department of physics - ucy€¦ · a product of original work of my own, unless otherwise...

Department of Physics

Hadron structure includingdisconnected quark loop

contributions

DOCTOR OF PHILOSOPHY DISSERTATION

KYRIAKOS HADJIYIANNAKOU

2015

Department of Physics

Hadron structure includingdisconnected quark loop

contributions

KYRIAKOS HADJIYIANNAKOU

A Dissertation Submitted to the University of Cyprus in Partial

Fulfillment of the Requirements for the Degree of Doctor of

Philosophy

June 2015

Doctoral Candidate: Kyriakos Hadjiyiannakou

Doctoral Thesis Title: Hadron structure including disconnected quark loop contri-butions

The present Doctoral Dissertation was submitted in partial fulfillment of the re-quirements for the Degree of Doctor of Philosophy at Department of Physics andwas approved on 11/05/2015 by the members of the Examination Committee.

Examination Committee:

Prof. Constantia Alexandrou, Supervisor

Prof. Chris Michael

Prof. Karl Jansen

Prof. Haralambos Panagopoulos

Associate Prof. Nicolaos Toumbas

The present doctoral dissertation was submitted in partial fulfillment of the re-quirements for the degree of Doctor of Philosophy of the University of Cyprus. It isa product of original work of my own, unless otherwise mentioned through references,notes, or any other statement.

Kyriakos Hadjiyiannakou

Acknowledgements

The work in this thesis could not have been possible if it were not for the constantsupport, knowledge and guidance of my supervisor Prof. Constantia Alexandrou. Ishall be forever grateful for the patience and trust she showed to me.

The results presented in this thesis are from collaborative efforts of several re-searchers whom I would like to thank. I would first like to thank Asst. Prof GiannisKoutsou who provided very useful discussions during my PhD studies on several as-pects of my research. Special thanks also to Martha Constantinou for the fruitful andinteresting conversations we shared. I would like also to thank Alejandro Vaquero forthe beneficial and advantageous collaboration. I am also grateful to Alexei Strelchenkofor my first steps in HPC programming.

Special thanks also to my dear friend Andreas Athenodorou for the interestingdiscussions we had on various issues. Thanks also to my fellow PhD student ChristosKallidonis for his company as a friend.

I would also like to express my gratitude to my family and especially my parentsfor their understanding and totally selfless support.

And last but definitely not least, I would like to thank Zorka for always being onmy side and the encouragement she provided for the fulfillment of this thesis.

Περίληψη

Σε αυτή την εργασία μελετούμε τη συνεισφορά των ασύνδετων διαγραμμάτων κουώρκς σε αδρο-

νικά στοιχεία πίνακα, όπως για παράδειγμα στοιχεία πίνακα τα οποία προσδιορίζουν τους παρά-

γοντες μορφής. Οι παράγοντες μορφής αποτελούν πολύ σημαντικές ποσότητες για τη μελέτη

σωματιδίων όπως το πρωτόνιο, και μπορούν τόσο να υπολογιστούν θεωρητικά όσο και να μελε-

τηθούν πειραματικά. Σε αυτή την εργασία αναπτύσσουμε διάφορες τεχνικές υπολογισμού των

ασύνδετων βρόγχων κουώρκς οι οποίοι αποτελούν ένα από τα δύο συστατικά που χρειάζομαστε

για να κτίσουμε τα ασύνδετα διαγράμματα. Το άλλο συστατικό είναι οι συναρτήσεις συσχέτισης

δύο σημείων. Οι ασύνδετοι βρόγχοι, σε συνδυασμό με τις συναρτήσεις συσχέτισης δύο σημεί-

ων, μας δίνουν τη δυνατότητα να υπολογίσουμε τη συνεισφορά των ασύνδετων διαγραμμάτων

στις διάφορες παρατηρήσιμες ποσότητες. Μέχρι τις μέρες μας τα πλείστα αποτελέσματα από την

κβαντική χρωμοδυναμική στο πλέγμα αγνοούσαν τη συνεισφορά των ασύνδετων διαγραμμάτων

επειδή αναμένεται να είναι μικρή και από την άλλη επειδή ο υπολογισμός τους απαιτεί τεράστια

υπολογιστική ισχύ. Με την ανάπτυξη νέων αλγορίθμων αλλά και τη συνεχή εξέλιξη των ηλεκτρο-

νικών υπολογιστών ο υπολογισμός τους στις μέρες μας γίνεται εφικτός. Στην παρούσα εργασία

εξηγούμε διάφορες στοχαστικές μεθόδους για τον υπολογισμό των ασύνδετων διαγραμμάτων.

Συζητείται η μέθοδος Truncated Solver Method (TSM) η οποία μας επιτρέπει να μειώσουμετο στοχαστικό σφάλμα στα ασύνδετα διαγράμματα μειώνοντας ταυτόχρονα και το υπολογιστικό

κόστος. Αυτή η μέθοδος μελετάται παράλληλα με άλλες μεθόδους όπως είναι το one-end tri-ck, διάφορα είδη αραίωσης (dilution) αλλά και μεθόδους υπολογισμού του αντιστρόφου πίνακαχρησιμοποιώντας ανάπτυγμα ως προς την ενεργό μάζα (Hopping Parameter Expansion) για ναβελτιώσουμε το σήμα. Οι μέθοδοι αυτές αξιολογούνται για 4 διαφορετικά κουάρκς (quarks) τοup, το down (light) το strange και το charm. Η επιλογή της κατάλληλης μεθόδου γίνεται μεκριτήριο την αποδοτικότητα της και εξαρτάται από τη μάζα των κουώρκς. Μετά απο την αξιο-

λόγηση των μεθόδων σε διάφορα παρατηρήσιμα μεγέθη, επιλέγουμε το συνδυασμό της μεθόδου

TSM μαζί με το one-end trick για τον υπολογισμό μιας μεγάλης συλλογής αποτελεσμάτων ταοποία θα χρησιμοποιηθούν για να μειώσουμε σημαντικά το στοχαστικό αλλά και το στατιστικό

σφάλμα προκειμένουν να εξάγουμε αξιόπιστα συμπεράσματα. Για την περίπτωση του παρατηρήσι-

μου μεγέθους σ-term βρίσκουμε αρκετά μεγάλη συνεισφορά από το ελαφρύ αλλά και το strangeκουάρκ, συνεισφορά η οποία δεν μπορεί να αγνοηθεί. Το άλλο μέγεθος στο οποίο βλέπουμε

σημαντική συνεισφορά είναι το αξονικό φορτίο gA του νουκλεονίου. Καθαρή συνεισφορά φαίνε-ται και από τα δύο κουάρκς (light, strange) αλλά το charm έχει μηδενική συνεισφορά. Επίσηςμελετούμε και την συνεισφορά στους ηλεκτρομαγνητικούς παράγοντες μορφής η οποία φαίνεται

να είναι αμελητέα. ΄Οσον αφορά το τανυστικό φορτίο gT το αποτέλεσμα δείχνει μια εξαιρετι-κά μικρή συνεισφορά. Μετέπειτα υπολογίζουμε και τις πρώτες ροπές των Generalized PartonDistributions (GPDs) του νουκλεονίου χρησιμοποιώντας παραγώγους στον τελεστή ρεύματος.Για την ποσότητα 〈x〉u+d παρατηρούμε μεγάλα σφάλματα και δεν μπορούμε να δώσουμε κάποια

αξιόπιστη εκτίμηση. Σε αντίθεση το 〈x〉∆u+∆d παρουσιάζει μια απρόσμενα μεγάλη τιμή, αλλά

χρειαζόμαστε μεγαλύτερη στατιστική προκειμένου να αποφανθούμε αν πράγματι το σήμα είναι

τόσο μεγάλο. Τέλος παρουσιάζουμε μια μέθοδο υπολογισμού της ηλεκτρικής διπολικής ροπής

του νετρονίου χρησιμοποιώντας ασύνδετα διαγράμματα σε πρώτη τάξη ως προς την παράμετρο

θ στη δράση της Κβαντικής Χρωμοδυναμικής (ΚΧΔ), που περιγράφει το σπάσιμο της συμμε-τρίας CP. Επίσης επεξηγούμε τον τρόπο με τον οποίο μπορεί να υπολογιστεί αυτή η ποσότηταμε ασύνδετα διαγράμματα αντί της συμβατικής μεθόδου η οποία εμπλέκει το τοπολογικό φορτίο.

Αφού έχουμε υπολογίσει την ηλεκτρική διπολική ροπή, τη συνδυάζουμε με πρόσφατα πειραματικά

δεδομένα για να δώσουμε μια εκτίμηση του μεγέθους της παραμέτρου θ.

Abstract

In this work we explore the contribution of disconnected diagrams in the study of hadronstructure for an important class of hadronic observables, namely the form factors, whichencapsulate very important information for particles such as the proton and the neutron.Nucleon form factors can be measured experimentally and studied phenomenologically. Inthis thesis we develop techniques enabling us to calculate disconnected quark loops, which areone of the two ingredients needed in order to evaluate the disconnected part of three-pointfunctions. Until recently most lattice Quantum Chromodynamics (QCD) studies ignored thedisconnected contributions either because are expected to be small or due to the enormouscomputational power one needs for their evaluation. After the development of improvedalgorithms and the tremendous increase of computational power of Central Processing Units(CPUs) and Graphics Processing Units (GPUs), the calculation of disconnected diagramshas become feasible. In this thesis we study various stochastic techniques for the evaluationof disconnected diagrams. We discuss the Truncated Solver Method (TSM), which allowsus to reduce the stochastic variance at reduced cost. This method is combined with othermethods the one-end trick for disconnected quark loops, various dilution schemes and theHopping Parameter Expansion (HPE) for the expansion of the Dirac matrix in terms ofthe hopping matrix. The combination of all these methods enables us to reduce the totaluncertainty from the disconnected contributions. We then assess the performance of thesemethods for three different masses for the light, strange and charm quark choosing the mostsuitable method based on its efficiency. We thoroughly assess the available methods forvarious observables, and employing a combination of TSM with the one-end trick and takinga very large number of measurements we evaluate various observables using an ensembleof Nf = 2 + 1 + 1 twisted mass fermions (TMFs) simulated at a pion mass of 373 MeV.About 150000 measurements are used in order to reduce both the stochastic and statisticalerror, extracting stable and reliable results for this ensemble. For the case of the σ-term wefind a large disconnected contribution from the light and strange quark sector, contributionsthat cannot be neglected. The disconnected contribution to the nucleon axial charge gA isalso non-zero for both light and strange quarks, whereas the contribution due to the charmquark is compatible with zero. In addition, we study disconnected contributions to theelectromagnetic form factors which seem to be negligible for this pion mass. Regarding thetensor charge gT the results show a tiny contribution, which can be ignored if we compareit with the connected contribution. Moreover, we study Generalized Parton Distributions(GPDs), which on the lattice can be computed using local operators, involving covariantderivatives. For the momentum fraction 〈x〉u+d we observe large errors, which forbid us fromgiving a reliable estimate of the disconnected result. In contrast, the helicity 〈x〉∆u+∆d showslarge but noisy disconnected contributions, which requires even more statistics in order forus to draw any reliable conclusion.

Another important quantity that we study is the neutron Electric Dipole Moment (nEDM).Due to the fact that this quantity is non-zero only, if we add a θ-term in the QCD action, weexpand to first order in θ. We discuss how we can use the pseudoscalar disconnected quarkloops in place of the topological charge that is necessary for the computation of the nEDM.We extract the nEDM in units of θ and use the most recent experimental result to give anupper limit to the magnitude of the CP-violating θ parameter.

Contents

List of Figures iv

List of Tables ix

1 Theoretical basis and fundamental concepts 11.1 Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Quantum Chromodynamics . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Path Integral Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3.1 Grassmann Variables . . . . . . . . . . . . . . . . . . . . . . . . 41.3.2 Analytical integration of fermion fields using Grassmann algebra 41.3.3 Relation between QCD and statistical mechanics . . . . . . . . 5

2 Lattice Quantum Chromodynamics 62.1 Naive discretization schemes . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Naive discretization for free fermions . . . . . . . . . . . . . . . 62.1.2 Link variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.3 Continuum limit of the Wilson fermion action . . . . . . . . . . 82.1.4 Gauge action on the lattice using link variables . . . . . . . . . 8

2.2 Improved discretization schemes . . . . . . . . . . . . . . . . . . . . . . 92.2.1 Wilson fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 Overlap fermions . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.3 Twisted Mass Fermions (TMFs) . . . . . . . . . . . . . . . . . . 12

3 Lattice Techniques for QCD 153.1 Hadron two-point correlation functions . . . . . . . . . . . . . . . . . . 15

3.1.1 Meson correlators . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1.2 Baryon correlators . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Signal improvement methods . . . . . . . . . . . . . . . . . . . . . . . . 193.2.1 Extended sources . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.2 Smoothing the gauge fields . . . . . . . . . . . . . . . . . . . . . 19

3.3 Extracting hadron masses . . . . . . . . . . . . . . . . . . . . . . . . . 213.4 Hadron three-point correlation functions . . . . . . . . . . . . . . . . . 23

3.4.1 Three-point functions . . . . . . . . . . . . . . . . . . . . . . . . 243.4.2 Three-point functions on the quark level . . . . . . . . . . . . . 26

4 Lattice methods for disconnected quark loops 294.1 Stochastic estimation using noise sources . . . . . . . . . . . . . . . . . 304.2 The Truncated Solver Method (TSM) . . . . . . . . . . . . . . . . . . . 314.3 Various dilution schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 344.4 Hopping Parameter Expansion (HPE) . . . . . . . . . . . . . . . . . . . 364.5 The one-end trick for disconnected quark loops . . . . . . . . . . . . . . 364.6 Simulation details and the QUDA package . . . . . . . . . . . . . . . . 404.7 Extracting matrix elements from lattice three-point functions . . . . . . 424.8 Comparing methods for the calculation of disconnected contributions . 43

4.8.1 Performance of the Truncated Solver Method . . . . . . . . . . 44

i

4.8.2 Combining time-dilution with HPE using TSM . . . . . . . . . 484.8.3 Comparing efficiency and performance of various methods . . . 494.8.4 Time-dilution plus HPE versus the one-end trick, using TSM . . 514.8.5 Summary of the performance for various methods . . . . . . . . 554.8.6 Review of the techniques . . . . . . . . . . . . . . . . . . . . . . 55

5 Disconnected contributions to hadron observables 575.1 σ-terms from lattice QCD . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.1.1 Scalar content of the nucleon . . . . . . . . . . . . . . . . . . . 585.1.2 Extracting the strangeness of the nucleon . . . . . . . . . . . . . 605.1.3 Scalar disconnected contributions to other hadrons . . . . . . . 61

5.2 Nucleon electromagnetic form factors . . . . . . . . . . . . . . . . . . . 655.3 Nucleon axial charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.4 Nucleon tensor charge . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.5 Nucleon moments of parton distributions . . . . . . . . . . . . . . . . . 745.6 Extracting flavour octet moments for nucleon . . . . . . . . . . . . . . 80

6 Neutron Electric Dipole Moment using Disconnected Diagrams 846.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846.2 Lattice Techniques for nEDM . . . . . . . . . . . . . . . . . . . . . . . 866.3 Results for nEDM using disconnected quark loops . . . . . . . . . . . . 88

7 Summary and Conclusions 91

Appendices 95

A Linear Algebra and Group Theory Definitions 95A.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95A.2 Lie Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95A.3 Generators of SU(2) and SU(3) groups . . . . . . . . . . . . . . . . . . 96

B Properties of γ-matrices and γ-bases 98B.1 Pauli Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98B.2 γ - matrices in the Dirac basis in Minkowski space . . . . . . . . . . . . 98B.3 Weyl or chiral basis in Minkowski space . . . . . . . . . . . . . . . . . . 99B.4 Chiral and non-relativistic bases in Euclidean space . . . . . . . . . . . 99B.5 Properties of the γ - matrices . . . . . . . . . . . . . . . . . . . . . . . 99B.6 Rotation from Minkowksi to Euclidean space . . . . . . . . . . . . . . . 99B.7 Charge Conjugation operator . . . . . . . . . . . . . . . . . . . . . . . 100

C Symmetries in the twisted basis 101

D Interpolating field operators 102

E Nucleon three-point function 104E.1 Fixed-sink method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105E.2 Fixed-current method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

F Code implementation in QUDA 107F.1 Implementing the twisted mass operator . . . . . . . . . . . . . . . . . 107F.2 Contracting disconnected quark loops . . . . . . . . . . . . . . . . . . . 108F.3 Interfaces and workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

ii

G Table with results 110

Bibliography 112

iii

List of Figures

2.1 A 2-D lattice where the blue lattice dots represent the fermion fields ψ(n)and the red lines the link variables U(n) defined in subsection 2.1.2. L isthe lattice extent, a is the lattice spacing and µ, ν are two unit vectorsin each direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 The connection of the fermion fields using link variables. . . . . . . . . 72.3 The elementary plaquette as a product of four link variables. . . . . . . 9

3.1 Example of meson two-point correlation function. Connected contribu-tion on the left and disconnected contributions on the right. . . . . . . 17

3.2 Two-point correlation function for the proton. . . . . . . . . . . . . . . 183.3 The link UAPE

µ (x) updated using the four neighbouring staples. On a4-D lattice, two additional staples in perpendicular directions contribute. 20

3.4 The 3 dimensional representation of HYP smearing. . . . . . . . . . . . 213.5 Effective mass for π+ on the left and for N on the right using 100 con-

figurations. The red band shows the extracted mass with its jackknifeerror. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.6 The effective mass of the nucleon as a function of the source-sink timeseparation a range of Gaussian smearing steps when α = 4.0. Thesmearing steps in APE smearing are kept fixed to n = 20 with α = 0.5. 23

3.7 The connected part (left) and disconnected part (right), of a three-pointfunction coupled to an operator Oµ(xins). . . . . . . . . . . . . . . . . . 24

3.8 On the left we show the backward and on the right the forward goingsequential propagator, for the case of π+. . . . . . . . . . . . . . . . . . 27

3.9 The ratio from which we extract GπE(0) as a function of tins/a for ts = 8

using 100 configurations. Green filled circles show the ratio using theconserved current, whereas red filled squares show results using the localcurrent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.10 The ratio from which we extract GpE(0) as a function of tins/a for ts = 12

using 100 configurations. Green filled circles show the ratio using theconserved current, whereas red filled squares show results using the localcurrent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.1 Tremendous evolution of GPUs compared to CPUs for various computerarchitectures [41]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2 GPUs speed tests for various scientific fields, namely Material Science,Physics, Earth Science and Molecular Dynamics [42]. . . . . . . . . . . 30

4.3 Absolute error of the operator iψγ3D3ψ with respect to NLP for variousNHP using 50 configurations. The insertion time is fixed at tins = 8a andsink time ts = 16a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.4 The error on the sigma term σπN and axial charge gA versus the numberof low precision (LP) inversions for a fixed number of high precisionNHP = 24 vectors [46] using 50 configurations. . . . . . . . . . . . . . . 33

4.5 The absolute error of gsA as we increase the number of HP noise vectors.Two sets of points are shown. The red squares are for NLP = 0, whileblue circles are for NLP = 300. . . . . . . . . . . . . . . . . . . . . . . . 34

iv

4.6 From top to bottom we show results obtained using color, spin, even-odd and cubic dilution. We show the absolute error on the disconnectedpart of the η′ correlator D(t) at t/a = 3 as a function of the number ofinversions for 200 configurations. . . . . . . . . . . . . . . . . . . . . . 35

4.7 The effective mass for the π (filled red squares) and the ρ (filled greencircles) mesons using the one-end trick with 3 noise vectors. . . . . . . 37

4.8 Left Panel: Strong scaling of the multi-GPU conjugate-gradient solverusing a 323 × 64 lattice for three floating point precision arithmetics.Right Panel: Weak scaling for a local volume of 244 [46]. . . . . . . . . 41

4.9 Strong scaling comparison between Fermi and Kepler architectures [61].The black circles show the performance as a function of the number ofGPUs when we use texture references for Fermi architectures, whereasthe red squares show the performance when we use texture objects in-troduced in the Kepler architecture. . . . . . . . . . . . . . . . . . . . 41

4.10 Left: strong scaling for double (green squares), single (red circles) andmixed precision (blue triangles) arithmetic. Right: weak scaling [62, 63].The solid lines show the theoretical scaling, whereas the dashed linesshow the experimental scaling as a function of the number of GPUs. . . 42

4.11 The connected (top) and the disconnected (bottom) contribution to thenucleon σπN . The gray band is the result obtained from the summationmethod while the coloured bands are results using the plateau method[46]. The ratio is shown for several source-sink time separations as afunction of the source-insertion time separation. . . . . . . . . . . . . . 44

4.12 Comparison of σπN obtained with and without TSM using ∼ 60000measurements. With the blue filled circles we show the results withoutTSM, whereas for the red filled squares we employ TSM with NLP = 200.The coloured bands show the extracted value with its jackknife error. . 45

4.13 Comparison of the performance of the TSM for σs (top) and σc (bottom).The notation is same as that of Fig. 4.12. . . . . . . . . . . . . . . . . 46

4.14 Results of the ratio of disconnected contributions to the isoscalar axialcharge from light quarks. The notation is the same as that of Fig. 4.12. 46

4.15 Results for ratio gsA when using TSM compared to HP inversions only.The notation is the same as that of Fig. 4.14 . . . . . . . . . . . . . . . 47

4.16 Results for ratio which yields gcA when using TSM compared to HPinversions only. The notation is the same as that of Fig. 4.14. . . . . . 47

4.17 Results for the ratio from which σs is extracted with respect to the sinktime when we use NHP = 24, NLP = 0 (blue filled circles) and TSMwith NLP = 300 and NHP = 8 (red filled squares). Top: Results usingonly time-dilution. Bottom: Results using time-dilution with the HPE.The total number of measurements is ∼ 20000. . . . . . . . . . . . . . . 48

4.18 Comparison for the ratio from which gsA is obtained between using HPonly and using TSM in combination with time-dilution. The notation isthe same as in Fig. 4.17. . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.19 Comparison of results for the ratio from which σs (top) and gsA (bottom)are extracted, when applying time dilution (red filled squares) and whentime dilution is combined with HPE (blue filled circles). . . . . . . . . . 50

v

4.20 Results for the ratio of σs (top) and gsA (bottom). The blue circles showresults using the one-end trick and the light blue error band shows theextracted value with its jackknife error. The magenta error band showsthe extracted value when using time-dilution plus HPE. The TSM withNHP = 24 and NLP = 300 is used in both methods with 18628 statistics. 52

4.21 Comparison of the plateau (above) and summation method (below) forσs. Using different colours and symbols we assess the behaviour of theresults for various source-sink time separations. The grey band on thetop panel shows the result from the summation method extracted fromfitting the slope of the summed ratio shown in the bottom panel. . . . . 53

4.22 The same as 4.21 but for gsA. . . . . . . . . . . . . . . . . . . . . . . . . 544.23 Results extracted from the slope of the summed ratio that yields σs and

gsA as a function of the lower fit range ti/a for various tf . . . . . . . . . 54

5.1 Results for the ratio from where we extract the connected part of σπN .We show results obtained for various source-sink time separations. Thevalue extracted from the plateau method (purple band) is comparedwith that extracted using the summation method (grey band). . . . . . 58

5.2 Disconnected contributions to nucleon σ-terms. Left panel: Results forthe ratios from where we extract σπN , σs and σc, using the plateau (yel-low band) and summation method (grey band). Right panel: The ex-tracted value is shown with respect to the lower fit range ti/a for severalupper fit ranges tf/a. The star symbol is the value shown with the greyband on the left panel. . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.3 Chiral behaviour of the yN parameter. Blue filled triangles correspondto a = 0.082 fm, while the green filled circle corresponds to a = 0.064 fm.We extrapolate to the physical value of the pion mass (marked by thevertical dotted line) using linear (solid line) and quadratic (dashed line)fits in m2

π. For the quadratic fit, we also show the corresponding errorband. Points represented by open symbols are only taken to estimatesystematic effects and are not included in our final analysis. The redfilled rhombus show the extrapolated values at the physical point. . . . 61

5.4 The baryon multiplets occuring from the decomposition of the SU(4) group. 625.5 Connected (right) and disconnected (left) contributions to the σ-terms

for Λ0 as a function of the appropriate time-separations. For the discon-nected part the ratio is shown as a function of (tins− ts/2)/a for severalsource-sink time separations. The results from the plateau and sum-mation method are compared. For the right panel the connected partis shown versus ts − tins for three cases tins/a = 5 (red filled squares),tins/a = 7 (green filled circles) and tins/a = 9 (blue filled triangles). . . 63

5.6 Sigma-terms of the Ω−. The notation is the same as in Fig. 5.5. . . . . 645.7 Sigma-terms of the Ω++

ccc . The notation is the same as in Fig. 5.5. . . . 655.8 Electric and magnetic nucleon form factors for the isovector combination

[29]. The different colours and symbols show the extracted value usingdifferent source-sink time separations, while the black star shows thevalue extracted from the summation method. The solid line shows theJ. Kelly’s parametrization of the experimental results. . . . . . . . . . . 66

vi

5.9 Results for the ratio showing disconnected contributions to isoscalarelectromagnetic form factors Gu+d

E and Gu+dM for the smallest non-zero

momentum transfer [97]. We show the ratio from where the electro-magnetic form factors are extracted for three different source-sink timeseparations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.10 Results for the axial charge from several ensembles using TMFs with (i)Nf = 2 twisted mass fermions including a clover term with a = 0.094fm and 1440 configurations (red cross) [102], (ii) Nf = 2 + 1 + 1 twistedmass fermions with a = 0.064 fm and 900 configurations (square witha cross), a = 0.082 fm and 950 configurations (open orange circle) [99],(iii) Nf = 2 twisted mass fermions, details are given in Ref. [103]. Theasterisk is the physical value as given in the PDG [104]. . . . . . . . . 68

5.11 The ratio yielding the connected contribution to the nucleon isoscalaraxial charge gu+d

A using various source-sink time separations, ts = 10a(red filled circles), ts = 12a (blue filled squares), ts = 14a (green emptysquares), ts = 16a (yellow filled triangles) and ts = 18a (purple emptystars). For the analysis of the connected part a statistics of 1200 config-urations has been used. . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.12 The disconnected contributions to the renormalized ratio which yieldsthe isoscalar axial charge of the nucleon, gu+d

A . The upper panel showsthe ratio as a function of the insertion time-slice with respect to themid-time separation (tins−ts/2) for source-sink time separations ts = 8a(red filled circles), ts = 10a (blue filled squares), ts = 12a (green opensquares) and ts = 14a (yellow filled triangles). The bottom panel showsthe results obtained for the fitted slope of the summation method forvarious choices of the initial and final fit time slices. The star shows thechoice of ti, which yields the gray band in the top plot. . . . . . . . . . 69

5.13 The strange and charm quark contributions to the renormalized ratioyielding the nucleon axial charge gsA (left) and gcA (right). The notationis the same as that of Fig. 5.12. . . . . . . . . . . . . . . . . . . . . . . 70

5.14 Disconnected contributions to the renormalized ratio yielding the isoscalaraxial-vector and pseudo-scalar form factors Gu+d

A and Gu+dp at the lowest

non-zero momentum transfer allowed for this lattice size. The notationis the same as that of Fig. 5.12. . . . . . . . . . . . . . . . . . . . . . . 71

5.15 The tensor charge, Eq. (5.16), measures the net light-front distributionof transversely polarised quarks inside a transversely polarized proton. . 72

5.16 The isovector (top) and isoscalar (bottom) tensor charge from variouspion masses taken from Ref. [107]. . . . . . . . . . . . . . . . . . . . . 72

5.17 Top: We show results on the disconnected Ru+dT versus (tins − ts/2)/a

for ts = 14a (red filled circles), ts = 16a (blue filled squares), ts = 18a(green empty squares) and ts = 20a (filled yellow triangles). Bottom:The value of gu+d

T extracted using the summation method for various tiand tf fit ranges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.18 The Q2 dependence of the isoscalar Gu+dA (Q2), Au+d

20 (Q2) and Bu+d20 (Q2)

for mπ = 213 MeV (blue filled squares) and mπ = 373 MeV (red filledcircles). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

vii

5.19 Results for the connected ratio of 〈x〉u+d using various source-sink timeseparations, namely ts = 10a (red filled circles), ts = 12a (blue filledsquares), ts = 14a (green empty squares), ts = 16a (yellow filled arrows)and ts = 18a (purple empty stars). The grey band shows the value of〈x〉u+d extracted using the summation method. . . . . . . . . . . . . . . 76

5.20 Results for the disconnected ratio of 〈x〉u+d using various source-sinktime separations, namely ts = 8a (filled red circles), ts = 10a (filled bluesquares) and ts = 12a (green empty squares). The grey band shows thevalue of 〈x〉u+d extracted using the summation method. . . . . . . . . 77

5.21 Behaviour of the summed ratio for 〈x〉u+d as a function of ts (left). Right:〈x〉u+d versus the lower fit range ti for various upper fit ranges tf . . . . 77

5.22 The behaviour of the total error of 〈x〉u+d is shown as a function of Nr forNc = 3000 for the B55.32 ensemble. The dotted black line is the resultof the fit to Eq. (5.26). For the first three points (red filled squares)we only use HP noise vectors, whereas for the last 3 points (blue filledcircles) we use TSM and the stochastic error decreases with the numberof noise vectors. The horizontal purple dotted line shows the gauge errorwhen extrapolating to Nr →∞ for Nc = 3000. . . . . . . . . . . . . . . 78

5.23 Results on the connected ratio of 〈x〉∆u+∆d using various source-sinktime separations. The notation is the same as that in Fig. 5.19. . . . . 79

5.24 Results for the disconnected contributions to 〈x〉∆u+∆d. The notation isthe same as that in Fig. 5.20 but for 〈x〉∆u+∆d. . . . . . . . . . . . . . 80

5.25 The ratio of Eq. (3.45) as a function of the source-sink time separation tsfor a fixed source operator time tins = 11a in the isovector (orange filledsquares) and octet case (red filled circles). The gray bands indicatethe results obtained from a fixed-sink calculation for ts = 12a. Theblue triangles show the disconnected contribution to Ra=8(ts, tins = 16).The results shown are extracted using the B55.32 ensemble with 23000measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.26 〈x〉(3,8) as a function of the pseudoscalar meson mass m2π. The phe-

nomenological estimates are represented by two black stars. . . . . . . . 825.27 〈x〉(3)/〈x〉(8) as a function of the pseudoscalar meson mass m2

π for twovalues of the lattice spacing. The result from a constant extrapolationis represented by an empty triangle. The systematic error on the ex-trapolated value is represented by a red error bar slightly shifted forreadability. The phenomenological estimate is represented by a blackstar [125]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.1 Disconnected contributions to nEDM. . . . . . . . . . . . . . . . . . . . 866.2 The ratio from which we extract the α1 parameter. . . . . . . . . . . . 896.3 Lattice results for the derivative acting on the ratio at zero momentum

transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.4 Top: The open circles show results for F3(Q2) versus Q2. The dashed

line shows the dipole fit and the filled blue circles show the extrapolatedvalue at zero momentum transfer. Bottom: Results on the derivativemethod after the subtraction of term proportional to α1, from where weextract directly F3(0). . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

viii

List of Tables

3.1 Quantum numbers of the most commonly used meson interpolating fields. 163.2 Interpolating fields for octet baryons. . . . . . . . . . . . . . . . . . . . 18

4.1 The ratio of computational cost between HP and LP inversions in orderto calculate the disconnected loops when we employ TSM, where ultra-local are operators of the general form ψΓψ and one-derivative ψΓDψ. 50

4.2 Comparison of the computational cost [78] for the σ-terms and the axialcharges using various methods for three quark sectors. . . . . . . . . . . 55

5.1 Representative results for the ratio of disconnected to connected contri-butions to σ-terms of baryons. The particle is given in the first column,in the second column we indicate the σ-term considered and in the rightcolumn we give the ratio as a percentage. The baryons are ordered withincreasing mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

D.1 Interpolating fields and quantum numbers for the 20′-plet of spin-1/2baryons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

D.2 Interpolating fields and quantum numbers for the 20-plet of spin-3/2baryons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

D.3 Additional interpolating fields for spin-1/2 and spin-3/2 baryons. . . . . 103

G.1 In the first column we give the observables, in the second column theconnected contribution, in the third column the disconnected contri-bution and in the fourth column is the total value. The results wereextracted from an Nf = 2 + 1 + 1 ensemble of twisted mass fermions atmπ = 373 MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

G.2 Estimation of the yN parameter and 〈x〉(3)

〈x〉(8) by extrapolation at the phys-

ical point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

ix

Chapter 1

Theoretical basis and fundamental con-cepts

This chapter summarizes the fundamental concepts of the standard model. We startwith a brief introduction for the electroweak sector of the standard model and thendiscuss several aspects of Quantum Chromodynamics (QCD). The formulation of thepath integral and the presentation of fermion fields using Grassmann variables are alsodiscussed.

