Gluon correlation functions from lattice quantum chromodynamics
aa r X i v : . [ h e p - l a t ] J a n Gluon Correlation Functions fromLattice Quantum Chromodynamics
Guilherme Telo Rodrigues Catumba
Supervisors:Orlando OliveiraPaulo Silva
A thesis submitted for the degree ofMaster in Physics
Departmento de FísicaFaculdade de Ciência e TecnologiaUniversidade de CoimbraOctober 2020 i bstract This dissertation reports on the work developed in the past year by the author and incollaboration with his supervisors, Prof. Dr. Orlando Oliveira and Dr. Paulo Silva. Themain topic of the thesis is the study of the gluon sector in pure Yang-Mills theories viathe computation of two, three and four point Landau gauge gluon correlation functionsevaluated using the lattice formalism of QCD. Monte-Carlo simulations reported hereinuse the Wilson gauge action for lattice QCD.The first goal was to understand and quantify the deviations, relative to the usualcontinuum description of lattice correlation functions, introduced by using appropriatelattice tensors. To achieve this we rely on different lattice tensor representations forthe gluon propagator in four dimensions to measure the deviations of the lattice prop-agator from its continuum form. We also identified classes of kinematic configurationswhere these deviations are minimal and the continuum description of lattice tensors isimproved. Other than testing how faithful our description of the propagator is, thesetensor structures also allow to study how the continuum Slavnov-Taylor identity for thepropagator is verified on the lattice for the pure Yang-Mills theory. We found that theSlavnov-Taylor identity is fulfilled, with good accuracy, by the lattice data for the twopoint function.A second goal was the lattice computation of the three gluon vertex using largeensembles of configurations. The so-called zero crossing, a property that is related withthe ghost dominance at the infrared mass scales and puts restrictions on the behaviourof the three gluon vertex, was investigated. In addition, we also explore the possibleexistence of a ghost mass preventing the infrared divergence of the vertex. In our studyof the three gluon correlation function we used functional forms to model the latticedata and explore the two different possibilities for the behaviour of the function. Forthe first case we provide an estimate of the mass scale associated with the zero-crossingand search for a possible sign of the divergence. On the other hand, for the second casewe study the possible occurrence of a sign change and the finite value of the three gluonvertex for vanishing momentum.A last topic is the computation of the four gluon vertex. On the lattice this is aparticularly difficult calculation that requires the subtraction of contributions from lowerorder correlation functions. A suitable choice of kinematics allows to eliminate suchunwanted contributions. Furthermore, large statistical fluctuations hinder the precisecomputation of this object. Our investigation is a proof of concept, we show that thelattice computation of the four gluon correlation function seems to be feasible withreasonable computational resources. Nonetheless, an increase in statistics is necessary toprovide a clearer and precise signal on the complete correlation function and to computethe corresponding one particle irreducible function.
Keywords:
Lattice QCD, Gluon propagator, Gluon correlation functions, Latticetensor representations, Three gluon vertex, Four gluon vertexv esumo
Esta dissertação é o resultado do trabalho desenvolvido ao longo do último ano peloautor e juntamente com os seus orientadores, Prof. Dr. Orlando Oliveira e Dr. PauloSilva. A dissertação consiste no estudo do sector gluónico em teorias de Yang-Millsatravés do cálculo de funções de correlação de dois, três e quatro gluões. Para istoutilizou-se o formalismo da QCD na rede usando simulações de Monte-Carlo com a açãode Wilson na gauge de Landau.O primeiro tópico de estudo passou por analisar os desvios, relativamente ao contínuo,introduzidos pela substituição do espaço-tempo por uma rede de quatro dimensões. Paraisso foram usadas representações tensoriais da rede para calcular o propagador de gluões ecomparadas com a descrição tensorial do contínuo. Com esta análise foram identificadasclasses de configurações cinemáticas para as quais os desvios relativamente à descriçãodo contínuo são reduzidos. Além de testar a integridade da descrição do propagador,é também possível investigar como a identidade de Slavnov-Taylor para o propagadoré validada nas simulações de Monte-Carlo. Os resultados das diferentes representaçõestensoriais mostram que a identidade de Slavnov-Taylor é satisfeita na rede.A função de correlação de três gluões também foi calculada usando dois conjuntos deconfigurações na rede. O objetivo principal foi a análise do comportamento da funçãode correlação no infra-vermelho, nomeadamente, a existência de uma possível troca desinal da função para baixos momentos. Esta propriedade relaciona-se com o domínio doscampos ghost para baixas escalas de momentos e que induz uma possível mudança desinal assim como uma possível divergência. Além desta hipótese, também a possibilidadeda existência de uma massa para o campo ghost que previne a divergência para baixosmomentos foi estudada. Com o objetivo de melhorar a análise, foram usadas formasfuncionais para modelar o vértice de três gluões e estudar as duas possibilidades noinfra-vermelho. Em particular, através dos modelos, a escala para a mudança de sinalfoi avaliada assim como o comportamento geral da função para baixos momentos.O último objetivo foi o cálculo do vértice de quatro gluões, que representa umadificuldade acrescentada, nunca tendo sido avaliado na rede. A dificuldade deve-se àcomplexidade tensorial e às contribuições de vértices de ordem menor que surgem nacomputação da função de correlação completa de quatro gluões. Estas contribuiçõesforam eliminadas através de uma escolha adequada da configuração cinemática. Alémdisso, as flutuações estatísticas são grandes e dificultam a análise. Os resultados demon-straram que o cálculo do vértice de quatro gluões é exequível com recursos computa-cionais acessíveis. No entanto, é fundamental aumentar a precisão no cálculo para obterum sinal mais definido e calcular o vértice sem propagadores externos.
Palavras-chave:
QCD na rede, Propagador do gluão, Funções de correlação degluões, Representações tensoriais na rede, Vértice de três gluões, Vértice de quatro gluõesi cknowledgements ‘A spectre is haunting Europe...’I would like to begin by thanking my supervisors for their exceptional support over thepast year. Both Prof. Dr. Orlando Oliveira and Dr. Paulo Silva were very patientand receptive towards my questions and their attentive guidance was certainly veryimportant. I am grateful for their insight and improvements towards the constructionof this dissertation.Moreover, I would like to thank all my cherished friends whose company throughoutthe past years was fundamental to my growth and without whom this journey wouldhave been much more tedious. A special thanks to all my friends in BiF for the company,affection and all the shared adventures. Likewise, to my childhood friends, thank youfor being caring and for the company throughout this journey.Finally, I wish to express my deepest gratitude to my mother for the strenuous careand dedication.This work was granted access to the HPC resources of the PDC Center for HighPerformance Computing at the KTH Royal Institute of Technology, Sweden, madeavailable within the Distributed European Computing Initiative by the PRACE-2IP,receiving funding from the European Community’s Seventh Framework Programme(FP7/2007–2013) under grand agreement no. RI-283493. The use of Lindgren has beenprovided under DECI-9 project COIMBRALATT. The author acknowledges that the re-sults of this research have been achieved using the PRACE-3IP project (FP7 RI312763)resource Sisu based in Finland at CSC. The use of Sisu has been provided under DECI-12project COIMBRALATT2.It is also important to acknowledge the Laboratory for Advanced Computing atUniversity of Coimbra for providing HPC resources that have contributed to the researchresults reported within this thesis.This work was supported with funds from Fundação para a Ciência e Tecnologiaunder the projects UID/FIS/04564/2019 and UIDB/04564/2020.iii ontents
List of Figures xList of Tables xvAcronyms xviiIntroduction 31 Quantum Field Theory 7
Conclusion 89Appendices 99A SU ( N ) generators and identities 101B Lattice tensors 103 B.1 Construction of the lattice basis . . . . . . . . . . . . . . . . . . . . . . . . 103B.1.1 Momentum polynomial under a transposition . . . . . . . . . . . . 103B.1.2 Second order tensors under H (4) symmetry . . . . . . . . . . . . . 104B.2 General construction for projectors . . . . . . . . . . . . . . . . . . . . . . 105B.2.1 Projectors for the lattice bases . . . . . . . . . . . . . . . . . . . . 106 C Results – Additional figures 109
C.1 Gluon propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109C.1.1 Continuum relations – mixed diagonal configurations . . . . . . . . 109x ist of Figures n , n + a ˆ µ and n − a ˆ µ . . . . . . . . . . . . . . . . 202.2 Schematic representation of the minimal planar lattice loop, plaquette inthe plane µ − ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.1 Diagrammatic representation of the connected and disconnected termscontributing for the full, four-gluon correlation function. . . . . . . . . . . 384.1 Gluon dressing function d ( p ) from the continuum basis as a function oflattice momentum (top left), and as a function of the improved momentum(top right). The momenta surviving cylindrical and conical cuts are shownfor the each plot. The comparison between the data in terms of theimproved and lattice momenta after complete momentum cuts againstthe H4 corrected data with lattice momentum is shown in the bottomplot. Results from the β = 6 . , lattice. . . . . . . . . . . . . . . . . . 454.2 p E ( p ), p J ( p ), and p A ( p ) dressing functions as a function of thelattice momentum after a p [4] extrapolation (left) and as a function ofthe improved momentum ˆ p after momentum cuts. The results come fromthe β = 6 . , lattice and the benchmark continuum dressing functionˆ p D (ˆ p ) is plotted as a function of the improved momentum. . . . . . . . 474.3 Dimensionless form factors p G ( p ) and p I ( p ). G is shown only afterthe correction methods. The original data is shown in the top row forthe lattice momentum p (left) and improved momentum ˆ p (right) for arestricted range of momenta. Below, p G ( p ) and p I ( p ) after the correc-tions are applied are presented, namely the H4 extrapolated results andmomentum cuts. All data from the β = 6 . , lattice. . . . . . . . . . . 48xi.4 Dressing functions for the different tensor bases as a function of thelattice momentum after a p [4] extrapolation (left) and as a function ofthe improved momentum ˆ p after momentum cuts. These come from the β = 6 . , lattice. The improved continuum tensor form factor D (ˆ p )is also shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.5 E ( p ), − p F ( p ), and − p H ( p ) from the improved momentum latticebasis (right) and from the normal momentum lattice basis (left). Datafrom the β = 6 . , lattice. The standard result for D (ˆ p ) is also shownas a function of the improved momentum. . . . . . . . . . . . . . . . . . . 504.6 Gluon dressing function d (ˆ p ) as a function of the improved momentum forthe continuum basis published in [73]. The left plot shows the completeset of data and the curve surviving momentum cuts. Additionally, theright plot shows the averaged data in each bin – description in the text. . 514.7 Dressing functions p E ( p ), p J ( p ), and p A ( p ) from the β = 6 . , lattice as a function of the lattice momentum after a p [4] extrapolation(left) and as a function of the improved momentum ˆ p . The data is shownafter a binning of 2 .
5% in momentum was performed. The continuumdressing function ˆ p D (ˆ p ) is shown with momentum cuts. . . . . . . . . . 524.8 Form factors for the higher order terms of the extended basis p G ( p )and p I ( p ) in terms of the usual momentum after the p [4] extrapolation(left) and as a function of the improved momentum (right) without anycorrection applied. Both cases are shown after a 2 .
5% binning is appliedin the momentum axis. Data from the β = 6 . , lattice. . . . . . . . . 534.9 β = 6 . , lattice non-metric dressing functions for three tensor basesas a function of the lattice momentum after a p [4] extrapolation (left) andas a function of the improved momentum ˆ p , both after a 2 .
5% binningprocedure applied to the momentum. The continuum dressing functionˆ p D (ˆ p ) is shown with momentum cuts. . . . . . . . . . . . . . . . . . . . 544.10 Reconstruction ratio for the normal momentum bases after the H4 extrap-olation. Each plot is labelled by the corresponding form factors for eachbasis. Data from the β = 6 . , lattice. . . . . . . . . . . . . . . . . . . 554.11 Reconstruction ratio R for various single scale momentum configurationsusing two lattice bases, eqs. (3.15) and (3.16), and the continuum tensor(1.40) using the improved momentum and lattice momentum. Resultsfrom the β = 6 . , ensemble. . . . . . . . . . . . . . . . . . . . . . . . . 564.12 Orthogonality condition, eq. (4.10) shown for the normal momentum basisafter H4 extrapolation from the β = 6 . , lattice. Right plot shows theresult using the improved basis result without corrections and also withmomentum cuts in terms of the improved momentum. For all data the p component was considered. . . . . . . . . . . . . . . . . . . . . . . . . . . 58xii.13 Reconstruction ratio for all four generalized diagonal configurations fromthe β = 6 . , lattice considering the most complete lattice basis (left)and the usual continuum tensor basis (right). Also shown is the recon-struction for the kinematics ( n, , ,
0) using the same two bases. . . . . . 594.14 Form factors from the lattice basis for the diagonal configuration p =( n, n, n, n ) (left) and for the on-axis momentum p = ( n, , ,
0) (right)both as a function of improved momentum. Results from the β = 6 . , lattice. Shown for comparison is the benchmark result d (ˆ p ). . . . . . . . 624.15 Reconstruction ratio for the extended lattice basis and the usual contin-uum description both in terms of the improved momentum. These areshown for the two different lattices with 80 and 64 sites, and same spac-ing 1 /a = 1 . − . Four distinct momentum configurations areshown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.16 Reconstruction ratio for all four generalized diagonal configurations con-sidering the most complete lattice basis for the (6 .
502 fm) lattice (left)and the (8 .
128 fm) lattice (right). Both lattices having the same latticespacing 1 /a = 1 . − . . . . . . . . . . . . . . . . . . . . . . . . 644.17 Three gluon correlation function from the β = 6 . , ensemble con-tracted with, and as a function of the improved momentum. All datais shown without correction methods using a partial Z4 averaging withpermutations only, and also for the complete Z4 averaging. . . . . . . . . 664.18 H4 extrapolated data for the gluon propagator dressing function d ( p )compared with full diagonal momenta ( n, n, n, n ) as a function of improvedmomentum. Data from the β = 6 . , ensemble. . . . . . . . . . . . . . 674.19 Original and p [4] extrapolated data for the three gluon correlation functionfrom the β = 6 . , ensemble as a function of the lattice momentum p . The H4 correction was applied for the full momentum range. Theconfiguration ( n, n, n, n ) is shown for comparison. . . . . . . . . . . . . . . 684.20 χ /d.o.f. obtained from the fit of the functional form (4.21) to the β =6 . , lattice data as a function of the momentum range cut off, p >p GeV. Left plot shows the result of the fit for the H4 corrected datawhile the right plot with diagonal momenta as a function of the improvedmomentum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.21 Three gluon correlation function G ( p ) after the H4 extrapolation as afunction of the lattice momentum (left) and as a function of the im-proved momentum after cuts for ˆ p > β = 6 . , ensemble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.22 Gluon propagator D ( p ) from the β = 6 . , lattice as a function ofthe improved momentum after cuts abover 1 GeV. The renormalizationgroup improved perturbative result, eq. (4.21) was fitted to the data for p ∈ [5 ,
8] GeV, resulting in a fit with χ /d.o.f. = 1 .
10. . . . . . . . . . . . 71xiii.23 Complete set of data from the β = 6 . , lattice for the three-gluon1PI, Γ( p ) as a function of the improved momentum. The data survivingmomentum cuts above 1 GeV is also shown. . . . . . . . . . . . . . . . . . 724.24 χ /d.o.f. of the three fits from eqs. (4.22) to (4.24) (top left, top rightand bottom, respectively) for the varying momentum range p ∈ [ p i , p f ].Both fits with and without momentum cuts were considered. . . . . . . . 734.25 Γ( p ) from the β = 6 . , ensemble as a function of improved mo-mentum. The data after momentum cuts is also shown. Two fits usingeq. (4.22) and p f = 1 . p ) from the complete set as a function of improved momentum fromthe β = 6 . , ensemble. The data after momentum cuts are appliedis also shown. The functional form in eq. (4.23) with range p f = 1 . p ) for the complete kinematics as a function of improved momentumfrom the β = 6 . , ensemble. The set of points surviving momentumcuts is also shown. The functional form in eq. (4.24) with p f = 0 .
85 GeVwas adjusted to the complete and partial data. . . . . . . . . . . . . . . . 764.28 Prediction for the sign change p from the fits using eq. (4.22) (left) andeq. (4.24) (right) for varying fitting ranges [0 , p f ]. . . . . . . . . . . . . . . 774.29 Γ( p ) from the β = 6 . , ensemble compared with the results from [21]using the β = 6 . , lattice with 2000 configurations. Above 1 GeV onlydata surviving momentum cuts is shown. . . . . . . . . . . . . . . . . . . . 784.30 Γ( p ) with momentum cuts above 1 GeV for the 80 and 64 lattice. Thecurves result from the fits with eq. (4.22) (top left), eq. (4.23) (top right),and eq. (4.24) (bottom plot) with fitting ranges p f = 1 . p f = 0 .
85 GeV for the latter. . . . . . . . . . . . . . . . . . . . . 804.31 Four gluon vertex form factor V Γ (0) ( p ) with external propagators from the β = 6 . , lattice. Only mixed diagonal configurations are considered.The smaller plot shows a restricted range of momentum to better visualizethe mid momentum region. All data was rescaled by a factor of 1000. . . 824.32 Four gluon vertex form factor V G ( p ) with external propagators from the β = 6 . , lattice. Only mixed diagonal configurations are considered.The smaller plot shows a restricted range of momentum to better visualizethe mid momentum region. All data was rescaled by a factor of 1000. . . 834.33 Four gluon vertex form factors V Γ (0) ( p ) and V G ( p ) with external propa-gators from the β = 6 . , lattice. Only mixed diagonal configurationsare shown and the lowest momentum points disregarded due to large fluc-tuations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.34 Four gluon vertex form factor V Γ (0) ( p ) with external propagators fromthe β = 6 . , (red) and 64 (green) ensembles. Only mixed diago-nal configurations are considered and the lowest momentum points weredisregarded. All data was rescaled by a factor of 1000. . . . . . . . . . . . 84xiv.35 Four gluon vertex form factor V G ( p ) with external propagators from the β = 6 . , (red) and 64 (green) ensembles. Only mixed diagonal config-urations are considered and the lowest momentum points were disregarded.All data was rescaled by a factor of 1000. . . . . . . . . . . . . . . . . . . 844.36 Original data from [31] for the DSE computation of the pure four gluonvertex associated with the tree-level tensor V ′ Γ (0) ( p ). The ‘total’ result inblack is the relevant structure for comparison. . . . . . . . . . . . . . . . . 864.37 Original data from [31] for the DSE computation of the pure four gluonvertex associated with the tree-level tensor V ′ G ( p ). The ‘total’ result inblack is the relevant structure for comparison. . . . . . . . . . . . . . . . . 86C.1 Form factors from the lattice basis for the mixed configurations p =( n, n, n,
0) (left) and for p = ( n, n, ,
0) (right) both as a function of im-proved momentum. Shown for comparison is the benchmark result d (ˆ p ). 110xvvi ist of Tables and 80 lattice using the three models ineqs. (4.22) to (4.24). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79xviiviii cronyms One particle irreducible.
DSE
Dyson-Schwinger equation(s). IR Infrared.
LQCD
Lattice Quantum Chromodynamics.
QCD
Quantum Chromodynamics. xixx nits and Conventions
In this dissertation we use natural units ~ = c = 1where ~ is the reduced Planck constant and c the speed of light in the vacuum. In theseunits energy, momentum and mass have the same units – expressed in MeV (1 . × − J). Length and time also have common units, inverse of energy. To re-establishunits, the following conversion factor is considered ~ c = 197 .
