Corrections to the One-Loop Fermion Propagator and O 38 Bilinear Operators

π 4 ◦ 2 2 d k kµ ln(a p ) 4 2 = apµ 0.008655827648 + 2 • π (2π) ˆ 2 " − 32π 1− k k + ap ) 0.0005107825(2) a2 p2 +0.001171329715 a2 p2 − µ 4 2 2 2 pµ ln(a p ) p a2 µ + a2 p2 + (a5 p5), (2.36) − 768π2 p2 384π2 2 − µ O ! 2 3*

2.4 Corrections to the Fermion Propagator

The fermion propagator of the interacting theory is given by the following 2-point correlation function (Green’s function):

π a 4 a,f b,g d p ab fg ip(x y) 1 1 1 ¯ − − Ψα (x)Ψβ (y) = 4 δ δ e Spert.(p) , (2.37) ⟨ ⟩ π (2π) i/p◦ + M(p) i/p◦ + M(p) 1− a 4 5αβ

2 r 1 M(p) m + sin2(ap /2),/p◦ γ sin(ap ), (2.38) ≡ f a µ ≡ µ a µ µ µ & & where α,β are Dirac indices, f, g are ﬂavor indices in the fundamental representation of SU(Nf ), and a, b are color indices in the fundamental representation of SU(Nc). The 1 inverse fermion propagator in momentum space, Spert− .(p), is given by the following 2-point amputated Green’s function:

1 a,f a,f amp. S− (p)= Ψ (p)Ψ¯ (p) . (2.39) pert. ⟨ α′ β′ ⟩

The correlation function Eq. (2.37) can be interpreted physically as the amplitude for propagation of a particle or excitation between y and x. The fermion propagator is the most common example of an off-shell quantity suffering from (a) effects. Capitani et al. [42] have calculated the first order terms in the lattice O spacing for massive fermions. We carried out this calculation beyond the first order correction, taking into account all terms up to (a2). Our results, to (a1), are in O O perfect agreement with those of Ref. [42]. The clover coefficient cSW has been considered to be a free parameter and our results are given as a polynomial of cSW.Moreover, the dependence on the number of colors Nc,thecouplingconstantg and the gauge fixing parameter λ, is shown explicitly. The Symanzik coefficients, ci, appear in a nontrivial way in the propagator and, thus, we tabulate these results for different choices of ci.Theone- loop Feynman diagrams that enter our 2-point amputated Green’s function calculation (Eq. (2.39)), are illustrated in Fig. 2.3. 2.4. Corrections to the Fermion Propagator 39

1 2 Figure 2.3: One-loop diagrams contributing to the fermion propagator. Wavy (solid) lines represent gluons (fermions).

1 Next, we provide the total expression for the inverse fermion propagator Spert− .(p)as afunctionofg, Nc,cSW and λ.Hereweshouldpointoutthatfordimensionalreasons, there is a global prefactor 1/a multiplying our expressions for the inverse propagator, and thus, the (a2) correction is achieved by considering all terms up to (a3p3): O O 2 1 a 2 a S− (p)=i p + p i p pert. ̸ 2 − 6 ̸ 3 i p g˜2 ε(0,1) 4.79200956(5) λ + ε(0,2) c + ε(0,3) c2 + λ ln(a2p2) − ̸ − SW SW ) * ap2 g˜2 ε(1,1) 3.86388443(2) λ + ε(1,2) c + ε(1,3) c2 − − SW SW ) 1 (3 2 λ 3 c ) ln(a2p2) − 2 − − SW * ia2 p g˜2 ε(2,1) +0.507001567(9) λ + ε(2,2) c + ε(2,3) c2 − ̸ 3 SW SW ) 101 11 λ + C ln(a2p2) 120 − 30 2 − 6 2 3 * ia2 p2 p g˜2 ε(2,4) +1.51604667(9) λ + ε(2,5) c + ε(2,6) c2 − ̸ SW SW ) 59 c C 1 3 + + 1 + 2 λ + c + c2 ln(a2p2) 240 2 60 − 4 2 SW SW 2 2 33 * p4 3 C 5 ia2 p µ µ g˜2 2 λ − ̸ p2 − 80 − 10 − 48 ! ) * + (a3,g4), (2.40) O where:

g2C N 2 1 g˜2 F ,C c − , p γ p , p γ p3 ,C c c c . (2.41) ≡ 16π2 F ≡ 2N ̸ ≡ µ µ ̸ 3 ≡ µ µ 2 ≡ 1 − 2 − 3 c µ µ & & The speciﬁc values λ =1(0)correspondtotheFeynman(Landau)gauge.Thequan- (i,j) 1 tities ε appearing in our results for Spert− .(p)arenumericalcoeﬃcientsdependingon the Symanzik parameters, calculated for each action we have considered and tabulated in Tables 2.2 - 2.5; the index i denotes the power of the lattice spacing a that they multiply. Chapter 2. (a2) Corrections to the One-Loop Fermion Propagator and O 40 Bilinear Operators

In all Tables, the systematic errors in parentheses come from the extrapolation over ﬁnite lattice size L . →∞ Action ε(0,1) ε(0,2) ε(0,3) Plaquette 16.6444139(2) -2.24886853(7) -1.39726711(7) Symanzik 13.02327272(7) -2.01542504(4) -1.24220271(2) TILW (8.60) 10.90082304(6) -1.85472029(6) -1.13919759(2) TILW (8.45) 10.82273528(9) -1.84838009(3) -1.13513794(1) TILW (8.30) 10.71525766(9) -1.83959982(6) -1.12951598(5) TILW (8.20) 10.6486809(1) -1.83412923(5) -1.12601312(2) TILW (8.10) 10.56292631(3) -1.82704771(6) -1.12147952(3) TILW (8.00) 10.45668970(6) -1.81821854(5) -1.11582732(3) Iwasaki 8.1165665(2) -1.60101088(7) -0.97320689(3) DBW2 2.9154231(2) -0.96082198(5) -0.56869876(4)

Table 2.2: The coeﬃcients ε(0,i) (Eq. (2.40)) for diﬀerent actions.

Action ε(1,1) ε(1,2) ε(1,3) Plaquette 12.8269254(2) -5.20234231(6) -0.08172763(4) Symanzik 10.69642966(8) -4.7529781(1) -0.075931174(1) TILW (8.60) 9.3381342(2) -4.4316083(2) -0.07178771(1) TILW (8.45) 9.2865455(1) -4.4186677(2) -0.07160078(1) TILW (8.30) 9.2153414(1) -4.40071157(1) -0.071339052(3) TILW (8.20) 9.17111769(1) -4.38950279(4) -0.07117418(3) TILW (8.10) 9.1140228(1) -4.37497018(8) -0.070959405(2) TILW (8.00) 9.0430829(2) -4.35681290(3) -0.070688697(3) Iwasaki 7.40724287(1) -3.88883584(9) -0.061025650(8) DBW2 3.0835163(2) -2.2646221(1) -0.03366740(1)

Table 2.3: The coeﬃcients ε(1,i) (Eq. (2.40)) for diﬀerent actions.

Action ε(2,1) ε(2,2) ε(2,3) Plaquette -4.74536466(2) 0.02028705(5) 0.10348577(3) Symanzik -4.2478783(2) 0.05136635(6) 0.07865292(7) TILW (8.60) -3.83065630(4) 0.05733870(8) 0.06695681(3) TILW (8.45) -3.8139475(2) 0.05751390(9) 0.06651692(3) TILW (8.30) -3.7907861(2) 0.05775197(7) 0.065909492(9) TILW (8.20) -3.7763441(2) 0.05789811(5) 0.06553180(2) TILW (8.10) -3.75762300(2) 0.05808114(5) 0.06504530(6) TILW (8.00) -3.7342556(1) 0.05830392(9) 0.06444077(4) Iwasaki -3.2018047(1) 0.08249970(7) 0.04192446(4) DBW2 -0.8678072(2) 0.1024452(2) -0.00343999(2)

Table 2.4: The coeﬃcients ε(2,1) ε(2,3) (Eq. (2.40)) for diﬀerent actions. − 2.4. Corrections to the Fermion Propagator 41

Action ε(2,4) ε(2,5) ε(2,6) Plaquette -1.5048070(1) 0.70358496(5) 0.534320852(7) Symanzik -1.14716212(5) 0.65343092(3) 0.49783419(2) TILW (8.60) -0.93394841(7) 0.62190916(4) 0.46915700(3) TILW (8.45) -0.92583451(6) 0.62061757(5) 0.467966296(9) TILW (8.30) -0.91463294(8) 0.61882111(4) 0.46630972(2) TILW (8.20) -0.9076739(2) 0.61769697(3) 0.46527307(3) TILW (8.10) -0.8986914(1) 0.61623801(3) 0.463925850(6) TILW (8.00) -0.8875297(1) 0.61441084(7) 0.462237852(9) Iwasaki -0.6202244(1) 0.55587473(6) 0.41846440(4) DBW2 -0.3202477(5) 0.34886590(2) 0.23968038(4)

Table 2.5: The coeﬃcients ε(2,4) ε(2,6) (Eq. (2.40)) for diﬀerent actions. −

The (a1) logarithms as well as all terms multiplied by λ,areindependentofthe O Symanzik coeﬃcients; on the contrary (a2) logarithms have a mild dependence on the O Symanzik parameters. A number of Lorentz non-invariant tensors ( p4 , p ) appear in µ µ ̸ 3 (a2)correctionterms,compatiblywithhypercubic invariance. Finally, our (a1) results O ! O for the Plaquette action, are in agreement with Eq. (37) of Ref. [42].

To enable cross-checks and comparisons, the per-diagram contributions d1(p),d2(p) are presented below. The tadpole diagram 1 of Fig. 2.3 is free of logarithmic terms and independent of cSW; its ﬁnal expression is:

d (p) 1 = i p ε˜(0,1) +3.050262540200(1) λ + ap2 ε˜(1,1) +1.529131270100(1) λ g˜2 ̸ 1 1 ) * ) * + ia2 p ε˜(2,1) 0.509710423367(1) λ + (a3,g2), (2.42) ̸ 3 1 − O ) * (i,1) where the numerical values for the Symanzik dependent coeﬃcientsε ˜1 are listed in Table 2.6.

(0,1) (1,1) (2,1) Actionε ˜1 ε˜1 ε˜1 Plaquette 9.174787621(1) 4.5873938103(5) -1.5291312701(2) Symanzik 7.071174701(5) 3.535587351(2) -1.1785291169(8) TILW (8.60) 5.903194489(4) 2.951597245(2) -0.9838657482(6) TILW (8.45) 5.86097856(2) 2.930489282(8) -0.976829761(3) TILW (8.30) 5.80296012(1) 2.901480060(5) -0.967160020(2) TILW (8.20) 5.767070636(3) 2.883535318(1) -0.9611784393(4) TILW (8.10) 5.720900703(3) 2.860450351(1) -0.9534834504(5) TILW (8.00) 5.663791993(4) 2.831895997(2) -0.9439653322(7) Iwasaki 4.423664730(5) 2.211832365(2) -0.7372774550(8) DBW2 1.86908767(4) 0.93454384(2) -0.311514612(6)

(0,i) Table 2.6: The coeﬃcients˜ε1 (Eq. (2.42)) for diﬀerent actions. Chapter 2. (a2) Corrections to the One-Loop Fermion Propagator and O 42 Bilinear Operators

The main contribution to the propagator correction comes from diagram 2, as can be seen from the following expression:

d (p) 2 = i p ε˜(0,1) 7.850272109(6) λ + ε(0,2) c + ε(0,3) c2 g˜2 ̸ 2 − SW SW ) + λ ln(a2p2) * + ap2 ε˜(1,1) 5.39301570(2) λ + ε(1,2) c + ε(1,3) c2 2 − SW SW ) 1 (3 2 λ 3 c ) ln(a2p2) − 2 − − SW * + ia2 p ε˜(2,1) +1.016711991(9) λ + ε(2,2) c + ε(2,3) c2 ̸ 3 2 SW SW ) 101 11 λ + C ln(a2p2) 120 − 30 2 − 6 2 3 * + ia2 p2 p ε(2,4) +1.51604667(9) λ + ε(2,5) c + ε(2,6) c2 ̸ SW SW ) 59 c C 1 3 + + 1 + 2 λ + c + c2 ln(a2p2) 240 2 60 − 4 2 SW SW 2 2 33 * p4 3 C 5 + ia2 p µ µ 2 λ + (a3,g2), (2.43) ̸ p2 −80 − 10 − 48 O ! ) * (i,1) (i,j) withε ˜2 provided in Table 2.7. The remaining terms with coeﬃcients ε are the same as in Eq. (2.40).

(0,1) (1,1) (2,1) Actionε ˜2 ε˜2 ε˜2 Plaquette 7.4696262(2) 8.2395316(2) -3.21623339(2) Symanzik 5.95209802(7) 7.16084231(8) -3.0693492(2) TILW (8.60) 4.99762855(6) 6.3865370(2) -2.84679055(4) TILW (8.45) 4.96175672(9) 6.3560562(1) -2.8371177(2) TILW (8.30) 4.91229754(9) 6.3138614(1) -2.8236261(2) TILW (8.20) 4.8816102(1) 6.287582370(9) -2.8151657(2) TILW (8.10) 4.84202561(3) 6.2535725(1) -2.80413954(2) TILW (8.00) 4.79289770(6) 6.2111869(2) -2.7902902(1) Iwasaki 3.6929018(2) 5.19541051(1) -2.4645273(1) DBW2 1.0463355(2) 2.1489724(2) -0.5562925(2)

(0,i) Table 2.7: The coeﬃcients˜ε2 (Eq. (2.43)) for diﬀerent actions.

Terms proportional to 1/a have been left out of Eq. (2.40) for conciseness; such terms represent (g2)correctionstothecriticalvalueofthefermionmass: O 2 g˜ ( 1,1) ( 1,2) ( 1,3) 2 1 4 m = ε − + ε − c + ε − c + (g ), (2.44) c − a m m SW m SW aO ) * 2.5. Corrections to Fermion Bilinear Operators 43

( 1,i) where the numerical values for the Symanzik dependent coeﬃcients εm− are listed in Table 2.8.

( 1,1) ( 1,2) ( 1,3) Action εm− εm− εm− Plaquette -51.4347118(1) 13.7331310(2) 5.71513853(1) Symanzik -40.44324021(3) 11.9482199(1) 4.66267212(2) TILW (8.60) -34.1774729(1) 10.7651652(1) 3.99834878(1) TILW (8.45) -33.9488669(2) 10.7194761(1) 3.97345187(1) TILW (8.30) -33.63443922(6) 10.6563262(1) 3.93913584(1) TILW (8.20) -33.4397933(1) 10.6170532(2) 3.91785126(1) TILW (8.10) -33.18922747(4) 10.5662931(1) 3.89040135(3) TILW (8.00) -32.87904078(6) 10.5031339(1) 3.85634587(1) Iwasaki -26.0729227(1) 9.0153353(1) 3.10613307(2) DBW2 -11.5127475(2) 4.9953067(2) 1.35177237(2)

( 1,1) ( 1,2) ( 1,3) Table 2.8: The coeﬃcients εm− ,εm− ,εm− (Eq. (2.44)) for diﬀerent actions.

2.5 Corrections to Fermion Bilinear Operators

In the context of this work we also study the (a2)correctionstoGreen’sfunctionsof O ¯ local fermion operators (currents) that have the form OΓ = z Ψ(z)ΓΨ(z). We restrict ourselves to forward matrix elements (i.e. 2-point Green’s! functions, zero momentum operator insertions). The symbol Γcorresponds to the following set of products of the Euclidean Dirac matrices:

1 Γ S, P, V, A, T 11,γ5,γ ,γ5γ ,γ5σ ; σ = [γ ,γ ], (2.45) ∈{ }≡{ µ µ µν } µν 2 µ ν for the scalar OS, pseudoscalar OP ,vectorOV , axial OA and tensor OT operator, respectively. We also considered the tensor operator O ,correspondingtoΓ=T ′ σ and T ′ ≡ µν checked that the Green’s function coincides with that of OT ; this is a nontrivial check for our calculational procedure. The exact relationship between the amputated 2-point

Green function ΛT and ΛT ′ is summarized in the following equation:

µν 1 µ′ ν′ Λ = ϵµνµ ν Λ . (2.46) T −2 ′ ′ T ′ &µ′ ν′ The matrix elements of the above set of fermion bilinear operators can be obtained by an analogous expression to Eq. (2.37):

π 4 a,f ¯ b,g a d p ab fg ip(x y) 1 pert. 1 Ψα (x)OΓΨβ (y) = 4 δ δ e − ΛΓ (p) , (2.47) ⟨ ⟩ π (2π) i/p◦ + M(p) i/p◦ + M(p) 1− a 4 5αβ Chapter 2. (a2) Corrections to the One-Loop Fermion Propagator and O 44 Bilinear Operators

pert. where ΛΓ (p) is the amputated 2-point Green’s function of each operator OΓ, in momentum space: Λpert.(p)= Ψa,f (p) Ψ¯ (p)ΓΨ(p) Ψ¯ a,f (p) amp.. (2.48) Γ ⟨ α′ β′ ⟩ / 0 The only one-particle irreducible Feynman diagram that enters the calculation of the above operators is shown in Fig. 2.4.

Figure 2.4: One-loop diagram contributing to the bilinear operators. A wavy (solid) line represents gluons (fermions). A cross denotes the Dirac matrices Γ.

In the next pages, we provide the (a) corrections to the one-loop amputated 2-point pert. O correlation functions ΛΓ (p). Our ﬁnal results are given as a polynomial of cSW, in a general covariant gauge. Since their dependence on the Symanzik parameters, ci,cannot be written in a closed form, as in the case of the quark propagator we will tabulate the numerical coeﬃcients for a variety of choices for ci, in order to cover a range of values that are used in both perturbative calculations and numerical simulations.

Scalar and Pseudoscalar: • We begin with the (a2)correctedexpressionforΛpert.(p); including the tree-level O S term, we obtain:

pert. 2 (0,1) (0,2) (0,3) 2 ΛS (p)=1+˜g εS +5.79200956(5) λ + εS cSW + εS cSW ) ln(a2p2)(3+λ) − * + aip g˜2 ε(1,1) 3.93575928(1) λ + ε(1,2) c + ε(1,3) c2 ̸ S − S SW S SW ) 3 3 + + λ + c ln(a2p2) 2 2 SW 2 3 * + a2 p2 g˜2 ε(2,1) 2.27358943(5) λ + ε(2,2) c + ε(2,3) c2 S − S SW S SW ) 1 3 3 + + λ + c ln(a2p2) −4 4 2 SW 2 3 * p4 13 C λ + a2 µ µ g˜2 + 2 p2 24 2 − 8 ! ) * + (a3g2,g4). (2.49) O

(0,i) (1,i) (2,i) The numerical coeﬃcients εS , εS and εS with their systematic errors are presented in Tables 2.9, 2.10 and 2.11, respectively. 2.5. Corrections to Fermion Bilinear Operators 45

(0,1) (0,2) (0,3) Action εS εS εS Plaquette 0.30799634(6) 9.9867847(2) 0.01688643(6) Symanzik 0.58345905(5) 8.8507071(1) -0.12521126(5) TILW (8.60) 0.7016277(1) 8.0838748(2) -0.20597818(2) TILW (8.45) 0.7049818(1) 8.0538938(2) -0.20881716(3) TILW (8.30) 0.7094599(1) 8.0124083(2) -0.21270530(4) TILW (8.20) 0.7121516(1) 7.9865805(2) -0.21510214(1) TILW (8.10) 0.7155260(1) 7.95316909(7) -0.21817689(4) TILW (8.00) 0.7195566(1) 7.9115477(2) -0.22196498(3) Iwasaki 0.74092360(2) 6.9016820(2) -0.29335071(4) DBW2 -0.0094234(5) 4.0385802(2) -0.35869680(4)

(0,i) Table 2.9: The coeﬃcients εS (Eq. (2.49)) for diﬀerent actions.

(1,1) (1,2) (1,3) Action εS εS εS Plaquette 0.6586287(1) -4.20298580(6) -1.286053869(4) Symanzik 0.33939970(4) -3.76353718(6) -1.150059945(4) TILW (8.60) 0.15294656(6) -3.44266838(3) -1.058108055(3) TILW (8.45) 0.1463203(2) -3.42960982(2) -1.054472092(1) TILW (8.30) 0.13723798(4) -3.41147256(2) -1.049435521(2) TILW (8.20) 0.1316347(2) -3.40014057(3) -1.046296634(2) TILW (8.10) 0.1244419(2) -3.38543682(5) -1.0422330688(8) TILW (8.00) 0.1155729(2) -3.36704753(5) -1.037165442(1) Iwasaki -0.05097214(7) -2.88571027(1) -0.909503374(3) DBW2 -0.1248521(3) -1.15247167(2) -0.53943631(1)

(1,i) Table 2.10: The coeﬃcients εS (Eq. (2.49)) for diﬀerent actions.

(2,1) (2,2) (2,3) Action εS εS εS Plaquette 2.60041308(7) -4.15080331(7) 0.17641091(2) Symanzik 2.3547298(2) -3.85277871(9) 0.196461884(5) TILW (8.60) 2.1940370(1) -3.6339313(1) 0.210560987(1) TILW (8.45) 2.1881285(8) -3.6249313(5) 0.21113016(1) TILW (8.30) 2.1800126(2) -3.61241893(3) 0.21191990(1) TILW (8.20) 2.1749945(5) -3.60459353(6) 0.21241288(1) TILW (8.10) 2.16854108(7) -3.59443262(3) 0.21305190(1) TILW (8.00) 2.1605653(8) -3.58171175(4) 0.21385016(2) Iwasaki 2.02123300(8) -3.23459547(4) 0.234502732(7) DBW2 2.3731619(3) -1.9332087(1) 0.2953480(3)

(2,i) Table 2.11: The coeﬃcients εS (Eq. (2.49)) for diﬀerent actions. Chapter 2. (a2) Corrections to the One-Loop Fermion Propagator and O 46 Bilinear Operators

pert. (i,j) Next, we turn to ΛP (p), where Symanzik dependent coeﬃcients, εP ,aretabulated in Table 2.12. The pseudoscalar operator is free of (a1)terms;moreover,allcontribu- O tions linear in cSW vanish:

pert. 5 5 2 (0,1) (0,2) 2 ΛP (p)=γ + γ g˜ εP +5.79200956(5) λ + εP cSW ) ln(a2p2)(3+λ) − * + a2 p2 γ5 g˜2 ε(2,1) 0.83810121(5) λ + ε(2,2) c2 P − P SW ) 1 λ + + ln(a2p2) −4 4 2 3 * p4 13 C λ + a2 µ µ γ5 g˜2 + 2 p2 24 2 − 8 ! ) * + (a3g2,g4). (2.50) O

(0,1) (0,2) (2,1) (2,2) Action εP εP εP εP Plaquette 9.95102761(8) 3.43328275(3) 0.84419938(7) -0.25823485(3) Symanzik 8.7100837(1) 2.98705498(3) 0.70640549(6) -0.27556247(3) TILW (8.60) 7.8777986(1) 2.69129130(3) 0.65172716(6) -0.28766479(1) TILW (8.45) 7.84510495(6) 2.67986902(3) 0.65030355(6) -0.28812231(2) TILW (8.30) 7.79983766(8) 2.66408156(3) 0.64842824(5) -0.28875327(2) TILW (8.20) 7.77163793(9) 2.65426331(3) 0.64731714(8) -0.28914474(2) TILW (8.10) 7.73514046(6) 2.64157327(3) 0.64594118(5) -0.28965017(1) TILW (8.00) 7.6896423(1) 2.62578350(2) 0.64432843(6) -0.29027771(3) Iwasaki 6.55611308(7) 2.25383382(3) 0.66990790(5) -0.30221183(3) DBW2 2.9781769(6) 1.24882665(4) 1.5569125(1) -0.3362271(2)

(0,i) (2,i) Table 2.12: The coeﬃcients εP and εP (Eq. (2.50)) for diﬀerent actions.

One might attempt to use the (a) corrections computed above in order to devise O an improved operator, with suppressed ﬁnite-a artifacts; it should be noted, however, that improvement by means of local operators, as permitted by Quantum Field Theory, is not suﬃcient to warrant a complete cancellation of (a2)termsinGreen’sfunctions, O since the latter contain also terms with non-polynomial momentum dependence, such as p4 /p2.Thus,atbest,onecanachievefull (a2) improvement only on-shell, or µ µ O approximate! improvement near a given reference momentum scale. Such non-polynomial terms are not present at (a1). This comment applies also to the remaining operators we O examine below. 2.5. Corrections to Fermion Bilinear Operators 47

Vector and Axial: • The (a2)correctedexpressionsforΛpert.(p)andΛpert.(p) are more complicated, com- O V A pared to the scalar and pseudoscalar amputated Green’s functions, in the sense that momentum dependence assumes a variety of functional forms; this fact also introduces several coeﬃcients which depend on the Symanzik parameters. The following expression concerns the vector operator:

pp Λpert.(p)=γ + ̸ µ g˜2 2 λ V µ p2 − ) * + γ g˜2 ε(0,1) +4.79200956(5) λ + ε(0,2) c + ε(0,3) c2 λ ln(a2p2) µ V V SW V SW − ) * + aip g˜2 ε(1,1) 0.93575928(1) λ + ε(1,2) c + ε(1,3) c2 µ V − V SW V SW ) +( 3+λ +3c ) ln(a2p2) − SW * λ + a2 γ p2 g˜2 ε(2,1) + + ε(2,2) c + ε(2,3) c2 µ µ V 8 V SW V SW ) 53 11 + + C ln(a2p2) −120 10 2 2 3 * + a2 γ p2 g˜2 ε(2,4) 0.8110353(1) λ + ε(2,5) c + ε(2,6) c2 µ V − V SW V SW ) 11 c C λ 5 c2 + 1 2 + c + SW ln(a2p2) 240 − 2 − 60 8 − 12 SW 4 2 3 * + a2 pp g˜2 ε(2,7) +0.2436436(1) λ + ε(2,8) c + ε(2,9) c2 ̸ µ V V SW V SW ) 149 C λ c c2 + c 2 + + SW + SW ln(a2p2) −120 − 1 − 30 4 6 2 2 3 * pp3 1 2 λ + a2 ̸ µ g˜2 + C + p2 − 60 5 2 12 ) * p p 101 11 λ + a2 ̸ 3 µ g˜2 + C + p2 − 60 15 2 3 ) * p4 3 C 5 + a2 γ ρ ρ g˜2 + 2 + λ µ p2 80 10 48 ! ) * 4 ppµ p 3 C 5 + a2 ̸ ρ ρ g˜2 2 λ (p2)2 − 40 − 5 − 24 ! ) * + (a3g2,g4). (2.51) O

(i,j) The numerical values of εV for diﬀerent Symanzik choices are given in Tables 2.13 - 2.17. Chapter 2. (a2) Corrections to the One-Loop Fermion Propagator and O 48 Bilinear Operators

(0,1) (0,2) (0,3) Action εV εV εV Plaquette 3.97338480(2) -2.49669620(4) 0.85409908(1) Symanzik 3.57961385(3) -2.21267683(2) 0.77806655(1) TILW (8.60) 3.33413815(2) -2.02096865(4) 0.72431736(1) TILW (8.45) 3.32483844(4) -2.01347343(2) 0.72217154(1) TILW (8.30) 3.31200609(4) -2.00310205(2) 0.719196712(9) TILW (8.20) 3.30403798(2) -1.99664511(3) 0.71734137(2) TILW (8.10) 3.29375434(2) -1.98829232(3) 0.71493754(1) TILW (8.00) 3.28098129(5) -1.97788691(3) 0.71193712(2) Iwasaki 2.98283189(2) -1.72542048(4) 0.63679613(2) DBW2 2.25812410(4) -1.00964505(3) 0.40188086(2)

(0,i) Table 2.13: The coeﬃcients εV (Eq. (2.51)) for diﬀerent actions.

(1,1) (1,2) (1,3) Action εV εV εV Plaquette 2.7109817(1) -1.84813992(2) -0.39052850(2) Symanzik 2.09743725(3) -1.51877201(8) -0.385127257(2) TILW (8.60) 1.66056122(8) -1.26832395(9) -0.37828578(1) TILW (8.45) 1.64290440(2) -1.2579161(2) -0.37793187(2) TILW (8.30) 1.61839348(4) -1.24343083(4) -0.377431044(6) TILW (8.20) 1.60308667(2) -1.23436329(2) -0.37711264(1) TILW (8.10) 1.58323443(3) -1.22257851(3) -0.376693884(4) TILW (8.00) 1.55841933(4) -1.20780827(3) -0.3761606511(4) Iwasaki 0.9074321(1) -0.80352187(4) -0.356005234(3) DBW2 -1.4498098(4) 0.8826550(3) -0.264655885(7)

(1,i) Table 2.14: The coeﬃcients εV (Eq. (2.51)) for diﬀerent actions.

(2,1) (2,2) (2,3) Action εV εV εV Plaquette 1.5541024(2) 0.32907377(4) -0.0060202576(6) Symanzik 1.6762868(2) 0.22601986(5) 0.02822949(2) TILW (8.60) 1.6366127(1) 0.16979077(1) 0.04247965(1) TILW (8.45) 1.63378530(3) 0.16772628(3) 0.04300929(5) TILW (8.30) 1.6297142(1) 0.1648845(3) 0.04374032(7) TILW (8.20) 1.62708722(6) 0.1631247(1) 0.04419483(2) TILW (8.10) 1.6235738(1) 0.16085909(8) 0.044779126(5) TILW (8.00) 1.6190247(1) 0.15805313(3) 0.04550457(5) Iwasaki 1.4573118(1) 0.0858961(2) 0.07934994(2) DBW2 -1.1604825(4) -0.0504803(3) 0.13992474(3)

Table 2.15: The coeﬃcients ε(2,1) ε(2,3) (Eq. (2.51)) for diﬀerent actions. V − V 2.5. Corrections to Fermion Bilinear Operators 49

(2,4) (2,5) (2,6) Action εV εV εV Plaquette 0.2500659(2) 0.8859920(1) -0.300364436(2) Symanzik 0.0214112(1) 0.8342659(2) -0.28736163(1) TILW (8.60) -0.1054509(1) 0.79278089(1) -0.27498942(2) TILW (8.45) -0.1100958(1) 0.791026749(4) -0.27444757(3) TILW (8.30) -0.1164845(1) 0.78858219(4) -0.27369011(3) TILW (8.20) -0.1204393(1) 0.78705002(2) -0.27321405(1) TILW (8.10) -0.1255286(1) 0.785056619(7) -0.27259292(1) TILW (8.00) -0.1318272(2) 0.78255494(3) -0.27181092(1) Iwasaki -0.2668492(1) 0.712786719(6) -0.25078366(2) DBW2 -0.1528741(6) 0.42190739(4) -0.13978037(7)

Table 2.16: The coeﬃcients ε(2,4) ε(2,6) (Eq. (2.51)) for diﬀerent actions. V − V

(2,7) (2,8) (2,9) Action εV εV εV Plaquette 1.27887765(9) 0.27776135(2) -0.35475044(2) Symanzik 1.03773908(9) 0.28969451(4) -0.302816648(5) TILW (8.60) 0.89908242(7) 0.2930171(2) -0.26063939(9) TILW (8.45) 0.89400856(7) 0.2930984(2) -0.25886703(3) TILW (8.30) 0.88703859(9) 0.29320522(1) -0.25639844(3) TILW (8.20) 0.88273007(3) 0.2932689(2) -0.2548522(1) TILW (8.10) 0.87719053(7) 0.2933471(2) -0.25284039(1) TILW (8.00) 0.87034685(8) 0.29343883(9) -0.25031691(5) Iwasaki 0.76263373(2) 0.29755270(5) -0.184270928(8) DBW2 1.7371355(5) 0.2960594(1) 0.10831780(4)

Table 2.17: The coeﬃcients ε(2,7) ε(2,9) (Eq. (2.51)) for diﬀerent actions. V − V Chapter 2. (a2) Corrections to the One-Loop Fermion Propagator and O 50 Bilinear Operators

The amputated Green’s functions of the axial operator is given by the expression below:

γ5 pp Λpert.(p)=γ5 γ + ̸ µ g˜2 2 λ A µ p2 − ) * 5 2 (0,1) (0,2) (0,3) 2 + γ γµ g˜ εA +4.79200956(5) λ + εA cSW + εA cSW ) λ ln(a2p2) − * + aiγ5 (γ p p )˜g2 ε(1,1) 2.93575928(1) λ + ε(1,2) c + ε(1,3) c2 µ ̸ − µ A − A SW A SW ) + λ ln(a2p2) * λ + a2 γ5 γ p2 g˜2 ε(2,1) + + ε(2,2) c + ε(2,3) c2 µ µ A 8 A SW A SW ) 53 11 + + C ln(a2p2) −120 10 2 2 3 * + a2 γ5 γ p2 g˜2 ε(2,4) 1.7465235(1) λ + ε(2,5) c + ε(2,6) c2 µ A − A SW A SW ) 109 c C 5 7 c2 + 1 2 + λ + c SW ln(a2p2) −240 − 2 − 60 8 12 SW − 4 2 3 * + a2 γ5 pp g˜2 ε(2,7) +1.1146200(1) λ + ε(2,8) c + ε(2,9) c2 ̸ µ A A SW A SW ) 91 C 3 5 c2 + c 2 λ c SW ln(a2p2) 120 − 1 − 30 − 4 − 6 SW − 2 2 3 * pp3 1 2 λ + a2 γ5 ̸ µ g˜2 + C + p2 − 60 5 2 12 ) * p p 101 11 λ + a2 γ5 ̸ 3 µ g˜2 + C + p2 − 60 15 2 3 ) * p4 3 C 5 + a2 γ5 γ ρ ρ g˜2 + 2 + λ µ p2 80 10 48 ! ) * 4 ppµ p 3 C 5 + a2 γ5 ̸ ρ ρ g˜2 2 λ (p2)2 − 40 − 5 − 24 ! ) * + (a3g2,g4). (2.52) O

Eq. (2.51) and Eq. (2.52) have many similar terms, among them the coeﬃcients

ε(0,2) = ε(0,2),ε(0,3) = ε(0,3). (2.53) A − V A − V

(i,j) The rest of the coeﬃcients εA appear in Tables 2.18 - 2.21. 2.5. Corrections to Fermion Bilinear Operators 51

(0,1) (1,1) (1,2) (1,3) Action εA εA εA εA Plaquette -0.84813073(8) 1.34274645(8) -1.71809242(4) 0.130176166(7) Symanzik -0.48369852(8) 0.92541220(1) -1.54604828(4) 0.128375752(4) TILW (8.60) -0.25394726(8) 0.65548476(2) -1.425683888(5) 0.1260952614(2) TILW (8.45) -0.2452231(1) 0.64518173(2) -1.42093097(4) 0.125977289(1) TILW (8.30) -0.2331828(1) 0.63095649(2) -1.41434862(2) 0.125810348(1) TILW (8.20) -0.22570522(9) 0.62211877(3) -1.41024744(6) 0.125704213(3) TILW (8.10) -0.21605288(9) 0.61070615(5) -1.40493790(6) 0.125564628(3) TILW (8.00) -0.20406156(8) 0.59652190(3) -1.39831769(4) 0.125386884(3) Iwasaki 0.0752372(1) 0.2684958(1) -1.238019617(7) 0.1186684108(9) DBW2 0.7643240(1) -0.56650487(5) -0.75581589(7) 0.088218628(3)

(0,1) (1,i) Table 2.18: The coeﬃcients εA and εA (Eq. (2.52)) for diﬀerent actions.

(2,1) (2,2) (2,3) Action εA εA εA Plaquette 0.3879068(1) 1.85116980(8) -0.093094486(8) Symanzik 0.29616583(7) 1.7629637(2) -0.11136345(3) TILW (8.60) 0.2504164(1) 1.66189795(8) -0.11840825(2) TILW (8.45) 0.2483248(2) 1.65782471(1) -0.118658311(8) TILW (8.30) 0.2453697(2) 1.6521811(1) -0.11900206(4) TILW (8.20) 0.24349132(9) 1.6486651(1) -0.11921512(2) TILW (8.10) 0.24102344(9) 1.64410143(6) -0.11948757(3) TILW (8.00) 0.2378781(2) 1.63840648(4) -0.11982438(1) Iwasaki 0.05917686(4) 1.5707047(2) -0.13932655(1) DBW2 -2.2341918(4) 1.22932319(6) -0.17119304(8)

Table 2.19: The coeﬃcients ε(2,1) ε(2,3) (Eq. (2.52)) for diﬀerent actions. A − A

(2,4) (2,5) (2,6) Action εA εA εA Plaquette 1.6350438(1) -1.59945524(6) 0.333900263(8) Symanzik 1.3008790(1) -1.48761993(3) 0.314172576(5) TILW (8.60) 1.0558520(1) -1.39733614(3) 0.298455333(3) TILW (8.45) 1.0461303(2) -1.39361896(4) 0.297787700(5) TILW (8.30) 1.0326658(1) -1.38845204(4) 0.296856720(5) TILW (8.20) 1.0242768(2) -1.385221720(6) 0.29627287(1) TILW (8.10) 1.0134156(1) -1.381025878(8) 0.295512788(4) TILW (8.00) 0.9998744(1) -1.37577372(4) 0.29455820(2) Iwasaki 0.6845753(1) -1.24800562(3) 0.26827353(2) DBW2 0.0967251(2) -0.735419342(8) 0.14738921(4)

Table 2.20: The coeﬃcients ε(2,4) ε(2,6) (Eq. (2.52)) for diﬀerent actions. A − A Chapter 2. (a2) Corrections to the One-Loop Fermion Propagator and O 52 Bilinear Operators

(2,7) (2,8) (2,9) Action εA εA εA Plaquette 0.41758917(4) 0.395847810(9) 0.31972188(2) Symanzik 0.596637529(2) 0.33473715(4) 0.27870681(2) TILW (8.60) 0.72468673(3) 0.29351532(2) 0.24270425(2) TILW (8.45) 0.73021636(8) 0.29171961(4) 0.24115534(2) TILW (8.30) 0.73795096(1) 0.28920817(4) 0.23899366(4) TILW (8.20) 0.74281724(8) 0.28762824(2) 0.237637074(4 TILW (8.10) 0.74916398(5) 0.28556947(3) 0.235869441(2 TILW (8.00) 0.75716237(6) 0.28297665(3) 0.233647603(8 Iwasaki 1.05772129(4) 0.18672220(2) 0.17428813(3) DBW2 3.4449465(4) -0.22085461(4) -0.10748502(5)

Table 2.21: The coeﬃcients ε(2,7) ε(2,9) (Eq. (2.52)) for diﬀerent actions. A − A 2.5. Corrections to Fermion Bilinear Operators 53

Tensor and Tensor ′: • The remaining Green’s functions that we computed are those corresponding to the 5 tensor bilinears (T = γ σµν , T ′ = σµν ), which are the most complicated of all the operators that we studied. Clearly, the Green’s functions Λpert.(p)andΛpert.(p), corresponding T T ′ to T and T ′, coincide numerically, even though this fact is not immediately apparent from their algebraic forms. In fact, we computed both Λpert.(p)andΛpert.(p) in two T T ′ distinct calculations; their numerical coincidence constitutes a rather nontrivial check of our results. For the reader’s convenience, we present below both tensor Green’s functions:

pert. 5 5 2 (0,1) (0,2) (0,3) 2 ΛT (p)=γ σµν + γ σµν g˜ εT +3.79200956(5) λ + εT cSW + εT cSW ) +(1 λ) ln(a2p2) − * (γ p γ p ) + aiγ5 ν µ − µ ν g˜2 ε(1,1) +3.87151852(5) λ + ε(1,2) c + ε(1,3) c2 2 T T SW T SW ) +(3 2λ c ) ln(a2p2) − − SW * 2 2 γµ γν p γν γµ p λ + a2 γ5 µ − ν g˜2 ε(2,1) + + ε(2,2) c + ε(2,3) c2 2 T 4 T SW T SW / 0 ) * (γ pp γ pp ) + a2 γ5 ν ̸ µ − µ ̸ ν g˜2 ε(2,4) +0.62097643(2) λ + ε(2,5) c + ε(2,6) c2 2 T T SW T SW ) +(2 λ c ) ln(a2p2) − − SW * + a2 γ5 σ p2 g˜2 ε(2,7) 0.7839694(1) λ + ε(2,8) c + ε(2,9) c2 µν T − T SW T SW ) 1 C c + c + 2 SW ln(a2p2) 12 − 1 3 − 2 2 3 * p4 1 C λ + a2 γ5σ ρ ρ g˜2 + 2 + µν p2 − 3 2 3 ! ) * 3 3 p pν p pµ 17 2 + a2 γ5 µ − ν g˜2 + C 2 p2 3 3 2 / 0 ) * (γ p p γ p p ) 17 C + a2 γ5 ν ̸ 3 µ − µ ̸ 3 ν g˜2 + 2 2 p2 6 3 ) * 3 3 γν pp γµ pp 1 λ + a2 γ5 ̸ µ − ̸ ν g˜2 + C + 2 p2 − 2 2 2 / 0 ) * 2 2 γµ pp pν γν ppµ p 7 4 λ + a2 γ5 ̸ µ − ̸ ν g˜2 C 2 p2 − 3 − 3 2 − 2 / 0 ) * + (a3g2,g4). (2.54) O Chapter 2. (a2) Corrections to the One-Loop Fermion Propagator and O 54 Bilinear Operators

(i,j) The coeﬃcients εT are tabulated in Tables 2.22 - 2.26.