1.1 Standard Model

The understanding of the universe relies on physical laws, which describe its dynamicalevolution. Four fundamental forces are believed to govern the physical world. For themicrocosmos there are three forces: electromagnetism, weak interactions and stronginteractions, gravity being negligible.

Electromagnetism is the most common force because it can be observed in oureveryday life. Originally electricity and magnetism were considered as two separateforces but after James Clerk Maxwell’s pioneer work we know that all electric andmagnetic phenomena are described by a unified force. Fundamental particles such aselectrons and muons carry electric charge and interact with other charged particles byinterchanging photons. Electromagnetic forces are of infinite range since photons aremassless particles.

The weak interactions are responsible for β-decays and nuclear fission. These in-teractions take place on a very small scale and can change the flavour of a quark.The mediators of this force are the Z0 and W± bosons with masses greater than thenucleon. In 1968, Sheldon Glashow, Abdus Salam and Steven Weinberg unified theelectromagnetic force with the weak interaction by showing them to be two aspects ofa single force, now termed as electro-weak force [1].

The strong interaction is the force which binds together protons and neutrons toform the nucleus of atoms [2]. This force has a range of around 1 fm and a scale ofabout 1 GeV. The carrier of this force is the gluon, which mediates the interactionbetween quarks that appear in six flavours. The strong interaction introduces a newkind of charge which is called color charge.

Classical gravity, the fourth interaction, is relevant on the macroscale. Gravitationexplains the behaviour of planets and stars. Around 1680 Newton formulated the firstprinciples regarding gravity. Later on, Albert Einstein’s general theory of relativity [3]completed our understanding of classical gravity.

The electro-weak and strong interactions can be described by a quantum field theoryin contrast to the gravitational force for which no consistent quantum field theory has

1

been derived so far. There have been attempts to unify all the fundamental forces.String theory [4] is a strong candidate for this unification, but several issues remainunresolved. In 2012 CERN announced the discovery of a particle with all the propertiesof the Higgs boson. In 2014, CERN [5] confirmed that the particle discovered in 2012is the Higgs particle of the standard model [6]. This completed the standard model ofparticle physics.

1.2 Quantum Chromodynamics

After the invention of spark and bubble chambers in 1950 a large number of hadronswas discovered. These particles were categorized according to their properties. Theirstrangeness was one trait they had. In 1963 Gell-Mann and George Zweig [7] proposedthat the structure of these classes can be explained if hadrons are not fundamentalparticles, but that they consist of 3 smaller particles called partons.

The first puzzle was the Ω− that consists of three strange quarks with alignedspins. This state is prohibited by the Pauli principle, so the only case that such aparticle could exist was if the three quarks carried an additional quantum number,the color charge. When first proposed, quarks were considered as a mathematicalconstruction, not physical particles. Richard Feynman argued that experiments athigh energy colliders showed evidences that hadrons are not elementary particles andthat quarks must exist in nature [8].

Quantum electrodynamics (QED) is the quantum field theoretical description ofelectromagnetism. It is based on the U(1) abelian gauge group. In contrast, quantumchromodynamics is based on an SU(3) theory [9] because quarks carry color chargestaking 3 different values. This introduces fundamental differences between QED andQCD, namely that the gluons can interact with each other, as opposed to photonswhich do not self-interact in the standard, non-compact formulation of QED. Thismeans QCD is a non-abelian theory, with two main characteristics that differentiate itfrom other theories: namely confinement and asymptotic freedom.

1. Confinement arises because the force between two quarks remains constant as thedistance between them goes to infinity. As we try to separate two quarks the en-ergy transferred to the system will permit the creation of a quark-antiquark pair.Thus, it is not possible to isolate a quark in order to study it experimentally. Themechanism of Confinement is still not understood but it has been demonstratedin lattice QCD.

2. Asymptotic freedom, implies that at high energy scale or equivalently when thedistance between quarks becomes small, they interact very weakly. This predic-tion from QCD was confirmed experimentally in the 1970s [10], by measuring thestrong coupling constant as a function of energy.

The QCD Langrangian has two parts: the first part describes the interactionsbetween quarks and gluons and the second part describes the interactions betweengluons. The Langrangian density is given by,

LQCD = Lgauge + Lquarks, (1.1)

where

Lgauge = −1

2Tr[GµνG

µν], (1.2)

2

and Gµν is the gluon tensor operator,

Gµν = ∂µAν − ∂νAµ − gfabcAbµAcν . (1.3)

The third term in Eq. (1.3) gives rise to interactions among gluons, where fabc are thestructure constants of the SU(3) algebra, defined in Appendix. A.

The fermion part is given by,

Lquarks =∑

f=u,d,s,c,t,b

ψf (i 6D −m)ψf (1.4)

where ψf is the quark fermion field in the fundamental representation of SU(3). Theoperator 6D is the covariant derivative given by,

6D ≡ γµ∂µ + igγµAµ. (1.5)

The Dirac algebra in Minkowski space obeys the relation

γµ, γν = 2gµν . (1.6)

The properties of γµ in the non-relativistic and chiral representations are given inAppendix. B. Gauge transformations of the fermion fields are given by,

ψ(x)→ ψ′(x) = Ω(x)ψ(x), ψ → ψ′(x) = ψ(x)Ω†(x). (1.7)

Under a gauge transformation the gluon field transforms as

Aµ(x)→ A′µ(x) = Ω(x)Aµ(x)Ω†(x) + i(∂µΩ(x))Ω†(x). (1.8)

and the QCD Langrangian remains invariant under such gauge transformations.

1.3 Path Integral Formulation

Observables in QCD are computed via the path integral formulation of the quantumfield theory. Within this picture, the vacuum expectation value of an operator O isgiven by,

〈Ω|O|Ω〉 =1

Z

∫Dψ(x)Dψ(x)DA(x)O[ψ, ψ, A]e−

ihS[ψ,ψ,A] (1.9)

where Dψ(x),Dψ(x) and DA(x) denote integration over all possible fermion and gluonfields (over all paths), while the partition function Z is given by,

Z =

∫Dψ(x)Dψ(x)DA(x)e−

ihS[ψ,ψ,A]. (1.10)

Wick rotating to a Euclidean space-time t −→ it we have,

〈Ω|O|Ω〉 =1

Z

∫Dψ(x)Dψ(x)DA(x)O[ψ, ψ, A]e−

1hS[ψ,ψ,A] (1.11)

which allows statistical treatment. The Wick rotation to Euclidean space enables us toperform numerical simulations using lattice QCD as will be explained in Chapter. 2.ψ(x) and ψ(x) are Grassmann variables, obeying Grassmann algebra. It is not possibleto represent such numbers on a computer, at least at a level low enough for such anintegration to be practical. The way around this is to integrate over the fermionicdegrees of freedom analytically.

3

1.3.1 Grassmann Variables

In order to simplify the path integral over the fermion fields we must introduce theGrassmann variables. We start by having an ensemble from independent Grassmannvariables ξ1, ξ2, ..., ξn. This algebra satisfies the anti-commuting relation

ξiξk + ξkξi = ξi, ξk = 0. (1.12)

What is important here, is the case when i = k then ξ2 = 0. For the partial derivativeswe can write

∂ξi∂ξj

= δij,∂

∂ξjξiξk = δijξk − δjkξi. (1.13)

Because of the properties of Grassmann variables derivation and integration are thesame. For the case in hand we require the gaussian integral of a square matrix whichgives ∫

[dξ∗dξ]e−ξ†Mξ = det(M), (1.14)

where [dξ∗dξ] ≡ dξ∗1dξ1 · · · dξ∗ndξn. Another important relation is∫[dξ∗dξ]ξjξ

∗ke−ξ†Mξ = M−1

jk det(M). (1.15)

1.3.2 Analytical integration of fermion fields using Grassmannalgebra

Observables in QCD in general can be computed using Eq. (1.11). The EuclideanQCD action from Eq. (1.11) depends on the fermion fields ψ,ψ and on the gauge fieldA. As a result we can write the equation which gives us the expectation value of anoperator as

〈O〉 =1

Z

∫Dψ(x)Dψ(x)DA(x) O[ψ, ψ, A] e−SQCD[ψ,ψ,A] (1.16)

where Dψ(x), Dψ(x) denote integration over all the fermion configurations and DU(x)denotes integration over all possible gluon configurations. In Eq. (1.16) we already per-formed the rotation from the Minkowski to Euclidean space. As we already discussed,the fermion fields obey Fermi statistics, which means their behaviour is described byGrassmann variables which cannot be represented on the computer, hence we mustperform these integrations analytically. The partition function of QCD is

Z =

∫Dψ Dψ DA e−SF−SG , (1.17)

where SF depends on both the fermion and gluon fields. Eq. (1.4) can be written in amore compact form if we define the Dirac operator as a matrix, having in mind thaton the lattice we have finite degrees of freedom, hence SF = ψM [A]ψ, where M [A] isthe Dirac matrix. Using Eq. (1.14) and performing the integration over the fermionfields, then the partition function becomes,

Z =

∫DA det(M [A])e−SG[A] (1.18)

4

and we have only one integration over the gauge fields. Raising the determinant to theexponential the partition function can be written in the form,

Z =

∫DA eln[det(M [A])]−SG[A]. (1.19)

The next step is to perform the integration over the fermion fields for the numeratorin 1.16. To simplify the algebra we take O(x, y) = ψ(x)ψ(y), then

〈O(x, y)〉 =1

Z

∫Dψ Dψ DA ψ(x)ψ(y)e−ψMψe−SG (1.20)

and by using 1.15,

〈O(x, y)〉 =1

Z

∫DA M−1(x; y)eln[det(M [A])]−SG[A], (1.21)

where the integration over the gluon fields cannot be done analytically.

1.3.3 Relation between QCD and statistical mechanics

There is a close relation between QCD in Euclidean space and statistical mechanics.In statistical mechanics a standard problem is the spin system. The degrees of freedomare classical spin variables si, which are located on 2-D, 3-D or 4-D lattices. Theenergy of the system is a functional over the spins (H[s]). From statistical mechanicsthe probability of finding the system in a particular configuration s is given by

P [s] =1

Ze−βH[s], (1.22)

and the partition function is

Z =∑s

e−βH[s], (1.23)

where the sum runs over all possible spin configurations. The expectation value of anoperator is given by

〈O〉 =1

Z

∑s

e−βH[s]O[s]. (1.24)

Comparing Eq. (1.24) with Eq. (1.21) we observe that, the weight function in the firstcase is the Boltzmann factor e−βH and in the second case is the Feynman weight factore−SQCD . The summation over the spin configurations becomes a summation over thegluon configurations. The partition function is the normalization factor.

This correspondence between statistical mechanics and QCD is very important sinceit allows using all the techniques known from statistical mechanics for simulating QCD.This allows us to create configurations of the gauge fields using Monte Carlo simulationswith probability eln[det(M)]−SG/Z, and by calculating the inverse of the Dirac operator,we can sum over all the available configurations to calculate observables in QCD. Inorder to perform this simulation a lattice formulation of QCD needs to be defined.

5

Chapter 2

Lattice Quantum Chromodynamics

Quantum Chromodynamics cannot be solved analytically because perturbative ap-proaches cannot be applied at nuclear energy scales tart are of O(1) GeV. Only at highenergies, where asymptotic freedom holds, can we employ a perturbative treatment.At low energy, chiral perturbation theory can be used, however at intermediate energyscales the only known non-perturbative method is lattice QCD.

2.1 Naive discretization schemes

Discretization of a continuum theory incurres loss of symmetries such as rotationalinvariance. Several lattice actions are constructed to decrease such lattice artifactsbut there is no lattice action which can recover all broken symmetries. All discretizedactions must converge to the same result as we take the continuum limit. The mostimportant symmetry to respect in order to obtain physical results is the gauge sym-metry. Kenneth Wilson [11] invented a discretization scheme in 1974, preserving thisimportant symmetry.

2.1.1 Naive discretization for free fermions

Let us begin by explaining how to discretize fermions using a naive approach introducedby Wilson [11]. In order to discretize QCD we consider a 4-D lattice with lattice spacinga, thus

Λ = Λ(n = 1, 2, · · · , L;nt = 1, 2, · · · , Lt), (2.1)

where L defines the spatial lattice extents, usually Lx = Ly = Lz = L, and Lt is thetemporal extent. We define fermions at each point of the lattice. They carry a colorindex for the 3 colors, a Dirac index for the spin components, and a lattice index forthe position on the lattice i.e ψaµ(n). A 2-D lattice is shown in Fig. 2.1.

LU(n)

Figure 2.1: A 2-D lattice where the blue lattice dots represent the fermion fields ψ(n) and thered lines the link variables U(n) defined in subsection 2.1.2. L is the lattice extent, a is thelattice spacing and µ, ν are two unit vectors in each direction.

6

The continuum action for a free fermion with Aµ = 0 is given by

S0F [ψ, ψ] =

∫d4x ψ(x)

(γµ∂µ +m

)ψ(x). (2.2)

We discretize the partial derivative using its symmetric form:

∂µψ(x)→ 1

2a

(ψ(n+ µ)− ψ(n− µ)

). (2.3)

Where a is the lattice spacing between two neighbouring sites and µ denotes the direc-tion in 4-D space. Hence, our lattice action for free fermions is given by

S0F [ψ, ψ] = a4

∑n

ψ(n)

(4∑

µ=1

γµψ(n+ µ)− ψ(n− µ)

2a+mψ(n)

). (2.4)

The free case has small physical interest, and thus the next sections are devoted infinding a way to define the gauge field on the lattice.

2.1.2 Link variables

Gauge symmetry requires that the fermion fields transform locally according to

ψ(n)→ ψ′(n) = Ω(n)ψ(n), ψ(n)→ ψ′(n) = ψ(n)Ω†(n). (2.5)

It is obvious that the mass term in Eq. (2.4) remains invariant under this transforma-tion. The term with the derivative transforms like,

ψ(n)ψ(n+ µ)→ ψ′(n)ψ′(n+ µ) = ψ(n)Ω†(n)Ω(n+ µ)ψ(n+ µ). (2.6)

For a general transformation Ω†(n)Ω(n + µ) 6= 1, which means that this term is notgauge invariant. But we can make it invariant if we introduce a field Uµ(n) with thefollowing transformation

Uµ(n)→ U ′µ(n) = Ω(n)Uµ(n)Ω(n+ µ)†. (2.7)

Then Eq. (2.6) becomes

ψ(n)Uµ(n)ψ(n+ µ)→ ψ′(n)U ′µ(n)ψ′(n+ µ) = ψ(n)Uµ(n)ψ(n+ µ), (2.8)

because the matrix Ω belongs to the SU(3) group. Since Uµ(n) connects ψ(n) toψ(n + µ) they are defined on the links of the lattice. For instance, Uµ(n) connectsthe lattice site n with n + µ, and U−µ(n + µ) links n + µ with n. The two are notindependent since they are defined on the same site. They are connected through therelation

U−µ(n+ µ) = U †µ(n). (2.9)

n n + µ

Uµ(n)

n n + µ

U†µ(n)

Figure 2.2: The connection of the fermion fields using link variables.

After we have introduced the link variables and their properties we can write theaction for the interacting fermions with an external gauge field Uµ(n),

S0F [ψ, ψ] = a4

∑n

ψ(n)

(4∑

µ=1

γµUµ(n)ψ(n+ µ)− U †µ(n− µ)ψ(n− µ)

2a+mψ(n)

).

(2.10)

7

2.1.3 Continuum limit of the Wilson fermion action

From Eq. (2.7) the transformation properties of the link variables depend on thematrices Ω(n) and Ω†(n+µ). In the continuum the object with the same transformationproperties is the well known gauge transporter defined by,

G(x, y) = Peig∫Cx,y A·ds (2.11)

where P means path-ordered and Cx,y is the path connecting the points x, y. The gaugetransporter has the transformation property

G(x, y)→ Ω(x)G(x, y)Ω†(y), (2.12)

which is of the same form as Eq. (2.7). This gauge transporter on the lattice takes thefollowing form when connecting two neighbouring sites,

G(n, n+ µ) = Uµ(n) = eiagAµ(n) +O(a). (2.13)

Because there is only one path which connects n with n + µ the integral is just equalwith aAµ(n) up to first order in lattice spacing. In the limit where a → 0 we canexpand the exponential in terms of the lattice spacing,

Uµ(n) = 1 + iagAµ(n) +O(a2), U †µ(n− µ) = 1− iagAµ(n− µ) +O(a2), (2.14)

where we made use of A†µ = Aµ. We plug this result into Eq. (2.10) and the fermionaction becomes,

SF [ψ, ψ, U ] = S0F [ψ, ψ] + ia4g

4∑µ=1

ψ(n)γµAµ(n)ψ(n) +O(a), (2.15)

where we have used that, ψ(n± µ) ' ψ(n) +O(a) and Aµ(n− µ) ' Aµ(n) +O(a) inthe limit where the lattice spacing goes to zero. This indeed recovers the continuumform of the fermion action.

2.1.4 Gauge action on the lattice using link variables

In this section we show that the Wilson gauge action recovers the continuum QCDgauge action as a → 0. The first non-trivial path of link variables, which has thecorrect continuum limit is the so-called fundamental plaquette Uµν(n) [11] defined as

Uµν(n) = Uµ(n)Uν(n+ µ)U †µ(n+ ν)U †ν(n). (2.16)

By taking the trace of this plaquette we can prove that it is gauge invariant when weuse the transformation from Eq. (2.7). To construct an action, which has the rightcontinuum limit we must sum over all plaquettes, where each plaquette has only oneorientation. This action is the Wilson gauge action and the elementary plaquette isdepicted in Fig. 2.3.

The Wilson gauge action is given by

SG[U ] =2

g2

∑n∈Λ

∑µ<ν

Re

Tr[11− Uµν(n)

], (2.17)

8

n n + µUµ(n)

n + µ + νn + ν

U†ν(n) Uν(n + µ)

U†µ(n + ν)

Figure 2.3: The elementary plaquette as a product of four link variables.

where the factor 2/g2 is included to match the continuum form of the gauge action. Inorder to establish the correct continuum limit we must expand the gauge links usingEq. (2.14):

Uµν(n) = exp

iaAµ(n) + iaAν(n+ µ)− a2

2[Aµ(n), Aν(n+ µ)]

−iaAµ(n+ ν)− iaAν(n)− a2

2[Aµ(n+ ν), Aν(n)]

+a2

2[Aµ(n+ µ), Aµ(n+ ν)] +

a2

2[Aµ(n), Aν(n)]

+a2

2[Aµ(n), Aµ(n+ ν)] +

a2

2[Aµ(n+ µ), Aµ(n)] +O(a3)

. (2.18)

The gauge fields with shifted arguments can be approximated by a Taylor expansionas

Aν(n+ µ) = Aν(n) + a∂µAν(n) +O(a2) (2.19)

and we keep terms up to O(a2). After this expansion several terms cancel and weobtain

Uµν(n) = exp

(ia2(∂µAν(n)− ∂νAµ(n) + i[Aµ(n), Aν(n)]

)+O(a3)

). (2.20)

According to the definition of the gluon tensor operator given in Eq. (1.3) then,

Uµν(n) = exp(ia2Gµν(n)

)+O(a3). (2.21)

Having the expression for the elementary plaquette in terms of the gluon tensor operatorwe plug it into Eq. (2.17) and by expanding the exponent we obtain

SG[U ] =a4

2g2

∑n∈Λ

∑µ<ν

Tr[G2µν(n)] +O(a2) a→ 0−−−−−→

1

2g2

∫d4x Tr[Gµν(x)Gµν(x)].

(2.22)Therefore in the limit where a→ 0 we regain the gluonic part of the QCD action.

2.2 Improved discretization schemes

There is a number of improvements we can carry out on the lattice in order to reducethe lattice artifacts, but we will first study one of the most severe problems of the naivediscretization, the so-called doubling problem which was solved by Kenneth Wilson.

9

2.2.1 Wilson fermions

Let us examine the properties of the Dirac operator discretized naively as presented inSec. 2.1. The Dirac operator can be written in matrix form as

D(n,m) =4∑

µ=1

γµUµ(n)δn+µ,m − U †µ(n− µ)δn−µ,m

2a+M, (2.23)

where we suppress the color and Dirac indices. In order to demonstrate the doublingproblem we set Uµ(n) = 11 i.e. we consider free fermions on the lattice. Taking theFourier transformation for n and m we have

D(p′, p) =1

V

∑n,m

e−ip′·naD(n,m)eip·ma

=1

V

∑n

e−i(p′−p)·na

(4∑

µ=1

eipµa − e−ipµa2a

+M

)= δ(p′ − p)D(p), (2.24)

where V is the volume and D(p) is given by

D(p) =i

a

4∑µ=1

(γµ sin(apµ)

)+M. (2.25)

In momentum space the free Dirac operator is diagonal due to the delta function.Hence, we can calculate the inverse matrix i.e. the Dirac propagator,

D(p)−1 =M − ia−1

∑4µ=1 γµ sin(apµ)

M2 + a−2∑4

µ=1 sin2(apµ). (2.26)

In the chiral limit when we have massless fermions the propagator reads,

D−1(p,M = 0) =−ia−1

∑µ γµ sin(apµ)

a−2∑

µ sin2(apµ)a→ 0−−−−−→

−i∑µ γµpµ

p2. (2.27)

In the continuum the propagator has only one pole when pµ = (0, 0, 0, 0), but on thelattice (a 6= 0) we have poles when pµ = 2π

a. The poles of the propagator count the

number of the fermions we have on the lattice. Specifically, for a 4-D lattice we have 24

poles, i.e 15 unwanted additional fermions. These fermions are the so-called doublers.In order to remove unwanted doublers which appear when pµ = π/a, Wilson added

an extra term in Eq. (2.25) that corresponds to a second derivative term,

−ar2ψ(n)∇µ∇µψ(m). (2.28)

With this modification the Dirac operator in momentum space becomes

D(p) = M +i

a

4∑µ=1

γµ sin(apµ) +1

a

4∑µ=1

(1− cos(apµ)

). (2.29)

This new term acts as an additional mass term

MDl = M +2j

a, (2.30)

10

where j counts the number of momentum components with value pµ = π/a. In thecontinuum limit where a → 0 the mass of the doublers goes to infinity and theydecouple from the theory. In position space the Wilson Dirac operator takes the form

D(n,m) = − 1

2a

4∑µ=1

((11− γµ)Uµ(n)δn+µ,m + (11 + γµ)U †µ(n− µ)δn−µ,m

)+

(M +

4

a

),

(2.31)where by defining

κ =1

2(M + 4a)

(2.32)

it can be written as

D(n,m) = −κ4∑

µ=1

((11− γµ)Uµ(n)δn+µ,m + (11 + γµ)U †µ(n− µ)δn−µ,m

)+ 11 (2.33)

and the fermion fields become

ψ → ψ√2aκ

. (2.34)

2.2.2 Overlap fermions

The doubling problem is thus solved by using the Wilson term, but another severeproblem has been introduced. Chiral symmetry, a very important symmetry for QCD,its spontaneous breaking which explains why pions have unexpectedly small masses, isexplicitly broken. This is because even in the limit where the fermions become massless,the fermion action has a term 4

awhich breaks the chiral symmetry.

In order to prove this we perform a chiral rotation of the fermion fields

ψ → ψ′ = eiαγ5ψ, ψ → ψ′ = ψeiαγ5 , (2.35)

where α is an arbitrary constant real number. The Langrangian density is invariantunder this transformation in the massless limit, but the introduction of this mass termbreaks chiral symmetry. This symmetry holds only in the chiral limit and can bewritten as

Dγ5 + γ5D = 0, (2.36)

where D is the massless Dirac operator. There is in fact a theorem by Nielsen andNinomiya [12] which states that on the lattice we cannot have a local theory, whichpreserves both, chiral symmetry and is free from doublers.

In 1982 [13], Wilson and Ginsparg formulated an equation which gives an indicationthat there are some remnants of chiral symmetry on the lattice. They proposed toreplace equation Eq. (2.36) by the so-called Ginsparg-Wilson equation

Dγ5 + γ5D = aDγ5D. (2.37)

This equation has the right continuum limit and preserves chiral properties. If onemodifies Eq. (2.35) by introducing a term, which vanishes in the continuum such as[14]

ψ′ = exp(iαγ5

(1− a

2D))

ψ, ψ′ = ψ exp(iαγ5

(1− a

2D))

(2.38)

then we partially recover chiral symmetry on the lattice. It was proven [15] that theLangrangian remains invariant under the new chiral transformation and Eq. (2.37)

11

holds. This is an extremely important result. What is now needed is to construct aDirac operator on the lattice that obeys the Ginsparg-Wilson equation.

One such operator is the overlap operator [14] which is defined by

Dov =1

a

(1 + γ5 sign[H]

), H = γ5 A, sign[H] ≡ H (H2)−1/2, (2.39)

where A must be a γ5-hermitian Dirac-type operator. One choice for A is

A = aD0W − (1 + s), (2.40)

where D0W is the massless Wilson-Dirac operator and s is a parameter, which can be

tuned to improve locality. In order to prove that the overlap operator satisfies theGinsparg-Wilson equation we consider

aDovD†ov =

1

a

(11 + γ5sign[H]

)(11 + sign[H]γ5

)=

1

a

(11 + γ5sign[H] + sign[H]γ5 + 11

)= Dov +D†ov, (2.41)

which is exactly Eq. (2.37). The overlap operator has restored chiral symmetry but isapplication requires the square root of the Dirac Wilson propagator which is extremelycomputationally demanding. Another action, which recovers chiral symmetry, withoutthe need of the inverse square root of the Dirac Wilson operator, is the so-called DomainWall Fermion (DWF) action defined in 5-dimensions [16]. Both overlap and domainwall actions are very expensive to simulate, but there has been tremendous progress inthe simulation of DWFs [17].

2.2.3 Twisted Mass Fermions (TMFs)

For many observables probing hadron structure the breaking of chiral symmetry doesnot seem to introduce significant lattice artifacts. Therefore, many lattice QCD col-laborations are using improved Wilson fermions. These improved actions use clover-improved or twisted mass fermion formulations.

The twisted mass operator [18] in its simplest form has two degenerate flavours.It uses the isospin degrees of freedom to introduce an additional term in the WilsonDirac operator. This additional term is the twisted mass term, which can be tuned forO(a) improvement.

Let us denote the new quark fields with χ and χ, which are doublets in flavourspace. The fermion action on the lattice for twisted mass fermions is given by

StmF [χ, χ, U ] = a4∑n,m

χ(n)(D0W (n;m) +M + iµγ5τ

3)χ(m), (2.42)

where we omit indices. The new term iµγ5τ3 is the so-called twisted mass term, where

µ is the twisted mass and τ 3 is the third Pauli matrix acting on flavour space. Onereason for introducing this term is to eliminate exceptional configurations, which mayappear when the mass term in the Wilson operator becomes small. The twisted massDirac operator is protected since

det[D0W +m+ iµγ5τ

3]

= det[D0W +m+ iµγ5

]det[D0W +m− iµγ5

]= det

[(D0

W +m+ iµγ5)(D0†W +m− iµγ5)

]= det

[(D0

W +m)(D0†W +m) + µ2

]> 0, (2.43)

12

if µ 6= 0, the determinant is always greater than zero and thus protected from excep-tional configurations [19]. The above expression shows that the twisted mass term maybe used as an alternative infrared regulator which can be combined with the standardmass term. When we have non-vanishing m and µ we can define the polar mass Mand the twist angle α as

M =√m2 + µ2, α = arctan(µ/m). (2.44)

With these definitions we can write the mass terms as,

Meiαγ5τ3

= m+ iµγ5τ3 (2.45)

where m = M cos(α) and µ = M sin(α). The case where α = π/2 when µ 6= 0 andm → 0, is called maximal twist and it was shown that automatic O(a) improvementoccurs [20].

After having introduced this new kind of infrared regulator we must prove that insome limit the twisted mass formulation reproduces conventional QCD. We begin bystudying the relation between them. In the continuum the action for two degeneratefermions is given by,

SF [ψ, ψ, A] =

∫d4x ψ(x)

(γµDµ(x) +M

)ψ(x) (2.46)

and the expectation value of an operator O,

〈O〉 =1

Z

∫D[ψ, ψ, A]e−SF [ψ,ψ,A]−SG[A]O[ψ, ψ, A]. (2.47)

If we transform the physical fields to the twisted ones

ψ = eiαγ5τ3/2χ, ψ = χeiαγ5τ

3/2 (2.48)

we get Eq. (2.42) and the expectation value of operator O in the twisted basis is givenby,

〈Otm〉 =1

Ztm

∫D[χ, χ, A]e−SF [χ,χ,A]−SG[A]Otm[χ, χ, A]. (2.49)

In order to study TMFs on the lattice we need to work out the relations in the twistedmass basis. Let us start with the axial-vector [18] operator

O[ψ, ψ] = A1µ(x)P 1(y) with Aaµ =

1

2ψγµγ5τ

aψ and P a =1

2ψγ5τ

aψ. (2.50)

Then we perform the twisted transformation on the fields,

A1µ → cos(α)A1tm

µ − sin(α)V 2tm

µ , P 1 → P 1tm (2.51)

with

Aatm

µ =1

2χγµγ5τ

aχ, V atm

µ =1

2χγµτ

aχ, P atm =1

2χγ5τ

aχ. (2.52)

Using these relations we can prove that

〈O〉 = cos(α)〈A1tm

µ P 1tm〉 − sin(α)〈V 1tm

µ P 1tm〉 = 〈Otm〉. (2.53)

A brief discussion about the symmetries of the twisted mass fermions is given in Ap-pendix. C. The above proof for a 1-point function can be generalized for an n-point

13

function. The discussion above focused on the continuum case, and for the lattice weneed to consider the renormalized theory [18]. In the renormalized theory we can usethe arguments given for the continuum discussion above and conclude that standardQCD correlation functions can be expressed as linear combinations of correlators intwisted mass QCD. This relation remains valid for finite lattice spacing up to dis-cretization errors.

Although the initial motivation for the introduction of the twisted mass term was toremove the exceptional configurations, the automatic O(a) improvement [21] achievedat maximal twist is central for the study of hadron structure. With the introductionof the twisted mass term we have two parameters, m and µ for the mass term. Thespecial role of α = π/2 can already be seen in the case of free fermions. In momentumspace the twisted mass operator becomes,

i

a

4∑µ=1

γµ sin(apµ) +1

a

4∑µ=1

(1− cos(apµ)) +M cos(α) + iM sin(α)γ5τ3. (2.54)

The propagator in momentum space is given by,

− ia

∑µ γµ sin(apµ) + 1

a

∑µ(1− cos(apµ)) +M cos(α)− iM sin(α)γ5τ

3

1a2

∑µ sin2(apµ) +

(1a

∑µ(1− cos(apµ)) +M cos(α)

)2

+M2 sin2(α). (2.55)

In order to find the energy for a particle with this propagator we must expand thedenominator in terms of the lattice spacing,

p2(1 + aM cos(α)) +M2 +O(a2). (2.56)

The energy is given from the poles of this equation, thus we find

E = ±√~p2 +M2 ∓ a cos(α)

M3

2√~p2 +M2

+O(a2). (2.57)

In the special case when α = π/2 the second term disappears and we get the energyup to O(a2). From an algebraic point of view this result can be understood by notingthat at maximal twist the Wilson term and the mass term are orthogonal in isospinspace and thus terms with O(a) dependence, cannot emerge.

