Padé Approximants and the analytic structure of the gluon and ghost propagators
PPadé Approximants and theanalytic structure of the gluonand ghost propagators
Alexandre Fonseca Falcão
A thesis submitted for the degree ofMaster in Physics
Department of PhysicsUniversity of CoimbraPortugalJuly 2020 a r X i v : . [ h e p - l a t ] S e p bstract In a Quantum Field Theory, the analytic structure of the 2-points correlationfunctions, i.e. the propagators, encloses information about the properties of the cor-responding quanta, particularly if they are or not confined. However, in QuantumChromodynamics (QCD), we can only have an analytic solution in a perturbativepicture of the theory. For the non-perturbative propagators, one resorts on numer-ical solutions of QCD that accesses specific regions of the Euclidean momentumspace, as, for example, those computed via Monte Carlo simulations on the lattice.In the present work, we rely on Padé Approximants (PA) to approximate thenumerical data for the gluon and ghost propagators, and investigate their analyticstructures.In a first stage, the advantages of using PAs are explored when reproducingthe properties of a function, focusing on its analytic structure. The use of PAsequences is tested for the perturbative solutions of the propagators, and a residueanalysis is performed to help in the identification of the analytic structure. Atechnique used to approximate a PA to a discrete set of points is proposed andtested for some test data sets. Finally, the methodology is applied to the Landaugauge gluon and ghost propagators, obtained via lattice simulations.The results identify a conjugate pair of complex poles for the gluon propagator,that is associated with the infrared structure of the theory. This is in line withthe presence of singularities for complex momenta in theories where confinement isobserved. Regarding the ghost propagator, a pole at p = 0 is identified. For bothpropagators, a branch cut is found on the real negative p -axis, which recovers theperturbative analysis at high momenta. Keywords:
Analytic structure, Padé Approximant, Gluon propagator, Ghostpropagator, Lattice QCD. esumo
Numa Teoria Quântica de Campos, a estrutura analítica das funções de corre-lação de 2 pontos, i.e. , os propagadores, contêm diversas informações acerca daspropriedades dos quanta da teoria, em particular se estes estão, ou não, confinados.No entanto, em Cromodinâmica Quântica (QCD), uma solução analítica é apenaspossível num quadro perturbativo da teoria. A obtenção dos propagadores deuma forma não perturbativa pode ser feita com recurso a soluções numéricas daQCD para momentos definidos no espaço Euclidiano. Estas soluções podem serconseguidas com base, por exemplo, em simulações de Monte Carlo na rede. Nestetrabalho baseamo-nos em Aproximantes de Padé (PA) para analisar os propa-gadores do gluão e do campo fantasma, dessa forma obtidos na gauge de Landau,e investigamos a sua estrutura analítica.Numa primeira fase, são exploradas as vantagens do uso de PAs para reproduziras propriedades de uma função, em especial a sua estrutura analítica. É testada autilização de sequências de PAs nas soluções não perturbativas dos propagadores,sendo feita uma análise de resíduos como auxílio à identificação da estruturaanalítica. É, também, proposta e testada uma nova técnica para aproximar umconjunto discreto de pontos a um PA, que é, por último, aplicada aos propagadoresdo gluão e do campo fantasma provindos de simulações na rede.Um par conjugado de polos complexos, associado à estrutura de infravermelhoda teoria, é identificado no propagador do gluão, estanto de acordo com a pre-sença de singularidades em momentos complexos em teorias nas quais se observaconfinamento. Quanto ao propagador do campo fantasma, é identificado um poloem p = 0. Em ambos os propagadores é identificada uma descontinuidade noeixo- p real negativo, sendo, desta forma, recuperada a análise perturbativa a altosmomentos. Palavras-chave:
Estrutura analítica; Aproximante de Padé; Propagador do gluão,Propagador do campo fantasma, QCD na rede. cknowledgements ontents
List of Figures xiList of Tables xviList of Abbreviations xvii1 Introduction 12 Elements of Padé Approximants 5 L and M . . . . . . . . . . . . . . . . . . . . . . . 193.3 The expansion point . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4 “Padé’s hint” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.5 Poles and zeros distribution . . . . . . . . . . . . . . . . . . . . . . 243.6 Residue analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.7 Adding mass generation terms . . . . . . . . . . . . . . . . . . . . . 293.8 The perturbative result for the ghost propagator . . . . . . . . . . . 32 x ist of Figures F ( x ) and the truncatedTaylor expansions F [4] ( x ) and F [100] ( x ); and between the originalfunction F ( x ), the rational function e F [4] ( w ( x )), and the PA F [2 | ( x ). 72.3 Key for the identification of poles, zeros and essential singularitiesin representations of complex functions. . . . . . . . . . . . . . . . . 112.4 Representation of the test functions’ analytic structure f ( z ), anddistribution, in the complex plane, of poles and zeros for the sequenceof diagonal PAs of orders [5 | |