(0,1) (0,2) (0,3) Action εT εT εT Plaquette 0.37366536(7) -1.66446414(3) -0.5750281973(1) Symanzik 0.51501972(4) -1.47511786(3) -0.4769739579(4) TILW (8.60) 0.62355617(4) -1.34731246(3) -0.4142188616(3) TILW (8.45) 0.62806240(5) -1.34231565(2) -0.411841977(2) TILW (8.30) 0.63433260(4) -1.33540138(3) -0.408562708(1) TILW (8.20) 0.63825692(2) -1.33109674(2) -0.4065268581(5) TILW (8.10) 0.64335655(7) -1.32552819(2) -0.40389939818(7) TILW (8.00) 0.64974666(4) -1.31859128(1) -0.4006364262(1) Iwasaki 0.82253993(3) -1.15028034(3) -0.3267471901(5) DBW2 1.5201736(4) -0.67309671(3) -0.1483549734(1)

(0,i) Table 2.22: The coeﬃcients εT (Eq. (2.54)) for diﬀerent actions.

(1,1) (1,2) (1,3) Action εT εT εT Plaquette -4.05372833(7) 1.866287582(5) -0.8573692476(6) Symanzik -3.0228493(1) 1.59558642(1) -0.7667066321(8) TILW (8.60) -2.31604591(8) 1.40180493(3) -0.7054053711(3) TILW (8.45) -2.28808611(8) 1.39406610(3) -0.7029813946(7) TILW (8.30) -2.2493499(1) 1.38333610(2) -0.699623681(1) TILW (8.20) -2.22520543(5) 1.37664335(3) -0.6975310900(4) TILW (8.10) -2.19394066(3) 1.36797022(1) -0.6948220459(5) TILW (8.00) -2.15494125(2) 1.357142525(7) -0.691443627(2) Iwasaki -1.17592792(3) 1.087913642(4) -0.6063355831(7) DBW2 2.0163147(3) 0.15488056(9) -0.359624204(3)

(1,i) Table 2.23: The coeﬃcients εT (Eq. (2.54)) for diﬀerent actions.

(2,1) (2,2) (2,3) Action εT εT εT Plaquette 2.3328621(2) -1.52209604(8) 0.23683195(1) Symanzik 2.4912319(2) -1.53694399(3) 0.26295051(2) TILW (8.60) 2.46020883(9) -1.49210713(6) 0.26752597(2) TILW (8.45) 2.4578347(1) -1.49009843(6) 0.26767225(1) TILW (8.30) 2.4544221(1) -1.48729616(1) 0.26787315(2) TILW (8.20) 2.45222528(7) -1.48554047(6) 0.26799811(3) TILW (8.10) 2.44929140(3) -1.48324231(5) 0.26815475(7) TILW (8.00) 2.44550431(4) -1.48035333(8) 0.26834751(3) Iwasaki 2.3441345(2) -1.4848088(1) 0.30172406(2) DBW2 1.3013094(2) -1.2798033(3) 0.3475909(1)

Table 2.24: The coeﬃcients ε(2,1) ε(2,3) (Eq. (2.54)) for diﬀerent actions. T − T 2.5. Corrections to Fermion Bilinear Operators 55

(2,4) (2,5) (2,6) Action εT εT εT Plaquette -2.02795509(9) 0.11808647(1) 0.07250824(3) Symanzik -1.55221265(9) 0.04504264(4) 0.07020813(1) TILW (8.60) -1.24840831(2) 0.00049825(7) 0.06338729(2) TILW (8.45) -1.2361662(1) -0.00137858(3) 0.06309676(1) TILW (8.30) -1.21916512(8) -0.0039972(1) 0.06269264(2) TILW (8.20) -1.20854210(4) -0.00564058(5) 0.06244010(6) TILW (8.10) -1.19476762(5) -0.00777755(2) 0.06211059(4) TILW (8.00) -1.17754207(2) -0.0104623(1) 0.06169762(2) Iwasaki -0.6509124(1) -0.11083048(4) 0.06071268(8) DBW2 1.4802111(2) -0.51691409(6) 0.03446776(7)

Table 2.25: The coeﬃcients ε(2,4) ε(2,6) (Eq. (2.54)) for diﬀerent actions. T − T

(2,7) (2,8) (2,9) Action εT εT εT Plaquette 0.3932905(2) 1.10184617(6) -0.02744360(1) Symanzik 0.1467998(1) 1.03762644(2) -0.03500219(3) TILW (8.60) -0.0037125(1) 0.97880659(1) -0.03818404(3) TILW (8.45) -0.0094312(1) 0.97633518(7) -0.038311849(6) TILW (8.30) -0.0173242(2) 0.9728945(1) -0.03848962(5) TILW (8.20) -0.0222263(1) 0.970740561(4) -0.03860084(3) TILW (8.10) -0.0285551(2) 0.9679382(1) -0.03874467(2) TILW (8.00) -0.0364177(1) 0.96442476(2) -0.03892456(3) Iwasaki -0.2049940(1) 0.88259379(5) -0.04896801(2) DBW2 -0.3118850(6) 0.51292384(2) -0.07146761(1)

Table 2.26: The coeﬃcients ε(2,7) ε(2,9) (Eq. (2.54)) for diﬀerent actions. T − T Chapter 2. (a2) Corrections to the One-Loop Fermion Propagator and O 56 Bilinear Operators

The amputated Green’s functions of the T ′ operator is given by the expression below:

pert. 2 (0,1) (0,2) (0,3) 2 Λ (p)= σµν + σµν g˜ ε +3.79200956(5) λ + ε cSW + ε c T ′ T ′ T ′ T ′ SW ) +(1 λ) ln(a2p2) − * (γν pµ γµ pν)+σµν p 2 (1,1) (1,2) (1,3) 2 + ai − ̸ g˜ ε 3.87151852(5) λ + ε cSW + ε c 2 T ′ − T ′ T ′ SW ) +( 3+2λ + c ) ln(a2p2) − SW * 2 2 2 γµ γν pµ γν γµ pν 2 (2,1) λ (2,2) (2,3) 2 + a − g˜ ε + + ε cSW + ε c 2 T ′ 4 T ′ T ′ SW / 0 ) * 2 (γν ppµ γµ ppν ) 2 (2,4) (2,5) (2,6) 2 + a ̸ − ̸ g˜ ε 1.12097643(1) λ + ε cSW + ε c 2 T ′ − T ′ T ′ SW ) +( 2+λ + c ) ln(a2p2) − SW * 2 2 2 (2,7) (2,8) (2,9) 2 + a σµν p g˜ ε 1.2194576(1) λ + ε cSW + ε c T ′ − T ′ T ′ SW ) 11 C λ + c + 2 + ln(a2p2) −12 − 1 3 2 2 3 * p4 1 C λ + a2 σ ρ ρ g˜2 + 2 + µν p2 − 3 2 3 ! ) * 3 3 p pν p pµ 17 2 + a2 µ − ν g˜2 + C 2 p2 3 3 2 / 0 ) * (γ p p γ p p ) 17 C + a2 ν ̸ 3 µ − µ ̸ 3 ν g˜2 + 2 2 p2 6 3 ) * 3 3 γν pp γµ pp 1 λ + a2 ̸ µ − ̸ ν g˜2 + C + 2 p2 − 2 2 2 / 0 ) * 2 2 γµ pp pν γν ppµ p 7 4 λ + a2 ̸ µ − ̸ ν g˜2 C 2 p2 − 3 − 3 2 − 2 / 0 ) * + (a3g2,g4). (2.55) O

Several coeﬃcients εT ′ can be written in terms of εT (Eqs. (2.56) - (2.58)), while the rest are given in Tables 2.27 - 2.28:

ε(0,1) = ε(0,1),ε(0,2) = ε(0,2),ε(0,3) = ε(0,3), (2.56) T ′ T T ′ T T ′ T

ε(1,1) = ε(1,1),ε(1,2) = ε(1,2),ε(1,3) = ε(1,3), (2.57) T ′ − T T ′ − T T ′ − T ε(2,2) = ε(2,2),ε(2,3) = ε(2,3) ,ε(2,5) = ε(2,5),ε(2,6) = ε(2,6). (2.58) T ′ − T T ′ − T T ′ − T T ′ − T 2.6. Renormalization Constants in the RI-MOM Scheme 57

Action ε(2,1) ε(2,4) ε(2,7) T ′ T ′ T ′ Plaquette 0.00047095(5) -0.30537822(7) 1.4070324(1) Symanzik -0.26900984(7) -0.67000961(5) 1.0574111(1) TILW (8.60) -0.31218378(2) -0.89961674(2) 0.7765836(1) TILW (8.45) -0.31308657(6) -0.90858179(3) 0.7651952(1) TILW (8.30) -0.31426732(9) -0.92098981(2) 0.7493919(2) TILW (8.20) -0.31496667(9) -0.92871655(3) 0.7395280(1) TILW (8.10) -0.31580945(8) -0.93871436(2) 0.7267335(1) TILW (8.00) -0.31678921(7) -0.95117312(1) 0.7107479(2) Iwasaki -0.45213470(9) -1.24108756(9) 0.3465297(2) DBW2 -0.8461093(2) -1.9354110(1) -0.6289363(3)

Table 2.27: The coeﬃcients ε(2,1),ε(2,4),ε(2,7) (Eq. (2.55)) for diﬀerent actions. T ′ T ′ T ′

Action ε(2,8) ε(2,9) T ′ T ′ Plaquette 0.28175492(1) 0.054718244(7) Symanzik 0.24663315(4) 0.06136902(2) TILW (8.60) 0.232503859(8) 0.063885306(7) TILW (8.45) 0.2319752(1) 0.06397589(2) TILW (8.30) 0.2312449(1) 0.06410069(3) TILW (8.20) 0.23079061(2) 0.06417815(2) TILW (8.10) 0.2302057(1) 0.06427742(1) TILW (8.00) 0.22947913(7) 0.064400395(7) Iwasaki 0.19560470(5) 0.07153771(1) DBW2 0.13147908(9) 0.08509400(2)

Table 2.28: The coeﬃcients ε(2,8),ε(2,9) (Eq. (2.55)) for diﬀerent actions. T ′ T ′

2.6 Renormalization Constants in the RI-MOM Scheme

An operator renormalization constant (RC) can be thought of as the link between its matrix element, regularized on the lattice, and its renormalized continuum counterpart. The renormalization constants (RCs) of lattice operators are necessary ingredients in the prediction of physical probability amplitudes from lattice matrix elements. In this section we present the multiplicative renormalization constants, in the RI MOM scheme, of − pert. pert. quark ﬁeld (Zq )andquarkbilinearoperators(ZΓ ) obtained by using the perturbative 1 expressions of S− (p)andΛΓ(p). The RI MOM renormalization scheme consists in imposing that the forward ampu- − tated Green function ΛΓ(p), computed in the chiral limit and at a given (large Euclidean) scale p2 = µ2, is equal to its tree-level value. In practice, the renormalization condition is Chapter 2. (a2) Corrections to the One-Loop Fermion Propagator and O 58 Bilinear Operators implemented by requiring in the chiral limit that1:

1 1 Z− Z (p) =1, (p) Tr Λ (p) P , (2.60) q Γ VΓ |pρ=µρ VΓ ≡ 4 Γ · Γ ) * where PΓ are the Dirac projectors deﬁned as follows:

P P ,P ,P ,P ,P ,P 11,γ5,γ, γ5γ , γ5σ , σ ;(2.61) Γ ∈{ S P V A T T ′ }≡{ µ − µ − µν − µν } they are chosen to obey the relation Tr[Γ P ] 1. The traces are always taken over the · Γ ≡ spin indices. The quark ﬁeld RC Zq, which enters Eq. (2.60), is obtained by imposing, again in the chiral limit, the condition2:

1 1 i a ρ γρ sin(apρ) 1 Z− (p) =1, (p) Tr S− (p) . (2.63) q Vq |pρ=µρ Vq ≡−4 1 sin2(ap ) · 6 a2! ρ ρ 7 ! A very important issue is that the (a2)termsdependnotonlyonthemagnitude,p2, O 4 but also on the direction of the momentum, pρ,asmanifestedbythepresenceof ρ pρ: ! a2 2 p4 pert. i p 6 p3 a ρ ρ 1 4 2 4 ̸ − ̸ − q (p)= Tr 2 1+ 2 Spert.(p) + (a g ,g ). (2.64) V −4 6 p 4 3 !p 5 · 7 O

As a consequence, different renormalization prescriptions, involving different directions of the renormalization scale µρ = pρ,treatlatticeartifactsdifferently. 1 By implementing the pertubative expressions of S− (p)andΛΓ(p) in Eqs. (2.60) and

(2.63), we obtain the following RCs (for the special choices: cSW =0,Landaugauge λ = 0, and for tree-level Symanzik improved gluons):

Zpert. =1+˜g2 13.02327272(7) q − " 73 + a2 µ2(1.14716212(5) ln(a2 µ2)) (2.65) − 360 ) 4 µρ 157 + ρ (2.1064995(2) ln(a2 µ2)) + (a4 g2,g4), µ2 − 180 O ! *#

1 A simpler version of Eq. (2.60) is given by the relation:

1 1 tree 1 tree tree Zq− ZΓ Tr ΛΓ(p) ΛΓ = Tr ΛΓ ΛΓ , (2.59) 4 · pρ=µρ 4 · ) * ) * tree where ΛΓ is the tree-level value of ΛΓ(p). 2 Strictly speaking, the renormalization condition of Eq. (2.63) defines the so called RI′ scheme. In the original RI MOM scheme the quark field renormalization condition reads: − 1 1 i ∂Sq(p)− Z− − Tr γ =1. (2.62) q 16 µ ∂p ' µ (p2=µ2 The two schemes differ in the Landau gauge at the N2 LO. 2.6. Renormalization Constants in the RI-MOM Scheme 59

Zpert. =1+˜g2 13.60673177(9) + 3 ln(a2 µ2) S − " 17 + a2 µ2( 1.2075677(2) + ln(a2 µ2)) (2.66) − 360 ) 4 µρ 157 + ρ (1.6064995(2) ln(a2 µ2)) + (a4 g2,g4), µ2 − 180 O ! *# Zpert. =1+˜g2 21.7333564(1) + 3 ln(a2 µ2) P − " 17 + a2 µ2(0.44075663(8) + ln(a2 µ2)) (2.67) 360 ) 4 µρ 157 + ρ (1.6064995(2) ln(a2 µ2)) + (a4 g2,g4), µ2 − 180 O ! *# Zpert. =1+˜g2 16.60288657(8) V − " 7 + a2 µ2(1.1257509(1) ln(a2 µ2)) − 24 ) 76 + µ2 ( 2.7140259(2) + ln(a2 µ2)) (2.68) µ − 45 4 ρ µρ 157 2 2 + 2 (2.0773329(2) ln(a µ )) !µ − 180 4 2 4 323 µ 7 µµ µρ + µ + ρ + (a4 g2,g4), 180 µ2 120 (µ2)2 O ! *# Zpert. =1+˜g2 12.5395742(1) A − " 5 + a2 µ2( 0.1537169(1) + ln(a2 µ2)) − 24 ) 14 + µ2 ( 0.89280336(7) ln(a2 µ2)) (2.69) µ − − 45 4 ρ µρ 157 2 2 + 2 (2.0773329(2) ln(a µ )) !µ − 180 4 2 4 323 µ 7 µµ µρ + µ + ρ + (a4 g2,g4), 180 µ2 120 (µ2)2 O ! *# Zpert. =1+˜g2 13.53829244(8) ln(a2 µ2) T − − " 41 + a2 µ2(1.0003623(2) ln(a2 µ2)) − 120 ) +(µ2 + µ2)( 2.0217223(1) + ln(a2 µ2)) (2.70) µ ν − 4 ρ µρ 157 2 2 + 2 (2.4814995(2) ln(a µ )) !µ − 180 10 (µ2 + µ2)2 + µ ν + (a4 g2,g4), 9 µ2 O *# Zpert. =1+˜g2 13.53829244(8) ln(a2 µ2) T ′ − − " 79 + a2 µ2(0.0897510(2) + ln(a2 µ2)) 120 ) +(µ2 + µ2)( 0.20049989(5) ln(a2 µ2)) (2.71) µ ν − − 4 ρ µρ 157 2 2 + 2 (2.4814995(2) ln(a µ )) !µ − 180 10 (µ2 + µ2)2 + µ ν + (a4 g2,g4). 9 µ2 O *# Chapter 2. (a2) Corrections to the One-Loop Fermion Propagator and O 60 Bilinear Operators 2.7 Subtraction of the (a2 g2) Discretization Effects O pert. pert. 1 pert. The renormalization functions Zq and ZΓ ,asobtainedfromSpert− .(p)andΛΓ (p), differ from the corresponding expressions evaluated at (a0), by lattice artifact, which are O functions of (aµ)(µ:renormalizationscale),andvanishasa 0. At the nonzero values of → a employed in numerical simulations, these factors are quite important. Non-perturbative renormalization is an essential ingredient of lattice QCD calculations which aim at a percent level accuracy. Ideally, one would prefer a non-perturbative determination of renormalization functions; while this is oftenpossible,severalsourcesoferrormustbe dealt with. A very effective way to proceed is through a combination of perturbative and non-perturbative results. This procedure is carried out and explained in detail in Ref. [79, 80]. Briefly stated, non-perturbative data are “corrected” by the perturbative expressions 1 pert. Spert− .(p)andΛΓ (p), and then extrapolated towards small a.Asanillustrationofthis mixed determination, we show in Fig. 2.5 non-perturbative data for Zq and ZΓ,determined with the RI MOM method of Ref. [81], before and after the perturbative corrections. − The simulation results are obtained by using the Symanzik tree-level improved gluon action at β 2 N /g2 =3.9(whereN =3)andtheN =2(dynamicalupanddown ≡ c c f degenerated flavours) twisted mass fermionicactionatmaximaltwist,whichguarantees automatic (a)-improvement. The gauge field configurations and quark propagators are O generated by the ETM Collaboration. In the remaining of this section we summarize the subtraction procedure of the (a2g2) O discretization effects from non-perturbative RI MOM estimates, carried out in Ref. [79, − 80]. In the chiral limit, the twisted mass action is related to the standard Wilson fermionic action by a chiral transformation, under which quark composite operators behave in a definite and simple way. Therefore the RCs computed in Ref. [79, 80] and presented in the next pages, can be also employed to renormalize bilinear quark operator matrix elements computed with the same gluon action but standard (un-twisted) Wilson quarks. Once the non-perturbative RCs have been extrapolated to the chiral limit, their dependence on the renormalization scale is being investigate by evolving, at fixed coupling, the RCs to a reference scale µ =1/a( 2GeV ). This is done using: 0 ∼

Zq(a, µ0)=Rq(µ0,µ) Zq(a, µ) ,ZΓ(a, µ0)=RΓ(µ0,µ) ZΓ(a, µ) , (2.72)

where the evolution (scale down) functions Rq and RΓ are expressed in terms of the beta function β(αs)andoftheanomalousdimensionfunctionsγq(αs)andγΓ(αs)by:

αs(µ) γ (α ) αs(µ) γ (α ) R (µ ,µ)=exp q s dα ,R(µ ,µ)=exp Γ s dα . (2.73) q 0 β(α ) s Γ 0 β(α ) s 61αs(µ0) s 7 61αs(µ0) s 7

This functions are known, in the RI MOM scheme, at the N2LO for Z [31] and at the − T 2.7. Subtraction of the (a2 g2) Discretization Eﬀects 61 O

3 N LO for Zq, ZS and ZP [30]. Since ZV , ZA and the ratio ZP /ZS are scale independent, they have vanishing anomalous dimensions; thus for these quantities we have RΓ =1.

In the non-perturbative calculation, the RCs evolved to the reference scale µ0 =1/a maintain a dependence on the renormalization scale µ2 = p2 at which they have been initially computed. This dependence, which (at large enough p2) is mostly due to discretization effects, is kept in track by denoting these RCs as Z(1/a; a2p2), where the first variable indicates the renormalization scale, µ0 =1/a,andthesecondthedependenceon the initial momentum. 2 2 2 2 In Fig. 2.5 we show the results for all ZΓ(1/a; a p )andZq(1/a; a p )atβ =3.9asa 2 function of a2p◦ sin2(ap )(emptysymbols).Theresiduala2p2 dependence which ≡ ρ ρ is observed in these! results in the large momentum region is practically linear, suggesting that leading discretization effects are (a2p2). As illustrated in the plots, this dependence O is particularly pronounced in the case of the pseudoscalar RC ZP . In order to reduce the size of discretization errors, the authors of Ref. [79, 80] analytically subtract from the quark propagator and the amputated vertex functions the perturbative (g2a2) contributions, presented in the previous sections. Their definition O for the form factor ˜ (p) on the lattice, which is equivalent to Eq. (2.63) up to terms of Vq (a2), is: O

1 i γρ S− (p) 1 ˜ ◦ q(p) Tr ′ ,pρ sin(apρ) , (2.74) V ≡−4N(p) p◦ ≡ a 6 ρ ρ 7 & 2 3 where the sum ρ′ only runs over the Lorentz indices for which pρ is diﬀerent from zero and N(p)= ′ 1. Using our perturbative expression for the inverse quark propagator !ρ 1 pert. S− (p) the following form factor ˜ (p) is obtained, from which the quark ﬁeld RC Z ! Vq q is evaluated:

pert. ˜ (p)=1+˜g2 b(1) + b(2) ln(a2 p2) Vq q q " p4 + a2 p2(c(1) + c(2) ln(a2 p2)) + c(3) ρ ρ + (a4 g2,g4) . (2.75) q q q p2 O ) ! *# For the lattice actions used in their simulations, the coeﬃcients of Eq. (2.75) take the values:

7 b(1) = 13.02327272(7) ,b(2) =0,c(3) = , q − q q 240 2.07733285(2) 73 157/180 c(1) =1.14716212(5) + ,c(2) = . (2.76) q N(p) q −360 − N(p)

AresultsimilartoEq.(2.75)alsoholdsforthe ˜ (p) deﬁned in the following lines. VΓ Summation and normalization over all independent values of Lorentz indices (if any), of Chapter 2. (a2) Corrections to the One-Loop Fermion Propagator and O 62 Bilinear Operators

(1) (2) (1) (2) (3) Γ bΓ bΓ cΓ cΓ cΓ S0.58345905(5)-32.3547298(2)-1/41/2 P8.7100837(1)-30.70640549(6)-1/41/2 V -0.48369852(8) 0 1.5240798(1) -1/3 -125/288 A3.57961385(3)00.6999177(1)-1/3-125/288 T0.51501972(6)10.9724758(2)-13/36-161/216 T′ 0.51501972(6) 1 0.9724758(2) -13/36 -161/216

(i) (i) Table 2.29: Values of the coeﬃcients bΓ and cΓ entering the 1-loop expression (2.77) of the amputated projected Green functions Γ(p). Results are presented for the case of the tree-level Symanzik improvedV gluon action in the Landau gauge and the tm fermionic action at maximal twist. the amputated projected Green functions (p), deﬁned in Eq. (2.60), yields the simple VΓ and general expression:

pert. 1 ˜ (p) pert.(p) VΓ ≡ N VΓ Γ Lorentz & 2 (1) (2) 2 2 =1+˜g bΓ + bΓ ln(a p ) " p4 + a2 p2(c(1) + c(2) ln(a2 p2)) + c(3) ρ ρ + (a4 g2,g4), (2.77) Γ Γ Γ p2 O ) ! *# where: N N ,N ,N ,N ,N ,N 1, 1, 4, 4, 6, 6 . (2.78) Γ ∈{ S P V A T T ′ }≡{ } (i) (i) The values of the coeﬃcients bΓ and cΓ are collected in Table 2.7 for the lattice action used in Ref. [79, 80], namely the tree-level Symanzik improved gluon action in the Landau gauge and the tm fermionic action at maximal twist 3. Using Eqs. (2.77) and (2.75), we can deﬁne the subtracted amputated projected Green sub sub functions ˜ (p)andthesubtractedformfactor˜ (p)as: VΓ Vq

4 ◦ sub 2 (1) (2) 2 (3) ρ p ρ ˜ ˜ 2 2 ◦ 2 ◦ Γ (p)= Γ(p) g˜ a p (cΓ + cΓ ln(a p )) + cΓ 2 , V V − !p◦ ) 4 *# ◦ sub 2 2 ρ p ρ ˜ ˜ 2 2 ◦ (1) (2) 2 ◦ (3) q (p)= q(p) g˜ a p (cq + cq ln(a p )) + cq 2 , (2.79) V V − p◦ ) ! *# which are free of (g2a2)effects.Inthenumericalevaluationoftheperturbativecorrection O in Eq. (2.79), g2 is replaced with a simple boosted coupling [84], defined as g2 g2/ P , → ⟨ ⟩ where the average plaquette P is computed non-perturbatively. ⟨ ⟩ The effect of the subtraction (2.79) is also illustrated in Fig. 2.5, which shows that the

3 The same results are also valid for the standard Wilson fermionic action (the one that we have used in the previous sections), except for the exchange of the values of the vector (V) and axial (A) coefficients. 2.7. Subtraction of the (a2 g2) Discretization Effects 63 O a2p2 dependence of the RCs is significantly reduced by the perturbative correction. By fitting the RCs as:

2 2 2 2 2 ◦ 2 2 2 ◦ Zq(1/a; a p )=Zq(1/a)+λq a p ,ZΓ(1/a; a p )=ZΓ(1/a)+λΓ a p , (2.80)

2 2 ◦ in the large momentum region a p " 1 (with λq and λΓ just constrained to depend smoothly on g2), we find that the slopes are reduced, after the perturbative subtraction, 2 at the level of 10− or smaller for ZV , ZA, ZT and Zq.ForZS we find that the slope, 2 which is also about 10− before implementing the correction, increases slightly after the subtraction. For ZP ,ontheotherhand,theslopeisrather large before the subtraction; 2 i.e. λ 5 10− . As shown in Fig. 2.5, the effect of the subtraction is beneficial also in P ≃ · this case, but inadequate for correcting the bulk of the observed a2p2 dependence. This is not completely unexpected: when discretization effects are large, the subtraction of only the leading (g2a2)termsmaybenotsufficienttoreducethemtoanegligiblelevel. O Similar results, with approximately equal values of the slopes, are obtained at all three values of the gauge coupling, namely β =3.80, 3.90 and 4.05, which correspond to inverse 1 lattice spacing a− 2.0, 2.3and2.9GeV. ≃ One other possible use of our results is in constructing improved versions of the operators OΓ, with reduced lattice artifacts. In doing so, however, one must bear in mind that, unlike the (a1)case,correctionsto (a2) include expressions which are non-polynomial O O in the external momentum and, therefore, cannot be eliminated by introducing admix- tures of local operators. Full improvement can be achieved at best for on-shell matrix elements only. Chapter 2. (a2) Corrections to the One-Loop Fermion Propagator and O 64 Bilinear Operators

β=3.90 β=3.90 0,95 0,85

Zq ZS 0,90 0,80 Z corrected Zq corrected S

0,85 0,75

0,80 0,70

0,75 0,65

0,70 0,60

0,65 0,55 0 0,5 1 1,5 2 2,5 0,0 0,5 1,0 1,5 2,0 2,5 2 o 2 2 o 2 a p a p

β=3.90 β=3.90 0,65 0,85

ZP ZV 0,60 0,80 ZP corrected ZV corrected

0,55 0,75

0,50 0,70

0,45 0,65

0,40 0,60

0,35 0,55 0 0,5 1 1,5 2 2,5 0,0 0,5 1,0 1,5 2,0 2,5 2 o 2 2 o 2 a p a p

β=3.90 β=3.90 0,90 0,95

ZA ZT 0,90 0,85 ZA corrected ZT corrected

0,85 0,80

0,80

0,75 0,75

0,70 0,70

0,65 0,65 0,0 0,5 1,0 1,5 2,0 2,5 0,0 0,5 1,0 1,5 2,0 2,5 2 o 2 2 o 2 a p a p

2 2 2 2 Figure 2.5: The RCs Zq(1/a; a p ) and ZΓ(1/a; a p ) at β =3.9,evaluatedat the reference scale µ0 =1/a,plottedagainsttheinitialrenormalizationscale 2 a2p◦ sin2(ap ).Filledsquares(emptycircles)areresultsobtainedwith ≡ ρ ρ (without) the subtraction of the (g2a2) discretization eﬀects, computed in ! perturbation theory. The solid linesO are linear ﬁts to the data. Chapter 3

(a2) Corrections to the One-Loop O Matrix Elements of Four-Fermion ∆S =2Operators

Anumberofﬂavour-changingprocessesarecurrentlyunderstudyinLatticesimulations. Among the most common examples are the decay K ππ and K0–K¯ 0 oscillations. → From experimental evidence, we know that these weak processes violate CP symmetry. In theory, the calculation of the amount of CP violation requires knowledge of the Kaon

BK parameter. The non-perturbative determination of the BK parameter, via lattice simulations, requires the evaluation of suitable matrix elements of composite operators, made out of 4-femrion fields. In this chapter we calculate the (a2)correctionstotheamputatedGreen’sfunctions O of 4-fermion operators, in 1-loop Lattice Perturbation theory. The novel aspect of our calculations is that they are carried out to second order in the lattice spacing, (a2). Even O (a0) results did not exist in the literature for some of the actions of current interest. O We employ the Wilson/clover action for massless fermions (also applicable for the twisted mass action in the chiral limit) and the Symanzik improved action for gluons. Our calculations have been carried out in a general covariant gauge. Results have been obtained for several popular choices of values for the Symanzik coefficients (Plaquette, Tree-level Symanzik, Iwasaki, TILW and DBW2 action). We pay particular attention to ∆F =2operators(F stands for flavour: S, C, B), both Parity Conserving and Parity Violating. We study the mixing pattern of these operators, to (a2), using appropriate projectors. Our perturbative results for the corresponding O renormalization matrices are given as a function of a large number of parameters: coupling constant g,cloverparameter cSW,numberofcolorsNc, lattice spacing a,external momentum p and gauge parameter λ.

65 Chapter 3. (a2) Corrections to the One-Loop Matrix Elements of O 66 Four-Fermion ∆S =2Operators 3.1 Introduction

Although a relativistic field theory must be invariant under Lorentz transformations, it may also be invariant under the discrete symmetries: parity (P ), time reversal (T ), and charge conjugation (C). Time reversal sends (t, ⃗x) ( t, ⃗x), interchanging in this way → − the forward and backward light cones. Parity sends (t, ⃗x) (t, ⃗x), reversing in this way → − the handedness of space. Therefore a left-handed electron (eL−) is transformed under P into a right-handed electron (eR−). Under charge conjugation particles and antiparticles are interchanged, by conjugating all internal quantum numbers (e.g., Q Q for the →− electric charge). From experimental evidence until today, we know that gravitational, electromagnetic, and strong interactions are symmetric under T , P ,andC. The weak interactions violate C and P separately in the strongest possible way. For example, charged W bosons couple + + to eL− and to their CP-conjugate eR,buttoneithertheirC-conjugate eL nor their P - conjugate eR−.WhileweakinteractionsviolateC and P separately, CP is preserved in almost all weak interaction processes. Certain rare weak processes, all of which involve charged or neutral K, D, B,orBs mesons, violate CP and T symmetries. All observations indicate that the combination CPT is a perfect symmetry of nature. This is consistent with the CPT theorem, which states that, one cannot build a relativistically invariant quantum field theory with a Hermitian Hamiltonian that violates CPT. Within the Standard Model (SM) the only way that CP is violated is through the Kobayashi-Maskawa mechanism [85]. More specifically the mixing matrix (defined below) that describes the charged current weak interactions of quarks contains a single CP- violating phase parameter. Let us briefly describe how the aforementioned mixing matrix is obtained. In doing so we will use the shorthand notation:

u d U = ⎛ c ⎞ ,D= ⎛ s ⎞ , (3.1) t b ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ where U stands for up type quarks and D for down type quarks. In order to diagonalize the mass terms (i.e., M U and M D non diagonal complex matrices) of the quark sector in the SM Lagrangian, we rewrite the interaction eigenstates

(UL,R and DL,R ﬁelds) in-terms of the mass eigenstates (UL,R′ and DL,R′ ﬁelds):

U D UL,R′ = VL,RUL,R,DL,R′ = VL,RDL,R, (3.2) where V U and V D are 3 3unitarymatricessuchthat: L,R L,R ×

U U U U D D D D = V M (V )†, = V M (V )†. (3.3) M L R M L R 3.1. Introduction 67

Now the matrices U and D are diagonal and real. In the mass eigenstates basis the M M U D remaining part of the SM Lagrangian is independent of VR and VR ,whiletheonlypart U D that depends on VL and VL is the interaction term between W -bosons and quarks:

q gW + µ µ = (Wµ J ′ + + Wµ−J ′ ), (3.4) −LW ± √2 W W − where: µ µ U D µ µ D U J ′ + = U¯ ′ γ V (V )†D′ ,J′ = D¯ ′ γ V (V )†U ′ . (3.5) W L L L L W − L L L L The unitary 3 3matrix: ×

U D D U V = V (V )†,V = V (V )†,V(V )† = 11, (3.6) CKM L L CKM† L L CKM CKM is the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix for quarks. The elements of

VCKM are written as follows:

Vud Vus Vub VCKM = ⎛ Vcd Vcs Vcb ⎞ (3.7) V V V ⎜ td ts tb ⎟ ⎝ ⎠ The CKM matrix takes into account the mismatch between mass and interaction eigenstates. Ageneral3 3unitarymatrixdependsonthreerealanglesandsixphases.Amultipli- × cation of the quark mass eigenstates by a phase factor (identical for left and right-handed quarks) can be used to remove ﬁve phases. Thesinglephysicalphase,theKobayashi- Maskawa phase, that is left is responsible for all CP violation in meson decays in the SM. In fact one can show that all terms in the SM Lagrangian are CP invariant, except for the charged currents in Eq. (3.4). Only if the CKM matrix is real can the theory be CP invariant:

+ µ CP µ T Wµ J ′W + Wµ−J ′W VCKM =(VCKM)† (i.e., VCKM 3 3). (3.8) ←→ − ⇐⇒ ∈ℜ×

But this in not the case. One of the many possible parameterizations of the CKM matrix is the standard choice [86], used by the Particle Data Group:

iδ c12 c13 s12 c13 s13 e− iδ iδ VCKM = s c c s s e c c s s s e s c (3.9) ⎛ − 12 13 − 12 23 13 − 12 13 − 12 23 13 23 13 ⎞ s s c c s eiδ c s s c s eiδ c c ⎜ 12 13 − 12 23 13 − 12 13 − 12 23 13 23 13 ⎟ ⎝ ⎠ where c cos θ , s sin θ .Theanglesθ , θ ,andθ are real parameters, while eiδ ij ≡ ij ij ≡ ij 12 23 13 is the Kobayashi-Maskawa phase. The above parameters can be determined experimen- Chapter 3. (a2) Corrections to the One-Loop Matrix Elements of O 68 Four-Fermion ∆S =2Operators tally. For more information see Ref. [6]. The CKM matrix elements are fundamental parameters of the SM, thus their precise determination is important. The unitarity of the CKM matrix VCKM(VCKM)† = 11, can also be expressed in twelve distinct relations between the matrix elements. The six equations expressing orthogonality among diﬀerent rows or columns can be represented geometrically as triangles in the complex plane, all with the same area. The most commonly used unitarity triangle arises from:

VudVub∗ + VcdVcb∗ + VtdVtb∗ =0. (3.10)

We rescale the above equation by dividing each term by the best-known one, V V ∗; − cd cb each term in the equation can now be interpreted as one side of a triangle in the complex plane with vertices at (0, 0), (1, 0), (¯ρ, η¯), where:

V V ud ub∗ ρ¯ + iη¯. (3.11) − VcdVcb∗ ≡

Figure 3.1: Graphical representation of the rescaled unitarity constraint (triangle) of Eq. (3.10).

The angles of the unitary triangle illustrated in Fig. 3.1 are deﬁned by the following relations:

V V α φ arg td tb∗ , ≡ 2 ≡ −V V 2 ud ub∗ 3 V V β φ arg cd cb∗ , (3.12) ≡ 1 ≡ − V V 2 td tb∗ 3 V V γ φ arg ud ub∗ . ≡ 3 ≡ − V V 2 cd cb∗ 3 An important goal of flavour physics is to over-constrain the CKM elements. The CKM matrix elements can be most precisely determined by a global fit that uses all 3.1. Introduction 69 available measurements and imposes the SM constraints. The constraints implied by the unitarity of the CKM matrix significantly reduce the allowed range of some of the CKM elements. Fig. 3.2 illustrates the constraints on theρ ¯ η¯ plane from various − measurements and the result of a global fit. The shaded regions (95% confidence level) all overlap consistently around the global fit region.

1.5 excluded at CL > 0.95 excluded area has CL > 0.95 γ 1.0 ∆md & ∆ms sin 2β 0.5 ∆md α εK γ β

η 0.0 α

Vub -0.5 α

εK -1.0 γ sol. w/ cos 2β < 0 (excl. at CL > 0.95)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 ρ

Figure 3.2: Constraints on the ρ¯, η¯ plane.

Because the CKM matrix is not diagonal, the W ± bosons couple to quarks (mass eigenstates) of diﬀerent generations. Within the SM, this is the only source of ﬂavour changing quark interactions, which gives rise to the Flavour Changing Neutral Currents (FCNC), such us the neutral Kaon oscillation:

0 0 K # K¯ .

In the above process the K0 (with quark content ds¯ and strangeness +1) can turn into its antiparticle K¯ 0 (with quark content sd¯ and strangeness 1) via weak interactions. To − lowest-order the K0 K¯ 0 oscillation are represented schematically by the diagrams of − Fig. 3.3. Note that K0 and K¯ 0 are ﬂavour eigenstates, therefore the particles we normally observe in the laboratory are not K0 and K¯ 0,butrathersomelinearcombinationofthe Chapter 3. (a2) Corrections to the One-Loop Matrix Elements of O 70 Four-Fermion ∆S =2Operators two.

✲ d u, c✲, t s✲

K0 WWK¯ 0 ✛ ✛ ✛ s¯ u¯,¯c, t¯ d¯

✲ d W s✲

0 u, c, tu✻, c, t ¯ 0 K ❄ K ✛ ✛ s¯ W d¯

Figure 3.3: Box diagrams contributing to neutral Kaon oscillation in the SM.