14

Chapter 3

Lattice Techniques for QCD

In this chapter we discuss techniques to extract observables using the lattice formu-lation. The simplest quantities involving fermionic degrees of freedom that one cancompute on the lattice are hadron masses. The first benchmark for the lattice QCDformulation is to reproduce correctly the low-lying mass spectrum, which is one of thetopics of this chapter. The second topic is to explain how to evaluate three-point cor-relation functions and extract various matrix elements probing hadron structure. Inthis work we will concentrate on baryons.

3.1 Hadron two-point correlation functions

The first step involved in the calculation of hadron masses is to identify the interpolat-ing fields with the quantum numbers of the particle of interest. We need to define anoperator J † which creates the particle state with the correct quantum numbers of theparticle we would like to study. Interpolating fields that reflect the internal intrinsicstructure of the hadron we want to study will have better signal to noise behaviour.Several examples are:

• Local operators, such as JM(x) = ψ(x)Γψ(x) for mesons, which consist of onequark and one anti-quark and for baryons JB(x) = εabc(ψa(x)Γψb(x))ψc(x) whichconsist of three quarks.

• Extended interpolators, involving gauge links, written as ψ(x)Uµ(x)Γψ(x+ µ) inthe case of mesons when studying quantities such as vacuum polarization andglueballs.

• Pure gauge operators, such as the plaquette, longer closed loops and more genericgluonic operators.

• Covariant derivative operators, involving the symmetrized covariant derivativethat connects neighbouring lattice points, namely ψΓDψ.

We consider the time correlator

C(~p, t) =∑~x

〈J (~x, t)J (~0, 0)〉e−i~p·~x (3.1)

where we take the initial position at x = 0. The time correlator can be written as

C(~p, t) =∑n

〈Ω|J |n〉〈n|J |Ω〉e−tEn(~p) = |Z|2e−tE0(~p)(1 +O(e−t∆E(~p))

)(3.2)

15

where ∆E is the energy difference between the ground state and the first excitedstate. In order to extract the ground state energy we require (t ∆E) 1, so that theexponential is negligible compared to unity. Extended information for the expansionof two-point correlation functions is given in section 3.3.

3.1.1 Meson correlators

Let us consider the simplest hadron, namely the pion, made of up and down quarks.The quantum numbers of the pion are isospin I = 1 and negative parity P = −1.Since the charge of a d-quark is −1/3 and the charge of a u-quark is 2/3, then theinterpolating field of π+ is

Jπ+(x) = d(x)γ5u(x) (3.3)

and that of π− isJπ−(x) = u(x)γ5d(x). (3.4)

Transformation under parity gives

Jπ+(~x, t) = d(~x, t)γ5u(~x, t)

P−−→ d(−~x, t)γ4γ5γ4u(−~x, t) = −d(−~x, t)γ5u(−~x, t)= −Jπ+(−~x, t), (3.5)

i.e Jπ+ has indeed negative parity. Applying charge conjugation gives,

Jπ+(x) = d(x)γ5u(x) C−−→− dT (x)Cγ5C

−1uT (x) = −d(x)TγT5 uT (x)

= u(x)γ5d(x) = Jπ−(x), (3.6)

where C is the charge conjugation matrix given in Appendix. B. The interpolatingfield for π0 with Iz = 0 is

Jπ0(x) =1√2

(u(x)γ5u(x)− d(x)γ5d(x)

). (3.7)

Table 3.1 lists the quantum numbers of the most commonly used mesons.

State J(spin) P(parity) C(charge conjugation) Γ (operator) ParticlesScalar 0 + + 11, γ4 f0, a0

Pseudoscalar 0 - + γ5,γ4γ5 π±, π0, ηVector 1 - - γi, γ4γi ρ±, ρ0, ω

Axial vector 1 + + γiγ5 a1, f1

Tensor 1 + - γiγj h1, b1

Table 3.1: Quantum numbers of the most commonly used meson interpolating fields.

The building blocks for the computation of time correlators are the quark prop-agators. We thus express the correlation function of Eq. (3.1) in terms of quarkpropagators,

〈J (y)J †(x)〉 = 〈d(y)γ5u(y)u(x)γ5d(x)〉 =

= (γ5)αβ(γ5)α′β′〈d(y)aαu(y)aβu(x)bα′d(x)bβ′〉= −(γ5)αβ(γ5)α′β′〈d(x)bβ′ d(y)aα〉〈u(y)aβu(x)bα′〉= −(γ5)αβ(γ5)α′β′Gd(x; y)baβ′αGu(y;x)abβα′

= −Tr[γ5Gd(x; y)γ5Gu(y;x)

]= −Tr

[|Gu(y;x)|2

]. (3.8)

16

Here we have used the anti-commutative properties of the fermion fields and per-formed the Wick contractions. In the last step we make use of the identity G†u(y;x) =γ5Gd(x; y)γ5, where the dagger acts on spin and color indices. This identity holds fortwisted mass fermions. By fixing the source point of the quark propagator we constructtwo-point functions that point from a fixed point x = 0 to a general point y. In such acase the computation needs only point-to-all quark propagators, which are obtained by12 inversions. Fig. 3.1 depicts diagrammatically an example of a connected two-pointfunction.

For π0 given in Eq. (3.7), Wick contractions lead to an additional diagram depictedin Fig. 3.1, which involves two fermionic loops. Such a diagram, in which the fermionlines are not connected, is known as a disconnected diagram. Note that gauge interac-tions still exist between the two closed quark loops. In such lattice QCD diagrams seagluons and sea quarks are typically not drawn, unless an operator couples to them.

xy

Gu(y; x)

Gd(x; y)

xy

Gu(x; x)

Gd(y; y)

Figure 3.1: Example of meson two-point correlation function. Connected contribution on theleft and disconnected contributions on the right.

3.1.2 Baryon correlators

Baryons are particles, which have half integer spin, with the most well known exam-ples being the proton and the neutron, made up of light quarks. The proton and theneutron belong to the same iso-doublet. Neglecting electromagnetic interactions andconsidering u- and d-quarks degenerate, the proton and the neutron are mass degener-ate and referred to as nucleons. The simplest interpolating field, which has an overlapwith the nucleon N is given by,

JN(x)γ = εabcu(x)aγ

[u(x)bα

(Cγ5

)αβd(x)cβ

]. (3.9)

The εabc antisymmetric tensor ensures that the interpolator is colorless and thus gaugeinvariant. The term in the square brackets has spin J = 0 and I = 0 which means thatthe interpolator has total spin J = 1/2. Since the nucleon has positive parity, we mustensure that the interpolating field projects to P = +1. Under parity transformationwe have

JN(~x, t) P−−→ εabcγ4u(−~x, t)auT (−~x, t)bγT4 Cγ5γ4d(−~x, t)c

= εabcγ4u(−~x, t)auT (−~x, t)bCγ5d(−~x, t)c= γ4JN(−~x, t). (3.10)

Instead of JN(x) we consider the combination

J ±N (x) =1

2

(JN(x)± J PN (x)

)= εabcP±u(x)a

(u(x)b Cγ5 d(x)c

), (3.11)

17

whereJ PN (x) = γ4 JN(−~x, t) (3.12)

and the parity projectors P± are defined as

P± ≡ 1

2(1± γ4), (3.13)

and hence, J ±N has definite parity.The creation operator for the nucleon is JN(x) ≡ J†N(x)γ0 and thus,

J ±N (x) = εabc(u(x)a Cγ5 d(x)b

)u(x)cP± (3.14)

where Cγ5 ≡ γ0(Cγ5)†γ0. We can construct interpolating fields for other J = 1/2hadrons by considering their flavour content. In Table 3.2 we give the interpolatingfields of the octet baryons, while a complete table is given in Appendix. D.

State Interpolating Fieldp+ εabcua

(ubCγ5d

c)

Σ+ εabcua(ubCγ5s

c)

Ξ0 εabcsa(sbCγ5u

c)

Λ0 1√6εabc(

2sa(ubCγ5dc) + da(ubCγ5s

c)− ua(dbCγ5sc))

Table 3.2: Interpolating fields for octet baryons.

The two-point time correlator for the nucleon is given by

CN(~p, t) =∑~x

e−i~p·~x〈Pγ′γJN(x)γJN(0)γ′〉 =∑~x

εabcεa′b′c′(Cγ5)α′β′(Cγ5)αβPγγ′Gd(x; 0)b

′bβ′β

×(Gu(x; 0)a

′aα′αGu(x; 0)c

′cγ′γ −Gu(x; 0)a

′cα′γGu(x; 0)c

′aγ′α

)e−i~p·~x

=∑~x

e−i~p·~xεabcεa′b′c′ ×

Tr

[[Cγ5Gd(x; 0)(Cγ5)T

]b′b[GTu (x; 0)

]a′a]Tr

[[Gu(x; 0)P

]c′c]−Tr

[[(Cγ5Gd(x; 0)(Cγ5)T

)T]b′b[Gu(x; 0)

]a′c [P Gu(x; 0)

]c′a]. (3.15)

Two-point correlation functions for baryons receive contributions only from connecteddiagrams shown in Fig. 3.2, for the case of the proton.

0x

Gu(x; 0)

Gu(x; 0)

Gd(x; 0)

Figure 3.2: Two-point correlation function for the proton.

18

3.2 Signal improvement methods

In order to extract the mass of the ground state we probe the behaviour of the two-pointfunction at long time separation between the source and the sink in order to suppressthe excited states contributions. The elements of the propagator decay exponentiallywith the source-sink time separation, however the gauge noise remains the same andthus the noise to signal ratio increases exponentially. As a result, it is very importantto increase the overlap of the interpolating field with the ground state of the particleof interest.

3.2.1 Extended sources

In order to suppress excited states we must find a way to improve the overlap of theinterpolating field with the ground state. One way to achieve this, is by constructingan interpolating field closer to the spatial wave function of the state of interest.

A well known method for increasing the overlap with the ground state and sup-pressing excited states is so-called Gaussian smearing, also known as Wuppertalsmearing [22]. A smeared quark field can be obtained from a point source via,

qsm(~x, t) =∑~y

F (~x, ~y;U(t))q(~y, t), (3.16)

whereF (~x, ~y;U(t)) ≡

(11 + αH(~x, ~y;U(t))

)n, (3.17)

with

H(~x, ~y;U(t)) =3∑

k=1

(Uk(~x, t)δ~x,~y−k + U †k(~x− k, t)δ~x,~y+k

). (3.18)

The parameters α and n can be tuned to increase the overlap with the ground state.The operator F (~x, ~y;U(t)) depends on the gauge links, which means we can use smearedlinks to improve the signal to noise behaviour.

3.2.2 Smoothing the gauge fields

One way to improve the signal to noise is to smooth the gauge links. When smoothingor smearing the configurations one typically replaces the original link variables bythe neighbouring averages. This procedure must be constructed in such a way thatit keeps the gauge field in the SU(3) group. These smeared configurations are thenused to evaluate the quark propagators with improved signal to noise behaviour. Weconsider three variants of smearing algorithms:

1. APE smearing [23]: In this case one takes the average over the given originallink and the six staples connecting its endpoints, as depicted in Fig. 3.3.

UAPEµ (x) = ProjSU(3)

((1− α)Uµ(x) +

1

α

∑ν 6=µ

Sµν(x)

), (3.19)

where

Sµν(x) = Uν(x)Uµ(x+ ν)U †ν(x+ µ) + U †ν(x− ν)Uµ(x− ν)Uν(x− ν + µ), (3.20)

and the real parameter α can be adjusted to optimize the reduction in the noise.In every iteration we perform in the smoothing procedure it is important toproject UAPE

µ (x) onto the SU(3) group.

19

UAPEµ (x)

Figure 3.3: The link UAPEµ (x) updated using the four neighbouring staples. On a 4-D lattice,two additional staples in perpendicular directions contribute.

2. HYP smearing [24]: This algorithm is more complicated than APE smearingsince it involves three ”APE” steps. Namely, in the last step, the original thinlink is updated using the staples, which are created from APE smearing in thehypercubic plane,

UHY Pµ (x) = ProjSU(3)

((1− ω1)Uµ(x) +

ω1

6

∑±ν 6=µ

Sν;µ(x)Sµ;ν(x+ ν)S†ν;µ(x+ µ)

)(3.21)

where links S are made of fat staples,

Sµ;ν(x) = ProjSU(3)

((1− ω2)Uµ(x) +

ω2

4

∑±ρ 6=ν,µ

Sρ;νµ(x)Sµ;ρν(x+ ρ)S†ρ;νµ(x+ µ)

)(3.22)

and the links S are made of thin links only,

Sµ;ν,ρ(x) = ProjSU(3)

((1− ω3)Uµ(x) +

ω3

2

∑±η 6=ρ,ν,µ

Uη(x)Uµ(x+ η)U †η(x+ µ)

).

(3.23)HYP smearing is depicted in Fig. 3.4.

3. Stout smearing [25]: This method uses a particular way of projection by defin-ing the new link after smearing as,

U ′µ(x) = eiQµ(x)Uµ(x), (3.24)

where Qµ(x) is a traceless hermitian matrix constructed from staples

Qµ(x) =i

2

(Ω†µ(x)− Ωµ(x)− 1

3Tr[Ω†µ(x)− Ωµ(x)

]), (3.25)

with

Ωµ(x) =

(∑ν 6=µ

ρµνSµν(x)

)U †µ(x) (3.26)

and Sµν is defined in Eq. (3.20). The real weight factors ρµν are tunable param-eters. The new links have gauge transformation properties like the original ones.The advantage of stout smearing is that U ′µ(x) is differentiable with respect tothe link variables. This is a very important property for the hybrid Monte Carlomethod for dynamical fermions, since it is needed for enabling the update of thelinks.

20

UHY Pµ (x)

Figure 3.4: The 3 dimensional representation of HYP smearing.

All the smearing techniques we showed are iterative, thus they affect not just theneighbouring sites, and as a result the asymptotic behaviour of the quark propagatorneeds to be checked to ensure that no long-range effects remain.

3.3 Extracting hadron masses

In subsection 3.1.1 we described how to build interpolators and calculate two-pointfunctions at the quark level. In this section we study two-point functions at thehadronic level and show how one can extract hadron masses.

If we use the Heisenberg picture we can write the interpolating field as

J (~x, t) = e−i~P·~xeHtJ (~0, 0)e−Htei

~P·~x, (3.27)

where ~P and H are the momentum and Hamiltonian operators respectively. Then weinsert a complete set of states as in Eq. (3.2),

11 =∑~k,n

|n,~k〉〈n,~k| (3.28)

and we find that at zero momentum the two-point function becomes

C(~0, ts) =∑n,~k

∑~x

∣∣∣〈Ω|J (~0, 0)|n,~k〉∣∣∣2e−En(~k)tsei

~k·~x, (3.29)

which yields

C(~0, ts) =∑n

∣∣∣〈Ω|J (~0, 0)|n,~0〉∣∣∣2e−En(~0)ts = Z0 e

−E0ts(

1+Z1e−ts∆E1 +Z2e

−ts∆E2 +· · ·),

(3.30)

where we have defined Zn ≡∣∣∣〈Ω|J (~0, 0)|n,~0〉

∣∣∣2 and ∆En ≡ En−E0. In the limit where

ts →∞ the contribution of the excited states is negligible compared to unity, thus we

21

obtain the ground state,

C(~0, ts) =∑n

∣∣∣〈Ω|J (~0, 0)|n,~0〉∣∣∣2e−En(~0) ts ts →∞−−−−−−−→

∣∣∣〈Ω|J (~0, 0)|H〉∣∣∣2e−E0(~0) ts ,

(3.31)which for a single particle state yields the mass mH = E0(~0) of the lowest state |H〉.The term Z0 ≡ 〈Ω|J (~0, 0)|H〉 is an overlap term which preserves invariance of thecorrelator under Lorentz transformations. Since we have a finite lattice one needs toapply boundary conditions. This periodicity of the lattice means that for mesons wehave

C(ts) =∣∣∣〈Ω|J (~0, 0)|H〉

∣∣∣2(e−mts + e−m(Lt−ts)), (3.32)

which can be written as

C(ts) =∣∣∣〈Ω|J (~0, 0)|H〉

∣∣∣2e−mLt/2 cosh(m(ts − Lt/2)

). (3.33)

In order to analyse in which range of t the contributions of the leading exponentialscan be neglected, one defines the so-called effective mass as

meff(ts) = logC(ts)

C(ts + 1). (3.34)

For mesons one must respect the periodicity of the lattice in the temporal direction,

C(ts)

C(ts + 1)=

cosh(m(ts − Lt/2)

)cosh

(m(ts + 1− Lt/2)

) . (3.35)

When the excited states are suppressed, is equal to the mass of the lowest hadron state.In the case of baryons, the backward propagating baryon corresponds to the particle

with opposite parity, which is not degenerate with the forward going one. For example,the nucleon two-point function on the lattice with periodic or anti-periodic boundaryconditions in the temporal direction is given by

CN(ts) =∣∣∣ZN(~0)

∣∣∣e−ts mN − ∣∣∣ZN(−)(~0)∣∣∣e(ts−Lt) mN(−) , (3.36)

where N (−) is the nucleon with negative parity.

0

0.05

0.1

0.15

0.2

0.25

0.3

0 2 4 6 8 10 12 14 16 18 20 22

mef

fπ+

(t s)

ts/a

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16 18 20 22

mef

fN(t s

)

ts/a

Figure 3.5: Effective mass for π+ on the left and for N on the right using 100 configurations.The red band shows the extracted mass with its jackknife error.

In Fig. 3.5 we show the effective mass using 100 configurations of Nf = 2 + 1 + 1TMFs simulated at a pion mass of 373 MeV and a lattice spacing a = 0.082 fm [26, 27]

22

on a lattice size of (323 × 64)a4. On the left panel we show the effective mass of thepseudoscalar meson π+ and on the right we show that of the nucleon. As we can see,for small time slices ts/a < 8 we have excited states contamination and the effectivemass changes with time as excited states gradually vanish. After ts/a = 8, meff(ts)becomes time independent and we observe a plateau and by fitting to a constant oneextracts the mass of the ground state. The band shows the fit to a constant with itsjackknife error. It is important to also study what happens to the signal to noise aswe increase the overlap with the ground state. For the pion noise to signal is constant,but for the nucleon increases rapidly with the source-sink time slice. This means thatif we want to keep the noise under control we must optimize the overlap to the nucleonstate in order to achieve faster convergence.

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16 18 20 22 24 26

mef

fN(t s

)

ts/a

n=0n=10n=20n=30n=40n=50

Figure 3.6: The effective mass of the nucleon as a function of the source-sink time separationa range of Gaussian smearing steps when α = 4.0. The smearing steps in APE smearing arekept fixed to n = 20 with α = 0.5.

In Fig. 3.6 we show results for the effective mass of the nucleon as a function of thesource-sink time separation ts/a. We assess the behaviour of the meff(ts) for a range ofGaussian smearing steps when the number of APE smearing steps is kept fixed. As canbe seen, if we do not employ any smearing (n = 0) the excited states contributions arelarge and a source-sink time separation with ts/a = 10 is needed in order to suppressthe excited states. For n = 10 we observe a better overlap with the ground state and afaster convergence is achieved. By increasing further the Gaussian smearing steps, theimprovement saturates when n = 50 where we observe a plateau for ts/a = 5, wherethe statistical error is minimal.

3.4 Hadron three-point correlation functions

We have so far presented the simplest correlator, namely the two-point function fromwhich we can extract the mass of a particle. There are also more complex correlationfunctions from which we extract more interesting quantities such as the hadron chargeradii, hadron decay amplitudes, coupling constants and form factors. In order to extractsuch quantities one needs to compute three-point functions.

23

3.4.1 Three-point functions

The general form of such a three-point function is given by

Gµ(xs;xins) = 〈Ω|J (xs)Oµ(xins)J (0)|Ω〉. (3.37)

The operator Oµ(xins) couples to a quark current and can carry a Lorentz index, asfor example in the case of the electromagnetic current or the axial-vector current. Onthe quark level, a three-point function can be composed in terms of two diagrams, asdepicted in Fig. 3.7. Details for the contractions are given in Appendix. E. In theso-called connected diagram the operator couples to a valance quark, whereas in thedisconnected one it couples to a sea quark.

J (0)J (xs)

Oµ(xins)

J (0)J (xs)

Oµ(xins)

Figure 3.7: The connected part (left) and disconnected part (right), of a three-point functioncoupled to an operator Oµ(xins).

On the hadronic level we consider the three-point function in momentum space. Wetake the initial state to be fixed at the origin, and we perform two Fourier transforms,one over ~xs and one over ~xins. The momentum at the source is automatically fixedfrom the momentum at the insertion and the sink, due to momentum conservation,thus

Gµ(~p′, ts; ~p1, tins) =∑~xs,~xins

Gµ(~xs, ts; ~xins, tins)e−i~xs·~p′e+i~xins·~p1 . (3.38)

The interpolating field and insertion operator can be expanded in the Heisenberg pic-ture, hence


e−i~xs·~p′e+i~xins·~p1 ×

〈Ω|J (~0, 0)e−Htsei~xs·~Pe−i~xins·

~PeHtinsOµe−Htinsei~xins·~PJ (~0, 0)|Ω〉. (3.39)

By inserting two complete sets of states we can write the three-point function in theform,


∑n,n′

∑~k,~k′

〈Ω|J |~k′, n′〉〈~k, n|J |Ω〉 ×

e−En′ (~k′)(ts−tins)e−i~xs·(

~p′−~k′)〈n′, ~k′|Oµ|n,~k〉e−En(~k)tinse−i~xins·(~k′−~k−~p1). (3.40)

Then, performing the sum over ~xs, ~xins and ~k,~k′, we finally take the large time limitwhere (ts−tins)→∞ and tins →∞, so that the excited states contribution is minimal.

24

The three-point function then takes the form,

Gµ(~p′, ts; ~p1, tins) = 〈Ω|J |H(~p′)〉〈H(~p1)|J |Ω〉〈H(~p′)|Oµ|H(~p′−~p1)〉e−E(~p′)(ts−tins)−E(~p1)tins .(3.41)

The overlap term can be written as

〈Ω|J |H(~p)〉 =

√M

E(~p)Z(~p) (3.42)

and the three-point function takes its final form,

Gµ(~p′, ts; ~p1, tins) (ts − tins)→∞, tins →∞−−−−−−−−−−−−−−−−−−−−→

Z(~p′)Z∗(~p1)

√M2

E(~p′)E(~p1)〈H(~p′)|Oµ|H(~p′ − ~p1)〉e−E(~p′)(ts−tins)−E(~p1)tins .(3.43)

The matrix element we are interested in is 〈H(~p′)|Oµ|H(~p′ − ~p1)〉.The direct way to isolate the matrix element is to take a ratio of the three-point

function with two-point functions in order to cancel the unwanted exponential termsand unknown overlaps Z(~p). A possible ratio is

Rµ(~p′, ts; ~p1, tins) =Gµ(~p′, ts; ~p1, tins)√

G(~p′, 2(ts − tins))G(~p1, 2tins). (3.44)

This simple ratio has an obvious problem, namely that it involves two-point functionsat twice the time separation, thus increasing the gauge noise in particular for baryons.There is a more optimal combination we can construct which has a better behaviour,namely [28]

Rµ(~p′, ts; ~p1, tins) =Gµ(~p′, ts; ~p1, tins)

G(~p′, ts)×√G(~p1, ts − tins)G(~p′, tins)G(~p′, ts)

G(~p′, ts − tins)G(~p1, tins)G(~p1, ts), (3.45)

which involves only time separations tins and (ts−tins). In the limit where (ts−tins)→∞ and tins →∞ the ratio becomes time independent and we observe a plateau.

In order to relate the lattice results with experimentally measured quantities weneed the decomposition on the hadronic level. We will restrict ourselves to the casewhere the original hadron is in the rest frame. Thus the momenta fulfil the relations~p′ = ~0 and q ≡ ~p′ − ~p = ~p1. Starting from Eq. (3.41), the overlap factor for the case ofspin-1/2 baryons is

〈Ω|JN |N(~p, s)〉 =

√M

E(~p)Z(~p)u(~p, s), (3.46)

with u(~p, s) the free Dirac spinor. For the two-point function we will use the parityprojector Γ0 ≡ 1

2(1 + γ0). Performing the trace algebra we find

G(ts, ~p,Γ0) =

EN(~p) +MN

2EN(~p)|ZN(~p)|2e−EN (~p)ts , (3.47)

where in Euclidean space the spin sum rule gives,∑s

u(~p, s)u(~p, s) =−i 6p+MN

2EN(~p). (3.48)

25

The matrix element in Eq. (3.41) can be decomposed [29] in the case of the electro-magnetic current Oµ = ψγµψ as

〈N(~p′, s′)|Oµ|N(~p, s)〉 =

√M2

N

EN(~p′)EN(~p)u(~p′, s′)jµu(~p, s), (3.49)

where

jµ = γµF1(q2) +iσµνq

ν

2MN

F2(q2), (3.50)

with F1 and F2 the Dirac and Pauli form factors respectively. These are related to theelectric and magnetic Sachs form factors via,

GE(q2) = F1(q2) + F2(q2), GM(q2) = F1(q2) +q2

(2MN)2F2(q2). (3.51)

The three-point function for a general projection matrix is given by

Gµ(~q, ts, tins; Γ) =ZN(~0)ZN(~q)

4MNEN(~q)Tr[Γ( 6p′+MN)jµ(6p+MN)

]e−MN (ts−tins)−E(~q)tins . (3.52)

We plug Eqs. (3.47) and (3.52) into Eq. (3.45) and after performing the trace algebrawe obtain the relations, connecting the lattice results with the nucleon electromagneticform factors [30],

Πi(~q,Γk) = C1

2MN

εijkqjGM(Q2) (3.53)

Πi(~q,Γ0) = Cqi

2MN

GE(Q2) (3.54)

Π0(~q,Γ0) = CEN +MN

2MN

GE(Q2) (3.55)

where Γk = Γ0iγ5γk and C =

√2M2

N

EN (EN+MN )is a kinematical term.

3.4.2 Three-point functions on the quark level

On the quark level the three-point function can be calculated in terms of quark propa-gators. As explained, a three-point function consists of two parts depicted in Fig. 3.7.Let us now explain the computation of the connected part for the simple case of π+.The method can be generalized for the nucleon, as done in Appendix. E.

For the case where the current is uΓu the connected part of the π+ three-pointfunction is given by,∑

~xs,~xins

Tr[γ5Gd(0;xs)γ5Gu(xs;xins)ΓGu(xins; 0)

]e+i~q·~xins . (3.56)

As can be seen from Eq. (3.56) one needs the all-to-all quark propagator Gu(xs;xins).There are two ways to compute the double sum in Eq. (3.56). One involves summingover ~xins (fixed-current) and the other over ~xs (fixed-sink).

The idea of the fixed-sink method is to construct an appropriate source b thatthrough inversion performs the summation over ~xs. Considering the expression,

b = γ5Gu(xs; 0)δ(ts − t′s) (3.57)

26

after solving the equation DΣdu = b we obtain,

Σdu(xins; t′s) =

∑xs

Gd(xins;xs)γ5Gu(xs; 0)δ(ts − t′s) (3.58)

where we define the sequential propagator [31] Σdu(xins; t′s). The sequential propagator

is depicted in Fig. 3.8. If we take the dagger of the backward sequential propagator and

0t′s

Σud(t′s; xins)

0t′s

Σud(xins; t′s)

Figure 3.8: On the left we show the backward and on the right the forward going sequentialpropagator, for the case of π+.

multiply with γ5 from the right and make use of the property Gu(x; y) = γ5G†d(y;x)γ5

we can create the forward going sequential propagator

Σdu(t′s;xins) =

∑xs

γ5Gd(0;xs)γ5Gu(xs;xins)δ(ts − t′s), (3.59)

which is schematically presented in Fig. 3.8. Then the connected part of the three-pointfunction for the pion can be written as,

G3pfconn(t′s, tins) =

∑~xins

Tr[Σdu(t

′s;xins)ΓGu(xins; 0)

](3.60)

where all the ingredients have been computed. The electric form factor given in Eq.(3.55) at zero momentum transfer yields the electric charge of p+ which must be equalto unity. For the case of π+ the electric charge [31] can be extracted from the pionelectromagnetic form factor Fπ using

Π0(~q) =Eπ(~q) +Mπ

2Eπ(~q)Fπ(q2), (3.61)

at zero momentum transfer. For the extraction of electromagnetic form factors, one canuse either the local current ψγµψ or the lattice conserved current in its symmetrizedform,

jµ(x) −→ 1

2[jµ(x) + jµ(x− µ)]. (3.62)

The advantage of the conserved current is that there is no need for renormalization.In Figs. 3.9 and 3.10 we show results for the ratio R as a function of the insertiontime. For comparison we also show results for the case of the nucleon using the same100 configurations. The fact that GE(0) = 1 for both π+ and p+ is guaranteed bysymmetry in the case of the conserved current. This quantity is thus used to check theconsistency of our code implementation.

27

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 1 2 3 4 5 6 7 8

R(t i

ns)

tins/a

local currentconserved current

Figure 3.9: The ratio from which we extract GπE(0) as a function of tins/a for ts = 8 using100 configurations. Green filled circles show the ratio using the conserved current, whereasred filled squares show results using the local current.

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 2 4 6 8 10 12

R(t i

ns)

tins/a

local currentconserved current

Figure 3.10: The ratio from which we extract GpE(0) as a function of tins/a for ts = 12 using100 configurations. Green filled circles show the ratio using the conserved current, whereasred filled squares show results using the local current.

28

Chapter 4

Lattice methods for disconnected quarkloops

In the previous chapter we discussed methods to calculate the connected part of three-point functions. In the present chapter we will present several methods to compute thedisconnected quark loops.

The evaluation of disconnected diagrams is very important if we want to have thecomplete description of hadronic matrix elements from lattice QCD. Disconnected con-tributions to flavour singlet quantities have been neglected in most calculations becausethey are computationally demanding quantities. For some quantities the disconnectedcontributions are expected to be small, but for others like the η′ mass, and σ-termsthey are essential. Some perturbative calculations for flavour singlet quantities dis-agree with the experimental values, which suggests that flavour singlet phenomena arelinked with non-perturbative properties of the vacuum. For instance the mass of theη′ [32, 33] can be calculated correctly only under a non-perturbative calculation, dueto the topological properties of lattice QCD.

Within the lattice QCD formulation, the calculation of the disconnected quarkloops requires the all-to-all or time-slice-to-all propagator which is computationallyextremely intensive to be calculated exactly [33]. Even if we use stochastic methods, theresources required are much larger than for connected contributions. Therefore, in mosthadron studies up to now the disconnected contributions were neglected, introducinguncontrolled systematic error.

Nowadays computational power has increased dramatically and the development ofnew algorithms have made the calculation of disconnected contributions feasible. In thealgorithmic side we have a variety of new techniques to estimate the disconnected quarkloops, such as the one-end trick [34–36], dilution schemes [33, 37, 38], the TruncatedSolver Method [33, 39] and the hopping parameter expansion [40]. These techniqueshave led to a significant reduction in both stochastic and gauge noise. The one-endtrick using the special properties of twisted mass fermions transforms the single volumesum we have in the calculation of the loop to an effective double sum, which reduces thenoise. On the hardware side new platforms such as Graphic Processing Units (GPUs)can accelerate both the contractions and the inversions needed for the calculation ofdisconnected quark loops. In Fig. 4.1 we show the tremendous evolution in GPUscompared to CPUs. The speed-up achieved can reach 30 times those achieved in CPUsfor scientific fields like material science, shown in Fig. 4.2.