326 MeV fm = 1and in SI units 1 MeV = 1 . × − kg1 fm = 3 . × − s . Greek indices ( µ, ν, ρ, etc) are associated with space-time indices going through(0 , , ,
3) or (1 , , ,
4) for Minkowski and Euclidean space, respectively. The g µν symbolis reserved for the Minkowski metric tensor g µν = diag(1 , − , − , −
1) while the Kro-necker symbol δ µν is the Euclidean metric tensor. Latin indices ( a, b, etc) are usuallyreserved for the colour degrees of freedom associated with the SU ( N ) algebra.The Einstein summation convention for repeated indices a µ b µ ≡ X µ a µ b µ (1)is used throughout the work, unless explicitly noted. This convention applies to bothspace-time and colour degrees of freedom. The position of the indices is irrelevant whenconsidering colour, or Euclidean metric. 1 ntroduction The modern description of the fundamental interactions in nature considers four in-teractions: gravitational, electromagnetic, weak, and strong. Apart from the gravita-tional interaction which does not have a proper quantum formulation, the last threeare described by quantum field theories. These three fundamental interactions definewhat is called the Standard Model, a gauge theory associated with the symmetry group SU (3) ⊗ SU (2) ⊗ U (1) describing current particle physics.The SU (2) ⊗ U (1) sector of the Standard Model contemplates the electromagnetic andweak interactions (electroweak) [2]. Perturbation theory accounts for most of the phe-nomena occurring in this sector. When the physical processes involve hadrons throughthe strong force (e.g. protons, neutrons, pions) for low energy processes, perturbationtheory fails. Hence, non-perturbative methods are necessary to study the SU (3) sec-tor which accounts for the dynamics of quarks and gluons. Quantum chromodynamics(QCD) is the current description of the strong interaction.Lattice field theory is a possible non-perturbative approach to formulate QCD. Theformulation of the theory on a discretized lattice with finite spacing and volume providesa regularization, which renders the theory finite. When combined with the Euclideanspace-time, lattice field theories become formally equivalent to classical statistical theo-ries. Hence, other than serving as a regularized formulation of the theory it also serves asa computational tool. In lattice quantum chromodynamics (LQCD), physical quantitiesare computed using Monte-Carlo simulations that require large computational power.Current simulations can reach a satisfying level of precision in the computation of sev-eral quantities such as the strong coupling constant, hadron masses, and also the studyof some properties such as confinement and chiral symmetry (see [3] for a summary ofthe current advances and investigations in the field).All of the work developed in this thesis uses the pure Yang-Mills theory, where thefermion dynamics is not taken into account – quenched approximation . This correspondsto disregarding quark loops in the diagrammatic expansion. Although this approxima-tion seems too radical, the systematic errors involved are small [4].A quantum field theory is defined by its correlation functions [5, 6], summarizingthe dynamics and interactions among fields. Despite not being physical observablesand not experimentally detectable, due to its gauge dependency, correlation functionsare important for they can be related to various phenomena of the theory. Indeed,in supposedly confining theories such as QCD whose quanta (quarks, gluons, and the3nphysical ghosts) do not represent physically observable states, correlation functionsshould encode information on this phenomenon [7, 8]. Vertices can also serve to computethe coupling constant and define a static potential between colour charges [9, 10], andalso explore properties of bound states [11]. Correlation functions are also the buildingblocks of other non-perturbative continuum approaches such as the Dyson-Schwingerequations (DSE) [12]. These frameworks usually partially rely on lattice data, and thusa good comprehension of these objects is important.This thesis addresses three different topics. Firstly, we investigate the lattice gluonpropagator relying on lattice tensor representations with the aim to understand thedeviations of correlation functions relative to the continuum theory [13, 14]. This hasbecome a relevant topic as modern computations of the gluon propagator use largestatistical ensembles of configurations.The second objective is to compute the three gluon vertex and study its infrared(IR) behaviour. The purpose of this analysis is to search for evidences and shorten theestimated interval of the zero-crossing , corresponding to a possible sign change of thethree gluon one particle irreducible (1PI) function for low momentum. This propertycan be traced back to the fundamental dynamics of the pure Yang-Mills theory, namelythe ghost dynamics as predicted by the DSEs [15, 16]. In this framework, the signchange is necessary for the finiteness of the equations assuming a tree level form of theghost-gluon, and four gluon vertex [17]. Various DSE investigations [17, 18] as well asother methods [19, 20] found the zero-crossing for the deep IR. Recent lattice SU (3)studies [21–23] as well as SU (2) [24, 25] predict the zero crossing for the deep infraredregion, around 150 −
250 MeV. Moreover, the exact momentum of the crossing seems tobe dependent on the group symmetry and dimensionality, being generally lower for thefour-dimensional case [15]. Additionally, general predictions come from pure Yang-Millstheories and thus unquenching the theory could spoil this behaviour. However, severalDSE based references [17, 19, 26] argue this is a pure gluon phenomenon, and that thepresence of light mesons [27, 28] only shifts the zero-crossing momentum to a lower IRregion.From the point of view of continuum frameworks, this property is highly dependent onthe approximations employed and thus should always be validated by lattice simulations.The latter usually suffer from large fluctuations, or from difficult access to IR momenta.Furthermore, a recent analytical investigation on both the gluon and ghost propagatorsfound evidence of the existence of a non-vanishing ghost mass which could regularizethe three gluon vertex, thus removing the divergence [29]. While the existence of adynamical gluon mass is properly established in previous investigations [30], the case ofthe ghost field is undetermined. The existence of a finite dynamical ghost mass wouldin principle remove the logarithmic divergence and thus we also explore this possibility.The last objective of this work is to perform a first lattice computation of the fourgluon correlation function. General predictions for the IR structure of this vertex existonly from continuum formulations [31, 32]. These are dependent on truncation schemesand other approximations and again lattice results are needed to validate the predictions.The four gluon vertex has four Lorentz indices and four colour indices, therefore its4ensor structure is rather complex, allowing for a large number of possible tensors. Theincreased statistical fluctuations are related to it being a higher order correlation function,involving fields at four distinct lattice sites. Besides, as a higher order function, itscomputation requires the removal of unwanted contributions from lower order correlationfunctions. These can be eliminated by a suitable choice of kinematics.The outline of this dissertation begins with a general introduction to the necessarytools and theoretical basis to understand the lattice formulation and results. Chapter1 begins with a brief description of the formalism for a general quantum field theorywith the QCD theory being introduced and its properties briefly reviewed. Correlationfunctions and other objects of the theory are introduced.The lattice formulation of QCD is presented in chapter 2. We motivate and constructthe discretization procedure and present the lattice version of various fundamental ob-jects. This chapter also includes some computational aspects needed to perform latticesimulations.In chapter 3 the main work of this dissertation begins with an analysis of the correctlattice symmetries and the construction of lattice adequate tensor bases. Additionally,details about discretization effects, possible correction methods and tensor bases for thethree and four gluon correlation functions are introduced.Results are shown in chapter 4 which is divided in three main sections, dedicated toeach of the three main objectives of this work. This is followed by final conclusions andpossible extensions for this work.Finally, the results obtained in this thesis regarding the tensor structure of the prop-agator were summarized in [1]. 5 hapter 1
Quantum Field Theory
Quantum Chromodynamics is a SU (3) gauge theory. Historically, the colour quantumnumber was introduced in order to reconcile Fermi statistics with the observed groundstate of strongly interacting particles. A new quantum number was needed to guaranteethe anti-symmetry of the wave-function [2]. Later, these new degrees of freedom werefound to be associated with a gauge theory.In this chapter we give a brief overview of QCD and how the theory arises from theprinciple of gauge invariance. Some important concepts in a quantum field theory arealso presented. Quantum field theories are well described in [6, 33, 34], and QCD isthoroughly exposed in [35]. The Lagrangian of QCD involves the matter, quark fields ψ and the gluon fields A µ .The first form a representation of the group symmetry, namely the fundamental repre-sentation of SU (3), while the latter are in the adjoint representation of the group (seeappendix A).The classical QCD Lagrangian arises when we impose gauge invariance to the DiracLagrangian L Dirac = ¯ ψ ( iγ µ ∂ µ − m ) ψ. (1.1)where ¯ ψ = ψ † γ with γ being the zeroth Dirac matrix, γ µ . For a general SU ( N )theory, the gauge principle requires the invariance of the Lagrangian under a local grouptransformation ψ ( x ) → ψ ′ ( x ) = V ( x ) ψ ( x ) (1.2)with V ( x ) an element of the fundamental representation of the group. When performinga local transformation, the kinetic term of the Lagrangian breaks the invariance since itcompares fields at different points with distinct transformation laws ψ ( y ) − ψ ( x ) → V ( y ) ψ ( y ) − V ( x ) ψ ( x ) . (1.3)7n order to make comparisons at different points we introduce the group valued com-parator U ( x, y ) satisfying U ( x, x ) = and the gauge transformation U ( x, y ) → V ( x ) U ( x, y ) V † ( y ) . (1.4)With this object we may define the covariant derivative , using the following difference, D µ ψ ( x ) ≡ lim ε µ → ε [ U ( x, x + ε ) ψ ( x + ε ) − ψ ( x )] . (1.5)with y = x + ε , and ε an infinitesimal. With this definition, the new derivative transformssimilarly to the fields, D µ ψ ( x ) → V ( x ) D µ ψ ( x ) . (1.6)Introducing a new field, the connection A µ ( x ), by U ( x, x + ε ) = − igε µ A µ ( x ) + O (cid:16) ε (cid:17) . (1.7)where g is the bare strong coupling constant, we write the covariant derivative as D µ ψ ( x ) = ( ∂ µ − igA µ ( x )) ψ ( x ) . (1.8)The transformation law for the newly introduced field A µ ( x ) is A µ ( x ) → V ( x ) A µ ( x ) V − ( x ) − ig ( ∂ µ V ( x )) V − ( x ) . (1.9)An arbitrary group element V ( x ) can be expressed by the Lie algebra elementsthrough the exponentiation mapping V ( x ) = exp( iα a ( x ) t a ) (1.10)with the algebra generators t a defined in appendix A and α a ( x ) a set of functionsparametrizing the transformation. The connection A µ ( x ) is thus an element of thealgebra which can be written in terms of the fields A aµ ( x ) A µ ( x ) = A aµ ( x ) t a . (1.11)Hence, to guarantee gauge invariance of the Dirac Lagrangian we replace normalderivatives by the covariant. Furthermore, we need to introduce a kinetic term for thenew field that must depend only on the gauge fields A µ and its derivatives. The usualconstruction is the field-strength tensor F µν = ig [ D µ , D ν ] = ( ∂ µ A ν − ∂ ν A µ ) − ig [ A µ , A ν ] (1.12)which can be written in terms of its components F µν = F aµν t a using the structure con-stants of the group f abc , F aµν = ( ∂ µ A aν − ∂ ν A aµ ) + gf abc A bµ A cν . (1.13)8he first equality in 1.12 gives a geometrical interpretation of the tensor, as it can beseen as the comparison of the field around an infinitesimal square loop in the µ − ν plane, indicating how much it rotates in the internal space when translated along thispath [6]. To obtain a gauge invariant scalar object from this tensor, we consider thetrace operation over the algebra elements and the following contractionTr h ( F aµν t a ) i = ( F aµν ) / . (1.14)With these elements we write the classical QCD Lagrangian L QCD = − F aµν F aµν + ¯ ψ (cid:16) iγ µ ( ∂ µ − igA aµ t a ) − m (cid:17) ψ (1.15)whose form, namely the gluon-quark interaction is restricted by gauge invariance . Thematter field ψ ( x ) is a vector of spinors for each flavour of quark ( f = u, d, s, c, t, b ).Each quark flavour has an additional colour index a = 1 , , SU (3) group. m is a diagonal matrix in flavour space containingthe bare quark masses for each flavour. The eight independent gluon fields associatedwith the group generators are the gauge fields A aµ ( x ) which also carry a Lorentz index,labelling the corresponding directions in space-time, µ = 0 , , , L YM = − F aµν F aµν . (1.16) In the path integral quantization for a general quantum field theory [5, 6, 36], describedby a set of fields φ a , the theory is defined by the generating functional Z [ J ] = Z D φe i R d x ( L + J a ( x ) φ a ( x )) (1.17)where J a ( x ) is an external source, and the condensed notation was employed D φ ≡ Y x,a dφ a ( x ) . (1.18)A quantum field theory is completely determined by its Green’s functions [5, 6] definedas G ( n ) i ,...,i n ( x , ..., x n ) = h | T h ˆ φ i ( x ) ... ˆ φ i n ( x n ) i | i (1.19) Gauge invariance also restricts the gauge fields to be massless since the term A aµ A aµ is not gaugeinvariant. The index a may represent independent fields, different members of a set of fields related by someinternal symmetry, or the components of a field transforming non-trivially under Lorentz transformation,e.g., a vector. n field operators atdistinct points. In this quantization procedure, Green’s functions are computed fromthe generating functional by functional differentiation with respect to the sources h | T h ˆ φ i ( x ) ... ˆ φ i n ( x n ) i | i = 1 i n Z [ J ] δ n Z [ J ] δJ i ( x ) ...δJ i n ( x n ) (cid:12)(cid:12)(cid:12)(cid:12) J =0 . (1.20)This vacuum expectation value can thus be written as D ˆ φ i ( x ) ... ˆ φ i n ( x n ) E = 1 Z [0] Z D φ (cid:16) φ i ( x ) ... ˆ φ i n ( x n ) (cid:17) e iS (1.21)with the notation D ˆ φ i n ( x n ) ... ˆ φ i n ( x n ) E ≡ h | T h ˆ φ i n ( x n ) ... ˆ φ i n ( x n ) i | i . Equation (1.21)shows that Green’s functions are accessed by performing a weighted average over allpossible configurations of the system.The path integral quantization carries some problems when applied to gauge theories.The generating functional Z = Z D Ae iS [ A ] . (1.22)involves the integral over the gauge fields A aµ ( x ). For any field configuration A µ we maydefine a gauge orbit to be the set of all fields related to the first by a gauge transformation α . All these configurations have the same contribution to the functional integral, andso constitute an infinite contribution.The over counting of these degrees of freedom need to be eliminated in order tohave a well defined theory. Faddeev and Popov [37] suggested the use of a hypersurfaceto restrict the integration in configuration space. This is achieved by a gauge fixingcondition of the form F a [ A ] − C a ( x ) = 0 . This way we isolate the contribution overrepeated configurations by factorizing it as R D α R D A µ exp iS [ A ] , being eliminated by thenormalization.To impose this integration restriction we insert the following expression in the gen-erating functional, 1 = Z D αδ ( F a [ A α ] − C a ( x )) det (cid:18) δF a [ A α ] δα (cid:19) (1.23)where A α represents the gauge transformed field A , δ ( F [ A α ]) is a Dirac δ over eachspace-time point, and the determinant is due to the change of variables. The generatingfunctional reads Z = Z D A Z D αδ ( F a [ A α ] − C a ( x )) det (cid:18) δF [ A α ] δα (cid:19) e iS [ A ] . (1.24)Performing a gauge transformation from A αµ to A µ we can eliminate the dependenceon the gauge transformation from the integrand. For this we use the gauge invariance F [ A ] is a field dependent term. C a ( x ) is a set of functions also determining the gauge fixing condition. F [ A ] = ∂ µ A µ ( x ) and C a ( x ) = 0 in the Landau gauge.
10f the action and of the volume element in group space D α [38]. Also, an unitarytransformation leaves the measure D A and the determinant unchanged Z = Z D α Z D Aδ ( F a [ A ] − C a ( x )) det (cid:18) δF [ A ] δα (cid:19) e iS [ A ] . (1.25)This way we factorized the infinite factor, which is eliminated by normalization. Inaddition, we may multiply Z by a constant factor Z D C exp (cid:20) − i ξ Z d xC a (cid:21) (1.26)corresponding to a linear combination of different Gaussian weighted functions C a . Thegenerating functional now reads Z = Z D A det (cid:18) δF [ A ] δα (cid:19) exp (cid:26) iS [ A ] − i ξ Z d xF [ A ] (cid:27) . (1.27)The Faddeev-Popov determinant is defined asdet M = det (cid:18) δF ([ A ] , x ) δα ( y ) (cid:19) , M ab ([ A ] , x, y ) = δF a ([ A ] , x ) δα b ( y ) (1.28)Using Grassmann, anti-commuting variables it is possible to define the Faddeev-Popovdeterminant as a functional integral over a set of anti-commuting fields – ghost fields¯ η, η det M = Z D ¯ η D η exp (cid:18) − i Z d x ¯ η a M ab η b (cid:19) . (1.29)With this, we have a final form for the generating functional, Z = Z D A µ D ¯ η D ηe i R d x L eff , (1.30)expressed with an effective Lagrangian L eff = L − F ξ − ¯ ηM η. (1.31)These new anti-commuting fields can be interpreted as new particles contributing to thedynamics of the system. However, being scalars under Lorentz transformations whileanti-commuting fields, ghosts do not respect the spin-statistics theorem [39] and cannotbe interpreted as physical particles – only contributing to closed loops in Feynmandiagrams and never as external fields. They are a mathematical artifact resulting fromthe gauge fixing procedure. 11 .3 Propagator and vertices The effective Yang-Mills Lagrangian is L = 12 ( ∂ µ A aν ∂ ν A aµ − ∂ µ A aν ∂ µ A aν ) − ξ ( ∂ µ A µ ) − gf abc A bµ A cν ( ∂ µ A aν − ∂ ν A aµ ) − g f abc f ade A bµ A cν A dµ A eν − ¯ η a ∂ µ ( ∂ µ − gf abc A aµ ) η b . (1.32)Analytically, the computation of the complete correlation functions (Green’s functions)is not possible. However, perturbation theory can provide some information on the formof these functions. For this we need to know the Feynman rules for the theory, whichcan be read off from the Lagrangian at tree level and are summarized in this section. Itsderivation can be consulted in [6, 40].The gluon propagator is read off from the quadratic terms in the gluon fields in theLagrangian. In momentum space, the propagator reads D abµν ( p ) = δ ab p (cid:20) g µν + ( ξ − p µ p ν p (cid:21) . (1.33)Note that ξ = 0 in the Landau gauge.The ghost fields also have associated Feynman rules. In the chosen gauge the func-tional derivative (1.28), obtained with the infinitesimal version of (1.9), A ′ aµ = A aµ + f abc A bµ α c + ∂ µ α a , (1.34)is of the form M ab = ∂ µ D µ , resulting in a lagrangian contribution L ghost = − ¯ η a ∂ µ ∂ µ η a + gf abc ¯ η a ∂ µ ( A bµ η c ) . (1.35)The ghost will have an associated tree-level propagator, fig. 1.1,∆ ab ( p ) = δ ab p (1.36)and a ghost-gauge field coupling vertex − gf abc p µ represented in figure 1.2.The gluon self ‘interaction’ vertices result from the second and third line of theLagrangian. Their form, however, is written considering the Bose symmetry of theobjects, which allow us to interchange each particle ( p i , a i , µ i ) without affecting its form.The Feynman rule for the three gluon vertex in momentum space, shown schematicallyin fig. 1.2, readsΓ (0) a a a µ µ µ ( p , p , p ) = gf a a a [ g µ µ ( p − p ) µ + g µ µ ( p − p ) µ + g µ µ ( p − p ) µ ](1.37) Note that D µ here is written in the adjoint representation with the generators ( t a ) bc = − if abc . (0) a a a a µ µ µ µ ( p , p , p , p ) = − g (cid:2) f a a m f a a m ( g µ µ g µ µ − g µ µ g µ µ ) f a a m f a a m ( g µ µ g µ µ − g µ µ g µ µ ) f a a m f a a m ( g µ µ g µ µ − g µ µ g µ µ ) (cid:3) . (1.38) pa bµ ν pa b Figure 1.1: Gluon and ghost propagators. pa q bc µ ( p a µ ) ( p a µ )( p a µ ) ( a µ ) ( a µ )( a µ ) ( a µ ) Figure 1.2: Ghost-gluon coupling vertex (top) and three and four gluon vertices with all momentadefined inwards.
In a non-perturbative framework, we aim to have access to the complete correlationfunctions whose tensor structure ought to be different from the simple bare verticesobtained at zero order in perturbation theory. Hence, we must build the most generalstructure for each correlation function under the symmetries of the theory.13he tensor structure for the gluon propagator is completely defined by the Slavnov-Taylor identity and the gauge condition – see [6, 40]. The Landau gauge Slavnov-Tayloridentity for the gluon propagator reads [41] ∂ µx ∂ νy D T { A aµ ( x ) A bν ( y ) } E = 0 (1.39)which fixes the orthogonal form of the propagator. Therefore, in the Landau gauge, thisresults in D abµν ( p ) = δ ab D ( p ) (cid:20) g µν − p µ p ν p (cid:21) (1.40)with its coefficient differing from the tree-level form by a form factor D ( p ).For higher order correlation functions we distinguish the gluon correlation functions G a ...a n µ ...µ n obtained with (1.20) from the pure gluon vertex Γ a ...a n µ ...µ n obtained with theremoval of the external propagators. For the three gluon vertex we thus define D A a µ ( p ) A a µ ( p ) A a µ ( p ) E = (2 π ) δ ( p + p + p ) G a a a µ µ µ ( p , p , p ) (1.41) G a a a µ µ µ ( p , p , p ) = D a b µ ν ( p ) D a b µ ν ( p ) D a b µ ν ( p )Γ a a a ν ν ν ( p , p , p ) . (1.42)Analogous expressions can be considered for the four gluon vertex. Notice that theΓ a a a ν ν ν ( p , p , p ) Γ a a a a ν ν ν ν ( p , p , p , p ) Figure 1.3: Three and four gluon vertices with external propagators removed. average for the three gluon correlation function is computed as D A a µ ( x ) A a µ ( x ) A a µ ( x ) E = R D AA a µ ( x ) A a µ ( x ) A a µ ( x ) e i R d x L R D Ae i R d x L . (1.43)To compute these higher order correlation functions we construct their tensor structuresby taking into account the symmetries of the system, namely Bose symmetry allowingto freely exchange each pair of indistinguishable particles and their associated quantumnumbers. Proceeding this way we construct the most general form for these objects.This construction will be presented in chapter 3. These are relations between the correlation functions which come from the gauge invariance of thetheory. They express the symmetries of the classical theory through the quantum expectation values.Also called generalized Ward identities.
14t is also important to make a further distinction between the pure (gluon) vertices G and the one particle irreducible (1PI) functions, Γ which do not have the contributionfrom disconnected diagrams and cannot be reduced to other diagrams by removing apropagator – see [6, 40]. These are the objects we are interested in obtaining from thelattice – further details will be given when considering the four gluon vertex in section 3.7. In general, quantum field theories involve divergences other than the ones solved by theFaddeev-Popov method. These divergences need to be taken care of.The theory is first regularized, making it finite. This is done, in general, by introduc-ing parameters in the theory which absorb the divergences. In a perturbative approach,this could be done by an ultraviolet momentum cut off or dimensional regularization forexample. The introduction of a finite space-time lattice with spacing a is a common reg-ularization procedure with the advantage of allowing to perform numerical simulations.The theory is then renormalized by rescaling the parameters and fields of the theoryin a way that the removal of the divergences is not spoiled when the regularizationparameter is eliminated.The rescaling is performed on a finite number of parameters such as the fields, andthe fundamental constants of the theory. Following [5] a possible rescaling procedure forQCD would be A aµ → Z / A A aµ , m → Z m Z − ψ m, (1.44) ψ → Z / ψ ψ, g → Z g g, (1.45) η a → Z / η η a , ξ − → Z ξ Z − A ξ − (1.46)where the various Z i are the necessary renormalization constants to render the theoryfinite.Green’s functions have associated rescaling rules constructed from the ones above.Considering gauge fields only, the Green’s functions renormalization involve Z A . Forinstance, the renormalized gluon propagator G (2) r relates to the bare object as G (2) r = Z A G (2) .Performing a renormalization procedure involves choosing a point where the quanti-ties are fixed by some given, standard values. The momentum subtraction MOM schemeis a usual choice, it fixes the renormalized Green’s function to match the tree level valuefor a given momentum scale µ . Again, using the gluon propagator, the constant Z A isfound from D ( p = µ ) = Z A D L ( µ ) = 1 µ (1.47)where D ( p ) is the renormalized form factor and D L ( p ) the non-renormalized formfactor. See [42] for more details, and [43] for a lattice dedicated description.156 hapter 2 Lattice quantum chromodynamics
In this chapter the formulation of quantum chromodynamics on a finite discretized latticewill be presented. Lattice QCD provides a formulation which allows to study the non-perturbative regime of QCD and a regularization of the theory. This framework preservesgauge invariance and serves as an explicit computational tool.This chapter begins with the introduction of the lattice formalism, constructing allobjects in the discretized framework. After this, attention will be given to some compu-tational aspects of this work which are necessary to compute lattice quantities. Latticetheories, with emphasis on LQCD are presented in [38, 43, 44].