Since K0 and K¯ 0 are pseudoscalars and antiparticles:

P K0 = K0 ,PK¯ 0 = K¯ 0 , (3.13) | ⟩ −| ⟩ | ⟩ −| ⟩ C K0 = K¯ 0 ,CK¯ 0 = K0 , (3.14) | ⟩ −| ⟩ | ⟩ | ⟩ we can form the eigenstates of CP as follows:

1 0 0 K1 = K K¯ ,CPK1 = K1 , (3.15) | ⟩ √2 | ⟩−| ⟩ | ⟩ | ⟩ 1 / 0 0 0 K2 = K + K¯ ,CPK2 = K2 . (3.16) | ⟩ √2 | ⟩ | ⟩ | ⟩ −| ⟩ / 0 Usually neutral Kaons decay into two or three Pion states. Since two (three)-Pion states have P =+1(P = 1) and C =+1(C =+1),then: −

K π0 + π0, 1 → + K π + π−, 1 → K π0 + π0 + π0, 2 → + 0 K π + π + π−. 2 →

The 2π decay is much faster than the 3π,becausetheenergyreleasedisgreater.Since there is experimental evidence of small CP violation eﬀects in K0 K¯ 0 oscillations, we − conclude that the mass eigenstates should depend on the CP eigenstates through the 3.1. Introduction 71 combination:

K1 + ϵK K2 K2 + ϵK K1 KS = | ⟩ | ⟩, KL = | ⟩ | ⟩, (3.17) | ⟩ 1+ ϵ 2 | ⟩ 1+ ϵ 2 | K| | K | > > where KS and KL are the short and long-lived neutral Kaons respectively. The departure of ϵ from 0 measures the amount of CP violation in the K0 K¯ 0 mixing. | K| − It can be seen (for more information see Refs. [87, 88]) that, modulo a phase, ϵK can put in the form: 1 ¯ 0 ∆S=2 0 ϵK = Im K eff K , (3.18) √2∆mK ⟨ |H | ⟩ where ∆m = m m is the mass difference between the long and short lived Kaons, K KL − KS and ∆S=2 is the effective Hamiltonian which describes the K0 K¯ 0 oscillations. The Heff − Hamiltonian ∆S=2 describes the electroweak interactions governing the process under Heff study, while the non-perturbative evaluation of the above matrix element takes into account the strong interactions. In general, effective Hamiltonians arise as a result of the Operator Product Expansion (OPE). The basic idea of the OPE was first proposed by Wilson and it states that: An interaction which is fundamentally a product of two or more current operators (as the ones in Fig. 3.3), can be expanded into a linear combination of local operators. Let us implement the OPE in the case of K0 K¯ 0 oscillations. At leading order, the − aforementioned process is described by the Feynman diagrams of Fig. 3.3. With full W boson and t quark propagators the diagrams represent the situation at very high energy (short distance) scales of (m ,m), while the true picture of a decaying hadron with O W t masses of (m ,m,m ) is more effectively described by point-like vertices represented O b c K by local operators. Due to the symmetries of SM, ∆S=2 can be expressed in-terms of Heff only one local operator, namely:

11 γ O∆S=2 =(¯sγLd)(¯sγLd),γL = γ − 5 , (3.19) µ µ µ µ 2 where in each parenthesis the spin and colour indices are meant to be summed. The crucial property of the weak interactions, shown explicitly in the above operator, is that the W bosons couple only to left-handed quarks. All physical information concerning the K0 K¯ 0 oscillations, can be extracted from − the following expectation value (see Ref. [87]):

2 2 ¯ 0 ∆S=2 0 GF mW ∆S=2 ¯ 0 ˆ ∆S=2 0 K eff K = 2 Cl,n (µ)λlλn K O (µ) K , (3.20) ⟨ |H | ⟩ 16π 6 7 ⟨ | | ⟩ l,m&=u,c,t

2 2 where we have used the OPE. In the above relation GF =(√2gw)/(8mW )andmW are the Chapter 3. (a2) Corrections to the One-Loop Matrix Elements of O 72 Four-Fermion ∆S =2Operators

1 Fermi constant and the W boson mass, respectively, while λi = Vis∗ Vid are combinations of the CKM matrix elements. The “hat” symbol represents renormalized operators. In ∆S=2 general, the Wilson coefficients Cl,n (µ)dependontheflavourofthequarksexchanged in the process under study. At next-to-leading order in QCD, the explicit form of the Wilson coefficients can be found in Refs. [88, 89, 90]. With the usage of OPE we have separated the short distance from the long distance physics. Due to asymptotic freedom the short distance contributions from scales higher ∆S=2 than µ,whicharecontainedintheWilsoncoefficientsCl,n (µ), can be calculated in perturbation theory provided that µ is not too small. Due to confinement the long distance physics, from scales lower than µ, is taken into account in the non-perturbative evaluation of K¯ 0 Oˆ ∆S=2(µ) K0 . ⟨ | | ⟩ The calculation of the amount of CP violation in K0 K¯ 0 oscillations requires the − knowledge of the Kaon BK parameter:

K¯ 0 (¯sγLd)(¯sγLd) K0 B ⟨ | µ µ | ⟩ . (3.21) K ≡ 8 K¯ 0 (¯sγLd) 0 0 (¯sγLd) K0 3 ⟨ | µ | ⟩⟨ | µ | ⟩

The above ratio implies that the value of BK is close to unity. The numerator is the desired ∆S = 2 matrix element, while the denominator is the same matrix element evaluated in the vacuum insertion approximation. Usually the renormalized BK parameter is expressed in terms of the Kaon decay constant fK and the Kaon mass mK:

¯ 0 ˆ ∆S=2 0 ˆ K O (µ) K BK(µ) ⟨ | 8 2 2 | ⟩. (3.22) ≡ 3 mK fK

Combining Eqs. (3.18), (3.20), (3.22) we obtain the following relation between ϵK and

BˆK (µ): ˆ ∆S=2 ϵK = CϵBK (µ)Im Cl,n (µ)λlλn , (3.23) 6 7 l,m&=u,c,t where: 2 2 2 2 GF mW fKmK Cϵ . (3.24) 2 ≡ 6√2π ∆mK

Since ϵK can be measured experimentally and BˆK (µ)canbeextractedfromlatticesim- ulations, one can use the above equation in order to constrain the (¯ρ, η¯)vertexofthe unitarity triangle defined by Eq. (3.10). More specifically the above equation specifies a hyperbola in theρ ¯ η¯ complex plane of the form: −

η¯(1 ρ¯)=constant, (3.25) −

1 2 2 2 The factor 1/mW comes from the W boson propagator, considered in the limit q << mW , (q is the momentum of the W boson). Note that the Fermi constant gives the strength of the weak interactions at energies much smaller then mW . 3.2. Amputated Green’s Functions of Four-Fermion ∆S =2Operators. 73 which is represented by light green colour in Fig. 3.2. Theoretical calculations using the Kobayashi-Maskawa mechanism agree with experimental measurements up to date. However the Kobayashi-Maskawa mechanism is unable to generate a baryon asymmetry of the observed amount in nature. The experimental accuracy and the theoretical uncertainties involved in the evaluation of CP-violating processes leave room for new physics waiting to be uncovered. The most promising extension of the SM seems to be the “Minimal Supersymmetric Standard Model” (MSSM). In this case we can extend the OPE of the eﬀective Hamiltonian in order to accommodate operators which give rise to supersymmetric particle exchanges. This will modify the constraint over the unitarity triangle.

Improving the accuracy of BK is essential in order to derive stringent bounds on the amount of non-SM CP violation in kaon decay. The aforementioned matrix elements are very sensitive to various systematic errors. Inordertoobtainreliablenon-perturbative estimates of physical quantities it is essential to keep under control the (a) systematic O errors in simulations or, additionally, reduce the lattice artifacts in numerical results. Such a reduction, regarding renormalizationfunctions,canbeachievedbysubtracting appropriately the (a2)perturbativecorrectiontermscalculatedinthischapter,from O respective non-perturbative results. In this chapter we evaluate the amputated Green’s functions and the renormalization matrices of the complete basis of 20 four-fermion operators of dimension six which do not need power subtractions (i.e. mixing occurs only with other operators of equal dimensions). The calculations are carried out up to 1-loop in Lattice Perturbation theory and up to (a2) in lattice spacing. Since we are dealing with a mass-independent renor- O malization scheme our results are immediately applicable to other ∆F =2heavy-ﬂavour processes of great phenomenological interest, such as D D¯ or B B¯ mixing. Let us − − also mention that in generic new physics models (i.e. beyond the standard model), the complete basis of 4-fermion operators contributes to neutral meson mixing amplitudes; this is the case for instance of SUSY models (see e.g. [88]).

3.2 Amputated Green’s Functions of Four-Fermion ∆S =2Operators.

The four-quark eﬀective interaction operator, O∆S=2, splits into parity-even and parity- odd parts; in standard notation:

1 O∆S=2 = (O∆S=2 O∆S=2 ), (3.26) 4 VV+AA − VA+AV Chapter 3. (a2) Corrections to the One-Loop Matrix Elements of O 74 Four-Fermion ∆S =2Operators where:

∆S=2 OVV+AA =(¯sγµd)(¯sγµd)+(¯sγµγ5d)(¯sγµγ5d) , (3.27)

∆S=2 OVA+AV =(¯sγµd)(¯sγµγ5d)+(¯sγµγ5d)(¯sγµd) . (3.28)

Since the weak process of oscillations is simulated in the framework of Lattice QCD, where Parity is a symmetry, the parity-odd part gives no contribution to the K0–K¯ 0 matrix element. Thus, we conclude that BK can be extracted from the correlator (x0 >0, y0 <0): C (x, y)= 0 (d¯γ s)(x)Oˆ ∆S=2 (0)(d¯γ s)(y) 0 , (3.29) KOK ⟨ | 5 VV+AA 5 | ⟩ where Oˆ ∆S=2 is the renormalized operator of O∆S=2 .Thepseudo-scalarstates(d¯γ s) 0 VV+AA VV+AA 5 | ⟩ and 0 (d¯γ s)playtheroleofinterpolatingfields,respectivelyfor K0 and K¯ 0 states. ⟨ | 5 | ⟩ ⟨ | In place of the operator in Eq. (3.27) it is advantageous to use a four-quark operator with a different flavour content (s, d, s′, d′), and with ∆S =∆s +∆s′ = 2, namely [91]:

∆S=2 =(¯sγ d)(¯s′γ d′)+(¯sγ γ d)(¯s′γ γ d′) OVV+AA µ µ µ 5 µ 5 +(¯sγµd′)(¯s′γµd)+(¯sγµγ5d′)(¯s′γµγ5d) , (3.30) where now the correlator is given by:

¯ ∆S=2 ¯ CK K (x, y)= 0 (dγ5s)(x)2 VV+AA(0)(d′γ5s′)(y) 0 . (3.31) O ′ ⟨ | O | ⟩

Making use of Wick’s theorem one checks the equality:

CK K (x, y)=CKOK(x, y), (3.32) O ′ which means that both correlators contain the same physical information.

In the present section we evaluate, up to (a2), the 1-loop matrix element of the O 4-fermion operators (the superscript letter F stands for Fierz.):

c c ¯ d d (¯sXd)(¯s′ Yd′) s¯ (z) X d (z) s′ (z) Y d′ (z) (3.33) OXY ≡ ≡ k1 k1k2 k2 k3 k3k4 k4 z & &c,d k1,k&2,k3,k4, -, - F c c ¯ d d (¯sXd′)(¯s′ Yd) s¯ (z) X d′ (z) s′ (z) Y d (z) (3.34) OXY ≡ ≡ k1 k1k2 k2 k3 k3k4 k4 z & &c,d k1,k&2,k3,k4, -, -

¯ a4 a3 a2 a1 with a generic initial state: d′ (p ) s′ (p ) 0 ,andagenericﬁnalstate: 0 d¯ (p ) s (p ). i4 4 i3 3 | ⟩ ⟨ | i2 2 i1 1 Spin indices are denoted by i, k, and color indices by a, c, d,whileX and Y correspond to the following set of products of the Dirac matrices:

1 X, Y = 11,γ5,γ ,γ γ5,σ ,γ5σ S, P, V, A, T, T˜ ; σ = [γ ,γ ]. (3.35) { µ µ µν µν }≡{ } µν 2 µ ν 3.2. Amputated Green’s Functions of Four-Fermion ∆S =2Operators. 75

Our calculations are performed using massless fermions described by the Wilson/clover action. By taking mf = 0, our results are identical also for the twisted mass action and the Osterwalder-Seiler action in the chiral limit (in the so called twisted mass basis). For gluons we employ a 3-parameter family of Symanzik improved actions, which comprises all common gluon actions (Plaquette, tree-level Symanzik, Iwasaki, DBW2, L¨uscher-Weisz). Conventions and notations for the actions used, as well as algebraic manipulations involving the evaluation of 1-loop Feynman diagrams (up to (a2)), have been described O in detail in chapter 2. To establish notation and normalization, let us ﬁrst write the tree-level expression for the amputated Green’s functions of the operators and F : OXY OXY ΛXY (p ,p ,p ,p ,r ,r ,r ,r )a1a2a3a4 = X Y δ δ , (3.36) tree 1 2 3 4 s d s′ d′ i1i2i3i4 i1i2 i3i4 a1a2 a3a4

F XY a1a2a3a4 (Λ ) (p1,p2,p3,p4,rs,rd,rs ,rd ) = Xi i Yi i δa a δa a , (3.37) tree ′ ′ i1i2i3i4 − 1 4 3 2 1 4 3 2 where r is the Wilson parameter, one for each ﬂavour. We continue with the 1-loop corrections. There are twelve 1-loop diagrams that enter our 4-fermion calculation, six for each operator , F .Thediagramsd d corre- OXY OXY 1 − 6 sponding to the operator are illustrated in Fig. 3.4. The other six diagrams, dF dF , OXY 1 − 6 involved in the Green’s function of F are similar to d d ,andmaybeobtainedfrom OXY 1 − 6 d d by interchanging the fermionic ﬁelds d and d′ along with their momenta, color and 1 − 6 spin indices, and respective Wilson parameters.

s’ s s’ s s’ s s’ s s’ s s’ s p p p p p p p p p p p p 3 1 3 1 3 1 3 1 3 1 3 1

Y X Y X Y X Y X Y X Y X p p p p p p p p p p p p 4 2 4 2 4 2 4 2 4 2 4 2 d’ d d’ d d’ d d’ d d’ d d’ d d1 d2 d3 d4 dd56 Figure 3.4: 1-loop diagrams contributing to the amputated Green’s function of the 4-fermion operator XY .Wavy(solid)linesrepresentgluons(fermions). O

The only diagrams that need to be calculated from ﬁrst principles are d1, d2 and d3, while the rest can be expressed in terms of the ﬁrst three. In particular, the expressions for the amputated Green’s functions ΛXY ΛXY can be obtained via the following relations: d4 − d6

XY a1a2a3a4 XY a2a1a4a3 ⋆ Λ (p1,p2,p3,p4,rs,rd,rs ,rd ) = Λ ( p2, p1, p4, p3,rd,rs,rd ,rs ) , d4 ′ ′ i1i2i3i4 d1 − − − − ′ ′ i2i1i4i3 / 0 ΛXY (p ,p ,p ,p ,r ,r ,r ,r )a1a2a3a4 =ΛYX(p ,p ,p ,p ,r ,r ,r ,r )a3a4a1a2, (3.38) d5 1 2 3 4 s d s′ d′ i1i2i3i4 d2 3 4 1 2 s′ d′ s d i3i4i1i2

ΛXY (p ,p ,p ,p ,r ,r ,r ,r )a1a2a3a4 =ΛYX(p ,p ,p ,p ,r ,r ,r ,r )a3a4a1a2. d6 1 2 3 4 s d s′ d′ i1i2i3i4 d3 3 4 1 2 s′ d′ s d i3i4i1i2 Chapter 3. (a2) Corrections to the One-Loop Matrix Elements of O 76 Four-Fermion ∆S =2Operators

Once we have constructed ΛXY ΛXY we can use the relation: d4 − d6

F XY a1a2a3a4 XY a1a4a3a2 (Λ ) (p1,p2,p3,p4,rs,rd,rs ,rd ) = Λ (p1,p4,p3,p2,rs,rd ,rs ,rd) , (3.39) dj ′ ′ i1i2i3i4 − dj ′ ′ i1i4i3i2 to derive the expressions for (ΛF )XY (j =1, 6). From the amputated Green’s functions dj ··· for all twelve diagrams we can write down the total 1-loop expressions for the operators F XY and XY : O O 6 6 XY XY F XY F XY Λ1 loop = Λd , (Λ )1 loop = (Λ )d . (3.40) − j − j &j=1 &j=1

XY XY In our algebraic expressions for the 1-loop amputated Green’s functions Λd1 ,Λd2 XY and Λd3 we kept the Wilson parameters for each quark field distinct, that is: rs, rd, rs′ , rd′ for the quark fields s, d, s′ and d′ respectively. For the required numerical integration of the algebraic expressions of the integrands, corresponding to each Feynman diagram, we are forced to choose the square of the value for each r parameter. As in all present day simulations, we set: r2 = r2 = r2 = r2 =1. (3.41) s d s′ d′ Since the Wilson action is reflection positive2 only for the special choice r =1,wehave set:

rs = rd = rs′ = rd′ =1, (3.42) for the following calculations.

Concerning the external momenta pi (shown explicitly in Fig. 3.4) we have chosen to evaluate the amputated Green’s functions at the renormalization point:

p = p = p = p p, (3.43) 1 2 3 4 ≡ where all external quark legs have the same momentum p. This choice is, of course, not unique but is the simplest way to regulate the infrared divergences. It is easy and not time consuming to repeat the calculations for other choices of Wilson parameters and for other renormalization prescriptions. The crucial point of our calculation is the correct extraction of the full (a2)de- O pendence from loop integrands with strong IR divergences (convergent only beyond 6 dimensions). The singularities are isolated using the procedure explained in chapter 2. In order to reduce the number of strong IR divergent integrals, appearing in diagram d1,we have inserted the identity below:

1 2 2 1= k"+ ap + k"ap 2kˆ 2 +16 sin(k )2 sin(ap )2 , (3.44) 2 − − σ σ ap σ , & - 2 Matrix elements computed? in Euclidean space do not always correspond to analytic continuation of matrix elements of a physical theory in Minkowski space. For this to happen, the lattice action has to satisfy the property of reﬂection positivity, which involves time reﬂections and complex conjugations 3.2. Amputated Green’s Functions of Four-Fermion ∆S =2Operators. 77 into selected 3-point functions of the form:

f(k, p) 2 2 , (3.45) (kˆ2)a (k"+ ap )b (k"ap )c −

2 2 qµ whereq ˆ =4 µ sin ( 2 ); k is the loop momentum and p is the external momentum. Repeated use of Eq. (3.44) reduces the 3-point functions to either 2-point functions or ! more convergent expressions. The common factor 1/ap2 in Eq. (3.44) can be treated by Taylor expansion. For our calculations it was necessary only to (a0): O ? 1 1 p4 = + σ σ + (a2 p2). (3.46) 2 a2 p2 (p2)2 O ap !

Here we present one of the four? integrals with strong IR divergences that enter in this calculation:

π 4 2 2 d k sin(kµ)sin(kν ) ln(a p ) 4 2 2 = δµν 0.002457072288 2 π (2π) ˆ 2 − 64π 1− k k"+ ap k"ap − ) +0.00055270353(6) a2 p2 0.0001282022(1) a2 p2 − µ 4 2 2 2 µ pµ 2 ln(a p ) 2 2 +0.000157122310 a 2 + a 2 2 pµ 3 p !p 1536π − − / 0* 1 ln(a2 p2 ) + a2 p p 0.001870841540 0.00029731225(4) + µ ν a2 p2 − 768π2 ) 2 2 4 (p + p ) pµ 0.000047949674 µ ν +0.000268598599 µ − p2 (p2 )2 ! * + (a4 p4 ). (3.47) O The results for the other three integrals can be found in chapter 2. Integrands with simple IR divergences (convergent beyond 4 dimensions) can be handled by well-known techniques. The final 1-loop expressions for ΛXY ,ΛXY and ΛXY ,upto (a2), are obtained as a d1 d2 d3 O function of: the coupling constant g,cloverparametercSW ,numberofcolorsNc, lattice spacing a,externalmomentump,andgaugeparameterλ.Thespecificvaluesλ =1(0) correspond to the Feynman (Landau) gauge. Due to lack of space we present in the next XY XY XY pages the results for Λd1 ,Λd2 and Λd3 for the special choices: rs = rd = rs′ = rd′ =1, and tree-level Symanzik action. Similar results exist for the rest Symanzik improved gluon actions that we have studied and for other choices of the Wilson parameters r. We note in passing that in diagram 3 the dependence on external momentum has the same terms as in diagram 2, with identical numerical coefficients; the difference between the two diagrams lies in the structure of color and gamma matrices multiplying each term. Chapter 3. (a2) Corrections to the One-Loop Matrix Elements of O 78 Four-Fermion ∆S =2Operators

Diagram d1

2 XY a1 a2 a3 a4 g δa1 a2 δa3 a4 0 XY 0 Λd (p)i i i i = δa1 a4 δa3 a2 a (Λ (a ))d 1 1 2 3 4 16π2 − N × O 1 2 c 3 @ 1 XY +a (Λ (a1))d O 1

2 XY +a (Λ (a2))d , (3.48) O 1 A

XY 2 0 (Λ (a ))d = Xi1 i2 Yi3 i4 0.05294144(3) + 0.737558970(1) cSW +0.238486988(3) cSW O 1 − ) 1 2.100573331(5) λ + ( 1+λ) ln(a2 p2 ) − 2 − * + (Xγµ) (Yγµ) 0.507914049(6) + 0.55316919(1) c 0.194516637(3) c2 i1 i2 i3 i4 − SW − SW µ & ) * + (Xγµγν ) (Yγµγν ) 0.018598520(2) 0.1843897425(8) c 0.0596217473(8) c2 i1 i2 i3 i4 − SW − SW µ,ν & ) 1 + ln(a2 p2 ) 8 * p p + (Xγµγρ) (Yγν γρ) µ ν 0.397715726853 + 0.147715726853 λ , (3.49) i1 i2 i3 i4 p2 µ,ν,ρ & ) , -*

XY µ µ 1 (Λ (a ))d = Xi1 i2 (Yγ )i3 i4 +(Xγ )i1 i2 Yi3 i4 ipµ 0.09460083(1) 0.065711182(4) cSW O 1 × − µ & , - ) , 0.059929106(1) c2 +0.438508366(3) λ − SW 1 + ( 1+ c λ) ln(a2 p2 ) 4 − SW − -* + (Xγµγν ) (Yγν ) +(Xγν ) (Yγµγν ) ip 0.1692905881(6) + 0.010283104(5) c i1 i2 i3 i4 i1 i2 i3 i4 × µ SW µ,ν & , - ) , 0.0680031615(8) c2 +0.073857863427 λ − SW 1 + (1 + 3 c ) ln(a2 p2 ) 16 SW -* + (Xγµγν γρ) (Yγν γρ) +(Xγνγρ) (Yγµγν γρ) ip 0.0279443091(3) c i1 i2 i3 i4 i1 i2 i3 i4 × µ − SW µ,ν,ρ & , - ) , 2 +0.0319830668(5) cSW 1 c ln(a2 p2 ) , (3.50) −16 SW -* 3.2. Amputated Green’s Functions of Four-Fermion ∆S =2Operators. 79

XY 2 2 2 (Λ (a ))d = Xi1 i2 Yi3 i4 p 1.32362251(5) 0.43684285(3) cSW 0.0208665277(5) cSW O 1 − − ) , 1 +0.64073441(3) λ + ( 17 + 9 c 9 λ) ln(a2 p2 ) 72 − SW − - p4 + σ σ 0.06213648(8) 0.07400055(8) λ p2 − ! , -* + (Xγµ) (Yγµ) p2 0.059895142(8) 0.241755150(3) c +0.114731816(7) c2 i1 i2 i3 i4 − SW SW µ & ) , 1 0.036928931713 λ + ( 7+11c 4 c2 ) ln(a2 p2 ) − 48 − SW − SW - +p2 1.01694823(2) 0.44474062(1) c 0.033265121(3) c2 µ − SW − SW , -* 3 pν p + (Xγµγν ) Y + X (Yγµγν ) µ 0.00592406(2) 0.00295805(2) λ i1 i2 i3 i4 i1 i2 i3 i4 × p2 − µ,ν & , - ) , -* + (Xγµ) (Yγν ) p p 0.19915360(1) + 0.212823513(3) c +0.033028338(2) c2 i1 i2 i3 i4 µ ν − SW SW µ,ν & ) , 1 +0.1600141922(8) λ + ( 4+5c +2c2 3 λ) ln(a2 p2 ) 24 − SW SW − -* + (Xγµγν) (Yγµγν ) p2 0.08962805(1) + 0.0769373498(3) c +0.0067184623(3) c2 i1 i2 i3 i4 − SW SW µ,ν & ) , 1 + (7 5 c ) ln(a2 p2 ) 240 − SW - +p2 +0.16608907(6) + 0.07446360(2) c 0.0087763322(3) c2 µ SW − SW , 29 ln(a2 p2 ) −180 - p4 + σ σ 0.048180849735 p2 − ! , -* + (Xγµγρ) (Yγν γρ) p p 0.21865904(4) + 0.054629909(8) c +0.00276900638(7) c2 i1 i2 i3 i4 µ ν − SW SW µ,ν,ρ & ) , 1 0.082411837(6) λ + (164 60 c +45λ) ln(a2 p2 ) − 1440 − SW - 3 3 (p pν + pµp ) + µ ν 0.110138789528 0.024619287809 λ p2 − − , - p4 +p p σ σ 0.140961390102 + 0.045240352404 λ µ ν (p2 )2 ! , - 2 pµpν p + ρ 0.477634781(8) 0.083831642(8) λ p2 − − , -* µ ν ρ µ ν ρ 2 2 + (Xγ γ γ )i1 i2 (Yγ γ γ )i3 i4 pµ 0.00385492408(3) cSW µ,ν,ρ & ) , -* 1 + (Xγµγργσ) (Yγν γργσ) p p 0.0209503296(6) c2 c2 ln(a2 p2 ) . (3.51) i1 i2 i3 i4 µ ν − SW − 32 SW µ,ν,ρ,σ & ) , -* Chapter 3. (a2) Corrections to the One-Loop Matrix Elements of O 80 Four-Fermion ∆S =2Operators Diagram d2

2 XY a1 a2 a3 a4 g 1 0 XY 0 Λd (p)i i i i = δa1 a2 δa3 a4 Nc a (Λ (a ))d 2 1 2 3 4 16π2 − N × O 2 2 c 3 @ 1 XY +a (Λ (a1))d O 2

2 XY +a (Λ (a2))d , (3.52) O 2 A

XY 2 0 (Λ (a ))d = Xi1 i2 Yi3 i4 1.2904478(4) + 0.737558970(1) cSW +0.238486988(3) cSW O 2 ) 1 +2.3960046(4) λ + ( 1 λ) ln(a2 p2 ) 2 − − * + X (γµYγµ) 0.507914047(8) + 0.55316917(2) c 0.194516638(9) c2 i1 i2 i3 i4 − SW − SW µ & ) * + X (γµγν Yγµγν ) 0.129117207(2) 0.1843897425(8) c 0.0596217473(8) c2 i1 i2 i3 i4 − − SW − SW µ,ν & ) 1 + ln(a2 p2 ) 8 * p p 1 + X (γµγρYγν γρ) µ ν λ , (3.53) i1 i2 i3 i4 p2 − 4 µ,ν,ρ & ) , -*

XY µ µ 1 (Λ (a ))d = Xi1 i2 (Yγ )i3 i4 + Xi1 i2 (γ Y )i3 i4 ipµ 0.37785613(9) 0.56675680(2) cSW O 2 × − µ & , - ) , 0.160026205(2) c2 0.48393977(9) λ − SW − 1 + ( 3+5c +2λ) ln(a2 p2 ) 8 − SW -* + X (γµγν Yγν ) + X (γν Yγν γµ) ip 0.073251555(1) 0.001704761(4) c i1 i2 i3 i4 i1 i2 i3 i4 × µ − − SW µ,ν & , - ) , 1 +0.032093938(2) c2 λ SW − 8 1 + (3 c ) ln(a2 p2 ) 16 − SW -* + X (γµγν γρYγν γρ) + X (γν γρYγν γργµ) ip 0.0459135542(5) c i1 i2 i3 i4 i1 i2 i3 i4 × µ SW µ,ν,ρ & , - ) , 2 +0.0319830668(5) cSW 1 c ln(a2 p2 ) , (3.54) −16 SW -* 3.2. Amputated Green’s Functions of Four-Fermion ∆S =2Operators. 81

XY 2 2 2 (Λ (a ))d = Xi1 i2 Yi3 i4 p 0.7374671(6) 0.24301094(4) cSW 0.0096054476(8) cSW O 2 − − ) , 1 0.4696085(6) λ + ( 23 + 5 c +15λ) ln(a2 p2 ) − 120 − SW - p4 1 + σ σ ( 77 + 15 λ) p2 90 − ! , -* + X (γµYγµ) p2 0.04610701(4) 0.19136171(3) c +0.02347831(4) c2 i1 i2 i3 i4 − SW SW µ & ) , 1 0.06249999(1) λ + (1 + c ) ln(a2 p2 ) − 16 SW - +p2 0.17251518(3) 0.19211806(3) c +0.01744902(2) c2 µ − SW SW , -* 3 pν p 101 + X (Yγµγν ) + X (γν γµY ) µ i1 i2 i3 i4 i1 i2 i3 i4 × p2 288 µ,ν & , - ) , -* + X (γµYγν ) p p 0.05513763(3) 0.005630284(8) c 0.072690409(7) c2 i1 i2 i3 i4 µ ν − SW − SW µ,ν & ) , 1 0.10887203(3) λ + ( 2+ c + c2 + λ) ln(a2 p2 ) − 8 − SW SW -* + X (γµγν Yγµγν ) p2 0.05064893(1) + 0.0394316274(6) c +0.00332360968(9) c2 i1 i2 i3 i4 − SW SW µ,ν & ) , 1 + (13 15 c ) ln(a2 p2 ) 720 − SW - +p2 +0.05383442(9) + 0.124311493(7) c 0.0051958638(1) c2 µ SW − SW , 1 ln(a2 p2 ) −15 - p4 1 + σ σ p2 − 240 ! , -* + X (γµγρYγργν ) p p 0.03270359(5) 0.039026988(4) c +0.0015068706(2) c2 i1 i2 i3 i4 µ ν − − SW SW µ,ν,ρ & ) , 1 0.06926696(4) λ + (28 + 60 c +45λ) ln(a2 p2 ) − 1440 SW - 3 3 (p pν + pµp ) 1 + µ ν (41 40 λ) p2 960 − , - p4 1 +p p σ σ (7 + 25 λ) µ ν (p2 )2 960 ! , - 2 pµpν p 1 + ρ ( 40 9 λ) p2 288 − − , -* µ ν ρ µ ν ρ 2 2 + Xi1 i2 (γ γ γ Yγ γ γ )i3 i4 pµ 0.00385492795(2) cSW µ,ν,ρ & ) , -* 1 + X (γµγργσYγν γργσ) p p 0.015978597(1) c2 c2 ln(a2 p2 ) . (3.55) i1 i2 i3 i4 µ ν SW − 32 SW µ,ν,ρ,σ & ) , -* Chapter 3. (a2) Corrections to the One-Loop Matrix Elements of O 82 Four-Fermion ∆S =2Operators Diagram d3

2 XY a1 a2 a3 a4 g δa1 a2 δa3 a4 0 XY 0 Λd (p)i i i i = δa1 a4 δa3 a2 a (Λ (a ))d 3 1 2 3 4 16π2 − N × O 3 2 c 3 @ 1 XY +a (Λ (a1))d O 3

2 XY +a (Λ (a2))d , (3.56) O 3 A

XY 2 0 (Λ (a ))d = Xi1 i2 Yi3 i4 1.2904478(4) + 0.737558970(1) cSW +0.238486988(3) cSW O 3 ) 1 +2.3960046(4) λ + ( 1 λ) ln(a2 p2 ) 2 − − * + (Xγµ) (γµY ) 0.507914047(8) + 0.55316917(2) c 0.194516638(9) c2 i1 i2 i3 i4 − SW − SW µ & ) * + (Xγµγν ) (γµγνY ) 0.129117207(2) 0.1843897425(8) c 0.0596217473(8) c2 i1 i2 i3 i4 − − SW − SW µ,ν & ) 1 + ln(a2 p2 ) 8 * p p 1 + (Xγµγρ) (γν γρY ) µ ν λ , (3.57) i1 i2 i3 i4 p2 − 4 µ,ν,ρ & ) , -*

XY µ µ 1 (Λ (a ))d = (Xγ )i1 i2 Yi3 i4 + Xi1 i2 (γ Y )i3 i4 ipµ 0.37785613(9) 0.56675680(2) cSW O 3 × − µ & , - ) , 0.160026205(2) c2 0.48393977(9) λ − SW − 1 + ( 3+5c +2λ) ln(a2 p2 ) 8 − SW -* + (Xγν ) (γµγν Y ) +(Xγν γµ) (γν Y ) ip 0.073251555(1) 0.001704761(4) c i1 i2 i3 i4 i1 i2 i3 i4 × µ − − SW µ,ν & , - ) , 1 +0.032093938(2) c2 λ SW − 8 1 + (3 c ) ln(a2 p2 ) 16 − SW -* + (Xγµγν γρ) (γν γρY ) +(Xγνγρ) (γν γργµY ) ip 0.0459135542(5) c i1 i2 i3 i4 i1 i2 i3 i4 × µ SW µ,ν,ρ & , - ) , 2 +0.0319830668(5) cSW 1 c ln(a2 p2 ) , (3.58) −16 SW -* 3.2. Amputated Green’s Functions of Four-Fermion ∆S =2Operators. 83

XY 2 2 2 (Λ (a ))d = Xi1 i2 Yi3 i4 p 0.7374671(6) 0.24301094(4) cSW 0.0096054476(8) cSW O 3 − − ) , 1 0.4696085(6) λ + ( 23 + 5 c +15λ) ln(a2 p2 ) − 120 − SW - p4 1 + σ σ ( 77 + 15 λ) p2 90 − ! , -* + (Xγµ) (γµY ) p2 0.04610701(4) 0.19136171(3) c +0.02347831(4) c2 i1 i2 i3 i4 − SW SW µ & ) , 1 0.06249999(1) λ + (1 + c ) ln(a2 p2 ) − 16 SW - +p2 0.17251518(3) 0.19211806(3) c +0.01744902(2) c2 µ − SW SW , -* 3 pν p 101 + (Xγµγν ) Y + X (γν γµY ) µ i1 i2 i3 i4 i1 i2 i3 i4 × p2 288 µ,ν & , - ) , -* + (Xγµ) (γν Y ) p p 0.05513763(3) 0.005630284(8) c 0.072690409(7) c2 i1 i2 i3 i4 µ ν − SW − SW µ,ν & ) , 1 0.10887203(3) λ + ( 2+ c + c2 + λ) ln(a2 p2 ) − 8 − SW SW -* + (Xγµγν ) (γµγν Y ) p2 0.05064893(1) + 0.0394316274(6) c +0.00332360968(9) c2 i1 i2 i3 i4 − SW SW µ,ν & ) , 1 + (13 15 c ) ln(a2 p2 ) 720 − SW - +p2 +0.05383442(9) + 0.124311493(7) c 0.0051958638(1) c2 µ SW − SW , 1 ln(a2 p2 ) −15 - p4 1 + σ σ p2 − 240 ! , -* + (Xγργµ) (γν γρY ) p p 0.03270359(5) 0.039026988(4) c +0.0015068706(2) c2 i1 i2 i3 i4 µ ν − − SW SW µ,ν,ρ & ) , 1 0.06926696(4) λ + (28 + 60 c +45λ) ln(a2 p2 ) − 1440 SW - 3 3 (p pν + pµp ) 1 + µ ν (41 40 λ) p2 960 − , - p4 1 +p p σ σ (7 + 25 λ) µ ν (p2 )2 960 ! , - 2 pµpν p 1 + ρ ( 40 9 λ) p2 288 − − , -* µ ν ρ µ ν ρ 2 2 + (Xγ γ γ )i1 i2 (γ γ γ Y )i3 i4 pµ 0.00385492795(2) cSW µ,ν,ρ & ) , -* 1 + (Xγµγργσ) (γν γργσY ) p p 0.015978597(1) c2 c2 ln(a2 p2 ) . (3.59) i1 i2 i3 i4 µ ν SW − 32 SW µ,ν,ρ,σ & ) , -* Chapter 3. (a2) Corrections to the One-Loop Matrix Elements of O 84 Four-Fermion ∆S =2Operators 3.3 Mixing and Renormalization of and F on OXY OXY the Lattice.

The matrix element K¯ 0 O∆S=2 K0 ,calculatednon-perturbatively,isverysensitiveto ⟨ | VV+AA| ⟩ various systematic errors which aﬀect its chiral behaviour. The main roots of this problem are:a) (a) systematic errors due to numerical integration, b) the operator O∆S=2 mixes O VV+AA with other 4-fermion ∆S =2operatorsofthesamedimensions3 but with “wrong naive chirality”. Mixing with operators with the “wrong naive chirality” is allowed even in the chiral limit. This is due to the presence of explicit chiral symmetry breaking, induced by the wilson term. In order to address these problems we have calculated the mixing pattern (renormalization matrices) of the Parity Conserving and Parity Violating 4-fermion ∆S =2operators (deﬁned below), by using the perturbative analytic expressions for the amputated Green’s functions obtained in the previous section. A more extensive theoretical background and non-perturbative results, concerning renormalization matrices of 4-fermion operators, can be found in Ref. [92] (see also [91, 93, 81]). Next we summarize all important relations from Ref. [92] needed for the present calculation. One can construct a complete basis of 20 independent operators which have the symmetries of the generic QCD Wilson lattice action (Parity P ,ChargeconjugationC, Flavour exchange symmetry S (d d′), Flavour Switching symmetries S′ (s d, s′ d′)and ≡ ↔ ≡ ↔ ↔ S′′ (s d′,d s′)), with 4 degenerate quarks. In Table 3.1 we classify the operators ≡ ↔ ↔ , F for all X and Y combinations of interest, according to the discrete symmetries OXY OXY P , C, S′ and S′′.