In the following sections our goal is to assess the recently developed methods for thecalculation of disconnected quark loops exploiting the power of GPUs. Our study willbe performed on one ensemble of twisted mass fermions. The ensemble is generatedusing Nf = 2 + 1 + 1 dynamical fermions where the masses for strange and charmquarks are tuned to their physical values. The lattice size is 323×64 and the pion mass

29

Figure 4.1: Tremendous evolution of GPUs compared to CPUs for various computer archi-tectures [41].

Figure 4.2: GPUs speed tests for various scientific fields, namely Material Science, Physics,Earth Science and Molecular Dynamics [42].

is 373 MeV [27] with lattice spacing a = 0.082 fm determined using the nucleon mass.From now on we will refer to this ensemble as B55.32.

4.1 Stochastic estimation using noise sources

The exact computation [33] of the all-to-all propagator or time-slice-to-all propagatoris computationally prohibited for lattice sizes such as 323× 64 or 483× 96. The typicalway to obtain a reliable estimate to the all-to-all propagator is by using sources withspecial properties [43]. These special sources are called noise vectors and they obeythe following properties

1

Nr

∑r

ξr = 0 +O(

1√Nr

), (4.1)

1

Nr

∑r

ξr(x)ξ∗r (y) = 11x,y +O(

1√Nr

). (4.2)

30

An essential ingredient of this method is to generate a set of Nr noise sources, withentries filled with numbers drawn from the roots of one, randomly. For instance Z2 is1,−1 and Z4 is 1,−1, i,−i. There are also other noise sets we can use, but it hasbeen shown that Z4 suppresses the stochastic noise better [44].

The first property in Eq. (4.1) is required to create an unbiased estimate of theall-to-all propagator and the second one in Eq. (4.2) allows us to construct the all-to-allpropagator by solving the equation

Mφr = ξr. (4.3)

Therefore, the diagonal elements of the all-to-all propagator can be estimated by

G(x;x) =1

Nr

∑r

φr(x) ξ†r(x) +O(

1√Nr

), (4.4)

where in the limit as the number of noise vectors goes to infinity the stochastic errorgoes to zero and we reconstruct the exact all-to-all propagator. In practice the ensembleof noise vectors is much smaller as compared to the volume, making this methodcomputationally feasible.

Since the quantities we measure have gauge noise, it is sufficient to reduce thestochastic error until it becomes comparable to the gauge error. This criterion ideallydetermines the number of noise vectors Nr we need, which is expected to be differentfor each observable. If on the other hand we are interested only in one observable wecan choose Nr for this specific quantity. However, in general we want to study manyobservables so Nr must be chosen in a satisfactory way for all of them.

4.2 The Truncated Solver Method (TSM)

The truncated solver method [33, 39] is a way to reduce the stochastic variance atreduced computational cost. The basic idea of this method is the following: Insteadof calculating the loop using high precision inversions, we can estimate it with lowprecision inversions where,

φr(x)LP = G(x; y)LP ξr(y). (4.5)

In order to do this we truncate the precision of the solver, for instance the ConjugateGradient Method (CG). The truncation criterion can be a relaxed condition for theresidual, for example |r| = 10−2 instead of |r| = 10−9, or a fixed number of iterationswhich is much smaller than the number of the iterations needed for high precisioninversions. Using this idea we can increase the number of noise vectors at low cost butbecause we truncate the precision of CG our results for the loops will be biased.

In order to correct the bias introduced from the reduced inversion precision, weneed to evaluate a sufficient number of high precision inversions. Usually this can bedone by creating a small set of noise vectors for which we invert for both high and lowprecisions. The correction CE is then calculated as

CE =1

NHP

∑r

(φrHP ξ

†r − φrLP ξ†r

), (4.6)

and thus the final estimate for the all-to-all propagator becomes,

G(x;x)TSM =1

NLP

NLP∑r=1

φrLP (x) ξ†r(x) + CE +O(

1√NLP

). (4.7)

31

The improved stochastic error [45] is expected to scale as

δ ' δ

√2(1− rc) +

NHP

NLP

, (4.8)

where rc depends on the correlation between low and high precision results. rc isexpected to be close to unity if the correlation between high and low precision inversionsis high. The bias introduced in the TSM does not only affect the error but also themean value, thus the correction CE needs fine tuning in order to fix the bias in allobservables. For instance, rc may be different when we calculate the loop with thescalar insertion or the axial-vector insertion operator.

In order to achieve good performance for TSM and unbiased results we must tunethe number of low and high precision noise sources. We have found that a reasonableselection for the residual is |r| = (10−2 − 10−4) when the high precision is |r| = 10−9.NHP depends on the number of low precision inversions, and thus in order to tune bothNHP and NLP we first create the low precision set of noise vectors and then increasethe number of the high precision vectors so that no bias is observed. NHP must bebig enough to correct the bias for a quantity which requires high statistics. The dis-connected contributions we are interested in can be computed from the local and theone-derivative operators. We have found that the nucleon σ-term does not need manynoise vectors, but the nucleon axial charge gA and the momentum fraction 〈x〉 needa much larger number. Therefore, it is reasonable to tune NHP and NLP using 〈x〉 orgA. In Fig. 4.3 we present the error of a matrix element, which is needed for the

0 200 400 600 800 1000LP Sources

0

5

10

15

20

25

30

Abso

lute

Err

or

HP= 4HP= 8HP=12HP=18HP=24HP=36HP=48HP=70

Figure 4.3: Absolute error of the operator iψγ3D3ψ with respect to NLP for various NHP using50 configurations. The insertion time is fixed at tins = 8a and sink time ts = 16a.

calculation of the nucleon momentum fraction, 〈x〉 versus NLP for different NHP. In thespecial case where NLP = 0 we do not actually use the TSM and the error decreases aswe increase the number of the noise vectors until it saturates for Nr = 36, for the givennumber of configurations used here. When we use the TSM the error converges whenNLP ∼ 200 but we observe that 4 HP noise vectors are not sufficient to correct the bias,but instead we need between 8 and 12 to correct it. If we increase further the NHP ,the error does not change, thus the bias has already been corrected for NHP ' 12.

As we have already mentioned, the number of noise vectors needed depends onthe observable. For instance in Fig. 4.4 we show what happens to the error for two

32

0 100 200 300 400 5000.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

Error/Error(LP=0)

πΝ , 24 HP

gA , 24HP

LP Sources

Figure 4.4: The error on the sigma term σπN and axial charge gA versus the number of lowprecision (LP) inversions for a fixed number of high precision NHP = 24 vectors [46] using 50configurations.

different observables as we increase NLP keeping NHP fixed. The first observable weprobe is the nucleon sigma term [47] which is defined by

σπN = ml〈N |uu+ dd|N〉, (4.9)

where ml = (mu + md)/2. The second quantity is the nucleon axial charge whichcan be extracted from the axial-vector current and is responsible for the intrinsic spinorientation. The nucleon isoscalar axial charge can be extracted using

〈N(p′, s′)|ψγµγ5ψ|N(p, s)〉 = uN(p′, s′)

[GA(q2)γµγ5 +

qµγ5

2mN

Gp(q2)

]uN(p, s), (4.10)

where at zero momentum transfer gA ≡ GA(0). In Fig. 4.4 we show the error normal-ized over the error for 24 HP noise vectors versus NLP . For σπN the error does notchange with the number of LP noise vectors, which means that the stochastic error hasalready reached the gauge noise when using NHP = 24. This means that the σ-termcan be estimated without the need of TSM. The situation is very different for gA, be-cause the error drops with the number of noise vectors, where even with NLP = 500 ithas not yet converged.

In Fig. 4.5 we show the contribution to the axial charge from the strange quark. Weshow the absolute error as we increase NHP for a fixed NLP = 300. When NHP > 20for TSM we observe that there is no bias.

33

5 10 15 20 25NHP

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Abs.E

rr. g

s A

TSM NLP =300No TSM

Figure 4.5: The absolute error of gsA as we increase the number of HP noise vectors. Two setsof points are shown. The red squares are for NLP = 0, while blue circles are for NLP = 300.

4.3 Various dilution schemes

As we have already explained the noise vectors must obey Eqs. (4.1) and (4.2). Wehave chosen noise vectors generated according to Z4 for all time-slices. Hence, the loopcan be constructed for all time-slices, according to

L(t; Γ) =∑~x

Tr[G(x;x)Γ

]. (4.11)

The stochastic error for some quantities like the axial charge dominates the signal andmakes our estimate unreliable. From Eq. (4.2) we know that the outer product of thenoise vectors must become a delta function as Nr → ∞. When we truncate the sumwe introduce uncertainties to the diagonal elements of the all-to-all propagator dueto mixing with off-diagonal contributions. In order to reduce the contamination fromthese uncertainties we either increase the number of noise vectors or perform dilutionto reduce the density of the noise vectors.

The most promising kind of dilution is time-dilution, where we set all the entriesof the noise vector to zero, except a specific time-slice. The diluted vector is given as

ξt′

r = ξr δ(t− t′), (4.12)

and the loop can be calculated for time-slice t′ only. As we mentioned, neighbouringpoints of the noise vector contribute to the stochastic error, but when we perform timedilution t′−1 and t′+1 do not pollute the loop at t′. The next to neighbouring points canalso contribute to the noise but the contribution decays exponentially as the distancefrom t′ increases. If we fill the entries for a few time-slices, which are far away fromeach other we can produce the loop at several time-slices without extra inversions. Thismethod is called the coherent method [48, 49] and it can reduce the stochastic errorby increasing the statistics without additional cost. Note that the coherent methodcan also be used without stochastic noise vectors, e.g in the sequential method. Thedisadvantage of time-dilution is that for one-derivative operators where we need theloop at neighbouring sites we need additional inversions.

Besides time-dilution we can also perform dilution in the spatial volume. Onepossibility is to decompose the spatial volume into even and odd points. This scheme

34

guarantees that the closest neighbours do not pollute our calculation. Another optionis to divide the spatial cube into smaller cubes with side much smaller than the latticeextent. This approach can reduce further the noise because it discards points which arefurther from the entry points. One more sophisticated method to reduce the stochasticvariance is the hierarchical probing [50]. It relies on the fact that the Dirac operator isa sparse matrix with a special form due to the derivative operator. The elements of thepropagator decay exponentially as we increase the distance due to this special form.Probing is a technique exploiting this sparsity in order to create noise vectors withspecial structure. These improved noise vectors can significantly reduce the stochasticerror for specific operators.

Until now we discussed the dilution schemes in the volume subspace, but the noisevector also consists of spin and color [33] subspaces. Therefore, we can also considerspin or/and colour dilution. The noise vector in this case can be written as

ξr(x)a1α1= ξr(x)aαδ

aa1δαα1 . (4.13)

In Eq. (4.13) we perform full dilution in both subspaces, but we need to invert forevery spin and color index, so we need 12 times more inversions in order to calculatethe disconnected quark loops. In Fig. 4.6 we show the absolute error of the disconnected

5 15 25 35

∆[D

(t=

3)] Color

5 15 25 35

∆[D

(t=

3)] Spin

5 15 25 35

∆[D

(t=

3)] Even Odd

5 15 25 35

200 400 600 800 1000 1200 1400 1600 1800 2000

∆[D

(t=

3)]

# Inversions

Cubic

Figure 4.6: From top to bottom we show results obtained using color, spin, even-odd andcubic dilution. We show the absolute error on the disconnected part of the η′ correlator D(t)at t/a = 3 as a function of the number of inversions for 200 configurations.

part of the η′ correlator D(t) at t/a = 3 for various dilution schemes. The error whichreceives contributions from both gauge and stochastic error decreases as we increase thenumber of the noise vectors, and thus all dilution schemes converge to the gauge error.For this quantity the even-odd dilution performs better compared to other dilutionschemes however this is not always the case since the improvement depends on theobservable.

35

4.4 Hopping Parameter Expansion (HPE)

There are also other methods we can combine with the dilution to reduce the noise,like the hopping parameter expansion. The idea is to expand the inverse of the WilsonDirac operator as a series of hopping terms. For instance, for the twisted mass fermionformulation the operator is given by

M = −κH + (11 + 2κµγ5τ3), (4.14)

where H is the hopping term. We define A ≡ κH and B ≡ (11 + 2κµγ5τ3), thus the

inverse matrix can be written as

M−1 =1

B − A =1

B(1−B−1A)=

1

(1− AB−1)B, (4.15)

and B−1 can be computed analytically in the case of twisted mass fermions as,

B−1 =1− 2κµγ5τ

3

1 + (2κµ)2. (4.16)

The inverse matrix can be expanded in terms of B−1A,

M−1 = B−1 −B−1AB−1 + (B−1A)2B−1 − · · · (4.17)

Because this is an infinite series we can rewrite Eq. (4.17) in the form,

M−1 = B−1 −B−1AB−1 + (B−1A)2B−1 − (B−1A)3B−1 + (B−1A)4M−1 (4.18)

where the rest sum is replaced by M−1. The first four terms in the expansion canbe computed exactly without any considerable cost but the fifth term is computedstochastically as,

Tr[Γ(B−1A)4M−1

]=

1

Nr

Nr∑r=1

Tr[Γ(B−1A)4φr ξ

†r

]+O

(1√Nr

). (4.19)

The idea is to calculate the first four terms precisely in order to reduce the variancewhich comes from the fifth term. The first term in Eq. (4.18) does not involve thegauge links and thus can be computed once. For the case when Γ in Eq. (4.19) isproportional to 11 or γ5 the first term gives a non-zero contribution. The rest of theterms involve the hopping matrix, which is traceless, so only the terms with even powerswill survive for ultra-local operators. Moreover, if Γ is proportional to 11 or γ5 also thethird term is zero. For one-derivative operators, which include a hopping term, onlythe odd powers contribute. In any case, since these terms can be computed in advancerequiring no inversion, they do not incur to any significant computational overhead.

4.5 The one-end trick for disconnected quark loops

The one-end trick is another method we can employ to reduce stochastic noise. Thismethod can be applied to calculate two-point functions but also to estimate discon-nected quark loops. For instance, for the simplest case which is the pion, the zeromomentum two-point function is given by,

C(t) =∑~x

Tr[Gu(x; 0)γ5Gd(0;x)γ5

], (4.20)

36

where we fix the source position and sum only over the spatial coordinates of the sink.In the more general case where we have the all-to-all propagator, the two-point functioncan be computed from a double sum over the volume

C(t, t0) =∑~x,~x0

Tr[Gu(x;x0)γ5Gd(x0;x)γ5

], (4.21)

reducing significantly the noise-to-signal ratio. In practice this is not feasible becausethe all-to-all propagator can not be computed exactly, but we can use stochastic meth-ods to estimate this double sum appearing in 4.21.

The one-end trick [51–53] effectively transforms a double sum over the volume to asingle sum at the price of introducing stochastic noise to our results. We consider theexpression

1

Nr

∑r

∑~x

φ†u;r(~x, t; t0)φu;r(~x, t; t0) =

=∑~x,~x′0,~x

′′0

Gu(x;x′0)∗abµν Gu(x;x′′0)acµκ1

Nr

∑r

ξ∗bν (x′0)rξcκ(x′′0)r

=∑~x,~x′0,~x

′′0

Gu(x;x′0)∗abµν Gu(x;x′′0)acµκδbcδνκδ(~x

′0 − ~x′′0)

=∑~x,~x′0

Tr[|Gu(x, x0)|2

](4.22)

where we used the properties of the noise vectors to transform the double sum over theall-to-all propagator to a single sum, which involves only the solution vectors φu;r. In

0.2

0.3

0.4

0.5

0.6

0.7

0.8

2 4 6 8 10 12 14 16

mef

f(ts)

ts/a

πρ

Figure 4.7: The effective mass for the π (filled red squares) and the ρ (filled green circles)mesons using the one-end trick with 3 noise vectors.

Fig. 4.7 we present results for the effective mass of the pion and rho meson using theone-end trick. As can be seen, the error is very small even with 3 noise vectors due tothe effective double sum over the volume.

The one-end trick can significantly reduce the error but it needs the product ofpropagators thus it is not applicable for the disconnected quark loops computed by thetrace of a single propagator. For the twisted mass fermion formulation there is a way

37

to rewrite the sum of propagators into a product. Consider the Dirac matrices Mu andMd. From the properties of the twisted mass action we have

Mu −Md = 2iµγ5. (4.23)

The subtraction of the two propagators can also be written in the form,

Gu −Gd = Gu(Md −Mu)Gd (4.24)

where G M = 11 , and by using Eq. (4.23) we can transform the subtraction ofpropagators into a multiplication, hence

Gu −Gd = −2iµGuγ5Gd. (4.25)

In order to calculate the disconnected quark loop, we take the trace

Lu−d(t; Γ) =∑~x

(Tr[Gu(x;x)Γ

]−Tr

[Gd(x;x)Γ

])= −2iµ

∑~x,y

Tr[Gu(x; y)γ5Gd(y;x)Γ

](4.26)

where Γ depends on the γ-structure of the current that couples to the quark loop andthe single sum and the subtraction transform to a double sum and multiplication. Thenew form of Eq. (4.26) allows us to apply the one-end trick [46, 54], as follows

Lu−d(t; Γ) = −2iµ

Nr

∑r

∑~x

Tr[φ†r(x)γ5Γφr(x)

]=

∑~x

(Tr[Gu(x;x)Γ

]− Tr

[Gd(x;x)Γ

])+O

(1√Nr

). (4.27)

From now on we will refer to the expression in Eq. (4.27) as the standard one-endtrick. For the special case, where we want to calculate the scalar disconnected quarkloop in the physical basis, in the twisted basis we have

Lu+dpb (t; Γ = 1) =

∑~x

(Tr[Gu(x;x)

]+ Tr

[Gd(x;x)

])=

∑~x

(Tr[Gtmu (x;x)iγ5

]− Tr

[Gtmd (x;x)iγ5

])= Lu−dtm (t; Γ = iγ5τ

3) (4.28)

and thus the standard one-end trick is applicable.There are two main advantages arising from the application of the standard one-

end trick for twisted mass fermions. Firstly, when we transform the subtraction intomultiplication a factor of µ emerges, which suppresses the fluctuations as it is a smallparameter. The second advantage is that Eq. (4.27) offers an effective double sum overthe volume, which means that the signal-to-noise goes as V/

√V 2 instead of 1/

√V thus

improving dramatically the signal. In Eq. (4.26) there are two sums over the volume,one is only for the spatial volume and the other is over the whole 4-D volume. Thesecond sum implies that the stochastic noise vectors must be volume sources withoutany dilution. On the one hand, this means that we cannot further reduce the noise usingdilution methods, but on the other hand it allows us to compute disconnected quarkloops for all the time-slices at once. For one-derivative operators where we need thecovariant derivative in the time direction it is automatically applicable. Furthermore,

38

the disconnected quark loops from various time-slices can be combined with two-pointfunctions from several source positions to reduce the gauge and stochastic variancesimultaneously.

The standard one-end trick can be applied only when the insertion operator in thetwisted basis carries a τ 3, acting in flavour space. For the other cases, we can use theidentity

Mu +Md = 2DW (4.29)

where DW is the massless Wilson-Dirac operator. The addition of the two propagatorcan be written as,

Gu +Gd = Gu(Mu +Md)Gd

and by using Eq. (4.29)Gu +Gd = 2GuDWGd.

Then the loop is computed from the trace as,

Lu+d = 2 Tr[GuDWGdΓ

]or

Lu+d = 2 Tr[Gu(x; z1)DW (z1; z2)γ5G

†u(x; z2)γ5Γ

].

For twisted mass fermions we can write these two relations

Du(x; y)γ5DW (y; z) = DW (x; y)γ5DW (y;x) + 2iµDW (x; y)

andDW (x; y)γ5Du(y; z) = DW (x; y)γ5DW (y;x) + 2iµDW (x; y)

thusDu(x; y)γ5DW (y; z) = DW (x; y)γ5Du(y; z),

multiplying both sides by Gu

γ5DWGu = GuDWγ5.

Then we can write

Lu+d(t) = 2∑~x

∑z1,z2

[(γ5)αβDW (x; z1)abβγGu(z1; z2)bcγδG

∗(x; z2)acεδ(γ5Γ)εα

]and by inserting the unity

Lu+d(t) = 2∑~x

∑z1,z2,z′2

[(γ5)αβDW (x; z1)abβγGu(z1; z′2)bc

′

γδ′δ(z′2−z2)δcc

′δδδ′G

∗u(x; z2)acεδ(γ5Γ)εα

].

Using the properties of the noise vectors we can write

Lu+d(t) = 2∑~x

∑z1,z2

[(γ5)αβDW (x; z1)abβγGu(z1; z′2)bc

′

γδ′

∑r

ξr(z′2)c′

δ′ξ∗r (z2)cδG

∗u(x; z2)acεδ(γ5Γ)εα

]and using that

φr = Gξ,

thusLu+d(t) = 2

∑r

∑~x

∑z1

[(γ5)αβDW (x; z1)abβγφu;r(z1)bγφ

∗u;r(x)aε (γ5Γ)εα

]39

and the final form is

Lu+d(t; Γ) =2

Nr

∑r

∑~x

Tr[φ†r(x)γ5Γγ5DWφr(x)

]=

∑~x

(Tr[Gu(x;x)Γ

]+ Tr

[Gd(x;x)Γ

])+O

(1√Nr

), (4.30)

which we call the generalized one-end trick. We can observe that in the generalizedone-end trick there is no µ factor which means that no automatic suppression of thestochastic variance is achieved.

4.6 Simulation details and the QUDA package

The evaluation of the disconnected quark loops will be done by analysing an ensemble ofNf = 2+1+1 dynamical fermions. The pion mass ismπ = 373 MeV and the strange andcharm quark masses are tuned to their physical values in order to reproduce kaon andD meson physical masses [55]. To obtain reliable estimates of the disconnected quarkloops a large number of statistics is required and thus we employ the computationalefficiency of GPUs to speed-up the inversions and contractions. The stochastic methodimplies that for each inverted noise vector we perform the contractions on the fly.

The QUDA package [56–59] is a library which is written using the CUDA API. Wemake use of these CUDA kernels in order to perform inversions and contractions onGPUs. We have developed new kernels [60], which carry out efficiently the calculationof disconnected quark loops for hundreds or even thousands of noise vectors. Detailsof the code implementation are given in Appendix. F.

After the contractions we Fourier transform the loops from position to momentumspace,

L(t, ~q) =∑~x

Tr[ΓG(x;x)

]e+i~x·~q. (4.31)

The Fourier transform is performed using the cuFFT library, where an improved fastFourier kernel takes care of the transformation.

The QUDA library allows for inter- and intra-node multi-GPU communication usingthe Message Passing Interface (MPI) library. Most GPU nodes consist of 1-4 GPU cardswith device memory from 6-12 Gbytes. For lattice sizes smaller or equal to 323×64 wecan use one GPU because 6 Gbytes of device memory are enough for the inversions andcontractions. In recent days, most simulations are being performed for bigger lattices,such as 483 × 96 or even 643 × 128, thus there is a demand for multi-GPU codes.

In parallel implementations, it is important to know the performance behaviour as afunction of the number of GPUs. There are two kinds of scaling that can be considered,namely i) weak scaling : measures the total performance when the local lattice is keptfixed as we increase the number of GPUs and ii) strong scaling : where the local latticesize decreases with the number of GPUs such that the global problem size does notchange. In Fig. 4.8 we show strong and weak scaling as a function of the number ofGPUs. Strong scaling is good for few GPUs, with about 90% increase in performancewhen adding one more GPU. The scaling is good until 8 GPUs, but by increasing thenumber beyond 8 GPUs we observe a rapid drop in performance. We thus restrict topartition sizes of 8 GPUs and less for our runs, unless memory requirements requiresus to work on larger partition sizes. This drop in performance can be understood as anunderutilization of the CUDA cores due to the smaller local volume. On the other hand,weak scaling is perfect because QUDA overlaps computations with communications.

40

0 2 4 6 8 10 12 14 16#GPUs

0

200

400

600

800

1000

1200

GFlo

ps

GPU Strong Scalingdouble, 323 ×64single, 323 ×64half, 323 ×64

0 2 4 6 8 10 12 14 16#GPUs

0

500

1000

1500

2000

2500

GFlo

ps

GPU Weak Scalingdouble, 244

single, 244

half, 244

Figure 4.8: Left Panel: Strong scaling of the multi-GPU conjugate-gradient solver using a323 × 64 lattice for three floating point precision arithmetics. Right Panel: Weak scaling fora local volume of 244 [46].

In modern NVIDIA architectures, such as the Kepler architecture, where the tex-

Figure 4.9: Strong scaling comparison between Fermi and Kepler architectures [61]. Theblack circles show the performance as a function of the number of GPUs when we use texturereferences for Fermi architectures, whereas the red squares show the performance when weuse texture objects introduced in the Kepler architecture.

ture binding is achieved using texture objects instead of texture references there is asignificant improvement in the strong scaling, as demonstrated in Fig. 4.9.

There are also other packages which utilize the power of GPUs for lattice gauge the-ories. The cuLGT package [62, 63] features Landau, Coulomb and Maximally Abeliangauge fixing with Overrelaxation. The strong scaling in Fig. 4.10 presents the sameperformance drop after a specific number of GPUs whereas the weak scaling is perfect.

41

0

500

1000

1500

2000

2500

3000

3500

0 4 8 12 16

Sus

tain

ed G

Flo

ps

Number of GPUs

SPMPDP

0

500

1000

1500

2000

2500

3000

3500

0 4 8 12 16

Sus

tain

ed G

Flo

ps

Number of GPUs

SPMPDP

Figure 4.10: Left: strong scaling for double (green squares), single (red circles) and mixedprecision (blue triangles) arithmetic. Right: weak scaling [62, 63]. The solid lines show thetheoretical scaling, whereas the dashed lines show the experimental scaling as a function ofthe number of GPUs.

4.7 Extracting matrix elements from lattice three-

point functions

One of the main advantages of the one-end trick for twisted mass fermions is thefact that, we obtain the disconnected quark loops for all insertion time-slices, sincethe noise vectors are non-zero on the whole lattice. Therefore, we can multiply withtwo-point functions to create the appropriate three-point functions with the insertiontime-slice but also the sink time-slice allowed to vary. This is very important becausewe can probe the excited states contamination by testing different source-insertion andsource-sink time separations. This feature enables us to use the summation method inaddition to the plateau method with no extra computational cost.

The summation method [64, 65] was proposed a long time ago and it has beenrevisited for the study of the nucleon axial charge [66]. In both plateau and summationapproaches, one constructs ratios of three- to two- point functions in order to cancelunknown overlaps and time-dependent exponentials. In the large time limit, the excitedstates contributions are negligible and the ratio can be used to isolate the matrixelement we are interested in.

A number of interesting quantities are extracted from lattice QCD at zero momen-tum transfer. In this special case the ratio takes the form

R(tins, ts) =G3pt(tins, ts)

G2pt(ts). (4.32)

This ratio becomes time independent, in the large time limit. In practice one can nottake an arbitrarily large source-sink time separation on the lattice because the noiseincreases rapidly. We must thus ensure the fastest possible suppression of the excitedstates. The leading terms of the ratio are,

R(tins, ts) = RGS +O(e−∆Etins

)+O

(e−∆E(ts−tins)

)(4.33)

where RGS is the matrix element of interest, which is time independent, and the otherterms are the excited states contributions which decay exponentially with the energygap ∆E between the ground state and the first excited state.

In the plateau method, one chooses the source-sink time separation as well as theinsertion-sink time separation to be large enough to suppress contributions from excited

42

states. In the summation method we perform a sum over all the insertions betweenthe source and the sink to obtain

Rsum(ts) =

tins=ts−1∑tins=1

R(tins, ts) = tsRGS + a+O(e−∆Ets

). (4.34)

As Eq. (4.34) shows the exponential contributions coming from the excited states decayas e−∆Ets as opposed to the plateau method where the excited states are suppressedlike e−∆E(ts−tins) or e−∆Etins . The advantage of the summation method is that thesuppression of excited states is better than the plateau method, because ts ≥ ts − tins.The main computational drawback of the summation method is that one requires theknowledge of three-point function for all the insertion times and for many sink times.Another drawback is that we must fit in two parameters instead of one and this increasesthe uncertainty of the extracted value of RGS.

4.8 Comparing methods for the calculation of dis-

connected contributions

In order to illustrate the extraction of matrix elements from the ratio of Eq. (4.32)we choose two quantities with very different behaviour. Regarding the number ofstochastic noise vectors needed for the computation of disconnected quark loops weconsider the nucleon σ-term where the stochastic error can be suppressed with fewnoise vectors and the other quantity is gA, which belongs to the class of observablesrequiring a large number of noise vectors in order to obtain reliable results for thedisconnected contributions.

These two quantities also have a different behaviour regarding excited states contri-butions. The σ-term is found to have large excited states contributions [67, 68] while gA[69–71] has been shown to be less affected, although the degree of contamination maydepend on the pion mass [72–74]. In particular, we note that the summation method asapplied in the extraction of isoscalar gA in Ref. [74] led to agreement with the physicalvalue after performing chiral extrapolation, while in Ref. [72] it was shown that verynear to the physical point the summation method produces a value incompatible withthe experimental value, with the authors suggesting thermal effects as a possible expla-nation [73]. In our work using very high statistics to analyse gA, we do not detect anysignificant excited state effects. In contrast, for the σ-term we found large excited stateeffects. In Fig. 4.11 we study the excited states contamination for several source-sinktime separations and we observe similar behaviour for the connected and disconnectedcontributions to σπN .

In the following sections we evaluate the light, the strange and the charm discon-nected contributions to both, σ-term and gA, using the standard and generalized one-end trick respectively. In addition, we calculate the strange quark contribution bothwhen we implement the HPE and without, and compare the resulting error. Regard-ing the renormalization of σ-terms, the twisted mass formulation has the advantage ofavoiding any mixing, even if we use Wilson-type fermions. For the case of the isoscalaraxial charge, the renormalization involves mixing from the light, strange and charmquarks. Perturbatively, it was shown that for the tree-level Symanzik improved gaugeaction the effect of the mixing is small [75]. We will neglect it in this work, since weexpect the mixing to also be small for the Iwasaki action used for the B55.32 ensemble.In the following analysis we will renormalize gA using the perturbatively corrected [76]renormalization constant ZA.

43

10 5 0 5 10(tins ts/2)/a

0

50

100

150

200

N [M

eV]

SMts =10ats =12a

ts =14ats =16ats =18a

10 5 0 5 10(tins ts/2)/a

0

5

10

15

20

25

30

35

N [M

eV]

SMts /a =14ts /a =16

ts /a =18ts /a =20

Figure 4.11: The connected (top) and the disconnected (bottom) contribution to the nucleonσπN . The gray band is the result obtained from the summation method while the colouredbands are results using the plateau method [46]. The ratio is shown for several source-sinktime separations as a function of the source-insertion time separation.

4.8.1 Performance of the Truncated Solver Method

Let us now turn to the study of the efficiency of TSM. For the σ-terms we expect thatTSM is not essential, but nevertheless we will use it since all the loops can be extractedat once when the low-precisions inversions are performed. The σ-terms are defined by,

σπN = µl〈N |uu+ dd|N〉 (4.35)

σs = µs〈N |ss|N〉 (4.36)

σc = µc〈N |cc|N〉 (4.37)

where σπN has also a connected contribution for the nucleon, whereas σs and σc haveonly disconnected contributions. For the calculation of disconnected quark loops weuse the Osterwalder-Seiler action [77], in which the strange and charm quarks are eachtreated as a degenerate flavor doublet, i.e. χs = (s+, s−) and χc = (c+, c−) usingthe same action as for the light quarks Eq. (2.42), but with µ replaced with µs orµc accordingly. The values of µs and µc are tuned in order to reproduce the kaonand D-meson masses of the unitary theory. In Fig. 4.12 we compare the results with

44

10 5 0 5 10(tins ts /2)/a

505

101520253035

N[M

eV]

No TSM 24HPTSM 8HP+200LP

Figure 4.12: Comparison of σπN obtained with and without TSM using ∼ 60000 measure-ments. With the blue filled circles we show the results without TSM, whereas for the red filledsquares we employ TSM with NLP = 200. The coloured bands show the extracted value withits jackknife error.

and without implementing the TSM method combined with the one-end trick, whenonly high precision inversions are used NHP = 24, while for the TSM method we useNLP = 200 and NHP = 8. When we use TSM we have approximately ten times morenoise vectors, and thus we expect smaller errors. However, this is not observed becausethe error is bounded by the gauge error. We choose NLP = 200 although it yields thesame error and is achieved with 34% reduced computational cost.