The Minkowski space-time is not convenient to study functional path integrals due tothe oscillatory behaviour of the exponential in the action. We use imaginary time thusbecoming an Euclidean space. This is accomplished by a Wick rotation, where the realtime t is rotated by π/ τ = it . The exponential becomes similarto the Boltzmann factor on the partition function of statistical mechanics, Z D φe iS [ φ ] → Z D e − S E [ φ ] . The object S E is the Euclidean version of the action, obtained by performing the changeof variables above. This transformation establishes the formal connection with statisticalmechanics, allowing its methods to be applied on lattice field theories, notably Monte-Carlo methods to obtain correlation functions. In the forthcoming analysis we considerthe Euclidean formulation of QCD and the metric is thus equivalent to δ µν . In the lattice formulation the continuous space-time is replaced by a 4-dimensionalEuclidean lattice Λ with spacing a whereby each point is labelled by four integers,17 = ( n , n , n , n ). We consider n to be the imaginary time direction. In this work weconsider hypercubic lattices, each side having the same number of points, n i ∈ [0 , N − φ , the degrees of freedomare the classical fields φ ( an ) in the discrete lattice sites. The lattice action must be builtin a way that preserves all possible properties of the continuum theory. However, thediscretization procedure is not unique which can be seen by the structure of the discretederivative, taking various possible forms, ∂ µ φ ( x ) = 1 a ( φ ( x + ˆ µa ) − φ ( x )) + O ( a ) (2.1) ∂ µ φ ( x ) = 12 a ( φ ( x + ˆ µa ) − φ ( x − ˆ µa )) + O (cid:16) a (cid:17) . (2.2)This freedom in obtaining the lattice form can be used to minimize the appearance oflattice artifacts .On the lattice, all possible space translations are restricted to be at least one latticeunit in size. This results in the discretization of the allowed momenta. To see this,consider the usual continuum Fourier transform, φ ( x ) = Z d p (2 π ) ˜ φ ( p ) e ipx . Since x = an is an integer multiple of the spacing a we get e ip µ x µ = e i ( p µ x µ +2 πn µ ) = e i ( p µ +2 π/a ) x µ , hence the momentum p µ is equivalent to p µ +2 π/a , allowing us to restrict the momentumintegration to the Brillouin zone, − π/a < p µ ≤ π/a . This removes high frequency modesand regularizes the theory. Thus, in infinite volume we would write φ ( x ) = Z π/a − π/a d p (2 π ) ˜ φ ( p ) e ipx . To perform numerical simulations, however, the volume of the lattice is finite, where weimpose boundary conditions, φ ( x + ˆ µN µ a ) = e iθ µ φ ( x ). The finite volume imposes theadditional discretization of momentum. Applying the Fourier transform to this condition Z π/a − π/a d p (2 π ) ˜ φ ( p ) e ip µ ( x µ +ˆ µN µ a ) = Z π/a − π/a d p (2 π ) ˜ φ ( p ) e ip µ x µ + iθ µ ⇔ e ip µ N µ = e iθ µ (no sum)where ˆ µ is an unitary lattice vector in the direction µ . We work with periodic boundaryconditions, thus θ µ = 0 and we get the discrete momentum values, p µ = 2 πn µ aN µ , n µ ∈ {− N µ / , ..., N µ / } . (2.3) This freedom opens the possibility for improvement schemes which modify the action in a way toreduce lattice artifacts [45] – these are not considered in this work. φ ( x ) = 1 V X n ∈ Λ ˜ φ ( p n ) e ip n · x where V = N is the volume of the space-time grid for the hypercubic lattice.Other than the discretized momentum (2.3), in this work we will also consider thelattice perturbation theory [46] improved momentum defined byˆ p µ = 2 a sin (cid:18) ap µ (cid:19) = 2 a sin (cid:18) πn µ N (cid:19) . (2.4)This form comes from the tree-level propagator of a massless scalar field on the lattice.The general path integral quantization scheme is built analogously to the continuumformulation. The partition function is constructed Z = Z D φe − S E ( ψ ) (2.5)with the field measure replaced by a finite product D φ = Y n ∈ Λ dφ ( n ) (2.6)and the expectation value of an observable is computed as hOi = 1 Z Z D φe − S E ( φ ) O ( φ ) . (2.7) We consider the discretization of the pure Yang-Mills sector of the QCD Lagrangian.On the lattice the gluon fields appear in order to preserve gauge invariance in localgauge transformations, ψ ( n ) → V ( n ) ψ ( n ), where V ( n ) are SU (3) group elements onthe lattice sites. In the continuum, we considered the covariant derivative to ensure thegauge invariance of the action, and this was implemented such that the comparison offields at different points was properly defined. To this end, we used the concept of acomparator.On the lattice, two fields in neighbouring points have corresponding transformations V ( n ) and V ( n + a ˆ µ ). We define the link variables as a comparator U µ ( n ), connectingboth points. These oriented group elements live in the links between sites and are thefundamental fields in this framework. These satisfy an analogous gauge transformationas the continuum counterpart U µ ( n ) → V ( n ) U µ ( n ) V † ( n + a ˆ µ ) . (2.8)19he inverse link from the same lattice point is given by the adjoint operator U † µ ( n − a ˆ µ )– see figure 2.1. U µ ( n ) U − µ = U † µ ( n − a ˆ µ ) n − a ˆ µ n + a ˆ µnn Figure 2.1: Link variables between n , n + a ˆ µ and n − a ˆ µ . The simplest lattice action, such that the Yang-Mills form is restored when the limit a → plaquette , fig. 2.2, which is the simplest loop on the lattice U µν ( n ) = U µ ( n ) U ν ( n + a ˆ µ ) U † µ ( n + a ˆ ν ) U † ν ( n ) . (2.9)The gauge transformation of this product depends on a single lattice point, U µν ( n ) → V ( n ) U µν ( n ) V † ( n ) . (2.10)Hence, applying the trace we obtain a gauge invariant termTr U ′ µν ( n ) = Tr (cid:16) V ( n ) U µν ( n ) V † ( n ) (cid:17) = Tr U µν ( n ) , (2.11) nn + a ˆ ν n + a ˆ µ + a ˆ νn + a ˆ µU † ν ( n ) U µ ( n ) U ν ( n + a ˆ µ ) U † µ ( n + a ˆ ν ) Figure 2.2: Schematic representation of theminimal planar lattice loop, plaquette in theplane µ − ν . Due to the form of the continuum action we need a relation between the link variablesand the continuum gauge fields A µ ( x ). Hence we establish a relation between lattice andcontinuum comparators U µ ( n ) = U ( n, n + ˆ µ ) + O ( a ). For this purpose, we introducealgebra valued lattice gauge A µ fields by U µ ( n ) = e iagA µ ( n + a ˆ µ/ + O ( a ) . (2.12)We rewrite eq. (2.9) using (2.12) to relate the plaquette with F µν ( n ) U µν = e ia ( ∂ µ A ν ( n )+ ∂ ν A µ ( n )+ i [ A µ ( n ) ,A ν ( n )])+ O ( a )= e iga F µν ( n )+ O ( a ) . (2.13) Using the Baker-Campbell-Hausdorff formula for the product of exponentials of matrices e A e B = e A + B + [ A,B ]+ ... . S G [ U ] = β N c X n X µ,ν Re Tr( − U µν ( n )) (2.14)= a g X n X µ,ν Tr (cid:16) F µν ( n ) (cid:17) + O (cid:16) a (cid:17) (2.15)where we defined the inverse bare lattice coupling β = 2 N c /g . This action was formu-lated by Wilson in 1974 – see [44].In this work we consider only the gauge part of the QCD action. This approximation,disregarding the quarks dynamics is called quenched approximation . Fermions are repre-sented by Grassmann variables and its contribution to the generating functional can bewritten as a fermion determinant. The quenched approximation consists in replacing thedeterminant by a constant which diagrammatically consists in neglecting fermion loopscontributions. Typically, quenched lattice calculations of the hadronic spectra showsdifferences around 10 to 20% relative to experimental data [4]. While physical observables are gauge independent, the computation of correlation func-tions requires to choose a gauge. In fact, they can be shown to vanish if no gauge isfixed – Elitzur’s theorem [47].In this work we consider the Landau gauge which in the continuum reads ∂ µ A µ ( x ) =0, or equivalently p µ A µ ( p ) = 0 in momentum space. On the lattice, it can be shown [38]that this is equivalent to finding a stationary point of the following functional F U [ V ] = 1 V N d N c X n,ν Tr h V ( n ) U µ ( n ) V † ( n + ˆ µ ) i , (2.16)where N d and N c the dimensions and colour number, respectively, and V is the volumeof the lattice – not to be confused with the gauge transformation V ( n ).However, in general the functional eq. (2.16) has many extrema – this problem arisesalready in the continuum formulation. Ideally, we want the gauge condition (hypersur-face defined in section 1.2) to intersect each gauge orbit uniquely, and thus a singlerepresentative is chosen from each gauge orbit.However, Gribov [48] found that the Faddeev-Popov procedure alone is not suffi-cient, and that there are multiple solutions for the gauge condition still related by agauge transformation. These multiple solutions due to the multiple intersections of thehypersurface within each orbit are the so called Gribov copies .The presence of the copies implies the existence of various stationary points of thefunctional. Gribov suggested additional constraints to the gauge field configuration Gribov considered non-abelian gauge theories in the Coulomb gauge ∂ i A i = 0. This was latergeneralized for a 4-dimensional hypercubic and periodic lattice for any SU ( N c ) gauge theory [49]. Gribov region isstill not free of Gribov copies. Further restrictions define a subspace containing onlythe global maxima of F U – called fundamental modular region . It can be shown thaton the lattice this restriction guarantees the absence of Gribov copies in this region [50].Numerically, the search is limited to a local maximum – in this work we used the steepestdescent method, described in [51]. The computer code uses both the Chroma [52] andPFFT [53] libraries.A review of the gauge fixing on the lattice can be found in [54]. It is worth referringthat the effect of the Gribov copies was studied for the gluon propagator on the lattice[55, 56] concluding that its effect are small – less than 10%. In this work we do notconsider the effect of the Gribov copies. We are interested in computing correlation functions involving gauge fields A µ . On thelattice, the gluon field can be computed from the links eq. (2.12) agA µ ( x + ˆ µ/
2) = 12 i h U µ ( n ) − U † µ ( n ) i − i Tr h U µ ( n ) − U † µ ( n ) i (2.17)up to O (cid:0) a (cid:1) corrections. The second term ensures that the field is traceless, Tr A µ = 0.The momentum space lattice gauge field is obtained with the discrete Fourier transformdefined before, A µ ( p ) = X x e − ip · ( x +ˆ µ/ A µ ( x + ˆ µ/
2) (2.18)with p = 2 πn/aN and x = an where n µ ∈ [ − N/ , N/ D A a µ ( p ) A a µ ( p ) E = D a a µ µ ( p ) V δ ( p + p ) . (2.19)In our numerical framework, we have access to algebra valued gauge fields A µ ( p ) fromeqs. (2.17) and (2.18). To form a scalar in the colour sector we consider a trace and asuitable Lorentz contraction for the space-time indices. Considering the usual continuumtensor description for the gluon propagator eq. (1.40), the form factor D ( p ) is obtainedby D ( p ) = 2( N c − N d − n ) X µ h Tr [ A µ ( p ) A µ ( − p )] i (2.20)where n = 0 if p = 0, or 1 otherwise. This subspace contains all local maxima of the functional.Ω = { A : ∂ µ A µ = 0 , M [ A ] ≥ } where M is the Faddeev-Popov matrix eq. (1.28). δ ab can be used. For the three and four gluon vertices we again access the product of gaugefields to which we apply the trace to obtain a scalar in colour space, h Tr [ A µ ( p ) A µ ( p ) A µ ( p )] i = V δ ( X i p i ) G µ µ µ ( p , p , p ) (2.21) h Tr [ A µ ( p ) A µ ( p ) A µ ( p ) A µ ( p )] i = V δ ( X i p i ) G µ µ µ µ ( p , p , p , p ) . (2.22)The G ’s represent the Green’s functions with colour indices absorbed by the trace op-eration and whose form depends on the Lorentz tensor basis considered – these will beproperly defined in chapter 3. In the Euclidean formulation of the theory, the expectation value of some field dependentoperator is given by hOi = 1 Z Z D U O ( U ) e − S E [ U ] . (2.23)To obtain numerical results we consider only a finite number of field configurations. Thisis done by importance sampling considering the weight of the Boltzmann factor in theEuclidean action, and the integrals estimated by Monte-Carlo methods, [57].A set of gauge field configurations { U i } , i = 1 , ..., n is generated according to theprobability distribution P ( U ) = e − S E ( U ) / Z . (2.24)The sequence is obtained by a Markov chain which generates the configurations, one afteranother according to a transition amplitude P ( U i → U j ) depending solely on the pre-decessor configuration. This transition amplitude should create a sequence distributedaccording to P ( U ) in the large n limit.When the set { U i } , i = m, ..., n is distributed according to P ( U ), it is said to be thermalized . From the thermalized set we chose N configurations, each separated fromthe former by k Markov steps in order to reduce correlations among them. The set { U i } , i = 1 , ..., N is the one used for the computation. The configurations consideredin this thesis [21] were obtained using a combination of the over-relaxation and the heatbath methods according to [38].Having a finite number of configurations following the exp( − S E ( U )) / Z probabilitydistribution, the expectation value (2.23) is estimated by the sample mean¯ O = 1 N N X i =1 O ( U i ) , (2.25) By a gauge field configuration we mean that each site of the lattice is attributed a value of the field U , i.e. a Lorentz vector of SU (3) matrices. The precise form of the amplitudes depends on the chosen method [43]. hOi in the large N limit.If all configurations in the sample are statistically independent, having no correla-tions, then the sample average is normally distributed around the true expectation value,and the error estimate would be hOi = ¯ O + 1 / √ N . To estimate the uncertainty of anaverage over the configurations without assuming a statistical distribution inherent tothe variables, we use the Bootstrap method defined below.
Setting the scale
Lattice quantities are, in general, dimensionless with the values given in terms of thelattice spacing a . To obtain physical values we need to set this scale by choosing asuitable value for a which is not an input parameter of the formulation.To do this we match a given dimensionless lattice object, am g , with an experimentalvalue ( m g, phys ). The lattice spacing is then obtained by a = am g m g, phys . (2.26)The lattice spacing of the configuration ensembles used in this work were computedfrom the string tension data in [9]. The string tension is defined from the quark-antiquarkpotential which is related to the large n behaviour of the lattice expectation value of aplanar rectangular loop (analogous to the square loop, eq. (2.9)), see [38]. In this thesis, all statistical errors from the simulations are estimated using the boot-strap method. The bootstrap is a distribution independent method that can be used toestimate the statistical error of any quantity S . A review of the method can be foundin [58].Considering a given initial sample of N elements { U i } , i = 1 , ..., N obtained froman unknown distribution (in our case the sample is the set of gauge field configurations).We are interested in obtaining the statistical error associated to a quantity S ( U ) whichin this work corresponds to a mean value of some quantity over the configurations.The method considers the empirical distribution for the original sample, assigningthe probability 1 /N to each of the observed elements. A bootstrap sample is constructedby random sampling with replacement from this probability distribution. We obtain N b random samples U jb = ( U j , ..., U jN ) from the original, of the same size N . For eachsample j , the quantity is computed to be S j ≡ S ( U j ). The idea of the method is thatnow, we have a proper random variable S j with a known distribution – the empirical.To obtain confidence intervals without assuming the underlying distribution, thebootstrap method provides asymmetric boundaries around the expectation value. Hav-ing N b values S j , from which we obtain ¯ S , the upper and lower errors are estimatedusing confidence intervals, σ up = S up − ¯ S , σ down = ¯ S − S down (2.27)24here S up and S down are found in a way that they satisfy {S j < S up } N b = 1 + C , {S j < S down } N b = 1 − C C is the coefficient chosen for the confidence interval, C ∈ [0 ,
1] and {} representsthe cardinality of a given set.In this work, C was chosen to be C = 0 .
675 representing a 67 .
5% probability of thetrue estimator falling in the interval. The uncertainty was taken to be the largest of thetwo errors. 256 hapter 3
Gluon tensor bases
In this chapter we describe how the discretization of space-time affects the tensor repre-sentations of the gluon propagator. Although we consider these structures for the gluonpropagator, we will find that there are special kinematic configurations for which thelattice structures provide similar results as those obtained using the continuum tensorbasis.Some general aspects of discretization effects and possible corrections methods willbe also introduced. Finally, the three and four gluon vertices will be discussed, andcorresponding tensor bases will be shown.
The O (4) symmetry of the Euclidean continuum theory is replaced by the H (4) groupwhen space-time is discretized using an hypercubic group. This group consists of powersof π/ H (4). Gluon correlation functions are tensors with respect to the H (4)group and, therefore, identifying the tensor bases for this group is crucial to achieve aproper description for the gluon Green’s functions. These tensor structures differ fromthe continuum tensors due to lessened symmetry restrictions.To see how this affects the construction of tensors we consider an N d -dimensionalvector space with a given transformation having matrix representation M . A givenvector p in this space transforms as p ′ = M p, p ′ µ = M µν p ν . (3.1)with components p µ defined with respect to a given coordinate basis. The generalizationto higher order vector spaces is given by the definition of tensors with respect to the given The summation convention over repeated indices is used throughout this chapter. k -rank tensor is a quantity described in general by N kd components T µ ...µ k in a given coordinate basis with the following transformation law T ′ µ ...µ k = M µ ν ...M µ k ν k T ν ...ν k . (3.2)This definition includes vectors ( k = 1), as well as scalars ( k = 0) which are unchangedby the group transformations.In an O ( N d ) symmetric space, scalar products of vectors are unchanged under thegroup transformations, employed by orthogonal N d × N d matrices, M µν = M − νµ . Tosee how the definition (3.2) restricts the form of tensors, we consider the case of ascalar quantity S depending on a vector p . As a scalar, it remains unchanged by thetransformation, S ( p ′ ) = S ( p ). These two transformations restrict the dependence of S on p through the scalar product, S ( p ), since p is an O ( N d ) group invariant.If instead of a scalar we consider a vector valued function ~V ( p ) also depending onthe vector p . By using its transformation law V ′ µ ( p ′ ) = M µν V ν ( p ) we conclude that themost general form for its components is V µ ( p ) = V ( p ) p µ (3.3)where V ( p ) is a scalar of the vector p , [59].An important case for this work are second rank tensors D µν ( p ) depending on asingle vector p . From (3.2) its transformation law is D ′ µν ( p ′ ) = M µρ M νσ D ρσ ( p ). Hence,the most general form for this quantity is of the form D µν ( p ) = A ( p ) δ µν + B ( p ) p µ p ν . (3.4)This tensor will be considered for the description of the gluon propagator to evaluate howthe Landau gauge Slavnov-Taylor identity, eq. (1.39), acts on the lattice. With thesethree examples we see that continuum vectors have a simple, linear structure imposedby the continuum symmetry. We are interested in performing a similar constructionconsidering the lattice symmetry.The H ( N d ) group is a discrete subgroup of O ( N d ) in an N d -dimensional space. Itconsists of π/ . The reason why it isworth to decompose the H ( N d ) group into these two smaller subgroups is that they aredisjoint , and thus can be analysed independently. Hence, to find objects transformingproperly under the H ( N d ) group it is sufficient to find those which transform properlyaccording to both permutations and inversions. This is seen by considering a 2-dimensional example: performing a clockwise π/ c = ( c , c ) to c ′ = ( c , − c ) can be achieved by the composition of the inversion of the firstcomponent followed by a permutation of both components. Generalizations for higher dimensional spacesare straightforward since these transformations may be independently applied to each hyperplane. In fact, permutations correspond to transformations with determinant +1 while inversions to trans-formations with determinant − .1.1 Scalars under the hypercubic group Proceeding as for the continuum case, we start with the scalar functions on the latticedepending on a single momentum vector p . We inspect the vector dependence of theseobjects which must be invariant under permutations and inversions of components. Itcan be easily seen that the class of objects p [2 n ] ≡ X µ p nµ , n ∈ N (3.5)satisfies this property, and each of them is an hypercubic invariant . Hence, we wouldthink that in general a momentum dependent scalar function would depend on all of theseobjects. It was shown in [60], however, that only N d invariants are linearly independent,thus creating a minimal set of invariants.The interesting cases for this work are the scalar functions depending on a 4-dimensionalvector p which will generally change to S ( p ) → S L ( p , p [4] , p [6] , p [8] ) (3.6)when passing to the lattice. The choice of the four lowest mass dimension independentinvariants is done for practical reasons, but is nonetheless arbitrary. We now generalize the vector notion for the hypercubic symmetric space. As referred,we find its properties by analysing the permutations and inversions independently.Starting with the permutations, and given that any general transformation of thiskind can be written as a product of exchanges of only two components – transpositions[59] – we focus on those. Hence, an object transforming as a vector under arbitrarytranspositions will also transform as a vector under a general permutation. Performinga transposition of components σ ↔ ρ , the transformation for the vector components p µ in an N d -dimensional space is p ′ ν = p ν , ν = σ, ρp ′ σ = p ρ ,p ′ ρ = p σ . (3.7)This is the fundamental transformation rule for a vector, however we are interested infinding the most general structure satisfying this rule. Indeed, any polynomial of thevector, ( p µ ) n also transforms as a vector under transpositions (a brief proof is shown inappendix B.1.1)However, to be a proper vector under H ( N d ) it also needs to satisfy the transfor-mation under inversions. Taking the same N d -dimensional vector p and applying an The case p [2] = p is the only invariant in the continuum, i.e. for O ( N d ). σ -th component, the transformed components are p ′ µ = p µ , µ = σ,p ′ σ = − p σ . (3.8)To be a vector, the polynomial should transform exactly as (3.8)( p ′ µ ) n = ( p µ ) n , µ = σ, ( p ′ σ ) n = − ( p σ ) n , (3.9)and for this to be true, n is necessarily an odd integer, otherwise an even integer wouldspoil the transformation by eliminating the minus sign of the inversion. Therefore themost general structure satisfying the vector transformation is v nν = p n +1 ν , n ∈ N . (3.10)Moreover, we also note that any linear combination of these vectors is also a vector(by linearity) and thus any function whose Taylor expansion includes only odd powersof a vector also constitutes a lattice vector. We now see that the sinusoidal, improvedmomentum ˆ p µ = 2 sin (cid:18) ap µ (cid:19) (3.11)arising from lattice perturbation theory is a proper lattice vector, since it transformscorrectly under the H (4) group.A general lattice vector is then composed of a linear combination of N d vectors fromthe infinite possible vectors of the form (3.10) V µ ( p ) = N d X n =1 V n v n +1 ν (3.12)where V n ( p ) are lattice scalar functions. The sum is limited by the dimension of spacesince in a N d -dimensional space only N d linearly independent basis vectors can be con-structed. We now consider the gluon propagator – a second order tensor depending on a singlevector, the momentum p . In colour space the lattice gluon propagator is a two dimen-sional tensor having the same form as in the continuum formulation. Indeed, δ ab is theonly second order SU (3) tensor available. Thus we focus on the space-time structure ofthe propagator. Being a second order tensor depending on a single momentum D µν ( p ),the gluon propagator transforms as D ′ µν ( p ) = M µσ M νρ D σρ ( p ) . (3.13)30here M ∈ H (4) is a matrix representation of an arbitrary group element.Following [13] we consider the splitting of the tensor basis in the diagonal and off-diagonal terms. This is related with the way the hypercubic transformations act on thelattice tensors, not mixing the aforementioned groups of elements D µµ and D µν , µ = ν (see appendix B.1.2 for a proof of this property). Accordingly, the diagonal and off-diagonal tensor elements will be parametrized differently, i.e. by different form factors.The most general objects to construct the tensor basis are { δ µν , p mµ p nν } . However, forthe second element, since the transformation rule for the tensor applies independentlyfor each momentum, a similar argument as the one used for the vectors in section 3.1.2restricts m and n to be odd integers. Thus, we obtain a set of the most general possibletensor basis elements { δ µν , p k +1 µ p s +1 ν } , k, s ∈ N . (3.14)For the propagator itself, notice that a symmetric second order tensor has only N d ( N d + 1) / . However, for reasons that will be evident when analysing the results, weconsider only two reduced bases for the propagator with three and five form factors.Consider the case of approximating the tensor by three form factors. The possiblechoices for diagonal and off-diagonal terms are { δ µµ , p µ , p µ , ... } , and { p µ p ν , p µ p ν , ... } ,respectively. Choosing the parametrization with the lowest mass dimension terms weobtain the form D µµ ( p ) = J ( p ) δ µµ + K ( p ) p µ , (no sum) D µν ( p ) = L ( p ) p µ p ν , µ = ν. (3.15)We also consider an extended tensor basis using five form factors. Performing thesame construction as before and considering an explicit symmetrization on the spaceindices for the higher order non-diagonal terms, we obtain D µµ ( p ) = E ( p ) δ µµ + F ( p ) p µ + G ( p ) p µ , (no sum) D µν ( p ) = H ( p ) p µ p ν + I ( p ) p µ p ν ( p µ + p ν ) , µ = ν (no sum) . (3.16)The extraction of the form factors involves the computation of its projectors, theseare built in appendix B.2. In chapter 4 these form factors will be obtained from thelattice and there we will introduce continuum relations among them that follow fromboth the Slavnov-Taylor identity and gauge condition on the lattice.Notice that the tensor basis can be built with normal momentum p µ or the latticeperturbation theory improved momentum ˆ p µ which may serve as a further improvement.However, structures mixing both types of momenta are not considered.Notice that the tensor parametrization by the bases is independent of the chosengauge, however this choice will entail different relations among the form factors. Wework with the Landau gauge, implying orthogonality of the gauge fields in the continuum, p µ A µ ( p ) = 0. In principle, however, further conditions implied by the Slavnov-Taylor identity and gauge fixingfurther reduce the number of independent parameters. eneralized diagonal kinematics Having the general form of the lattice basis, it is important to consider configurationsfor which the basis is reduced to a simpler form, closer to the continuum tensor basis.To those we call generalized diagonal kinematics and its form is specified by a singlescale or vanishing components. Of this group belong the full diagonal, ( n, n, n, n ), themixed configurations ( n, n, n,
0) and ( n, n, , n, , , n, n, n, n ) we get p µ = n δ µµ . Therefore only a reduced number ofform factors is extracted. Details on the changes of the lattice basis for these kinematicsand how the form factors are extracted are shown in appendix B. To analyse how accurately a tensor basis describes the correlators from the lattice, weperform a reconstruction procedure [13, 14]. This consists in extracting a given set ofform factors, associated to the corresponding basis element, from the lattice correlationfunction and with these functions rebuild the original tensor. If the rebuilt function isdifferent from the original we can infer that the basis is not complete and informationwas lost during the projection process. To do this we consider the following quotient R = P µν | Γ orig µν | P µν | Γ rec µν | (3.17)given by the sum of absolute values of the original tensor and the reconstructed one. Avalue of R = 1 indicates that the basis is complete.The procedure follows by assuming that the correlator is described by its basis ele-ments τ j with corresponding form factor γ j Γ = N X j =1 γ j τ j . (3.18)One starts by computing each form factor γ j using the respective projector – this stepis the one where information may be lost if the basis is not complete, since in thiscase there are not enough form factors to fully represent the object. This extraction isperformed on the original vertex Γ orig , which in the case of this work comes from thelattice simulation. Using eq. (3.18) we reconstruct the vertex and obtain Γ rec . In the continuum formulation, having rotational invariance means that the form factorsdepend only on the magnitude of the momenta, i.e., that exists some sort of rotational The absolute value was considered in order to prevent possible unintentional cancellations amongthe tensor components. π/ H (4) invari-ants which label the orbits of the group, and are invariant under the transformations.Therefore, these points should have the same contribution when computing lattice cor-relation functions .Hence, to help suppressing statistical fluctuations we consider equally the contribu-tion from all points in the subspace defined from all possible group transformations ona given lattice point. This is accomplished by averaging all computed quantities overall points in the same orbit which amounts to 4! × = 384 points for each momentumconfiguration in four dimensions. In order to properly evaluate the form factors that characterize the correlation functionsit is necessary to account for the artifacts arising from the discretization of space. Thesesystematic errors become noticeable when the precision associated with a computationbecomes high enough such that the statistical errors are small compared with these‘defects’. Since the gluon propagator is computed with a good degree of precision, theremoval of these artifacts becomes relevant.We distinguish two types of artifacts related to the introduction of the lattice. Firstly,finite size effects due to the use of a finite spacing a as well as volume V . These werestudied in [62] where it was found for the gluon propagator that the interplay betweenthese two effects were far from trivial. Secondly, what we call hypercubic artifacts arisefrom the breaking of O (4) symmetry, and the appearance of multiple H (4) orbits fromeach O (4) orbit. We consider the latter in this section.Since we are interested in extracting scalar form factors, we consider the behaviourof lattice scalar functions and how they relate to the corresponding continuum objects.Any scalar function with respect to a given symmetry group is invariant along the orbitgenerated by the corresponding group symmetry applied to a given point. For the H (4)group each orbit is specified by the four group invariants { p [2] , p [4] , p [6] , p [8] } . The simplest example of this is given by comparing with the continuum symmetry. Inthis case, an orbit is simply labelled by the invariant p . For instance, both momenta p = (2 , , ,
0) and p = (1 , , ,
1) have p = p = 4 in the same O (4) orbit. However,these two points have different H (4) invariants, p = 16 and p = 4 belongingto distinct H (4) orbits, thus should not be averaged equivalently. We see that thedependence of the scalars on the p [4] invariant spoils the continuum symmetry. The contribution of these points may not be exactly the same due to statistical fluctuations. n > p dependence as in the continuum . Anotherway to understand why the finiteness of the higher order invariants relates to hypercubicartifacts is seen by considering the improved momentum arising from lattice perturbationtheory. By looking at the improved invariant ˆ p expanded in orders of a ˆ p = (2 sin( ap/ = p − a p [4] + a p [6] + ... (3.19)we see that it differs from the naively discretized continuum momentum by terms whichare proportional to the invariants. Therefore, we can minimize the lattice invariants inorder to suppress hypercubic artifacts depending on non O (4) group invariants, i.e. byreducing the first higher order invariant p [4] we are effectively reducing the artifacts. Toperform this correction two distinct methods are considered. The simplest method consists in applying cuts to the momenta. This arises by noticingthat the further a momentum is from the diagonal, the higher are its non O (4) invariantsfor a fixed O (4) invariant p . This was seen for the example considered before with(2 , , ,
0) being on-axis momentum with higher p [4] .An empirical way to deal with higher invariants coming from these kinematics isto directly discard these momenta from the data. The usual choice is to consider onlymomenta inside a cylinder directed along the diagonal of the lattice as defined in [63].This selects the largest momenta with the smallest components, i.e. with the lowest H (4)invariants. The radius of the cylinder is chosen as to maintain a good amount of datawhile reducing the artifacts, and in general a radius of one momentum unit ( ap = 2 π/N )is considered.This cut, however, does not remove low momentum on-axis points. To improve themethod we consider further conical cuts, i.e. we consider only momenta falling insidea conical region around the diagonal of the lattice (1 , , , The H4 method [64, 65] is more involved as it attempts to entirely eliminate the contri-bution of the invariants p [ n ] with n > p [4] , however, this method can Note that finite size effects still affect the result after this correction.