PCS′ CS′′ CPS′ CPS′′ OXY VV +1 +1 +1 +1 +1 O +1 +1 +1 +1 +1 OAA +1 +1 +1 +1 +1 OPP SS +1 +1 +1 +1 +1 O +1 +1 +1 +1 +1 OTT VA 1 1 AV +1 + AV O −1 −1 −O +1 +O OAV − − −OVA OVA SP 1+1+PS 1 PS O −1+1+O −1 −O OPS − OSP − −OSP ˜ 1+1+1 1 1 OT T − − − Table 3.1: Classiﬁcation of four-fermion operators according to P and OXY useful products of their discrete symmetries C, S′ and S′′.Fortheoperators F ,wemustexchangetheentriesofthecolumnsCS′ CS′′ and CPS′ OXY ↔ ↔ CPS′′.Notethat ˜ ˜ = and ˜ = ˜ . OT T OTT OT T OTT

3 Mixing with operators of lower dimensionality is impossible because there is no candidate ∆S =2 operator. 3.3. Mixing and Renormalization of and F on the Lattice. 85 OXY OXY

This basis can be decomposed into smaller independent bases according to the discrete symmetries P, S, CPS′,CPS′′. Following the notation of Ref. [92] we have 10 Parity Conserving operators, Q,(P =+1,S= 1) and 10 Parity Violating operators, ,(P = ± Q − 1,S= 1): ±

S= 1 1 F 1 F Q ± + , 1 ≡ 2 OVV ±OVV 2 OAA ±OAA S= 1 1 $ F % 1 $ F % ⎫ Q2 ± VV VV AA AA , ≡ 2 O ±O − 2 O ±O ⎪ ⎪ CPS =+1, S= 1 1 $ F % 1 $ F % ⎪ ′ Q3 ± 2 SS SS 2 PP PP , ⎪ (3.60) ⎪ CPS′′ =+1, ≡ O ±O − O ±O ⎪ S= 1 1 $ F % 1 $ F % ⎬ Q4 ± 2 SS SS + 2 PP PP , ≡ O ±O O ±O ⎪ S= 1 1 $ F % $ % ⎪ Q5 ± 2 TT TT , ⎪ ≡ O ±O ⎪ ⎪ $ % ⎭⎪

S= 1 1 F 1 F CPS′ =+1, 1 ± 2 VA VA + 2 AV AV , Q ≡ O ±O O ±O CPS′′ =+1, S= 1 1 $ F % 1 $ F % # 2 ± 2 VA VA 2 AV AV , CPS =undeﬁned, Q ≡ O ±O − O ±O ′ (3.61) S= 1 1 F 1 F ⎫ CPS′′ =undeﬁned, ± $ % $ %, Q3 ≡ 2 OPS ±OPS − 2 OSP ±OSP ⎬ S= 1 1 $ F % 1 $ F % ± PS + SP , ⎭ 4 2 PS 2 SP CPS′ = 1, Q ≡ O ±O O ±O − S= 1 1 F ⎫ CPS′′ = 1. ± $ ˜ % , $ % − Q5 ≡ 2 OT T ±OT T˜ ⎬ $ % Summation over all independent Lorentz indices (if any),⎭ of the Dirac matrices, is implied. The operators shown above are grouped together according to their allowed mixing pattern. This implies that on the basis of CPS symmetries the renormalization matrices S= 1 S= 1 Z ± ( ± ), for the Parity Conserving (Violating) operators, have the form: Z S= 1 ± Z11 Z12 Z13 Z14 Z15 ⎛ Z21 Z22 Z23 Z24 Z25 ⎞ S= 1 Z ± = Z Z Z Z Z , (3.62) ⎜ 31 32 33 34 35 ⎟ ⎜ ⎟ ⎜ Z41 Z42 Z43 Z44 Z45 ⎟ ⎜ ⎟ ⎜ Z Z Z Z Z ⎟ ⎜ 51 52 53 54 55 ⎟ ⎝ ⎠ S= 1 0000 ± Z11 ⎛ 0 22 23 00⎞ S= 1 Z Z ± = 0 00 . (3.63) Z ⎜ 32 33 ⎟ ⎜ Z Z ⎟ ⎜ 00044 45 ⎟ ⎜ Z Z ⎟ ⎜ 000 ⎟ ⎜ Z54 Z55 ⎟ ⎝ ⎠ As explained in Ref. [92] the explicit chiral symmetry breaking on the lattice does not induce extra renormalizationmixingpatterns,withrespecttothecontinuum,forthe parity violating sector. In contrast to the above, the renormalization matrix for the parity Chapter 3. (a2) Corrections to the One-Loop Matrix Elements of O 86 Four-Fermion ∆S =2Operators conserving case can be separated into two classes4:

S= 1 S= 1 S= 1 Z ± = Z ± (11+∆ ± ). (3.64) χ ·

The ﬁrst part: S= 1 χ ± Z11 0000 χ χ ⎛ 0 Z22 Z23 00⎞ S= 1 χ χ Z ± = 0 Z Z 00 , (3.65) χ ⎜ 32 33 ⎟ ⎜ χ χ ⎟ ⎜ 000Z44 Z45 ⎟ ⎜ ⎟ ⎜ 000Zχ Zχ ⎟ ⎜ 54 55 ⎟ ⎝ ⎠ is the one that survives in the continuum limit, while the second part:

S= 1 ± 0∆12 ∆13 ∆14 ∆15 ⎛ ∆21 00∆24 ∆25 ⎞ S= 1 ∆ ± = ∆ 00∆∆ , (3.66) ⎜ 31 34 35 ⎟ ⎜ ⎟ ⎜ ∆41 ∆42 ∆43 00⎟ ⎜ ⎟ ⎜ ∆ ∆ ∆ 00⎟ ⎜ 51 52 53 ⎟ ⎝ ⎠ is the one induced by the breaking of chiral symmetry due to the wilson term of the action. S= 1 Next we defined the renormalized Parity Conserving (Violating) operators, Qˆ ± S= 1 ( ˆ ± ), via the equations: Q S= 1 ˆ ± S= 1 S= 1 Ql = Zlm ± Qm ± , · (3.67) S= 1 S= 1 S= 1 ˆ ± = ± ± , Ql Zlm ·Qm where l, m =1,...,5(asumoverm is implied). The renormalized amputated Green’s S= 1 S= 1 S= 1 S= 1 functions Lˆ ± (ˆ ± )correspondingtoQ ± ( ± ), are given in terms of their L Q S= 1 S= 1 bare counterparts L ± ( ± )through: L S= 1 ˆ ± 2 S= 1 S= 1 Ll = Zq− Zlm ± Lm ± , · (3.68) S= 1 2 S= 1 S= 1 ˆ ± = Z− ± ± , Ll q Zlm ·Lm where Zq is the quark field renormalization constant. S= 1 S= 1 The amputated Green’s functions Lˆ ± (ˆ ± )arehigh-ranktensors,fromwhich L more manageable functions of the external momenta p can be obtained by projecting over all the possible Dirac structures. By using the appropriate Parity Conserving (Violating) S= 1 S= 1 Projectors P ± ( ± ), defined below, we can compute the renormalization matrices P

4 S= 1 S= 1 S= 1 4 In perturbation theory Z ± = Z ± +(11 +∆ ± )+ (g ). χ O 3.3. Mixing and Renormalization of and F on the Lattice. 87 OXY OXY

S= 1 S= 1 Z ± ( ± ): Z

S= 1 ΠVV +ΠAA P ± + , 1 ≡ 64N (N 1) c c ±

S= 1 ΠVV ΠAA ΠSS ΠPP P ± + − − , 2 ≡ 64(N 2 1) ± 32N (N 2 1) c − c c −

S= 1 ΠVV ΠAA ΠSS ΠPP P ± − + − , (3.69) 3 ≡±32N (N 2 1) 16(N 2 1) c c − c −

S= 1 ΠSS +ΠPP ΠTT P ± + 2 , 4 32Nc(N 1) 2 ≡ c − ∓ 32Nc(Nc 1) 2Nc 1 ± −

S= 1 ΠSS +ΠPP ΠTT P ± + 2 , 5 2 96Nc(N 1) ≡∓32Nc(Nc 1) c − 2Nc 1 − ∓

S= 1 ΠVA +ΠAV ± , P1 ≡−64N (N 1) c c ±

S= 1 ΠVA ΠAV ΠSP ΠPS ± − − , P2 ≡−64(N 2 1) ∓ 32N (N 2 1) c − c c −

S= 1 ΠVA ΠAV ΠSP ΠPS ± − − , (3.70) P3 ≡∓32N (N 2 1) − 16(N 2 1) c c − c −

S= 1 ΠSP +ΠPS ΠT T˜ ± + 2 , 4 32Nc(N 1) 2 P ≡ c − ∓ 32Nc(Nc 1) 2Nc 1 ± −

S= 1 ΠSP +ΠPS ΠT T˜ ± + 2 , 5 2 96Nc(N 1) P ≡∓32Nc(Nc 1) c − 2Nc 1 − ∓ where Π (X Y )δ δ .Again,summationisimpliedoverallindependent XY ≡ i2i1 ⊗ i4i3 a2a1 a4a3 Lorentz indices (if any) of the Dirac matrices. The above Projectors are chosen to obey the following orthogonality conditions:

S= 1 S= 1 Tr(Pl ± Lm (tree± ))=δlm, · (3.71) S= 1 S= 1 Tr( ± ± )=δ , Pl ·Lm (tree) lm

S= 1 S= 1 where the trace is taken over spin and color indices, and L ± , ± are the tree-level (tree) L(tree) S= 1 S= 1 amputated Green’s functions of the operators Q ± , ± respectively. Q We now define the renormalization conditions for the operators under study. The general principle is to impose “suitable” renormalization conditions, which are satisfied S= 1 S= 1 by the renormalized (projected amputated) Green functions Lˆ ± and ˆ ± at a fixed L scale µ. The renormalization condition is arbitrary. A simple choice is to impose that S= 1 S= 1 the fully interacting Lˆ ± (and ˆ ± ), at a given scale µ, are equal to their tree level L Chapter 3. (a2) Corrections to the One-Loop Matrix Elements of O 88 Four-Fermion ∆S =2Operators values. Consistently with the RI MOM schemes, one may impose the renormalization − conditions5: S= 1 S= 1 ˆ ± Tr(Pl ± Lm )=δlm, · (3.72) S= 1 S= 1 Tr( ± ˆ ± )=δ . Pl · Lm lm By inserting Eqs. (3.68) in the above relations, we obtain the renormalization matrices S= 1 S= 1 Z ± , ± in terms of known quantities: Z 1 S= 1 2 S= 1 T − Z ± = Zq D ± , 1 (3.73) S= 1 2 )/ S= 10T *− ± = Z ± , Z q D )/ 0 * where: S= 1 S= 1 S= 1 Dlm ± Tr(Pl ± Lm ± ), ≡ · (3.74) S= 1 S= 1 S= 1 ± Tr( ± ± ). Dlm ≡ Pl ·Lm S= 1 S= 1 S= 1 S= 1 Note that D ± and ± have the same matrix structure as Z ± and ± respec- D Z tively. When using perturbation theory it is more convenient to express them as:

d11± d12± d13± d14± d15± ± ± ± ± ± 2 ⎛ d21 d22 d23 d24 d25 ⎞ S= 1 g 4 D ± = 11+ d± d± d± d± d± + (g ), (3.75) 16 π2 ⎜ 31 32 33 34 35 ⎟ O ⎜ ⎟ ⎜ d41± d42± d43± d44± d45± ⎟ ⎜ ⎟ ⎜ d± d± d± d± d± ⎟ ⎜ 51 52 53 54 55 ⎟ ⎝ ⎠

δ11± 0000 ± ± 2 ⎛ 0 δ22 δ23 00⎞ S= 1 g 4 ± = 11+ 0 δ± δ± 00 + (g ). (3.76) D 16 π2 ⎜ 32 33 ⎟ O ⎜ ⎟ ⎜ 000δ44± δ45± ⎟ ⎜ ⎟ ⎜ 000δ± δ± ⎟ ⎜ 54 55 ⎟ ⎝ ⎠ In the parity violating case, as explained in Ref. [92] (Section 5.3), an equality holds between four pairs of matrix elements:

+ δ22 =+δ22− , (3.77) + δ = δ− , (3.78) 23 − 23 + δ = δ− , (3.79) 32 − 32 + δ33 =+δ33− . (3.80)

5 In (a2 ) perturbative calculations these conditions should also be imposed at a given renormalization O 4 scale, µ. Note however that due to the presence of Lorentz non-invariant quantities, such as ρ pρ, in the perturbative expressions for the Green’s functions, a choice of renormalization scale requires also a choice of the direction for the external momentum. ! 3.3. Mixing and Renormalization of and F on the Lattice. 89 OXY OXY

+ In addition, for the parity conserving projection the matrix elements d53, d53− give zero at the one-loop of perturbative theory:

+ d53 =0, (3.81)

d53− =0. (3.82)

S= 1 S= 1 With aim the perturbative calculation of the matrix elements of D ± ( ± ), we D XY F XY make use of our analytic expressions (Λ)dj and (Λ )dj to construct the amputated S= 1 S= 1 Green’s functions L ± ( ± ). By implementing these functions in Eq. (3.74) we L obtain the following “simple” and general expressions for each matrix element dl,m± (δl,m± ):

(0,1) (0,2) 2 (0,3) (0,4) dl,m± = ϵdl,m± + cSW ϵdl,m± + cSW ϵdl,m± + λϵdl,m±

(0,5) (0,6) 2 2 +(ϵdl,m± + λϵdl,m± ) ln(a p )

2 2 (2,3) (2,4) 2 (2,5) (2,6) + a p (ϵdl,m± + cSW ϵdl,m± + cSW ϵdl,m± + λϵdl,m± )(3.83) ) 2 2 2 (2,7) (2,8) 2 (2,9) (2,10) + p ln(a p )(ϵdl,m± + cSW ϵdl,m± + cSW ϵdl,m± + λϵdl,m± )

4 µ pµ (2,1) (2,2) + (ϵd± + λϵd± ) p2 l,m l,m ! * + (a3), O

(0,1) (0,2) 2 (0,3) (0,4) δl,m± = ϵδl,m± + cSW ϵδl,m± + cSW ϵδl,m± + λϵδl,m±

(0,5) (0,6) 2 2 +(ϵδl,m± + λϵδl,m± ) ln(a p )

2 2 (2,3) (2,4) 2 (2,5) (2,6) + a p (ϵδl,m± + cSW ϵδl,m± + cSW ϵδl,m± + λϵδl,m± )(3.84) ) 2 2 2 (2,7) (2,8) 2 (2,9) (2,10) + p ln(a p )(ϵδl,m± + cSW ϵδl,m± + cSW ϵδl,m± + λϵδl,m± )

4 µ pµ (2,1) (2,2) + (ϵδ± + λϵδ± ) p2 l,m l,m ! * + (a3). O

(i,j) (i,j) The quantities ϵdl,m± (ϵδl,m± )appearinginourresultsforthematrixelementsdl,m± (δl,m± ) are numerical coeﬃcients depending on the number of colors Nc,theWilsonparameters r,andtheSymanzikparametersforeachactionwehaveconsidered;theindexi denotes the power of the lattice spacing a that they multiply. Due to extremely lengthy results (i,j) (i,j) we provide the quantities ϵdl,m± , ϵδl,m± (Tables 3.2 - 3.9) only for the special choices:

Nc =3,rs = rd = rs′ = rd′ = 1, and tree-level Symanzik action. In all Tables the Chapter 3. (a2) Corrections to the One-Loop Matrix Elements of O 90 Four-Fermion ∆S =2Operators systematic errors in parentheses come from the extrapolation (L ) over finite lattice →∞ sizes. The (a2) correction terms of Eqs. (3.84) and (3.85), along with our previous (a2) O O calculation of Zq, are essential ingredients for minimizing the lattice artifacts which are S= 1 S= 1 present in non-perturbative evaluations of renormalization matrices Z ± and ± Z with the RI MOM method. − Asubtractionprocedure(similartothatofchapter2)oftheaforementioned (a2g2) O discretization effects from respective non-perturbative RI MOM estimates is an ongoing − project by members of the ETM Collaboration [94]. Some preliminary result can be found in Ref. [95]. In Figure 3.5 (taken from Ref. [95]) the behaviour of the renormalisation constant is shown as a function of the momentum squared in lattice units a2p2 for Z11 β =3.90 at the chiral limit. Discretization effects of (a2g2)havebeensubtracted O from the relevant correlation functions. Thus, the leading discretization effects on the renormalization constant are of (g4a2,g2a4). The amount of the subtraction depends O on the choice of the value for the gauge coupling. Two cases for the gauge coupling have been considered, the naive (g )andtheboostedone(g ). Result for the without 0 b Z11 considering any perturbative subtractions (indicated as “uncorrected” in the figure) are also shown.

11 Z Uncorrected 2 2 O(g 0 a ) 2 2 0,76 O(g b a )

0,72

0,68

0,64

0,6

0,56 0 0,5 1 1,5 2 2 2 p a Figure 3.5: RI MOM computation of the multiplicative renormalization fac- − tor at β =3.90. Z11 3.3. Mixing and Renormalization of and F on the Lattice. 91 OXY OXY 2) , +(2 ϵ d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.83396857(3) 0.39295395(5) 0.08962151(2) 0.57016434(2) 0.02849782(3) 0.92098603(9) -0.06601462(2) -0.27358566(2) -0.76054387(7) 1) , +(2 ϵ d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.89270365(2) 1.92846910(2) 2.01567488(3) 1.70275904(9) -3.06215787(3) -2.79899092(5) -0.98220341(2) -0.87361423(2) -1.07855468(7) 6) , . 2) , +(0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -8/3 -8/3 -8/3 -8/3 -8/3 ϵ d +(2 l,m ϵ d − 5) , 1) , 0 6 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 +(0 -8 -5 -5 1/3 +(2 l,m 17/3 ϵ d ϵ d and 4) , 6) , +(0 +(0 l,m ϵ d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ϵ d 0.1590523(6) 3.477157(2) -0.1931470(4) -3.772588(1) 12.293750(2) 10.134698(2) 10.816593(2) 15.316593(2) 14.816594(2) − 1) , +(0 l,m ϵ d 3) , +(0 ϵ d 0 0.129677758(6) 0.129677758(2) 0.51871103(2) 2.0748441(1) 3.6309772(1) 2.8529107(1) 0.30109257(1) 4.59385837(5) 2.38486989(5) 2.38486988(5) -0.25935552(2) -0.259355516(6) -0.77806655(2) -0.38903328(2) -2.34313283(3) -4.1496883(2) -0.51871103(2) -2.3341997(1) -1.03742206(3) -1.03742206(2) -0.15899133(2) -0.95394796(3) -3.3716217(2) -2.70285255(5) The coe ffi cients 2) Parity Conserving, Flavour Exchange Symmetry Plus , +(0 ϵ d 0 0.73755889(5) 0.73755892(2) 2.21267677(7) 1.10633834(5) 6.6380301(3) 2.9502357(1) 2.9502357(1) 1.47511779(6) 9.58826675(6) 9.5882656(6) 7.37558970(4) 7.37558970(3) -0.36877945(2) -5.9004712(3) -1.47511779(6) -0.368779461(8) -8.1131478(4) -0.49170598(2) -2.95023588(3) -3.68779471(5) -8.35900166(4) 11.8009423(6) -10.3258245(4) -10.32582548(4) Table 3.2: 1) , +(0 ϵ d 0 7.607190(2) 5.41774985(8) 7.4494060(1) 4.331175(2) 0.338609366(5) 1.35443746(2) 9.4810622(1) 9.732709(2) 1.1783609(6) 2.297078(2) -0.67721873(1) -0.67721873(1) -2.03165620(4) -0.8545378(4) -1.01582810(1) -9.615778(1) -8.8038435(2) -6.09496858(8) -2.70887493(5) -2.70887493(5) -1.35443746(2) 13.627614(2) 10.269733(2) -10.8354997(2) =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m Element =1 =1 =1 =1 =1 =2 =2 =2 =2 =2 =3 =3 =3 =3 =3 =4 =4 =4 =4 =4 =5 =5 =5 =5 =5 l l l l l l l l l l l l l l l l l l l l l l l l l Chapter 3. (a2) Corrections to the One-Loop Matrix Elements of O 92 Four-Fermion ∆S =2Operators 10) , 0 +(2 1/24 1/16 1/16 9/16 1/12 3/8 2/3 -1/24 -1/24 -1/8 -1/48 -1/6 -1/6 -1/12 -7/12 -1/3 41/48 17/12 35/48 59/48 25/24 13/24 ϵ d -11/24 -13/24 9) , +(2 ϵ d 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -1 -1 1/16 3/8 3/8 1/4 1/4 -1/4 -3/4 -1/8 11/8 8) , 2 0 -1 +(2 3/8 9/8 3/2 3/2 1/8 3/16 7/8 5/4 5/4 1/4 9/8 -3/16 -1/16 -1/12 -1/2 -1/4 -9/4 -7/4 13/8 -11/8 -17/12 -11/8 ϵ d 7) , 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 +(2 1/18 1/18 5/3 . -1/4 -3/4 -7/9 11/24 11/36 -17/54 -19/18 -23/27 ϵ d 10) , +(2 l,m ϵ d − 6) , 3) , +(2 +(2 l,m ϵ d ϵ d 0 0.0040994899(4) 0.05981201(1) 0.012298470(1) 0.016397959(2) 0.016397959(2) 0.23924803(5) 0.9569921(2) 1.3158642(3) 0.870210(3) 1.6747362(3) 0.009222(1) -0.0214962(6) -0.11962402(4) -0.17943602(4) -4.273362(4) -2.739903(4) -4.174196(4) -0.928529(3) -0.23924803(5) -1.0766161(2) -3.289500(4) -1.5551122(4) -1.9139843(4) -2.661259(4) 5) , +(2 ϵ d The coe ffi cients 0 0.06172133(2) 0.11866942(8) 0.18516399(5) 0.05900298(7) 0.24688532(7) 0.24688532(6) 0.0786706(1) 0.3540179(4) 0.5113591(7) 0.6293651(7) 0.08346609(1) 0.009152736(3) 0.12704697(1) -0.065051477(5) -0.01966766(2) -0.2373388(1) -0.9493554(4) -1.3053636(4) -0.5506945(6) -0.37712545(3) -0.023092586(8) -0.18110545(2) -0.49577614(3) -0.05164224(1) Table 3.3: 4) , Parity Conserving, Flavour Exchange Symmetry Plus +(2 ϵ d 0 0.16053253(1) 2.5685206(3) 3.5317158(3) 0.199822951(3) 0.64213013(6) 4.4949109(3) 4.6520712(2) 1.7473718(2) 1.84106755(5) 0.2526361(1) 4.0632560(3) -0.32106507(4) -0.399645902(8) -1.19893771(2) -0.48159760(4) -1.59858361(4) -1.59858361(3) -2.8895856(3) -0.64213013(6) -2.8633160(3) -3.9381018(3) -4.1738459(5) -5.1370411(5) -3.0948716(3) 3) , +(2 ϵ d 0 2.642227(4) 0.13736030(9) 0.35245840(2) 1.05737519(6) 0.789419(4) 0.20604045(9) 2.045124(4) 1.785683(1) 1.2362427(5) 1.40983359(8) 2.197764(1) 4.602710(3) 1.40983359(8) 0.2747206(1) 1.255191(1) 0.828945(4) -1.0988822(5) -1.5109631(5) -1.6447718(6) -0.06868015(3) -0.2747206(1) -7.416224(3) -1.9230442(7) -0.286209(4) =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m Element =1 =1 =1 =1 =1 =2 =2 =2 =2 =2 =3 =3 =3 =3 =3 =4 =4 =4 =4 =4 =5 =5 =5 =5 =5 l l l l l l l l l l l l l l l l l l l l l l l l l 3.3. Mixing and Renormalization of and F on the Lattice. 93 OXY OXY 2) , (2 − ϵ d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.68000502(3) 0.57016434(2) 0.46409210(5) 0.47726124(7) -0.57566914(3) -0.02841292(2) -0.08962151(2) -0.27358566(2) -0.92098603(9) 1) , (2 − ϵ d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.84126332(3) 1.34020247(2) 0.16287196(3) 1.92846910(2) 2.12575961(5) 2.15255184(7) -0.87361423(2) -0.89270365(2) -1.70275904(9) 6) , . (0 2) , − 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (2 -8/3 -8/3 -8/3 -8/3 -8/3 ϵ d − l,m ϵ d − 5) , 1) (0 , − 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (2 -8 -6 -4 − l,m -11 -5/3 -1/3 ϵ d ϵ d and 4) , 6) , (0 (0 − − l,m ϵ d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ϵ d 0.1931470(4) 9.748573(3) 3.772588(1) 8.771247(2) -3.068020(2) -0.0226732(6) 15.816593(2) 15.316593(2) 10.816593(2) − 1) , (0 − l,m ϵ d 3) , (0 − ϵ d 0 0.129677758(6) 0.51871103(2) 0.518711032(7) 1.55613310(2) 0.129677758(2) 0.51871103(2) 2.0748441(1) 1.2967776(1) 2.3341997(1) 2.07484413(3) 5.24671376(5) 2.07484413(3) 3.03772527(5) 0.79495663(2) 1.90789592(4) 3.38055490(3) 0.15899132(5) -0.38903328(2) -4.9277548(2) -4.1496883(2) -0.51871103(2) -0.47697397(5) -2.5935552(1) -1.25504053(1) The coe ffi cients 2) , Parity Conserving, Flavour Exchange Symmetry Minus (0 − ϵ d 0 1.10633834(5) 1.47511779(6) 7.3755889(4) 2.45852990(2) 7.37558979(4) 5.90047176(3) 0.49170598(4) 0.73755883(5) -0.36877945(2) -5.9004712(3) -1.47511779(6) -1.47511784(3) -4.42535353(9) -3.6877945(3) -1.47511779(6) -6.6380301(3) -5.9004714(1) -5.9004714(1) -0.368779461(8) -1.47511794(3) 14.0136190(6) 11.8009423(6) 16.22629734(4) 14.01362029(6) Table 3.4: 1) , (0 − ϵ d 0 5.41774985(8) 1.35443746(2) 1.35443746(2) 4.06331239(5) 3.38609366(9) 0.267862(2) 1.5317566(4) 0.338609366(5) 1.35443746(2) 9.564301(2) 6.09496858(8) 5.41774986(5) 5.41774986(6) -2.830716(3) -1.01582810(1) -6.7721873(1) -9.555272(2) -1.35443746(2) -1.1192154(6) -3.777376(2) 12.324653(1) 12.922182(2) -12.8671559(2) -10.8354997(2) =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m Element =1 =1 =1 =1 =1 =2 =2 =2 =2 =2 =3 =3 =3 =3 =3 =4 =4 =4 =4 =4 =5 =5 =5 =5 =5 l l l l l l l l l l l l l l l l l l l l l l l l l Chapter 3. (a2) Corrections to the One-Loop Matrix Elements of O 94 Four-Fermion ∆S =2Operators 10) , (2 − 0 1/12 1/12 1/4 7/6 1/16 1/3 1/3 1/12 5/12 5/12 2/3 ϵ d -1/48 -1/16 -1/12 -1/12 -5/24 -9/16 -3/8 -1/3 29/48 65/48 17/48 17/24 19/24 9) , (2 − ϵ d 0 0 0 0 0 2 2 0 0 0 0 0 0 1 0 0 1/2 1/16 3/2 1/4 1/4 1/4 5/8 -3/8 -3/8 8) , (2 1 2 0 − -3 -3 -1 3/16 5/12 1/12 7/8 3/4 1/4 5/4 -3/4 -9/4 -3/16 -1/16 -1/4 -1/4 -1/4 -5/8 -9/8 11/4 25/8 19/8 ϵ d 7) , (2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 − 1/2 3/2 5/9 . -5/24 -5/9 -2/3 13/54 -17/18 -13/9 -23/27 -43/36 ϵ d 10) , (2 − l,m ϵ d − 6) , 3) (2 , − (2 − l,m ϵ d ϵ d 0 0.0173967(6) 0.05981201(1) 0.031675(1) 0.23924803(5) 0.23924803(5) 0.9569921(2) 0.5981201(2) 1.0766161(2) 1.051220(3) -0.0081989797(6) -0.024596939(2) -0.032795919(2) -0.17943602(4) -0.032795919(2) -4.050432(4) -2.043017(4) -2.715306(4) -0.886608(3) -0.23924803(5) -1.755961(4) -4.149599(4) -1.1962402(3) -2.2728563(4) -1.9139843(4) 5) , (2 − ϵ d The coe ffi cients 0 0.05900298(7) 0.0786706(1) 0.40349227(3) 0.3933532(6) 0.28484158(3) 0.7473711(7) 0.6293651(7) -0.12344266(2) -0.37032798(6) -0.01966766(2) -0.49377065(7) -0.065051477(5) -0.2373388(1) -0.2373388(1) -0.49377065(8) -0.9493554(4) -0.5933471(4) -0.3540179(4) -0.045763706(3) -0.16693217(1) -0.33930637(2) -0.187841964(8) -0.05819067(1) -0.09480166(1) Table 3.5: 4) , Parity Conserving, Flavour Exchange Symmetry Minus (2 − ϵ d 0 0.79929180(1) 2.39787541(3) 0.16053253(1) 3.19716722(4) 2.5685206(3) 0.64213013(6) 1.6053254(3) 0.64213013(6) 2.8895856(3) 3.19716722(4) 0.199822951(3) 0.0632400(3) -0.48159760(4) -3.2106507(3) -0.64213013(6) -1.2631800(1) -3.4947437(3) -0.55680786(5) -3.0534875(2) -6.3359772(3) -6.1002363(5) -5.1370411(5) -7.4107631(3) -0.9473033(3) 3) , (2 − ϵ d 0 3.611581(4) 2.904169(4) 1.2923134(6) 0.20604045(9) 6.006391(3) 4.159875(4) 2.609845(1) 1.3736030(7) 2.197764(1) 6.491207(4) 0.2747206(1) 3.734320(4) -1.0988822(5) -0.2747206(1) -0.70491680(2) -2.11475039(7) -0.6868013(5) -0.06868015(3) -0.2747206(1) -1.2362427(5) -2.81966719(8) -2.042489(3) -2.8196672(1) -0.401784(1) =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m Element =1 =1 =1 =1 =1 =2 =2 =2 =2 =2 =3 =3 =3 =3 =3 =4 =4 =4 =4 =4 =5 =5 =5 =5 =5 l l l l l l l l l l l l l l l l l l l l l l l l l 3.3. Mixing and Renormalization of and F on the Lattice. 95 OXY OXY 2) , +(2 ϵδ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.83396857(3) 0.57016434(2) 0.39295395(5) 0.02849782(3) 0.92098603(9) 0.08962151(2) -0.27358566(2) -0.06601462(2) -0.76054387(7) 1) , +(2 ϵδ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.92846910(2) 2.01567488(3) 1.70275904(9) 0.89270365(2) -3.06215787(3) -0.87361423(2) -2.79899092(5) -0.98220341(2) -1.07855468(7) 6) , . 2) , +(0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -8/3 -8/3 -8/3 -8/3 -8/3 ϵδ +(2 l,m ϵδ − 5) , 1) , 1 6 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 +(0 -8 -5 -5 1/3 +(2 l,m 17/3 ϵδ ϵδ and 4) , 6) , +(0 +(0 l,m ϵδ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ϵδ 0.1590523(6) 3.477157(2) -3.772588(1) -0.1931470(4) 10.816593(2) 15.316593(2) 12.293750(2) 10.134698(2) 14.816594(2) − 1) , +(0 l,m ϵδ 3) , +(0 ϵδ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3.81579182(4) 2.38486989(5) 2.38486988(5) -2.86184387(3) -0.476973978(5) -0.15899133(2) -0.95394796(3) -2.70285255(5) The coe ffi cients Parity Violating, Flavour Exchange Symmetry Plus 2) , +(0 ϵδ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7.37558970(4) 7.37558970(3) -8.85070764(3) -0.49170598(2) -2.95023588(3) -1.475117940(5) -8.35900166(4) Table 3.6: 11.80094352(4) 1) , +(0 ϵδ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7.607190(2) 2.299519(2) 9.732709(2) 1.1783609(6) 2.297078(2) -1.1931472(4) 11.595958(2) 10.269733(2) -10.970215(1) =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m Element =1 =1 =1 =1 =1 =2 =2 =2 =2 =2 =3 =3 =3 =3 =3 =4 =4 =4 =4 =4 =5 =5 =5 =5 =5 l l l l l l l l l l l l l l l l l l l l l l l l l Chapter 3. (a2) Corrections to the One-Loop Matrix Elements of O 96 Four-Fermion ∆S =2Operators 10) , +(2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1/16 9/16 ϵδ -5/8 -1/16 35/48 35/48 59/48 25/24 31/24 9) , +(2 ϵδ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8) , 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 +(2 5/4 5/4 -1/4 -1/2 -1/12 -3/2 -17/12 ϵδ 7) , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 +(2 1/18 7/6 1/3 . -4/9 -7/9 -25/36 -17/54 -19/18 -23/27 ϵδ 10) , +(2 l,m ϵδ − 6) , 3) , +(2 ϵδ +(2 l,m ϵδ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.878409(3) 0.009222(1) -2.727604(4) -4.273362(4) -4.161897(4) -0.928529(3) -3.289500(4) -2.661259(4) -0.0194464(6) 5) , +(2 ϵδ The coe ffi cients 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0131834100(7) 0.079100460(4) 0.08346609(1) 0.009152736(3) 0.12704697(1) -0.023092586(8) -0.105467280(7) -0.05164224(1) Table 3.7: 4) Parity Violating, Flavour Exchange Symmetry Plus , +(2 ϵδ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3.8527794(2) 1.7473718(2) 0.64212984(5) 0.2526361(1) 4.0632560(3) -2.8633160(3) -5.1370395(3) -3.0948716(3) 3) , +(2 ϵδ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2.642227(4) 1.846794(4) 3.102499(4) 4.602710(3) 1.255191(1) 0.828945(4) -1.4685426(6) -6.711307(3) -0.286209(4) =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m Element =1 =1 =1 =1 =1 =2 =2 =2 =2 =2 =3 =3 =3 =3 =3 =4 =4 =4 =4 =4 =5 =5 =5 =5 =5 l l l l l l l l l l l l l l l l l l l l l l l l l 3.3. Mixing and Renormalization of and F on the Lattice. 97 OXY OXY 2) , (2 − ϵδ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.57016434(2) 0.68000502(3) 0.46409210(5) 0.47726124(7) -0.57566914(3) -0.27358566(2) -0.02841292(2) -0.92098603(9) -0.08962151(2) 1) , (2 − ϵδ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.84126332(3) 1.92846910(2) 1.34020247(2) 0.16287196(3) 2.12575961(5) 2.15255184(7) -0.87361423(2) -1.70275904(9) -0.89270365(2) 6) , . (0 2) , − 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (2 -8/3 -8/3 -8/3 -8/3 -8/3 ϵδ − l,m ϵδ − 5) , 1) (0 , − 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (2 -8 -6 -4 − l,m -11 -5/3 -1/3 ϵδ ϵδ and 4) , 6) , (0 (0 − − l,m ϵδ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ϵδ 9.748573(3) 3.772588(1) 8.771247(2) 0.1931470(4) -3.068020(2) -0.0226732(6) 10.816593(2) 15.316593(2) 15.816593(2) − 1) , (0 − l,m ϵδ 3) , (0 − ϵδ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3.81579182(4) 5.24671376(5) 0.79495663(2) 1.90789592(4) 2.86184387(3) 0.15899132(5) -0.476973978(5) -0.47697397(5) The coe ffi cients 2) Parity Violating, Flavour Exchange Symmetry Minus , (0 − ϵδ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2.45852990(2) 8.85070764(3) 5.90047176(3) 0.49170598(4) -1.47511794(3) -1.475117940(5) Table 3.8: 11.80094352(4) 16.22629734(4) 1) , (0 − ϵδ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2.299519(2) 1.1931472(4) -2.830716(3) -9.555272(2) -1.1192154(6) -3.777376(2) 10.970215(1) 11.595958(2) 12.922182(2) =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m Element =1 =1 =1 =1 =1 =2 =2 =2 =2 =2 =3 =3 =3 =3 =3 =4 =4 =4 =4 =4 =5 =5 =5 =5 =5 l l l l l l l l l l l l l l l l l l l l l l l l l Chapter 3. (a2) Corrections to the One-Loop Matrix Elements of O 98 Four-Fermion ∆S =2Operators 10) , (2 − 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5/8 5/12 1/16 ϵδ -1/16 -9/16 35/48 65/48 17/48 31/24 9) , (2 − ϵδ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8) , (2 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 5/12 1/12 3/2 -1/4 -1/4 11/4 ϵδ 7) , (2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 5/9 . -7/6 -4/9 -5/9 -1/3 13/54 -25/36 -17/18 -23/27 ϵδ 10) , (2 − l,m ϵδ − 6) , (2 3) , − (2 ϵδ − l,m ϵδ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.031675(1) 1.051220(3) 0.0194464(6) -2.727604(4) -0.878409(3) -4.161897(4) -4.050432(4) -2.043017(4) -1.755961(4) 5) , (2 − ϵδ The coe ffi cients 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0131834100(7) -0.045763706(3) -0.16693217(1) -0.079100460(4) -0.105467280(7) -0.187841964(8) -0.05819067(1) -0.09480166(1) Table 3.9: 4) , Parity Violating, Flavour Exchange Symmetry Minus (2 − ϵδ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0632400(3) 0.64212984(5) -1.2631800(1) -3.4947437(3) -3.8527794(2) -5.1370395(3) -7.4107631(3) -0.9473033(3) 3) , (2 − ϵδ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3.611581(4) 1.846794(4) 1.4685426(6) 6.711307(3) 3.102499(4) 6.491207(4) 3.734320(4) -2.042489(3) -0.401784(1) =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 =1 =2 =3 =4 =5 ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m ,m Element =1 =1 =1 =1 =1 =2 =2 =2 =2 =2 =3 =3 =3 =3 =3 =4 =4 =4 =4 =4 =5 =5 =5 =5 =5 l l l l l l l l l l l l l l l l l l l l l l l l l Chapter 4

Renormalization Constants for Fermion Field and Ultra-Local Operators in Twisted Mass QCD

In this chapter we present perturbative and non-perturbative results on the renormalization constants of the fermion ﬁeld and six ultra-local fermion bilinears: scalar, pseudoscalar, vector, axial, tensor (with and without γ5). The perturbative computation is performed for general fermion action, which includes the clover term and the twisted mass parameter, so that the results are applicable for Wilson/Clover/Twisted mass actions. In the gluon sector we use the Symanzik improved action and we evaluate all quantities for ten sets of values for the Symanzik coeﬃcients (Wilson, tree-level Symanzik, Iwasaki, TILW, DBW2). Non-perturbative results are obtained using the twisted mass Wilson fermion formulation employing two degenerate dynamical quarks and the tree- level Symanzik improved gluon action. The simulations have been performed for pion masses in the range of about 450-260 MeV and at three values of the lattice spacing, a, corresponding to β =3.9, 4.05, 4.20. The perturbative work is carried out at 1-loop level and up to second order in the lattice spacing, and we use these lattice correction terms to improve the non-perturbative estimates. In particular, for each renormalization factor we subtract its perturbative (a2)termsfromthenon-perturbativeresultsofthesamequantity,sothatweeliminate O unwanted lattice artifacts.

The renormalization conditions are deﬁned in the RI′-MOM scheme, for both perturbative and non-perturbative results. The renormalization factors, obtained for diﬀerent values of the renormalization scale, are evolved perturbatively to a reference scale µ =2 GeV. In addition, they are translated to MS at 2 GeV using 3-loop perturbative results for the conversion factors. Our results depend on a large number of parameters: coupling constant, number of col-

99 Chapter 4. Renormalization Constants for Fermion Field and Ultra-Local 100 Operators in Twisted Mass QCD ors, lattice spacing, external momentum, lagrangian mass, twisted mass parameter, clover parameter, Symanzik coeﬃcients, gauge parameter. The dependence on these parameters is shown explicitly in our expressions, except the dependence on Symanzik parameters, which cannot shown analytically. We thus provide the perturbative expressions for ten diﬀerent sets of gluon actions.

4.1 Introduction

This chapter is organized as follows: in Section 4.2 we give the expressions for the fermion and gluon actions we employed, and define the operators. Sections 4.3 and 4.4 concentrate on the perturbative procedure, and the (a2)-corrected expressions for the renormaliza- O tion constants Zq and Z . Moreover, in Section 4.5 we provide the renormalization O prescription of the RI′-MOM scheme, and we discuss alternative ways for its application, while in Section 4.8 we provide all necessary formulae for the conversion to MS and the evolution to a reference scale of 2 GeV. Section 4.9 focuses on the non-perturbative computation, where we explain the different steps of the calculation. Moreover, we provide the renormalization prescription of the RI′-MOM scheme, and we discuss alternative ways for its application. The main results of this work are presented in Section 4.10: the reader can find numerical values for the Z-factors of the fermion field and fermion operators, which are computed non-perturbatively and corrected using the perturbative (a2) O terms presented in Section 4.5. Since in general Z-factors depend on the renormalization scale, we also provide results in the RI′-MOM scheme at a reference scale, µ 1/a.For ∼ comparison with phenomenological and experimental results, we convert the Z-factors to the MS scheme at 2 GeV. In Section 4.11 we give our conclusions.

4.2 Formulation

4.2.1 Lattice actions

Our perturbative calculation makes use of the twisted mass fermion action including the usual clover (SW) term with general clover parameter. For NF ﬂavor species this action reads, in standard notation:

a3 S = ψ¯ (x)(r γ ) U ψ (x + aµ)+ψ¯ (x + aµ)(r + γ ) U ψ (x) F − 2 f − µ x, x+aµ f f µ x+aµ,x f x,& f, µ ' ( 4r + a4 ( + mf + iµ γ τ 3)ψ¯ (x)ψ (x) a 0 0 5 f f &x, f a5 rc ψ¯ (x)σ F (x)ψ (x), (4.1) − 4 SW f µν µν f x,& f, µ, ν 4.2. Formulation 101

where the Wilson parameter r is set to r =1,f is a flavor index, σµν =[γµ,γν]/2and the clover coefficient cSW as well as the twisted mass parameter µ0 are kept as a free parameters throughout. The fermionic action we employ in the non-perturbative calculation is the purely twisted mass action (no clover term), which for two degenerate flavors of quarks in twisted mass QCD is given by:

4 3 SF = a χ(x)(DW [U]+m0 + iµ0γ5τ )χ(x) , (4.2) x & 3 with τ the Pauli matrix acting in the isospin space, µ0 the bare twisted mass and DW the massless Wilson-Dirac operator (deﬁned in Ref. [96]). Maximally twisted Wilson quarks are obtained by setting the untwisted bare quark mass m0 to its critical value mcr,while the twisted quark mass parameter µ0 is kept non-vanishing in order to give the light quarks their mass. In Eq. (4.2) the quark ﬁelds χ are in the so-called “twisted basis”. The “physical basis” is obtained for maximal twist by the simple transformation:

iω iω π ψ(x)=exp γ τ 3 χ(x), ψ(x)=χ(x)exp γ τ 3 ,ω= . (4.3) 2 5 2 5 2 2 3 2 3 In terms of the physical ﬁelds the action is given by:

1 ar Sψ = a4 ψ(x) γ [−→ + −→∗ ] iγ τ 3 −→ −→∗ + m + M ψ(x). (4.4) F 2 µ ∇ µ ∇ µ − 5 − 2 ∇ µ ∇ µ cr x & 2 , - 3 One can check that this action is equivalent from the action in the twisted basis Eq. (4.2), just by performing the rotations of Eq. (4.3) and identifying m0 = mcr and M = µ. For gluons we employ the Symanzik improved action, involving Wilson loops with 4 and 6 links (1 1 plaquette,1 2 rectangle,1 2 chair,and1 1 1 parallelogram × × × × × wrapped around an elementary 3-d cube), which is given by the relation:

2 S = c Re Tr 1 U + c Re Tr 1 U G g2 0 { − plaq.} 1 { − rect.} 0 rect. ) plaq&. & + c Re Tr 1 U + c Re Tr 1 U . (4.5) 2 { − chair} 3 { − paral.} chair& paral&. *

The coeﬃcients ci can in principle be chosen arbitrarily, subject to the following normalization condition, which ensures the correct classical continuum limit of the action:

c0 +8c1 +16c2 +8c3 =1. (4.6)

Some popular choices of values for ci used in numerical simulations will be considered in this work, and are itemized in Table 2.1; they are normally tuned in a way as to Chapter 4. Renormalization Constants for Fermion Field and Ultra-Local 102 Operators in Twisted Mass QCD ensure (a2) improvement in the pure gluon sector. In our non-perturbative computation O presented here we employ the tree-level Symanzik action (c =5/3, c = 1/12, c = 0 1 − 2 c3 =0).Our1-loopFeynmandiagramsdonotinvolvepuregluonvertices,andthegluon propagator depends only on three combinations of the Symanzik parameters:

C c +8c +16c +8c =1, 0 ≡ 0 1 2 3 C c + c , (4.7) 1 ≡ 2 3 C c c c . 2 ≡ 1 − 2 − 3

Therefore, with no loss of generality all these sets of values have c2 =0.