As the quark mass increases, the computational cost for TSM for similar errorsbecomes comparable to that of using only HP inversions. In Fig. 4.13 we show resultsfor the strange σ-term. We find that the errors when using TSM are comparablewith when using only HP inversions with the same computational effort. For σc theinversions become very fast and thus there is no need for TSM. Specifically TSM needs5 times more computational effort as compared to only HP inversions to achieve only a33% reduction in error. Therefore, for heavy quark masses the inversion becomes fastbut the contraction time remains the same independently of the quark mass and thuswe spend most of the computational time on contractions instead of inversions.

Then, the next observable is the isoscalar gA, which has a very different convergencepattern compare to σ-term. In Fig. 4.14 we compare results both when we use TSMand when we do not for the same number of noise vectors we used for the computationof the σ-term. It is apparent that the Truncated Solver Method provides a two-foldreduction in the stochastic error for about 66% reduced computational cost. For thestrange and charm quark, shown in Figs. 4.15 and 4.16 respectively, we still observe animprovement in the results when using the TSM. Specifically for gcA there is a four-foldreduction in the error for 5 times more computational cost but nevertheless the gaincompensates the additional effort. The conclusion for gA for all three flavours, is thatthe TSM can significantly reduce the error.

45

10 5 0 5 10(tins ts/2)/a

0.00

0.01

0.02

0.03

0.04

0.05

s[G

eV]

No TSM 24HP

TSM 8HP+300LP

6 4 2 0 2 4 6(tins ts /2)/a

0.05

0.00

0.05

0.10

0.15

0.20

c[Ge

V]


Figure 4.13: Comparison of the performance of the TSM for σs (top) and σc (bottom). Thenotation is same as that of Fig. 4.12.

6 4 2 0 2 4 6(tins ts/2)/a

0.4

0.2

0.0

0.2

0.4

g A

No TSM 24HP

TSM 8HP+200LP

Figure 4.14: Results of the ratio of disconnected contributions to the isoscalar axial chargefrom light quarks. The notation is the same as that of Fig. 4.12.

46

6 4 2 0 2 4 6(tins ts /2)/a

0.20

0.15

0.10

0.05

0.00

0.05

0.10

0.15

gs A


Figure 4.15: Results for ratio gsA when using TSM compared to HP inversions only. Thenotation is the same as that of Fig. 4.14

6 4 2 0 2 4 6(tins ts /2)/a

0.20

0.15

0.10

0.05

0.00

0.05

0.10

0.15

gc A


Figure 4.16: Results for ratio which yields gcA when using TSM compared to HP inversionsonly. The notation is the same as that of Fig. 4.14.

47

4.8.2 Combining time-dilution with HPE using TSM

In this subsection, we assess the performance of time-dilution in the case where wecombine it with the HPE. As we have already mentioned, we cannot combine time-dilution with the one-end trick, hence we study them independently, in combinationwith the TSM. The comparison is performed for the same observables we used in theprevious subsection. Here, we will limit ourselves only in the strange quark sector, inan effort to conserve the computational resources at our disposal. We will also studythe case when only time-dilution is performed. For the one-end trick the applicationof the HPE is no longer straightforward and it will not be considered. The hoppingparameter expansion is expected to behave better for heavy masses where the hoppingterm becomes less important compared to the mass term. As already explained, theoverhead in computer time for computing the hopping terms is insignificant, since itonly needs few applications of the Wilson-Dirac hopping matrix.

8 10 12 14 16ts /a

0.00

0.01

0.02

0.03

0.04

0.05

0.06

s[GeV

]


8 10 12 14 16ts /a

0.00

0.01

0.02

0.03

0.04

0.05

s[GeV

]


Figure 4.17: Results for the ratio from which σs is extracted with respect to the sink time whenwe use NHP = 24, NLP = 0 (blue filled circles) and TSM with NLP = 300 and NHP = 8 (redfilled squares). Top: Results using only time-dilution. Bottom: Results using time-dilutionwith the HPE. The total number of measurements is ∼ 20000.

In Fig. 4.17 we present the ratio from which we extract σs with and without HPEand TSM. It is crucial to mention here that, due to the time-dilution, we obtain theloop for one or only few time slices, so the study of the excited states contributionsrequires more computational resources. The conclusion drawn from Fig. 4.17 is thatthe TSM improves the errors by a factor of two. The hopping parameter expansiondoes not seem to bring any improvement for this observable.

48

The next observable we consider is gsA which is more demanding in terms of noisevectors required. In Fig. 4.18 we show the equivalent results as in Fig. 4.17 but

8 10 12 14 16ts /a

0.3

0.2

0.1

0.0

0.1

0.2

0.3

gs A


8 10 12 14 16ts /a

0.10

0.05

0.00

0.05

gs A


Figure 4.18: Comparison for the ratio from which gsA is obtained between using HP only andusing TSM in combination with time-dilution. The notation is the same as in Fig. 4.17.

for gsA. We observe an inconsistency between TSM with NLP = 300, NHP = 8 andwhen using NHP = 24, which means that the latter results are still biased due to thestochastic error. In contrast, the application of HPE improves the situation makingthe results compatible. The conclusion of this study is that employing TSM and HPEcombined with the time-dilution seems essential in order to obtain reliable results forgA. To make things clear we compare directly the results on gsA and σs side by sidewhen employing TSM and time dilution to what is obtained when using TSM withHPE and time dilution. As can be seen from Fig. 4.19 the σ-term does not improve byincluding HPE, whereas by contrast the results on gsA show a clear improvement thatcomes without any additional computational effort.

4.8.3 Comparing efficiency and performance of various meth-ods

We now turn to the more technical aspects in order to compare the efficiency of thevarious methods. It is important for this discussion to clarify that the generation ofstochastic sources, the inversions and all the contractions are carried out on GPUsso that the communication between CPU and GPU is kept minimal. In order toaccomplish this, contractions are done on the fly on GPU and then the solution vectorsare discarded in order to free memory for the next set of inversions. The only datatransfer done between the device memory and the host memory is right at the endwhen we store the contractions to disk.

49

8 10 12 14 16ts /a

0.00

0.01

0.02

0.03

0.04

0.05

s[GeV

]

Time Dilution + HPETime Dilution

8 10 12 14 16ts /a

0.10

0.05

0.00

gs A

Time Dilution + HPETime Dilution

Figure 4.19: Comparison of results for the ratio from which σs (top) and gsA (bottom) areextracted, when applying time dilution (red filled squares) and when time dilution is combinedwith HPE (blue filled circles).

Even with such a setup, for quark masses larger than the strange mass the dif-ferences in computer time between high and low precision inversions become smallcompared to the time spent for the contractions. This is due to the fact that the pre-and post-processing computational costs are independent of the quark mass and there-fore more time is required for carrying out the TSM where an order of magnitude morenoise vectors are used. Therefore, this reduces the efficiency of the TSM for the case ofheavy quarks. In Table 4.1 we give a summary of the computer time required for the

Method Quark sector Rultra−localHP/LP (cost) ROne−Deriv.

HP/LP (cost)

One-end trick Light ∼ 26.7 ∼ 10One-end trick Strange ∼ 16.9 ∼ 5.8One-end trick Charm ∼ 2.9 ∼ 1.4Time-Dilution Strange ∼ 20.7 -

Time-Dilution + HPE Strange ∼ 19.1 -

Table 4.1: The ratio of computational cost between HP and LP inversions in order to calculatethe disconnected loops when we employ TSM, where ultra-local are operators of the generalform ψΓψ and one-derivative ψΓDψ.

computation of fermion loops within the various methods. We give the ratio RHP/LP

which is the computer time required to compute a fermion loop for one noise vectorusing HP over the corresponding time when using LP. This ratio takes into accountthe total time needed, i.e. it includes the creation of the noise vectors, the inversion,the contractions and taking the traces. A large value of this ratio indicates that the

50

TSM is more efficient, since we can use more LP noise vectors to reduce the stochasticnoise at low cost. When the ratio is near unity, it means that the TSM is no longeradvantageous, since instead of inverting to LP we can switch to HP with a very smalloverhead. In the case of the strange quark loops with ultra-local operators definedas ψΓψ the ratio is much larger than unity when we use time-dilution plus HPE andthus TSM is very efficient. When the one-end trick is used the ratio is large for bothlight and strange quarks and for both ultra-local and one-derivative operators. As thequark mass increases further and approaches the charm quark mass, TSM becomescomparable with using only HP inversions.

We note that the one-derivative operator has not been computed using time-dilutionbecause the time derivative needs disconnected quark loops from two neighbouringtime-slices, which triples the computational cost. Thus, for derivative operators theone-end trick is preferable.

Our main conclusion from this study is that TSM is the method of choice for thelight and the strange sectors. For the charm sector even if it is marginally advantageous,one may still use it since there is no drawback. For observables like the axial charge,the TSM can significantly reduce the stochastic variance, while for the σ-term, becausethe stochastic noise can be easily suppressed, it is not necessary. However, if one wantsto study all the observables at once without additional inversions, TSM is the mostappropriate approach, at least for the range of quark masses studied here.

4.8.4 Time-dilution plus HPE versus the one-end trick, usingTSM

In the previous subsections, we extensively studied combining TSM with both time-dilution and the one-end trick. In addition we assessed the combination of time-dilutionwith and without the HPE and we showed that HPE reduces the error without consid-erable additional computational effort. In this section, we will evaluate the performancewhen using one-end trick, and when using time-dilution plus HPE when we employ theTSM.

In Fig. 4.20 we show ratios from where we extract σs and gsA. For the case oftime-dilution, as we calculate the loop for only one time-slice, we only have a singlesource-insertion time separation. For σs we compare the standard one-end trick withthe time-dilution plus HPE, while for gsA we compare the generalized one-end trick withthe time-dilution plus HPE. As can be seen for σs, the standard one-end trick yieldssmaller errors than time-dilution plus HPE for the same number of measurements.

On the other hand, for gsA time-dilution yields slightly smaller errors. However, inthe case of the one-end trick one obtains the fermion loops at all time-slices withoutany further inversions, while for time-dilution we can create loops for 4 time-slicesusing the coherent method. As a consequence, using the one-end trick we constructthe disconnected three-point function for all the insertion time-slices and for all source-sink time separations. Due to this advantage we can perform a constant fit to a plateauand vary both the insertion and sink time-slices, in contrast with the time-dilution wesearch for a plateau as we vary only the insertion-sink time-slice. Fitting the plateaufor gsA yields a result with the same error as that obtained when using the time-dilutionwith HPE. Thus, this comparison shows that both standard and generalized one-endtrick are preferable for the calculation of disconnected quark loops.

An additional advantage of the one-end trick is that, having results for multiplesource-sink time separations allows us to assess excited states contributions, as well asapplying the summation method without requiring additional inversions. In contrast,

51

10 5 0 5 10

0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

s[G

eV]

TD+HPE ts= 16a

(tins ts/2)/a

8 6 4 2 0 2 4 6 8(tins ts/2)/a

0.10

0.05

0.00

0.05

gs A

TD + HPEts =14a

Figure 4.20: Results for the ratio of σs (top) and gsA (bottom). The blue circles show resultsusing the one-end trick and the light blue error band shows the extracted value with its jack-knife error. The magenta error band shows the extracted value when using time-dilution plusHPE. The TSM with NHP = 24 and NLP = 300 is used in both methods with 18628 statistics.

time-dilution needs additional inversions for every insertion time slice we want to add.Furthermore, with the one-end trick, since we have the disconnected quark loops forall the time-slices, we can multiply them with two-point functions evaluated at severalsource positions because the quark loop does not depend on the source position. Theadvantage of using multiple source-sink time separations can be seen in Fig. 4.21 wherewe probe the excited states contribution as we increase the source-sink time separation.As we can see, the summation method and the plateau method give compatible resultswhile the summation method gives bigger errors because we fit in two parametersinstead of one.

In Fig. 4.22 we perform the same analysis for the axial charge. Again the sum-mation method produces compatible results with the plateau method. In both cases,we computed two-point functions for 16 source positions per configuration on 2,300gauge-field configurations, where we average over proton and neutron and backwardand forward propagating nucleons, which results in total approximately 150000 mea-surements. In order to check that the range where we perform the linear fit for thesummation does not suffer from excited states, we vary the lower fit range and plot theresulting values in Fig. 4.23. As we can see, varying the lower point in the fit of theslope we obtain compatible results, while the error increases as we increase the lowerfit range as expected.

The summation method, which serves as an alternative way of extracting thehadronic matrix elements, can only be applied if we have multiple source-sink time

52

15 10 5 0 5 10 15(tins ts /2)/a

10

5

0

5

10

15

20

25

30

35

s [M

eV]

SM ts =14ats =16a

ts =18ats =20a

0 5 10 15 20 25ts /a

2

0

2

4

6

8

10

12

14

∑ t ins

R(t in

s,t s

)

Figure 4.21: Comparison of the plateau (above) and summation method (below) for σs. Usingdifferent colours and symbols we assess the behaviour of the results for various source-sinktime separations. The grey band on the top panel shows the result from the summation methodextracted from fitting the slope of the summed ratio shown in the bottom panel.

separations, although a noticeable improvement in statistical accuracy is not observedwhen using the summation method. We consider it as an additional check of theground state dominance, especially for the case of the σ-term, where excited stateeffects appear to be large.

In our comparison between time-dilution with HPE and the one-end trick we lim-ited ourselves to the strange quark loops. However, we expect that the one-end trickperforms better for light quarks since HPE is better suited for heavier masses wherethe hopping term is less important. Another reason to favour the one-end trick methodis in the case of one-derivative operators. To compute such derivative operators in timedirection one requires the computation of fermion loops at three neighbouring time-slices, tripling the computational effort as compared to the effort needed for ultra-localoperators. For the one-end trick this requires no further effort since one obtains theloops for all time-slices.

53

8 6 4 2 0 2 4 6 8(tins ts /2)/a

0.05

0.04

0.03

0.02

0.01

0.00

0.01

gs A

SMts =8ats =10ats =12a

0 2 4 6 8 10 12 14 16ts /a

0.5

0.4

0.3

0.2

0.1

0.0

0.1

∑ t ins

R(t in

s,t s

)

Figure 4.22: The same as 4.21 but for gsA.

2 4 6 8 10 12 14 16 18 20ti /a

10

5

0

5

10

15

20

25

30

35

s [M

eV]

tf =21atf =22atf =23atf =24a

2 3 4 5 6 7 8 9 10 11ti /a

0.05

0.04

0.03

0.02

0.01

0.00

0.01

gs A


Figure 4.23: Results extracted from the slope of the summed ratio that yields σs and gsA as afunction of the lower fit range ti/a for various tf .

54

4.8.5 Summary of the performance for various methods

We summarize the outcome of our comparison for all the methods we tested, in Table4.2. The cost, in units of GPU-node seconds ( -2 GPUs per node- ), is given for the

Method Obs. Abs. Error OH Cost Cost × Error2

One-end trick σπN 4.3 MeV 64 2234 0.032One-end trick + TSM σπN 3.8 MeV 290 1471 0.027

One-end trick σs 5.1 MeV 65 754 0.019One-end trick + TSM σs 4.9 MeV 409 809 0.019

Time-dil. σs 13 MeV 31 745 0.126Time-dil. + TSM σs 7.5 MeV 281 710 0.040Time-dil. + HPE σs 8.0 MeV 34 750 0.048

Time-dil. + HPE + TSM σs 6.2 MeV 322 750 0.029One-end trick σc 95 MeV 65 144 1.30

One-end trick + TSM σc 61 MeV 409 692 2.57One-end trick gA 0.19 65 2234 80.6

One-end trick + TSM gA 0.081 409 1471 9.65One-end trick gsA 0.076 65 754 4.36

One-end trick + TSM gsA 0.023 409 809 0.43Time-dil. gsA 0.132 31 721 5.08

Time-dil. + TSM gsA 0.049 281 676 1.62Time-dil. + HPE gsA 0.040 34 725 1.16

Time-dil. + HPE + TSM gsA 0.024 322 692 0.40One-end trick gcA 0.076 65 144 0.83One-end trick gcA 0.0215 409 692 0.32

Table 4.2: Comparison of the computational cost [78] for the σ-terms and the axial chargesusing various methods for three quark sectors.

computation of the quark loops for one configuration, using NHP = 24, NLP = 0,and for NHP = 8 with NLP = 200 or NLP = 300 depending on the quark mass. Forfair comparison we use the same statistics, approximately (-20000-) measurements, fortime-dilution and the one-end trick. The sink was set to ts = 16a for the one-end trickand for the time dilution the insertion was fixed to tins = 8a. The column labelled asOH gives the overhead i.e the time for pre- and post-processing spent in generating thesources and performing the contraction. The column with the cost gives the total timewhen we add the inversion time as well. It can be seen, that the overhead time dependsonly on the number of noise sources and it becomes increasingly more important, asthe inversions become faster for heavier quark masses. The last column determines theefficiency of the methods. The smaller this number is, the better the performance ofthe method.

4.8.6 Review of the techniques

As we have seen, the computation of disconnected contributions has become feasibledue to the development of new techniques to reduce the gauge and stochastic noise,but also due to the increase in computational power and resources. In this chapterwe explored the recent methods for the calculation of disconnected quark loops. Theexploitation of GPUs is of particular importance, due to its efficiency in the evaluation

55

of disconnected quark loops using TSM, since GPUs can yield a large speed-up whenwe utilize half and single precision arithmetic for the LP inversions.

Among all the algorithms we assess, the one-end trick seems to perform better inmost of the cases by reducing the noise of the disconnected quark loops at the samecomputational cost for a number of observables. It also delivers the fermion loops forall the possible insertion times at no extra cost, so we can use the summation method inthe analysis. Furthermore, it enables us to compute one-derivative operators withoutadditional inversions, while time-dilution needs triple the time required as comparedto that needed for loops with ultra-local operators.

The TSM improves the performance of the one-end trick for quark masses no heavierthan the strange quark mass. For heavier masses, TSM becomes less effective and itcan be omitted, depending on the observable. In our case we observe a performancedegradation when we calculate σc, but a clear improvement for the axial charges. Inthe next chapter we will utilize the one end-trick combined with the TSM methodto calculate the disconnected contributions to various hadron observable using highstatistics.

56

Chapter 5

Disconnected contributions to hadronobservables

In recent days lattice QCD simulations are performed very close to or at the physicalpion mass. This allows us to study hadron structure with direct contact to experimentand impact to phenomenology. However, many observables were up to now computedby neglecting disconnected contributions, with the penalty of an undetermined sys-tematic error. The evaluation of disconnected quark loops is therefore of paramountimportance if we want to eliminate this source of systematic uncertainty. In this chap-ter we will present results on hadron observables including disconnected contributions,enables by the methods developed in the previous chapter. Numerical values for theresults are include in Appendix. G.

5.1 σ-terms from lattice QCD

Astronomical observations pointing to the existence of dark matter have led to the de-velopment of dedicated experiments to detect dark matter. Some of these experimentsrely on measuring the recoil of atoms when they are hit by a dark matter candidate.There is a class of dark matter models, which describe the interaction between a weaklyinteracting massive particle (WIMP, dark matter candidate) [79–83] and a nucleon me-diated by a Higgs boson. As a result, the scalar content (σ-term) of the nucleon is afundamental ingredient in determining the scattering amplitudes between a WIMP anda nucleon. In other words, the uncertainties of the scalar content translate into theaccuracy constrains enter experiments for beyond the standard model physics. Sinceit is well-known that the coupling between a Higgs boson and a quark is proportionalto the quark mass, it is important to measure how large scalar matrix elements of thenucleon are, for light, strange and charm quarks.

Lattice QCD can measure the parameters entering the relevant cross section, theso-called σ-terms of the nucleon,

σπN ≡ ml〈N |uu+ dd|N〉 and σs ≡ ms〈N |ss|N〉, (5.1)

where ml(ms) denotes the average light (strange) quark mass. To quantify the scalarstrange content (strangeness) of the nucleon we define the parameter yN , as:

yN ≡2〈N |ss|N〉〈N |uu+ dd|N〉 . (5.2)

The direct lattice QCD calculation of the above matrix elements is challenging fortwo main reasons: i) The numerator in Eq. (5.2) receives contributions from only

57

disconnected diagrams and thus requires improved techniques as well as large compu-tational resources in order to compute it. The denominator has both connected anddisconnected contributions. ii) The discretization breaks explicitly chiral symmetry,therefore it is difficult to treat iso-singlet operators from a mixing between the lightand the strange sector using non-perturbative approaches. As discussed in Chapter4, new methods and the increased computational resources enable us to evaluate thedisconnected contributions where we have shown that the one-end trick is particularlysuited for twisted mass fermions. In addition, the twisted mass formulation avoids thechirally violating contributions that are responsible for the mixing under renormaliza-tion [84].

5.1.1 Scalar content of the nucleon

In this section, we present a high statistics analysis for the scalar content of the nucleon.Specifically, we use approximately 150000 measurements for Nf = 2+1+1 twisted massfermions for an ensemble with pion mass mπ = 373 MeV, lattice spacing a = 0.082fm and 323 × 64 lattice size. The disconnected quark loops are computed using theone-end trick combined with the truncated solver method, and the two-point functionsare created for 16 source positions.

10 5 0 5 10(tins ts/2)/a

0.00

0.05

0.10

0.15

0.20

R(t in

s,t s

)→N

[GeV

]

SMts =10ats =12a


Figure 5.1: Results for the ratio from where we extract the connected part of σπN . Weshow results obtained for various source-sink time separations. The value extracted from theplateau method (purple band) is compared with that extracted using the summation method(grey band).

In Fig. 5.1 we show results for the connected contribution to the ratio from whichσπN is extracted. As we can see, this quantity suffers from large excited state effects,and therefore, one must increase the source-sink time separation ts, until the resultsconverge. We employ two methods to extract the σ-term, the plateau and the summa-tion method, where we find consistent results after a sufficiently large source-sink timeseparation.

In order to obtain the complete value of σ-terms we need the disconnected contri-butions from the light, strange and charm sectors. In Fig. 5.2 we show on the leftpanel the ratio for various source-sink time separations, as well as the result extracted

58

15 10 5 0 5 10 15(tins ts /2)/a

10

5

0

5

10

15

20

25

30

35

R(t in

s,t s

)→N

[MeV

]

SMts =14ats =16a

ts =18ats =20a

2 4 6 8 10 12 14 16 18 20ti /a

10

5

0

5

10

15

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35

N [M

eV]


15 10 5 0 5 10 15(tins ts /2)/a

10

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30

35

R(t in

s,t s

)→s [

MeV

]

SMts =14ats =16a

ts =18ats =20a

2 4 6 8 10 12 14 16 18 20ti /a

10

5

0

5

10

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20

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35

s [M

eV]


15 10 5 0 5 10 15(tins ts /2)/a

200

150

100

50

0

50

100

150

200

R(t in

s,t s

)→c [

MeV

]

ts =14ats =16a

ts =18ats =20a

2 4 6 8 10 12 14 16 18 20ti /a

200

150

100

50

0

50

100

150

200

c [M

eV]


Figure 5.2: Disconnected contributions to nucleon σ-terms. Left panel: Results for the ratiosfrom where we extract σπN , σs and σc, using the plateau (yellow band) and summation method(grey band). Right panel: The extracted value is shown with respect to the lower fit range ti/afor several upper fit ranges tf/a. The star symbol is the value shown with the grey band onthe left panel.

from fitting the slope of the summed ratio. On the right panel we show the extractedσ-terms when varying the lower fit ranges in the summation method for several choicesof the upper fit range. We observe similar excited states contamination in the discon-nected three-point functions as those observed in the connected part. As we increasethe source-sink time separation the ratio increases, converging finally to a value that isin agreement with the one extracted using the summation method. For the light andthe strange quarks this behaviour is clearly visible. In contrast, for the charm quarksector the ratio becomes very noisy and the signal is consistent with zero. From theseresults we conclude that disconnected contributions to the σ-term cannot be neglected

59

for the light quark, making ∼ 10% of the total value for the specific pion mass. Thestrange σ-term is also of the same order and cannot be omitted, while σc is consistentwith zero with our current statistics.

5.1.2 Extracting the strangeness of the nucleon

In the following subsection, we examine the strangeness of the nucleon using the yNparameter defined in Eq. (5.2). Defining

σ0 ≡ ml〈N |uu+ dd− 2ss|N〉 (5.3)

the yN parameter can be written as,

yN = 1− σ0

σπN. (5.4)

The value of σπN can be extracted from the pion-nucleon cross section data at an un-physical kinematics, known as the Cheng-Dashen point. The phenomenological resultsextracted from the analysis of πN phase shifts give σπN = 45± 8 MeV [85] (GLS) andσπN = 64± 7 MeV from [86] (GWU). In a more recent analysis using baryon covariantchiral perturbation theory from Ref. [87], the value of σπN = 59± 7 MeV is obtained.The value of σ0 can be estimated through the SU(3) flavour symmetry breaking ob-served in the baryons spectrum. An estimate of this quantity is given in Ref. [88] andreads σ0 = 36± 7 MeV.

If we use Eq. (5.4) we obtain

yGLSN = 0.20(21), yGWUN = 0.44(13), yAMO

N = 0.39(14) (5.5)

depending on which value of σπN is used. The value for yN ranges, leading to acorrespondingly large ambiguity to the cross-section, from about 0.55 (GWU) to beconsistent with zero (GLS) for the dark matter detection. Using methods introducedin the previous chapter we evaluate yN using Nf = 2 + 1 + 1 twisted mass fermionsfor two lattice spacing, a = 0.082 fm and a = 0.064 fm for several volumes and pionmasses ranging from about 490 MeV to 210 MeV. Our results for yN are shown in Fig.5.3 as a function of the pion mass squared m2

π. We perform an extrapolation to thephysical pion mass employing a linear fit in m2

π as well as a quadratic one. The linearfit is depicted with the solid line, and the quadratic fit with the dashed line. The blueand green filled symbols correspond to results extracted from lattices with a = 0.082fm and a = 0.064 fm respectively.

In the Osterwalder-Seiler setup the mass of the strange quark is tuned to its physicalvalue, by reproducing the physical kaon mass. In principle the strange quark mass hasa small uncertainty due to the matching conditions, but this is not visible due to thelarge error we get for yN . Since σ-terms are particularly sensitive to excited stateeffects we need a careful analysis to extract the correct value.

In order to study the systematic effects from excited state contributions we com-puted fits to plateaus using two different source-sink time separations. In Fig. 5.3,with filled triangles we show the value extracted from the plateau when a source-sinktime separation ts = 1 fm is used and with open symbols when ts = 1.5 fm for thesame lattice spacing a = 0.082. In particular, we find that yN = 0.061(4) for ts = 1 fmand yN = 0.081(9) for ts = 1.5 fm, which indicates that the systematic error from theexcited states is around ∼ 32%. The statistics used is high, making the computationvery expensive. It is for this reason that we can not repeat it for the other pion masses.

60

Figure 5.3: Chiral behaviour of the yN parameter. Blue filled triangles correspond to a = 0.082fm, while the green filled circle corresponds to a = 0.064 fm. We extrapolate to the physicalvalue of the pion mass (marked by the vertical dotted line) using linear (solid line) andquadratic (dashed line) fits in m2

π. For the quadratic fit, we also show the corresponding errorband. Points represented by open symbols are only taken to estimate systematic effects andare not included in our final analysis. The red filled rhombus show the extrapolated values atthe physical point.

We assume that this systematic error is independent of the quark mass and latticespacing. We thus assign a 30% error to all our results. We note that to obtain thevolume effects we work with two lattice sizes on the same pion mass. From Fig. 5.3we observe that volume effects and cut-off effects are small. If we consider that thesources of systematic error are the excited state contributions, the volume and cut-offeffects then our final value for the yN parameter is

yN = 0.135(22)(33)(22)(9) (5.6)

where the value is extracted from the quadratic fit. The first error is the statistical errorand the last three errors estimate the systematic uncertainties, namely the chiral ex-trapolation, the excited states contamination and the discretization error, respectively.Adding these errors quadratically we find yN = 0.135(46). This value is compatiblewith the value obtained in other works [89–93] for Nf = 2 + 1 fermions.

5.1.3 Scalar disconnected contributions to other hadrons

Nucleon observables are benchmark quantities for lattice QCD simulations becausethey have been studied for many years. The methods used for the nucleon can beextended to study properties of other hadrons. In lattice QCD, baryon states arecreated from the vacuum with the use of interpolating fields, which are constructedsuch that they have the quantum numbers of the baryon of interest. Typically, goodinterpolating fields for the low-lying states reduce to the quark model wave functionsin the non-relativistic limit.

If we include the strange and charm flavours with the light quark flavours we havea four-dimensional flavour space. Baryons consisting of three quarks, therefore belongto SU(3) sub-groups of SU(4) group. From 4 ⊗ 4 ⊗ 4 = 20 ⊕ 20 ⊕ 20 ⊕ 4 overall, thebaryon states split into two 20-plets, one of spin-1/2 states and one of spin-3/2 states.Light, strange and charm baryons can be classified according to their transformationproperties under SU(4) in flavour space and their charm content. This can be shownschematically in Fig. 5.4.

61

Hyperons and charmed baryons

We use Nf = 2 + 1 + 1 twisted mass fermions with dynamical strange andcharm quark masses fixed to their physical values

SU(4)flavour representations4 4 4 = 20 20 20 4

=) A 20-plet with SU(3) octet and a 20-plet with SU(3) decuplet.

K. Hadjiyiannakou ( University of Cyprus) Hadron Structure 1 August , 2013 3 / 20

Figure 5.4: The baryon multiplets occuring from the decomposition of the SU(4) group.

The spin-1/2 20′-plet is decomposed into three SU(3) subgroups. The charmlesssubgroup is the standard octet of the SU(3) symmetry. The c = 1 particles split intoa 6 and a 3, and c = 2 baryons transform as a 3 under SU(3). In a similar way,the 20-plet of spin-3/2 baryons consists of the standard c = 0 decuplet, c = 1, whichtransform as a 6 under SU(3), the c = 2, which transform as a 3 and a c = 3 singletat the top of the pyramid. The interpolating fields for the two 20-plet are given inAppendix. D.

In this section we are interested in the σ-terms of hyperons and charmed baryons.We show results for the σ-terms receiving connected or/and disconnected contributions.While in our study of nucleon observables it was advantageous to use the fixed-sinkmethod to calculate the connected three-point functions, this is no longer the case herewhere we want to compute the σ-terms for various particles. The fixed-sink methodallows us to calculate the matrix elements for all operators for a specific particle statewithout additional inversions. Here we want to calculate a specific observable (σ-term)for several particles at once. This can be done if we use the so-called fixed-insertionmethod (given in Appendix. E ), that fixes the insertion operator instead of the stateof the particle. Both fixed-sink and fixed-insertion methods have positive and negativecharacteristics. Another approach is to use a stochastic method [94] for the evaluationof the connected three-point functions. In this method the all-to-all propagator iscomputed stochastically, with the drawback of the introduction of stochastic noise.This method has the flexibility of both fixed-sink and fixed-insertion method for theevaluation of the connected diagrams. The disconnected diagrams will be computedusing the methods developed for the nucleon and only require two-point functions fromdifferent particles.