10] and [20 | f ( z ), anddistribution, in the complex plane, of poles and zeros for the sequenceof diagonal PAs of orders [5 | |
10] and [20 | f ( z ), anddistribution, in the complex plane, of poles and zeros for the sequenceof diagonal PAs of orders [5 | |
10] and [20 | f ( z ), anddistribution, in the complex plane, of poles and zeros for the sequenceof diagonal PAs of orders [5 | |
10] and [20 | D gl ( p ), and of itsdressing function d gl ( p ). . . . . . . . . . . . . . . . . . . . . . . . . 18xi.2 Analytic structures of the gluon propagator D gl ( p ), and of its dress-ing function d gl ( p ). . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Graphical representation of the asymptotic behaviour of D gl ( p ) and d gl ( p ), together with PAs of orders [ N | N ], [ N − | N ], and [ N − | N ]. 203.4 Representation of d gl ( p ), together with the respective PA of order[1 |
1] using, as expansion point, p = 5 . d gl ( p ), together with the respective PAs of order[1 |
1] using, as expansion points, p = 1 GeV and p = 10 GeV, andthe approximation error. . . . . . . . . . . . . . . . . . . . . . . . . 213.6 Evolution of the approximation error at the endpoints p = 1 GeVand p = 10 GeV with N , using p = 5 . , , . N within thediagonal PA sequence of order [ N | N ], for the perturbative gluondressing function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.8 Distribution of poles and zeros for some values of N within thenear-diagonal PA sequence of order [ N − | N ], for the perturbativegluon propagator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.9 Distribution of poles and zeros for the PA of order [50 |
50] of d gl ( p ),and the absolute value of the residues | A k | for each pole k . . . . . . 283.10 Distribution of poles and zeros for the PA of order [50 |
50] of d gl ( p ),with a cut in the residues at | A k | = 10 − . . . . . . . . . . . . . . . . 283.11 Distribution of poles and zeros for the PA of order [50 |
51] of D gl ( p ),with a cut in the residues at | A k | = 10 − . . . . . . . . . . . . . . . . 293.12 Distribution of poles and zeros for the elements of the PA sequencesof order [ N | N ] with N = 50 of d gl ( p ), obtained for m = − , , − i i
10 GeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30xii.13 Distribution of poles and zeros for the elements of the PA sequencesof order [ N − | N ] of D gl ( p ) with N = 50, obtained for m = − , , − − i
10 and 5 + 10 GeV and m = 0. . . . . . . . . . . . . 314.1 Achieved values of e χ for the test data generated from f ( p ), using m = 0 and m = 0 .
5, and for both methods of minimisation, DEand SA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Representation of the test data points generated from the f ( p ),which is also represented, using m = 0 . |
4] and [17 | f ( p ), using m = 0 and m = 0 .
5, and for both methods ofminimisation, DE and SA. . . . . . . . . . . . . . . . . . . . . . . . 384.4 Evolution of the off-axis poles and zeros with N in the PA sequence,obtained from the test data generated from f ( p ), using m = 0and m = 0 .
5, and for both methods of minimisation, DE and SA. . 404.5 Evolution of the on-axis poles and zeros with N in the PA sequence,obtained from the test data generated from f ( p ), using m = 0and m = 0 .
5, and for both methods of minimisation, DE and SA. . 414.6 Achieved values of e χ for the test data generated from f ( p ), using m = 0 and m = 0 .
5, and for both methods of minimisation, DEand SA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.7 All-poles representation obtained for the test data generated from f ( p ), using m = 0 and m = 0 .
5, and for both methods ofminimisation, DE and SA. . . . . . . . . . . . . . . . . . . . . . . . 424.8 Evolution of the off-axis poles and zeros with N in the PA sequence,obtained from the test data generated from f ( p ), using m = 0and m = 0 .
5, and for both methods of minimisation, DE and SA. . 43xiii.9 Evolution of the on-axis poles and zeros with N in the PA sequence,obtained from the test data generated from f ( p ), using m = 0and m = 0 .
5, and for both methods of minimisation, DE and SA. . 454.10 Achieved values of e χ for the test data generated from f ( p ), using m = 0 and m = 0 .
5, and for both methods of minimisation, DEand SA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.11 All-poles representation obtained for the test data generated from f ( p ), using m = 0 and m = 0 .
5, and for both methods ofminimisation, DE and SA. . . . . . . . . . . . . . . . . . . . . . . . 464.12 Evolution of the off-axis poles and zeros with N in the PA sequence,obtained from the test data generated from f ( p ), using m = 0and m = 0 .