4.2.2 Deﬁnition of operators and Renormalization condition

The ultra-local fermion operators considered here are the following:

ψτ¯ aψa=1, 2 a a S =¯χτ χ = (4.8) O ⎧ iψγ¯ 11ψa=3 ⎨ − 5 ψγ¯ τ aψa=1, 2 a a ⎩ 5 P =¯χγ5τ χ = (4.9) O ⎧ iψ¯11ψa=3 ⎨ − ¯ 2 ⎩ ψγ5γµτ ψa=1 a a ⎧ 1 V =¯χγµτ χ = ψγ¯ 5γµτ ψa=2 (4.10) O ⎪ − ⎨⎪ 3 ψγ¯ µτ ψa=3

⎪ ¯ 2 ⎩⎪ ψγµτ ψa=1 a a ⎧ 1 A =¯χγ5γµτ χ = ψγ¯ µτ ψa=2 (4.11) O ⎪ − ⎨⎪ ¯ 3 ψγ5γµτ ψa=3 ⎪ ⎪ ψσ¯ τ aψa=1, 2 a a ⎩ µν T =¯χσµν τ χ = (4.12) O ⎧ iψγ¯ σ 11ψa=3 ⎨ − 5 µν ψγ¯ σ τ aψa=1, 2 a a ⎩ 5 µν Tp =¯χγ5σµν τ χ = (4.13) O ⎧ iψσ¯ 11ψa=3 ⎨ − µν ⎩a a a a a a We denote the corresponding Z-factors by ZS ,ZP,ZV,ZA,ZT,ZTp.Inamasslessrenor- malization scheme the renormalization constants are deﬁned in the chiral limit, where iso-spin symmetry is recovered. Hence independent of a =1, 2, 3thesameZ-factors are obtained and we drop the a index on the Z-factors from here on. Still note that for 1 3 instance the physical ψγ¯ µτ ψ is renormalized with ZA while ψγ¯ µτ ψ needs the ZV, which diﬀer from each other even in the chiral limit. 4.3. Corrections to the Fermion Propagator 103

The renormalization constants are computed both perturbatively and non-perturbatively in the RI′-MOM scheme at diﬀerent renormalization scales. We translate them to the MS scheme at (2 GeV)2 using a conversion factor computed in perturbation theory to (g6) O as described in Section 4.8. The Z-factors are determined by imposing the following conditions in the massless theory, i.e., at critical mass and vanishing twisted mass

1 L 1 (0) Zq = Tr (S (p))− S (p) (4.14) 12 p2=µ2 I $ %I 1 µν 1 L (0) 1 I Zq− Z Tr Γµν (p)Γµν − (p) =1, (4.15) O 12 p2=µ2 I $ %I where the trace is taken over spin and color indices, µI is the renormalization scale, while (0) (0) SL and ΓL correspond to the perturbative or non-perturbative results, and S , Γµν are deﬁned as below:

i γρ sin(pρ) S(0)(p)=− ρ (4.16) sin(p )2 !ρ ρ (0) !˜ Γµν (p)= i µ sin(pν ) . (4.17) − O{ }

The trace is taken over spin and color indices. For alternative renormalization prescriptions the reader can refer to Ref. [96]. (0) (0) The choices for S and Γµν given in Eqs. (4.16) - (4.17) are optimal, since we obtain

Zq =1,Z =1whenthegaugeﬁeldissettounity.Similarly,intheperturbative O computation this condition leads to Zq =1,andZ =1attree-level. O

4.3 Corrections to the Fermion Propagator

The fermion propagator of the interacting theory is given by the following 2-point correlation function (Green’s function), with the various quantities computed in perturbation theory:

π fg a 4 a,f b,g d p ab ip(x y) 1 − − χα (x)¯χβ (y) = 4 δ e Stree Spert.(p) Stree , (4.18) ⟨ ⟩ π (2π) · · αβ 1− a 2 3

1 with Spert− .(p) being the inverse propagator in momentum space computed perturbatively up to a desired order

1 f g f ′ g′ amp. S− (p)=δ ′ ′ χ (p)¯χ (p) . (4.19) pert. ⟨ α′ β′ ⟩

Stree is the tree-level propagator for the twisted mass action:

1 Stree = , (4.20) 5 3 i/p◦ + M(p)+iµ0γ τ Chapter 4. Renormalization Constants for Fermion Field and Ultra-Local 104 Operators in Twisted Mass QCD 2 r 1 M(p) mf + sin2(ap /2),/p◦ γ sin(ap ), (4.21) ≡ 0 a µ ≡ µ a µ µ µ & & where α,β are Dirac indices, f, g are ﬂavor indices in the fundamental representation of

SU(NF ), and a, b are color indices in the fundamental representation of SU(Nc). The dot product runs over ﬂavor and Dirac indices. Due to the diagonal form of the τ 3 matrix, and since we are studying the case of only two degenerate quarks (up/down) we can simplify 3 the expression of Stree and omit τ by giving a ﬂavor index to the twisted mass parameter, and at the same time we take mf m : 0 → 0 1 Stree = (f) , (4.22) ◦ 5 i/p + M(p)+iµ0 γ where now µ(1) =+µ for the up quark propagator and µ(2) = µ for the down quark 0 0 0 − 0 propagator. The correlation function of Eq. (4.18) can be interpreted physically as the amplitude for propagation of a particle or excitation between y and x.The1-loopFeyn- man diagrams that enter our 2-point amputated Green’s function calculation (Eq. (4.19)), are illustrated in Fig. 2.3.

For the algebraic operations involved while evaluating Feynman diagrams, we make use of our symbolic package in Mathematica. In a nutshell, the required steps for the computation of a Feynman diagram are the following (the reader can ﬁnd more details in Ref. [97]): A preliminary expression for each diagram is obtained by contracting the appropriate • vertices, which is performed automatically once the algebraic expression of the vertices and the topology (“incidence matrix”) of the diagram are speciﬁed. To limit the proliferation of the algebraic expressions we exploit symmetries of the theory, and we simplify the color dependence, Dirac matrices and tensor structures. The (a2) computation introduces several complications, especially when isolating • O logarithms and Lorentz non-invariant terms, which leads to a whole family of infrared divergent integrals. These can be reduced to a minimal set of approximately 250 primitive integrals. This is achieved by using two kinds of subtractions among the propagators:

2 4 sin4(q /2) 4 sin2(q /2) 4 m sin2(q /2) 1 1 µ µ − µ µ − 0 µ µ 2 = 2 + 2 2 (4.23) q˜ qˆ + ! ,! q˜ qˆ - ! .

D(q)=D (q)+ D(q) D (q) (4.24) plaq − plaq " 1 # 1 = D (q)+D (q) D− (q) D− (q) D(q) plaq plaq plaq − " # 4.3. Corrections to the Fermion Propagator 105

The denominator of the fermion propagator,q ˜2, is deﬁned as

r 2 q q˜2 = sin2(q )+ m + qˆ2 +(µ(f))2, qˆ2 =4 sin2( µ ) . (4.25) µ 0 2 0 2 µ µ & , - &

1 1 D is the 4 4 Symanzik gluon propagator; the expression for the matrix D− (q) D− (q) , × plaq − which is (q4), is independent of the gauge parameter, λ,anditcanbeeasilyobtained O / 0 in closed form. Moreover, we have

δ qˆ qˆ (D (q)) = µν (1 λ) µ ν (4.26) plaq µν qˆ2 − − (ˆq2)2

Terms in curly brackets of Eqs. (4.23) and (4.25) are less IR divergent than their unsubtracted counterparts, by two powers in the momentum. These subtractions are performed iteratively until all primitively divergent integrals (initially depending on the fermion and the Symanzik propagator) are expressed in terms of the Wilson gluon propagator. The computation of the divergent integrals is performed in a noninteger number of dimensions D>4. Ultraviolet divergences are explicitly isolated à la Zimmermann and evaluated as in the continuum. The remainders are D-dimensional, parameter-free, zero external momentum lattice integrals which can be recast in terms of Bessel functions, and finally expressed as sums of a pole part plus numerical constants. A small subset of the infrared divergent integrals, shown in Section 4.7, contains the most demanding ones in the list; they must be evaluated to two further orders in a,beyond the order at which an IR divergence initially sets in. As a consequence, their evaluation requires going to D>6 dimensions. A correct way to evaluate strong divergent integrals is given in detail in Ref. [97]. The required numerical integrations of the algebraic expressions for the loop inte- • grands are performed by highly optimized Fortran programs; these are generated by our Mathematica ‘integrator’ routine. Each integral is expressed as a sum over the discrete Brillouin zone of finite lattices, with varying size L (44 L4 1284), and evaluated for ≤ ≤ all values of the Symanzik coefficients listed inTable2.1(correspondingtothePlaquette, Symanzik, Iwasaki, TILW and DBW2 action). The last part of the evaluation is the extrapolation of the numerical results to infinite • lattice size. This procedure entails a systematic error, which is reliably estimated, using asophisticatedinferencetechnique;forone-loop quantities we expect a fractional error 7 smaller than 10− .

1 Next, we provide the total expression for the inverse fermion propagator Spert− .(p), computed up to 1-loop in perturbation theory. Here we should point out that for dimensional reasons, there is a global prefactor 1/a multiplying our expressions for the inverse propagator, and thus, the (a2) correction is achieved by considering all terms up to (a3). O O Chapter 4. Renormalization Constants for Fermion Field and Ultra-Local 106 Operators in Twisted Mass QCD

The most general expression for the inverse propagator appears in the m-ﬁle.m, under the name:

1 f g 3 4 S− = δ ′ ′ propagator[Action, csw, beta, g2tilde, aL, m, mu] + (a ,g ) . (4.27) pert. O L 2 3 where the variable Action can take the a value from 1 to 10 which correspond to one of the actions of Table 2.1. The Lagrangian mass is denoted by m, the twisted mass parameter by mu, aL is the lattice spacing and beta is the gauge ﬁxing parameter (beta=1-λ). In the main text we provide a more compact expression, for special values of the various parameters, that is tree-level Symanzik improved gluon action, cSW = 0, landau gauge (beta=1), but we keep the Lagrangian mass and the twisted mass parameter general. 4.3. Corrections to the Fermion Propagator 107

1 f ′ g′ 3 4 S− = δ propagator[2, 0, 1, g2tilde, aL, m, mu] + (a , g ) (4.28) pert. O L 2 3 a2 ip3 γρ1 5 ρ1 aLδα′ β′ p2 L ρ1 α′ β′ = mδα′ β′ + iµγα β + ipρ1 γ + ′ ′ α′ β′ 2 − 6

2 p2 2 ρ1 2 2 2 3M ln[1 + M 2 ] + g˜ 13.0232725(2)ipρ1γα β + δα′ β′ m 0.5834586(2) 3 ln[aLM + aLp2] 4− ′ ′ 4 − − p2 5

2 p2 5 2 2 2 3M ln[1 + M 2 ] + iµγα β 8.7100834(2) 3 ln[aLM + aLp2] + aL δα′ β′ 10.69642965(5)p2 ′ ′ 4 − − p2 5 4 4−

2 2 p2 2 6M m ln[1 + 2 ] 3p2 3M 0.8530378(3)m2 1.842911859(4)M 2 + M + +3m2 + ln[a2 M 2 + a2 p2] − − p2 2 2 L L 2 3 5

2 4 p2 ρ1 3M 3 2 2 2 3M ln[1 + M 2 ] + impρ1 γα β 0.3393996(2) + + ln[aLM + aLp2] 2 ′ ′ 4 2p2 2 − 2p2 5

2 p2 5 2 2 2 6M ln[1 + M 2 ] + iµmγα β 6.68582372(4) + 3 ln[aLM + aLp2] + ′ ′ 4− p2 5 5

4 6 2 2 2 2 M M 3m p2 + aL mδα′ β′ 2.3547298(1)p2+2.3562747(1)m +3.46524146(4)M + 2 2 4 4 − 6p2 3p2 − 2(M + p2)

2 6 2 p2 p2 11M p2 M M ln[1 + 2 ] +3m2 + ln[a2 M 2 + a2 p2] + 9m2 2M 2 M − 4 3 L L 3 − − − 3p22 p2 2 3 2 3 5

M 4 M 6 3m2 p2 + iµγ5 0.70640552(8)p2+6.79538844(2)m2 +1.16985307(3)M 2 + α′ β′ 2 2 4 − 6p2 3p2 − 2(M + p2)

2 2 6 2 p2 p2 2M p2 M M M ln[1 + 2 ] +3m2 + ln[a2 M 2 + a2 p2] + 9m2 M − 4 3 L L 3 − − 2 − 3p22 p2 2 3 2 3 5

5 2 4 6 6 2 p2 p4(mδα β + iµγ ) 1 2M M 2M 1 2M M ln[1 + 2 ] + ′ ′ α′ β′ + + + M p2 2 − 9p2 3p22 − 3p23 −3 3p23 p2 4 2 3 5

2 2 4 6 8 ρ1 2 2 9m M 209M M 7M +ipρ γ 1.1471643(7)p2 0.2145514(2)m +1.15904388(6)M + 1 α′ β′ 2 3 4 − − 2p2 − 360p2 − 240p2 40p2

2 2 2 2 4 6 4 p2 73p2 3m 2M 1 9m 43M M 7M M ln[1 + 2 ] + + ln[a2 M 2 + a2 p2] + + + M − 360 2 3 L L 24 2p2 72p2 − 12p22 − 40p23 p2 2 3 2 3 5

67M 2 M 4 8M 6 7M 8 157 + ip3 γρ1 4.2478764(2) + + ln[a2 M 2 + a2 p2] ρ1 α β 2 3 4 L L ′ ′ 4 − 120p2 120p2 − 15p2 30p2 − 180

2 4 6 4 p2 ρ1 2 4 6 1 5M 5M 7M M ln[1 + 2 ] ip4pρ1 γ 7 M 67M 13M + + + M + α′ β′ + + + 2 18p2 12p22 − 30p23 p22 p2 240 48p2 72p22 24p23 2 3 5 4

8 2 4 6 4 p2 7M 5 5M M 7M M ln[1 + 2 ] + + M + (a3 , g4 ) − 12p24 −12 − 4p2 − 4p22 12p23 p22 O L 2 3 5 55 To make the above expressions less complicated we deﬁned m m and M 2 = m2+µ2.We ≡ 0 0 0 Chapter 4. Renormalization Constants for Fermion Field and Ultra-Local 108 Operators in Twisted Mass QCD would like to point out that the up quark propagator is obtained by the choice µ =+µ0, 2 2 g CF while for the down quark propagator one should choose µ = µ0.Moreover,˜g 16π2 2 − ≡ Nc 1 and CF − . A summation over the indices ρi is implied. ≡ 2Nc Another byproduct of this part of the computation is the additive critical fermion mass; its general expression depends on cSW and the Symanzik parameters. These are terms proportional to 1/a that have been left out of Eq. (4.27) for conciseness:

1 m =mcritical[Action, csw, g2tilde, aL] + (g4)(4.29) c aO 2 g˜ ( 1,1) ( 1,2) ( 1,3) 2 1 4 = ε − + ε − c + ε − c + (g ) . − a m m SW m SW aO ) * (i,j) The quantities εm (listed in Table 4.1) are numerical coeﬃcients depending on the Symanzik parameters; the index i denotes the power of the lattice spacing a that they multiply, thus i = 1. −

( 1,1) ( 1,2) ( 1,3) Action εm− εm− εm− Plaquette -51.4347118(2) 13.73313097(5) 5.71513853(1) Symanzik -40.44324019(7) 11.94821988(5) 4.662672112(4) TILW (8.60) -34.17747288(3) 10.76516514(3) 3.998348778(3) TILW (8.45) -33.9488671(1) 10.71947605(3) 3.97345187(1) TILW (8.30) -33.6344391(1) 10.65632621(4) 3.939135834(8) TILW (8.20) -33.43979350(6) 10.61705314(7) 3.917851255(1) TILW (8.10) -33.1892274(1) 10.56629305(3) 3.890401337(1) TILW (8.00) -32.87904072(9) 10.50313393(3) 3.856345868(2) Iwasaki -26.07292275(7) 9.01533524(3) 3.1061330684(3) DBW2 -11.5127475(2) 4.9953066(1) 1.351772367(3)

( 1,1) ( 1,2) ( 1,3) Table 4.1: Numerical results for the coefficients εm− ,εm− ,εm− (Eq. (4.29)) for different actions. The systematic errors in parentheses come from the extrapolation over finite lattice size, L . →∞

4.4 Corrections to Fermion Bilinear Operators

In the context of this work we also study the perturbative (a2)correctionstoGreen’s O functions of local fermion operators (currents) that have the form:

f ′′ g′′ OΓ = χ¯ (z)Γα β χ (z) . (4.30) α′′ ′′ ′′ β′′ z & α&′′ β′′ , - We restrict ourselves to forward matrix elements (i.e. 2-point Green’s functions, zero momentum operator insertions). The symbol Γcorresponds to the following set of products 4.4. Corrections to Fermion Bilinear Operators 109 of the Euclidean Dirac matrices:

5 5 5 1 Γ S, P, V, A, T, T ′ 11,γ,γ ,γγ ,γσ ,σ ; σ = [γ ,γ ], (4.31) ∈{ }≡{ µ µ µν µν } µν 2 µ ν

for the scalar OS, pseudoscalar OP ,vectorOV , axial OA,tensorOT and tensor prime OT ′ operator, respectively. The matrix element of OT ′ ,canberelatedtothematrixelementof

OT ; this is a nontrivial check for our calculational procedure [97]. The exact relationship between the amputated 2-point Green function ΛT and ΛT ′ is summarized in the following equation:

µν 1 µ′ ν′ Λ = ϵµνµ ν Λ . (4.32) T −2 ′ ′ T ′ &µ′ ν′

The matrix elements of the above set of fermion bilinear operators can be obtained by an analogous expression to Eq. (4.18):

π fg a 4 a,f b,g d p ab ip(x y) pert. χα (x)OΓχ¯β (y) = 4 δ e − Stree ΛΓ (p) Stree , (4.33) ⟨ ⟩ π (2π) · · αβ 1− a 2 3

pert. where ΛΓ (p) is the amputated 2-point Green’s function of each operator OΓ, in momentum space, which upon indices contraction becomes:

pert. f f g g amp. Λ (p)= χ (p) χ¯ (p)Γα β χ (p) χ¯ (p) . (4.34) Γ ⟨ α α′ ′ ′ β′ β ⟩ α&′ β′ , -

The only 1-particle irreducible Feynman diagram that enters the calculation of the above operators is shown in Fig. 2.4.

In this diagram there are two fermion propagator, for which we allowed diﬀerent µ- values, in order to have more general results. In other words, the fermion lines on the left and on the right of the current insertion (see Fig. 2.4) can represent the up or down propagator, that is

OΓ =¯u(z)Γu(z) ,

OΓ =¯u(z)Γd(z) ,

OΓ = d¯(z)Γu(z) , ¯ OΓ = d(z)Γd(z) .

For the evaluation of the Z-factors, one need the combination: OΓ =¯u(z)Γd(z) .

The amputated Green’s functions of the OΓ operator are given in the m-ﬁle.m by the Chapter 4. Renormalization Constants for Fermion Field and Ultra-Local 110 Operators in Twisted Mass QCD expressions:

Λpert. = δ δ scalar[Action, csw, beta, g2tilde, aL, m, mu1, mu2] + (a3 ,g4 ) , (4.35) S f ′ f ′′ g′ g′′ O L 2 3 Λpert. = δ δ pseudoscalar[Action, csw, beta, g2tilde, aL, m, mu1, mu2] + (a3 ,g4 ) , (4.36) P f ′ f ′′ g′ g′′ O L 2 3 Λpert. = δ δ vector[Action, csw, beta, g2tilde, aL, m, mu1, mu2] + (a3 ,g4 ) , (4.37) V f ′ f ′′ g′ g′′ O L 2 3 Λpert. = δ δ axial[Action, csw, beta, g2tilde, aL, m, mu1, mu2] + (a3 ,g4 ) , (4.38) A f ′ f ′′ g′ g′′ O L 2 3 Λpert. = δ δ tensor[Action, csw, beta, g2tilde, aL, m, mu1, mu2] + (a3 ,g4 ) , (4.39) T f ′ f ′′ g′ g′′ O L 2 3 pert. 3 4 Λ = δf f δg g tensorprime[Action, csw, beta, g2tilde, aL, m, mu1, mu2] + (a ,g ) , (4.40) T ′ ′ ′′ ′ ′′ O L 2 3 where most of the variables are deﬁned below Eq. (4.27). As mentioned above, one may choose the two fermion propagators of the diagram to correspond either to the up or down quark, thus there are two twisted mass parameters mu1, and mu2. Theses can have any sign and the only restriction is: µ = µ . | 1| | 2|

4.5 Quark Field and Quark Bilinear Renormalization

Constants in the RI′-MOM Scheme

The RI′-MOM renormalization scheme consists in imposing that the forward amputated Green function ΛΓ(p), computed in the chiral limit and at a given (large Euclidean) scale p2 = µ2, is equal to its tree-level value. In practice, the renormalization condition is implemented by requiring in the chiral limit that1:

1 1 Z− Z (p) =1, (p) Tr Λ (p) P , (4.42) q Γ VΓ |pρ=µρ VΓ ≡ 4 Γ · Γ ) * 1 A simpler version of Eq. (4.42) is given by the relation:

1 1 tree 1 tree tree Zq− ZΓ Tr ΛΓ(p) ΛΓ = Tr ΛΓ ΛΓ , (4.41) 4 · pρ=µρ 4 · ) * ) * tree where ΛΓ is the tree-level value of ΛΓ(p). 4.5. Quark Field and Quark Bilinear Renormalization Constants in the RI′-MOM Scheme 111 where PΓ are the Dirac projectors deﬁned as follows:

P P ,P ,P ,P ,P ,P 11,γ5,γ, γ5γ , γ5σ , σ ;(4.43) Γ ∈{ S P V A T T ′ }≡{ µ − µ − µν − µν } they are chosen to obey the relation Tr[Γ P ] 1. The traces are always taken over the · Γ ≡ spin indices. The quark ﬁeld RC Zq, which enters Eq. (4.42), is obtained by imposing, again in the chiral limit, the condition2:

1 1 i a ρ γρ sin(apρ) 1 Z− (p) =1, (p) Tr S− (p) . (4.45) q Vq |pρ=µρ Vq ≡−4 1 sin2(ap ) · 6 a2! ρ ρ 7 ! We compute Zq in the RI′-MOM renormalization scheme, deﬁned in Eq. (4.14), which can be Taylor expanded in order to take into account all (a2) terms. This leads to: O

a2 3 4 i γρ(pρ p ) a2 p ρ − 6 ρ ρ ρ 1 4 2 4 Zq = Tr 2 1+ 2 S1− loop(p) + (a g ,g )(4.46) −4 ρ pρ 3 ρ pρ · − O 6! 4 ! 5 7pρ=µρ ! ! 2 3 i p 1 a 1 p pp4 1 4 2 4 = Tr ̸ 2 S1− loop(p) ̸ 2 ̸ 2 2 S1− loop(p) + (a g ,g ). −4 p · − − 3 2 p − (p ) · − O 6 4 5 7pρ=µρ

1 The trace is taken only over spin indices and S1− loop is the inverse fermion propagator − that we computed up to 1-loop and up to (a2). We make the following deﬁnitions for O convenience: p2 p2, p4 p4, p = γ p and p3 γ p3. ≡ ρ ρ ≡ ρ ρ ̸ ρ ρ ρ ̸ ≡ ρ ρ ρ A very important! issue is that! the (a2)termsdependnotonlyonthemagnitude,! ! p2, O 4 but also on the direction of the momentum, pρ,asmanifestedbythepresenceof ρ pρ: ! a2 2 p4 pert. i p 6 p3 a ρ ρ 1 4 2 4 ̸ − ̸ − q (p)= Tr 2 1+ 2 Spert.(p) + (a g ,g ). (4.47) V −4 6 p 4 3 !p 5 · 7 O

As a consequence, alternative renormalization prescriptions, involving diﬀerent directions of the renormalization scale µρ = pρ,treatlatticeartifactsdiﬀerently.

1 By implementing the pertubative expressions of S− (p)andΛΓ(p) in Eqs. (4.42) and

2 Strictly speaking, the renormalization condition of Eq. (4.45) deﬁnes the so called RI′ scheme. In the original RI′-MOM scheme the quark ﬁeld renormalization condition reads:

1 1 i ∂Sq(p)− Z− − Tr γ =1. (4.44) q 16 µ ∂p ' µ (p2=µ2 The two schemes diﬀer in the Landau gauge at the N2 LO. Chapter 4. Renormalization Constants for Fermion Field and Ultra-Local 112 Operators in Twisted Mass QCD

(4.45), we obtain the following RCs:

Zpert. = zq[Action, csw, beta, g2tilde, aL, m, mu, p2, p4] + (a3 ,g4), (4.48) q O L Zpert. = zs[Action, csw, beta, g2tilde, aL, m, mu1, mu2] + (a3 ,g4), (4.49) S O L Zpert. = zp[Action, csw, beta, g2tilde, aL, m, mu1, mu2] + (a3 ,g4), (4.50) P O L Zpert. = zv[Action, csw, beta, g2tilde, aL, m, mu1, mu2] + (a3 ,g4), (4.51) V O L Zpert. =za[Action, csw, beta, g2tilde, aL, m, mu1, mu2] + (a3 ,g4), (4.52) A O L ν1= ν2 Zpert. ̸ =zt[Action, csw, beta, g2tilde, aL, m, mu1, mu2] + (a3 ,g4), (4.53) T O L pert. ν1= ν2 /Z 0 ̸ = ztp[Action, csw, beta, g2tilde, aL, m, mu1, mu2] + (a3 ,g4), (4.54) T O L / 0 The additional variables in Eqs. (4.48) - (4.54), as compared to Eqs. (4.35) - (4.40), are 2 4 p2andp4whicharedeﬁnedas:p2= ρ pρ, p4= ρ pρ. For the special choices: tree-level! Symanzik gluons! (Action=2), cSW =0,Landau gauge (beta=1), and general masses m, mu, M=√m2 +mu2 the results of the RCs under study are:

Zpert. = zq[2, 0, 1, g2tilde, aL, m, mu, p2, p4] + (a3 g2, g4) (4.55) q O L 2 2 2 2 2 3 ln[aLM + aLp2] 3M =1+g˜ 13.0232725(2) + aLm 0.3393996(2) + + 4− 4 2 2p2

4 p2 3M ln[1 + M 2] 2 2 2 2 + aL 1.1471643(7)p2 0.2145514(2)m +1.15904388(6)M − 2p2 5 4 − 2.1064977(2)p4 9m2M 2 209M 4 M 6 7M 8 + + p2 − 2p2 − 360p2 − 240p22 40p23 73p2 3m2 2M 2 157p4 + + + ln[a2 M 2 + a2 p2] − 360 2 3 180p2 L L 2 3 2 2 4 6 4 p2 2 4 1 9m 43M M 7M M ln[1 + 2] p4 43M 169M + + + M + + 24 2p2 72p2 − 12p22 − 40p23 p2 p2 − 80p2 180p22 2 3 4

6 8 2 4 6 4 p2 M 7M 1 35M M 7M M ln[1 + 2] + + + + M 120p23 − 20p24 12 − 36p2 6p22 20p23 p22 2 3 5 55 + (a3 g2, g4) O L 4.5. Quark Field and Quark Bilinear Renormalization Constants in the RI′-MOM Scheme 113

Zpert. = zs[2, 0, 1, g2tilde, aL, m, 0, 0] + (a3 , g4) (4.56) S O L 2 p2 2 2 2 2 9m ln[1 + m2] =1+g˜ 13.606731(1) + 3 ln[aLm + aLp2] + + aLm 2.7312983(2) 4− p2 4

2 2 2 2 p2 15 3m 6m 3m m ln[1 + 2] ln[a2 m2 + a2 p2] + + 24 + m − 2 L L 2p2 m2 + p2 − 2p2 p2 2 3 5

4 6 8 4 2 2 1289m 721m 7m 18m + aL 1.207563(2)p2 10.853390(2)m 2 + 3 2 4− − − 360p2 − 240p2 40p2 − m + p2 3m6 107m2 17p2 + + + ln[a2 m2 + a2 p2] (m2 + p2)2 6 360 L L 2 3 1321m2 367m4 35m6 7m8 p2 p4 + 1+ + + m2 ln[1 + ]+ 0.3935023(2) − 24p2 72p22 12p23 − 40p24 m2 p2 − 2 3 4 157 ln[a2 m2 + a2 p2] 117m2 371m4 721m6 7m8 2p2 L L + + + − 180 80p2 − 180p22 120p23 − 20p24 m2 + p2

2 4 6 8 2 p2 m 35m 35m 7m m ln[1 + 2] + 1+ + m + (a3 , g4) 12p2 − 36p22 − 6p23 20p24 p2 O L 2 3 555

Zpert. = zp[2, 0, 1, g2tilde, aL, m, 0, 0] + (a3 , g4) (4.57) P O L 2 p2 2 2 2 2 3m ln[1 + m2] =1+g˜ 21.733356(1) + 3 ln[aLm + aLp2] + 4− p2

2 2 2 p2 3 3m 3m m ln[1 + 2] + a m 7.0252230(2) ln[a2 m2 + a2 p2] + 6+ m L − 2 L L 2p2 − 2p2 p2 4 2 3 5

4 6 8 4 2 2 1769m 27m 7m 3m + aL 0.440762(2)p2 5.520750(2)m 2 + 3 2 4 − − 360p2 − 80p2 40p2 − 2(m + p2) 3m2 17p2 + + ln[a2 m2 + a2 p2] 2 360 L L 2 3 1 229m2 367m4 m6 7m8 p2 + + + + m2 ln[1 + ] −3 24p2 72p22 4p23 − 40p24 m2 2 3 2 2 2 2 4 6 8 p4 157 ln[aLm + aLp2] 227m 109m 27m 7m + 1.6064977(2) + 2 + 3 4 p2 4 − 180 − 720p2 180p2 40p2 − 20p2

2 4 6 8 2 p2 1 m 35m m 7m m ln[1 + 2] + + + m + (a3 , g4) 3 12p2 − 36p22 − 2p23 20p24 p2 O L 2 3 5 55 Chapter 4. Renormalization Constants for Fermion Field and Ultra-Local 114 Operators in Twisted Mass QCD

Zpert. = zv[2, 0, 1, g2tilde, aL, m, 0, 0] + (a3 , g4) (4.58) V O L 2 2 2 2 = δν1ν1 1+g˜ 16.6028865(8) + aLm 2.2261230(2) + 3 ln[aLm + aLp2] 4 4 − 4

2 2 2 2 2 p2 p 6m 6m p m ln[1 + 2] + ν1 3+ + 3 ν1 m + a2 1.125750(1)p2 p2 − p2 − p22 p2 L 2 3 2 3 5 4 65m4 m6 25m2 76p2 7p2 +1.102770(2)m2 + + + + ν1 ln[a2 m2 + a2 p2] 48p2 8p22 − 4 45 − 24 L L 2 3 5017m2 9401m4 m6 7m8 6m2 + p2 2.714031(1) + + ν1 − 360p2 − 360p22 − 10p23 10p24 − m2 + p2 2 3 p4 323 59m2 35m4 14m6 14m8 + ν1 + p2 180 − 18p2 − 18p22 − 3p23 3p24 2 3 41 17m2 m4 p2 11 236m2 m4 7m6 + + ν1 + − 4 − 12p2 − 8p22 p2 −12 9p2 − 4p22 − 10p23 2 2 3 4 2 4 6 4 p2 p 11 14m 7m 14m m ln[1 + 2] + ν1 + + m p22 3 3p2 3p22 − 3p23 p2 2 33 2 2 2 2 4 6 8 p4 157 ln[aLm + aLp2] 67m m 8m 7m + 2.0773310(2) + 2 3 + 4 p2 4 − 180 − 120p2 120p2 − 15p2 30p2

p2 7 m2 77m4 6m6 7m8 + ν1 + + + p2 120 12p2 12p22 p23 − p24 2 3 2 4 6 2 2 4 6 4 p2 1 5m 5m 7m p 5 10m 5m 7m m ln[1 + 2] + + + + + ν1 + m 2 18p2 12p22 − 30p23 p2 −2 − p2 − 2p22 p23 p22 2 2 33 5 555 + (a3 , g4) O L 4.5. Quark Field and Quark Bilinear Renormalization Constants in the RI′-MOM Scheme 115

Zpert. = za[2, 0, 1, g2tilde, aL, m, 0, 0] + (a3 , g4) (4.59) A O L 2 2 2 2 2m 8pν1m = δν1ν1 1+g˜ 12.5395741(8) + 2 4 4 − p2 − p2

2 2 2 2 p2 2m p 8m m ln[1 + 2] + 2 + ν1 4+ m − p2 p2 p2 p2 2 2 33 5m2 4m2 p2 26m2 4p2 + a m 1.4208394(2) + + ν1 1+ + L − − p2 m2 + p2 p2 − p2 m2 + p2 4 2 3

2 2 2 2 p2 5m p 26m m ln[1 + 2] + 3m + ν1 12 + m + a2 0.153718(1)p2 − p2 − p2 p2 p2 L − 2 2 33 5 4 557m4 13m6 3m8 10m4 2m6 +1.290617(2)m2 + + + 48p2 − 12p22 4p23 − m2 + p2 (m2 + p2)2

23m2 14p2 5p2 + ν1 + ln[a2 m2 + a2 p2] 12 − 45 24 L L 2 3 2707m2 23201m4 3m6 23m8 11m2 2m4 +p2 0.892808(1)+ + ν1 − 360p2 − 360p22 − 5p23 − 10p24 − m2 + p2 2 2 2 (m + p2) 3 p4 323 5m2 145m4 8m6 46m8 + ν1 + + + + p2 180 18p2 18p22 p23 − 3p24 2 3 2 73m2 11m4 17m6 3m8 p2 4 99m2 581m4 7m6 23m8 + + + + ν1 + + + + −3 12p2 − p22 24p23 − 4p24 p2 3 4p2 9p22 4p23 10p24 2 2 3 p4 3m2 40m4 m6 46m8 p2 p4 + ν1 2 + m2 ln[1 + ]+ 0.7439977(2) p22 − − p2 − 3p22 − 3p23 3p24 m2 p2 2 33 4

2 2 2 2 4 6 8 2 157 ln[aLm + aLp2] 19m 13m 163m 34m 4p2 2 m + 2 + 3 4 + 2 + + − 180 60p2 − 40p2 60p2 − 15p2 3(m + p2) 43 6p2

4 6 8 2 2 4 6 8 2 p2 11m 19m 34m p 2 11m 14m 5m 23m m ln[1 + 2] + + ν1 + + m − 9p22 − 12p23 15p24 p2 3 2p2 p22 − 2p23 − p24 p2 2 3 5 p2 167 4p2 41m2 91m4 9m6 23m8 4p22 + ν1 + + + p2 120 − 3m2 − 12p2 − 12p22 − p23 p24 3(m2 + p2) m2 2 35 555 + (a3 , g4) O L Chapter 4. Renormalization Constants for Fermion Field and Ultra-Local 116 Operators in Twisted Mass QCD

ν1= ν2 Zpert. ̸ =zt[2, 0, 1, g2tilde, aL, m, 0, 0] + (a3 , g4) (4.60) T O L , - 2 2 2 2 2 2m = δν1ν1δν2ν2 1+g˜ 13.5382926(8) ln[aLm + aLp2] 4 4 − − − p2

2 2 p2 2 2 2 2 2 p2 2m m ln[1 + 2] p + p 4m 4m m ln[1 + 2] + 1+ m + ν1 ν2 2+ m p2 p2 p2 p2 − p2 p2 2 3 / 0 4 2 3 5

2 4 p2 7 2 2 2 13m 13m ln[1 + m2] + aLm 0.4107689(2) + ln[aLm + aLp2] + 2 4 2 2p2 − 2p2

2 2 2 2 2 p2 p + p 10m 2p2 10m m ln[1 + 2] + ν1 ν2 1 + 6+ m p2 − − p2 − m2 + p2 p2 p2 / 0 4 2 3 55

4 6 8 4 2 2 628m 407m 23m 3m + aL 1.000358(2)p2+1.509337(2)m + 2 3 + 2 4 − 45p2 720p2 − 40p2 2(m + p2)

55m2 41p2 p2 p2 20 38m2 4m4 16m6 + ln[a2 m2 + a2 p2] + ν1 ν2 + + − 9 120 L L p2 9 − 9p2 p22 3p23 2 3 2 3 1 65m2 109m4 5m6 23m8 p2 p2 8m2 20m4 16m6 p2 + + + + ν1 ν2 m2 ln[1+ ] 3 − 24p2 8p22 − 18p23 40p24 p22 3p2 − 3p22 − 3p23 m2 2 2 33 383m2 103m4 29m6 3m8 + p2 + p2 2.0217225(2) + ln[a2 m2 + a2 p2] + + + ν1 ν2 L L 2 3 4 4 − 72p2 6p2 − 12p2 2p2 / 0 2 4 2 4 6 8 2 p2 5m m 2 85m 95m 5m 3m m ln[1 + 2] + + + m 2(m2 + p2) − 2 2 −3 − 6p2 − 6p22 3p23 − 2p24 p2 (m + p2) 2 3 5

4 4 2 4 6 8 pν1 + pν2 10 35m 3m 11m 10m + 2 3 + 4 / p2 04 9 − 9p2 − p2 − 3p2 p2

2 4 6 8 2 p2 14m 17m 4m 10m m ln[1 + 2] p4 + 1+ + m + 2.4814977(2) 3p2 3p22 − 3p23 − p24 p2 p2 2 3 5 4 157 ln[a2 m2 + a2 p2] 497m2 73m4 43m6 43m8 L L + − 180 − 720p2 − 90p22 − 40p23 20p24

2 4 8 2 p2 2 2 2 4 1 11m 55m 43m m ln[1 + 2] p + p 2 2p2 7m 7m + + + m + ν1 ν2 + + + −3 12p2 36p22 − 20p24 p2 p2 − 3 3m2 4p2 p22 2 3 / 04

6 8 2 4 8 2 p2 15m 15m 2p2 1 4m 12m 15m m ln[1 + 2] + + + m 2p23 − p24 − 3(m2 + p2) − 3 − p2 − p22 p24 p2 2 3 555 55 + (a3 , g4) O L 4.5. Quark Field and Quark Bilinear Renormalization Constants in the RI′-MOM Scheme 117

ν1= ν2 pert. ̸ 3 4 Z =ztp[2, 0, 1, g2tilde, aL, m, 0, 0] + (aL, g ) (4.61) T ′ O , - 2 2 2 2 2 2m = δν1ν1δν2ν2 1+g˜ 13.5382926(8) ln[aLm + aLp2] + 4 4 − − p2

2 2 p2 2 2 2 2 2 p2 2m m ln[1 + 2] p + p 4m 4m m ln[1 + 2] 1+ m + ν1 ν2 + 2+ m − p2 p2 p2 − p2 p2 p2 2 3 / 0 4 2 3 5

2 2 2 2 p2 7 7m 2m 7m m ln[1 + 2] +a m 2.5892311(2)+ ln[a2 m2 +a2 p2] + + 6+ m L − 2 L L − 2p2 m2 + p2 2p2 p2 4 2 3

2 2 2 2 2 p2 p + p 10m 2p2 10m m ln[1 + 2] + ν1 ν2 1+ + 6+ m p2 p2 m2 + p2 − p2 p2 / 0 4 2 3 55

4 6 8 4 2 2 469m 587m 37m 2m + aL 0.089747(2)p2+7.217670(2)m + + 2 + 3 2 4 90p2 720p2 40p2 − m + p2

m6 79p2 55m2 p2 p2 20 38m2 4m4 16m6 + + ln[a2 m2 + a2 p2] + ν1 ν2 + + 2 2 120 − 9 L L p2 9 − 9p2 p22 3p23 (m + p2) 2 3 2 3 1 373m2 133m4 23m6 37m8 p2 p2 8m2 20m4 16m6 p2 + + ν1 ν2 m2 ln[1+ ] −3 − 24p2 − 24p22 − 18p23 − 40p24 p22 3p2 − 3p22 − 3p23 m2 2 2 33 79m2 127m4 35m6 3m8 + p2 + p2 0.2004998(2) ln[a2 m2 + a2 p2] ν1 ν2 L L 2 3 4 4 − − − 72p2 − 6p2 − 12p2 − 2p2 / 0 2 4 2 4 6 8 2 p2 5m m 2 23m 45m 11m 3m m ln[1 + 2] + + + + + + m − 2(m2 + p2) 2 2 3 2p2 2p22 3p23 2p24 p2 (m + p2) 2 3 5

4 4 2 4 6 8 pν1 + pν2 10 m 7m 9m 10m + + 2 + 3 4 / p2 04 9 − 3p2 p2 p2 − p2

2 4 6 8 2 p2 2m 37m 4m 10m m ln[1 + 2] p4 + 1 + m + 1.8148310(2) − − p2 − 3p22 − p23 p24 p2 p2 2 3 5 4 157 ln[a2 m2 + a2 p2] 517m2 107m4 11m6 57m8 2p2 L L + + + − 180 − 720p2 90p22 120p23 − 20p24 3(m2 + p2)

2 4 6 8 2 p2 2 2 2 4 1 m 53m 4m 57m m ln[1 + 2] p + p 2 2p2 7m 7m + + + + m + ν1 ν2 3 4p2 − 36p22 3p23 20p24 p2 p2 3 − 3m2 − 4p2 − p22 2 3 / 04

6 8 2 2 4 8 2 p2 15m 15m 2p2 1 4m 12m 15m m ln[1 + 2] + + + + + m − 2p23 p24 3(m2 + p2) m2 3 p2 p22 − p24 p2 2 3 555 55 + (a3 , g4) O L

where: 2 f 2 2 M =(m0 ) + µ0,ν1 = µ, ν2 = ν. (4.62) Chapter 4. Renormalization Constants for Fermion Field and Ultra-Local 118 Operators in Twisted Mass QCD 4.6 Analytic expressions for one-derivative operators

Here we present the Z-factors for the one-derivative vector, axial and tensor operators published separately in Ref. [96], deﬁned as follows

2 ψγ5γ µ←D→ν τ ψa=1 { } µν a ⎧ 1 DV{ } = χγ µ←D→ν τ χ = ψγ5γ µ←D→ν τ ψa=2 (4.63) O { } ⎪ − { } ⎨⎪ 3 ψγ µ←D→ν τ ψa=3 { } ⎪ ⎩⎪ 2 ψγ µ←D→ν τ ψa=1 { } µν a ⎧ 1 DA{ } = χγ5γ µ←D→ν τ χ = ψγ µ←D→ν τ ψa=2 (4.64) O { } ⎪ − { } ⎨⎪ 3 ψγ5γ µ←D→ν τ ψa=3 { } ⎪ ⎩⎪ a µ νρ ψγ5σµ ν ←D→ρ τ ψa=1, 2 a { } DT{ } = χγ5σµ ν ←D→ρ τ χ = . (4.65) O { } ⎧ i ψσµ ν ←D→ρ 11ψa=3 ⎨ − { } The above operators are symmetrized over two⎩ Lorentz indices and are made traceless

στ 1 στ τσ 1 στ λλ { } + δ . O ≡ 2 O O − 4 O , - &λ The one derivative operators fall into diﬀerent irreducible representations of the hypercubic group, depending on the choice of indices:

= with µ = ν ODV1 ODV = with µ = ν ODV2 ODV ̸ = with µ = ν ODA1 ODA = with µ = ν ODA2 ODA ̸ = with µ = ν = ρ ODT1 ODT ̸ = with µ = ν = ρ = µ. ODT2 ODT ̸ ̸ ̸