34e improved with higher order corrections (given that enough data is available). Exam-ples of the applications, improvements and general considerations on the method can befound in [64, 66, 67].We consider a given scalar function under the lattice symmetry Γ L ( p [ n ] ) , n = 2 , , , O ( p [ n ] ),Γ L (cid:16) p , p [4] , p [6] , p [8] (cid:17) = 1 N O X p ∈ O ( p [ n ] ) Γ( p ) (3.20)where N O corresponds to the cardinality of the orbit. We want to study how it relatesto the continuum counterpart Γ( p ).Assuming that the scalar is a smooth function of the invariants, we may extrapolateto the continuum by Γ( p ) ≡ lim p [4] → Γ L ( p , p [4] ) (3.21)neglecting higher order invariants which vanish as O (cid:0) a (cid:1) . In fact, to O (cid:0) a (cid:1) the sameextrapolation is possible for the improved momentumlim p [4] → Γ L ( p , p [4] ) = lim ˆ p [4] → Γ L (ˆ p , ˆ p [4] ) (3.22)although in practice this extrapolation is not easily feasible.To implement the extrapolation in practice, we assume that the dependence on theinvariants is smooth, and also that the lattice is close to the continuum limit (small a )to use the expansionΓ L (cid:16) p , p [4] , p [6] , p [8] (cid:17) = Γ L ( p , , ,
0) + ∂ Γ L ∂p [4] ( p , , , p [4] + O (cid:16) a (cid:17) . (3.23)Thus we may identify Γ L ( p , , ,
0) as the continuum function Γ( p ) in finite volumeand up to higher order lattice artifacts. In practice this is applied only when several H (4) orbits exist with the same O (4) invariant p . The extrapolation is done by a linearregression in p [4] at fixed p , taking the results as p [4] → H (4) orbits should exist, this restricts the range of momentum to whichthe method is applicable. Normally, only the mid range of momentum contains enoughdata to perform the extrapolation, thus the deep infrared and high ultraviolet are notconsidered in this correction. The H4 method can be generalized for cases with morethan a single independent momentum. In this work, both for the propagator and threegluon vertex, the simplest case of a single scale momentum is considered. While the gluon propagator in the continuum is described by a single scalar function, D ( p ), under the symmetries of the theory, higher order correlation functions admit anincreased number of form factors for a general kinematic configuration. Thus we mustconsider the most general form under the required symmetries.35or the three gluon vertex the colour structure is restricted to be antisymmetricΓ abcµ µ µ ( p , p , p ) = f abc Γ µ µ µ ( p , p , p ) (3.24)due to the charge invariance of the QCD Lagrangian [68, 69]. This guarantees thevanishing contribution from the symmetric term d abc . We then require that the completeobject obeys Bose symmetry, and since the colour structure is established by the anti-symmetric structure constants, this requires Γ µ µ µ ( p , p , p ) to be anti-symmetric tothe interchange of any pair ( p i , µ i ).For the space-time part of the tensor representing the three gluon vertex we considera continuum basis which consists of 14 independent tensors. Throughout the work weuse the basis constructed in [70] which considers a separation between terms orthogonalto all momenta, and longitudinal terms. The general tensor is given by the transverseand longitudinal termsΓ µ µ µ ( p , p , p ) = Γ ( T ) µ µ µ ( p , p , p ) + Γ ( L ) µ µ µ ( p , p , p ) . (3.25)The first consists of four tensorsΓ ( T ) µ µ µ ( p , p , p ) = F ( p , p ; p ) (cid:2) g µ µ ( p · p ) − p µ p µ (cid:3) B µ + H ( p , p , p ) (cid:2) − g µ µ B µ + 13 ( p µ p µ p µ − p µ p µ p µ ) (cid:3) + cyclic permutations, (3.26)with the definition, B µ = p µ ( p · p ) − p µ ( p · p ) . (3.27)The scalar form factors F ( p , p ; p ) are symmetric under interchange of the first twoarguments, evidenced by the used of the semi-colon, while H ( p , p , p ) is symmetricunder the interchange of any of its arguments. The remaining 10 longitudinal elementsare of the formΓ ( L ) µ µ µ ( p , p , p ) = A ( p , p ; p ) g µ µ ( p − p ) µ + B ( p , p ; p ) g µ µ ( p + p ) µ + C ( p , p ; p )( p µ p µ − g µ µ p · p )( p − p ) µ + 13 S ( p , p , p )( p µ p µ p µ + p µ p µ p µ )+ cyclic permutations (3.28)where both A ( p , p ; p ) and C ( p , p ; p ) are symmetric in their first two argumentswhile B ( p , p ; p ) is anti-symmetric. S ( p , p , p ) is completely anti-symmetric.With this form we have a proper description of the correlation function extractedfrom the lattice, with the right hand side of (2.21) being replaced by G µ µ µ ( p , p , p ) = N c ( N c − D µ ν ( p ) D µ ν ( p ) D µ ν ( p ) ×× (Γ ( L ) ν ν ν ( p , p , p ) + Γ ( T ) ν ν ν ( p , p , p )) (3.29)36here the colour factor comes from the trace operation and N c = 3. The extraction of ageneral form factor is done by suitable projectors built analogously to those consideredfor the propagator. Kinematical configuration ( p, , − p ) The kinematics used in this work is defined by ( p , p , p ) = ( p, , − p ) which due tohaving a single scale p allows only the longitudinal terms. This is because contractionswith external propagators eliminate the transverse terms with p µ i Γ ( T ) µ µ µ ( p , p , p ) = 0 (3.30)for any i = 1 , ,
3. The explicit expression for eq. (3.29) becomes G µ µ µ ( p, , − p ) = V N c ( N c − D ( p ) D (0)Γ( p ) p µ (cid:18) δ µ µ − p µ p µ p (cid:19) (3.31)with Γ( p ) = 2 (cid:16) p C ( p , p ; 0) − A ( p , p ; 0) (cid:17) (3.32)a dimensionless form factor. We see that for this specific configuration, only a combina-tion of form factors can be extracted. Finally, the 1PI form factor Γ( p ) can be projectedby the following contractionΓ( p ) p = 4 p µ δ µ µ G µ µ µ ( p, , − p ) V N c ( N c − D ( p ) D (0)( N d −
1) (3.33)for non-vanishing momentum.
The four point correlation function in QCD is the most complex elementary correlationfunction arising in the Yang-Mills theory. Having three independent momenta, fourLorentz and colour indices, it generates a large amount of possible structures [71]. Onthe other hand, being a higher order correlation function, its signal from the Monte-Carlo simulations is strongly affected by noise. This last problem justifies the absenceof previous four gluon lattice studies.A further complication arises for this higher order correlation function. We are inter-ested in computing the four gluon 1PI function, i.e. the pure four gluon vertex. Whilefor the three gluon vertex this is simply obtained by the removal of external propagatorsfrom the complete correlation function, the four gluon correlation function carries ad-ditional contributions from lower order Green’s functions. Namely, disconnected termsand the three gluon vertex enter in the computation of the complete correlation function– see fig. 3.1. Thus the object we have access in the lattice for a general momentum37onfiguration reads G (4) a a a a µ µ µ µ ( p , p , p , p ) = D µ ν ( p ) D µ ν ( p ) D µ ν ( p ) D µ ν ( p )¯Γ (4) a a a a ν ν ν ν ( p , p , p , p ) − iD µ ν ( p ) D µ ν ( p )Γ (3) ma a σν ν ( p + p , p , p ) ×× D σρ ( p + p )Γ (3) ma a ρν ν ( p + p , p , p ) D µ ν ( p ) D µ ν ( p )+ D a a µ µ ( p ) D a a µ µ ( p ) δ ( p + p ) δ ( p + p )+ cyclic permutations. (3.34)Only the first term, that includes the four gluon 1PI function is of interest to us and theremaining ought to be removed.= +3 +3 Figure 3.1: Diagrammatic representation of the connected and disconnected terms contributingfor the full, four-gluon correlation function.
We wish to remove lower order contributions without affecting the quality of thesignal. Hence, we do not directly subtract the unwanted contributions in the simulationssince other than requiring a heavier computation, the statistical fluctuations would beincreased. To carry out this extraction we consider a suitable choice of kinematics.To see how this removes the unwanted contributions we notice that momentum con-servation constrains the possible kinematic configuration for each vertex. Moreover, theorthogonality of external gluon propagators eliminates terms when contracted with thecorresponding momentum p µ D µν ( p ) = 0 . (3.35)The disconnected terms without interaction (last line in eq. (3.34)) are eliminated bya suitable kinematic configuration, that while allowed by momentum conservation for thefour gluon vertex, it is not permitted for the two propagators. Whereas the cancellationof disconnected terms is straightforward, the three gluon contributions requires to noticethat the most general rank-3 continuum tensor necessarily involves a momentum factor.They are either linear, g µ µ p µ or cubic in the momenta p µ p µ p µ – see section 3.6.Therefore we can eliminate the three gluon contribution by eliminating each of theseterms appearing in Γ (3) above. If we choose a single scale momentum configuration( p , p , p , p ) = ( ap, bp, cp, dp ) , each external propagator will be of the form D µν ( p )thus eliminating each of the three gluon tensor structures by orthogonality. Of the coefficients a, b, c, d only three are independent, by momentum conservation.
38e see that a proper choice of kinematic configuration provides access to the purefour gluon vertex in the lattice G (4) a a a a µ µ µ µ ( ap, bp, cp, dp ) = D µ ν ( ap ) D µ ν ( bp ) D µ ν ( cp ) D µ ν ( dp )¯Γ (4) a a a a ν ν ν ν ( ap, bp, cp, dp )) (3.36)using the complete correlation function only, i.e. without additional operations involvinglower order functions. Having access to the four gluon 1PI function we need to construct a tensor basis in whichthis function will be projected. This basis involves a large number of possible structures.At the level of Lorentz tensors, there are three types of structures allowed that are builtwith the metric tensor and momenta. These are linear, quadratic or quartic in momenta, { g µ µ g µ µ , g µ µ p µ q µ , p µ q µ r µ k µ } . (3.37)which for a general momentum configuration make up 138 possible structures [72]. How-ever, due to practical reasons, in the present work we consider a reduced basis limitedto the first elements using the metric tensor only . With this choice, only a smallernumber of independent tensors will contribute to the vertex.For the colour sector we can use the SU (3) antisymmetric structure constants f abc ,the symmetric terms d abc as well as δ ab to construct all possible structures { f ma a f ma a , d ma a d ma a , d ma a f ma a , δ a a δ a a } . (3.38)However, various group identities reduce the number of possible terms, see appendix A.Due to the complexity associated with the tensor basis for a general kinematic con-figuration, in the following we restrict the construction to a specific, single scale config-uration. Kinematical configuration ( p, p, p, − p ) We work with the configuration ( p, p, p, − p ) which was considered in the continuuminvestigations [31, 32]. The most complete basis within our approximation to metricstructures consists of three possible Bose symmetric tensors. These are the tree-leveltensor, written again for convenienceΓ (0) a a a a µ µ µ µ = − g (cid:2) f a a m f a a m ( g µ µ g µ µ − g µ µ g µ µ ) f a a m f a a m ( g µ µ g µ µ − g µ µ g µ µ ) f a a m f a a m ( g µ µ g µ µ − g µ µ g µ µ ) (cid:3) , (3.39) Although this approximation cuts a large number of possible tensor structures, previous investiga-tions found that the tree-level tensor seems to provides the leading contribution in comparison with therest of tensor structures [32]. This behaviour is also found in the three gluon correlation function [17].
39 fully symmetric tensor (in both colour and Lorentz sectors) G a a a a µ µ µ µ = ( δ a a δ a a + δ a a δ a a + δ a a δ a a )( g µ µ g µ µ + g µ µ g µ µ + g µ µ g µ µ )(3.40)which is orthogonal to Γ (0) in both spacesΓ (0) b b b b µ µ µ µ G a a a a µ µ µ µ = 0 , Γ (0) a a a a ν ν ν ν G a a a a µ µ µ µ = 0 . (3.41)And finally, the third independent tensor is X a a a a µ µ µ µ = g µ µ g µ µ (cid:18) δ a a δ a a − d ma a d ma a (cid:19) + g µ µ g µ µ (cid:18) δ a a δ a a − d ma a d ma a (cid:19) + g µ µ g µ µ (cid:18) δ a a δ a a − d ma a d ma a (cid:19) . (3.42)With this tensor basis, we construct the general structure with three symmetric formfactors asΓ a a a a ν ν ν ν = V ′ Γ (0) ( p )Γ (0) a a a a µ µ µ µ + V ′ G ( p ) G a a a a µ µ µ µ + V ′ X ( p ) X a a a a µ µ µ µ . (3.43)with scalar form factors V ′ i depending on the single momentum scale p . This in turnis related to the complete correlation function by the contraction with four externalpropagators. To extract each form factor from the lattice we again apply the traceoperation in the colour space. This operation involves the structures in eq. (3.38) whichmake for more intricate operations than the one found for the three gluon vertex. Forthese the group identities in appendix A were used. Using the notationTr [ G µ µ µ µ ] = D µ ν ( p ) D µ ν ( p ) D µ ν ( p ) D µ ν ( p ) X a i i ∈ , , , Tr ( t a t a t a t a ) Γ a a a a ν ν ν ν (3.44)with the arguments of G µ µ µ µ ( p , p , p , p ) and Γ µ µ µ µ ( p , p , p , p ) omitted, andafter performing the three non-vanishing Lorentz contractions we obtain g µ µ g µ µ Tr [ G µ µ µ µ ] = 6 A n V Γ (0) + 15 G n V G + 3(4 X n + X ′ n ) V X (3.45) g µ µ g µ µ Tr [ G µ µ µ µ ] = − A n V Γ (0) + 15 G n V G + 3(2 X n + 3 X ′ n ) V X (3.46) g µ µ g µ µ Tr [ G µ µ µ µ ] = 6 A n V Γ (0) + 15 G n V G + 3(4 X n + X ′ n ) V X (3.47)where the V i are related to the pure vertex form factors by V i ( p ) = V ′ i ( p ) D ( p ) D (9 p ) , (3.48)40nd the following colour coefficients resulting from the trace and sum operation are A n = N c ( N c − , (3.49) G n = N c − N c (2 N c − , (3.50) X n = 13 ( N c − N c − ( N c − N c − N c , (3.51) X ′ n = −
13 ( N c − N c − ( N c − N c − N c . (3.52)Our interest is to obtain each form factor V independently, however by looking ateqs. (3.45) to (3.47) we see that only two contractions are linearly independent andthus only two objects can be extracted. Hence, following [31] the X structure will bedisregarded. With this further approximation the equations simplify to g µ µ g µ µ Tr [ G µ µ µ µ ] = 6 A n V Γ (0) + 15 G n V G (3.53) g µ µ g µ µ Tr [ G µ µ µ µ ] = − A n V Γ (0) + 15 G n V G (3.54)and each form factor is obtained by V Γ (0) = 118 A n ( g µ µ g µ µ Tr [ G µ µ µ µ ] − g µ µ g µ µ Tr [ G µ µ µ µ ]) , (3.55) V G = 145 AG n (2 g µ µ g µ µ Tr [ G µ µ µ µ ] + g µ µ g µ µ Tr [ G µ µ µ µ ]) . (3.56)These complete form factors are obtained in lattice Monte-Carlo simulations by comput-ing the corresponding linear combinations of the complete correlation function G µ µ µ µ .In section 4.3, Monte-Carlo results for this kinematic configurations will be presented.412 hapter 4 Results
In this chapter we investigate lattice tensor representations of the gluon propagatorby considering the tensor structures introduced in the previous chapter. In additionwe study the IR behaviour of the three gluon correlation function and report a firstcomputation of the lattice four gluon correlation function. All results were obtained ina Landau gauge, 4-dimensional pure SU (3) Yang-Mills theory from the Wilson action,eq. (2.15). a (fm) 1 /a (GeV) β N V (fm ) p min (GeV)0.1016(25) 1.943(47) 6.0 80 (8 .
550 0 . . . Table 4.1: Lattice setup for both ensembles used in the computation of the gluon correlationfunctions.