5, and for both methods of minimisation, DE and SA. . 474.13 Evolution of the on-axis poles and zeros with N in the PA sequence,obtained from the test data generated from f ( p ), using m = 0and m = 0 .
5, and for both methods of minimisation, DE and SA. . 485.1 Landau gauge lattice ghost propagator used in the analysis. . . . . 525.2 Achieved values of e χ for the Landau gauge lattice ghost propagator,for both methods of minimisation, DE and SA. . . . . . . . . . . . 535.3 All-poles representation obtained for the ghost propagator data, forboth methods of minimisation, DE and SA. . . . . . . . . . . . . . 535.4 Evolution of the on-axis poles and zeros within the PA sequenceobtained for the ghost propagator data, for both methods of min-imisation, DE and SA. . . . . . . . . . . . . . . . . . . . . . . . . . 545.5 Evolution of the off-axis poles within the PA sequence obtained forthe ghost propagator data, for both methods of minimisation, DEand SA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.6 Gluon propagator used in the analysis, obtained with four differentlattice volumes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56xiv.7 Achieved values of e χ for the Landau gauge lattice gluon propagator,and for both methods of minimisation, DE and SA. . . . . . . . . . 575.8 All-poles representation obtained for the gluon propagator data with32 , 64 , 80 and 128 lattices, for both methods of minimisation,DE and SA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.9 Evolution of the off-axis poles within the PA sequence, obtainedfor the gluon propagator data with 32 and 64 lattices, for bothmethods of minimisation, DE and SA. . . . . . . . . . . . . . . . . 605.10 Evolution of the off-axis poles within the PA sequence, obtainedfor the gluon propagator data with 80 and 128 lattices, for bothmethods of minimisation, DE and SA. . . . . . . . . . . . . . . . . 615.11 Position of the poles present in the analytic structure of the gluonpropagator obtained in the present work and in other studies. . . . 635.12 Evolution of the on-axis poles and zeros within the PA sequence,obtained for the gluon propagator data with 32 and 64 lattices,for both methods of minimisation, DE and SA. . . . . . . . . . . . 645.13 Evolution of the on-axis poles and zeros within the PA sequence,obtained for the gluon propagator data with 80 and 128 lattices,for both methods of minimisation, DE and SA. . . . . . . . . . . . 65xv ist of Tables w → /
2. . . . 83.1 Padé coefficients obtained for the orders of approximation [3 |
3] and[3 |
4] of the dressing function d gl ( p ). . . . . . . . . . . . . . . . . . 233.2 Padé coefficients obtained for the orders of approximation [3 |
4] and[3 |
5] of the propagator D gl ( p ). . . . . . . . . . . . . . . . . . . . . 24xvi ist of Abbreviations DE Differential Evolution
DSE
Dyson-Schwinger Equations PA Padé Approximant
QCD
Quantum Chromodynamics
QED
Quantum Electrodynamics
QFT
Quantum Field Theory
RGZ
Refined Gribov-Zwanziger SA Simulated Annealing
SPM
Schlessinger Point Method xvii hapter 1Introduction
The current theoretical picture of the electromagnetic interaction, a componentof the electroweak part of the standard model, and the strong interaction, betweenquarks and gluons, boils down to Quantum Electrodynamics (QED) and QuantumChromodynamics (QCD), respectively. Both are gauge theories associated withdifferent gauge groups. QED is an abelian gauge theory associated with the sym-metry group U(1), whilst QCD is a non-abelian gauge theory with the symmetrygroup SU(3). The fundamental quanta of QED, e.g. the electron and the photon,are experimentally observed particles, whereas the quanta of QCD are not. Indeed,single particle states associated with quarks and gluons were never observed exper-imentally. It is believed that quarks and gluon states do not belong to the Hilbertspace of the physical states. Therefore, quarks and gluons can only be present inNature as components of other particles, i.e. , they are confined particles.In Quantum Field Theories (QFT), the 2-point correlation functions, i.e. , thepropagators, summarise the dynamical information of the theory. In the QED,that can be solved via perturbation theory, these propagators are well known, see, e.g. , [22, 26]. Unfortunately, the same approach cannot be followed in QCD, whereperturbative techniques can only be applied to the ultraviolet (UV) momentumregion. Additionally, since these quanta cannot be experimentally observed alone,their behaviour and properties cannot be directly measured. Hence, the full knowl-edge of the gluon, quark and the unphysical ghost dynamics has to be acquired bymeans of theoretical ab initio non-perturbative methods.There are mainly two such methods that are commonly applied to investi-1ate the non-perturbative regime of QCD: the Dyson-Schwinger Equations (DSE),and lattice regularised Monte Carlo simulations (lattice QCD). Both offer non-perturbative solutions for QCD in the whole range of momentum, but both havelimitations. Although the DSE promise us an exact solution for the theory, aninfinite system of coupled integral equations has to be solved, and a self-consistenttruncation scheme needs to be applied in such a way that the important propertiesand quantities are not compromised. Regarding the lattice calculations, they arelimited by the finite volume of the