Thus, ZDV1,ZDA1 will be different from ZDV2,ZDA2,respectively.Moredetailsonthe one-derivative renormalization factors can be found in Ref. [96]. We have computed, to (a2), the forward matrix elements of these operators for gen- O eral external indices µ, ν (and ρ for the tensor operator), external momentum p, m, g, Nc, a, cSW and gauge fixing. Our final results were obtained for the 10 sets of Symanzik coefficients given in Table 2.1. The amputated Green’As functions of the operator appear in the file m-file.m as ODΓ 4.6. Analytic expressions for one-derivative operators 119 below:

L( pert. = δ δ LDV[Action, csw, beta, g2tilde, aL, m] + (a3 ,g4) , DV f ′ f ′′ g′ g′′ O L 2 3 L( pert. = δ δ LDA[Action, csw, beta, g2tilde, aL, m] + (a3 ,g4) , DA f ′ f ′′ g′ g′′ O L 2 3 L( pert. = δ δ LDT[Action, csw, beta, g2tilde, aL, m] + (a3 ,g4) . DT f ′ f ′′ g′ g′′ O L 2 3

In order to deﬁne Z ,wehaveusedarenormalizationprescriptionwhichismost O amenable to non-perturbative treatment:

1 O O O O Zq− Z Tr L (p) Ltree(p) =Tr Ltree(p) Ltree(p) (4.66) O · pλ=µλ · pλ=µλ ) * ) * where LO denotes the amputated 2-point Green’s function of the operators up to 1-loop and up to (a2). These Z-factors appear in electronic form with the name: O ν =ν Zpert. 1 2 =ZDV1[Action, csw, beta, g2tilde, aL, m] + (a3 ,g4), DV1 O L pert. ν1= ν2 /Z 0 ̸ =ZDV2[Action, csw, beta, g2tilde, aL, m] + (a3 ,g4), DV2 O L pert. ν1=ν2 /Z 0 =ZDA1[Action, csw, beta, g2tilde, aL, m] + (a3 ,g4), DA1 O L pert. ν1= ν2 /Z 0 ̸ =ZDA2[Action, csw, beta, g2tilde, aL, m] + (a3 ,g4), DA2 O L pert. ν1=ν3= ν2 Z/ 0 ̸ =ZDT1[Action, csw, beta, g2tilde, aL, m] + (a3 ,g4), DT1 O L pert. ν1= ν2= ν3= ν1 Z / ̸ 0 ̸ ̸ =ZDT2[Action, csw, beta, g2tilde, aL, m] + (a3 ,g4) . DT2 O L / 0

Due to very lengthy expressions we only show the results for speciﬁc choices of the action parameters, that is Landau gauge, tree-level Symanzik gluons, cSW =0,m =0:

ν = ν Zpert. 1 2 = ZDV1[2, 0, 1, g2tilde, aL, 0] + (a4 , g4 ) (4.67) DV 1 O L 2 2 4 / 0 8 2pν 6pν 21pν = δ 1+g˜ 2 1.41698(1) ln[a2 p2]+ 1 1 +a2 1.62067(6)p2 6.4175(7)p2 + 1 ν1 ν1 3 L 3p2 8p2 + p2 L ν1 10p2 4 4 − − ν1 4 −

4 6 2 4 23.328(6)pν1 16pν1 19p2 334pν1 232pν1 2 2.0544143(2) + + + ln[aLp2] + p4 2 2 2 2 8pν1 + p2 − − 180 − 45 3 8pν + p2 p2 8pν1 + p2 4 1 5 4 / 0 2 / 0 2 29pν1 2.0100(1) 2pν1 157 79 2 4 4 + + + ln[aLp2] + (aL, g ) 2 2 2 2 2 180p2 − 8pν1 + p2 − −180p2 60 8p + p2 O 8pν1 + p2 4 ν1 5 5555 / 0 / 0 Chapter 4. Renormalization Constants for Fermion Field and Ultra-Local 120 Operators in Twisted Mass QCD

ν = ν Zpert. 1̸ 2 = ZDV2[2, 0, 1, g2tilde, aL, 0] + (a4 , g4 ) (4.68) DV 2 O L 2 2 2 2 / 0 8 4pν pν 209pν pν = δ δ 1+g˜ 2 2.02248(1) ln[a2 p2] + 1 2 + a2 1.01505(3)p2+ 1 2 ν1 ν1 ν2 ν2 − 3 L 2 2 L 90p2 4 4 3p2 pν1 + pν2 4

/ 0 2 2 4 4 2 2 0.2456(3)pν1 pν2 8pν1 pν2 2.1276(1) pν + pν + − 1 2 p2 + p2 − 2 2 2 ν1 ν2 9p2 pν + pν / 0 1 2 / 0 22p2 19 421p2 p2 + + p2 + p2 + ν1 ν2 ln[a2 p2] − 45 40 ν1 ν2 2 2 L 4 90 pν1 + pν2 5 / 0 / 0 2.5967755(2) 157 ln[a2 p2] 29p2 p2 + p4 L + ν1 ν2 + (a4 , g4 ) p2 − 180p2 2 2 2 O L 4 90p2 pν1 + pν2 5 555 / 0

ν = ν Zpert. 1 2 = ZDA1[2, 0, 1, g2tilde, aL, 0] + (a4 , g4 ) (4.69) DA1 O L 2 2 4 / 0 2 8 2 2pν1 6pν1 2 2 21pν1 = δν ν 1+g˜ 3.48606(1) ln[a p2]+ +a 0.46577(6)p2+3.8584(7)p + 1 1 3 L 3p2 8p2 + p2 L ν1 10p2 4 4 − − ν1 4

4 6 2 4 62.787(6)pν1 16pν1 199p2 866pν1 488pν1 2 2.0544143(2) + + ln[aLp2]+p4 2 2 2 2 − 8pν1 + p2 − − 180 45 − 3 8pν + p2 p2 8pν1 + p2 4 1 5 4 / 0 2 / 0 2 29pν1 1.2140(1) 2pν1 157 79 2 4 4 + + + ln[aLp2] + (aL, g ) 2 2 2 2 2 180p2 − 8pν1 + p2 − −180p2 60 8pν + p2 O 8pν1 + p2 4 1 5 5555 / 0 / 0

ν = ν Zpert. 1̸ 2 = ZDA2[2, 0, 1, g2tilde, aL, 0] + (a4 , g4 ) (4.70) DA2 O L 2 2 2 2 / 0 8 4pν pν 209pν pν = δ δ 1+g˜ 2 3.07868(1) ln[a2 p2] + 1 2 + a2 0.38848(3)p2+ 1 2 ν1 ν1 ν2 ν2 − 3 L 2 2 L 90p2 4 4 3p2 pν1 + pν2 4

/ 0 2 2 4 4 2 2 1.1283(3)pν1 pν2 8pν1 pν2 1.8613(1) pν + pν + − − 1 2 p2 + p2 − 2 2 2 ν1 ν2 9p2 pν + pν / 0 1 2 / 0 8p2 19 179p2 p2 + + p2 + p2 ν1 ν2 ln[a2 p2] 45 40 ν1 ν2 − 2 2 L 4 90 pν1 + pν2 5 / 0 / 0 2.5967755(2) 157 ln[a2 p2] 29p2 p2 + p4 L + ν1 ν2 + (a4 , g4 ) p2 − 180p2 2 2 2 O L 4 90p2 pν1 + pν2 5 555 / 0 4.6. Analytic expressions for one-derivative operators 121

ν = ν = ν Zpert. 1 3̸ 2 = ZDT1[2, 0, 1, g2tilde, aL, 0] + (a4 , g4 ) (4.71) DT1 O L 2 2 / 0 18.7832(1)pν 12pν = δ δ 1+g˜ 2 3.88296(1) 1 + 4+ 1 ln[a2 p2] ν1 ν1 ν2 ν2 − 8p2 p2 + p2 − 8p2 p2 + p2 L 4 4 ν1 − ν2 2 ν1 − ν2 3 78p2 p2 7058p4 28.163(3)p2 p2 + a2 0.7730(3)p2 26.269(8)p2 3.3610(1)p2 ν1 ν2 ν1 + ν1 L − − ν1 − ν2 − 5p2 − 45p2 8p2 p2 + p2 4 ν1 − ν2 2.4703(2)p22 302.33(1)p4 3800p6 6.2611(1)p2 p22 + + ν1 + ν1 + ν1 2 2 2 2 2 2 2 2 2 8pν1 pν2 + p2 8pν1 pν2 + p2 3p2 8pν pν + p2 8p p + p2 − − 1 − 2 ν1 − ν2 / 0 / 0 100.1772(9)p4 p2 350.620(3)p6 80p2 82p2 739p2 293p2 p2 + ν1 + ν1 + + ν2 ν1 ν1 2 2 2 2 2 2 90 45 − 180 − 2 2 8p p + p2 8p p + p2 4 45 8pν1 pν2 + p2 ν1 − ν2 ν1 − ν2 − / 0 / 193p022 / 6580p4 4p2 p22 64p4 p2 + ν1 ν1 ν1 2 2 2 2 2 2 − 180 8pν pν + p2 9 8pν pν + p2 − 8p2 p2 + p2 − 8p2 p2 + p2 1 − 2 1 − 2 ν1 − ν2 ν1 − ν2 / 0 / 0 6 / 0 / 2 0 224pν1 2 2.4606643(2) 2.4703(1) 131pν1 ln[aLp2] + p4 − 2 2 2 p2 − 8p2 p2 + p2 − 2 2 8p p + p2 5 4 ν1 ν2 180p2 8pν1 pν2 + p2 ν1 − ν2 − − / 0 / 2 0 2 6.26108(4)pν1 157 193 4pν1 2 + + + ln[aLp2] − 2 2 2 −180p2 2 2 2 2 2 8p p + p2 4 180 8pν1 pν2 + p2 8p p + p2 5 55 55 ν1 − ν2 − ν1 − ν2 / 0 + /(a4 , g4 ) 0 / 0 O L

ν = ν = ν = ν Zpert. 1̸ 2̸ 3̸ 1 = ZDT2[2, 0, 1, g2tilde, aL, 0] + (a4 , g4 ) (4.72) DT2 O L 2 2 / 0 2 2 2 2 67pν1 pν3 = δν1 ν1 δν2 ν2 δν3 ν3 1+g˜ 2.82413(1) 3 ln[aLp2] + aL 0.92582(3)p2 0.73604(2)pν2 + 4 4 − 4 − 45p2

67p4 1.2403(3)p2 p2 67p2 p2 p2 2.1124(1) p2 + p2 + ν2 + ν1 ν3 + ν1 ν2 ν3 ν1 ν3 90p2 p2 + p2 2 2 − ν1 ν3 15p2 pν1 + pν3 / 0 / 0 p2 301p2 331 71p2 p2 + + ν2 + p2 + p2 + ν1 ν3 ln[a2 p2] − 2 360 720 ν1 ν3 2 2 L 4 20 pν1 + pν3 5 / 0 / 0 2.1064977(2) 41 157 ln[a2 p2] + p4 + L + (a4 , g4 ) p2 60p2 − 180p2 O L 2 3 555 where for ZDV1, ZDV2, ZDA1, ZDA2:

ν1 = µ, ν2 = ν, and for ZDT1, ZDT2:

ν1 = ρ,ν2 = µ, ν3 = ν. Chapter 4. Renormalization Constants for Fermion Field and Ultra-Local 122 Operators in Twisted Mass QCD 4.7 Strong IR divergent integrals

The integrals, with strong IR divergences (convergent only beyond D>6), encountered in the present calculation are listed below along with their results. For completeness we also include the integrals that appeared in our related publication [96] for the matrix elements of twist-2 operators. All these integrals can be found in electronic form in the ASCII ﬁle m-ﬁle.m, with the names: IntegralPropagator1-IntegralPropagator3,IntegralBilinears1 -IntegralBilinears6,andIntegralExtendedBilinears1 - IntegralExtendedBilinears2.

Propagator

π 4 d k 1 4 4 2 =IntegralPropagator1+ aL (4.73) • π (2π) ˆ 2 2 2 O 1 k k#+¯p + a M1 − L / 0 p2 , - 2 2 2 2 ln[a M + a p2] M1 ln[1 + M 2] =0.03667832907475711(1) L 1 L 1 − 16π2 − 16π2p2

M 2 ln[a2 M 2 + a2 p2] M 4 + a2 0.00007524033(9)p2 0.00396328514(4)M 2 + 1 L 1 L 1 L ⎛ − 1 128π2 − 128π2p2 ⎝ p2 2 4 1 M M1 ln[1 + M 2] + + 1 1 64π2 128π2p2 p2 2 3 p2 2 4 2 4 2 p4 1 M M 1 M M M1 ln[1 + M 2] + + 1 + 1 + 1 + 1 1 p2 ⎛384π2 128π2p2 64π2p22 − 192π2 64π2p2 64π2p22 p2 ⎞⎞ 2 3 ⎝ ⎠⎠ + a4 O L / 0 4.7. Strong IR divergent integrals 123

π 4 ◦ d k kν1 5 4 2 =IntegralPropagator2+ aL (4.74) • π (2π) ˆ 2 2 2 O 1 k k#+¯p + a M1 − L / 0 , - ln[a2 M 2 + a2 p2] M 2 = a p 0.008655827647937295(1) + L 1 L 1 L ν1 ⎛− 32π2 − 32π2p2 ⎝ p2 2 2 2 4 1 M M1 ln[1 + M 2] p p4 1 M 3M + + 1 1 + a3 ν1 1 1 16π2 32π2p2 p2 ⎞ L ⎛ p2 ⎛−768π2 − 96π2p2 − 128π2p22 2 3 ⎠ ⎝ ⎝ p2 6 2 4 6 2 M 1 M M M M1 ln[1 + M 2] 1 + + 1 + 1 + 1 1 − 64π2p23 192π2 48π2p2 32π2p22 64π2p23 p2 ⎞ 2 3 ⎠ p2ln[a2 M 2 + a2 p2] 13M 4 + p 0.0005107794(2)p2+ L 1 L 0.00028240872(9)M 2 + 1 ν1 ⎛− 768π2 − 1 1536π2p2 ⎝ p2 6 2 4 4 M 1 M M M1 ln[1 + M 2] + 1 + 1 + 1 1 256π2p22 − 128π2 96π2p2 256π2p22 p2 ⎞ 2 3 ⎠ ln[a2 M 2 + a2 p2] 5M 2 13M 4 + p3 0.0011713297148098348(1) L 1 L + 1 + 1 ν1⎛ − 384π2 384π2p2 768π2p22 ⎝ p2 6 2 4 6 2 M 1 M M M M1 ln[1 + M 2] + 1 + 1 + 1 + 1 1 + a5 128π2p23 − 96π2 48π2p2 48π2p22 128π2p23 p2 ⎞⎞ O L 2 3 / 0 ⎠⎠ Chapter 4. Renormalization Constants for Fermion Field and Ultra-Local 124 Operators in Twisted Mass QCD

π d4k k◦ k◦ ν1 ν2 =IntegralPropagator3+ a4 (4.75) 4 2 2 L • (2π) 2 2 O 1 π kˆ 2 k#+¯p + a M − L 1 / 0 p2 , - , - 2 2 2 2 4 ln[a M + a p2] M M1 ln[1 + M 2] = δ 0.004327913823968648(1) L 1 L 1 + 1 ν1ν2 ⎛ − 64π2 − 64π2p2 64π2p22 ⎞ ⎝ ⎠ p2 2 2 2 p p 1 M 1 M M1 ln[1 + M 2] + ν1 ν2 + 1 + 1 1 p2 ⎛32π2 16π2p2 − 16π2 16π2p2 p2 ⎞ 2 3 ⎝ ⎠ p2ln[a2 M 2 + a2 p2] + a2 δ 0.00025539124(4)p2 L 1 L +0.00010358434(2)M 2 L ⎛ ν1ν2 ⎛ − 1536π2 1 ⎝ ⎝ p2 4 6 2 6 5M M 1 M M1 ln[1 + M 2] + 1 + 1 + 1 1 3072π2p2 512π2p22 − 384π2 512π2p2 p22 ⎞ 2 3 ⎠ ln[a2 M 2 + a2 p2] M 2 23M 4 3M 6 + p p 0.00037885376(9) + L 1 L 1 1 1 ν1 ν2⎛− 768π2 − 768π2p2 − 1536π2p22 − 256π2p23 ⎝ p2 2 4 4 1 M 3M M1 ln[1 + M 2] + + 1 + 1 1 128π2 48π2p2 256π2p22 p22 ⎞ 2 3 ⎠ ln[a2 M 2 + a2 p2] 5M 2 31M 4 + δ p2 0.00013565411323668763(1) + L 1 L + 1 + 1 ν1ν2 ν1⎛− 768π2 768π2p2 1536π2p22 ⎝ p2 6 2 4 4 3M 1 5M 3M M1 ln[1 + M 2] + 1 + 1 + 1 1 256π2p23 − 64π2 192π2p2 256π2p22 p22 ⎞ 2 3 ⎠ 2 2 2 4 6 pν1pν2 pν1 + pν2 1 M1 3M1 M1 + 2 2 2 2 2 3 /p2 0 ⎛−384π − 48π p2 − 64π p2 − 32π p2 ⎝ p2 2 4 6 2 1 M M M M1 ln[1 + M 2] δ p4 1 + + 1 + 1 + 1 1 + ν1ν2 96π2 24π2p2 16π2p22 32π2p23 p2 ⎞ p2 ⎛1536π2 2 3 ⎠ ⎝ p2 2 4 6 2 4 4 M M M 1 M M M1 ln[1 + M 2] 1 1 1 + + 1 + 1 1 − 768π2p2 − 256π2p22 − 128π2p23 384π2 128π2p2 128π2p22 p22 ⎞ 2 3 ⎠ p p p4 1 5M 2 11M 4 5M 6 + ν1 ν2 + 1 + 1 + 1 p22 ⎛768π2 192π2p2 128π2p22 64π2p23 ⎝ p2 2 4 6 2 1 M M 5M M1 ln[1 + M 2] + 1 + 1 + 1 1 + a4 − 96π2 16π2p2 8π2p22 64π2p23 p2 ⎞⎞ O L 2 3 / 0 ⎠⎠ 4.7. Strong IR divergent integrals 125

Bilinears

π 4 d k 1 2 4 2 2 =IntegralBilinears1+ aL (4.76) • π (2π) ˆ 2 2 2 2 2 O 1 k k#+¯p + a M1 k#+¯p + a M2 − L L / 0 p2 , -,2 - 2 2 2 M 2 ln[1 + ] p4 ( 1)j+1 ln[a M + a p2] j M 2 =0.0039632853(1) + L j L + j 2 2 − 2 2 2 2 2 2 2 − 128π p2 ⎧ aL ⎛16π M1 M2 16π p2 M1 M2 ⎞ &j=1 ⎨ − − ⎝ ⎠ / 0 / p02 2 2 2 ⎩2 2 2 M 4 ln[1 + ] M M ln[a M + a p2] 1 M j M 2 + j +( 1)j j L j L +( 1)j + j j 128π2p2 − 128π2 M 2 M 2 − 64π2 128π2p2 p2 M 2 M 2 1 − 2 4 5 1 − 2 / 0 2 p2 / 0 2 2 4 Mj ln[1 + 2] p4 Mj j+1 1 Mj Mj Mj 2 + 2 2 +( 1) 2 + 2 + 2 2 2 2 + aL p2 ⎛−64π p2 − 4192π 64π p2 64π p2 5 M1 M2 ⎞⎫ O − ⎬ / 0 ⎝ / 0 ⎠ ⎭ π 4 d k sin (kν1 +¯pν1) 3 4 2 2 =IntegralBilinears2+ aL (4.77) • (2π) 2 2 2 2 2 O 1 π kˆ k#+¯p + a M k#+¯p + a M − L 1 L 2 / 0 1 p , p p4 1-, M 2M 2 - M 2M 2 = ν1 + a ν1 + 1 2 p 0.0002071688(1) + 1 2 a 32π2p2 L p22 384π2 64π2p22 − ν1 256π2p22 L 2 2 3 2 3 p3 1 M 2M 2 ν1 + 1 2 − p2 384π2 128π2p22 2 33 p2 2 2 2 2 M 4 ln[1 + ] p ln[a M + a p2] j M 2 + ( 1)j+1 ν1 L j L j 2 2 2 2 2 2 2 ⎧ − aL ⎛32π M1 M2 − 32π p2 M1 M2 ⎞ &j=1 ⎨ − − ⎝ / 0 / 0⎠ ⎩ 2 4 pν1p4 Mj Mj + aL 2 2 + 2 2 ⎛ p2 4128π p2 64π p2 ⎝ 4 p2 2 4 Mj ln[1 + 2] 2 4 j 1 Mj Mj Mj 5Mj Mj +( 1) + + + pν − 192π2 64π2p2 64π2p22 p2 M 2 M 2 1 − 1536π2p2 − 256π2p22 4 5 1 − 2 5 4 / 0 p2 2 2 2 2 M 6 ln[1 + ] p2ln[a M + a p2] 1 M j M 2 +( 1)j+1 L j L +( 1)j+1 + j j − 768π2 M 2 M 2 − 192π2 256π2p2 p22 M 2 M 2 1 − 2 4 5 1 − 2 5 / 0 / 0 p3 5M 2 M 4 p2ln[a2 M 2 + a2 p2] + ν1 j j +( 1)j L j L p2 − 768π2p2 − 128π2p22 − 384π2 M 2 M 2 4 1 − 2 4 p2 / 0 2 4 Mj ln[1 + 2] j+1 1 Mj Mj Mj 3 +( 1) 2 + 2 + 2 2 2 2 + aL − 4192π 96π p2 128π p2 5 p2 M1 M2 5⎞⎫ O − ⎬ / 0 / 0 ⎠ ⎭ π 4 d k sin (kν1 +¯pν1)sin(kν2 +¯pν2) 4 4 2 2 =IntegralBilinears3+ aL (4.78) • π (2π) ˆ 2 2 2 2 2 O 1 k k#+¯p + a M1 k#+¯p + a M2 − L L / 0 , -, - Chapter 4. Renormalization Constants for Fermion Field and Ultra-Local 126 Operators in Twisted Mass QCD

π d4k k◦ k◦ ν1 ν2 =IntegralBilinears4+ a2 (4.79) 4 2 2 2 L • (2π) 2 2 2 2 O 1 π kˆ 2 k#+¯p + a M k#+¯p + a M − L 1 L 2 / 0 , - , -, -

π d4k k◦ k◦ sin (k +¯p ) ν1 ν2 ν3 ν3 =IntegralBilinears5+ a3 (4.80) 4 2 2 2 L • (2π) 2 2 2 2 O 1 π kˆ 2 k#+¯p + a M k#+¯p + a M − L 1 L 2 / 0 , - , -, -

π d4k k◦ k◦ sin (k +¯p )sin(k +¯p ) ν1 ν2 ν3 ν3 ν4 ν4 =IntegralBilinears6+ a4 (4.81) 4 2 2 2 L • (2π) 2 2 2 2 O 1 π kˆ 2 k#+¯p + a M k#+¯p + a M − L 1 L 2 / 0 , - , -, -

Extended Bilinears

π d4 k sin (k +¯p ) sin (k +¯p ) sin (k +¯p ) ν1 ν1 ν2 ν2 ν3 ν3 =IntegralExtendedBilinears1+ a5 (4.82) 4 2 2 L • π (2π) 2 2 2 2 2 O 1− kˆ k#+¯p + a M k#+¯p + a M L 1 L 2 / 0 2 32 3

π d4 k sin (k +¯p ) sin (k +¯p ) sin (k +¯p ) k◦ k◦ ν1 ν1 ν2 ν2 ν3 ν3 ν4 ν5 =IntegralExtendedBilinears2+ a5 4 2 2 2 L • π (2π) 2 2 2 2 2 O 1− kˆ k#+¯p + a M k#+¯p + a M L 1 L 2 / 0 , - 2 32 3 (4.83)

where:

2 f 2 2 Mj =(m0 ) + µj ,

p¯ν = aL pν, 2 p2= pρ, ρ & 4 p4= pρ, ρ & q qˆ = 2 sin( ν ), ν 2 q qˆ2 =4 sin2( ρ ), 2 ρ & ◦ qν = sin(qν), where q stands for k or k +¯p,whilek is the loop momentum and p is the external momentum. No summation over the indices νi is implied. 4.8. Conversion to the contimuum MS scheme 127

4.8 Conversion to the contimuum MS scheme

4.8.1 Conversion factors

In this section we provide the expressions for the conversion factors to the MS scheme, given in Ref. [31]. In our analysis we use the 2-loop formulae since the 3-loop correction for the particular expressions are 1 per cent. Note that our deﬁnition of Cq is equivalent to 1/Cq of Ref. [31].

g2 C C g2 2 C =1+λ F F 8λ2 +5 C q 16 π2 − 8 16 π2 F 2 3 9λ2 24ζ(3)λ +52λ 24$/ζ(3) + 820 C +28T N λ2C2 (4.84) − − − A F F − F / g2 C g2 2 0 % C =1 (λ +4)C + F 24λ2 +96λ 288ζ(3) + 57 C S,P − F 16 π2 24 16 π2 − F 2 3 +332T N 18λ2 +84λ 432ζ(3)$/ + 1285 C 0 (4.85) F F − − A 8 / 0 % CA,V =1+O(g )(4.86)

g2 C g2 2 C =1+λC + F 216λ2 +4320ζ(3) 4815 C T,T′ F 16 π2 216 16 π2 − F 2 3 1252T N + 162λ2 +756λ $/3024ζ(3) + 5987 C 0 (4.87) − F F − A / 0 % The variable λ is related to the gauge ﬁxing parameter beta via λ =1 beta, thus λ =0 − for Landau gauge. The deﬁnitions of CA,TF are:

a b ab Tr[T T ]=Tf δ (4.88) acd bcd ab f f = CA δ (4.89) and ζ(n) is the Riemann zeta function. It is worth mentioning that the coupling constant is not aﬀected by the conversion to the MS scheme up to the order that we investigate. For completeness we include in the m-ﬁle.m the conversion factors to the MS scheme for the one-derivative RCs, studied in Ref. [96].

C =CDV1[alphaRIprime, lambdaRIprime, CA, Cf, Nf, Tf] + (g8)(4.90) DV 1,DA1 O C =CDV2[alphaRIprime, lambdaRIprime, CA, Cf, Nf, Tf] + (g8)(4.91) DV 2,DA2 O alphaRIprime g2/(16π2) ≡ lambdaRIprime λ 1 beta ≡ ≡ −

Setting λ =1(λ =0)correspondstotheFeynman(Landau)gauge.Thevariables alphaRIprime, lambdaRIprime correspond to the RI′ scheme coupling constant and covariant gauge parameter (deﬁned in Ref. [31]). Chapter 4. Renormalization Constants for Fermion Field and Ultra-Local 128 Operators in Twisted Mass QCD 4.8.2 Evolution to a reference scale

All our Z-factors have been evaluated for a range of renormalization scales. In this subsection we use 3-loop perturbative expressions to extrapolate to a scale µ =2GeV.Thus, each result is extrapolated to 2 GeV, maintaining information of the initial renormalization scale at which it was computed. The scale dependence is predicted by the renormalization group (at ﬁxed bare parameters), that is MS MS Z (µ)=R (µ, µ0) Z (µ0)(4.92) O O O with g¯2(µ2) exp F 16π2 R (µ, µ0)= 2 2 (4.93) O , g¯ (µ0) - exp F 16π2 , - with

γ0 β0γ2 β2γ0 2 F (x)= ln(x)+ − ln( β0 + β1x + β2x ) 2β0 4β0β2 2β β γ β β γ β β /γ β +20 β x + 0 2 1 − 1 2 0 − 0 1 2 arctan 1 2 . (4.94) 2 2 2β0β2 4β0β2 β1 4 4β0β2 β1 5 − − > > To 3 loops, the running coupling, β-function and anomalous dimension γ are as follows [30, 98, 99, 100, 32]:

g¯2(µ2) 1 β ln(ln(µ2/Λ2)) = 1 (4.95) 2 2 2 3 2 2 2 16π β0 ln(µ /Λ ) − β0 ln (µ /Λ )

1 + β2 ln2(ln(µ2/Λ2)) β2 ln(ln(µ2/Λ2)) + β β β2 + 5 3 2 2 1 1 2 0 1 β0 ln (µ /Λ ) − − ··· / 0

2 β =11 N (4.96) 0 − 3 F 38 β =102 N (4.97) 1 − 3 F 2857 5033 N 325 N 2 β = F + F (4.98) 2 2 − 18 54

q γ0 =0 (4.99)

67 4 γq = 2 N (4.100) 1 − 3 − 3 F 2 3 4.9. Non-perturbative calculation 129

20729 79 550 20 γq = 2 ζ(3) N + N 2 (4.101) 2 − 36 − 2 − 9 F 27 F 2 3 3 CF γS,P = 2 (4.102) 0 − 4 202 20 γS,P = 2 N (4.103) 1 − 3 − 9 F 2 3 4432 320 280 γS,P = 2498 + + ζ(3) N + N 2 (4.104) 2 − 27 3 F 81 F 2 3 8 γT,T′ = (4.105) 0 3 4 γT,T′ = (26 N 543) (4.106) 1 −27 F − 2 γT,T′ = 36 N 2 +1440ζ(3) N +5240N +2784ζ(3) 52555 (4.107) 2 −81 F F F − / 0 Eqs. (4.99) - (4.107) diﬀer by numerical factors compared to Refs. [30, 98, 99, 100, 32] due to alternative deﬁnitions of the factor R (µ, µ0). O

4.9 Non-perturbative calculation

In the literature there are two main approaches that have been employed for the non- perturbative evaluation of the renormalizationconstants.Theybothstartbyconsidering that the operators can all be written in the form

(z)= u(z) (z, z′)d(z′) , (4.108) O J &z′ where u and d denote quark ﬁelds in the physical basis and denotes the operator we J are interested in, e.g. (z, z′)=δ γ would correspond to the local vector current. For J z,z′ µ each operator we deﬁne a bare vertex function given by

12 a ip(x y) G(p)= e− − u(x)u(z) (z, z′)d(z′)d(y) , (4.109) V ⟨ J ⟩ x,y,z,z& ′ where p is a momentum allowed by the boundary conditions, V is the lattice volume, and the gauge average is performed over gauge-fixed configurations. We have suppressed the Dirac and color indices of G(p). The first approach relies on translation invariance to shift the coordinates of the correlators in Eq. (4.109) to position z =0[80,79,101]. Having shifted to z =0allowsonetocalculatetheamputatedvertexfunctionforagiven operator for any momentum with one inversion per quark flavor. J Chapter 4. Renormalization Constants for Fermion Field and Ultra-Local 130 Operators in Twisted Mass QCD

In this work we explore the second approach, introduced in Ref. [98], which uses directly Eq. (4.109) without employing translation invariance. One must now use a source that is momentum dependent but can couple to any operator. For twisted mass fermions, u d we use the symmetry S (x, y)=γ S †(y,x)γ between the u and d quark propagators. 5 5 − − Therefore with a single inversion one can extract the vertex function for a single momentum. The advantage of this approach is a high statistical accuracy and the evaluation of the vertex for any operator including extended operators at no signiﬁcant additional computational cost. Since we are interested in a number of operators with their associated renormalization constants we use the second approach. We ﬁx to Landau gauge using a stochastic over-relaxation algorithm [102], converging to a gauge transformation which minimizes the functional

F = Re tr U (x)+U †(x µˆ) . (4.110) µ µ − x,µ & $ % Questions related to the Gribov ambiguity will not be addressed in this work. The propagator in momentum space, in the physical basis, is deﬁned by

8 8 u a ip(x y) d a ip(x y) S (p)= e− − u(x)u(y) ,S(p)= e− − d(x)d(y) . (4.111) V ⟨ ⟩ V x,y x,y & & J K An amputated vertex function is given by

u 1 d 1 Γ(p)=(S (p))− G(p)(S (p))− . (4.112) and the corresponding renormalized quantities are

1 SR(p)=ZqS(p) , ΓR(p)=Zq− Z Γ(p) , (4.113) O

In the twisted basis at maximal twist, Eq. (4.109) takes the form

12 a ip(x y) G(p)= e− − (11+iγ )u(x)u(z)(11+iγ ) (z, z′)(11 iγ )d(z′) d(y)(11 iγ ) . 4V 5 5 J − 5 − 5 x,y,z,z′ & J (4.114) K After integration over the fermion ﬁelds, and using u(x, z)=γ d†(z, x)γ this becomes S 5S 5

12 G a ˘d† d G(p)= (11 iγ ) (z, p)(11 iγ ) (z, z′)(11 iγ )˘ (z, p)(11 iγ ) , 4V − 5 S − 5 J − 5 S − 5 z L M & (4.115) where ... G is the integration over gluon ﬁelds, and ˘(z, p)= eipy (z, y) is the Fourier ⟨ ⟩ S y S transformed propagator on one of its argument on a particular! gauge background. It can 4.10. Non-perturbative results 131 be obtained by inversion using the Fourier source

a ipx bα(x)=e δαβ δab , (4.116) for all Dirac α and color a indices. The propagators in the physical basis given in Eq. (4.111) can be obtained from

d 1 ipz d G S (p)= e− (11 iγ )˘ (z, p)(11 iγ ) 4 ⟨ − 5 S − 5 ⟩ z 1 & Su(p)= e+ipz (11 iγ ) ˘d†(z, p)(11 iγ ) G , (4.117) −4 ⟨ − 5 S − 5 ⟩ z & which evidently only need 12 inversions despite the occurrence of both u and d quarks in the original expression. We evaluate Eq. (4.114) and Eq. (4.117) for each momentum separately employing Fourier sources over a range of a2p2 for which perturbative results can be trusted and ﬁnite a corrections are reasonably small.

4.10 Non-perturbative results

We perform the non-perturbative calculation of renormalization constants for three values of the lattice spacing, corresponding to β =3.9, 4.05 and 4.20. The lattice spacing as determined from the nucleon mass is 0.089 fm, 0.070 fm and 0.056 fm respectively. To extract the renormalization constants reliably one needs to consider momenta in the range Λ

β a(fm) aµ m (GeV) L3 T 0 π × 3.9 0.089 0.0040 0.3021(14) 243 48 3.9 0.089 0.0064 0.37553(80) 243 × 48 × 3.9 0.089 0.0085 0.4302(11) 243 48 3.9 0.089 0.01 0.4675 243 × 48 × 4.05 0.070 0.003 0.2925 323 64 4.05 0.070 0.006 0.4082(31) 243 × 48 4.05 0.070 0.006 0.404(2) 323 × 64 × 4.05 0.070 0.008 0.465(1) 323 64 4.20 0.055 0.0065 0.476(2) 323 × 64 × Table 4.2: Action parameters used in the simulations.

β =3.9 β =4.05 β =4.20 (n ,2,2,2), n :4 8, 10 14 (n ,2,2,2), n :4 8, 10, 13 14 (n ,2,2,2), n :4 8, 10, 13 14 t t − − t t − − t t − − (nt,3,3,3), nt :2 6, 8 9(nt,3,3,3), nt :2 6, 8 11, 13 (nt,3,3,3), nt :2 6, 8 11, 13 (n ,4,4,4), n :4− 9(− n ,4,4,4), n :8− 10− (n ,4,4,4), n :7− 11 − t t − t t − t t − (3,3,3,2) (3,3,3,2) (nt,5,5,5), nt :2 4 (3,3,3,2) −

Table 4.3: Momenta list for the various ensembles at β =3.9, 4.05, 4.20.

4.10.1 Pion mass dependence

For β =3.9weconsiderfourdiﬀerentquarkmasses,correspondingtomπ =0.302 GeV

(aµ0 =0.004), mπ =0.376 GeV (aµ0 =0.0064), mπ =0.430 GeV (aµ0 =0.0085) and mπ =0.468 GeV (aµ0 =0.01), in order to explore the dependence of the Z-factors on the pion mass. At β =4.05 we consider three ensembles: mπ =0.293 GeV (aµ0 =0.003), mπ =0.403 GeV (aµ0 =0.006) and mπ =0.466 GeV (aµ0 =0.008). For the highest

β value we have only one ensemble, that is mπ =0.470 GeV (aµ0 =0.0065). The pion mass dependence of Zq,ZV,ZA,ZT is demonstrated in Fig. 4.1, which is shown to be insignificant. Allowing a slope and performing a linear extrapolation to the data shown in Fig. 4.1 yields a slope consistent with zero. This behavior is also observed at the other β values. Thus, it would be sufficient to obtain the results on the aforementioned RCs at one pion mass value; like in the case of β =4.20. The necessity of having results on various pion masses for each beta value regards the RCs of the scalar and pseudoscalar operators that suffer from a pion pole that needs to be subtracted. This is discussed in Subsection 4.10.3.