To study the spin-3/2 particles the interpolating field must be projected out fromspin-1/2 and spin-3/2 mixed states. In order to project to spin-3/2 state we use thespin-3/2 projector P µν

3/2 defined by [95],

P µν3/2 = δµν − 1

3γµγν − 1

3p2(6pγµpν + pµγν 6p). (5.7)

for a general momentum. To create the projector for the spin-1/2 state we use P µν1/2 =

δµν − P µν3/2 and thus

P µν1/2 =

1

3γµγν +

1

3p2( 6pγµpν + pµγν 6p). (5.8)

For spin-3/2 baryons the matrix element decomposition reads

〈B(p′, s′)|ψψ|B(p, s)〉 = vσ(p′, s′)Fστvτ (p, s) (5.9)

62

and the Rarita-Schwinger spinors satisfy the spin sum relation given by [95],

Λστ3/2 ≡

3/2∑s=−3/2

vσ(p, s)vτ (p, s) = −−i 6p+MB

2MB

(δστ − γσγτ

3+

2pσpτ

3M2B

− ipσγτ − pτγσ

3MB

).

(5.10)Then, Eq. 3.41 in the rest frame becomes,

Gµν3/2(~q, ts, tins) =

MB√MBEB(~q)

|Z3/2|2e−MB(ts−tins)e−EB(~q)(tins)

× Tr[Γ0P στ

3/2(0)Λτρ3/2(0)FρπΛπκ

3/2(~q)P κσ3/2(~q)

]. (5.11)

For zero momentum transfer and for spatial indices i.e Gij3/2(ts, tins), the three-point

function simplifies as,

Gij3/2(ts, tins) = |Z3/2|2e−MN (ts)Tr

[P il

3/2(0)Γ0Λlk3/2(0)FktΛtj

3/2(0)]

(5.12)

where for the specific projector Γ0 = (11 + γ0)/2, the projectors satisfy the relation

P ijΓ0 = P ikΓ0P kj. (5.13)

0

0.01

0.02

0.03

σ lΛ[G

eV]

Disconnected Part

Summ. [6-12]Plateau

ts=8

ts=12ts=14ts=16

0.08

0.1

0.12

σ lΛ[G

eV]

Connected Part

tins=5 tins=7 tins=9

-0.01

0

0.01

0.02

0.03

0.04

-10 -5 0 5 10

σ sΛ[G

eV]

(tins -ts/2)/a

0.08

0.1

0.12

0.14

0 2 4 6 8 10 12 14 16

σ sΛ[G

eV]

ts-tins

Figure 5.5: Connected (right) and disconnected (left) contributions to the σ-terms for Λ0 asa function of the appropriate time-separations. For the disconnected part the ratio is shownas a function of (tins− ts/2)/a for several source-sink time separations. The results from theplateau and summation method are compared. For the right panel the connected part is shownversus ts − tins for three cases tins/a = 5 (red filled squares), tins/a = 7 (green filled circles)and tins/a = 9 (blue filled triangles).

63

In Fig. 5.5 we present results for the σ-term of Λ0 which consists of two light and onestrange quarks. For the connected contributions, results are shown for three differentinsertion-sink time separations. For both contributions, excited state effects play animportant role. In the case of the disconnected contributions, where we use the one-endtrick, we can also employ the summation method to compare with the results obtainedusing the plateau method. In contrast, for the connected part we calculate threesource-insertion time separations, thus needing three times more computational effort.The summation method cannot be used because the source-insertion time separation isfixed due to the fixed-current method. From the evaluation of the disconnected partswe find that we need a large source-sink time separation for the excited states to besatisfactorily suppressed. In the connected part, a source-insertion time separation oftins = 5a is not enough to suppress the excited states and we thus need tins = 7a ortins = 9a.

0

0.01

0.02

0.03

0.04

σ lΩ

- [GeV

]

Disconnected Part

Summ. [6-12]Plateau

ts=8

ts=12ts=14ts=16

Connected Part

-0.01

0.01

0.03

0.05

-10 -5 0 5 10

σ sΩ

- [GeV

]

(tins -ts/2)/a

0.2

0.25

0.3

0.35

0.4

0 2 4 6 8 10 12

σ sΩ

- [GeV

]

ts-tins

tins=5 tins=7 tins=9

Figure 5.6: Sigma-terms of the Ω−. The notation is the same as in Fig. 5.5.

In Fig 5.6 we show selected results for a representative particle from the decuplet,taking the Ω−, as an example. The light σ-term for Ω− has only disconnected contri-butions, while for the strange we have both connected and disconnected contributions.

Excited states contributions are also observed for both the light and the strangeσ-terms, σl and σs of Ω−. As can be seen, for the Ω− the connected part dominates.For Λ0 and Ω− baryons we do not show the disconnected contributions from the charmquark because they are very noisy and compatible with zero. In order to study thecharm σ-term, σc, we consider Ω++

ccc consisting of 3 charm quarks. This particle has notbeen observed yet but recent lattice QCD studies estimate its mass to be approximatelyat ∼ 4.7 GeV [27]. In Fig. 5.7 we show results for the σ-terms of Ω++

ccc . Two importantaspects can be observed: From the behaviour of the disconnected contributions we seethat the excited state effects are minimal as compared to what we found in the case

64

0

0.01

0.02

0.03σ l

Ω+

+cc

c [G

eV]

Disconnected Part Connected Part

0.01

0.02

0.03

0.04

σ sΩ

++

ccc [

GeV

]

0

0.1

0.2

0.3

-10 -5 0 5 10

σ cΩ

++

ccc [

GeV

]

(tins -ts/2)/a

3

3.1

3.2

3.3

3.4

0 2 4 6 8 10 12

σ cΩ

++

ccc [

GeV

]

ts-tins

Figure 5.7: Sigma-terms of the Ω++ccc . The notation is the same as in Fig. 5.5.

of Λ0 and Ω−. In contrast, the connected part shows a similar behaviour as far as theexcited states contributions are concerned. The other observation is that, there is areliable non-zero disconnected contribution for the charm σ-term. The magnitude of thedisconnected one is about 10% for σc, while only about 1% for σl and σs. This findingindicates that for the very heavy particles the relative contribution of the disconnectedto connected part of the σ-term becomes smaller.

Particle Observable Disc./Conn.N σπN ∼ 15%Λ0 σπΛ0 ∼ 16%Λ0 σs ∼ 23%Ω− σs ∼ 11%

Ω++ccc σc ∼ 6%

Table 5.1: Representative results for the ratio of disconnected to connected contributions toσ-terms of baryons. The particle is given in the first column, in the second column we indicatethe σ-term considered and in the right column we give the ratio as a percentage. The baryonsare ordered with increasing mass.

In Tab. 5.1 we summarize the ratio of disconnected to connected contributions tothe σ-terms for different particles. For the nucleon we give the ratio for σπN , for Λ0

for σπΛ and σs, for Ω− for σs and for Ω++ccc for σc. It shows that the ratio reduces

for disconnected contributions containing heavier quarks as compared to connectedcontributions to the σ-terms.

5.2 Nucleon electromagnetic form factors

The electromagnetic form factors of the proton and neutron are fundamental quan-tities probing the internal structure of the nucleon. They are measured in scatter-

65

ing experiments and used to study basic properties, like the charge distribution andthe anomalous magnetic moment. Recent precise experimental studies have revealeddiscrepancies in the charge radius of the proton. Namely, the proton radius, whenmeasured recently via Lamb shifts in muonic hydrogen, is 7.7σ smaller than the valueextracted from electron scattering experiments [96].

Lattice QCD can in principle provide an insight in the origin of this discrepancy.Until now, however, many studies using heavier than physical pion mass have showna persistent underestimation in the proton charge radius [28, 30, 49]. Thus in orderto be able to provide input from lattice QCD as to the nature of the discrepancyobserved in the charge radius, one needs to be able to compute the form factors withcontrolled systematics. One source of systematic error is by neglecting the disconnectedcontributions. We investigate the disconnected contributions to the electromagneticform factors using the B55.32 ensemble. The matrix element decomposition is givenin Eq. (3.49) and the electromagnetic form factors can be extracted from Eqs. (3.53),(3.54) and (3.55).

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6Q2 [GeV2]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

GV E

(Q2 )

Experiment

ts [fm]0.50.70.91.0

1.21.4sm

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Q2 [GeV2]

0

1

2

3

4

5

GV M(Q

2 )

Figure 5.8: Electric and magnetic nucleon form factors for the isovector combination [29].The different colours and symbols show the extracted value using different source-sink timeseparations, while the black star shows the value extracted from the summation method. Thesolid line shows the J. Kelly’s parametrization of the experimental results.

In Fig. 5.8 we show results for the electromagnetic form factors with respect to theEuclidean square momentum transfer. As we can see, by increasing the source-sinktime separation, the form factors become steeper, which means that there is contami-nation from excited states. However, these results are still in contradiction with exper-iments, shown by the solid curved line which represents J. Kelly’s parametrization ofthe experimental results. Figure 5.8 shows the isovector combination where the lightdisconnected contributions are cancelled.

We know that the disconnected contributions for GE(Q2 = 0) are zero becauseonly the connected part contributes at this case. In the case of GM(Q2 = 0) wecannot extract any results because Eq. (3.53) has a kinematical factor that is zerofor Q2 = 0. Therefore, disconnected contributions to electromagnetic form factors willbe investigated for the smallest non-zero momentum transfer. In Fig. 5.9 we showthe ratio from where the isoscalar Gu+d

E and Gu+dM are extracted. We obtain results

that are small and consistent with zero. They can be used to put a bound on the sizeof the disconnected contributions, which are of the order of 0.01 or about 1% of theconnected.

66

8 6 4 2 0 2 4 6 8(tins ts /2)/a

0.03

0.02

0.01

0.00

0.01

0.02

0.03

R(t in

s,t s

)→G

u+d

M(q

2=1

)ts =8ats =10ats =12a

8 6 4 2 0 2 4 6 8(tins ts /2)/a

0.04

0.03

0.02

0.01

0.00

0.01

0.02

0.03

0.04

R(t in

s,t s

)→G

u+d

E(q

2=1

)


Figure 5.9: Results for the ratio showing disconnected contributions to isoscalar electromag-netic form factors Gu+d

E and Gu+dM for the smallest non-zero momentum transfer [97]. We

show the ratio from where the electromagnetic form factors are extracted for three differentsource-sink time separations.

5.3 Nucleon axial charge

Beyond the interest in the nucleon axial charge gA the isoscalar gu+dA determines the in-

trinsic spin carried by the quark in the nucleon. Most major lattice QCD collaborationshave computed the axial charge; for a recent review see [98].

For the isovector combination the disconnected contributions cancel in the isospinlimit, while for the isoscalar combination these contributions must be evaluated. Letus first discuss gA which is known up to high accuracy and thus serves as a benchmarkquantity for lattice QCD calculations. The majority of the recent studies show anunderestimate [74, 99–101] to gA. Recent results from ETM collaboration are shownin 5.10, taken from Ref. [98, 102].

Our interest here is the study of the isoscalar combination gu+dA . In order to assess

the disconnected contributions we first compute the connected contribution. In Fig.5.11 we show the connected isoscalar contribution to gu+d

A using various source-sinktime separations. The results show that gu+d

A does not suffer from any excited statescontamination and this can be verified also from the summation method. We expect

67

Figure 5.10: Results for the axial charge from several ensembles using TMFs with (i) Nf = 2twisted mass fermions including a clover term with a = 0.094 fm and 1440 configurations(red cross) [102], (ii) Nf = 2 + 1 + 1 twisted mass fermions with a = 0.064 fm and 900configurations (square with a cross), a = 0.082 fm and 950 configurations (open orangecircle) [99], (iii) Nf = 2 twisted mass fermions, details are given in Ref. [103]. The asteriskis the physical value as given in the PDG [104].

10 5 0 5 10(tins ts/2)/a

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

R(t in

s,t s

)→gu

+dA


ts =16ats =18aSM

Figure 5.11: The ratio yielding the connected contribution to the nucleon isoscalar axial chargegu+dA using various source-sink time separations, ts = 10a (red filled circles), ts = 12a (blue

filled squares), ts = 14a (green empty squares), ts = 16a (yellow filled triangles) and ts = 18a(purple empty stars). For the analysis of the connected part a statistics of 1200 configurationshas been used.

that the same statement also holds for the disconnected part.For the computation of the disconnected part we use the B55.32 ensemble using

approximately 150000 measurements. We employ the one-end trick and TSM withNLP = 500 and NHP = 24 for the light quarks and NLP = 300 with NHP = 24 for thestrange and charm quarks. In Fig. 5.12 we show results for the ratio from where gu+d

A

is extracted. As can be seen, there is a non-zero contribution from the disconnected

68

8 6 4 2 0 2 4 6 8(tins ts /2)/a

0.20

0.15

0.10

0.05

0.00

0.05

R(t in

s,t s

)→gu

+dA

SMts =8ats =10ats =12ats =14a

2 3 4 5 6 7 8 9 10 11ti /a

0.20

0.15

0.10

0.05

0.00

0.05

gu+d

A


Figure 5.12: The disconnected contributions to the renormalized ratio which yields theisoscalar axial charge of the nucleon, gu+d

A . The upper panel shows the ratio as a function ofthe insertion time-slice with respect to the mid-time separation (tins − ts/2) for source-sinktime separations ts = 8a (red filled circles), ts = 10a (blue filled squares), ts = 12a (greenopen squares) and ts = 14a (yellow filled triangles). The bottom panel shows the results ob-tained for the fitted slope of the summation method for various choices of the initial and finalfit time slices. The star shows the choice of ti, which yields the gray band in the top plot.

diagrams. In order to extract the matrix element of interest we employ both plateauand summation method and we find consistent results between the two methods. Fig.5.12 shows the dependence of the results as we vary the upper and lower fit range ofthe summation method. The results plotted as a grey band on the top panel is shownon the bottom panel by the star. As we can see, by varying ti and tf yields compatiblevalues within the error bands. If we compare the connected part from Fig. 5.11 with thedisconnected one from Fig. 5.12 we observe that the two contributions have oppositesign [97, 105] with the disconnected being about 15% of the connected. The relativesign between the connected and disconnected contribution is very important and adetailed proof is given in Appendix. E.

The axial charge of the nucleon also takes contributions from the strange and charmquark loops in the sea sector. Even if the contribution from heavier quarks is expectedto be small it is of great interest to estimate and compare with the light contribution.In Fig. 5.13 we show results for gsA and gcA. For the case of gsA we observe small excited

69

8 6 4 2 0 2 4 6 8(tins ts /2)/a

0.05

0.04

0.03

0.02

0.01

0.00

0.01

R(t in

s,t s

)→gs A


2 3 4 5 6 7 8 9 10 11ti /a

0.05

0.04

0.03

0.02

0.01

0.00

0.01

gs A


8 6 4 2 0 2 4 6 8(tins ts /2)/a

0.03

0.02

0.01

0.00

0.01

0.02

0.03

R(t in

s,t s

)→gc A

ts =8ats =10ats =12ats =14a

2 3 4 5 6 7 8 9 10 11ti /a

0.03

0.02

0.01

0.00

0.01

0.02

0.03

gc A


Figure 5.13: The strange and charm quark contributions to the renormalized ratio yieldingthe nucleon axial charge gsA (left) and gcA (right). The notation is the same as that of Fig.5.12.

states effects and the summation method coincides with the result from the plateaumethod at source-sink time separation ts = 12a. Shifting the lower fit range ti enteringin the summation method, the value remains unchanged, showing indeed that excitedsates do not contribute to the determination of the slope.

For gcA the results are consistent with zero even for small ts. This is shown in Fig.5.13 where it is clearly demonstrated that gcA ' 0 to accuracy better than 1%.

While the axial charge is extracted from the axial form factor at zero momentumtransfer, one can study GA(Q2) as a function of Q2. For non-zero Q2 the matrix elementdecomposition includes two form factors and a more sophisticated analysis is neededto extract the form factors as a function of Q2. The matrix element decomposition hasbeen defined in Eq. (4.10). Performing the analysis for finite Q2 we obtain results forGA(Q2) using the smallest non-zero momentum transfer, namely 2π/L. In Fig. 5.14we show results for the disconnected part of the ratio which contributes to Gu+d

A (Q2),taking into account only the light quark loops. These results show for the first time anon-zero value for the disconnected contributions for non-zero momentum transfer.

The accuracy of these results was improved by using symmetries in order to reducethe stochastic noise. The disconnected loop with the axial-vector current can be writtenas

Lu+dA = Tr

[(Gu(x;x) +Gd(x;x)

)γ5γi

]= Tr

[γ5

(G†d(x;x) +G†u(x;x)

)γ5γ5γi

](5.14)

70

8 6 4 2 0 2 4 6 8(tins ts /2)/a

0.06

0.04

0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

R(t in

s,t s

)→Ru

+dA

(q2

=1)


Figure 5.14: Disconnected contributions to the renormalized ratio yielding the isoscalar axial-vector and pseudo-scalar form factors Gu+d

A and Gu+dp at the lowest non-zero momentum

transfer allowed for this lattice size. The notation is the same as that of Fig. 5.12.

where we used Gu(x;x) = γ5G†d(x;x)γ5. If we use that (γ5)2 = 1 and take the dagger

out of the trace we can write,

Lu+dA = Tr

[(Gu(x;x) +Gd(x;x)

)†γiγ5

]= Tr

[(Gu(x;x) +Gd(x;x)

)γ†5γ

†i

]∗=

(Lu+dA

)∗(5.15)

where we used the properties of γ-matrices, i.e that they are hermitian in Euclideanspace. This tells us that if the axial-vector disconnected quark loop is computed exactlywe expect a purely real number. When we compute the disconnected quark loopstochastically the imaginary part is only noise. Therefore, we take the real part of theloop when we correlate it with the nucleon two-point function in order to create thenucleon three-point function.

5.4 Nucleon tensor charge

The transverse spin structure of the nucleon has been receiving a lot of attention in therecent years from both theory and experiment as it provides a potential probe of newtensor interactions at TeV scale. Another quantity of interest is the quark transversitydistribution hq1T (x). The proton’s tensor charge for a quark q is defined by,

δqT ≡∫ 1

−1

dx hq1T (x) =

∫ 1

0

dx(hq1T (x)− hq1T (x)

)(5.16)

and illustrated in Fig. 5.15. The light-front number-density of quarks for transversepolarization parallel to that of the proton, minus that of quarks with anti-parallelpolarisation, measures any bias in quark transverse polarization induced by a polariza-tion of the parent proton. The charges δqT represent a close analogue of the nucleon’s

71

Figure 5.15: The tensor charge, Eq. (5.16), measures the net light-front distribution oftransversely polarised quarks inside a transversely polarized proton.

flavour separated axial-charges, which measures the difference between the light-frontnumber-density of quarks with helicity parallel to that of the proton and the densityof quarks with helicity anti-parallel [106]. In non-relativistic systems the helicity andthe transversity distributions are identical because boosts and rotations commute withthe Hamiltonian.

0 0.05 0.1 0.15 0.2

mπ

2 (GeV2)

0.4

0.6

0.8

1.0

1.2

1.4

gTu

-d

ETMC (Nf =2) TMF/CloverETMC (Nf =2+1+1) TMFLHPC (Nf =2+1) CloverPNDME (Nf =2+1+1) HISQ/CloverRBC/UK (Nf =2+1) DWF

0 0.05 0.1 0.15 0.2

mπ

2 (GeV2)

0

0.2

0.4

0.6

0.8

1

gTu

+d

ETMC (Nf =2) TMF/CloverETMC (Nf =2+1+1) TMFPNDME (Nf =2+1+1) HISQ/Clover

Figure 5.16: The isovector (top) and isoscalar (bottom) tensor charge from various pionmasses taken from Ref. [107].

The transversity distribution is measurable using Drell-Yan processes in which atleast one of the two colliding particles is transversely polarized [108], but such data

72

is not yet available. Alternatively, the transversity distribution is accesible via semi-inclusive deep-inelastic scattering using transversely polarized targets and also in unpo-larized e−e+ processes, by studying azimuthal correlations between produced hadronsthat appear in opposing jets (e+e− → h1h2χ).

The tensor charge associated with a given quark flavour in the proton is defined viathe matrix element

〈N(p′, s′)|ψσµνψ|N(p, s)〉 = δqT u(p′, s′)σµνu(p, s) (5.17)

where the isoscalar and isovector combination are given as,

gu+dT ≡ δuT + δdT , gu−dT ≡ δuT − δdT . (5.18)

8 6 4 2 0 2 4 6 8(tins ts /2)/a

0.03

0.02

0.01

0.00

0.01

0.02

0.03

R(t in

s,t s

)→gu

+dT

ts =14ats =16ats =18ats =20a

2 3 4 5 6 7 8 9 10 11ti /a

0.03

0.02

0.01

0.00

0.01

0.02

0.03

gu+d

T

tf =12atf =14atf =16a

Figure 5.17: Top: We show results on the disconnected Ru+dT versus (tins − ts/2)/a for

ts = 14a (red filled circles), ts = 16a (blue filled squares), ts = 18a (green empty squares) andts = 20a (filled yellow triangles). Bottom: The value of gu+d

T extracted using the summationmethod for various ti and tf fit ranges.

There have been many attempts from lattice QCD to estimate precisely the tensorcharge [106, 107, 109, 110]. In Fig. 5.16 we collect recent results for the tensor charge.Our results are computed at different lattice spacings ranging from a ∼ 0.1 fm toa ∼ 0.06 fm, and at different lattice sizes. As can be seen, there are no sizeable cut-offeffects. A comparison with other lattice discretizations shows that all lattice results

73

are in good agreement [109, 111–113]. We show both the isovector, which receives nodisconnected contributions and isoscalar tensor charge when neglecting disconnectedcontributions.

In Fig. 5.17 (top) we show results for the disconnected contributions to gu+dT for

the B55.32 ensemble. The results for the ratio are compatible with zero for severalsource-sink time separations. As we increase this separation we do not see any excitedstates effects. In this respect gT behaves similarly to gA. The conclusion is thatdisconnected contributions are not larger than 0.1% of the connected and can thus besafely neglected.

In Fig. 5.17 (bottom) we also show results from the summation method as we varyti and tf . As can be seen, for small values of fit ranges the results are compatible withzero in agreement with those extracted from the plateau method. For larger values thevalue extracted from the slope becomes very noisy and cannot be used.

5.5 Nucleon moments of parton distributions

The Generalized Parton Distributions (GPDs) encode information about nucleon struc-ture which complements the knowledge we obtain from the form factors [114–116].They enter in several physical processes such as deeply virtual Compton scatteringand deeply virtual meson production. In the forward limit they coincide with theusual parton distributions, and by using Ji’s sum rule [116], we can determine the con-tribution of a specific parton to the nucleon spin. The proton spin puzzle, which refersto the unexpectedly small fraction of the total spin of the nucleon carried by quarks,has triggered intensive theoretical and experimental studies [117–120].

How much of the proton spin is carried by the quarks is a question which has beenreceiving a lot of attention ever since the results of the European Muon Collaboration(EMC) claimed that the quarks carry only a small fraction of the proton spin [121].This discovery became known as the -proton spin crisis-. In order to understand thispuzzle one requires to take into account the non-perturbative structure of the proton[122].

In order to obtain information from lattice QCD for the spin content of the nucleonwe need to evaluate the isoscalar moments Au+d

20 and Bu+d20 , since the total angular

momentum of a quark in the nucleon is given by,

Jq =1

2

(Aq20(0) +Bq

20(0)). (5.19)

The total angular momentum Jq can be decomposed into an orbital angular momentumLq and into a spin component ∆Σq as,

Jq =1

2∆Σq + Lq. (5.20)

The spin carried by u- and d- quarks is determined using ∆Σu+d = Au+d10 = gu+d

A , andtherefore, we need the isoscalar axial charge studied in section 5.3. At zero momentumtransfer the first moments can be extracted from the generalized form factors, e.g〈x〉u+d = Au+d

20 (0) defined by

〈x〉u+d =

∫ 1

−1

dx x[u(x, µ2) + d(x, µ2)

], (5.21)

74

and 〈x〉∆u+∆d = Au+d20 (0). These quantities can be extracted directly from lattice data

at zero momentum transfer, but Bu+d20 (0) can only be obtained by extrapolation to zero

momentum transfer.These generalized form factors can be extracted from matrix elements of local op-

erators only. For the quantities of interest here we consider one-derivative operators.More specifically, for the vector and axial-vector current the operators are,

OµνV ≡ uγµ

←→D νu+ dγµ

←→D νd (5.22)

andOµνA ≡ uγ5γµ

←→D νu+ dγ5γµ

←→D νd (5.23)

correspondingly. The notation γµ←→D ν means we perform symmetrization over the

γ-index matrix and derivative direction and then we make it traceless. The derivative

operator←→D corresponds to the symmetrized version

←→D ≡ (

−→D −←−D)/2. The matrix

elements of the one-derivative operators are parametrized in terms of the GFFs for thevector and axial-vector operators respectively, according to

〈N(p′, s′)|OµνV |N(p, s)〉 = u(p′, s′)

[A20(Q2)γµP ν +B20(Q2)

iσµαqαPν

2m

+ C20(Q2)1

mqµqν

]u(p, s) (5.24)

and

〈N(p′, s′)|OµνA |N(p, s)〉 = u(p′, s′)

[A20(Q2)γµP νγ5 + B20(Q2)

qµP ν

2mγ5

]u(p, s)

(5.25)where P = (p′ + p)/2.

Due to the fact that we need the isoscalar combination we need to assess the con-tribution from the disconnected diagrams which up to recently were neglected in moststudies. In Fig. 5.18, [99] we show the isoscalar combination for Gu+d

A (Q2), Au+d20 (Q2)

and Bu+d20 (Q2) for Nf = 2 + 1 + 1 twisted mass fermions for mπ = 213 MeV and

mπ = 373 MeV . Figure 5.18 shows only the contributions from connected diagrams.We have already seen that for Gu+d

A (0) if we neglect the disconnected part we introduce(10− 15)% systematic error. While Au+d

20 (0) can be extracted directly from the matrixelement at Q2 = 0, Bu+d

20 (0) can only be extracted by extrapolating the results at zeroQ2.

Let us first examine the excited states contamination of Au+d20 (0) or equivalently

〈x〉u+d. To address this question we perform an analysis for the B55.32 ensemble usingvarious source-sink time separations. The results shown in Fig. 5.19 show that thisquantity is affected by exited-states effects and in order to obtain a reliable estimate, alarge source-sink time separation is needed. We observe that only at ts = 18a we obtaina value from the plateau method which becomes consistent with the one extracted fromthe summation method.

In Fig. 5.20 we show the disconnected contribution to 〈x〉u+d using high statistics,namely approximately 150000 measurements. The noise seems to be very large and theratio gives a value which is compatible with zero for all source-sink time separations.We have to mention at this point that for the calculation of this quantity we used amoving frame, where the initial and final momentum are equal and non-zero. Thisallows us to increase the statistics and we observed that we can reduce the noise. The

75

0.2

0.3

0.4

0.5

0.6

GAu

+d

TMF: 213MeVTMF: 373MeV

0.2

0.3

0.4

0.5

0.6

A20

u+

d

0 0.5 1 1.5

Q 2

(GeV 2

)

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

B20

u+

d

Figure 5.18: The Q2 dependence of the isoscalar Gu+dA (Q2), Au+d

20 (Q2) and Bu+d20 (Q2) for

mπ = 213 MeV (blue filled squares) and mπ = 373 MeV (red filled circles).

10 5 0 5 10(tins ts/2)/a

0.0

0.2

0.4

0.6

0.8

1.0

1.2

R(t in

s,t s

)→⟨ x⟩ u

+d


ts =16ats =18aSM

Figure 5.19: Results for the connected ratio of 〈x〉u+d using various source-sink time sepa-rations, namely ts = 10a (red filled circles), ts = 12a (blue filled squares), ts = 14a (greenempty squares), ts = 16a (yellow filled arrows) and ts = 18a (purple empty stars). The greyband shows the value of 〈x〉u+d extracted using the summation method.

76

8 6 4 2 0 2 4 6 8(tins ts /2)/a

1.0

0.5

0.0

0.5

1.0

R(t in

s,t s

)→⟨ x⟩ u

+d

SMts =8ats =10ats =12a

Figure 5.20: Results for the disconnected ratio of 〈x〉u+d using various source-sink time sepa-rations, namely ts = 8a (filled red circles), ts = 10a (filled blue squares) and ts = 12a (greenempty squares). The grey band shows the value of 〈x〉u+d extracted using the summationmethod.

most important improvement comes when the moving frame has the first non-zeromomentum because if we go for higher momenta the noise starts to increase rapidly.

To study further the source of this noise we investigate the behaviour of the summa-tion method for various fit ranges. In Fig. 5.21 we explore the behaviour of the summed

0 2 4 6 8 10 12 14 16ts /a

5

4

3

2

1

0

1

2

∑ t ins

R(t in

s,t s

)

2 3 4 5 6 7 8 9 10 11ti /a

0.4

0.2

0.0

0.2

0.4

⟨ x⟩ u+d


Figure 5.21: Behaviour of the summed ratio for 〈x〉u+d as a function of ts (left). Right:〈x〉u+d versus the lower fit range ti for various upper fit ranges tf .

ratio. As can be seen, the summation method gives flat data, which means that theslope is zero. In Fig. 5.21 we also show the extracted value for various fit ranges. Weconfirm from the plot that the results are consistently zero. If we take the result shownwith black star in Fig. 5.21 and compare it with the connected contribution shown inFig. 5.19 we can put a bound on the maximum disconnected contribution. We findthat is less than ∼ 17% of the connected one. This is not a small bound and therefore

77

the uncertainty in the value of the disconnected contribution represents a rather largesystematic error.

We do not know a priori what percentage of the error is due to the gauge noiseand what due to the stochastic noise. This needs an extensive error analysis in orderto understand its origin better. We will attempt here to model this error using theobservation that as the number of noise vectors Nr → ∞ the stochastic error goes tozero and when the number of configurations Nc →∞ the total error goes to zero. Wethus write the total error as

δ =

√a

Nc

+b

NcNr

=1√Nc

√a+

b

Nr

. (5.26)

The constants a and b are parameters, which depend on the observable the error ofwhich we want to model. Note that for fixed Nr, δ → 0 as Nc →∞ irrespective of thevalue of Nr. The assumption we make is that the bias also goes to zero as Nc →∞.

We investigate the error of 〈x〉u+d, δ〈x〉u+d, as we increase the number of noisevectors for a fixed number of configurations Nc = 3000. This is shown in Fig. 5.22. The

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300 350 400 450 500

δ<x

> u+d

Nr

Figure 5.22: The behaviour of the total error of 〈x〉u+d is shown as a function of Nr forNc = 3000 for the B55.32 ensemble. The dotted black line is the result of the fit to Eq.(5.26). For the first three points (red filled squares) we only use HP noise vectors, whereasfor the last 3 points (blue filled circles) we use TSM and the stochastic error decreases withthe number of noise vectors. The horizontal purple dotted line shows the gauge error whenextrapolating to Nr →∞ for Nc = 3000.

asymptotic behaviour shows that the total error falls with the number of noise vectorsbut it saturates at some point when we approach the gauge noise, for a fixed numberof configurations, as expected. Fitting to these results we extract the parameters aand b that are different for each observable. For instance, in the case of the σ-term,the term with the a parameter dominates the contribution to the total error whereas,for the axial charge both terms contribute equally. This means that for gA we need touse a large number of configurations and noise vectors. For δ〈x〉u+d shown in Fig. 5.22we observe that 500 noise vectors are not enough to sufficiently reduce the stochasticnoise and we expect around 2000 noise vectors to suppress it satisfactorily. To quantifythis statement we quote the parameters extracted from Fig. 5.22. For a fixed numberof Nc = 3000, a = 16.9(4.9) and b = 15884(269) and the total error is

δ〈x〉u+d '√

0.00563 +5.29

Nr

(5.27)

78

thus for Nr = 2000 the contribution of the stochastic noise is half of the gauge noisefor Nc = 3000. Also, the gauge noise is high even with ∼ 150000 measurements and ifwe are to obtain an error comparable with the connected part we need to increase ourstatistics by about 10 times.