The lattice setup used in this work can be seen in table 4.1. We used two ensembleswith the same lattice spacing but different volumes. The smaller volume lattice also hasa larger number of configurations.The results shown are either dimensionless or expressed in terms of lattice units.However, these are shown as a function of the physical momentum, p = p lat a − GeVwith a − = 1 . H (4) group averaging is applied for all quantities as defined in section 3.4.An average of the quantity is taken over all group equivalent points for each gauge fieldconfiguration. Only then the ensemble average is taken. Also, the reader should beaware that scalar functions on the lattice have the four H (4) invariants as argumentsalthough represented herein with p only. The exception is the case of the extrapolatedvalues where the dependence is partially corrected.The error bars shown correspond to a tenfold bootstrap sampling from the originalset of configurations. For H4 corrected data, error bars result from an initial bootstrap,43ollowed by the linear regression propagation. Regarding the correction methods, wewill use the following convention through all results (unless explicitly stated) – p [4] ex-trapolated data is shown always as a function of the usual lattice momentum p whilemomentum cuts are generally reserved for the improved momentum data ˆ p . In this section we consider the lattice description of the gluon propagator, comparedwith the usual continuum tensor structure. For most of this section we analyse the 80 lattice exclusively. The 64 lattice will be considered in the end in order to search forpossible finite volume effects on the results. We begin by illustrating the correction methods defined in the previous chapter to il-lustrate its advantages and setbacks. We use the gluon propagator as a test, but theconclusions should be applicable to other correlation functions as well as other tensorstructures.All results shown in this analysis are for the continuum tensor eq. (1.40) with formfactor D ( p ) and dimensionless dressing function d ( p ) = p D ( p ). D ( p ) = 1( N c − N d − X µ D µµ ( p ) . (4.1)Notice that the extraction of D ( p ) is independent of the use of the normal or improvedmomentum for the basis.In fig. 4.1 results for the correction methods are shown – use of the improved mo-mentum; momentum cuts; and the H4 extrapolation. In a ) and b ) the complete dataand after momentum cuts is shown in terms of lattice and improved momentum, respec-tively. The complete set of data shows structures created by the hypercubic artifactswhich are much more pronounced when using lattice momentum. This is expected since,as introduced in the section 3.5, ˆ p partially accounts for hypercubic errors up to O (cid:0) a (cid:1) .The use of the complete momentum cuts (cylindrical and conical) are also shown, andcreate a much smoother curve.The curves in terms of lattice and improved momentum after cuts do not agree formomenta above ∼ . c ). In this plot, the p [4] extrapolateddata is also shown, and we see that it matches the data with cuts as a function ofimproved momentum for a large range. An advantage from the extrapolation method isthat it offers a higher density of points for a large range when compared with the curvesurviving the cuts. However, other than the loss of information for lower momentum,the high momentum region is also problematic due to the lack of different H (4) orbits,hence the extrapolation is not reliable. This becomes noticeable for p ∼ a ) b ) c ) d ( p ) p (GeV) p D ( p ) p D ( p )+cuts ˆ p (GeV) p D ( p ) p D ( p )+cuts d ( p ) p (GeV) H4 p D ( p )+cutsˆ p D (ˆ p )+cuts Figure 4.1: Gluon dressing function d ( p ) from the continuum basis as a function of latticemomentum (top left), and as a function of the improved momentum (top right). The momentasurviving cylindrical and conical cuts are shown for the each plot. The comparison betweenthe data in terms of the improved and lattice momenta after complete momentum cuts againstthe H4 corrected data with lattice momentum is shown in the bottom plot. Results from the β = 6 . , lattice. In this section we compare the behaviour of the usual continuum tensor, eq. (1.40), withtwo lattice descriptions given in eq. (3.16) and eq. (3.15). The most general continuumbasis, eq. (3.4), will also be considered. We disregard, for now, the generalized diagonalconfigurations and other kinematics for which the extraction of all form factors is notpossible (details in appendix B.2).The dimensionless form factors p Γ( p ) will be considered due to their appearancein the continuum relations, defined below. These are p E ( p ), p F ( p ), p H ( p ) forthe larger basis. The only exception is for the terms p G ( p ), and p I ( p ) which areexpressed in lattice units. Continuum relations
To probe the accuracy of our results we consider a benchmark result. We use the datapublished in [73] from a precise continuum basis computation of the propagator using45mproved momentum and additional cuts. This result comes from a partial Z4 averagingprocedure, i.e. only using momentum permutations. This data will always be referredas D (ˆ p ) or d (ˆ p ) = ˆ p D (ˆ p ) and shown as a function of improved momentum only.In addition to this benchmark, we consider continuum relations that relate formfactors among themselves and also with the continuum tensor basis result, D ( p ). Theserelations are expected to be properly satisfied for the infrared region where hypercubiceffects are smaller . The reproduction of the continuum basis, eq. (1.40), by the extendedbasis, eq. (3.16), for low momentum implies E ( p ) → D ( p ) (4.2) − p F ( p ) , − p H ( p ) → D ( p ) (4.3) G ( p ) , I ( p ) → . (4.4)while for the reduced lattice basis, eq. (3.15), the continuum relations are J ( p ) → D ( p ) (4.5) − p K ( p ) , − p L ( p ) → D ( p ) (4.6)In addition, for the most general continuum second order tensor, eq. (3.4), we obtain A ( p ) , − p B ( p ) → D ( p ) . (4.7)The reproduction of these relations can be verified in figs. 4.2 to 4.4 where the formfactors are reported as a function of lattice momentum p after a p [4] extrapolation (leftcolumn), and as a function of improved momentum with momentum cuts (right). Infig. 4.2, we compare only the form factors associated with the metric tensor E ( p ), J ( p ), and A ( p ).The functions represented in figs. 4.2 and 4.4 are such that in the continuum limitthey all should become equal, thus satisfying eqs. (4.4), (4.6) and (4.7). It can be seenfor figs. 4.2 and 4.4 that within one standard deviation, continuum relations are satis-fied for improved momentum with additional cuts, although with increased fluctuationswhen compared with the H4 corrected data on the left. The latter, however, have arestricted range of compatibility with the benchmark result. In addition, for fig. 4.4 thetwo H4 form factors F and H for the extended basis seem to deviate from the expectedbehaviour. The same happens for the smaller lattice basis, and this should be relatedto the limitations of the extrapolation for low and high momentum. Despite the fluctua-tions, the fact that the continuum relations are satisfied for a large range of momentumindicates that the lattice is fine and large enough to obtain results close to continuum.In fig. 4.3, the form factors p G ( p ) /a and p I ( p ) /a are reported. In the bottomrow, results are shown after the correction methods are applied for both form factors.The appearance of the larger fluctuations for G and I are expected due to its valuesbeing closer to zero and the increased mixing among a larger number of form factors Note that this does not guarantee that we are extracting proper continuum physics for the IR region.There are still finite volume and finite spacing effects – see [62]. p Γ ( p ) p E ( p ) d (ˆ p ) p E (ˆ p ) d (ˆ p ) p Γ ( p ) p J ( p ) d (ˆ p ) ˆ p J (ˆ p ) d (ˆ p ) p Γ ( p ) p (GeV) p A ( p ) d (ˆ p ) ˆ p (GeV) ˆ p A (ˆ p ) d (ˆ p ) Figure 4.2: p E ( p ), p J ( p ), and p A ( p ) dressing functions as a function of the lattice mo-mentum after a p [4] extrapolation (left) and as a function of the improved momentum ˆ p aftermomentum cuts. The results come from the β = 6 . , lattice and the benchmark continuumdressing function ˆ p D (ˆ p ) is plotted as a function of the improved momentum. when extracting each function. This is also why I ( p ), which only mixes with H ( p ),shows less fluctuations when compared with G ( p ).For low momentum, both correction methods and functions satisfy the continuumrelations within statistical fluctuations in fig. 4.3. However, for momenta above ∼ p I ( p ) is shown for all available47-1.0-0.50.51.01 1 . . . . . . . . . . . . . . . . p Γ ( p ) / a p I ( p ) ˆ p I (ˆ p ) p Γ ( p ) / a p (GeV) p G ( p ) + H4 p I ( p ) + H4 ˆ p (GeV) ˆ p G (ˆ p )+Cutsˆ p I (ˆ p )+Cuts Figure 4.3: Dimensionless form factors p G ( p ) and p I ( p ). G is shown only after the correctionmethods. The original data is shown in the top row for the lattice momentum p (left) and im-proved momentum ˆ p (right) for a restricted range of momenta. Below, p G ( p ) and p I ( p ) afterthe corrections are applied are presented, namely the H4 extrapolated results and momentumcuts. All data from the β = 6 . , lattice. configurations without corrections, but for a restricted range of momentum ( p G ( p )was disregarded due to having large fluctuations). p I ( p ) is much closer to zero for theimproved momentum basis than for lattice momentum before any correction is applied.This result can be viewed as a another improvement in the tensor description after thechange of variables to the momentum ˆ p when building the tensor basis.In fact, the change of variables from p to ˆ p also provides an improvement for theremaining form factors E ( p ), − p F ( p ), and − p H ( p ). However, this is concealed bythe complete set of data, thus specific momentum configurations are helpful in exposingthis effect. In fig. 4.5 these three form factors are shown for two different kinematics forboth the normal and improved momentum bases in the left and right columns, respec-tively. The continuum relations are much better satisfied for the improved momentumcase. In regards to reproducing the expected result, D (ˆ p ), the form factor E ( p ) showsthe best results for lattice momentum.The combination of the results from figs. 4.2 to 4.4 means that the continuum rela-tions are properly reproduced for a large range of momenta. This can be interpreted asthe survival (at least to some extent) of the Slavnov-Taylor identity and Landau gauge480.51.01.52.02.53.03.54.00 1 2 3 4 5H4 extrapolation 01.02.03.04.05.06.00 1 2 3 4 5 6 7 8Momentum cuts00.51.01.52.02.53.03.54.00 1 2 3 4 5 00.51.01.52.02.53.03.54.00 1 2 3 4 5 6 7 800.51.01.52.02.53.03.54.00 1 2 3 4 5 00.51.01.52.02.53.03.54.00 1 2 3 4 5 6 7 8 p Γ ( p ) − p F ( p ) − p H ( p ) d (ˆ p ) − ˆ p F (ˆ p ) − ˆ p H (ˆ p ) d (ˆ p ) p Γ ( p ) − p K ( p ) − p L ( p ) d (ˆ p ) − ˆ p K (ˆ p ) − ˆ p L (ˆ p ) d (ˆ p ) p Γ ( p ) p (GeV) − p B ( p ) d (ˆ p ) ˆ p (GeV) ˆ p B (ˆ p ) d (ˆ p ) Figure 4.4: Dressing functions for the different tensor bases as a function of the lattice momentumafter a p [4] extrapolation (left) and as a function of the improved momentum ˆ p after momentumcuts. These come from the β = 6 . , lattice. The improved continuum tensor form factor D (ˆ p ) is also shown. condition on the lattice that fix the form of the gluon propagator to be orthogonal. Thisalso confirms the improvement obtained from the change of variables p → ˆ p with respectto the description of lattice correlation functions.Other than allowing to check the continuum relations, figs. 4.2 to 4.4 allow to com-pare the three extended tensor bases from the point of view of the general descriptionof the gluon propagator. With this analysis we inspect the difference between the re-49-0.20.20.40.60.81.01.21.41.61.8 . . a ) (20 ,n,n, . . b ) (20 ,n,n, . . . . . . c ) p =( n +6 ,n,n,n − . . . . . . d ) p =( n +6 ,n,n,n − Γ ( p ) / a D (ˆ p ) E ( p ) − p F ( p ) − p H ( p ) D (ˆ p ) E (ˆ p ) − ˆ p F (ˆ p ) − ˆ p H (ˆ p ) Γ ( p ) / a p (GeV) ˆ p (GeV) Figure 4.5: E ( p ), − p F ( p ), and − p H ( p ) from the improved momentum lattice basis (right)and from the normal momentum lattice basis (left). Data from the β = 6 . , lattice. Thestandard result for D (ˆ p ) is also shown as a function of the improved momentum. duced and extended lattice bases in regards to reproducing the gluon propagator – thiswill be complemented by the reconstruction analysis below. Turning again to figs. 4.2and 4.4, all results portray ˆ p D (ˆ p ) within one standard deviation, although with in-creased fluctuations as one increases the basis elements (bottom to top in the rightcolumns). Nonetheless, all three sets of functions seem define a single curve compatiblewith the benchmark result when represented in terms of the improved momentum ˆ p .However, even with the momentum cuts large fluctuations appear for the larger tensorbasis, due to the mixing of different elements in the projection of form factors. In fact, for p F ( p ) in terms of improved momentum in fig. 4.4 the fluctuations are present througha larger range, starting around 1 . p [4] extrapolation also reproduce the benchmark result d (ˆ p ) although in a limited range. The H4 extrapolation seems to remove most of thestatistical fluctuations when compared to the data in the right column. For this methodthere is a clear distinction between the metric, p E ( p ), p J ( p ), and p A ( p ) in fig. 4.2and the remaining non-vanishing form factors in fig. 4.4. The range of agreement withthe benchmark result is larger for the metric form factors with the deviation appearingfor p ∼ { A, B } ) with deviations starting forlower momenta.Regarding the fluctuations appearing for larger tensor bases, this problem can beovercome by using a binning procedure, where points inside each momentum bin areaveraged using a weighted average. Although with this we are summing non equivalentpoints with respect to the group symmetry, this procedure is allowed by noting that theuncertainty in the scale setting (choice of a ) is around 2 . p ∼ − d ( ˆ p ) ˆ p (GeV)All dataCuts ˆ p (GeV) CutsBins Figure 4.6: Gluon dressing function d (ˆ p ) as a function of the improved momentum for thecontinuum basis published in [73]. The left plot shows the complete set of data and the curvesurviving momentum cuts. Additionally, the right plot shows the averaged data in each bin –description in the text. The binned versions of figs. 4.2 to 4.4 are shown in figs. 4.7 to 4.9. The binning ofthe data defines smoother curves with smaller statistical errors which allow for betteranalysis of the deviations from the benchmark result. For fig. 4.9 some small fluctuationsare noticed for p ∼ . d (ˆ p ) while the large statistical fluctuations havebeen absorbed by the averaging procedure. The visible deviation for the mid range ofmomentum do not appear in non-binned results and should be associated with the bin-510.51.01.52.02.53.03.54.00 1 2 3 4 5H4 extrapolation 00.51.01.52.02.53.03.54.00 1 2 3 4 5 6 7 800.51.01.52.02.53.03.54.00 1 2 3 4 5 00.51.01.52.02.53.03.54.00 1 2 3 4 5 6 7 800.51.01.52.02.53.03.54.00 1 2 3 4 5 00.51.01.52.02.53.03.54.00 1 2 3 4 5 6 7 8 p Γ ( p ) p E ( p ) d (ˆ p ) p E (ˆ p ) d (ˆ p ) p Γ ( p ) p J ( p ) d (ˆ p ) ˆ p J (ˆ p ) d (ˆ p ) p Γ ( p ) p (GeV) p A ( p ) d (ˆ p ) ˆ p (GeV) ˆ p A (ˆ p ) d (ˆ p ) Figure 4.7: Dressing functions p E ( p ), p J ( p ), and p A ( p ) from the β = 6 . , lattice asa function of the lattice momentum after a p [4] extrapolation (left) and as a function of theimproved momentum ˆ p . The data is shown after a binning of 2 .
5% in momentum was performed.The continuum dressing function ˆ p D (ˆ p ) is shown with momentum cuts. ning procedure. For the H4 corrected data, the binning procedure results in a reductionof fluctuations and allows to better recognize the deviations from the benchmark result.In general, for the extrapolated data, the best agreement with the expected result seemsto be obtained by the smaller tensor basis { A, B } . For the improved momentum basesthe situation is not so clear, the best match with d (ˆ p ) seems to be obtained for p H ( p ).For the form factors G ( p ) and I ( p ), also shown after a binning procedure in fig. 4.8,52-2.0-1.5-1.0-0.50.51.01.52.01 1 . . . . . . . . . . Γ ( p ) / a p (GeV) p G ( p ) + H4 p I ( p ) + H4 ˆ p (GeV) ˆ p G (ˆ p )ˆ p I (ˆ p ) Figure 4.8: Form factors for the higher order terms of the extended basis p G ( p ) and p I ( p ) interms of the usual momentum after the p [4] extrapolation (left) and as a function of the improvedmomentum (right) without any correction applied. Both cases are shown after a 2 .
5% binningis applied in the momentum axis. Data from the β = 6 . , lattice. the interpretation given for fig. 4.3 is now much clearer. Large fluctuations for lowmomentum are expected due to the smallness of ∆ , ∆ in the extraction of bothterms – appendix B.2. The improved momentum basis shows a better agreement withthe continuum relations, while the normal momentum after the extrapolation showsdeviations for higher momenta.From the above analysis we would conclude that the use of larger bases does notimprove the description of the gluon propagator. In fact, the use of larger bases intro-duces fluctuations in the computations. This, together with the fact that the continuumrelations are obtained through the complete range of momentum restrains us from con-sidering further additions to the lattice basis. The use of a more complete tensor basiswould require an increase in the statistics to counteract the fluctuations coming fromthe mixing with a larger number of terms.Regarding the results obtained in [13] using a similar approach, the continuum rela-tions are only satisfied for low momentum (or close to diagonal configurations) while inour case the relations are satisfied through all range of momentum, namely when usingˆ p . Note, however that the referred work uses only 2 and 3-dimensional SU (2) latticeswith a larger lattice spacing, and thus the comparison is to be taken with care. Completeness of the tensor bases
The analysis of the form factors alone does not offer the full picture for how the latticebases affect the description of the tensor . Indeed, form factors alone do not allow toperceive how faithful the tensor description with a given basis is. The most evident caseis for the continuum description which returns the exact same form factor using normalor improved momentum while the latter reproduces the original tensor with greater It is important to distinguish the description of the gluon propagator D ( p ), from the description ofthe original lattice tensor D µν ( p ) which is the focus when exploring the completeness of a basis. p Γ ( p ) − p F ( p ) − p H ( p ) d (ˆ p ) − ˆ p F (ˆ p ) − ˆ p H (ˆ p ) d (ˆ p ) p Γ ( p ) − p K ( p ) − p L ( p ) d (ˆ p ) − ˆ p K (ˆ p ) − ˆ p L (ˆ p ) d (ˆ p ) p Γ ( p ) p (GeV) − p B ( p ) d (ˆ p ) ˆ p (GeV) ˆ p B (ˆ p ) d (ˆ p ) Figure 4.9: β = 6 . , lattice non-metric dressing functions for three tensor bases as a functionof the lattice momentum after a p [4] extrapolation (left) and as a function of the improvedmomentum ˆ p , both after a 2 .
5% binning procedure applied to the momentum. The continuumdressing function ˆ p D (ˆ p ) is shown with momentum cuts. accuracy. This will be analysed below.We consider the reconstruction introduced in section 3.3 applied to the tensor basesthat have been studied, namely the extended and reduced lattice bases, eqs. (3.15)and (3.16), and also the continuum basis with a single form factor D ( p ). The re-construction ratio R = P µν | Γ orig µν | P µν | Γ rec µν | (4.8)54s computed using the previously shown form factors.We begin by consider H4 corrected data shown in fig. 4.10. From its analysis wenotice an improvement in the reconstruction when adding tensor elements. In fact, thelarger basis has the best result when compared to the other three structures, with theresults being in general closer to one. The comparison between the two continuumtensors is not very informative since the differences appear to be negligible.To understand the differences in tensor descriptions from the lattice bases we considerspecific momentum configurations to evaluate the reconstruction ratio in eq. (4.8). Theuse of specific momentum configurations also helps to reinforce the existence of specialkinematics for which the continuum description is approached.10.800.850.900.951.051.101.151.20 R { E, F, G, H, I } {
J, K, L } R p (GeV) { A, B } p (GeV) { D } Figure 4.10: Reconstruction ratio for the normal momentum bases after the H4 extrapolation.Each plot is labelled by the corresponding form factors for each basis. Data from the β = 6 . , lattice. In fig. 4.11 the ratio for six different momentum configurations is shown. The range ofmomentum was chosen for each plot in order to evidence the essential behaviour for eachkinematics. The continuum basis { A, B } is not shown since the results exactly matchthe ones from the single form factor basis. This could be explained by the orthogonalityof the propagator on the lattice, that further restricts the { A, B } basis, ending with asingle effective form factor. In addition, to study the differences in using improved orlattice momentum we consider the usual continuum basis in terms of both momenta.Conversely, both lattice tensors are shown as a function of ˆ p only.55.201.201.201.211.211.221.221.231.23 . . . a ) p =(2 n,n,n, . . . b ) p =(4 n,n,n, . . . . c ) p =( n +1 ,n,n,n − . . . . d ) p =( n +6 ,n,n,n − . . e ) p =(40 ,n,n, . . . . . . f ) p =( n, , , RR { E,F,G,H,I }{ J,K,L }{ D (ˆ p ) }{ D ( p ) } R ˆ p (GeV) ˆ p (GeV) Figure 4.11: Reconstruction ratio R for various single scale momentum configurations usingtwo lattice bases, eqs. (3.15) and (3.16), and the continuum tensor (1.40) using the improvedmomentum and lattice momentum. Results from the β = 6 . , ensemble. The general behaviour in fig. 4.11 shows that the most complete lattice basis is betterat portraying the original tensor, having lower ratios across most of the configurationsand for a large range of momentum. There are, however, special kinematic points forwhich the remaining tensor bases match the result from this basis.Another striking feature comes from the comparison between the two continuumbases using normal and improved momentum. The latter shows better ratios and thus56 better description of original tensor .The first row in Figure 4.11 displays two similar kinematics, only distinguished byits distance from the diagonal, with (4 n, n, n,
0) being farther from it. The same generalbehaviour is obtained for both kinematics, although with a significant improvement forthe left case whose R values are closer to 1 for the whole range of momenta.The second row in fig. 4.11 also represents two similar configurations, again withthe one on the left being closer to the diagonal, thus having an overall better ratioamong all bases. Additionally, there is an effect common to both, namely the angle fromthe diagonal is not constant through all momenta. Instead, it depends on n like θ =arccos p / (1 + 1 / (2 n )). This dependence dictates the behaviour of the ratio, decreasingfor increasing n .The bottom row shows two distinct configurations. The case (40 , n, n,
0) has anexpected minimum for large n , when approaching the configuration (40 , , ,
0) fromthe left. The one on the right has a constant ratio, but very different descriptions amongthe basis with the extended lattice basis having a much lower ratio.In general, we conclude that with respect to the description of the gluon propagatortensor, D µν ( p ) the use of more complete bases provides a better result. In addition, theimproved momentum is again reinforced as the better momentum vector to use. Notethat the purpose of considering larger bases is not only to obtain a better description ofthe scalar functions characterizing the propagator, but also to properly understand itslattice tensor structure, and how it deviates from the continuum form (these deviationsshould be more evident for coarser lattices, with a larger lattice spacing).In addition, our analysis provides results differing from those in [13]. Namely, in thiswork the reconstruction from the three form factor lattice basis shows better reconstruc-tion results than in our case. This, however is related to the use of a lower dimensionallattice for which the tensor is fully described by less form factors . This results in thestructure { J, K, L } being a more complete basis for N d < Orthogonality of the tensor basis
The Landau gauge condition is expressed by the orthogonality of the gluon field, p µ A µ ( p ) =0. This condition, together with the Slavnov-Taylor condition, constrains the tensor formof the gluon propagator in the continuum. It is important to study how this conditionaffects the form of the two gluon correlation function on the lattice.It is also relevant to notice that the gauge fixing on the lattice cannot be implementedwith infinite precision. In our simulations the condition satisfies | ∂A | . − . It isalso worth referring that we have explicitly tested orthogonality of the gluon fields by Notice that although the extraction of D ( p ) is independent of the use of p ou ˆ p , the use of bothmomenta changes the description of the full tensor. The extended tensor basis with five form factors was not considered in this previous work. The gluon propagator is described in general by N d ( N d + 1) / N d . A ort µ = (cid:18) δ µν − p µ p ν p (cid:19) A ν ( p ) (4.9)where A µ ( p ) are the original gauge fields. Yet, the analysis after this demand does notchange neither the form factors nor the ratios R . This serves as a good test of theorthogonality on the lattice.Also, in lattice simulations for general kinematics the Landau gauge condition is muchbetter realized for the improved momentum rather than normal momentum, ˆ p µ A µ ( p ) ≪ p µ A µ ( p ), with the results differing by several orders of magnitude. The exception occursfor kinematics having a single momentum scale for which we can establish ˆ p µ A µ ( p ) ∝ p µ A µ ( p ), with the proportionality constant given by sin( n ) /n .In the continuum, the orthogonality of the propagator is ensured by its tensor struc-ture by the transverse form ( δ µν − p µ p ν /p ). However, for the extended bases this is notthe case, and the orthogonality should manifest in relations among the form factors. Forthe extended lattice basis, the following relation is expected X µ p µ D µν ( p ) = 0= E ( p ) + p ν F ( p ) + p ν G ( p ) + ( p − p ν ) H ( p ) + (cid:16) p [4] + p p ν − p ν (cid:17) I ( p ) (4.10)for momentum p ν = 0.0-0.15-0.10-0.050.050.100.150.20 1 1 . . . . . . p µ D µ ν ( p ) / a p (GeV) p +H4 p (GeV) ˆ p ˆ p +Cuts Figure 4.12: Orthogonality condition, eq. (4.10) shown for the normal momentum basis after H4extrapolation from the β = 6 . , lattice. Right plot shows the result using the improved basisresult without corrections and also with momentum cuts in terms of the improved momentum.For all data the p component was considered. We look for deviations from this relation which, following the previous discussion,are expected to be more perceptible for the lattice momentum p . In fig. 4.12 the orthog-onality condition is shown for the fourth component of momentum, p (the conclusions58rom the remaining components are the same). The orthogonality relation, eq. (4.10),is shown for the H4 extrapolated data (left) where we see that the condition is satisfiedonly for lower momenta although with increased fluctuations. Contrarily, the improvedbasis (right) shows a much better realization of the orthogonality for the full momentumrange. The low momentum region involves higher statistical fluctuations that can bepartially eliminated by cutting momenta farther from the diagonal.Note that this analysis of the orthogonality serves also as a complementary verifi-cation of the continuum relations and the completeness of the basis. Indeed, imposing G, I → − p F, − p H → E the relation (4.10) is immediately satisfied. Throughout the previous analysis we excluded the generalized diagonal kinematics forwhich the complete set of lattice form factors is not possible to obtain. However, it washinted that these are special regarding the description by the continuum tensor and forthe orthogonality condition. In this section these configurations are studied, and somequantitative arguments are laid to support previous claims. The generalized diagonalconfigurations were introduced in section 3.1. These are defined by a single scale, thusinclude on-axis momenta with a single non-vanishing component, full diagonal momenta( n, n, n, n ), and mixed configurations ( n, n, ,
0) and ( n, n, n,
Lattice
Continuum R ˆ p (GeV) ( n, , , ˆ p (GeV) ( n, , , n,n, , n,n,n, n,n,n,n ) Figure 4.13: Reconstruction ratio for all four generalized diagonal configurations from the β =6 . , lattice considering the most complete lattice basis (left) and the usual continuum tensorbasis (right). Also shown is the reconstruction for the kinematics ( n, , ,
0) using the same twobases.