4.10.2 Volume dependence

We perform the evaluation of the RCs at β =4.05 and µ =0.006 for two volumes, 243 48 and 323 64 in order to check for ﬁnite volume eﬀects. For this comparison we × × 4.10. Non-perturbative results 133

Z 0.85 q Z 0.8 V Z A Z 0.75 T

0.7

0.65

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 m (GeV) π

Figure 4.1: Zq,ZV,ZA,ZT at β =3.9,asafunctionofthepionmass:mπ = 0.302 GeV (aµ0 =0.004), mπ =0.375 GeV (aµ0 =0.0064), mπ =0.429 GeV (aµ0 =0.0085)andmπ =0.468 GeV (aµ0 =0.01). used momenta that correspond to the same renormalization scale: For the small lattice we use (ap)=2π(3/48, 3/24, 3/24, 3/24), in lattice units, whereas for the larger one we employ (ap)=2π(4/64, 4/32, 4/32, 4/32). The volume eﬀects appear to be in the worst case 0.1%, as can be seen from Table 4.4. We would like to point out that Z and ∼ S ZP show the larger volume dependence, but this becomes smaller when one subtracts the pion pole contribution (Subsection 4.10.3, Fig. 4.2).

lattice Zq ZS ZP ZV ZA ZT 243x48 0.82315(7) 0.743(2) 0.512(2) 0.7068(1) 0.7935(2) 0.7759(1) 323x64 0.82303(3) 0.744(1) 0.508(1) 0.7069(1) 0.7935(1) 0.7759(1)

Table 4.4: Renormalization constants at β =4.05,µ0 =0.006 using two lattice sizes: 323 64 for (4,4,4,4) and 243x48 for (3,3,3,3), and at a scale of (ap)2 2. × ∼

4.10.3 Pion pole subtraction

The correlation functions of the scalar and pseudoscalar operators have contamination from the presence of a Goldstone boson and thus need to be treated carefully 3. We will try to subtract the pole contribution by using the following ﬁt

2 cS(P ) ΛS(P ) = aS(P ) + bS(P ) mp + 2 , (4.118) mp which we apply in our data that were produced for ensembles that diﬀer only in the value of the pion mass. Once we have the ﬁtting parameters we subtract the pion pole using

3 The coupling with the Goldstone boson in the scalar case only appears for twisted mass fermions, due to the explicit breaking of parity. Chapter 4. Renormalization Constants for Fermion Field and Ultra-Local 134 Operators in Twisted Mass QCD the ﬁtting parameter cS(P ), sub cS(P ) ΛS(P ) =ΛS(P ) 2 . (4.119) − mp

We expect an appreciable dependence on the parameter cS(P ) and a very small contribution of term with the bS(P ) coefficient, as shown in Subsection 4.10.1 for the other RCs. This is indeed verified by our data: by subtracting the pion pole term that we obtain from 2 the fitting, the remaining pion mass dependence (bS(P ) mp) is almost zero for the various ensembles (see Fig. 4.2). To reliably obtain the three fitting parameters of Eq. (4.118) we need the RCs of the scalar and pseudoscalar operators for at least 4 pion masses. This is feasible for β =3.9, while for β =4.05 we have data for only three pion masses. One way to test the efficiency of the fitting we used data on three of the four pion masses at β =3.9andcompare them with the fitting using all available data. The conclusion is that both fittings are compatible within error bars. This testing at β =3.9allowsustoperformthefitting at β =4.05 using the three pion masses. One may observe the reliability of the pion pole subtraction in Fig. 4.2, where in the lower plots of each figure we show that the subtracted data (Eq. (4.119)) fall on top of each other. The curverature of the data in the upper plots (Eq. (4.118)) indicates the presence of the Goldstone pole. The errors shown in Fig. 4.2 come from the fitting and for their estimation we tried two methods: super jackknife error analysis [103, 104] and the least-squares analysis. We find that both methods lead to similar error estimations. According to theoretical arguments the Goldstone pole contaminations for the ratio

ZP/ZS should cancel out [105]. In practice, our data still shows signiﬁcant dependence on the pion mass, as indicated also in Ref. [106], and thus to form the aforementioned ratio we used the subtracted data of Fig. 4.2 which we compute in the chiral limit, although the pion mass dependence is insigniﬁcant after the subtraction. This procedure leads to data that are plotted in Fig. 4.3. With black circles we show the purely non-perturbative data extracted from the pion pole subtracted ZS,ZP of Eq. (4.119). The magenta diamonds show the non-perturbative data of Eq. (4.119) to which we have perform the subtraction of the perturbative (a2)contributions,presentedinSections4.3-4.4.TheratioZ /Z is O P S renormalization scale independence, thus onemaytakethecontimuunlimitdirectlyfrom the (a2) subtracted data that are shown with the magenta diamond points of Fig. 4.3. O

4.10.4 Results in RI′-MOM scheme

In this section we present results in the RI′-MOM scheme for Zq,ZS,ZP, as well as for the scale-independent RCs Z and Z .Theleadingdiscretizationeﬀectsof (a2)that V A O we computed in 1-loop perturbation theory are subtracted from our non-perturbative estimates. All renormalization constants are evaluated at the three β values. In fact, for β =3.9, 4.05 we have performed a chiral extrapolation using the data in diﬀerent pion 4.10. Non-perturbative results 135

1.00 0.90

0.90 0.80 0.80 RI RI S S

Z Z 0.70 0.70 mpi = 303 MeV mpi = 293 MeV mpi = 377 MeV mpi = 404 MeV 0.60 mpi = 432 MeV mpi = 465 MeV 0.60 mpi = 468 MeV 0.50 0.90 0.90

0.80 0.80 RI RI S 0.70 S Z Z mpi = 377 MeV 0.70 mpi = 293 MeV 0.60 mpi = 468 MeV mpi = 404 MeV mpi = 303 MeV mpi = 465 MeV 0.60 0.50 mpi = 432 MeV 12345 0.5 1 1.5 2 2.5 3 3.5 (a p)2 (a p)2

0.60 0.60

0.50 0.50 RI P RI P

Z 0.40 0.40 Z m = 303 MeV pi mpi = 293 MeV m = 377 MeV pi mpi = 404 MeV m = 432 MeV 0.30 0.30 pi mpi = 465 MeV mpi = 468 MeV 0.20 0.70 0.60 0.60 RI RI P P 0.50 Z Z mpi = 377 MeV mpi = 293 MeV 0.50 mpi = 468 MeV mpi = 404 MeV mpi = 303 MeV 0.40 mpi = 465 MeV m = 432 MeV 0.40 pi 0.30 12345 0.5 1 1.5 2 2.5 3 3.5 (a p)2 (a p)2

Figure 4.2: ZS,ZP at β =3.9 (left panel) and β =4.05 (right panel) for various masses. The upper plot shows the results before the pion pole subtraction (Eq. (4.118)), while the lower ﬁgure the results upon the appropriate subtraction (Eq. (4.119)). Chapter 4. Renormalization Constants for Fermion Field and Ultra-Local 136 Operators in Twisted Mass QCD

1.10 unsubtracted 1.00 subtracted 0.90 s 0.80 / Z

p 0.70 Z 0.60 0.50 0.40 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 (a p)2

1.10 unsubtracted 1.00 subtracted 0.90 s 0.80 / Z

p 0.70 Z 0.60 0.50 0.40 0 0.5 1 1.5 2 2.5 3 3.5 (a p)2

Figure 4.3: ZP/ZS at β =3.9 and β =4.05 as a function of the momentum. In each plot we demonstrate the eﬀect of the (a2)-terms subtraction by plot- O ting the pure non-perturbative results (black circles) and the subtracted ones (magenta diamonds). The Goldstone boson contamination was also removed from the plotted data.

masses; the fitting is consistent to a constant as demonstrated in Fig. 4.1. The results for β =4.02 are presented at one pion mass, but we also expect a flat dependence on the pion mass. For this β value we don’t show results for ZS,ZP,ZP/ZS,sincetherearenot enough ensembles to subtract the pion pole contribution. The first results we present regard the renormalization of the fermion field, and the results are demonstrated in Fig. 4.4

Although the analysis of the pure non-perturbative Zq for all available momenta re- veales a non smooth behavior on the momentum, this becomes smoother once we subtract the (a2) perturbative terms. We would like to point out that in all our non-perturbative O results we have some data that don’t follow the general behavior and these correspond to momenta that lead to large non-Lorentz invariant contributions in the perturbative ex- 4 2 pressions of Sections 4.3-4.4. Such terms are ( ρ pρ)/( ρ pρ). Clearly, upon subtraction these non-Lorentz invariant contributions are! being removed.! Next we present the scale-independent RCs ZV and ZA,forwhichthenon-perturbative results show a slope proportional to (ap)2 that can be explained as discretization effects, as can be seen in Fig. 4.5. This is justified by the fact that if one subtracts the appropriate (a2) terms this dependence becomes almost invinsible. We remove the remaining lattice O artifacts by performing an extrapolation to the continuum limit, represented by dashed lines in the figure. The unsubtracted data have been fitted to a line with a slope, while 4.10. Non-perturbative results 137

0.95 unsubtracted 0.9 subtracted 0.85 RI q 0.8 Z 0.75

0.7

0.65 012345 (a p)2

0.95 unsubtracted 0.9 subtracted 0.85 RI q 0.8 Z 0.75

0.7

0.65 0123 (a p)2

0.95 unsubtracted 0.9 subtracted 0.85 RI q 0.8 Z 0.75

0.7

0.65 0123 (a p)2

Figure 4.4: Non-perturbative results on Zq for β =3.9, β =4.05 and β =4.20. The upper plots are the results of a chiral extrapolation procedure, while the lower one is presented at mπ =0.470 GeV. Black circles represent the non- perturbative data, and magenta diamonds the (a2) subtracted ones. O the (a2)subtracteddatahaveaslopeconsistenttozero.Itisimportanttonotethat O the continuum limit of the non-perturbative data diﬀers from the continuum limit of the subtracted ones. This is an indication of large systematic errors in the non-perturbative evaluation. For the continuun extrapolation we choose the same momentum range in physical units for all β values and we thus extract all renormalization constants using the same physical momentum range, p2 15 32 (GeV)2.Thismomentumrangehasbeenchosen ∼ − in our previous work on the one-derivative RCs [96] so that we are in a region where an approximate plateau is seen at each β. The momentum range in lattice units at each β is as follows: β =3.9: (ap)2 =3 5, β =4.05: (ap)2 =1.8 3.1, β =4.20: − − (ap)2 =1.2 2.5. The choice for the momentum range is irrelevant for Z and Z since − V A the subtracted data are almost ﬂat. Chapter 4. Renormalization Constants for Fermion Field and Ultra-Local 138 Operators in Twisted Mass QCD

0.80

0.70 V Z

0.60 0.85

0.80 A Z 0.75

0.70 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 (a p)2

0.80 0.75

V 0.70 Z 0.65

0.85

A 0.80 Z 0.75

0.70 0 0.5 1 1.5 2 2.5 3 3.5 (a p)2

0.80 0.75

V 0.70 Z 0.65

0.85

A 0.80 Z 0.75

0.70 0 0.5 1 1.5 2 2.5 3 3.5 (a p)2

Figure 4.5: The scale-independent ZV and ZA for β =3.9, β =4.05 and β =4.20.Theupperplotsaretheresultsofachiralextrapolationprocedure, while the lower one is presented at mπ =0.470 GeV. Black circles represent the non-perturbative data, and magenta diamonds the (a2) subtracted ones. O

4.10.5 MS scheme

In this Section we present our results on Zq,ZS,ZP and ZT converted in the continuun

MS scheme and at a reference scale of µ =2GeV.FortheconversionfromRI′-MOM to MS scheme we use the formulae given in Eqs. (4.84) - (4.87). We use the 3-loop formulae of Eqs. (4.93) to evolve the scale from µ to 2 Gev. 2 2 2 A“renormalizationwindow”shouldexistforΛQCD << µ << 1/a where perturbation theory holds and ﬁnite-a artifacts are small, leading to scale-independent results (plateau). In practice such a condition is hard to satisfy: The right inequality is extended to (2 5)/a2 leading to lattice artifacts in our results that are of (a2p2). Fortunately − O our perturbative calculations allow us to subtract the leading perturbative O(a2) lattice 4.10. Non-perturbative results 139

βZq ZS ZP ZP/ZS ZV ZA ZT 3.90 0.757(10)(21) 0.584(22)(37) 0.385(22)(08) 0.666(42)(17) 0.624(01)(04) 0.759(01)(01) 0.761(25)(42) 4.05 0.778(04)(06) 0.590(12)(19) 0.406(13)(20) 0.681(29)(53) 0.659(01)(03) 0.7727(4)(1) 0.808(07)(15) 4.20 0.804(05)(18) — — — 0.686(01)(01) 0.7867(7)(1) 0.831(06)(11)

Table 4.5: Final results of the renormalization constants Zq,ZS,ZP,ZT in the MS scheme, as well as for the scale-independent ZP/ZS,ZV and ZA.The above values have been obtained by performing a chiral extrapolation and then by extrapolating linearly in a2p2. Statistical errors are are shown in the first parenthesis. The error in the second parenthesis is the systematic error due to the continuum extrapolation, the difference between results using the fit range p2 15 32 (GeV)2 and the range p2 17 24 (GeV)2.Anerrorsmaller than∼ the− last digit given for the mean value∼ is− not quoted.

artifacts which alleviates the problem. To remove the remaining O(a2p2)artifactsweex- trapolate linearly to a2p2 = 0 as demonstrated in Figs. 4.6 -4.8. The statistical errors are negligible and therefore an estimate of the systematic errors is important. We note that, in general, the evaluation of systematic errors is difficult. The largest systematic error comes from the choice of the momentum range to use for the extrapolation to a2p2 =0. One way to estimate this systematic error is to vary the momentum range where we perform the fit. Another approach is to fix a range and then eliminate a given momentum in the fit range and refit. The spread of the results about the mean gives an estimate of the systematic error. In the final results we give as systematic error the largest one from using these two procedures which is the one obtained by modifying the fit range.As already mentioned ee choose the same momentum range in physical units for the three β’s and extract all renormalization constants using the same physical momentum range, p2 15 32 (GeV)2;withinthisrangethedatafallonastraightlineofasmallslope. ∼ − We note that in most cases the (a2)perturbativetermswhichwesubtract,decreaseas O β increases, as expected. In all the following plots the statistical errors are too small to be visible, while the error bars in Fig. 4.7 come from the pion pole subtraction procedure. Black circles represent the non-perturbative data, and magenta diamonds the (a2)subtractedones.The O corresponding dashed lines show the extrapolation to the continuum limit and with a full diamond we denote the final value of the RCs in the continuum. Our final results for the Z-factors in the MS-scheme at 2 GeV are given in Table 4.5, which have been obtained in the chiral limit for β =3.9, 4.05, and by extrapolating linearly in a2p2, using the fixed momentum range p2 15 32 (GeV)2.Thesystematic ∼ − error due to the continuum extrapolation, are estimated from the difference between results using the fit range p2 15 32 (GeV)2 and the range p2 17 24 (GeV)2.The ∼ − ∼ − results at β =3.9andβ =4.05 agree with the results of Ref. [80], where the authors use adifferentmethodfortheevaluationofthecorrelationfunctions. Chapter 4. Renormalization Constants for Fermion Field and Ultra-Local 140 Operators in Twisted Mass QCD

1.00

0.90 _ MS q 0.80 Z 0.70

0.60 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 (a p)2

1.00

0.90 _ MS q 0.80 Z 0.70

0.60 0 0.5 1 1.5 2 2.5 3 3.5 (a p)2

1.00

0.90 _ MS q 0.80 Z 0.70

0.60 0 0.5 1 1.5 2 2.5 3 3.5 (a p)2

Figure 4.6: Results on Zq for β =3.9, β =4.05 and β =4.20.Theupper plots are the results of a chiral extrapolation procedure, while the lower one is presented at mπ =0.470 GeV.

4.11 Conclusions

The values of the renormalization factors forthefermionﬁeldandasetoflocaloperators

(Zq,ZS,ZP,ZV,ZA,ZT)havebeencalculatednon-perturbatively.Themethodofchoice is to use a momentum dependent source and extract the renormalization constants for all the relevant operators. This leads to a very accurate evaluation of these renormalization factors using a small ensemble of gauge configurations. The accuracy of the results allows us to check for any light quark mass dependence. For most of the renormalization constants studied in this work we do not find any light quark mass dependence within our small statistical errors. Therefore it would suffices to calculate them at a given quark mass, which is the case for our ensemble on the finer lattice (β =4.20). Despite the weak mass dependence for β =3.9andβ =4.05 we perform chiral extrapolation using a constant fitting. For the RCs of the scalar and pseudoscalar operators, ZS,ZP,wefind aquarkmassdependenceduetoaGoldstoneboson contamination, which we subtract using an appropreate fitting. Once the pole is subtracted, the behavior of the data allow a linear extrapolation to the chiral limit with a zero slope. We also show that, despite 4.11. Conclusions 141

0.80

_ 0.70 MS S Z 0.60

0.50 _ MS P Z 0.40

0.30 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 (a p)2

0.70 _ MS S

Z 0.60

0.50

0.50 _ MS P Z 0.40

0.30 0 0.5 1 1.5 2 2.5 3 3.5 (a p)2

Figure 4.7: Chiral results on ZS and ZP for β =3.9 and β =4.05. of using lattice spacing smaller than 1 fm, (a2)effectsaresizable,andthusweperform O aperturbativesubtractionof (a2) terms. This leads to a smoother dependence of the O renormalization constants on the momentum values at which they are extracted. Residual (a2p2) effects are removed by extrapolating to zero. In this way we can accurately de- O termine the renormalization constants in the RI′-MOM scheme. In order to compare with experiment we convert our values to the MS scheme at a scale of 2 GeV. The statistical errors are in general smaller than the systematic. The latter are estimated by changing the window of values of the momentum used to extrapolate to a2p2 =0.Ourfinalvalues are given in Table 4.5. Chapter 4. Renormalization Constants for Fermion Field and Ultra-Local 142 Operators in Twisted Mass QCD

1.00 0.90 _ MS T 0.80 Z 0.70 0.60 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 (a p)2

1.00 0.90 _ MS T 0.80 Z 0.70 0.60 0 0.5 1 1.5 2 2.5 3 3.5 (a p)2

Figure 4.8: Results on ZT for β =3.9, β =4.05 and β =4.20.Theupper plots are the results of a chiral extrapolation procedure, while the lower one is presented at mπ =0.470 GeV. Chapter 5

Renormalization Constants for Twist-2 Operators in Twisted Mass QCD

Perturbative and non-perturbative results on the renormalization constants of the fermion field and the twist-2 fermion bilinears are presented with emphasis on the non-perturbative evaluation of the one-derivative twist-2 vector and axial vector operators. Non-perturbative results are obtained using the twisted massWilsonfermionformulationemployingtwo degenerate dynamical quarks and the tree-level Symanzik improved gluon action. The simulations have been performed for pion masses in the range of about 450 260 MeV ∼ and at three values of the lattice spacing a corresponding to β =3.9, 4.05, 4.20. Subtraction of (a2)termsiscarriedoutbyperformingtheperturbativeevaluation O of these operators at 1-loop and up to (a2). The renormalization conditions are defined O in the RI′ MOM scheme, for both perturbative and non-perturbative results. The − renormalization factors, obtained for different values of the renormalization scale, are evolved perturbatively to a reference scale set by the inverse of the lattice spacing. In addition, they are translated to MS at 2 GeV using 3-loop perturbative results for the conversion factors.

5.1 Introduction

Simulations in lattice QCD have advanced remarkably in the past couple of years reaching within 100 MeV of the physical pion mass. This progress is due to theoretical improvements in deﬁning the theory on the lattice and to algorithmic improvements that give a better scaling behavior as the quark mass decreases. These developments, combined with the tremendous increase in computational power, have made ab initio calculations of key observables on hadron structure in the chiral regime feasible enabling comparison with

143 Chapter 5. Renormalization Constants for Twist-2 Operators in Twisted 144 Mass QCD experiment. The hadron mass spectrum [16, 107] illustrates the good quality of lattice results that can be obtained. The agreement with experiment is a validation of the lattice approach, and justiﬁes the computation of hadron observables beyond hadron masses, such as form factors and parton distributionfunctions.Bothformfactorsandparton distribution functions can be obtained from theso-calledgeneralizedpartondistributions (GPDs) in certain limiting cases. GPDs provide detailed information on the internal structure of hadron in terms of both the longitudinal momentum fraction and the total momentum transfer squared. Beyond the information that the form factors yield, such as size, magnetization and shape, GPDs encode additional information, relevant for experimental investigations, such as the decomposition of the total hadron spin into angular momentum and spin carried by quarks and gluons.

GPDs are single particle matrix elements of the light-cone operator [108, 109]:

dλ f λ λ/2 λ f iλx ig λ/2dαn A(αn) f Γ(x)= e ψ ( n)Γ n e − · ψ ( n) , (5.1) O 4π − 2 · P ! 2 1 where n is a light-cone vector, and denotes a path-ordering of the gauge ﬁelds in the P exponential. Such matrix elements cannot be calculated directly in lattice QCD. However, (x) can be expanded in terms of local twist-2 operators: O

f µ2 µn f, µ1µ2 µn µ1 } f { ··· } = ψ Γ{ i←D→ i←D→ ψ , (5.2) OΓ ··· where ←D→ = 1 (−→D ←D−)and µ , ,µ denotes symmetrization of indices and subtrac- 2 − { 1 ··· n} µ µ 5 µν tion of traces. In this work we focus on the Dirac structures Γ= γ ,γ5 γ and γ σ , which are referred to as vector f (x), axial-vector f (x)andtensor f (x)operators, OV OA OT respectively. In lattice QCD we consider matrix elements of such bilinear operators. A number of lattice groups are producing results on nucleon form factors and ﬁrst moments of structure functions closer to the physical regime both in terms of pion mass as well as in terms of the continuum limit [110, 111, 112, 113, 114, 115, 116]. While experiments are able to measure convolutions of GPDs, lattice QCD allows us to extract hadron matrix elements for the twist-2 operators, which can be expressed in terms of generalized form factors.

In order to compare hadron matrix elements of these local operators to experiment one needs to renormalize them. Our aim in this chapter is to calculate non-perturbatively the renormalization factors of the above twist-2 fermion operators within the twisted mass formulation. We show that, although the lattice spacings considered in this work are smaller than 1 fm, (a2) terms are non-negligible and signiﬁcantly larger than statistical O errors. We therefore compute the (a2)-terms perturbatively and subtract them from O the non-perturbative results. This subtraction suppresses lattice artifacts considerably depending on the operator under study and leads to a more accurate determination of 5.2. Formulation 145 the renormalization constants.

The chapter is organized as follows: in Section 5.2 we give the expressions for the fermion and gluon actions we employed, and define the twist-2 operators. Section 5.3 concentrates on the perturbative procedure, and the (a2)-corrected expressions for the O renormalization constants Z . Section 5.4 focuses on the non-perturbative computation, O where we explain the different steps of the calculation. Moreover, we provide the renormalization prescription of the RI′ MOM scheme, and we discuss alternative ways for − its application. The main results of this work are presented in Section 5.5: the reader can find numerical values for the Z-factors of the twist-2 operators, which are computed non-perturbatively and corrected using the perturbative (a2) terms presented in Sec- O tion 5.3. Since in general Z-factors depend on the renormalization scale, we also provide results in the RI′ MOM scheme at a reference scale, µ 1/a. For comparison with − ∼ phenomenological and experimental results, we convert the Z-factors to the MS scheme at 2 GeV. In Section 5.6 we give our conclusions.

5.2 Formulation

5.2.1 Lattice action

For the gauge ﬁelds we use the tree-level Symanzik improved gauge action [117], which 1 1 1 2 includes besides the plaquette term U × also rectangular (1 2) Wilson loops U × : x,µ,ν × x,µ,ν

4 4 β 1 1 1 2 S = b 1 Re Tr(U × ) + b 1 Re Tr(U × ) , (5.3) g 3 0 − x,µ,ν 1 − x,µ,ν x 4 µ,ν=1 µ,ν=1 5 & 1&µ<ν &µ=ν ≤ N O ̸ N O with β =2N /g2, b = 1/12 and the (proper) normalization condition b =1 8b . c 0 1 − 0 − 1 Note that at b1 =0thisactionbecomestheusualWilsonplaquettegaugeaction. The fermionic action for two degenerate ﬂavors of quarks in twisted mass QCD is given by: 4 3 SF = a χ(x)(DW [U]+m0 + iµ0γ5τ )χ(x) , (5.4) x & 3 with τ the Pauli matrix acting in the isospin space, µ0 the bare twisted mass and DW the massless Wilson-Dirac operator deﬁned as:

1 ar D [U]= γ (−→ + −→∗ ) −→ −→∗ , (5.5) W 2 µ ∇ µ ∇ µ − 2 ∇ µ ∇ µ Chapter 5. Renormalization Constants for Twist-2 Operators in Twisted 146 Mass QCD where:

1 −→ ψ(x)= U (x)ψ(x + aµˆ) ψ(x) , (5.6) ∇ µ a µ − ' ( ∗ 1 −→ ψ(x)= U †(x aµˆ)ψ(x aµˆ) ψ(x) . (5.7) ∇ µ −a µ − − − ' ( For completeness we also provide the deﬁnition of the backward derivatives:

1 ψ(x)←− = ψ(x + aµˆ)U †(x) ψ(x) , (5.8) ∇ µ a µ − ' ( 1 ψ(x)←−∗ = ψ(x aµˆ)U (x aµˆ) ψ(x) . (5.9) ∇ µ −a − µ − − ' ( Maximally twisted Wilson quarks are obtained by setting the untwisted bare quark mass m0 to its critical value mcr, while the twisted quark mass parameter µ0 is kept non- vanishing in order to give the light quarks their mass. In Eq. (5.4) the quark ﬁelds χ are in the so-called “twisted basis”. The “physical basis” is obtained for maximal twist by the simple transformation:

iω iω π ψ(x)=exp γ τ 3 χ(x), ψ(x)=χ(x)exp γ τ 3 ,ω= . (5.10) 2 5 2 5 2 2 3 2 3 In terms of the physical ﬁelds the action is given by:

1 ar Sψ = a4 ψ(x) γ [−→ + −→∗ ] iγ τ 3 −→ −→∗ + m + M ψ(x). (5.11) F 2 µ ∇ µ ∇ µ − 5 − 2 ∇ µ ∇ µ cr x & 2 , - 3 One can check that this action is equivalent from the action in the twisted basis Eq. (5.4), just performing the rotations of Eq. (5.10) and identifying m0 = mcr and M = µ. In this work we consider twist-2 operators with one derivative, which are given in the twisted basis as follows:

2 ψγ5γ µ←D→ν τ ψa=1 { } µν a ⎧ 1 DV{ } = χγ µ←D→ν τ χ = ψγ5γ µ←D→ν τ ψa=2 (5.12) O { } ⎪ − { } ⎨⎪ 3 ψγ µ←D→ν τ ψa=3 { } ⎪ ⎩⎪ 2 ψγ µ←D→ν τ ψa=1 { } µν a ⎧ 1 DA{ } = χγ5γ µ←D→ν τ χ = ψγ µ←D→ν τ ψa=2 (5.13) O { } ⎪ − { } ⎨⎪ 3 ψγ5γ µ←D→ν τ ψa=3 { } ⎪ ⎩⎪ a µ νρ ψγ5σµ ν ←D→ρ τ ψa=1, 2 a { } DT{ } = χγ5σµ ν ←D→ρ τ χ = (5.14) O { } ⎧ i ψσµ ν ←D→ρ 11ψa=3 ⎨ − { } ⎩ 5.3. Perturbative Procedure 147 with the covariant derivative deﬁned as:

∗ ∗ 1 (−→µ + −→ ) (←−µ + ←− ) ←D→ = ∇ ∇ µ ∇ ∇ µ . (5.15) 2 2 − 2 ) * The above operators are symmetrized over two Lorentz indices and are made traceless:

στ 1 στ τσ 1 στ λλ { } + δ . (5.16) O ≡ 2 O O − 4 O , - &λ This definition avoids mixing with lower dimension operators. We denote the correspond- a a a ing Z-factors by ZDV, ZDA, ZDT .Inamasslessrenormalizationschemetherenormaliza- tion constants are defined in the chiral limit, where isospin symmetry is exact. Hence, the same value for Z is obtained independently of the value of the isospin index a and therefore we drop the a index on the Z-factors from here on. However, one must note that, 1 3 for instance, the physical ψγ µ←D→ν τ ψ is renormalized with ZDA,whileψγ µ←D→ν τ ψ { } { } requires the ZDV,whichdifferfromeachothereveninthechirallimit. The one derivative operators fall into different irreducible representations of the hypercubic group, depending on the choice of indices. Hence, we distinguish between:

= with µ = ν, (5.17) ODV1 ODV = with µ = ν, (5.18) ODV2 ODV ̸ = with µ = ν, (5.19) ODA1 ODA = with µ = ν, (5.20) ODA2 ODA ̸ = with µ = ν = ρ, (5.21) ODT1 ODT ̸ = with µ = ν = ρ = µ. (5.22) ODT2 ODT ̸ ̸ ̸

Thus, ZDV1 will be diﬀerent from ZDV2,butrenormalizedmatrixelementsofthetwocor- responding operators will be components of the same tensor in the continuum limit. ODT1 is suﬃcient to extract all generalized form factors of the one derivative tensor operators. Although, in this work, we will calculate non-perturbatively only the renormalization constants for the vector and axial-vector operator, we will provide the perturbative (a2) O terms also for the tensor operators.

5.3 Perturbative Procedure

Here we present our results for the renormalization factors ZDV, ZDA, ZDT of the twist-2 operators , , , respectively. Our calculation is performed in 1-loop pertur- ODV ODA ODT bation theory to (a2). Extending the calculation up to (a2) brings in new diﬃculties, O O compared to lowers order in a;forinstance,thereappearnewtypesofsingularities.The Chapter 5. Renormalization Constants for Twist-2 Operators in Twisted 148 Mass QCD procedure to address this issue is extensively described in the previous chapters. Many IR singularities encountered at (a2)wouldpersistevenupto6dimensions,making O their extraction more delicate. In addition to that, there appear Lorentz non-invariant contributions in (a2)terms,suchas p4 /p2,wherep is the external momentum; as O µ µ 2 a consequence, the Z-factors also depend! on such terms. The knowledge of the order a terms is a big advantage for non-perturbative estimates, since they can eliminate possible large lattice artifacts, once the (a2)perturbativetermsaresubtracted. O For all our perturbative results we employ a general fermion action, which includes the clover parameter, cSW,andnon-zeroLagrangianmass,m.ForgluonsweuseSymanzik improved actions (Plaquette, Tree-level Symanzik, Iwasaki, TILW, DBW2) [97]. The purpose of using such general fermion and gluonactionsistomakeourresultsapplicable to a variety of actions used nowadays in simulations. Our results are given in a general covariant gauge, and their dependence on the coupling constant, the external momentum, the mass and the clover parameter is shown explicitly.

5.3.1 Renormalization of Twist-2 Operators

Here we present the computation of the amputated Green’s functions for the following three twist-2 operators:

µν 1 1 { } = Ψ γ ←D→ Ψ+Ψ γ ←D→ Ψ δ Ψ γ ←D→ Ψ, (5.23) ODV 2 µ ν ν µ − 4 µν τ τ τ ) * &

µν 1 1 { } = Ψ γ γ ←D→ Ψ+Ψ γ γ ←D→ Ψ δ Ψ γ γ ←D→ Ψ, (5.24) ODA 2 5 µ ν 5 ν µ − 4 µν 5 τ τ τ ) * &

µ νρ 1 1 { } = Ψ γ σ ←D→ Ψ+Ψ γ σ ←D→ Ψ δ Ψ γ σ ←D→ Ψ, (5.25) ODT 2 5 µν ρ 5 µρ ν − 4 νρ 5 µτ τ τ ) * & which, being symmetrized and traceless, have no mixing with lower dimension operators. The Feynman diagrams that enter our calculation are illustrated in Fig. 5.1, where the insertion of the twist-2 operator is represented by a cross. We have computed, to (a2), O the forward matrix elements of these operators for general external indices µ, ν (and ρ for the tensor operator), external momentum p, m, g, Nc,a,cSW and λ.Settingλ =1 corresponds to the Feynman gauge, whereas λ =0correspondstotheLandaugauge.Our final results were obtained for the 10 sets of Symanzik coefficients itemized in Table 2.1. In order to define Z ,wehaveusedarenormalizationprescriptionwhichismost O amenable to non-perturbative treatment:

1 Zq− Z Tr LO(p) LtreeO (p) =Tr LtreeO (p) LtreeO (p) , (5.26) O · pλ=µλ · pλ=µλ ) * ) * 5.3. Perturbative Procedure 149

Figure 5.1: One-loop diagrams contributing to the computation of the twist-2 operators. A wavy (solid) line representsgluons(fermions).Acrossdenotes an insertion of the operator under study.

where Zq is the fermion ﬁeld renormalization calculated in the previous chapter, and LO denotes the amputated 2-point Green’s function of the operators in Eqs. (5.23) - (5.25), up to 1-loop and up to (a2). The tree-level expressions of the operators including the O (a2)termsare: O p3 i p3 LDV1(p)=iγ p a2 µ γ p a2 τ + (a4) , (5.27) tree µ µ − 6 − 4 τ τ − 6 O τ , - & , - DA1 DV1 Ltree (p)=γ5 Ltree (p), (5.28) i p3 p3 LDV2(p)= γ p a2 ν + γ p a2 µ + (a4) , (5.29) tree 2 µ ν − 6 ν µ − 6 O 2 , - , -3 DA2 DV2 Ltree (p)=γ5 Ltree (p), (5.30) p3 i p3 LDT1(p)=iγ σ p a2 ν γ σ p a2 τ + (a4), (5.31) tree 5 µν ν − 6 − 4 5 µτ τ − 6 O τ , - & , - i p3 p3 LDT2(p)= γ σ p a2 ρ + σ p a2 ν + (a4). (5.32) tree 2 5 µν ρ − 6 µρ ν − 6 O 2 , - , -3 We perform a Taylor expansion up to (a2) in the right hand side of the renormalization O condition and it leads to the following:

1 1 2 Tr LDV1(p) LDV1(p) = 2 p2 p2 + a2( p4+ p4 )+ (a4), (5.33) tree · tree − µ − 4 12 3 µ O ) * Tr LDA1(p) LDA1(p) = Tr LDV1(p) LDV1(p) , (5.34) tree · tree − tree · tree ) * ) * a2 Tr LDV2(p) LDV2(p) = p2 p2 + (p4 + p4)+ (a4), (5.35) tree · tree − µ − ν 3 µ ν O ) * Tr LDA2(p) LDA2(p) = Tr LDV2(p) LDV2(p) , (5.36) tree · tree − tree · tree ) * ) * Chapter 5. Renormalization Constants for Twist-2 Operators in Twisted 150 Mass QCD

p2 p2 p4 2 p4 p4 Tr LDT1(p) LDT1(p) = +2p2 µ a2 + ν µ + (a4), (5.37) tree · tree 4 ν − 4 − 12 3 − 12 O ) * 2 3 a2 Tr LDT2(p) LDT2(p) = p2 + p2 p4 + p4 + (a4), (5.38) tree · tree ν ρ − 3 ν ρ O ) * / 0 where p2 p2 and p4 p4. For the special choices: m =0,c =0,r =1,λ =0 ≡ ρ ρ ≡ ρ ρ SW (Landau gauge), and for tree-level Symanzik gluons, we obtain for the left hand side of ! ! Eq. (5.26):

1 1 2 Tr LDV1 (p) LDV1 (p) = 2 p2 p2 + a2 ( p4+ p4 ) · tree − µ − 4 12 3 µ ) * 4 p4 2 16 +˜g2 µ + p2 (3.610062(3) ln(a2 p2 )) + p2 (27.54716(3) ln(a2 p2 )) 3 p2 − 3 µ − 3 " 7 299 + a2 (p2 )2 (0.11838(2) + ln(a2 p2 )) + p2 p2 ( 0.6573(1) ln(a2 p2 )) 288 µ − − 180 ) 397 43 p2 + p4( 1.71886(3) + ln(a2 p2 ) µ ) − 720 − 360 p2 94 29 p4 169 p2 + p4 ( 16.1049(5) + ln(a2 p2 )+ + µ ) µ − 15 90 (p2 )2 45 p2 *# + (a4 ,g4 ), (5.39) O

a2 Tr LDV2 (p) LDV2 (p) = p2 p2 + (p4 + p4 ) · tree − µ − ν 3 µ ν ) * 4 p2 p2 8 +˜g2 µ ν +(p2 + p2 )(15.04575(1) ln(a2 p2 )) 3 p2 µ ν − 3 " 491 + a2 (p4 + p4 )( 7.1429(1) + ln(a2 p2 )) µ ν − 360 ) 103 353 p4 +(p2 + p2 ) p2 ( 0.13212(3) ln(a2 p2 )) + µ ν − − 360 720 p2 , - 1013 29 p4 169 (p2 + p2 ) + p2 p2 ( 4.0096(1) + ln(a2 p2 )+ + µ µ ) µ ν − 180 90 (p2 )2 90 p2 *# + (a4 ,g4 ), (5.40) O 1 1 2 Tr LDA1 (p) LDA1 (p) =2p2 + p2 + a2 ( p4 p4 ) · tree µ 4 −12 − 3 µ ) * 4 p4 2 16 +˜g2 µ + p2 ( 4.127332(3) + ln(a2 p2 )) + p2 ( 31.68532(3) + ln(a2 p2 )) −3 p2 − 3 µ − 3 " 65 541 + a2 (p2 )2 (0.17035(2) + ln(a2 p2 )) + p2 p2 (0.3982(1) ln(a2 p2 )) 288 µ − 180 ) 397 43 p2 + p4(1.69230(3) ln(a2 p2 )+ µ ) − 720 360 p2 2 29 p4 169 p2 + p4 (18.4613(5) + ln(a2 p2 ) µ ) µ 5 − 90 (p2 )2 − 45 p2 *# + (a4 ,g4 ), (5.41) O 5.3. Perturbative Procedure 151

a2 Tr LDA2 (p) LDA2 (p) = p2 + p2 (p4 + p4 ) · tree µ ν − 3 µ ν ) * 4 p2 p2 8 +˜g2 µ ν +(p2 + p2 )( 16.10196(1) + ln(a2 p2 )) −3 p2 µ ν − 3 " 491 + a2 (p4 + p4 )(7.2286(1) ln(a2 p2 )) µ ν − 360 ) 137 353 p4 +(p2 + p2 ) p2 (0.75869(3) ln(a2 p2 )) µ ν − 360 − 720 p2 , - 187 29 p4 169 (p2 + p2 ) + p2 p2 (4.8509(1) + ln(a2 p2 ) µ µ ) µ ν 180 − 90 (p2 )2 − 90 p2 *# + (a4 ,g4 ), (5.42) O

p2 p2 p4 p4 2 p4 Tr LDT1 (p) LDT1 (p) = − µ +2p2 + a2 ( µ − ν ) · tree 4 ν 12 − 3 ) * +˜g2 (p2 p2 )(4.226559(3) ln(a2 p2 )) + p2 ( 29.11666(2) + 5 ln(a2 p2 )) µ − − ν − " 43 + a2 p4 ( 0.14754(2) ln(a2 p2 )) − − 1440 ) 433 379 p2 17 p2 + p4(1.93789(3) ln(a2 p2 ) ν + µ ) − 720 − 720 p2 192 p2 61 227 + p2 (1.7215(1) p2 + ln(a2 p2 ) p2 +0.37022(2) p2 ln(a2 p2 ) p2 ) ν 48 ν µ − 1440 µ 71 721 p2 + p4 (14.9155(4) ln(a2 p2 ) µ ) ν − 15 − 90 p2 881 39 p2 + p2 p2 (2.4896(1) ln(a2 p2 ) µ ) ν µ − 240 − 10 p2 71 134 p6 + p4 ( 2.24911(4) + ln(a2 p2 )) ν µ − 90 − 45 p2 *# + (a4 ,g4 ), (5.43) O

p4 p4 Tr LDT2 (p) LDT2 (p) = p2 + p2 + a2 ( ρ ν ) · tree ρ ν − 3 − 3 ) * +˜g2 (p2 + p2 )( 15.84740(1) + 3 ln(a2 p2 )) ν ρ − " 107 41 p4 + a2 (p2 + p2 )(0.22134(3) p2 + ln(a2 p2 ) p2 ν ρ 360 − 60 p2 ) 301 67 p4 +0.73604(2) p2 ln(a2 p2 ) p2 µ ) µ − 360 µ − 90 p2 67 p2 p2 p2 1051 ρ µ ν +(p4 + p4 )(7.3949(1) ln(a2 p2 )) − 15 p2 ν ρ − 720 1609 67 p4 p2 + p2 p4 + p2 p2 (2.98450(8) ln(a2 p2 )) ρ ν ρ ν ρ ν − 360 − 45 p2 *# + (a4 ,g4 ), (5.44) O

whereg ˜2 g2C /(16π2)andC =(N 2 1)/(2N ). Our results for general value of ≡ F F c − c λ, cSW,mand the 10 sets of Symanzik gluon actions, are far too lengthy to be included in paper form (vector, axial, tensor: 2667, 2854, 7052 terms respectively); we provide them in electronic form along with Ref. [118].

The (a2) terms shown in Eqs. (5.39)-(5.42) are used to correct our non-perturbative O Chapter 5. Renormalization Constants for Twist-2 Operators in Twisted 152 Mass QCD

2 results for ZDV1,ZDV2,ZDA1,ZDA2, in order to better control a artifacts. For the subtraction procedure we use the boosted coupling [84] instead of the bare one:

g2 g2 = bare , (5.45) boosted u ⟨ plaq⟩ where u is the plaquette mean value. ⟨ plaq⟩

5.4 Non-Perturbative Calculation

5.4.1 Evaluation of Correlators

(z)= u(z) (z, z′)d(z′) , (5.46) O J &z′ where u and d denote quark ﬁelds in the physical basis and denotes the operator we J are interested in, e.g. (z, z′)=δ γ would correspond to the local vector current. For J z,z′ µ each operator we deﬁne a bare vertex function given by:

12 a ip(x y) G(p)= e− − u(x)u(z) (z, z′)d(z′)d(y) , (5.47) V ⟨ J ⟩ x,y,z,z& ′ where p is a momentum allowed by the boundary conditions, V is the lattice volume, and the gauge average is performed over gauge-fixed configurations. We have suppressed the Dirac and color indices of G(p). The first approach relies on translation invariance to shift the coordinates of the correlators in Eq. (5.47) to position z =0[79,119].Havingshifted to z =0allowsonetocalculatetheamputatedvertexfunctionforagivenoperator for J any momentum with one inversion per quark flavor. In this work we explore the second approach, introduced in Ref. [98], which uses directly Eq. (5.47) without employing translation invariance. One must now use a source that is momentum dependent but can couple to any operator. For twisted mass fermions, we use the symmetry S (x, y)=γ S†(y,x)γ between the u and d quark propagators. u 5 d 5 − − Therefore with a single inversion one can extract the vertex function for a single momentum. The advantage of this approach is a high statistical accuracy and the evaluation of the vertex for any operator including extended operators at no significant additional computational cost. Since we are interested in a number of operators with their associated renormalization constants we use the second approach. We fix to Landau gauge using a stochastic over-relaxation algorithm [102], converging to a gauge transformation which 5.4. Non-Perturbative Calculation 153 minimizes the functional:

F = Re tr U (x)+U †(x µˆ) . (5.48) µ µ − x,µ & $ % Questions related to the Gribov ambiguity will not be addressed in this work. The propagator in momentum space, in the physical basis, is deﬁned by:

8 8 a ip(x y) a ip(x y) S (p)= e− − S (x, y) ,S(p)= e− − S (x, y) , (5.49) u V u d V d x,y x,y & & where S (x, y)= u(x)u(y) and S (x, y)= d(x)d(y) .Anamputatedvertexfunctionis u ⟨ ⟩ d given by: J K 1 1 Γ(p)=Su− (p) G(p) Sd− (p) , (5.50) and the corresponding renormalized quantities are:

1 SR(p)=ZqS(p) , ΓR(p)=Zq− Z Γ(p) , (5.51) O

In the twisted basis at maximal twist, Eq. (5.47) takes the form:

12 a ip(x y) G(p)= e− − (11+iγ )u(x)u(z)(11+iγ ) (z, z′) 4V ⟨ 5 5 ×J x,y,z,z& ′ (11 iγ )d(z′)d(y)(11 iγ ) . (5.52) × − 5 − 5 ⟩

After integration over the fermion ﬁelds, and using Su(x, z)=γ5Sd†(z, x)γ5 this becomes:

12 a G G(p)= (11 iγ )S˘ †(z, p)(11 iγ ) (z, z′)(11 iγ )S˘ (z′,p)(11 iγ ) , (5.53) 4V ⟨ − 5 d − 5 J − 5 d − 5 ⟩ &z,z′ where ... G is the integration over gluon ﬁelds, and S˘(z, p)= eipyS(z, y) is the Fourier ⟨ ⟩ y transformed propagator on one of its argument on a particular! gauge background. It can be obtained by inversion using the Fourier source:

a ipx bα(x)=e δαβ δab , (5.54) for all Dirac α and color a indices. The propagators in the physical basis given in Eq. (5.49) can be obtained from:

1 ipz ˘ G Sd(p)=+4 z e− (11 iγ5)Sd(z, p)(11 iγ5) , ⟨ − − ⟩ (5.55) S (p)= 1 ! e+ipz (11 iγ )S˘ †(z, p)(11 iγ ) G, u − 4 z ⟨ − 5 d − 5 ⟩ ! which evidently only need 12 inversions despite the occurrence of both u and d quarks in Chapter 5. Renormalization Constants for Twist-2 Operators in Twisted 154 Mass QCD the original expression. We evaluate Eq. (5.52) and Eq. (5.55) for each momentum separately employing Fourier sources over a range of a2p2 for which perturbative results can be trusted and ﬁnite a corrections are reasonably small.