Another interesting quantity is the isoscalar nucleon polarized moment 〈x〉∆u+∆d.

8 6 4 2 0 2 4 6 8(tins ts /2)/a

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40R(

t ins,

t s)→

⟨ x⟩ u+

dts =10ats =12ats =14a

ts =16ats =18aSM

Figure 5.23: Results on the connected ratio of 〈x〉∆u+∆d using various source-sink time sep-arations. The notation is the same as that in Fig. 5.19.

Fig. 5.23 shows the contribution of the connected part to 〈x〉∆u+∆d. For thisquantity the excited states do not pollute the results and we get compatible resultsfrom both small and large source-sink time separations using the plateau method. Theresult coincides with the one extracted using the summation method.

The disconnected contributions to 〈x〉∆u+∆d are shown in Fig. 5.24. We fit toobtain a non-zero value albeit with large errors. If we fit the plateau for source-sinktime separation ts = 14a we obtain a non-zero contribution. That is in agreement withthe results extracted from the summation method but with a larger error. Results usingdifferent fit ranges, however, yield incompatible results which are only stabilized whenthe lower fit range is larger than ti = 8a. We find a sizeable disconnected contributionwhich is, however, very noisy. It is thus important to use even larger statistics orimprovements to reduce the noise in order to have a better bound on the disconnectedcontributions.

The final results for the total angular momentum of a quark in the nucleon, Ju andJd are given in Appendix. G.

79

8 6 4 2 0 2 4 6 8(tins ts /2)/a

0.20

0.15

0.10

0.05

0.00

0.05

0.10

0.15

0.20

R(t in

s,t s

)→⟨ x⟩ u

+d


2 3 4 5 6 7 8 9 10 11ti /a

0.20

0.15

0.10

0.05

0.00

0.05

0.10

0.15

0.20

⟨ x⟩ u+

d


Figure 5.24: Results for the disconnected contributions to 〈x〉∆u+∆d. The notation is thesame as that in Fig. 5.20 but for 〈x〉∆u+∆d.

5.6 Extracting flavour octet moments for nucleon

In the previous section we discussed the disconnected contribution to 〈x〉u+d. In thissection we go one step further and we also include the strange quark in the calculation.

The moments of Parton Distribution Functions (PDFs) are very important bench-mark quantities for lattice calculations, but they also provide insight into the structureof hadrons. As such they are of great interest in lattice QCD. For recent lattice QCDresults see [30, 99, 103, 123]. The average quark momentum fraction 〈x〉u−d has beenstudied extensively because it is protected from disconnected contributions. Anotherquantity of interest is the flavour octet combination 〈x〉u+d−2s which is also protectedfrom disconnected contributions in the isospin limit.

The flavour octet moment is defined as,

〈x〉(8)

µ2 ≡ 〈x〉u+d−2s =

∫ 1

−1

dx x[u(x, µ2) + d(x, µ2)− 2s(x, µ2)

](5.28)

and the isovector combination is,

〈x〉(3)

µ2 ≡ 〈x〉u−d =

∫ 1

−1

dx x[u(x, µ2)− d(x, µ2)

]. (5.29)

80

At a specific renormalization scale µ2 = 4 GeV 2 the two moments can be estimatedphenomenologically using PDFs determined from deep inelastic scattering. From nowon we will drop the index µ2, since everything will be computed at this energy scale.Using the data from PDFs [124], the two moments read,

〈x〉(3) = 0.153(4), 〈x〉(8) = 0.470(7). (5.30)

Taking the ratio 〈x〉(3)/〈x〉(8) cancels the renormalization factor making it indepen-dent of any uncertainty in the determination of the renormalization factor. This allowsa direct comparison with the phenomenological results and also allows us to check ifthe ratio is in agreement with the experimental value even though 〈x〉(3) is not.

For this study we will use pion masses in the range of (300-500) MeV. In addition,we have results using lattices with two different lattice spacings in order to study cut-off effects. All the results presented in this chapter are produced using the one-endtrick to evaluate the light and the strange quark loop contributions. We can modifythe one-end trick in order to create the octet combination in the same contraction. Wethen consider the following property,

Ml,± −Ms,± = ±iγ5(µl − µs) (5.31)

where the ± stands for the sign in front of the twisted mass term. Then the quarkpropagators can be written as,

Gl,± −Gs,± = ∓i(µl − µs)Gs,±γ5Gl,±. (5.32)

In order to calculate the disconnected quark loops of interest we need to add, in theinsertion operator of Eq. (5.22), the contribution of the strange quark, hence

Oµν = uγµ←→D νu+ dγµ

←→D νd− 2sγµ

←→D νs (5.33)

and the disconnected quark loop can be estimated stochastically as,

Lµν〈x〉(8) = 2

(Tr[Gl(x;x)Oµν

]− Tr

[Gs(x;x)Oµν

])= −2i(µl − µs)

1

Nr

∑r

Tr[φ∗r,lγ5O

µνφr,s

]+O

(1√Nr

). (5.34)

The loop is proportional to the mass difference (µl − µs), which vanishes in the SU(3)flavour limit as expected. Because this is an SU(3) flavour symmetry breaking effectwe expect that the disconnected contributions will be very small and can be neglected.

The renormalization of the operator Oµν can be obtained non-perturbatively usingthe methodology explained in Ref. [99]. In the chiral limit where the quark massesgo to zero the isovector and octet flavour operators O(3) and O(8), share the samerenormalization and the ratio is thus free from renormalization factors.

The ratio for the two cases is presented in Fig. 5.25, where the connected anddisconnected contributions for the flavour octet case are shown independently. As canbe seen, the disconnected part is compatible with zero as expected. Next, we performthe same procedure using 5 different pion masses in order to probe the chiral behaviourof these two quantities separately before we take the ratio.

In Fig. 5.26 we plot our results for 〈x〉(3) and 〈x〉(8) as a function of the pionmass mπ. As can be seen, cut-off effects are small when comparing results obtainedat a ' 0.082 fm and a ' 0.064 fm. For both quantities and for heavier pion masses

81

ts /a

RXa

(ts

,tin

s=

11a)

ts /a

RXa

(ts

,t=

11a)

ts /a

RXa

(ts

,t=

11a)

14 16 18 20

0.0

0.2

0.4

0.6

0.8

1.0

Rfulla=8

Rfulla=3

Rdisc.a=8

Figure 5.25: The ratio of Eq. (3.45) as a function of the source-sink time separation ts fora fixed source operator time tins = 11a in the isovector (orange filled squares) and octet case(red filled circles). The gray bands indicate the results obtained from a fixed-sink calculationfor ts = 12a. The blue triangles show the disconnected contribution to Ra=8(ts, tins = 16).The results shown are extracted using the B55.32 ensemble with 23000 measurements.

m [GeV2 ]

<x>(a

=3,8

)

m [GeV2 ]

<x>(a

=3,8

)

m [GeV2 ]

<x>(a

=3,8

)

m [GeV2 ]

<x>(a

=3,8

)

2π

0.00 0.05 0.10 0.15 0.20 0.25 0.30

0.0

0.2

0.4

0.6

0.8

<x>(8) β = 1.95 ts /a =12<x>(3) β = 1.95 ts /a =12

<x>(8) β = 2.10 ts /a =16<x>(3) β = 2.10 ts /a =16

Figure 5.26: 〈x〉(3,8) as a function of the pseudoscalar meson mass m2π. The phenomenological

estimates are represented by two black stars.

82

than the physical ones, the lattice results lay consistently above the phenomenologicalvalues. This behaviour is well known for 〈x〉(3) and now we find that 〈x〉(8) shows thesame behaviour.

The ratio 〈x〉(3)/〈x〉(8) shown in Fig. 5.27 is consistent with the phenomenologicalresults, even for heavy pion masses. This suggests that systematic uncertainties cancelout in the ratio. Therefore, we conclude that for the individual quantities either thenon-perturbative renormalization factor or/and the chiral extrapolation are the mostprobable sources of the discrepancy. Studies directly at the physical point can shedlight on whether the discrepancy is due to heavier pion mass. The systematic errorin the ratio is estimated by the difference between the largest and smallest valuespresented with the red line. For the ratio [125] we find the value

〈x〉(3)

〈x〉(8)= 0.39(1)(4) (5.35)

where the first error is statistical and the second is systematic.

m [GeV2 ]

<x>(3

)/<

x>(8

)

m [GeV2 ]

<x>(3

)/<

x>(8

)

m [GeV2 ]

<x>(3

)/<

x>(8

)

m [GeV2 ]

<x>(3

)/<

x>(8

)

m [GeV2 ]

<x>(3

)/<

x>(8

)

2π

0.00 0.05 0.10 0.15 0.20 0.25 0.30

0.0

0.1

0.2

0.3

0.4

0.5

0.6 <x>(3) / <x>(8) β = 1.95 ts /a =12<x>(3) / <x>(8) β = 2.10 ts /a =16<x>(3) / <x>(8) constant fit

Figure 5.27: 〈x〉(3)/〈x〉(8) as a function of the pseudoscalar meson mass m2π for two values

of the lattice spacing. The result from a constant extrapolation is represented by an emptytriangle. The systematic error on the extrapolated value is represented by a red error barslightly shifted for readability. The phenomenological estimate is represented by a black star[125].

83

Chapter 6

Neutron Electric Dipole Moment us-ing Disconnected Diagrams

6.1 Introduction

Symmetries are very important properties of the physical world. Discrete symmetrieslike parity P, charge conjugation C, and time-reversal T play a crucial role in the con-struction of the Standard Model. The strong and electroweak interactions break P andT symmetries. If the neutron would have a persistent electric dipole moment it wouldindicate a violation of both parity and time-reversal symmetries. Several experimentalmeasurements yield an upper limit on the neutron Electric Dipole Moment (nEDM)

denoted as ~dN . The most recent upper limit [126] reads,

|~dN | < 2.9× 10−26e cm. (6.1)

The QCD Langrangian as written in Eq. (1.1) is invariant under P and T trans-formations, which does not allow for a nEDM. However, there is no a priori reasonwhy the Standard Model would preclude a CP-violating term in the QCD action. TheQCD Langrangian including a CP-violating term is:

LQCD(x) = LQCD(x) + θ Q(x) (6.2)

where θ is the so-called theta parameter which controls the strength of this CP-violatingterm and Q(x) is the topological charge density defined by

Q(x) ≡ i

32π2εµνρσTr

[Gµν(x)Gρσ(x)

]. (6.3)

There are several model studies which provide an estimate of the value of nEDM.According to these studies [127–129] its value is of the order of

|~dN | ∼ θ O(10−2 − 10−3)e fm, (6.4)

providing a very small value of θ ∼ 10−10. According to these models, θ can be smalleror even zero leading to a vanishing nEDM. The nEDM can be extracted from theCP-odd form factor F3(Q2) at zero momentum transfer as,

|~dN | = θF3(0)

2MN

. (6.5)

The observables in a theory with a θ-term can be calculated as,

〈O〉θ =1

Z

∫D[ψ]D[ψ]D[U ]O[ψ, ψ, U ] e−SQCD−iθQ (6.6)

84

where Q is the topological charge which is defined as,

Q ≡∫Q(x)d4x. (6.7)

This, however, is extremely challenging to compute directly as it requires configurationssimulated with the a complex probability, which suffers from a sign problem. Instead,one treats the θ-term as perturbation. Since θ is expected to be small, the expectationvalues with the θ-term can be expanded in terms of those without the θ-term as,

〈O〉θ = 〈O〉+ 〈O(−iθQ)〉+ · · · . (6.8)

In the case where no index θ appears it means that θ is set to zero. The main ingredientneeded is the topological charge, which can in principle be calculated using Eqs. (6.7)and (6.3). The calculation of the gluonic topological charge involves the gluon tensoroperator. We refer to Ref. [130] and references for a discussion on the lattice definitionof the topological charge. Here, we will examine an alternative way to calculate thetopological charge using disconnected quark loops. One can prove that, by employingthe axial Ward Identity and performing the non-singlet axial rotations of the quarkfields, following Ref. [131]

ψ(x) −→(

1 + iαa(x)λaγ5)ψ(x), ψ(x) −→ ψ(x)

(1 + iαa(x)λaγ5

)(6.9)

where λa are the Gell-Mann matrices, given in Appendix. A. If we take variations withrespect to α0, we obtain the relation,⟨

O δS

δ(iα0(x))

⟩= −∂µ〈OAµ(x)〉+ 〈Oψ(x)2mγ5ψ(x)〉+ 2Nf〈Oq(x)〉, (6.10)

where

m ≡ 3mlms

2ms +ml

. (6.11)

Integrating over the whole space the divergence term vanishes, and by defining thepseudoscalar density PS ≡ uγ5u+ dγ5d+ sγ5s, we can write the following relation,

Nf〈OQ〉 = −m∫d4x〈OPS(x)〉. (6.12)

For the specific case when the operator O is

O = J (xs)j0(xins)J (0), (6.13)

where j0 is the electromagnetic current, we can perform the contractions for 〈OPS(x)〉and classify the results into connected and disconnected diagrams. The connecteddiagrams do not contribute [131] because the chiral variations cancel out their contri-butions. There are several disconnected diagrams, but it was shown that due to SU(3)flavour symmetry their contributions are negligible while the rest of them are zero dueto other symmetries [132].

The final result allows us to substitute the topological charge with the pseudoscalardisconnected quark loops,

〈OQ〉 = − m

Nf

∫d4x〈[OPS(x)]disc〉 (6.14)

where the disconnected diagram is depicted in Fig. 6.1. The relation in Eq. (6.14) isexact only in the continuum limit, because on the lattice we expect that the left-handside of Eq. (6.14) has different lattice artifacts from the right-hand side. Also, thedisconnected quark loops requires renormalization as Z = ZP/ZS [130].

85

J (0)J (xs)

j0(xins)

LS

Figure 6.1: Disconnected contributions to nEDM.

6.2 Lattice Techniques for nEDM

Knowing how the pseudoscalar density can be used to substitute the gluonic definitionof the topological charge, we examine how to extract the CP-odd form factor F3(Q2).As aforementioned, we will treat the θ-term as a perturbation and express quantitieswith the θ-term in terms of what we measure without the θ-term. The spin sum rulein the presence of the θ-term gives

Λθ1/2(p) ≡ −i 6p+M θ

Ne2iα(θ)γ5

2EθN

' −i 6p+MN

2EN+θα1MNγ5

EN, (6.15)

where we keep terms up to first order in θ. Here we used that,

α(θ) ' α1θ, M θN 'MN , Eθ

N ' EN , (6.16)

to leading order.The matrix element decomposes into form factors which are even or odd with respect

to θ. Writing the matrix element in the form

〈N(~pf , sf )|jµ|N(~pi, si)〉 = uθN(~pf , sf )Wθµ(q)uθN(~pi, si), (6.17)

whereW θµ(q) = W even

µ (q) + iθW oddµ (q), (6.18)

we isolate the even and odd parts as

W evenµ (q) = γµF1(Q2) +

F2(Q2)

2MN

qνσµν , (6.19)

and

W oddµ (q) =

F3(Q2)

2MN

qνσµνγ5 + FA(Q2)(qµ 6q − γµq2)γ5. (6.20)

F1,2(Q2) are the CP conserving Sachs form factors and FA(Q2) is the anapole formfactor, while F3(Q2) is the one we are interested in. The neutron matrix elementcan be extracted from an appropriately defined ratio between three- and two- pointfunctions. Using Eq. (6.15) we can write the two-point functions to zero and first orderin θ for zero momentum as

G(ts; Γ0) = |ZN |2e−EN ts , (6.21)

GQ(ts; γ5) = |ZN |2e−EN tsα1. (6.22)

86

In order to isolate F3(Q2) we must first extract α1. It can be obtained from the ratioof two-point functions from Eqs. (6.21) and (6.22):

Rα(ts) =GQ(ts; γ5)

G(ts; Γ0)−→ α1, (6.23)

where for large source-sink time separation the ratio yields a plateau and thus a con-stant fit yields the value of α1.

The three-point function to first order in θ is given by,

GµQ(ts, tins, p

′, p; Γk) = |ZN |2e−EN (ts−tins) × Tr

[Γk

(α1MNγ5

ENW evenµ (q)Λ1/2(p)

+ Λ1/2(p′)W evenµ (q)

α1MNγ5

EN+ Λ1/2(p′)W odd

µ (q)Λ1/2(p))]

(6.24)

where Γk = 1/4(11 + γ0)iγ5γk. Using Eq. (3.45) and for sufficient source-sink timeseparations we can relate the lattice results with the form factors in the continuum.Using the ratio in Eq. (3.45) and performing the trace algebra, then

Π0(Γk) = Cqk

[α1F1(Q2)

2MN

+(EN + 3MN)α1F2(Q2)

4M2N

+(EN +MN)F3(Q2)

4M2N

](6.25)

or,

Πj(Γk) = C

[(EN −mN)α1F1(Q2)δj,k

2MN

+qkqjF3(Q2)

4M2N

+ FA

((E2

N −M2N)δkj − qkqj

)+

α1F2(Q2)(

(2ENMN − 2M2N)δkj + qkqj

)4M2

N

, (6.26)

where C is a kinematic factor given as,

C =

√2M2

N

EN(EN +MN). (6.27)

In Eqs. (6.25) and (6.26) the F3(0) form factor cannot be extracted directly from thelattice results because the kinematical factor multiplying F3 vanishes at zero momen-tum transfer. One option we have is to calculate F3(Q2) for several values of Q2 andthen we extrapolate to zero momentum transfer using a dipole fit, of the form

F3(Q2) =F3(0)

(1 +Q2/m2)2(6.28)

treating F3(0) and m2 as fitting parameters.The second option is to apply a continuum derivative with respect to the momentum

transfer in both sides of Eq. (6.25),

∂Π0(~q; Γk)

∂qj

∣∣∣∣∣qj→0

= δkj

[F3(0)

2MN

+α1

2MN

(F1(0) + 2F2(0)

)]. (6.29)

The derivative on the right hand side of Eq. (6.29) is well-defined because it is appliedin the continuum, but on the left hand side we must apply its lattice version. Thederivative acting on two-point functions in the ratio of Eq. (3.45) vanishes exactly in

87

the limit of zero momentum transfer. The only term which survives is the one wherethe derivative acts on the three-point function,

∂R0(ts, tins, ~q; Γk)

∂qj

∣∣∣∣∣qj→0

=

∂∂qjG0Q(ts, tins, ~q; Γk)

G(ts; Γ0). (6.30)

The three-point function in the momentum space is

G(~q, t) =∑~x

G(~x, t)e−i~x·~q (6.31)

and thus the derivative introduces a factor of −i~x, thus

∂R0(ts, tins, ~q; Γk)

∂qj

∣∣∣∣∣qj→0

=

∑~x

G0Q(ts, tins, ~x; Γk)(−i~x)

G(ts; Γ0). (6.32)

The continuum derivative is obtained only when the lattice spatial length L −→ ∞,but for finite lattice size it may carry large volume effects. This approximation ofthe continuum derivative may introduce a systematic error which needs to be checked.It is expected that for large L this will be small. In finite volume this expressionapproximates the derivative of a δ-distribution in momentum space

a3∑~x

(−i~x)G0Q(ts, tins; ~x) =

1

V

∑~k

(a3∑~x

(−i~x)ei~k·~xG0

Q(ts, tins;~k)

)

L→∞−−−−−−→1

(2π)3

∫d3~k

∂

∂kjδ(3)(~k)G0

Q(ts, tins;~k), (6.33)

which implies a residual time-dependence of e−∆Etins [133] in the three-point function,where ∆E = E(~q) −mN is the momentum transfer between the final and the initialstate and ∆E → 0 for L→∞.

6.3 Results for nEDM using disconnected quark

loops

In order to extract the CP-odd form factor F3(0) from Eq. (6.29) we have to subtractthe term proportional to α1. The quantity α1 can be extracted directly from the ratioof two-point functions as given in Eq. (6.23). This ratio is plotted in Fig. 6.2, versusthe source-sink time separation ts/a. As we can see, at some point the ratio becomestime independent and we can extract the α1 parameter by performing a constant fit.The determination of α1 through Rα1(t) is rather precise.

The next quantity needed is the derivative of the ratio, given in Eq. (6.32). Theresults are plotted in Fig. 6.3 with respect to the source-insertion time separationwhen we keep fixed the sink at ts/a = 12. The plateau is rather stable albeit with anerror at 33% of the mean value. In addition, the ratio appears to be time independent,which means that the volume is large enough and the lattice derivative approaches thecontinuum derivative within the statistical error of this calculation.

As we explained, the derivative method is not the only option for extracting F3(0).One can also use Eq. (6.26) to extract F3(Q2) for several Q2 and perform a dipole

88

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0 2 4 6 8 10 12 14 16 18

Rα1(t s

)

ts/a

Figure 6.2: The ratio from which we extract the α1 parameter.

0

0.04

0.08

0.12

0.16

0.2

0 2 4 6 8 10 12

dR/d

q

tins/a

Figure 6.3: Lattice results for the derivative acting on the ratio at zero momentum transfer.

fit using Eq. (6.28) to extrapolate to zero momentum transfer. This method has twodrawbacks: first, we need to fit the Q2-dependence to an ansatz such as a dipole, andsecond, fitting with two parameters yields a result with larger error compared to aconstant fit. The comparison between the two methods is presented in Fig. 6.4. Inthe top panel we show the dipole behaviour of F3(Q2) for several Q2. The dipole fit isperformed over the 5 lowest momenta and with extrapolation we obtain the value ofF3(0). The bottom panel shows results for the derivative of the ratio after we subtractthe term proportional to α1 using Eq. (6.29). The two methods produce compatibleresults with the error being (15 − 20)% of the mean value. In terms of statistics weuse ∼ 150000 measurements for the extraction of α1, while for the extraction of F3(0)we use 15000 measurements. The final value of the neutron electric dipole momentextracted for the B55.32 ensemble is

|~dN |latt = −0.205(45) θ (6.34)

89

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Q2 (GeV2)

-0.25

-0.2

-0.15

-0.1

-0.05

0

F 3 (Q

2 )

n

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0 2 4 6 8 10 12

F3(

0)

tins/a

Figure 6.4: Top: The open circles show results for F3(Q2) versus Q2. The dashed line showsthe dipole fit and the filled blue circles show the extrapolated value at zero momentum transfer.Bottom: Results on the derivative method after the subtraction of term proportional to α1,from where we extract directly F3(0).

in lattice units, while in physical units is

|~dN | = −0.0168(37) θ e fm. (6.35)

This value is compatible with the recent work in Ref. [134]. If we use the most recentexperimental result from Eq. (6.1) we can give an upper bound to the θ angle,

|θ| . 2.2× 10−11. (6.36)

In order to extract the nEDM using the pseudo-scalar disconnected quark loops,we made use of the axial Ward identity, since it allows us to replace the topologicalcharge with the pseudo-scalar density. On the lattice the relation between the twodefinitions of the topological charge is susceptible to lattice artifacts. This systematicuncertainty vanishes only if we extrapolate our results to the continuum limit wherethe two definitions (gluonic and fermionic) coincide. Nevertheless the disconnectedquark loops provide an alternative way of measuring the nEDM with lattice QCD.

90

Chapter 7

Summary and Conclusions

After a brief introduction of the standard model we presented a discussion of thepath integral formulation used in the description of gauge field theories. The latticeQCD formulation was explained in chapter 2 with a detailed discussion of the latticesymmetries. The discretization of the QCD action and the recovery of the continuumlimit was presented paying attention to discretization artifacts, as for example thefermion doubling problem. We introduced Wilson fermions which provide a solution tothe doubling problem at the cost of explicitly breaking chiral symmetry. Discretizationschemes that preserve chiral symmetry at finite lattice spacing were introduced. Dueto the fact that chiral fermion actions are computationally demanding we performedour analyses using the twisted mass fermion formulation which has the advantage ofproviding O(a) improvement. We discussed the twisted mass transformation of thefermion fields and how we can relate them to the physical fields. We showed howautomatic O(a) improvement is achieved for the extracted observables when employingmaximally twisted fermions.

In chapter 3 we introduced various interpolating fields, with the quantum numbersof the particles we studied. We explained how discrete symmetries such as parity andcharge conjugation are used to construct interpolating fields and how the appropriatetwo-point correlation functions yield the mass of mesons and baryons. Improved meth-ods were employed to achieve a better signal, such as APE smearing for smoothingthe gauge link variables and Gaussian smearing producing broader fermion fields, inorder to accomplish better overlap with the ground state. Utilizing these techniqueswe demonstrated how one obtains the mass spectrum for the low-lying mesons andbaryons. Furthermore, we discussed how we can study matrix elements using three-point correlation functions. We explained extensively how we can isolate the matrixelements and relate lattice results with observed quantities in the continuum. Tech-niques such as the fixed-sink and the fixed-insertion methods, were discussed and usedto carry out the calculation of three-point functions. We included a discussion of theconnected diagrams since these contribute significantly in the calculation of the quan-tities of interest.

The calculation of disconnected diagrams is nowadays feasible. Several methodssuitable for the evaluation of disconnected diagrams were assessed in chapter 4. Theall-to-all propagator needed for the calculation of disconnected quark loops is a verycomputationally demanding quantity. Therefore, we exploited the power of GPUs inorder to speed up their computation. We explored new noise reduction methods, suchas the truncated solver method which reduces the computational cost. This allowedus to invert for many noise vectors, though care was taken in order to correct thebias introduced by using low precision for the inversions. We showed how to tunethe parameters entering the TSM in order to have this bias under control. Severaldilution schemes are also studied, including the most commonly used one, time-dilution.

91

Spin and color dilutions were also considered and compared to volume dilutions. Thehopping parameter expansion, which expands the inverse Dirac operator as a series withrespect to the hopping term, was also investigated. HPE is effective when the mass termdominates the hopping term as is the case for heavy quarks, since it can be combinedwith time-dilution without any considerable overhead. Special emphasis was given inthe introduction of the one-end trick for the specific case of the twisted mass fermions,where it was used to reduce the noise of the disconnected quark loops. We introducedtwo different kinds of one-end trick and we explained when each one is applicable.Generalizing, the one-end trick enabled us to apply this approach to calculate thedisconnected loop for all time-slices. This allowed us to employ the summation methodas an alternative method to the plateau method for the extraction of matrix elements.Having an additional method to investigate the excited states contamination ensuredthat systematic effects due to excited states were controlled. These studies were carriedout for a pion mass of about 373 MeV. TSM was found to be a very effective methodfor the computation of quark loops involving the light and strange quarks. For heavierquark masses such as the charm its effectiveness was not as significant. Observables,such as the σ-term, was found to require only few noise vectors, while on the contrary,was observed that gA slowly converged with the number of noise vectors. For the caseof time-dilution, our investigation showed that only after combining with HPE couldwe obtain reliable results. After considering both the computational cost as well asthe impact on the stochastic error, we opted for TSM in combination with the one-endtrick as the method of choice for all three quark sectors. Although time-dilution wasfound to be advantageous in certain cases the fact that it produces the fermion loopat a single time-slice only, means that one-derivative operators cannot be computedwithout additional inversions, incurring additional computational cost as compared tothe one-end trick. One-end trick was therefore chosen as the most versatile approachsince it produces loops for all time-slices allowing us to combine them with two-pointfunctions from several source positions to reduce the statistical error.

Computation of disconnected contributions is essential if we want to have controllederrors to the extracted observables. For some observables disconnected contributionsare negligible while for others they are very important and must be taken into account.Using TSM in combination with the one-end trick in chapter 5 we discussed physicalresults on all nucleon observables involving ultra-local and one-derivative operators.The results were computed using high statistics for the B55.32 ensemble. Specifically,we computed the disconnected loops involving the light quark using NLP = 500 andNHP = 24, while for the strange and charm quark we used NLP = 300 and NHP = 24.These disconnected quark loops were combined with two-point functions from 16 sourcepositions to create three-point functions for 3000 configurations. Averaging over pro-ton, neutron, forward and backward propagation, statistics of approximately 150000measurements were obtained. We presented results on the nucleon σ-term, includingcontributions from the light, strange and charm quark sectors. This observable wasfound to be strongly affected by excited states contributions requiring us to increasethe source-sink time separation beyond 1.5 fm in order to suppress them. We alsofound that disconnected contributions to σπN and σs are significant and thus cannotbe neglected. In contrast, a very noisy signal was obtained for σc, compatible withzero. Results on the yN parameter are obtained at several pion masses and by chirallyextrapolating we obtained a value of yN = 0.135(46). The σ-terms of hyperons andcharmed baryons are also extracted. We have shown representative results, namely,Λ0,Ω− and Ω++

ccc . Both connected and disconnected contributions were evaluated inorder to probe the importance of the disconnected contribution compared to the con-

92

nected one. We observed that the disconnected contributions are (20 − 25)% of theconnected, for baryons made of light quarks such as the nucleon and Λ0. For veryheavy baryons, like Ω++

ccc , the connected part dominates, with the disconnected beingabout 6% of the connected one. Disconnected contributions to the electromagneticform factors were examined for the first non-zero momentum transfer and found tobe compatible with zero. The nucleon axial charge is another quantity of interest.While the isovector combination has no disconnected diagrams in the isospin limit, theisoscalar combination receives contributions from disconnected diagrams and a non-zero value is obtained for the disconnected contributions to gu+d

A and gsA. The light andstrange disconnected parts contribute with a value of opposite sign from the connectedone, which is ∼ 15% of the total contribution and thus must be taken into account. Wehighlight that for the case of the nucleon axial form factor we also observed a clear non-zero signal for the disconnected contributions to the first non-zero momentum transfer.Regarding the tensor charge, the results showed a very small or even zero contributionfrom the disconnected diagrams and thus can be neglected.

Matrix elements of one-derivative operators enter in the evaluation of observablessuch as 〈x〉u+d and 〈x〉∆u+∆d. For 〈x〉u+d we performed an improved analysis usingmeasurements from different moving frames, but the noise was still found to be largeand did not allowed extracting any reliable value for the disconnected contributions tothis quantity. The only conclusion that we can draw is that the disconnected contri-bution is smaller than 17% of the connected one. Regarding 〈x〉∆u+∆d the statisticalerrors are large, giving a bound for the disconnected contribution which is significant.

The ratio 〈x〉(3)/〈x〉(8) was also investigated in an attempt to understand the sourceof the discrepancy between the phenomenological and lattice results for 〈x〉(3) and〈x〉(8). We used a number of Nf = 2 + 1 + 1 twisted mass fermion ensembles. Namely,we used ensembles spanning pion masses in the range of (300-500) MeV with two latticespacings and two volumes. For this study, we concluded that for the 〈x〉(3) there is adiscrepancy with the experiment, but that the ratio agrees with the experimental value.

In chapter 6 we presented our results for the nEDM using disconnected diagrams.A CP-violating term was added to the QCD Langrangian giving rise to the CP-oddform factors F3(Q2) and FA(Q2). The neutron electric dipole moment was extracted

from |~dN | = F3(0)2MN

. The introduction of the θ-term in the Langrangian would make theaction complex, which would suffer from a sign problem if simulated. Since we expectthe θ parameter to be small we treated it as a perturbation. The θ-term involves thetopological charge which is defined as the trace of the gluon tensor operator squared.The topological charge using the gluonic definition can be calculated on the lattice usingcooling or the Wilson flow method. Instead of using this purely gluonic definition wechose here to substitute the topological charge with the pseudoscalar density. In orderto compute the topological charge using the pseudoscalar density we used disconnectedquark loops. Keeping terms only at first order in θ we isolated F3(Q2). Due to thefact that F3(0) cannot be extracted directly from the lattice results we explained twomethods that we can employ in order to obtain its value at Q2 = 0 on the lattice. In thefirst one we assume that the Q2-dependence of F3(Q2) is of a dipole form, and fitted tothis form and extrapolated to zero momentum transfer to extract F3(0). In the secondmethod a continuum derivative is employed to eliminate the kinematical term thatvanishes at Q2 = 0. Using these two approaches we determined the nEDM for a pionmass of 373 MeV in units of the θ parameter. Its value reads |~dN | = −0.0168(37) θ e fm.Using the most recent experimental upper bound for the value of nEDM we obtainedan upper bound for |θ| . 2.2× 10−11.