We start by analysing the reconstruction results for the four generalized diagonalconfigurations in fig. 4.13. Firstly, there is a clear hierarchy in the faithfulness in thedescription among the four configurations. The closer to the diagonal, the better de-scription. This should be related to softer discretization artifacts along the diagonal, asopposite to the ones farther from it. The ratio deviates considerably from unity, reaching59ifferences of about 40% for on-axis momenta.The other striking feature is the correspondence between both bases. Although nei-ther basis is complete, it would be expected that having more independent terms wouldresult in a better description. This apparent conflict can be explained by the specialproperties of these kinematics. Although we are using five form factors, the degener-acy of the tensor allows only to extract a reduced number (two or three depending onthe configuration – see appendix B) hence reducing the freedom in the tensor descrip-tion. In addition, the combination of the gauge condition and Slavnov-Taylor identityon the lattice further restricts the tensor by establishing relations among the form fac-tors. Therefore, for these kinematics, both bases provide the same effective degrees offreedom.In fig. 4.13 a momentum configuration close to on-axis momentum is also shown.It represents the same configuration as in fig. 4.11 f ). It should be noticed that forthis kinematic configuration, the complete extraction of 5 form factors is possible. Theratio for ( n, , ,
0) is much smaller when using the lattice basis than for the continuumstructure which is closer to the result from ( n, , ,
0) and again shows that the latticebasis is better at describing the original tensor for a general configuration.
Continuum relations
In the above analysis we referred that the diagonal kinematics are special regardingits reproduction of the continuum relations. To sustain these claims, we verify thatthese are exactly satisfied for these kinematics. We consider the full diagonal momenta p = ( n, n, n, n ), for which only two objects may be extracted, E ( p ) + n F ( p ) + n G ( p ) = 1 N d X µ D µµ ( p ) (4.11) n H ( p ) + 2 n I ( p ) = 1 N d ( N d − X µ = ν D µν ( p ) . (4.12)Since we want to establish relations among the continuum and lattice parametriza-tions, we consider the right side of eqs. (4.11) and (4.12) expressed by the continuumtensor D cµν = D ( p )( δ µν − p µ p ν /p ). By carrying out this replacement, the expressionsreduce to, 4 E ( p ) + p F ( p ) + p G ( p ) = 3 D ( p ) (4.13) − p H ( p ) − p I ( p ) = D ( p ) (4.14)which by considering G, I → E ( p ) , − p F ( p ) , − p H ( p ) = D ( p ) . (4.15)In fact, this last step was unnecessary since due to the form of the basis, p F ( p )+ p G ( p )could just be replaced by a new form factor p F ′ ( p ). In this case it is irrelevant how60he form factor is defined since only the combination of the two can be extracted. Ananalogous argument can be made for the off-diagonal terms. Thus, for diagonal momenta,the extended lattice basis exactly reduce to the continuum description. In fact, this isthe rationale for the argument given above on the decrease in independent form factorsin the case of diagonal kinematics.For on-axis momenta only diagonal terms can be attained D µµ ( p ) = E ( p ) + p µ F ′ ( p ) (4.16)where we used the simpler notation, F ′ ( p ) = F ( p ) + n G ( p ). For this configurationthe continuum parametrization has the following form D cµµ = ( D ( p ) µ = 2 , , µ = 1 . Extracting each lattice form factor with eqs. (B.47) and (B.48) and replacing the tensorelements by the continuum parametrization gives E ( p ) = 13 X µ D cµµ ( p ) = D ( p ) p F ( p ) = D c ( p ) − E ( p ) = − D ( p ) , thus confirming the continuum relations for this configuration. The treatment for themixed configurations ( n, n, ,
0) and ( n, n, n,
0) is analogous and does not alter the con-clusions – it can be seen in appendix C.1.1.We confirm that the continuum relations are satisfied for single scale configurationsand thus the description with the lattice or continuum tensor is equivalent. Hence, wesee that if we want to have a proper description of lattice objects the continuum tensorbasis provides a good result if one focus on the diagonal kinematics. This serves alsoto again validate the conventional approach to the computation of the propagator usingmomentum cuts.We confirm this numerically in fig. 4.14 which shows the previous continuum relations.The three expressions show a very good agreement. The left plot shows the two possibleform factors for ( n, n, n, n ) which other than satisfying the continuum relations amongthem also have a very good agreement with the benchmark result d (ˆ p ). For on-axismomentum the continuum relations are also confirmed among the two lattice form factorsand the continuum scalar D ( p ). However, hypercubic artifacts render this configurationproblematic from the perspective of the reproducing the expected result .Regarding the orthogonality for generalized diagonal configurations, these are thesame as continuum relations. In fact, for the case ( n, n, n, n ) the orthogonality conditionis p µ D µν ( p ) = n ( E ( p ) + n F ( p ) + n G ( p )) + 3 n ( H ( p ) + 2 n I ( p )) = 0 Note that the benchmark result consists of data surviving momentum cuts, and on-axis momentado not survive the cuts. This is the reason the result deviates quite considerably for momentum above ∼ . . . . . ( n,n,n,n ) . . . ( n, , , p Γ ( p ) ˆ p (GeV) d (ˆ p )( E (ˆ p )+ n F (ˆ p )+ n G (ˆ p )) ˆ p − p H (ˆ p ) − p I (ˆ p ) ˆ p (GeV) d (ˆ p )ˆ p D (ˆ p )ˆ p E (ˆ p ) − ˆ p F (ˆ p ) − ˆ p G (ˆ p ) Figure 4.14: Form factors from the lattice basis for the diagonal configuration p = ( n, n, n, n )(left) and for the on-axis momentum p = ( n, , ,
0) (right) both as a function of improvedmomentum. Results from the β = 6 . , lattice. Shown for comparison is the benchmarkresult d (ˆ p ). which is the same as obtained above for the continuum relations. Thus, following theprevious conclusions, both orthogonality and continuum relations are guaranteed whenstudying the generalized diagonal kinematics. We explore possible finite volume effects by analysing results from a 64 lattice with thesame inverse coupling, β = 6 .
0. Having a larger ensemble (2000 configurations) resultsin lessened statistical fluctuations. On the other hand, a smaller volume restricts theaccess to low momenta.Due to the momentum restriction on the extraction of the five form factors for ageneral kinematics, we cannot reach the lowest momentum points where the finite volumeeffects should be noticeable. For the rest of momentum range the continuum relationsfor the form factors show the same general behaviour as the 80 lattice, figs. 4.2 to 4.4,as thus we do not consider its analysis.We turn our attention to the reconstruction – the finite volume of the lattice is nottaken into account in the basis construction and thus it could affect the reconstructionof the original tensor. The comparison among the two lattices is shown in fig. 4.15 withthe extended and continuum basis shown in terms of the improved momentum. Thefirst thing to notice is that the reconstruction is better for the 80 lattice, showing asmaller ratio, except for special points such as diagonal kinematics. This is perceptiblefor the high momentum region of a ), c ), and d ). In b ), both lattices show the same ratiofor the extended basis while the continuum basis shows a slight difference with the 80 ensemble having a higher ratio.Despite both lattices provide similar results for special kinematic points, the remain-62ng configurations differ, and the completeness of the bases seems to be reduced for thesmaller volume lattice. In fact, in fig. 4.15 c ) and d ) even the 80 continuum tensorprovides a better reconstruction than the 64 extended lattice basis.1.181.201.221.241.261.281.301.321.341.361.38 . . . . a ) p =(32 ,n,n, . . . . . . b ) p =( n, , , . . . . c ) p =( n +1 ,n,n,n − . . . . d ) p =( n +6 ,n,n,n − R −{ E,F,G,H,I } −{ E,F,G,H,I } −{ D } −{ D } R ˆ p (GeV) ˆ p (GeV) Figure 4.15: Reconstruction ratio for the extended lattice basis and the usual continuum descrip-tion both in terms of the improved momentum. These are shown for the two different latticeswith 80 and 64 sites, and same spacing 1 /a = 1 . − . Four distinct momentumconfigurations are shown. To complete the reconstruction analysis it is worth to reproduce fig. 4.13 for the twodifferent lattices, see fig. 4.16. We consider only the largest basis and confirm that thereconstruction for diagonal kinematics is independent of the lattice volume. Therefore,other than having a better description by the continuum form, these kinematics seem alsoto be insensitive to the volume of the lattice regarding its tensor description. With thisanalysis we confirm that the momentum cuts, namely choosing the diagonal momentaseems to be an appropriate methodology for lattice computation of correlation functions.
The focus of this section is the analysis of the three gluon correlation function. Inparticular, we look for a possible sign change and subsequent logarithmic divergencewhich are expected to occur in the infrared region for some specific kinematic limits631.051.101.151.201.251.301.351.400 1 2 3 4 5 6 β =6 . , β =6 . , R ˆ p (GeV) ˆ p (GeV) ( n, , , n,n, , n,n,n, n,n,n,n ) Figure 4.16: Reconstruction ratio for all four generalized diagonal configurations considering themost complete lattice basis for the (6 .
502 fm) lattice (left) and the (8 .
128 fm) lattice (right).Both lattices having the same lattice spacing 1 /a = 1 . − . and for some form factors of the three gluon correlation function. The zero-crossingand IR divergence are related to the concept of dynamical mass generation [15, 74, 75]whereby the gluon acquires an effective momentum dependent mass m ( p ), while theghost seems to be transparent to this process thus remaining effectively massless. Thisproperty should also affect different gluon correlation functions, particularly the IR formof the gluon propagator [12, 76].This behaviour has been predicted by various DSE analysis employing different trun-cation schemes and approximations for the three gluon vertex [17, 19, 26, 68]. The basicmechanism for the appearance of the zero-crossing and subsequent logarithmic diver-gence in the three gluon vertex is reviewed in [15]. It boils down to the appearance of adiverging ghost loop in the Dyson-Schwinger equation for the propagators which in turnaffects the three gluon vertex – see [17] for a thorough analysis. From a qualitative pointof view we can justify the divergence due to the supposedly ghost masslessness and itsloop contributing with a term of the form ∼ ln (cid:0) q (cid:1) , which diverges for p →
0. On theother hand, the gluon loop is associated with a term ∼ ln (cid:0) q + m (cid:1) , remaining IR finitedue to the momentum dependent effective gluon mass , m (0) > SU (2) theory, its degree of divergence seems to be lower than the one Note that in these schemes the divergence occurs in a theory with a finite gluon propagator D (0) ≥ SU (2) and SU (3) and in three and four dimensions [21–24] suggest the presence of the zero-crossingalbeit failing to observe the divergence. Contrarily, a recent analytical study of the gluonand ghost propagators using lattice data suggest the presence of a mass regularizing theghost propagator in the deep IR [29]. This could in turn remove the infrared divergencefor the three gluon vertex.The zero-crossing provides a non-trivial constraint on the behaviour of gluon ver-tices which due to its logarithm divergence makes the effect difficult to observe insmall volume lattices. This effect also strongly depends on the kinematic configuration.In this work we focus on the ‘asymmetric’ configuration with a vanishing momentum( p , p , p ) = ( p, , − p ) for which we extract a single form factor Γ( p ) that is expected todisplay the sign change in the IR region. This kinematic was considered in other latticestudies [10, 22, 23] as well as continuum approaches [16, 17]. In [15] the ratio R ( p ) = Γ (0) a a a µ µ µ ( p, , − p ) G a a a µ µ µ ( p, , − p )Γ (0) a a a µ µ µ ( p, , − p ) D a b µ ν ( p ) D a b µ ν (0) D a b µ ν ( p )Γ (0) b b b ν ν ν ( p, , − p ) = Γ( p )2(4.17)was related to the diverging ghost loop appearing in the DSE for the gluon propagator(under the chosen truncation scheme).Other than ( p, , − p ), other kinematics are generally considered in the literature,namely the ‘symmetric’ configuration ( p i = p , p i · p j = − p / , i = j ) [22, 23] for whichthe zero-crossing is easier to observe due to smaller fluctuations, thus having a moredefined range for the sign change. The asymmetric configuration, on the other hand,is associated with increased statistical fluctuations due to the vanishing momentumcomponent p = 0 [23].Therefore we aim at investigating the possible occurrence of the zero-crossing andnarrowing the range of momentum where it is expected to occur under both possiblehypothesis for the ghost behaviour, namely the existence or absence of a dynamical ghostmass that regularizes the vertex. In addition we look for possible signs of the divergencefor vanishing momentum. This work follows the investigation from [21] albeit withincreased statistics due to the use of a larger configuration ensemble and also due to theuse of the full group symmetry – complete Z4 averaging.For the three gluon vertex we restrict the analysis to the larger lattice, with 550configurations, see table 4.1. The reason is the need of deep IR momentum points tostudy the structures introduced before. The larger ensemble has a smaller volume andthus its smallest momentum is higher than the corresponding for the 80 lattice. Thisensemble will be considered as comparison for the general behaviour of the data in the IR.The reader should also be aware that all quantities shown below are not renormalized,which again amounts to a constant factor. In three dimensions the corresponding effect is a ∼ /p divergence favouring its detection [77, 78] .2.1 Three gluon correlation function We start by analysing the complete correlation function, i.e. the vertex with externalpropagators, extracted with the following contraction G ( p ) ≡ δ µ µ p µ h Tr [ A µ ( p ) A µ (0) A µ ( − p )] i = V N c ( N c − D ( p ) D (0) D ( p )Γ( p ) p . (4.18)It is important to notice the difference in the statistical accuracy obtained by consid-ering the complete Z4 averaging as opposed to the partial (permutation only) case. Alook at fig. 4.17 allows to perceive the change induced by the use of all H (4) equivalentpoints for the averaging, which enhances the signal to noise ratio. Statistical fluctua-tions are lessened through all range of momentum for the complete Z4 case and the datadefines a smoother curve, with decreased error bars. Given the lessened statistical pre-cision found in lattice computation of vertices when comparing with the results for thegluon propagator in the last section, it is crucial to consider possible ways of increasingthe statistics. For this reason, the rest of this section considers the complete Z4 averageddata. − G ( p ) / a ˆ p (GeV) Partial Z4Full Z4 Figure 4.17: Three gluon correlation function from the β = 6 . , ensemble contracted with,and as a function of the improved momentum. All data is shown without correction methodsusing a partial Z4 averaging with permutations only, and also for the complete Z4 averaging. Regarding the p [4] extrapolation, we notice that this procedure can be extended to ahigher momentum than the one used for the gluon propagator without loss of integrityof the method. The H4 method uses the H (4) orbits to ‘reconstruct’ the continuumobject – extrapolating data to p [4] →
0. While for the gluon propagator the structuresformed by the orbit points are well defined and with small uncertainty associated, the66hree gluon orbit structures are concealed by large fluctuations. Hence, the extrapolatedfunction for the three gluon maintains a momentum dependence close to the originaldata but with increased precision. Notice, however that this is not an advantage of themethod for the three gluon vertex, but a consequence of the reduced precision associatedwith this vertex which allows us to extend the range, within the original uncertainty.To support these claims on the extension of the method we compare the effect of ex-tending the extrapolation for both the gluon propagator and the three gluon vertex. Infig. 4.18 the H4 extrapolation for the propagator was extended to all momentum and com-pared with diagonal configurations due to its lessened hypercubic artifacts. The dressingfunction for ( n, n, n, n ) momentum is shown as a function of improved momentum asit was observed in the previous section to produce a better match with the expectedbehaviour. We see that for momenta above p ∼ p ∼ . . . .
54 0 1 2 3 4 5 6 7 8 d ( p ) p (GeV) H4 ( n,n,n,n ) Figure 4.18: H4 extrapolated data for the gluon propagator dressing function d ( p ) comparedwith full diagonal momenta ( n, n, n, n ) as a function of improved momentum. Data from the β = 6 . , ensemble. Contrarily to this case, if we extend the p [4] extrapolation for the three gluon vertex,the disagreement is only obtained for larger momenta. In fig. 4.19 the H4 corrected vertexis again plotted against the diagonal kinematics. We see that the general behaviourof the curve is maintained after the correction (with additional precision), and thatit follows the diagonal curve. Therefore, for the three gluon vertex an extension ofthe extrapolation is possible within the statistical accuracy. Notice however that theextension is not complete since for momenta above p ∼ H (4) orbit elements, analogously to the IR region.0100020003000400050006000 0 2 4 6 8 10 12 − − . . . G ( p ) / a p (GeV) G ( p ) G ( p )+H4( n,n,n,n ) Figure 4.19: Original and p [4] extrapolated data for the three gluon correlation function from the β = 6 . , ensemble as a function of the lattice momentum p . The H4 correction was appliedfor the full momentum range. The configuration ( n, n, n, n ) is shown for comparison. Perturbative UV prediction
Although we are interested in the infrared behaviour of the correlation function, we beginby probing how the continuum perturbative predictions match lattice results for highmomenta. To perform this comparison we apply the H4 extrapolation as well as conicalcuts with improved momentum. Following [21], to study the ultraviolet region of ourresults we use the one-loop renormalization group improved result for the propagator D ( p ) = Zp " ln p µ ! − γ (4.19)with Z a global constant, µ = 0 .
22 GeV and γ = 13 /
22 the gluon anomalous dimension.For the three gluon vertex a similar expression is obtained,Γ( p ) = Z ′ " ln p µ ! γ g (4.20)with the anomalous dimension γ g = 17 /
44. These two expressions can be combined toconstruct the corresponding three gluon correlation function computed above, eq. (4.18) G UV ( p ) = Z ′′ p " ln p µ ! γ ′ (4.21)68ith γ ′ = γ g − γ = − /
44 the overall anomalous dimension. This result is expectedto be valid for high momentum.10.900.951.051.101.151.201.251.301.35 2 3 4 5 6 7 10.80.91.11.21.31.4 1 2 3 4 5 6 7 χ / d . o . f . p (GeV) H4 ˆ p (GeV) Cuts Figure 4.20: χ /d.o.f. obtained from the fit of the functional form (4.21) to the β = 6 . , lattice data as a function of the momentum range cut off, p > p GeV. Left plot shows the resultof the fit for the H4 corrected data while the right plot with diagonal momenta as a function ofthe improved momentum. G ( p ) / a p (GeV) G ( p ) + H4 G UV ( p ) ˆ p (GeV) G (ˆ p ) G UV (ˆ p ) Figure 4.21: Three gluon correlation function G ( p ) after the H4 extrapolation as a function of thelattice momentum (left) and as a function of the improved momentum after cuts for ˆ p > β = 6 . , ensemble. To better understand the validity of the perturbative prediction, the fits were per-formed with
Gnuplot [79] for various momentum ranges [ p ,
8] GeV with varying p . Theupper bound at 8 GeV is considered also for H4 corrected data due to large errors in thelattice data. The fit was applied to H4 corrected data as a function of lattice momenta,and also for the data as a function of improved momentum. To evaluate its quality wecompute the χ /d.o.f. taking into account the uncertainty in the data, and which oughtto be minimized for various values p , this is shown in fig. 4.20. This function measures the deviation of the approximated curve obtained by the fit to the data p ∼ . p ∼ . χ /d.o.f. below ∼ .
15. The fit for p = 2 . χ /d.o.f. = 1 .
14. The data seems to followthe perturbation theory prediction for p above ∼ . χ /d.o.f. values for most fitting ranges. However the values seem to oscillateless smoothly, and in fact become high for p above 6 GeV. In the right plot of fig. 4.21the fit for p > χ /d.o.f. = 1 .
09. This curve also shows agood agreement with the lattice data thus validating the perturbative prediction forhigh momenta.To compute the pure three gluon vertex we need to explicitly remove the contributionof the external propagators by dividing by its form factor D ( p ), eq. (4.18). Hence, wealso compare the lattice computation of D ( p ) with the perturbative result, eq. (4.19).The increase in accuracy for this object allows only a fit to higher momenta and in addi-tion, we do not consider the extrapolated data due to its restrictions to high momentumfor the propagator. This is shown in fig. 4.22 as a function of the improved momentum.A good match with the lattice data is obtained, with χ /d.o.f. = 1 .
10 for the range p >
Although the possible sign change associated with the three gluon vertex should benoticeable for the complete correlation function shown before, this carries high statisticalfluctuations for momenta below p ∼ p, , − p ) kinematicsand the tensor basis considered is described by Γ( p ), eq. (3.33).Firstly, we notice that the comparison with the UV perturbative prediction fromeq. (4.20) is not possible for Γ( p ) due to large statistical fluctuations dominating the highmomentum region. These arise due to the high momentum form of the gluon propagators,where for a general kinematic configuration they behave as D ( p ) ∼ /p . This induces a p factor in Γ( p ) when dividing by D ( p ) . In turn, this factor enlarges the uncertainty points. It is defined as, χ = X i (cid:18) G i − f ( p i ) δG i (cid:19) where G i and δG i are the data points and corresponding error, while f ( p i ) is the fitted curve evaluatedat the momentum of G i . The degrees of freedom ( d.o.f. ) are the number of data points to be adjusteddeducted by the number of adjustable parameters. A good fit to the data is obtained by a reduced χ close to unit, i.e. χ /d.o.f. ∼ The poor signal to noise ratio for Γ( p ) for high momentum is a common complication for generallattice computed 1PI functions with more than two external legs. This problem is not completely solvedby the increase in the number of configurations since it is inherently associated with the high momentum . . . .
540 1 2 3 4 5 6 7 8 d ( p ) ˆ p (GeV) ˆ p D (ˆ p )ˆ p D UV (ˆ p ) Figure 4.22: Gluon propagator D ( p ) from the β = 6 . , lattice as a function of the improvedmomentum after cuts abover 1 GeV. The renormalization group improved perturbative result,eq. (4.21) was fitted to the data for p ∈ [5 ,
8] GeV, resulting in a fit with χ /d.o.f. = 1 . associated with Γ( p ) – this can be noticed by a simple Gaussian error propagation, see[21]. For the kinematics in consideration the factor is softened to p due to the vanishingmomentum p = 0, D (0) >
0. However, the p factor combined with large fluctuationsin D (0) create strong fluctuations in the ratio p µ G νµν ( p, , − p ) /D ( p ) D (0) for highmomenta.Regarding the detection of the zero-crossing this is not a problem since D ( p ) isessentially constant for the deep IR region and thus the signal has a more stable behaviourand higher precision. Additionally, the H4 extrapolation is not useful for it disregardspoints in this region.In fig. 4.23 both the complete set of data for Γ( p ), and the points surviving mo-mentum cuts after 1 GeV are shown as a function of improved momentum. This re-sult matches the momentum dependence obtained in other lattice studies, namely itfollows the results from [21] although with an improved signal to noise ratio. As ex-pected, large statistical fluctuations arise for momenta above ∼ . p = 0 .