5.4.2 Renormalization Condition

The renormalization constants are computed both perturbatively and non-perturbatively in the RI′ MOM scheme at diﬀerent renormalization scales.We translate them to the − MS-scheme at (2 GeV)2 using a conversion factor computed in perturbation theory to (g6)asdescribedinSection5.5.TheZ-factors are determined by imposing the following O conditions:

1 L 1 (0) Zq = Tr (S (p))− S (p) , (5.56) 12 p2=µ2 I $ %I 1 µν 1 L (0) 1 I Zq− Z Tr Γµν (p)Γµν − (p) =1, (5.57) O 12 p2=µ2 I $ %I where µ is the renormalization scale, while SL andI ΓL correspond to the perturbative or non-perturbative results. The trace is now taken over spin and color indices. These conditions are imposed in the massless theory, i.e. at critical mass and vanishing twisted mass. At ﬁnite lattice spacing there are two choices for S(0) and Γ(0) entering Eqs. (5.56) and (5.57). One can take either the tree level or the continuum results for S(0) and Γ(0),whichdiﬀerby (a2)-terms. The continuum free propagator in terms of continuum O momentum is:

i γ p (0) − ρ ρ ρ S (p)= 2 , (5.58) !p (0) ˜ Γµν (p)= i µ pν . (5.59) − O{ }

We refer to this choice as method 1.Adiﬀerentchoiceistodeﬁnethefreepropagator using the lattice momentum [98, 119] :

i γρ sin(pρ) S(0)(p)=− ρ , (5.60) sin(p )2 !ρ ρ (0) !˜ Γµν (p)= i µ sin(pν ) , (5.61) − O{ } which we will refer to as method 2 used in Ref. [80]. (0) (0) Only for this second choice for S and Γµν do we obtain Zq =1,Z =1whenthe O gauge ﬁeld is set to unity. Similarly, in the perturbative computation only method 2 gives

Zq =1,andZ =1attree-level.Onthecontrary,theZ-factorsobtainedfrommethod O 5.4. Non-Perturbative Calculation 155

1 have lattice artifacts even at tree-level. Obviously, the renormalization constants using the two methods differ only in their lattice artifacts. We find that, for the cases considered here, non-perturbative results using method 2 lead to Z-factors with smaller lattice effects. We demonstrate this by examining the following case: Let us consider the momenta as given in Table 5.2, which fall into two sets, those with spatial components 2 π (2, 2, 2)/L × and those with 2 π (3, 3, 3)/L;thereisonlyonenondemocraticmomentum,withspatial × components 2 π (3, 3, 2)/L,butthisbehavessimilarlytothesecondsetmentionedabove. × The implementation of momenta corresponding to the first set lead to Z-factors with 0ν different behavior than momenta of the second set. In Fig. 5.2 we present ZDA at β =3.9 and µ0 =0.0085 using the two methods (upper plot). The statistical errors in Fig. 5.2 and the rest of the graphs are too small to be visible. Clearly the two sets give different results, with method 1 giving larger discrepancies. However, it is important to note that, after subtracting the (g2 a2) perturbative contributions (also computed with the two O alternative ways respectively), one obtains values that are consistent between the two methods (lower plot). Moreover, in method 1 the jump between the two sets of momenta disappears.

1.35 Z A,unsub tree 1.30 Z B,unsub tree Z A,unsub 1.25 cont Z B,unsub 1.20 cont

1.15 ν

0 1.10 DA 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 Z sub 1.25 tree Z sub 1.20 cont

1.15

1.10

1.05

1.00 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 Z (a p)2

0ν 1 Figure 5.2: ZDA for β =3.9 (a− =2.217 GeV) and mπ =0.430 GeV for method 1(opensymbols)andmethod2(ﬁlledsymbols).Theupperplotcorresponds to non-perturbative results, where the index A, B represents the set of momenta with spatial components 2 π/L(3, 3, 3) and 2 π/L(2, 2, 2),respectively. The lower plot shows the non-perturbative results after subtracting the perturbative (g2 a2)-terms, where the two methods give almost identical results. O Chapter 5. Renormalization Constants for Twist-2 Operators in Twisted 156 Mass QCD

0ν AsimilarbehaviourcanbeseeninFig.5.3forZDV. The same pattern appears in all Z-factors of the one-derivative operators, as well as for Zq.Thelatterhasanimpact on all other renormalization constants discussed here. This eﬀect, as expected, becomes less pronounced at β =4.05 and 4.20, and disappears for small a2p2 as demonstrated in the next section. The results presented in all Tables correspond to the Z-factors obtained using method 2.

1.35 Z A,unsub tree 1.30 Z B,unsub tree 1.25 Z A,unsub cont 1.20 Z B,unsub cont 1.15 1.10 ν 0

DV 1.05 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 Z sub 1.20 tree Z sub 1.15 cont

1.10

1.05

1.00

0.95 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 Z (a p)2

0ν Figure 5.3: As in Fig. 5.2 but for ZDV.

5.5 Results

5.5.1 RI′-MOM Condition

We perform the calculation of renormalization constants for three values of the lattice spacing corresponding to β =3.9, 4.05 and 4.20. The lattice spacing as determined from the nucleon mass is 0.089 fm, 0.070 fm and 0.056 fm respectively. For β =3.9we consider three different quark masses, corresponding to mπ =0.302 GeV (aµ0 =0.004), mπ =0.376 GeV (aµ0 =0.0064) and mπ =0.430 GeV (aµ0 =0.0085), in order to explore the dependence of the Z-factors on the pion mass. At β =4.05 we consider two volumes, 243 48 and 323 64 in order to check for finite volume effects. To extract the × × renormalization constants reliably one needs to consider momenta in the range ΛQCD < p<1/a.Werelaxtheupperboundtobe 2/a to 3/a,whichisjustifiedbythelinear ∼ 5.5. Results 157 dependence of our results on a2. Therefore, for each value of β we consider momenta spanning the range 1

βa(fm) aµ m (GeV) L3 T 0 π × 3.9 0.089 0.0040 0.3021(14) 243 48 3.9 0.089 0.0064 0.37553(80) 243 × 48 3.9 0.089 0.0085 0.4302(11) 243 × 48 4.05 0.070 0.0060 0.4082(31) 243 × 48 4.05 0.070 0.0060 0.404(2) 323 × 64 4.05 0.070 0.0080 0.465(1) 323 × 64 4.20 0.055 0.0065 0.476(2) 323 × 64 × Table 5.1: Action parameters used in the simulations.

In Table 5.2 we present the statistical sample for the parameters and momenta we used in the simulations. Using the number of conﬁgurations shown in Table 5.2 leads to results with very high statistical accuracy, easily below 0.5%.

β =3.9 β =3.9 β =3.9 β =4.05 β =4.05 β =4.05 β =4.20 3 3 3 3 3 3 3 (nt,nx,ny,nz)2448 24 48 24 48 24 48 32 64 32 64 32 64 × × × × × × × µ0 =0.004 µ0 =0.0064 µ0 =0.0085 µ0 =0.0060 µ0 =0.0060 µ0 =0.0080 µ0 =0.0065

(4,2,2,2) 100 50 80 — 50 50 15 (5,2,2,2) 100 60 60 — — 33 15 (6,2,2,2) 100 50 50 — — 50 15 (3,3,3,2) — — 27 — — 15 15 (7,2,2,2) — — 20 — — 15 15 (2,3,3,3) — — 20 — — 15 15 (8,2,2,2) — — 20 — — 15 15 (3,3,3,3) 100 50 80 15 — 50 15 (4,4,4,4) — — — — 15 — — (4,3,3,3) 100 60 60 — — 50 15 (5,3,3,3) 100 60 60 — — 50 15 (6,3,3,3) — — 15 — — 15 15 (10,2,2,2) — — 15 — — 15 15 (8,3,3,3) — — — — — 15 15 (9,3,3,3) — — — — — 15 15 (10,3,3,3) — — — — — 15 15 (13,2,2,2) — — — — — 15 15 (11,3,3,3) — — — — — 15 15 (14,2,2,2) — — — — — 15 15

Table 5.2: Statistical sample at β =3.9, 4.05, 4.20 for various momenta.

The results for the subtracted Z-factors (method 2) at β =3.9aretabulatedinTable 5.3 for the highest and lowest twisted mass parameter used (for the lowest mass we have obtained the Z-factors only for 6 momenta). Comparison between the Z factors for − two diﬀerent masses shows that any dependence on the pion mass is within the small statistical errors. This negligible dependence is not a result of the (a2)subtraction, O as demonstrated in Fig. 5.4. The left plot illustrates the pion mass dependence of the 2 2 unsubtracted ZDV1 for three renormalization scales ranging from 5.75 GeV to 11.75 GeV , Chapter 5. Renormalization Constants for Twist-2 Operators in Twisted 158 Mass QCD

aµ0 =0.0040 aµ0 =0.0085

(nt,nx,ny,nz) ZDV1 ZDV2 ZDA1 ZDA2 ZDV1 ZDV2 ZDA1 ZDA2 (4,2,2,2) 1.1274(1) 1.1836(4) 1.2044(2) 1.2387(4) 1.1283(2) 1.1846(5) 1.2051(2) 1.2395(5) (5,2,2,2) 1.1058(1) 1.1548(4) 1.1792(2) 1.2094(4) 1.1067(2) 1.1558(5) 1.1800(2) 1.2102(5) (6,2,2,2) 1.0854(1) 1.1283(3) 1.1567(2) 1.1820(3) 1.0864(1) 1.1291(4) 1.1576(2) 1.1829(4) (3,3,3,2) ————1.0740(1) 1.1088(4) 1.1418(2) 1.1626(4) (7,2,2,2) ————1.06613(8) 1.1042(4) 1.1363(2) 1.1568(4) (2,3,3,3) ————1.0587(1) 1.0869(4) 1.1331(2) 1.1391(4) (8,2,2,2) ————1.04684(6) 1.0830(3) 1.1176(1) 1.1341(3) (3,3,3,3) ————1.05045(6) 1.0756(2) 1.11066(9) 1.1321(2) (4,3,3,3) 1.04985(7) 1.0750(3) 1.1100(1) 1.1315(3) 1.04204(6) 1.0625(2) 1.09097(9) 1.1229(2) (5,3,3,3) 1.04152(6) 1.0620(2) 1.09035(9) 1.1223(2) 1.03367(6) 1.0487(2) 1.07377(9) 1.1124(2) (6,3,3,3) 1.03327(5) 1.0482(2) 1.07332(8) 1.1120(2) 1.02513(5) 1.0346(2) 1.05817(6) 1.1009(1) (10,2,2,2) ————1.00804(9) 1.0482(2) 1.08561(9) 1.0945(2)

Table 5.3: The renormalization constants at β =3.9 with µ0 =0.0040, 0.0085 for lattice size: 243 48. ×

while the subtracted ZDV1 is shown in the right plot. The same behavior is observed for all renormalization constants considered here. The subtracted Z-factors (method 2) for β =4.05 and β =4.20 are presented in Tables 5.4-5.5, respectively. In order to see possible

volume effects we compute the renormalization constants at β =4.05,µ0 =0.0060, for two lattices with different size, namely for 243 48 and for 323 64. For this comparison × × we used momenta that correspond to the same renormalization scale: For the small lattice we use 2π (3/48, 3/24, 3/24, 3/24), in lattice units, whereas for the larger one we employ × 2π (4/64, 4/32, 4/32, 4/32). The volume effects appear to be 0.1%, as can be seen × ∼ from Table 5.6.

2 2 1.14 2 2 1.16 µ = 5.75 GeV µ = 5.75 GeV µ 2= 7.60 GeV2 1.12 µ 2= 7.60 GeV2 1.14 2 2 2 2 µ = 11.75 GeV 1.10 µ = 11.75 GeV 1.12 sub unsub

DV1 1.08 DV1 Z Z 1.10 1.06 1.08 1.04 1.06 1.02 0.30 0.35 0.40 0.45 0.50 0.55 0.30 0.35 0.40 0.45 0.50 0.55 m (GeV) m (GeV) π π

Figure 5.4: ZDV1 at β =3.9,asafunctionofthepionmass:mπ =0.302 GeV (aµ0 =0.004), mπ =0.375 GeV (aµ0 =0.0064)andmπ =0.429 GeV (aµ0 = 0.0085). The left plot regards the unsubtracted non-perturbative results and the right one corresponds to the subtracted data.

Given the small statistical errors one may carefully examine the systematic errors. As already noted a systematic effect comes from the choice of S(0) and Γ(0). To give an example, at β =3.9,µ =0.004,µ2 5.75 GeV2 method 1 leads to Z =0.76606(7) 0 ≈ q while method 2 gives Z =0.80514(7), before any subtraction of (a2) is carried out. q O This systematic effect is removed after perturbative subtraction is applied. Another, 5.5. Results 159 much smaller, systematic effect comes from the asymmetry of our lattices both because they are larger in their time extent and because of the antiperiodic boundary conditions in the time direction.

(nt,nx,ny,nz) ZDV1 ZDV2 ZDA1 ZDA2 (4,2,2,2) 1.1960(1) 1.2644(3) 1.2749(1) 1.3126(3) (5,2,2,2) 1.1718(2) 1.2324(5) 1.2483(2) 1.2794(5) (6,2,2,2) 1.1491(1) 1.2016(2) 1.2244(1) 1.2475(2) (3,3,3,2) 1.1336(1) 1.1805(3) 1.2069(1) 1.2260(3) (7,2,2,2) 1.1280(2) 1.1745(2) 1.2025(2) 1.2200(2) (2,3,3,3) 1.1188(1) 1.1555(3) 1.1948(1) 1.1998(3) (8,2,2,2) 1.1086(1) 1.1493(2) 1.1826(2) 1.1931(2) (3,3,3,3) 1.10592(6) 1.1458(1) 1.17497(7) 1.1914(2) (4,3,3,3) 1.09370(5) 1.1339(1) 1.15739(6) 1.1807(1) (5,3,3,3) 1.08232(5) 1.1206(1) 1.14186(6) 1.1683(1) (6,3,3,3) 1.07144(9) 1.1068(2) 1.1278(1) 1.1551(2) (10,2,2,2) 1.0724(1) 1.1090(2) 1.1477(1) 1.1505(2)

Table 5.4: Renormalization constants at β =4.05,aµ0 =0.0080 for lattice size 323 64. ×

(nt,nx,ny,nz) ZDV1 ZDV2 ZDA1 ZDA2 (4,2,2,2) 1.1585(4) 1.215(1) 1.2266(5) 1.257(1) (5,2,2,2) 1.1387(4) 1.189(1) 1.2052(5) 1.230(1) (6,2,2,2) 1.1203(3) 1.1642(9) 1.1853(4) 1.2040(9) (3,3,3,2) 1.1069(2) 1.1459(9) 1.1702(3) 1.1855(9) (7,2,2,2) 1.1028(2) 1.1413(8) 1.1668(3) 1.1804(8) (2,3,3,3) 1.0943(2) 1.1257(8) 1.1599(3) 1.1643(8) (8,2,2,2) 1.0853(1) 1.1197(4) 1.1488(2) 1.1579(4) (3,3,3,3) 1.0841(1) 1.1177(8) 1.1438(3) 1.1577(8) (4,3,3,3) 1.0743(2) 1.1079(7) 1.1293(2) 1.1491(7) (5,3,3,3) 1.0651(2) 1.0968(6) 1.1163(2) 1.1390(6) (6,3,3,3) 1.0511(2) 1.0812(5) 1.0983(3) 1.1274(5) (10,2,2,2) 1.0528(1) 1.0856(3) 1.1189(1) 1.1223(3)

Table 5.5: Renormalization constants at β =4.20,µ0 =0.0065 for lattice size: 323 64. ×

lattice ZDV1 ZDV2 ZDA1 ZDA2 243x48 1.0700(2) 1.0923(2) 1.1190(2) 1.1117(2) 323x64 1.07123(6) 1.0928(2) 1.12037(7) 1.1122(2)

Table 5.6: Renormalization constants at β =4.05,µ0 =0.0080 using method 2 and two lattice sizes: 323 64 for (4,4,4,4) and 243x48 for the rest of the momenta. ×

For instance, using the same β and µ as in the previous example, method 1 at µ2 7.6 GeV2, 0 ≈ in the temporal direction of the current gives ZDV1 =1.1387(2) while the average from Chapter 5. Renormalization Constants for Twist-2 Operators in Twisted 160 Mass QCD the three spatial directions leads to ZDV1 =1.1006(2). This effect can be seen in Fig. 5.5 where we plot separately the renormalization constant ZDV1 determined from the temporal indices, the spatial indices and the average of those two. In the same figure we also show that upon subtraction this systematic effect disappears (lower plot). For Tables 5.3 00 νν - 5.5 we use for ZDV1 the average of ZDV,ZDV with ν =1, 2, 3, while for ZDV2 the average of Z0ν ,Zνρ with ν = ρ =1, 2, 3. We apply the same procedure for the twist-2 axial DV DV ̸ operator.

1.25 Z mm 1.20 unsub Z 00 1.15 unsub Z aver 1.10 unsub

1.05

1.00

DV1 1.25 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 Zmm 1.20 sub Z00 1.15 sub Zaver 1.10 sub

1.05

1.00

0.95 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 Z (a p)2

µµ 00 aver Figure 5.5: ZDV (squares), ZDV (circles), ZDV (crosses), for β =3.9 1 (a− =2.217 GeV), mπ =0.430 GeV using method 1. The upper plot corresponds to the purely non-perturbative results, while the lower plot shows the non-perturbative results after subtracting the perturbative terms of (a2). O

Chiral extrapolations are necessary to obtain the renormalization factors in the chiral limit. As already pointed out the dependence on the pion mass is insignificant. Allowing aslopeandperformingalinearextrapolationtothedatashowninFig.5.4yieldsaslope consistent with zero. This behavior is also observed at the other β-values and therefore the renormalization constants are computed at one quark mass, given in the Tables 5.3- 5.5. Figures 5.6, 5.7, 5.8 demonstrate the effect of subtraction, for all three β values, as a function of the renormalization scale (in lattice units). For all cases we observe a significant correction upon subtraction; the lattice artifacts for ZDA2 turn out to be very small for most values of the momentum. In addition, the lattice artifacts decrease by employing higher values for β (finer lattice), as expected. 5.5. Results 161

unsub Z Zsub 1.20 1.25

1.15 1.20

1.10 1.15 DV1 DV2 Z 1.05 Z 1.10

1.00 1.05

0.95 1.00 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 (a p)2 (a p)2 1.25 1.30

1.20 1.25

1.15 1.20 DA2 DA1 Z Z 1.10 1.15

1.05 1.10

1.00 1.05 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 (a p)2 (a p)2 Figure 5.6: Renormalization scale dependence for the Z-factors at β =3.9 and mπ =0.430 GeV

unsub Z Zsub 1.25 1.30

1.20 1.25

1.15 1.20 DV1 1.10 DV2 1.15 Z Z 1.05 1.10

1.00 1.05

0.95 1.00 0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 2.5 (a p)2 (a p)2 1.30 1.30

1.25 1.25 1.20 1.20 DA2 DA1 1.15 Z Z 1.15 1.10

1.05 1.10

1.00 1.05 0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 2.5 (a p)2 (a p)2 Figure 5.7: Renormalization scale dependence for the Z-factors at β =4.05 and mπ =0.465 GeV Chapter 5. Renormalization Constants for Twist-2 Operators in Twisted 162 Mass QCD

unsub Z Zsub 1.25 1.30

1.20 1.25

1.15 1.20 DV1 1.10 DV2 1.15 Z Z 1.05 1.10

1.00 1.05

0.95 1.00 0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 2.5 (a p)2 (a p)2 1.30 1.30

1.25 1.25 1.20 1.20 DA2 DA1 1.15 Z Z 1.15 1.10

1.05 1.10

1.00 1.05 0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 2.5 (a p)2 (a p)2 Figure 5.8: Renormalization scale dependence for the Z-factors at β =4.20 and mπ =0.476 GeV

5.5.2 RI′-MOM at a Reference Scale

All our Z-factors have been evaluated for a range of renormalization scales. In this subsection we use 2-loop perturbative expressions to extrapolate to a scale µ =1/a (the values for a are taken from Table 5.1). Thus, each result is extrapolated to 1/a,maintaining the information of the initial renormalization scale at which it was computed. Although the 3-loop formula is available for the following expressions, the (g6)correctionsare O insigniﬁcant compared to the lower order results. The scale dependence is predicted by the renormalization group (at ﬁxed bare parameters), that is:

RI′ RI′ Z (µ)=R (µ, µ0) Z (µ0), (5.62) O O O with: 1 γ1O γ0O γ0O 2 2 2 β β 2 2 β1 g (µ ) 1 − 0 g (µ ) 2β0 1+ 2 " # R (µ, µ )= β0 16π . (5.63) 0 2 2 g2(µ2) O g (µ ) ⎛ β1 0 ⎞ 0 1+ 2 2 3 β0 16π ⎝ ⎠ To 2 loops, the running coupling, β-function and anomalous dimension γ are as follows:

g2(µ2) 1 β ln ln(µ2/Λ2) = 1 + (5.64) 2 2 2 3 2 2 2 16π β0 ln(µ /Λ ) − β0 ln (µ /Λ ) ··· 5.5. Results 163

2 38 β =11 N ,β =102 N (5.65) 0 − 3 F 1 − 3 F g2 g2 2 γO(g)=γO + γO + (5.66) 0 16π2 1 16π2 ··· 2 3 The expressions for the anomalous dimension of the fermion ﬁeld and the twist-2 vector/axial operators are given in Ref. [32],

g2 γ RI′ (a)=λC + 9λ3 +45λ2 +223λ +225 C ψ F 16 π2 A ) / 0C g2 2 54C (80λ +72)T N F , (5.67) − F − F F 36 16 π2 * 2 3 2 RI′ 8 g 2 2 γ µ ν (a)= CF + CF 27λ +81λ +1434 CA ψγ{ D }ψ 3 16 π2 54 ) / 0 g2 2 224C 504 T N , (5.68) − F − F F 16 π2 * 2 3 where TF =1/2,CA = Nc. Using Eqs. (5.62) - (5.68) we obtain the Z-factors at µ =1/a for β =3.9, 4.05, and 4.20, which are plotted in Figs. 5.9 - 5.11.

1.3 1.2 DV2 Z 1.1 1

1.3 1.2 DA2 Z 1.1 1

1.3 1.2 DV1 Z 1.1 1

1.3 1.2 DA1 Z 1.1 1 567 8 9 10 11 12 13 14 p2 (GeV2)

Figure 5.9: Renormalization factors in the RI′ MOM scheme at renormal- − ization scale 1/a,forβ =3.9,µ0 =0.0085.Theblackcirclescorrespond to the unsubtracted results, while the magenta diamonds to the results with perturbatively subtracted one loop (a2) artifacts. O Chapter 5. Renormalization Constants for Twist-2 Operators in Twisted 164 Mass QCD

1.3 1.2 DV2 Z 1.1 1 1.3 1.2 DA2

Z 1.1 1

1.3 1.2 DV1 Z 1.1 1

1.3 1.2 DA1 Z 1.1 1 567 8 9 10 11 12 13 14 15 16 17 18 19 20 21 p2 (GeV2)

Figure 5.10: Same as Fig 5.9, but for β =4.05 and µ0 =0.0080.

1.3 1.2 DV2 Z 1.1 1 1.3 1.2 DA2

Z 1.1 1

1.3 1.2 DV1 Z 1.1 1

1.3 1.2 DA1 Z 1.1 1 8 10 12 14 16 18 20 22 24 26 28 30 32 34 p2 (GeV2)

Figure 5.11: Same as Fig 5.9, but for β =4.20 and µ0 =0.0065. 5.5. Results 165

5.5.3 Conversion to MS

The passage to the continuum MS-scheme is accomplished through use of a conversion factor which is computed up to 3 loops in perturbation theory. By deﬁnition, this conversion factor is the same for the vector and axial twist-2 renormalization constant, but will diﬀer for the cases ZDV1 (ZDA1)andZDV2 (ZDA2), that is:

MS MS ZDV ZDV CDV1 CDA1 = ,CDV2 CDA2 = . (5.69) RI′ RI′ ≡ ZDV1 ≡ ZDV2

This requirement for diﬀerent conversion factors results from the fact that the Z-factors in the continuum MS-scheme do not depend on the external indices, µ, ν (see Eq. (2.5) of

Ref. [32]), while the results in the RI′ MOM scheme do depend on µ and ν.Ofcourse − the conversion factors take a diﬀerent value for each renormalization scale; actually, the direction of the momentum is required to be known (Eqs. (5.85)-(5.86)).

The 3-loop expressions for the conversion factors from our RI′ MOM scheme − (Eq. (5.57)) to the MS do not appear directly in the literature, but can be extracted using results from Ref. [32]. In the latter publication the reader can find the conversion factor from an alternative definition of RI′ MOM (which we denote by RI) to the usual − MS, CψγµDν ψ.Thisalternativedefinitionreads:

RI RI (1) lim Zψ ZψγµDν ψΣ µ ν (p) =1, (5.70) ϵ 0 ψγ D ψ p2= µ2 → ) *I I where Σ(1) can be extracted from the bare amputatedI Green’s function as follows:

µν µ ν G µ ν (p)= ψ(p)[ψγ{ D }ψ](0) ψ( p) ψγ{ D }ψ ⟨ − ⟩ (1) µ ν ν µ 2 µν =Σ µ ν (p) γ p + γ p pη ψγ{ D }ψ − d̸ 2 2 3 (2) 1 µ ν p µν +Σ µ ν (p) p p p pη , (5.71) ψγ{ D }ψ p2 ̸ − d ̸ 2 3 and p = γ p . ̸ ρ ρ ρ The author! of Ref. [32] provides the 3-loop expression for the renormalized Σ(2) in the scheme of Eq. (5.70) (note that by deﬁnition the renormalized Σ(1) equals 1 at p2 = µ2). These elements can be used to reconstruct the renormalized Green’s function:

µν,R µ ν ν µ 2 µν G µ ν (p) = 1 γ p + γ p pη ψγ{ D }ψ p2=µ2 6 · − d̸ I 2 3 I 2 I (2) RI′ ﬁnite 1 µ ν p µν +Σ µ ν (p) p p p pη , (5.72) ψγ{ D }ψ p2 ̸ − d ̸ 2 3 7p2=µ2 Chapter 5. Renormalization Constants for Twist-2 Operators in Twisted 166 Mass QCD in which we apply our RI′ MOM condition in order to obtain: − RI RI Z µ ν Z µ ν ψγ{ D }ψ , ψγ{ D }ψ . (5.73) RI′ RI′ ZDV1 ZDV2

Once we have these two elements we extract the conversion factor of Eq. (5.69) up to 3 loops:

RI RI MS Z µ ν Z µ ν ψγ{ D }ψ 1 ψγ{ D }ψ ZDV CDV1(µ)= C µ ν − = , (5.74) RI′ ψγ D ψ RI′ RI ZDV1 · ZDV1 · Z µ ν ψγ{ D }ψ RI / 0 RI MS Zψγ µDν ψ 1 Zψγ µDν ψ Z { } − { } DV CDV2(µ)= CψγµDν ψ = . (5.75) RI′ RI′ ZRI ZDV2 · ZDV2 · ψγ µDν ψ / 0 { } The conversion to the MS is then given by:

ZMS (µ)=C (µ) ZRI′ (µ) , (5.76) DV1 DV1 · DV1 ZMS (µ)=C (µ) ZRI′ (µ) , (5.77) DA1 DV1 · DA1 ZMS (µ)=C (µ) ZRI′ (µ) , (5.78) DV2 DV2 · DV2 ZMS (µ)=C (µ) ZRI′ (µ) , (5.79) DA2 DV2 · DA2 which correspond to the Z-factors at the same renormalization scale in the RI′.Onewants to obtain the renormalization constants at the scale of 2 GeV, and to do this we use the 2-loop formula in Eq. (5.63)-(5.64) to evolve the scale from µ to 2 GeV. In these formulas we need to insert the anomalous dimension in the MS-scheme which read [32]:

MS 8 8CF 2 γ µ ν (a)=2 CF a +2 [47CA 14CF 16TF NF ] a . (5.80) ψγ{ D }ψ 3 27 − −

The additional factor of 2 that we included, comes from the diﬀerent deﬁnition of the anomalous dimension that leads to Ref. [120]. To summarize, the Z-factors in the continuum MS -scheme at µ = 2 GeV are given by:

ZMS (2GeV )=R (2GeV, µ) C (µ) ZRI′ (µ), (5.81) DV1 DV · DV1 · DV1 ZMS (2GeV )=R (2GeV, µ) C (µ) ZRI′ (µ), (5.82) DV2 DV · DV2 · DV2 ZMS (2GeV )=R (2GeV, µ) C (µ) ZRI′ (µ), (5.83) DA1 DV · DV1 · DA1 ZMS (2GeV )=R (2GeV, µ) C (µ) ZRI′ (µ). (5.84) DA2 DV · DV2 · DA2

For the SU(Nc =3)colourgroup(CA =3,CF =4/3, TF =1/2), Landau gauge (λ = 0), and general quark ﬂavours, we have the following conversion factors: 5.5. Results 167

µ4 MS µ2 µ ZDV 136 64 µ − µ2 CDV1 =1+α + ≡ RI′ − 2 2 ZDV1 $ 27 9 µ +8µµ %

4 4 2 µµ 2 µµ 128096 3208 320 µµ 2 248 µµ 2 17792 320 + α2 + N − µ + ζ(3) + − µ + ζ(3) F ⎛ 2 2⎞ 2 2 $ − 729 243 − 9 µ +8µµ 9 µ +8µµ 27 9 % , - ⎜ ⎟ ⎝ ⎠ 4 2 µµ 627867571 64 π4 5588641 149552 77440 µµ 2 256 + α3 + ζ(3) + N 2 + − µ ζ(3) F ⎛ 2 2 ⎞ $ − 78732 − 729 2187 − 6561 729 µ +8µµ − 243 ⎜ ⎟ ⎝ ⎠ µ4 4 2 µ 19947676 64 π 1600 µµ 2 121024 9856 + N + ζ(3) + − µ + ζ(3) F ⎛ 2 2 ⎞ 19683 243 − 27 µ +8µµ − 27 81 , - ⎜ ⎟ ⎝ 4 ⎠ 2 µµ 19420 µµ 2 270701210 2993992 349600 ζ(5) + − µ ζ(3) + ζ(5) + (α4), (5.85) 2 2 − 27 µ +8µµ 6561 − 243 81 % O , -

MS 2 2 ZDV 124 16 µµµν CDV2 =1+α ≡ RI′ − − 2 2 2 ZDV1 $ 27 9 µ (µµ + µν ) %

98072 2668 80 µ2µ2 268 µ2µ2 4448 80 + α2 + N + µ ν + ζ(3) + µ ν ζ(3) F 2 2 2 2 2 2 $ − 729 . 243 9 µ (µµ + µν ) / 9 µ (µµ + µν ) − 27 − 9 % , - 849683327 64 π4 7809041 105992 19360 µ2µ2 256 + α3 + ζ(3) + N 2 µ ν ζ(3) F 2 2 2 $ − 157464 − 729 4374 .− 6561 − 729 µ (µµ + µν ) − 243 /

14433520 64 π4 4184 µ2µ2 30256 2464 + N + ζ(3) + µ ν ζ(3) F 2 2 2 . 19683 243 − 81 µ (µµ + µν ) 27 − 81 / , - 36410 µ2µ2 135350605 748498 87400 ζ(5) + µ ν + ζ(3) ζ(5) + (α4), (5.86) 2 2 2 − 81 µ (µµ + µν ) − 13122 243 − 81 % O , - where α = g2/(16π2)andζ(n) is the Riemann Zeta function. 2 2 2 A“renormalizationwindow”shouldexistforΛQCD << µ << 1/a where perturbation theory holds and finite-a artifacts are small, leading to scale-independent results (plateau). In practice such a condition is hard to satisfy: The right inequality is extended to (2 3)/a2 leading to lattice artifacts in our results that are of (a2p2). Fortunately ∼ O our perturbative calculations allow us to subtract the leading perturbative (a2) lattice O artifacts which alleviates the problem. To remove the remaining (a2p2)artifactswe O extrapolate linearly to a2p2 = 0 as demonstrated in Fig. 5.12. The statistical errors are negligible and therefore an estimate of the systematic errors is important. The largest systematic error comes from the choice of the momentum range to use for the extrapolation to a2p2 = 0. One way to estimate this systematic error is to vary the upper range from a2p2 =2.2toa2p2 =2.7. Another approach is to fix a range and then eliminate a given momentum in the fit range and refit. The spread of the results about the mean gives an estimate of the systematic error. In the final results we give as systematic error the largest one from using these two procedures which is the one obtained by increasing the upper fit range from a2p2 =2.2toa2p2 =2.7. Figures 5.13-5.14 show ZMS for β =4.05 O and β =4.20. Chapter 5. Renormalization Constants for Twist-2 Operators in Twisted 168 Mass QCD

1.10

DV2 1.05 Z 1.00

1.15 1.10 DA2

Z 1.05 1.00

1.10

DV1 1.05 Z 1.00

1.15

DA1 1.10 Z 1.05 1.00 0 0.5 1 1.5 2 2.5 3 (a p)2

Figure 5.12: Renormalization factors at β =3.9, µ0 =0.0085 in the MS- scheme at renormalization scale 2 GeV.Blackcirclescorrespondtotheun- subtracted results, while magenta diamonds correspond to the results with perturbatively subtracted 1-loop (a2) artifacts. The lines show extrapolations to a2p2 =0using the subtractedO results within the range a2p2 =1 2.2. ∼

1.25 1.20 1.15 DV2

Z 1.10 1.05

1.25 1.20 DA2

Z 1.15 1.10

1.15 1.10 DV1

Z 1.05 1.00

1.20 1.15 DA1 Z 1.10 1.05 0 0.5 1 1.5 2 2.5 (a p)2

Figure 5.13: Same as Fig 5.12, but for β =4.05 and µ0 =0.0080. 5.6. Conclusions 169

1.25 1.20 1.15 DV2

Z 1.10 1.05

1.25 1.20 DA2

Z 1.15 1.10

1.15 1.10 DV1

Z 1.05 1.00

1.20 1.15 DA1 Z 1.10 1.05 0 0.5 1 1.5 2 2.5 (a p)2

Figure 5.14: Same as Fig 5.12, but for β =4.20 and µ0 =0.0065.

Our ﬁnal results for the Z-factors in the MS-scheme at 2 GeV are given in Table 5.7, which have been obtained by extrapolating linearly in a2p2.

βZDV1 ZDV2 ZDA1 ZDA2 3.90 1.038(10)(20) 1.129(7)(3) 1.174(8)(11) 1.153(6)(16) 4.05 1.097(5)(4) 1.110(14)(26) 1.147(13)(24) 1.159(7)(16) 4.20 1.114(11)(17) 1.103(21)(42) 1.139(21)(40) 1.159(9)(20)

Table 5.7: Renormalization constants ZDV and ZDA in the MS scheme. The above values have been obtained by extrapolating linearly in a2p2. The error in the second parenthesis is the systematic error due to the extrapolation, namely the largest difference between results using the fit range a2p2 =1.2 2.7 and ∼ one of the ranges a2p2 =1 2.7, a2p2 =1.2 2.2. Statistical errors are in most cases smaller and are shown∼ in the first parenthesis.∼

5.6 Conclusions

The values of the renormalization factors fortheone-derivativetwist-2operatorsare calculated non-perturbatively. The method of choice is to use a momentum dependent source and extract the renormalization constants for all the relevant operators. This leads to a very accurate evaluation of these renormalization factors using a small ensemble of Chapter 5. Renormalization Constants for Twist-2 Operators in Twisted 170 Mass QCD gauge configurations. The accuracy of the results allows us to check for any light quark mass dependence. For all the renormalizationconstantsstudiedinthisworkwedonot find any light quark mass dependence within our small statistical errors. Therefore it suffices to calculate them at a given quark mass. We also show that, despite of using lattice spacing smaller than 1 fm, (a2) effects are sizable. We perform a perturbative O subtraction of (a2) terms. This leads to a smoother dependence of the renormalization O constants on the momentum values at which they are extracted. Residual (a2p2)effects O are removed by extrapolating to zero. In this way we can accurately determine the renormalization constants in the RI′-MOM scheme. In order to compare with experiment we convert our values to the MS scheme at a scale of 2 GeV. The statistical errors are in general smaller than the systematic. The latter are estimated by changing the window of values of the momentum used to extrapolate to a2p2 = 0. Our final values are given in Table 5.7. Appendix A: Notation

In this Appendix we present a brief introduction to the basic equations that govern QCD. We begin with the continuous Lagrangian, explaining all mathematical symbols. The expression for the action and the partition function are also provided, as well as the interaction vertices. Then we switch to Lattice QCD and show how the discretization is performed and its consequences on the action of QCD.

A.1 Continuum QCD

In the continuous Lagrangian, quarks (antiquarks) are Dirac 4-spinors denoted by Ψf (x) f f (Ψ (x)=(Ψ (x))†γ0)andcanhaveoneofthe6ﬂavors,denotedbyNf ;theyareSU(3) triplets in color space. Gluons are gauge bosons represented by 8 (more general N 2 1) − B B B gauge ﬁelds Aµ(x) ϵSU(3). In particular, they can be written as Aµ = Aµ T where T are traceless hermitian 3 3matricesandB =1, .., N 2 1. The Lagrangian density is × − the sum of a fermionic and a purely gluonic part:

Nf f 1 (x)= Ψ (x)(iγµD (x) mf )Ψf (x) Tr[G (x)Gµν (x)]. (A.1) LQCD µ − 0 − 2 µν &f=1 The trace of the second term is taken over color indices. The gluon ﬁeld strength tensor appearing in the Lagrangian is deﬁned by:

i Gµν (x)= [Dµ(x),Dν(x)] ,Dµ(x)=∂µ ig0 Aµ(x), (A.2) g0 − where Dµ is the covariant derivative and g0 is the bare coupling of the strong interaction. The indices µ, ν label the space time coordinates, and a summation over repeated indices is implied. Moreover, the bare quark mass m0 differs for quarks with different flavor. Both terms of Eq. (A.1) are invariantunderalocalgaugetransformation:

Ψ(x) Λ(x)Ψ(x) → Ψ(x) Ψ(x)Λ†(x) → 1 Aµ(x) Λ(x)Aµ(x)Λ†(x) ∂µΛ(x)Λ†(x), (A.3) → − g0

171 172 171 where Λ(x) ϵSU(3). There are 3 interaction vertices arising from Eq. (A.1): The quark-antiquark-gluon vertex, the 3- and 4-gluon vertex, as demonstrated in Fig. A.1.

Figure A.15: The interaction vertices of quarks and gluons. The interaction vertices of quarks and gluons. Solid (wavy) lines represent fermions (gluons).

For the introduction of the lattice formulation it is necessary to switch to Euclidean space, by performing a Wick’s rotation: t it, so that time is purely imaginary. The → product of two 4-vectors is now given by:

µ xµ y = x0 y0 + x1 y1 + x2 y2 + x3 y3,

with x0,y0 the time coordinates. In what follows and in the main body of the Thesis the metric of the Euclidean space is applied. One must deﬁne the partition function Z, necessary for the calculation of Green’s functions and the normalization of expectation values, deﬁned by the path integral:

S[Ψ,Ψ,A] Z = DΨ] DΨ DA e− , (A.4) 1 where S is the QCD action in Euclidean space:

S[Ψ, Ψ,A]= d4x (x). (A.5) LQCD 1 The expectation values for physical quantities, ,canberepresentedbyoperatorsbuilt O from quark and gluon ﬁelds:

1 S[Ψ,Ψ,A] < >= DΨ DΨ DA e− . (A.6) O Z O 1 4 Clearly, the action is dimensionless and thus the Lagrangian has dimensions [length]− , or equivalently [mass]4.Fromthemasstermwenotethatthefermionﬁeldshavedimen- 3/2 sion [length]− and using the pure gluon term one can see that the coupling constant is 1 dimensionless; thus the gauge ﬁelds have dimension [length]− . A.2. Lattice QCD 173

A.2 Lattice QCD

In a hypercubic space-time lattice (ultraviolet regulator), the continuum Euclidean coor- dinate xµ is replaced by a variable having discrete values:

xµ nµa, nµ ϵ Z ,a: lattice spacing. →

This discretization introduces a momentum cutoffwhich is inverse to the lattice spacing, since the momenta are restricted in the finite interval π/a p π/a (first Brillouin − ≤ ≤ zone). Thus, the integrals transform to finite sums:

d4x a4 , → n 1 & and all quantities calculated in the lattice are finite. The first thing that needs to be done 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 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)definedas:

iag0Aµ(n) Uµ(n)=e . (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. 3/2 For instance, the quark ﬁeld has to be multiplied by a− :

3/2 Ψ(n) a− Ψ(n). →

In order to write a lattice version of the QCD action, we discretize the derivative using the naive diﬀerences:

1 −→ Ψ(n)= [U (n)Ψ(n + aµˆ) Ψ(n)] (A.8) ∇ µ a µ − 1 1 ←− Ψ(n)= [Ψ(n) U (n aµˆ)− Ψ(n aµˆ)], (A.9) ∇ µ a − µ − − whereµ ˆ is the unit vector in direction µ. One of the desired properties of the lattice action is the gauge invariance. Gauge symmetry is required in order to make QCD a renormalizable theory (i.e., one in which the calculated predictions of all physically 174 171 measurable quantities are ﬁnite). The lattice gauge transformations take the form:

Ψ(n) Λ(n)Ψ(n) → Ψ(n) Ψ(n)Λ†(n) → U (n) Λ(n)U (n)Λ†(n). (A.10) µ → µ

For this purpose, the pure gluonic part of the action must be constructed by gauge invariant 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. 2.5). The ‘naive’ − lattice action can thus be written as:

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