There is work in progress using Nf = 2 twisted mass fermions including the clover

93

term at physical pion mass. Due to the fact that inversions require a large numberof solver iterations to converge as the pion mass is decreased, new methods are beingemployed to efficiently reduce the number of iterations. One such method involvescalculating the lowest eigenvectors in order to deflate the initial vector given in theCG iterative procedure. At the physical point after calculating several hundred vectorswe observe between 10- to 20-times speed-ups. This enables us to calculate two-pointfunctions for a large number of source positions, as well as disconnected quark loopsfor many noise vectors in order to reduce the stochastic error for quantities such as theaxial charge.

In future work we plan to use the calculated low-mode eigenvectors for the eval-uation of the disconnected quark loops. This can be done by expressing the inversematrix in two parts, one that can be calculated exactly and one that is estimatedstochastically, thus

M−1 = UΛ−1U † +M−1def (7.1)

where U is a matrix holding the lowest eigenvectors and Λ is a diagonal matrix with thecorresponding eigenvalues. The first term in Eq. (7.1) can be estimated exactly fromthe set of the computed eigenvectors, whereas the second term M−1

def is the deflatedoperator which can be computed stochastically, i.e. with the one-end trick. Thistechnique is expected to reduce the stochastic error since it is the low modes of the Diracoperator which are expected to be more susceptible to fluctuations due to stochasticnoise.

94

A. Linear Algebra and Group Theory Definitions

Appendix A

Linear Algebra and Group TheoryDefinitions

In this appendix we collect basic definitions and conventions of the Lie groups SU(N)- the Special Unitary Groups - and their corresponding Lie algebras SU(N).

A.1 Basic properties

One representation of an SU(N) group consists of N ×N complex matrices whichare unitary and have the determinant equal to 1, i.e. if U1 and U2 are elements of SU(N)they obey U †i = U−1

i and det(Ui) = 1. Using standard linear algebra manipulationsone can obtain

(U1U2)† = U †2U†1 = U−1

2 U−11 = (U1U2)−1 ,

det(U1U2) = det(U1) det(U2) = 1 . (A.1)

These relations establish that the product of two SU(N) elements is an element ofSU(N). The unit matrix 1N also belongs to SU(N) and for each matrix in SU(N) thereexists its hermitian conjugate matrix. Thus, the set SU(N) forms a group. Since thegroup operation - in this case the matrix multiplication - is not commutative in general,the groups SU(N) for N > 1 are non-abelian groups.

A.2 Lie Algebra

In order to describe the matrices in SU(N) one needs to define the generators of thealgebra of SU(N). These generators are matrices that form a basis and therefore they arelinearly independent from each other. The number of generators of an SU(N) algebradepends on how many independent parameters are needed to describe the matricesin SU(N). A complex N × N matrix consists of 2N2 real parameters in total. Therequirement of unitarity immediately reduces the number of independent parametersto N2. One more parameter needs to be fixed so that the determinant condition isobeyed. Thus, one needs a total of N2 − 1 real parameters for describing SU(N)matrices and therefore needs N2 − 1 generators.

A convenient way of representing an SU(N) matrix is to express it as an exponentialof the generators. In particular, an element U of SU(N) is written as

U = exp

(i

N2−1∑j=1

φ(j)Tj

), (A.2)

95


where φ(j) are used to parametrize U and Tj are the generators of the algebra. Weremark that parameters φ(j) can take continuous values, making elements of the SU(N)Lie-groups to depend continuously on these parameters. The generators Tj are chosenas traceless, complex and hermitian N × N matrices which are related among eachother by the commutation relations

[Tj, Tk] = ifjklTl , (A.3)

where we sum over the index l. The real numbers fjkl are the so-called structureconstants. Moreover, one can choose the generators so that they obey the normalizationcondition

tr(TjTk) =1

2δjk . (A.4)

If the above relation is satisfied, the structure constants coincide with the fully anti-symmetric tensor Levi-Civita, εjkl.

A.3 Generators of SU(2) and SU(3) groups

As discussed, an SU(N) group is described by N2 − 1 generators. Thus, for SU(2)we need 3 generators. The standard representation of the SU(2) generators is relatedto the Pauli matrices (see B.1) and the generators are given by

Tj =1

2τj . (A.5)

For the Pauli matrices, Eq. A.4 is satisfied by definition, and therefore the structureconstants of SU(2) are related to the Levi-Civita tensor.

For SU(3) we need 8 generators given by

Tj =1

2λj , (A.6)

where λj are the 3× 3 generalizations of the Pauli matrices, known as the Gell-Mannmatrices and they are defined as

λ1 =

0 1 01 0 00 0 0

, λ2 =

0 −i 0i 0 00 0 0

, λ3 =

1 0 00 −1 00 0 0

,

λ4 =

0 0 10 0 01 0 0

, λ5 =

0 0 −i0 0 0i 0 0

, λ6 =

0 0 00 0 10 1 0

,

λ7 =

0 0 00 0 −i0 i 0

, λ8 =1√3

1 0 00 1 00 0 −2

. (A.7)

These obey the relations[T a, T b] = ifabcT c (A.8)

and

T a, T b =1

3δab + dabcT c. (A.9)

96


The structure constants are given by:

f 123 = 1

f 147 = −f 156 = f 246 = f 257 = f 345 = −f 367 =1

2

f 458 = f 678 =

√3

2(A.10)

and all other fabc not related to these by permutation are zero. The d take the values:

d118 = d228 = d338 = −d888 =1√3

d448 = d558 = d668 = d778 = − 1

s√

3

d146 = d157 = −d247 = d256 = d344 = d355 = −d366 = −d377 =1

2. (A.11)

97

B. Pauli matrices and γ - matrices, Charge Conjugation Operator

Appendix B

Properties of γ-matrices and γ-bases

B.1 Pauli Matrices

Pauli matrices have the following form

τ1 =

(0 11 0

), τ2 =

(0 −ii 0

), τ3 =

(1 00 −1

). (B.1)

B.2 γ - matrices in the Dirac basis in Minkowski

space

In the Dirac representation, the γ - matrices are defined as

γ0 ≡ β , γi ≡ βαi , (B.2)

where

αi =

(0 τiτi 0

), β =

(12 00 −12

)(B.3)

and 12 is the 2× 2 unit matrix. Explicitly, the γ - matrices in the Dirac basis have thefollowing form

γ0 =

1 0 0 00 1 0 00 0 −1 00 0 0 −1

, γ1 =

0 0 0 10 0 1 00 −1 0 0−1 0 0 0

, (B.4)

γ2 =

0 0 0 −i0 0 i 00 i 0 0−i 0 0 0

, γ3 =

0 0 1 00 0 0 −1−1 0 0 00 1 0 0

, (B.5)

while γ5 is defined as

γ5 ≡ iγ0γ1γ2γ3 =

0 0 1 00 0 0 11 0 0 00 1 0 0

, (B.6)

and σµν is defined as

σµν =i

2[γµ, γν ]. (B.7)

98


B.3 Weyl or chiral basis in Minkowski space

Another commonly used basis for the γ - matrices is the chiral or Weyl basis, inwhich the matrices γi, i = 1, 2, 3 have the same form as those in the Dirac basis, butγ0 and γ5 acquire a different form according to

γW0 =

0 0 1 00 0 0 11 0 0 00 1 0 0

, γW5 = iγW0 γ1γ2γ3 =

−1 0 0 00 −1 0 00 0 1 00 0 0 1

. (B.8)

B.4 Chiral and non-relativistic bases in Euclidean

space

In Euclidean space there are two commonly used representations of the γ - matrices,the chiral representation and the non-relativistic representation. In both representa-tions γ1, γ2 and γ3 have the following form

γ1,2,3 =

(0 −iτ1,2,3

iτ1,2,3 0

). (B.9)

In the chiral representation γ4 and γ5 matrices are defined as

γ4 =

(0 12

12 0

), γ5 = γ1γ2γ3γ4 =

(12 00 −12

), (B.10)

whereas in the non-relativistic representation they have the following form

γ4 =

(12 00 −12

), γ5 =

(0 −12

−12 0

). (B.11)

B.5 Properties of the γ - matrices

In Minkowski space γ - matrices obey

γµ, γ5 = 0, γµ, γν = 2gµν14, µ, ν = 1, . . . , 4 (B.12)

where gµν is the Minkowski metric with signature (+ − −−) and 14 is the 4 × 4 unitmatrix. Furthermore,

(γ0)† = γ0 and (γi)† = −γi, i = 1, 2, 3 . (B.13)

In Euclidean space the metric gµν in Eq. (B.12) is replaced with the Kronecker δµν andEq. (B.13) becomes

(γµ)† = γµ, µ = 1, 2, 3, 4, 5 . (B.14)

B.6 Rotation from Minkowksi to Euclidean space

Minkowski and Euclidean metrics are equivalent if one permits the coordinate t to takeon imaginary values. Then, we can write the relations:

xµ = (−xE0 , ~xE), ∇µ = (−i∇E0 , ~∇E), pµ = (−ipE0 , ~pE), pµ = (−ipE0 ,−~pE) (B.15)

99


γ0 = γ0 = γE0 , γk = −γk = iγEk , 6p = −i 6pE, γ5 = γE0 γE1 γ

E2 γ

E3 = γE5 (B.16)

σabM =i

2[γa, γb], σabE =

1

2[γa, γb], σkapa =

i

2[γEk , γ

Ea ]pa = iσEkap

Ea (B.17)

εabcdpcpd =1

4Tr[γEa γ

Eb γ

Ec γ

Ed γ

E5 ]pEc p

Ed = εEabcdp

Ec p

Ed , εabcdpcγd = iεEabcdpcγ

Ed (B.18)

B.7 Charge Conjugation operator

Throughout, we use the Euclidean chiral representation of the γ - matrices, andtherefore it is convenient to define the Charge Conjugation operator in this space. Itis given by

C = iγ2γ4 (B.19)

and it has the following properties

C = C−1 = C† = −CT ,

γ5 = Cγ5C−1 ,

CγµC−1 = −γTµ , µ = 1, . . . , 4 . (B.20)

100

C. Symmetries in the twisted basis

Appendix C

Symmetries in the twisted basis

Twisted mass QCD and the standard QCD actions are exactly related by the trans-formation given in Eq. (2.48) and therefore, share all the symmetries. However, in thetwisted basis the symmetry transformations may take a somewhat unusual, twistedform. Here we list the form of the relevant symmetries for a generic angle ω. TheSU(2) axial vector transformations take the form

SUV (2)ω :

χ(x) −→ exp(−iω

2γ5τ

3) exp(iαV2τa) exp(iω

2γ5τ

3)χ(x)

χ(x) −→ χ(x) exp(iω2γ5τ

3) exp(−iαV2τa) exp(−iω

2γ5τ

3)

(C.1)

SUA(2)ω :

χ(x) −→ exp(−iω

2γ5τ

3) exp(iαA2τa) exp(iω

2γ5τ

3)χ(x)

χ(x) −→ χ(x) exp(iω2γ5τ

3) exp(−iαA2τa) exp(−iω

2γ5τ

3)

. (C.2)

The twisted mass discrete symmetries that involve axis reflections (parity P and timereversal T ) are:

Pω :

U(x0,x; 0) −→ U(x0,−x; 0), U(x0,x, k) −→ U−1(x0,−x− ak; k)

χ(x0,x) −→ γ0 exp(iωγ5τ3)χ(x0,−x)

χ(x0,x) −→ χ(x0,−x) exp(iωγ5τ3)γ0

, (C.3)

Tω :

U(x0,x; 0) −→ U−1(−x0 − a,x; 0), U(x0,x, k) −→ U(−x0,x; k)

χ(x0,x) −→ iγ0γ5 exp(iωγ5τ3)χ(−x0,x)

χ(x0,x) −→ iχ(−x0,x) exp(iωγ5τ3)γ5γ0

. (C.4)

The charge conjugation takes a form which is invariant under the twist transformation

C

U(x;µ) −→ U(x;µ)∗

χ(x) −→ C−1χ(x)T

χ(x) −→ −χ(x)TC

. (C.5)

101

D. Interpolating field operators

Appendix D

Interpolating field operators

In the following tables we give the interpolating fields for baryons. Throughout,C denotes the charge conjugation matrix and the transposition symbol refers to Diracindices which are suppressed.

Charm Strange Baryon Content Interpolating field

c = 2s = 0

Ξ++cc ucc εabc

(cTaCγ5ub

)cc

Ξ+cc dcc εabc

(cTaCγ5db

)cc

s = 1 Ω+cc scc εabc

(cTaCγ5sb

)cc

c = 1

s = 0

Σ++c uuc εabc

(uTaCγ5cb

)uc

Σ+c udc 1√

2εabc

[(uTaCγ5cb

)dc +

(dTaCγ5cb

)uc]

Σ0c ddc εabc

(dTaCγ5cb

)dc

s = 1Ξ′+c usc 1√

2εabc

[(uTaCγ5cb

)sc +

(sTaCγ5cb

)uc]

Ξ′0c dsc 1√2εabc

[(dTaCγ5cb

)sc +

(sTaCγ5cb

)dc]

s = 2 Ω0c ssc εabc

(sTaCγ5cb

)sc

s = 0 Λ+c udc 1√

6εabc

[2(uTaCγ5db

)cc +

(uTaCγ5cb

)dc −

(dTaCγ5cb

)uc]

s = 1Ξ+c usc 1√

6εabc

[2(sTaCγ5ub

)cc +

(sTaCγ5cb

)uc −

(uTaCγ5cb

)sc]

Ξ0c dsc 1√

6εabc

[2(sTaCγ5db

)cc +

(sTaCγ5cb

)dc −

(dTaCγ5cb

)sc]

c = 0

s = 0p uud εabc

(uTaCγ5db

)uc

n udd εabc(dTaCγ5ub

)dc

s = 1

Λ uds 1√6εabc

[2(uTaCγ5db

)sc +

(uTaCγ5sb

)dc −

(dTaCγ5sb

)uc]

Σ+ uus εabc(uTaCγ5sb

)uc

Σ0 uds 1√2εabc

[(uTaCγ5sb

)dc +

(dTaCγ5sb

)uc]

Σ− dds εabc(dTaCγ5sb

)dc

s = 2Ξ0 uss εabc

(sTaCγ5ub

)sc

Ξ− dss εabc(sTaCγ5db

)sc

Table D.1: Interpolating fields and quantum numbers for the 20′-plet of spin-1/2 baryons.

102

D. Interpolating field operators

Charm Strange Baryon Content Interpolating fieldc = 3 s = 0 Ω++

ccc ccc εabc(cTaCγµcb

)cc

c = 2s = 0

Ξ?++cc ucc 1√

3εabc

[2(cTaCγµub

)cc +

(cTaCγµcb

)uc]

Ξ?+cc dcc 1√

3εabc

[2(cTaCγµdb

)cc +

(cTaCγµcb

)dc]

s = 1 Ω?+cc scc 1√

3εabc

[2(cTaCγµsb

)cc +

(cTaCγµcb

)sc]

c = 1

s = 0Σ?++c uuc 1√

3εabc

[(uTaCγµub

)cc + 2

(cTaCγµub

)uc]

Σ?+c udc

√23εabc

[(uTaCγµdb

)cc +

(dTaCγµcb

)uc +

(cTaCγµub

)dc]

Σ?0c ddc 1√

3εabc

[(dTaCγµdb

)cc + 2

(cTaCγµdb

)dc]

s = 1Ξ?+c usc

√23εabc

[(uTaCγµsb

)cc +

(sTaCγµcb

)uc +

(cTaCγµub

)sc]

Ξ?0c dsc εabc

(sTaCγµdb

)cc

s = 2 Ω?0c ssc

√23εabc

[(dTaCγµsb

)cc +

(sTaCγµcb

)dc +

(cTaCγµdb

)sc]

c = 0

s = 0

∆++ uuu εabc(uTaCγµub

)uc

∆+ uud 1√3εabc

[2(uTaCγµdb

)uc +

(uTaCγµub

)dc]

∆0 udd 1√3εabc

[2(dTaCγµub

)dc +

(dTaCγµdb

)uc]

∆− ddd εabc(dTaCγµdb

)dc

s = 1Σ?+ uus 1√

3εabc

[(uTaCγµub

)sc + 2

(sTaCγµub

)uc]

Σ?0 uds√

23εabc

[(uTaCγµdb

)sc +

(dTaCγµsb

)uc +

(sTaCγµub

)dc]

Σ?− dds 1√3εabc

[(dTaCγµdb

)sc + 2

(sTaCγµdb

)dc]

s = 2Ξ?0 uss 1√

3εabc

[2(sTaCγµub

)sc +

(sTaCγµsb

)uc]

Ξ?− dss 1√3εabc

[2(sTaCγµdb

)sc +

(sTaCγµsb

)dc]

s = 3 Ω− sss εabc(sTaCγµsb

)sc

Table D.2: Interpolating fields and quantum numbers for the 20-plet of spin-3/2 baryons.

Charm Strange Baryon Content Interpolating field

Spin-1/2 baryons

c = 1 s = 1Ξ+c usc εabc

(uTaCγ5sb

)cc

Ξ0c dsc εabc

(dTaCγ5sb

)cc

Spin-3/2 baryons

c = 0 s = 2Ξ?0 uss εabc

(sTaCγµub

)sc

Ξ?− dss εabc(sTaCγµdb

)sc

c = 1s = 1

Ξ?+c usc εabc

(sTaCγµub

)cc

Ξ?0c dsc εabc

(sTaCγµcb

)sc

s = 2 Ω?0c ssc 1√

3εabc

[2(sTaCγµcb

)sc +

(sTaCγµsb

)cc]

c = 2s = 0

Ξ?++cc ucc εabc

(cTaCγµub

)cc

Ξ?+cc dcc εabc

(cTaCγµdb

)cc

s = 1 Ω?+cc scc εabc

(cTaCγµsb

)cc

Table D.3: Additional interpolating fields for spin-1/2 and spin-3/2 baryons.

103

E. Fixed-sink method

Appendix E

Nucleon three-point function

In this section we show explicitly how to generate the nucleon three-point function(connected part) and how to construct the sequential source, needed to compute severalnucleon matrix elements. As an illustration we use the isovector electromagnetic localcurrent as an insertion operator,

jρ(x) = u(x)γρu(x)− dγρd(x). (E.1)

The three-point function is defined by,

Gραβ(x2, x1) = 〈Ω|J N

α (x2)jρ(x1)J Nβ (0)|Ω〉, (E.2)

where the interpolating operator for the nucleon is taken as

J Nα (x) = εabc

(uaκ(x)(Cγ5)κπd

bπ(x)

)ucα(x). (E.3)

Expanding the three point function we have,

Gραβ(x2, x1) = 〈Ω|εabcuaµ(x2)(Cγ5)µνd

bν(x2)ucα(x2)×[

ufκ(x1)(γρ)κλufλ(x1)− df ′κ′(x1)(γρ)κ′λ′d

f ′

λ′(x1)]×

εa′b′c′uc

′

β (0)(Cγ5)ν′µ′ db′

ν′(0)ua′

µ′(0)|Ω〉. (E.4)

We split the three-point function into the terms that give the coupling of the currentnamely with the up quark, Uραβ , and those where the current couples with the downquark, Dραβ ,

Gραβ = Uραβ −Dραβ (E.5)

where

Uραβ(x2, x1) = 〈Ω|εabcuaµ(x2)(Cγ5)µνdbν(x2)ucα(x2)ufκ(x1)(γρ)κλu

fλ(x1)×

εa′b′c′uc

′

β (0)(Cγ5)ν′µ′ db′

ν′(0)ua′

µ′(0)|Ω〉. (E.6)

There are four possible ways to contract the up quark fields and only one for thedown quarks. Disconnected terms are omitted here. Rearranging the color indices

104

E. Fixed-sink method

appropriately while changing the sign of the antisymmetric tensor εabc we have,

Uραβ(x2, x1) = εabcεa′b′c′[Cγ5Gd(x2; 0)Cγ5

]aa′µν×[

Gu(x2;x1)γρGu(x1; 0)

]bb′αβ

Gu(x2; 0)cc′

µν

+

[Gu(x2;x1)γρGu(x1; 0)

]bb′αν

Gu(x2; 0)cc′

µβ

+


]bb′µν

Gu(x2; 0)cc′

αβ

+


]bb′µβ

Gu(x2; 0)cc′

αν

. (E.7)

For the case where the current is coupled to the down quark,

Dραβ(x2, x1) = 〈Ω|εabcuaµ(x2)(Cγ5)µνdbν(x2)ucα(x2)df

′

κ′(x1)(γρ)κ′λ′ ×εa′b′c′uc

′

β (0)(Cγ5)ν′µ′ db′

ν′(0)ua′

µ′(0)|Ω〉, (E.8)

where there is only one combination for the up quark and two for the down,

Dραβ(x2, x1) = εabcεa′b′c′[Cγ5Gd(x2;x1)γρGd(x1; 0)Cγ5

]aa′µν×

Gu(x2; 0)bb′

αβGu(x2; 0)cc′

µν +Gu(x2; 0)bb′

µβGu(x2; 0)cc′

αν

. (E.9)

The disconnected diagram is given as

Dαβ(x2;x1) = −Tr

[(Gu(x1;x1)−Gd(x1;x1)

)γρ

]εabcεa

′b′c′[Cγ5Gd(x2; 0)Cγ5

]bb′µµ′

×(Gu(x2; 0)cc

′

αβGu(x2; 0)aa′

µµ′ −Gu(x2; 0)ca′

αµ′Gu(x2; 0)ac′

µβ

). (E.10)

E.1 Fixed-sink method

In order to generate the fixed sink sequential propagator we need to rearrange theexpressions above. The steps will be explained for the case of Dραβ but a similarprocedure can be followed for Uραβ. For Dραβ we have∑

~x2

Tr

[Dρ(x2, x1)Γ

]=∑~x2

[γρGd(x1; 0)

]ra′µν

Gd(x2;x1)arκµ ×

εabcεa′b′c′

[CγT5 Gu(x2; 0)CγT5

]cc′κν

Gu(x2; 0)cc′

αβΓβα

+

[CγT5 Gu(x2; 0)

]bb′κβ

[Gu(x2; 0)ΓβαCγ

T5

]cc′αν

. (E.11)

We define the expression inside the big curly bracket to be

P (x2; 0)aa′

κν = εabcεa′b′c′

[CγT5 Gu(x2; 0)CγT5

]cc′κν

Gu(x2; 0)cc′

αβΓβα+

[CγT5 Gu(x2; 0)

]bb′κβ

[Gu(x2; 0)ΓβαCγ

T5

]cc′αν

, (E.12)

105

E. Fixed-current method

then∑~x2

Tr [Dρ(x2, x1)Γ] =∑~x2

[γρGd(x1; 0)

]ra′µν

[γ5G

∗u(x1;x2)

]raµκ

[γ5P (x2; 0)

]aa′κν

. (E.13)

We can now identify the sequential source needed as,

K(x2; 0)abµν ≡[γ5P (x2; 0)

]∗abµν

. (E.14)

After we have the sequential source the sequential propagator is,

S(y; 0)abµν ≡∑~x

G(y;x)acµκK(x; 0)cbκν . (E.15)

Finally we have,∑~x2

Tr

[Dρ(x2, x1)Γ

]= Tr

[γρGd(x1; 0)

(γ5S(x1; 0)

)∗]. (E.16)

We can similarly construct the Uραβ and then subtract it from the Dραβ to createthe connected three-point function. As the sequential source must be calculated forevery color and spin index we need in total 12 inversions to construct the sequentialpropagator for a given projector.

E.2 Fixed-current method

The fixed-current method is the second approach for the calculation of the connectedthree-point function. Using Eq. (E.9) the sequential propagator is given by,

S(x2; 0)ρ = Gd(x2;x1)γρGd(x1; 0) (E.17)

where the sequential source is,

K(x1; 0)ρ = γρGd(x1; 0). (E.18)

Then the Dραβ can be written in terms of the sequential propagator as,

Dραβ(x2, 0) = εabcεa′b′c′[Cγ5S(x2)ρCγ5

]aa′µν×

Gu(x2; 0)bb′

αβGu(x2; 0)cc′

µν +Gu(x2; 0)bb′

µβGu(x2; 0)cc′

αν

. (E.19)

Similarly to the fixed-sink method, the fixed-current method needs in total 12 in-versions to construct the sequential propagator for a given γρ.

106

F. Details on implementation in QUDA

Appendix F

Code implementation in QUDA

In this Appendix we explain in detail how we implement the CUDA kernels in theQUDA package and what strategies we follow in order to calculate and store discon-nected quark loops in an efficient way.

F.1 Implementing the twisted mass operator

In this subsection we provide implementation aspects of the twisted mass code inQUDA. The Wilson twisted mass operator for the degenerate flavour doublet reads,

6DTM = 6DW + iµγ5τ3 (F.1)

where 6DW stands for the Wilson Dirac operator, τ 3 denotes the third Pauli matrix,which acts in flavour space and µ is the twisted mass parameter. QUDA adopts twodifferent gamma bases, a chiral basis for calculations taking place on host and the non-relativistic basis for the CUDA kernels executing on GPUs. Namely the DeGrand-Rossibasis is a chiral basis where the gamma matrices are given by

γ1 =

0 0 0 +i0 0 +i 00 −i 0 0−i 0 0 0

, γ2 =

0 0 0 −10 0 +1 00 +1 0 0−1 0 0 0

, (F.2)

γ3 =

0 0 +i 00 0 0 −i−i 0 0 00 +i 0 0

, γ4 =

0 0 +1 00 0 0 +1

+1 0 0 00 +1 0 0

. (F.3)

For the internal computations we switch to the non-relativistic UKQCD basis wherethe γ-matrices are,

γ1 =

0 0 0 +i0 0 +i 00 −i 0 0−i 0 0 0

, γ2 =

0 0 0 +10 0 −1 00 −1 0 0

+1 0 0 0

, (F.4)

γ3 =

0 0 +i 00 0 0 −i−i 0 0 00 +i 0 0

, γ4 =

+1 0 0 00 +1 0 00 0 −1 00 0 0 −1

. (F.5)

The basic difference is in the temporal direction where γ4 becomes diagonal allowingus to reduce memory traffic on GPUs while applying the hopping term in this direction[57].

107


For the QUDA twisted mass iterative solvers, one can employ two types of (even-odd) preconditioning. The asymmetric and symmetric preconditioned systems aregiven by,

(Ree − k2 6DeoR−1oo 6Doe)ψe = be − 6DeoR

−1oo bo (F.6)

(Iee − k2R−1ee 6DeoR

−1oo 6Doe)ψe = R−1

ee (be − 6DeoR−1oo bo) (F.7)

where Ree, Roo represent the twisted diagonal term with a trivial inverse matrix. Thispreconditioning allows us to reduce the size of the problem by a factor of 2, defining abetter conditioned operator. The other half of the solution can be reconstructed by,

ψo = R−1oo (bo − 6DoeR

−1ee ψe). (F.8)

All local operators are merged into dslash kernels and computed on the fly reducingextensive access to GPU global memory. In order to include the operators in the wholeinterface, we add two new classes, the DiracTwistedMass and the DiracTwisted-MassPC which encapsulate all the necessary information for the application of thetwisted mass operator. More details about the QUDA library can be found in Refs.[59, 135, 136].

F.2 Contracting disconnected quark loops

A fundamental step in the calculation of quark loops is the contraction of the invertedsources. To this end, we developed an efficient GPU code yielding ∼ 300 GFlops ina single Tesla m2070 GPU in double precision, showing almost perfect scaling withincreasing the number of GPUs.

Traces were taken in color space, leaving the Dirac and volume indices open. Thevolume indices are used later for the fast Fourier transform (FFT), so we obtain solu-tions for different momenta, while the open Dirac indices are there in order to deal withthe different inserted gamma matrix. We calculated the outer product in Dirac spaceof both sources to be contracted, and consequently a 4× 4 matrix was obtained, withenough information to reconstruct any arbitrary γ insertion just by transposition andmultiplication. Therefore, our contraction code automatically outputs all the possibleinsertions for ultra-local operators. A covariant derivative kernel was also developedto allow the calculation of the one-derivative insertions.

F.3 Interfaces and workflow

Since QUDA already implements most of the code needed for computing disconnecteddiagrams, which is the inversion part, we implemented an interface for the contractions,FFT and the creation of the stochastic sources. Those were designed to calculate anyultra-local and one-derivative operator insertion using several variance reduction meth-ods, namely TSM, the one-end trick (only for twisted mass fermions), time-dilution andhopping parameter expansion, and all the possible combinations of these.

The interface generates random stochastic sources using the ranlux [137] randomnumber generator on the host, then we send the noise source to the device for inversionand contraction and finally we send them back to the host in order to store them onthe hard disk. This process is repeated many times. After we accumulate enough noisevectors we send the data to the GPU and call the cuFFT library [138] for the fastFourier transformation. The transformation returns all momenta, however only a few

108


of them are finally stored on disk. The results are written to disk in binary form inorder to reduce storage requirements.

109

G. Table with results

Appendix G

Table with resultsResults on nucleon observables involving matrix elements of ultra-local and one-derivativeoperators are presented in Table. G.1.

Observable Connected Disconnected total

Results at zero momentum transfer (Q2 = 0)σπN [MeV] 164.6(7.2) 16.6(2.4) 181.3(7.6)σs [MeV] - 21.7(3.6) 21.7(3.6)σc [MeV] - 16(30) 16(30)gu+dS 6.30(27) 0.639(95) 6.94(29)gsS - 0.246(41) 0.246(41)gu+dA 0.576(13) -0.0699(89) 0.506(15)gsA - -0.0227(34) -0.0227(34)gu+dT 0.673(13) -0.0016(14) 0.671(13)〈x〉u+d 0.586(22) 0.027(76) 0.614(80)〈x〉∆u+∆d 0.1948(51) -0.058(22) 0.136(23)

Ju 0.2781(94) -0.076(77) 0.202(78)Jd -0.0029(94) -0.076(77) -0.078(78)

∆Σu/2 0.4273(50) -0.0174(75) 0.4098(55)∆Σd/2 -0.1389(50) -0.0174(75) -0.1564(55)

Results for ~q2 = (2π/L)2 or (Q2 ' 0.19GeV 2)

Gu+dE 2.2698(78) 0.024(21) 2.293(22)

Gu+dM 2.088(49) -0.066(75) 2.022(89)

Gu+dA 0.5155(94) -0.0564(72) 0.459(11)

Gu+dp 9.81(65) -1.90(35) 7.90(74)

Bu+d20 -0.035(16) -0.33(29) -0.36(29)GpE 0.7453(32) 0.0040(58) 0.7493(47)

GnE 0.0113(32) 0.0040(48) 0.0153(47)

GpM 1.847(28) -0.011(42) 1.836(31)

GnM -1.151(28) -0.011(42) -1.162(31)

|~dN | - - −0.0168(37) θ e fmθ - - . 2.2× 10−11

Table G.1: In the first column we give the observables, in the second column the connectedcontribution, in the third column the disconnected contribution and in the fourth column isthe total value. The results were extracted from an Nf = 2 + 1 + 1 ensemble of twisted massfermions at mπ = 373 MeV.

110

G. Table with results

Observable valueyN 0.135(56)〈x〉(3)〈x〉(8) 0.39(4)

Table G.2: Estimation of the yN parameter and 〈x〉(3)

〈x〉(8) by extrapolation at the physical point.

111

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