216 GeV) = 0 . p = 0 .
152 GeV) = 0 . behaviour of the propagators. − . . . . .
54 0 0 . . . . . . . . Γ ( p ) ˆ p (GeV) Γ(ˆ p )Γ(ˆ p )+cuts Figure 4.23: Complete set of data from the β = 6 . , lattice for the three-gluon 1PI, Γ( p )as a function of the improved momentum. The data surviving momentum cuts above 1 GeV isalso shown. consider three different functional forms to fit the data in fig. 4.23,Γ ( p ) = a + z ln p µ ! , ( a , z ) (4.22)Γ ( p ) = a + z ln p + m µ ! , ( a , z , m ) (4.23)Γ ( p ) = 1 + cp − d , ( c, d ); (4.24)the adjustable parameters appear in parenthesis. The first functional form, eq. (4.22),comes from a simple Landau gauge, four-dimensional QCD toy model for asymptoticallylow momentum [15, 23]. The second logarithm, eq. (4.23) has an additional constant m to account for the possible dynamical ghost mass predicted in [29]. This mass could inprinciple remove the three gluon divergence by regularizing the ghost loop, nonethelessa sign change is possible depending on the value of the parameters. Both constants a , a serve to partially take into account the non-leading terms which become relevantfor higher momenta.The third form for Γ( p ), eq. (4.24), is a power law ansatz [25] which allows to studythe degree of the possible divergence in the IR and also estimate the position of thezero-crossing. In [15, 22, 23] more appropriate curves, obtained by solving the DSEs forthis momentum configuration are considered and fitted to lattice data.To better understand the validity of the functional forms, the range of the fit wastested for the limits [ p i , p f ] with variable p f while p i is the lowest, non-zero momentumvalue. The value of p f was restricted to 2 GeV, above which Γ( p ) is involved in large72uctuations, in fact these are noticeable already in the upper momenta of fig. 4.23. Asa lower bound, we consider p f above 0 . p f we consider theanalysis for the complete set of data in fig. 4.23. In addition, we compare the result ofthe fits with the data surviving momentum cuts above 1 GeV to try to overcome theproblem of large fluctuations for higher momenta. The quality of the fit was controlledwith the χ /d.o.f. shown for all functional forms and both sets of data in fig. 4.24.0 . . . . . . . . . . . . . . . . ( p ) 0 . . . . . . . . . . . . . . . . ( p )11 . . . . . . . . . . ( p ) χ / d . o . f . CompleteCuts χ / d . o . f . ˆ p f (GeV) Figure 4.24: χ /d.o.f. of the three fits from eqs. (4.22) to (4.24) (top left, top right and bottom,respectively) for the varying momentum range p ∈ [ p i , p f ]. Both fits with and without momentumcuts were considered. The results for the χ /d.o.f as a function of the fitting range, in fig. 4.24 are similarfor both logarithms, Γ and Γ . The quality of the fit seems to be highly dependent onthe range for p f below 0 . χ rapidly oscillating. Above 1 GeV the momentumcuts are applied and thus the results for both sets of data become different. The reduced χ oscillates around χ /d.o.f = 1 . p f ∼ − . p f > . χ values closer to one, indicating a better match to the data.Although the quality of the fit has a similar behaviour for both logarithms, theone with an additional mass shows χ /d.o.f values closer to unity. This value remainsbetween 0 . − . p f > . while for the form Γ χ stabilizes around 1 . χ around χ /d.o.f = 1 . p f > is different than the onedescribed above. From the bottom panel in fig. 4.24 we see that the best fit is obtained for p f in the range 0 . − . χ grows rapidly for momenta abovethis region. Since the quality of the fit becomes worse above 0 . p > . p f = 1 . p ) it should be valid forlower momentum when compared with the first two models. This is why the qualityof the fit rapidly decreases when reaching p f ∼ . χ /d.o.f = 1 . p f . We choose p f above 1 GeV in order to distinguish between the complete data and the one sur-viving momentum cuts. For the Γ logarithm the choice p f = 1 . χ /d.o.f. = 1 .
14 and χ /d.o.f. = 1 .
28 for the complete and the data after cuts,respectively. It is important to refer that the parameters of this curve and the corre-sponding uncertainty do not vary significantly for the range 1 . < p f < p ) for the lowest momentum range, namelyfor p ∼ . − . p = 0 . p = 0 . p f . The range p f = 1 . χ /d.o.f. =0 .
984 and χ /d.o.f. = 1 .
21 for the complete set and the data after cuts, respectively. Thecorresponding curves are shown in fig. 4.26. Although the quality of the fit indicatedby the χ seems to be better for the logarithm with additional mass, the uncertainty inthe parameters is larger. Nonetheless, both curves in fig. 4.26 have a similar form andsuggest a good match with the data for the full range of momenta.Regarding the possible sign change, the fit with the complete data suggests a positive This was thoroughly explored in [25] for both 3 and 4-dimensional cases and found that the powerlaw is compatible with the data for momenta below ∼ . . . . . . . . Γ ( p ) ˆ p (GeV) Γ(ˆ p )Γ(ˆ p )+cutsCutsComplete Figure 4.25: Γ( p ) from the β = 6 . , ensemble as a function of improved momentum. Thedata after momentum cuts is also shown. Two fits using eq. (4.22) and p f = 1 . IR value for Γ(0) and an absent sign change, within the uncertainty of the curve. On theother hand the curve using momentum cuts allows for a possible sign change. However,although we predict that within this model p should occur below 0 .
35 GeV, the existenceof a sign change is not guaranteed by the predictions made from the curve and thesubstantial uncertainty carried by the resulting curve does not allow further conclusions.For the power law form, eq. (4.24), a good balance in the quality of the fit anda reasonable uncertainty is obtained for p f = 0 .
85 GeV for which the complete dataprovides a better fit with χ /d.o.f. = 1 .
12 as opposed to χ /d.o.f. = 1 .
29 for the datasurviving momentum cuts. The analysis of the corresponding curves in fig. 4.27 showsthat both fits have a comparable form, barely changed by the change in the set of data(this is expected due to the small range considered above 0 . p = 0 . p = 0 . ∼ d from the fits are d = 0 . d = 1 . SU (2) and SU (3) lattice investigations [25, 80]. However, since we donot find a clear numerical evidence for the divergence due to the lack of points in thedeep IR region, this result is not reliable and should be taken with care.Both the first and last functional forms, eqs. (4.22) and (4.24), are considered in order75 . . . . . . . . Γ ( p ) ˆ p (GeV) Γ(ˆ p )Γ(ˆ p )+cutsCutsComplete Figure 4.26: Γ( p ) from the complete set as a function of improved momentum from the β =6 . , ensemble. The data after momentum cuts are applied is also shown. The functionalform in eq. (4.23) with range p f = 1 . − . . . . . . . . Γ ( p ) ˆ p (GeV) Γ(ˆ p )Γ(ˆ p )+cutsCutsComplete Figure 4.27: Γ( p ) for the complete kinematics as a function of improved momentum from the β = 6 . , ensemble. The set of points surviving momentum cuts is also shown. The functionalform in eq. (4.24) with p f = 0 .
85 GeV was adjusted to the complete and partial data. . . . . . . . ( p ) 00.050.100.150.200.250.300.350 . . . . . . . ( p ) p ( G e V ) ˆ p f (GeV) CutsComplete ˆ p f (GeV) Figure 4.28: Prediction for the sign change p from the fits using eq. (4.22) (left) and eq. (4.24)(right) for varying fitting ranges [0 , p f ]. to study the possible zero-crossing with subsequent divergence. Despite not having aclear signal on the divergence, we can study how the estimated position and uncertaintyfor p varies with different fitting ranges . The p values for Γ and Γ are shown infig. 4.28 as a function of p f for both the complete and partial sets of data. From theanalysis of this figure we notice that p is associated with smaller uncertainty whencomputed with the first form, eq. (4.22) and using the complete set of data. In addition,the complete data seems to shift the position of the zero-crossing for higher momentumwhen compared to the partial data.For the logarithmic case, p varies very little for the range p f < . − .
15 GeV. Above 1 GeV the data surviving momentum cuts maintains aconstant value around p = 0 .
15 GeV. This prediction lies in a region where in fact thelattice results are compatible with zero within the uncertainty. On the other hand theprediction from the complete data grows for p f > p = 0 .
25 GeV above p f = 1 . isobserved for p , the uncertainty in this model is much larger. The result from the datasurviving cuts seems to remain constant for the whole range of momenta, while thecomplete result increases for larger p f . However, in this case the intervals predicted bythe two sets are compatible within the uncertainty. The combination of these resultsindicates a possible value for the zero-crossing position at an interval 0 . − .
25 GeV.Although a similar analysis for the form Γ is not possible, it is important to referthat the fit with eq. (4.23) maintains a stable behaviour, similar to the one found infig. 4.26 for a large range of p f . This is a good indication of the model describing thedata. However, an increase in the precision of the results is needed to better understand Although a sign change can also be observed for the form (4.23), as seen in fig. 4.26, its existencestrongly depends on the momentum range of the fit. Besides, the uncertainty associated is much largerand therefore its explicit computation as a function of p f is not shown. Finite volume effects − . . .
52 0 0 . . . . . . . . Γ ( p ) ˆ p (GeV) +cuts64 +cuts Figure 4.29: Γ( p ) from the β = 6 . , ensemble compared with the results from [21] using the β = 6 . , lattice with 2000 configurations. Above 1 GeV only data surviving momentum cutsis shown. To complete the analysis of the three gluon vertex we compare the results obtainedfrom the 80 lattice using 550 configurations and those from the 64 lattice with 2000configurations and partial Z4 averaging . Since both lattices have the same spacing,this comparison allows to search for possible finite volume effects for the three gluonvertex.The dimensionless form factor Γ( p ) is shown for both lattices in fig. 4.29 wheremomentum cuts were applied above 1 GeV. Although the 80 lattice data is noisierand shows larger error bars, as a result of the difference in the size of the ensembles,both sets of data seem to have the same general behaviour approaching the infraredregion. However, the current data suggests a possible shift enhancing the Γ( p ) for the80 lattice in comparison with the 64 results. The curve produced by the 80 latticedata seems to be above the 64 results for momenta below 1 . a . However, The data from the 64 was previously computed in [21] using momentum cuts above 1 GeV. G ( p ), the pure vertexΓ( p ) is enhanced for low momentum when dividing by the product D ( p ) D (0). Indeed,the lattice data seems to be compatible with an increase for low momentum, howeverthis is a rather rough estimate of the effect and we should have in mind that the finitevolume can also directly affect the complete correlation function.64 χ /d.o.f. p (GeV) χ /d.o.f. p (GeV)Γ Table 4.2: Fit parameters for the 64 and 80 lattice using the three models in eqs. (4.22)to (4.24). Regarding the position of a possible sign change, assuming the previous hypothesis forthe finite volume effect, the change in the propagator amounts to an overall multiplicativefactor and thus the position of the zero-crossing is untouched. However, again we noticethat the complete effect on the three gluon correlation function may induce furtherchanges and can in fact change this value. Besides, since no statistically relevant signalof the zero-crossing is found for neither of the ensembles, we cannot probe how thevolume affects this property.To better understand the possible finite volume effect we reproduce the fits with thethree models, eqs. (4.22) to (4.24) for the same momentum ranges as in the previousanalysis for each corresponding model. The results are shown in fig. 4.30 for the threemodels and the fit parameters are summarized in table 4.2. We see that in generalthe χ is lower for the 64 due to the smoothness of the data computed from a largerensemble. Moreover, the position of the possible zero-crossing for both Γ and Γ seemto be shifted for slightly higher momenta in the 64 lattice, however both estimatesfor the sign change are compatible within the uncertainty. The form Γ seems to havelower p for the 64 lattice, however a large uncertainty is associated with the results formomenta below ∼ . In this section we report on the four gluon correlation function computed from the twoensembles in table 4.1. As referred in section 3.7, on a lattice simulation we have accessto the full Green’s functions only. However, the four point correlation function involves,besides the pure four gluon 1PI function, also the disconnected terms contributions and79-1.0-0.50.51.01.52.02.50 0 . . . . ( p ) 0-1.0-0.50.51.01.52.02.50 0 . . . . ( p )-1.0-0.50.00.51.01.52.02.5 0 0 . . . . ( p ) Γ ( p ) Γ ( p ) ˆ p (GeV) Figure 4.30: Γ( p ) with momentum cuts above 1 GeV for the 80 and 64 lattice. The curvesresult from the fits with eq. (4.22) (top left), eq. (4.23) (top right), and eq. (4.24) (bottom plot)with fitting ranges p f = 1 . p f = 0 .
85 GeV for the latter. those associated with the three gluon irreducible diagrams. All these contributions canbe removed by a proper choice of the kinematics.Even after discarding these contributions, a lattice simulation returns the four gluonGreen function that combines the corresponding irreducible diagram with external gluonpropagators, eq. (3.36). Then, to measure the four point 1PI function the full Green’sfunction requires the removal of the gluon propagators. However, this operation enhancesthe fluctuations, specially at large momenta, where the propagator becomes small, andadds a further difficulty to the measurement that we aim to perform. Due to increasedfluctuations for the pure vertex we only show the complete correlation function.Regarding previous investigations on the IR properties of the four gluon vertex onlycontinuum studies have been conducted [31, 32], also establishing a possible zero-crossingfor some form factors. Some qualitative relations may be established between lattice andcontinuum results. However, these comparisons should be considered with care due to aweak signal conveyed by the lattice four gluon correlation function.In general, the fluctuations of higher order functions in a Monte-Carlo simulation arelarger and the computation necessarily calls for the use of large ensembles of configura-tions. To try to overcome the problem of statistical fluctuations, in all cases we performa Z4 average, as done in the previous sections. Unfortunately, although increasing the80uality of the Monte-Carlo signal, the Z4 averaging is not sufficient to produce resultswith small or relatively small statistical errors for the statistics that we are using. Cer-tainly, an increase in the number of gauge configurations will allow to overcome, at leastpartially, the problem of the statistical fluctuations.Additionally, only a restricted class of momentum points will be shown, namely thegeneralized diagonal kinematics. These allow to reach lower momentum values and carrylessened hypercubic artifacts. However, of the four types of diagonal momenta only themixed cases will be shown. The reason is again related with the effort to increase thesignal to noise ratio. On-axis momenta are disregarded for involving higher hypercubicartifacts, and generally larger error bars due to smaller statistics. On the other hand,fully diagonal kinematics of the form ( n, n, n, n ) are disregarded due to having a smallerset of possible distinct H (4) averaging points. Both ( n, n, n,
0) and ( n, n, ,
0) retain agood balance in ‘non–equivalent’ Z4 averaging points while not being strongly affectedby H (4) artifacts when compared with on-axis momenta.As a starting point we are interested only in obtaining a proper signal of the fourgluon correlation function. A detailed analysis of the infrared behaviour of the functionsis difficult due to the uncertainty associated with the data. The 64 lattice with 2000configurations provides a much better result and will be analysed. The 80 lattice with550 configurations allows access to lower momenta, however substantial fluctuations inthe data inhibit its analysis. For the latter, only points above a given momentum willbe shown and compared with the results from the larger ensemble. We now show the results for the four gluon correlation function from the β = 6 . , and80 ensembles. As introduced in section 3.7, for the configuration ( p, p, p, − p ) only twoform factors are possible to extract, V Γ (0) ( p ) and V G ( p ) associated with the tree-leveland the G tensor, respectively.For this particular kinematics the results for the 64 lattice are shown in figs. 4.31and 4.32. Only the two mixed diagonal configurations are shown with V G ( p ) and V Γ (0) ( p ) on the first and second figure, respectively. Notice these are not the pure,dimensionless form factors due to the presence of the external propagators, i.e. we areusing V i ( p ) = V ′ i ( p )( D ( p )) D (9 p ) , (4.25)where V ′ i ( p ) corresponds to the pure vertex form factor, as defined in section 3.7.A smaller plot is shown in each figure with a narrower range to facilitate the anal-ysis of the behaviour of the function for the mid-momentum range. Both sets of data( n, n, ,
0) and ( n, n, n,
0) seem to follow a similar curve although with enlarged statis-tical fluctuations in the IR region. The fact that two sets of non-equivalent kinematicsproduce similar curves should be an evidence of this result being a proper signal of thefour gluon correlation function. V Γ (0) ( p ) shown in fig. 4.31 seems to oscillate quite smoothly near 1 . − . . . − . . . V Γ ( ) ( p ) / a ˆ p (GeV) − ( n,n, , − ( n,n,n, Figure 4.31: Four gluon vertex form factor V Γ (0) ( p ) with external propagators from the β =6 . , lattice. Only mixed diagonal configurations are considered. The smaller plot showsa restricted range of momentum to better visualize the mid momentum region. All data wasrescaled by a factor of 1000. finite value near the origin. However, the considerable amount of uncertainty associatedwith the first two points hinders the interpretation of the IR behaviour.The values for V G ( p ) in fig. 4.32 have larger uncertainty compared to V Γ (0) ( p ).Nonetheless, both kinematics seem to follow the same behaviour, suggesting a localmaximum for p ∼ p = 0 . V Γ (0) ( p ) is slightly larger than thecontribution from V G ( p ) for the range 0 . − . lattice. In figs. 4.34 and 4.35both V Γ (0) ( p ) and V G ( p ) are shown for mixed diagonal configurations ( n, n, ,
0) and( n, n, n,
0) and for both lattices. A smaller range of momentum was considered discardingthe two lowest momenta (these show large fluctuations, mainly for the larger lattice).The form factor V Γ (0) ( p ) is compared for both lattices in fig. 4.34. Looking only atthe 80 data we notice a possible similar structure to that found in fig. 4.31 (see the smallplot). The 80 results suggest a decrease for negative values and a subsequent growth82 − . . . − − − − − − − . . . . V G ( p ) / a ˆ p (GeV) − ( n,n, , − ( n,n,n, Figure 4.32: Four gluon vertex form factor V G ( p ) with external propagators from the β =6 . , lattice. Only mixed diagonal configurations are considered. The smaller plot showsa restricted range of momentum to better visualize the mid momentum region. All data wasrescaled by a factor of 1000. − − . . . . . . . V i ( p ) / a ˆ p (GeV) V Γ(0) ( p ) V G ( p ) Figure 4.33: Four gluon vertex form factors V Γ (0) ( p ) and V G ( p ) with external propagatorsfrom the β = 6 . , lattice. Only mixed diagonal configurations are shown and the lowestmomentum points disregarded due to large fluctuations. − . . . . . . . V Γ ( ) ( p ) / a ˆ p (GeV) Figure 4.34: Four gluon vertex form factor V Γ (0) ( p ) with external propagators from the β =6 . , (red) and 64 (green) ensembles. Only mixed diagonal configurations are consideredand the lowest momentum points were disregarded. All data was rescaled by a factor of 1000. − − . . . . . . . V G ( p ) / a ˆ p (GeV) Figure 4.35: Four gluon vertex form factor V G ( p ) with external propagators from the β =6 . , (red) and 64 (green) ensembles. Only mixed diagonal configurations are consideredand the lowest momentum points were disregarded. All data was rescaled by a factor of 1000. p = 0 . lattice. Inaddition, if we compare both sets of data in fig. 4.33 from both lattices we notice a shiftin the momentum scales where these structures are found. The possible minimum occursfor higher momentum in the 64 lattice. Although the general structure of the curveseems to provide the same oscillation, the shift in the data and the large uncertainty inthe 80 results could be a sign of inconsistent data and restrains us from making furtherclaims.The data for V G ( p ) in fig. 4.35 also suggests an agreement between the results fromboth lattices. However, albeit the curves created by both sets of data are compatibleand have the same general structure within the uncertainty, the error bars associatedwith the 80 lattice are large and thus this comparison is unreliable. In this case, we donot observe a shift in the structure of the curve . Both the local highest point, around p = 0 . p = 0 . lattice seems to have anegative value, the same cannot be claimed for the larger lattice due to the large errorbars.Despite the large uncertainty, it is remarkable that two distinct lattices seem toprovide the same general behaviour for the form factors with similar structures for thecurves. This should be an evidence that we are indeed computing a valid (albeit weak)signal of the four gluon correlation function. Nonetheless, a significant increase in theprecision of the signal is required to establish reliable conclusions. Comparison with continuum results
Despite the large statistical fluctuations, we try to compare our results with previouscontinuum predictions – these are currently the only source of possible comparison. Forthis we compare only the smaller, 64 lattice having a higher precision.The four gluon vertex was studied in a DSE analysis employing the same tensor basisand kinematic configuration, [31] where it was argued that only the form factor V G ( p )shows a possible divergent behaviour in the IR, while V Γ (0) ( p ) remains finite. Theoriginal data for the pure vertex form factors V ′ i ( p ) from this investigation is shown infigs. 4.36 and 4.37. We are interested in comparing our results with the black curves,representing the complete contribution (within the truncation scheme) .Although on the lattice we can only access the complete vertex with some reason-able statistical accuracy, we can establish some general comparisons with the continuumresults by considering the smooth, and practically constant behaviour of the gluon propa-gators in the IR. In addition to this approximation, both the large uncertainty associatedwith lattice results and the approximations involved in the DSE approach call for carefulconclusions from the following comparisons. Notice that the momentum points do not perfectly match due to the different lattice size, N . Thedefinition of lattice momentum is ap = 2 πn/N . The remaining curves are the individual contributions from one-loop diagrams in the DSE formalism. TotalFishnetTriangle 2Triangle 1Gluon boxTree-level p [GeV] V (cid:1) ( ) c o n t r i b u t i o n s Figure 4.36: Original data from [31] for the DSE computation of the pure four gluon vertexassociated with the tree-level tensor V ′ Γ (0) ( p ). The ‘total’ result in black is the relevant structurefor comparison. -4-3-2-1012310 -4 -3 -2 TotalFishnetTriangle 2Triangle 1Gluon boxGhost box p [GeV] V G c o n t r i b u t i o n s Figure 4.37: Original data from [31] for the DSE computation of the pure four gluon vertexassociated with the tree-level tensor V ′ G ( p ). The ‘total’ result in black is the relevant structurefor comparison. Comparing the results for the tree-level form factor in figs. 4.31 and 4.36 we notice adiscrepant shift in the overall functions, namely the DSE curve sets in at unit values forlarge momenta, while the lattice data seems to approach zero. Notice, however that thiscould be an effect of the external propagators. Nonetheless, the general structure of thelattice data seems to follow the behaviour of the continuum prediction within the large86ncertainty. Namely, the pattern of oscillations is similar, showing what seems like alocal minimum for p ∼ (0) and G , eq. (3.41),the results assuming only the tree-level tensor for the basis should have the same generalbehaviour as the one found in fig. 4.31. Therefore, this serves as a further connectionbetween lattice and continuum results due to the same qualitative structure in V Γ (0) ( p ).The results for V G ( p ) in figs. 4.32 and 4.37 are also compatible within the large un-certainty of the lattice results. In this case no shift is observed between continuum andlattice data. The form factor computed from the lattice shows a decrease to negativevalues for p ∼ . ensemble, which is also noticeable in the DSE resultaround the same momentum scales. For lower momentum the data suggests a possiblesign change and subsequent IR growth, again compatible with previous continuum re-sults. Notice, however that the error bars for low momenta provide limited confidencein these observations. Also, the finite volume of the lattice does not allow to reachsufficiently low momenta to better evaluate the IR behaviour.878 onclusion In this thesis we computed and analysed three different gluon correlation functions ofthe pure Yang-Mills theory in Landau gauge using the lattice formalism of QCD. Twolattices were considered with the same lattice spacing and different physical volumes –see table 4.1.In the first part of the work we investigated the gluon propagator to understand howthe use of continuum tensor bases affects the knowledge of lattice computed tensors ina 4-dimensional theory. To date, only 2 and 3-dimensional studies have been conductedon this topic [13, 14]. To this end we constructed suitable lattice tensor bases respectingthe corresponding lattice symmetries.Continuum relations among lattice and continuum form factors were identified andevaluated for every tensor structure. We found that, within the uncertainty, continuumrelations are satisfied for a large range of momentum which seems to indicate that thelattice data is compatible with the Slavnov-Taylor identity. Furthermore, to probe thequality of our results we used the data from a precise lattice computation [73] as a com-parison. The results obtained with various bases match this benchmark result althoughwith increased fluctuations for larger bases.The completeness of each tensor basis in describing the lattice tensor D µν ( p ) wasstudied. Specific kinematics were considered independently for a detailed analysis andwe found that, in general, the most complete bases (larger number of form factors)provide a better reproduction of the original lattice tensor and the use of a continuumtensor basis for the propagator leads to non-negligible loss of information of the latticecorrelation function. The orthogonality of the propagator using lattice tensors was alsostudied and it serves as a complementary analysis of the completeness for each basis.The analysis of the reconstruction for specific kinematics hinted about the existenceof special points for which the continuum basis matches the description from lattice bases.These are single scale momenta which were then investigated exclusively. Although forthese points the continuum and lattice tensors provide the same quality in the descriptionof the tensor, the results are substantially better for configurations closer to the diagonalof the lattice. Moreover, continuum relations are exactly satisfied by these kinematicsand constrain the number of independent form factors describing the tensor. This is inturn related with the similar completeness from lattice and continuum bases.With this work we provide additional validation for the traditional method to com-pute vertex functions using points near the diagonal of the lattice. We conclude that89iagonal data not only reduce hypercubic artifacts in the form factors (lattice scalars)but also in the tensor structures that form the basis. This is noticeable in the goodreconstruction results obtained for diagonal configurations. We also confirm that, ingeneral, the use of improved momentum provides a better description of lattice objectsthan the naively discretized lattice momentum. In fact, this change of variables improvesalso the fulfilment of both continuum and orthogonality conditions, as well as the matchwith the benchmark result.Although we did not consider a fully complete tensor to describe the gluon propagator,we found that an increase in the degrees of freedom is accompanied by a considerable risein statistical fluctuations in the form factors. This restricts the number of independenttensor structures used due to limited statistics.The effect of a finite volume lattice was also explored. We found that the gener-alized diagonal configurations seem to be insensible to the finite volume regarding itsreconstruction. For the remaining configurations we observed that, in general, the largerlattice provides lower ratios for the reconstruction for both continuum and lattice bases.The finiteness of the space was not taken into account in the construction of latticetensors, and the search for proper bases with respect to the symmetries as well as the sizeof the lattice should improve the description of the propagator. Moreover, mixed termsinvolving both improved and lattice momentum could be considered as well as continuumvanishing terms, depending explicitly on the lattice spacing. Identically, the behaviourof different tensor bases with varying spacings could be explored. Finally, proper tensorstructures respecting lattice symmetries for higher order correlation functions are yet tobe constructed, and would allow to probe how the use of continuum bases affects itsdescription.In the second part we analysed the three gluon correlation function from the 80 lattice. We began by showing that the use of the complete set of group transformations(Z4 average) provides an improved signal to noise ratio. This is crucial for the computa-tion of higher order functions. A comparison with the perturbative prediction for highmomenta was performed for both two and three gluon correlation functions, and wasconfirmed by fitting both curves for sufficiently high momentum.We analysed the IR behaviour for the three gluon 1PI function. Two different hypoth-esis were considered, namely a possible zero-crossing occurring for low momenta witha subsequent IR divergence. The effect is interpreted using the concept of dynamicalmass generation for the gluon which acquires a momentum dependent mass, whereas theghost is supposed to remain massless thus inducing a possible divergence. This hypoth-esis is advocated by various continuum studies, however it is highly dependent on theapproximations employed. Conversely, an analytic investigation of the gluon and ghosttwo point functions suggest a possible dynamical ghost mass which should regularize thevertex and thus remove the IR divergence [29].Since the IR data provides no clear evidence of the sign change, let alone the possibledivergence for lower momenta, we analysed the behaviour of the data by consideringthree different functional models. The first form contains an IR unprotected logarithm,eq. (4.22), which other than the zero-crossing also allows an subsequent divergence. Both90he complete set of data, and the points surviving momentum cuts above 1 GeV providegood quality fits. The results of the fits for various ranges indicate a zero-crossing around0 . − .
25 GeV from this functional form. Notice, however, that while we try to modelthe zero-crossing, the divergence is not sustained by lattice data, hence predictions forthis property are less reliable.The second functional form, eq. (4.23), represents the case of a non-vanishing dy-namical ghost mass which is included in the logarithm and removes the IR divergencewhile still allowing for a sign change. In this case the complete data provides a goodfit with the curve for the range of momenta considered. It is consistent with a positiveIR value for the vertex and an absent sign change. On the other hand, although thedata after momentum cuts also matches the data, this curve is associated with a largeruncertainty. In this case a sign change is possible below 0 . ∼ and64 lattice data was conducted to search for possible finite volume effects. The resultsfrom the 80 lattice seem to be enhanced relatively to those from the 64 lattice, creatinga shift for momenta below ∼ . lattice. While the curves are modifieddue to the shift in the data, remarkably the prediction for the zero-crossing seems toremain unchanged within the uncertainty. This is compatible with the finite volumeeffect amounting to a multiplicative factor such as the one induced by the division of theexternal propagators. However, in order to properly understand the effect, a detailed91nalysis of both the complete and pure three gluon functions is necessary for differentlattice volumes. For the second model, eq. (4.23), the fit with the 64 data follows thesame behaviour but with increased precision. The sign change seems to be predicted forlower momenta, however this is not unambiguously confirmed within the error bars.While we explored a single kinematic configuration, additional configurations couldbe considered to analyse its IR behaviour. The use of different volume lattices for otherkinematics would also allow to improve the knowledge on the possible finite volume ef-fect. Another extension of this work could be related to the large statistical fluctuationsaffecting the high momentum region of the three gluon 1PI function. However, as dis-cussed in section 4.2 this is not achievable by an increase in the number of gauge-fieldconfigurations and thus other alternatives should be envisioned.For the final topic we computed the four gluon correlation function. As a higher orderfunction, it is associated with larger statistical fluctuations which hinder the attainmentof a discernible signal. In fact, current precision allows only to study the completecorrelation function, while the 1PI function carries large fluctuations. Using a suitablekinematic configuration we isolated the contribution of the pure four gluon 1PI functionwith external propagators. In addition to the choice of kinematics, an approximation ofthe Lorentz tensor basis reduced the number of possible structures to three. However,for the kinematics ( p, p, p, − p ) and the approximation employed, only two form factorsare possible to extract.To improve the signal quality, we analysed the correlation function only for config-urations ( n, n, n,
0) and ( n, n, , V Γ (0) ( p ) formfactor, which show some discrepancies for the 80 data. Notice, however, that the 80 lattice provides reduced statistics and the comparison is to be taken with care.To complete the analysis we compared lattice results against the pure four gluonvertex from previous continuum investigations [31, 32]. This is a very delicate compar-ison due to the impossibility of the computation of the lattice four gluon 1PI function.Hence, only a very qualitative connection between the continuum and lattice curves wasestablished. Nonetheless, this should be a good indication of the signal obtained.Although the results are an evidence that we are indeed peeking at the four gluoncorrelation function, the statistical relevancy of the signal is still very small and thesignal should be improved in order to properly analyse the vertex. From the previousanalysis, the main structures observed in the form factors should be noticeable for areasonable range of momentum achievable by our current lattices. Thus, an increase instatistics for the current ensembles should help providing a clearer curve. Besides, thepure 1PI form factors may only be computed accurately with increased precision.92 ibliography [1] Guilherme T.R. Catumba, Orlando Oliveira, and Paulo J. Silva. “ H (4) tensorrepresentations for the lattice Landau gauge gluon propagator and the estimationof lattice artefacts”. In: (Jan. 2021). arXiv: .[2] F. Halzen, A.D. Martin, and John Wiley & Sons. Quarks and Leptons: An Intro-ductory Course in Modern Particle Physics . Wiley, 1984.[3]
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LATTICE2016 (2017). doi : .978 ppendices ppendix A S U ( N ) generators and identities SU ( N ) is the special unitary group of degree N whose elements U are N × N unitarymatrices, U † U = , satisfying det( U ) = 1. It is a Lie group, with its elements beingcontinuously generated by real parameters θ a ∈ R . Each element can be written as U = e iθ a t a (A.1)where t a are the N − θ a . Thegenerators are hermitian and traceless matrices( t a ) † = t a , tr( t a ) = 0 , (A.2)that span a vector space underlying the corresponding Lie algebra, su ( N ). The genera-tors obey the commutation relation [ t a , t b ] = if abc t c (A.3)where f abc are the antisymmetric structure constants, specific for each group and non-zero for a non-abelian group. A fundamental property of Lie groups is the Jacobi identity [ t a , [ t b , t c ]] + [ t b , [ t c , t a ]] + [ t c , [ t a , t b ]] = 0 (A.4)implying f ade f bcd + f bde f cad + f cde f abd = 0 . (A.5)There are two main irreducible representations of the groups SU ( N ). The fundamen-tal representation consists of N -dimensional complex vectors, with the group as well asthe algebra elements being N × N matrices. For QCD, N = 3, this corresponds to therepresentation of the 3-spinor quark field. The usual choice of the normalization of thegenerators is f acd f bcd = N δ ab (A.6)from which we can derive for the fundamental representation,Tr (cid:16) t a t b (cid:17) = δ ab . (A.7)101he structure constants may be written as f abc = − i tr (cid:16) [ t a , t b ] t c (cid:17) (A.8)and the product of two generators has the general form, t a t b = δ ab N + 12 d abc t c + 12 if abc t c (A.9)where the totally symmetric object is defined as d abc = 2 Tr (cid:16) t a { t b , t c } (cid:17) , making use ofthe anti-commutator defined as { t a , t b } = δ ab N + d abc t c . (A.10)Additional identities may be obtainedTr (cid:16) t a t b t c (cid:17) = 14 ( d abc + if abc ) (A.11) f abc f abc = N ( N −
1) (A.12) f abm f cdm = 2 N (cid:16) δ ac δ bd − δ ad δ bc (cid:17) + d acm d dbm − d adm d bcm (A.13) f abm d cdm + f acm d dbm + f adm d bcm = 0 (A.14)with a further relation for N = 3, δ ab δ cd + δ ac δ bd + δ ad δ bc = 3 (cid:16) d abm d cdm + d acm d dbm + d adm d bcm (cid:17) . (A.15)The other important representation is the adjoint representation to which the gener-ators belong and acts on the vector space spanned by the generators themselves – it isan N − SU (3) and transform accordingly. The representation matricesof the generators are given by the structure constants( t b ) ac = if abc . (A.16)A useful relation is the trace of four generators in the adjoint representationTr (cid:16) t a t b t c t d (cid:17) = δ ad δ bc + 12 (cid:16) δ ab δ cd + δ ac δ bd (cid:17) + N (cid:16) f adm f bcm + d adm d bcm (cid:17) . (A.17)In this representation, the covariant derivative D µ η ( x ) = ( ∂ µ − igA aµ t a ) η ( x ) (A.18)takes the component form( D µ η ( x )) a = ∂ µ η a ( x ) − igA bµ ( t b ) ac η c ( x ) (A.19)= ∂ µ η a ( x ) + gf abc A bµ η c ( x ) . (A.20)102 ppendix B Lattice tensors
B.1 Construction of the lattice basis
B.1.1 Momentum polynomial under a transposition
We consider a brief proof of the transformation of a polynomial of a vector p under atransposition is given. A transposition is defined by an exchange of two components ofa vector, σ ↔ ρ , under the operation T σρ . A matrix form for this operator is T ( σρ ) µν = δ µν , µ = σ, ρT ( σρ ) σν = δ ρν T ( σρ ) ρν = δ σν (B.1)which reproduces the correct transformation on the vector p : p ′ ν = p ν , ν = σ, ρp ′ σ = p ρ ,p ′ ρ = p σ . (B.2)Considering the transformation for an arbitrary order of p ( p ′ µ ) n = p ′ µ ...p ′ µ = T ( σρ ) µν p ν ...T ( σρ ) µν n p ν n (B.3)and considering the case µ = σ, ρ the correct transformation is immediate since allcomponents are left unchanged,( p ′ µ ) n = p µ ...p µ = ( p µ ) n = T ( σρ ) µν ( p ν ) . (B.4)For µ = σ, ρ , the transformation is( p ′ σ ) n = T ( σρ ) σν p ν ...T ( σρ ) σν n p ν n = δ σν p ν ...δ σν n p ν n = T ( σρ ) σν ( p ν ) n = ( p ρ ) n . (B.5)This is the same transformation as for the vector p , and thus the polynomial transformsaccordingly. 103 .1.2 Second order tensors under H (4) symmetry Here we show that there is no mixing among the diagonal and off-diagonal elementsunder a general H (4) transformation, using the fact that these transformations can beformed by products of transpositions and inversions.The transposition operator for the exchange of components σ ↔ ρ was defined inB.1. For the inversion of the component ρ , we define the operator as P ρµν = δ µν , µ = ρ (B.6) P ρρν = − δ ρν . (B.7)The transformation for a second order tensor under transpositions and inversions is D ′ µν = T ( σρ ) µτ T ( σρ ) µε D τε , (B.8) D ′ µν = P ( ρ ) µτ P ( ρ ) µε D τε . (B.9)Now we consider the transformation of diagonal elements µ = ν . For transpositionsthere are three distinct situations, D ′ σσ = δ ρτ δ ρε D τε = D ρρ D ′ ρρ = D ′ σσ D ′ µµ = D µµ , µ = ρ, σ (B.10)and we see that no off-diagonal terms appear.A similar analysis can be considered for the inversions using B.9 ( D ′ ρρ = ( − δ ρτ )( − δ ρε ) D τε = D ρρ D ′ µµ = D µµ , µ = ρ (B.11)and again for this transformation, no off-diagonal terms appear for the diagonal trans-formation.We now consider the off-diagonal transformation, µ = ν . For the transpositions thereare again three distinct cases D ′ σν = P τ,ε δ ρτ δ νε = D ρν D ′ ρν = D ′ σν D ′ ρσ = D σρ (B.12)and no diagonal terms are involved. On the other hand for inversions there are two cases ( D ′ µν = − D µν , µ = ρ ∧ ν = ρD ′ µµ = D µµ , µ = ρ ∧ ν = ρ. (B.13)We conclude that a general H (4) transformation does not mix the diagonal and off-diagonal elements for second order tensors.104 .2 General construction for projectors The projectors P k are necessary to extract form factors corresponding to each basiselement. Here we describe the general form of constructing projectors, for an arbitraryvector space.Given a general tensor Γ, this object will be described by a basis of N tensor elements τ j , Γ = N X j =1 γ j τ j (B.14)where γ j are the corresponding dressing functions. Suppose we want to extract one ofthe form factors γ k by acting on Γ with an operator P k (this operation involves thenecessary index contractions to build a scalar). The operation is of the form, P k Γ = P k N X j =1 γ j τ j = γ k . (B.15)From this we may extract the relation P k τ j = δ kj . (B.16)using the completeness of the basis, and the linearity of the operator. Considering themost general form of the projector P k , constructed from basis elements P k = N X i =1 A ki τ i (B.17)and substitute this into eq. B.16, to obtain N X i =1 A ki τ i τ j = δ kj ⇔ A ki = ( τ k τ i ) − . (B.18)This reduces the extraction of the form factors to a matrix inversion problem. We needonly to build the matrix with elements A − ki = τ k τ i , where the contraction of indicesreferred before is assumed, and obtain its inverse A P k = N X i =0 ( τ k τ i ) − τ i . (B.19)With this mechanism, it is straightforward to understand why it is impossible to buildwell defined projectors when there are redundant basis elements that can be written asa linear combination of the remaining elements. In this case, not all rows will be linearlyindependent, and it is know from linear algebra that matrices with this property aresingular, i.e. non-invertible, and the projectors cannot be defined.105 .2.1 Projectors for the lattice bases We use the previous mechanism to build the projectors for the tensor bases consideredthroughout the work. We begin with the general form for second order tensors in thecontinuum D µν ( p ) = A ( p ) δ µν + B ( p ) p µ p ν (B.20)with the elements τ = δ µν and τ = p µ p ν . The matrix A − for a N d dimensional spaceis A − = N d p p p ! , (B.21)and its inverse A = 1 p ( N d − p − p − p N d ! . (B.22)The projectors are built with eq. (B.17) P µν = 1 N d − (cid:18) δ µν − p µ p ν p (cid:19) (B.23) P µν = 1 N d − (cid:18) − δ µν p + N d p µ p ν p (cid:19) , (B.24)and the extraction of the respective form factors follows immediately A ( p ) = 1 N d − X µ D µµ ( p ) − p X µν p µ p ν D µν ( p ) ! (B.25) B ( p ) = 1 N d − − p X µ D µµ ( p ) + N d p X µν p µ p ν D µν ( p ) ! . (B.26)This procedure can be simplified when considering the tensor form D µν ( p ) = D ( p ) (cid:18) δ µν − p µ p ν p (cid:19) , (B.27)with the form factor extracted with D ( p ) = 1 N d − X µ D µµ ( p ) . (B.28)We consider now the lattice basis 3.16. As referred in the construction of the basis,the diagonal elements do not mix with off-diagonal, which allow us to analyse themindependently. The reducibility of the group representation splits the five dimensionalmatrix into two square matrices of size two and three. It is thus important to use twodifferent index contractions, one considering only diagonal terms, P µ τ iµµ τ jµµ , and the106econd considering only off-diagonal elements P µ = ν τ iµν τ jµν . Starting with the diagonalelements τ = δ µµ , τ = p µ and τ = p µ . The contraction matrix A − is A − = N d p p [4] p p [4] p [6] p [4] p [6] p [8] . (B.29)Hence, the diagonal form factors are E ( p ) = 1∆ (cid:20) X µ D µµ ( p [4] p [8] − ( p [6] ) ) + X µ p µ D µµ ( p [4] p [6] − p p [8] )+ X µ p µ D µµ ( p p [6] − ( p [4] ) ) (cid:21) (B.30) F ( p ) = 1∆ (cid:20) X µ D µµ ( p [4] p [6] − p p [8] ) + X µ p µ D µµ ( N d p [6] − ( p [4] ) )+ X µ p µ D µµ ( p p [4] − N d p [6] ) (cid:21) (B.31) G ( p ) = 1∆ (cid:20) X µ D µµ ( p p [6] − ( p [8] ) ) + X µ p µ D µµ ( p p [4] − N d p [6] )+ X µ p µ D µµ ( p p [4] − N d p [6] ) (cid:21) (B.32)with∆ = N d (cid:16) p [4] p [8] − ( p [6] ) (cid:17) + p (cid:16) p [4] p [6] − p p [8] (cid:17) + p [4] (cid:16) p p [6] − ( p [4] ) (cid:17) . (B.33)Similarly we can repeat the procedure for the two dimensional, off-diagonal case,obtaining both form factors, H ( p ) = 2∆ (cid:20) X µ = ν p µ p ν D µν ( p [4] p [6] − p [10] ) − X µ = ν p µ p ν D µν ( p p [4] − p [6] ) (cid:21) (B.34) I ( p ) = 1∆ (cid:20) X µ = ν p µ p ν D µν ( p [8] − ( p [4] ) ) + X µ = ν p µ p ν D µν ( p − p [4] ) (cid:21) (B.35)with ∆ = 2 (cid:16) p p [4] − p [6] (cid:17) (cid:16) p [8] − ( p [4] ) (cid:17) + 2 (cid:16) p − p [4] (cid:17) (cid:16) p [4] p [6] − p [10] (cid:17) . (B.36)Having all projectors for the lattice basis, we need to consider the case of the general-ized diagonal kinematics where these projectors are not possible to obtain. This analysisis done for each individual configuration. Starting with the diagonal, ( n, n, n, n ), thegluon propagator is D µµ ( p ) = ( E ( p ) + n F ( p ) + n G ( p )) δ µµ D µν ( p ) = n H ( p ) + 2 n I ( p ) , µ = ν (B.37)107nd in this case we can only extract two form factors, for the diagonal and off-diagonalterms. These are extracted with E ( p ) + n F ( p ) + n G ( p ) = 1 N d X µ D µµ ( p ) , (B.38) n H ( p ) + 2 n I ( p ) = 1 N d ( N d − X µ = ν D µν ( p ) . (B.39)The mixed configurations, ( n, n, ,
0) and ( n, n, n,
0) have non-diagonal terms and thegluon propagator reads D µµ ( p ) = E ( p ) δ µµ + ( F ( p ) + n G ( p )) p µ D µν ( p ) = ( H ( p ) + 2 I ( p ) n ) p µ p ν , µ = ν. (B.40)For these configurations we consider the parameter k representing the number of non-vanishing components. The contractions of tensor basis elements are summarized by A − = N d kn kn kn ! , A − = k ( k − n (B.41)with corresponding inverses A diag = 1 kn ( N d − k ) kn − kn − kn N d ! , A Off-diag = 1 k ( k − n (B.42)With this, the form factors follow easily E ( p ) = 1 kn ( N d − k ) X µ D µµ ( p ) (cid:16) kn δ µµ − kn p µ (cid:17) (B.43) F ( p ) + n G ( p ) = 1 kn ( N d − k ) X µ D µµ ( p ) (cid:16) − kn δ µµ + N d p µ (cid:17) (B.44) H ( p ) + 2 n I ( p ) = 1 k ( k − n X µ = ν D µν ( p ) p µ p ν . (B.45)Lastly, for on-axis momenta, ( n, , , D µµ ( p ) = E ( p ) + ( F ( p ) + n G ( p )) p µ , (B.46)and the form factors are extracted with E ( p ) = 13 X µ =1 D µµ ( p ) , (B.47) n F ( p ) + n G ( p ) = D ( p ) − E ( p ) . (B.48)108 ppendix C Results – Additional figures
C.1 Gluon propagator
C.1.1 Continuum relations – mixed diagonal configurations
In this section the continuum relations for the momentum configurations ( n, n, n,
0) and( n, n, ,
0) are computed. The procedure follows similarly as the other two diagonalkinematics. For both cases the lattice gluon propagator reads D µµ = E ( p ) δ µµ + ( F ( p ) + n G ( p )) p µ D µν = ( H ( p ) + 2 n I ( p )) p µ p ν , µ = ν. (C.1)Using the extraction for the form factors built in appendix B.2 and also the continuumparametrization D cµν ( p ) = D ( p ) (cid:18) δ µν − p µ p ν p (cid:19) the proof of the continuum relations is follows simply, E ( p ) = D ( p ) kn ( N d − k ) X µ (cid:18) δ µµ − p µ p ν p (cid:19) (cid:16) kn δ µµ − kn p µ (cid:17) = D ( p ) kn ( N d − k ) X µ kn − kn p µ − kn p p µ + kn p p µ ! = D ( p ) F ( p ) + n G ( p ) = D ( p ) kn ( N d − k ) X µ (cid:18) δ µµ − p µ p ν p (cid:19) (cid:16) − kn δ µµ + N d p µ (cid:17) = D ( p ) kn ( N d − k ) X µ − kn + N d p µ − kn p p µ + N d p p µ ! = − D ( p ) p ( p ) + 2 n I ( p ) = D ( p ) k ( k − n X µ = ν − p µ p ν p p µ p ν = − D ( p ) p . Notice that p = kn with the parameter k defined in appendix B.2 and N d = 4 thedimensionality of the lattice. In addition, this result is independent of the use o latticeor improved momentum.00 . . . .
540 0 . . . . ( n,n,n, . . . .
540 0 . . . ( n,n, , p Γ ( p ) ˆ p (GeV) d (ˆ p )ˆ p E (ˆ p )ˆ p ( n F (ˆ p )+ n G (ˆ p )) − ˆ p H (ˆ p ) − p I (ˆ p ) ˆ p (GeV) d (ˆ p )ˆ p E (ˆ p )ˆ p ( n F (ˆ p )+ n G (ˆ p )) − ˆ p H (ˆ p ) − p I (ˆ p ) Figure C.1: Form factors from the lattice basis for the mixed configurations p = ( n, n, n,
0) (left)and for p = ( n, n, ,
0) (right) both as a function of improved momentum. Shown for comparisonis the benchmark result d (ˆ p ). The analysis of the continuum relations for these two configurations is seen in fig. C.1.The continuum relations are exactly satisfied among all three form factors for bothconfigurations. The benchmark result was shown for comparison, and it is noticeable thatthe further from the diagonal, the worse the correspondence becomes. The configuration( n, n, ,
0) deviates from the gluon propagator dressing function for higher momentum,while the result for ( n, n, n,n, n, n,