lattice. Notwithstanding, these non-perturbativemethods offer valuable information in the infrared (IR) momentum region, a regionof momenta that is not accessed with perturbation theory.Most non-perturbative methods, including the ones above, are formulated in theEuclidean space. However, the real theory lives in the Minkowski space, where theobservables are to be computed. Hence, the Wightman functions (Minkowski spacecorrelation functions) must be obtained from the Schwinger functions (Euclideanspace correlation functions) via a Wick rotation. This can only be done if theanalytic structures of these functions are known.In general, the analytic structure, i.e. , the set of zeros, singularities and branchcuts, of a propagator have a well defined physical interpretation. For example,in QED, the electron propagator has a singularity at the physical mass of thisparticle. For a typical theory, the analytic structure of a propagator is shown inFigure 1.1. In the complex p -plane, a pole that corresponds to the one particlestate should appear, as well as a branch cut associated with two or more freeparticles; poles related to bound states also appear in the analytic structure [22].All of these structures occur in the real p -axis. When a calculation is made, onecan choose the integration path to go around these singularities. This also allowsto perform a Wick rotation when going from the Euclidean space to the Minkowskispace of momenta. However, this rotation is impracticable if complex singularitiesare present. While for four-dimensional QED, we do not find such singularities ,in non-perturbative QCD it is a different story, since the propagators acquire adifferent analytic structure.In fact, in a theory that displays confinement, which is believed to be the case To the author’s best knowledge, complex singularities were found in QED only when formu-lated in lower dimensions, where it shows confinement, see, e.g. , [20, 28]. igure 1.1: Analytic structure of a propagator, for a typical theory, obtained in theMinkowski space. Image from [22]. of QCD, the analytic structure of the propagators of confined particles may havesingularities that are not associated with physical states [19]. This has to do witha violation of the local axioms of QFT by theories that exhibit confinement [1, 5,16].Thereby, knowing the analytic structure of a propagator reveals to be crucial,not only because it gives information about the physical particle states of the theory,but also because it may bring new insight into the confinement mechanism. It isalso indispensable to know the analytic structure when one attempts to go from theEuclidean space to the Minkowski space, whenever non-perturbative calculationsare made. In this sense, many studies have been done with the main objectiveof finding the analytic structure of the full propagators for the QCD quanta, i.e. ,the gluon, the quark, and the unphysical ghost. Some predictions and studiesaround the existence of complex poles in the gluon propagator were made in, e.g. ,[2, 16, 30–32, 40]. Notably, the tree level solution for the propagator in the RefinedGribov-Zwanziger (RGZ) framework, that describes the lattice data extremely well,predicts the existence of a conjugate pair of complex poles at Euclidean momenta[8, 10]. These two complex poles were found using a global fit to lattice data,in [11]. Poles at similar positions in the complex p -plane were also found in [5],using Padé Approximants (PA) to reconstruct the gluon and ghost propagatorsobtained via DSE and lattice simulations. Regarding the ghost propagator, itsanalytic structure seems to be similar to the one obtained perturbatively, i.e. , withno complex poles, see, e.g. , [5, 16]. 3n the other hand, additional studies, e.g. [33], found no evidence of suchcomplex singularities. Further studies were undertaken in other types of theoriesto investigate the consequences of confinement in the analytic structure of therespective propagators, see e.g. [20, 25, 28].In order to complement the already known results, this work focuses on theinvestigation of the analytic structure of the fundamental propagators in pureSU(3) Yang-Mills theory, obtained non-perturbatively via lattice calculations inthe Landau gauge. No particular theoretical or empirical models to describe thelattice data will be considered. Instead, we rely on PAs to investigate the analyticstructure of the QCD propagators, since it provides a general approach to studyfunctions with singularities across the complex plane [4, 6, 18, 39].This work is organised as follows: in the next chapter, we will start by intro-ducing and defining Padé approximants in the context of the analytic continuationproblem. The use of Padé approximants will, then, be tested for the perturbativepropagators in the third chapter, in order to know how their analytic structuresare reproduced, and how faithful this reproduction is. Considering that the fullpropagators come in the form of discrete sets of data points, a method of recon-structing the former with Padé approximants will be introduced and tested in thefourth chapter. In the fifth chapter, the obtained analytic structures for the gluonand ghost propagators will be investigated and discussed, followed by the finalconclusions and ideas for possible future works.4 hapter 2Elements of Padé Approximants The identification of the analytic structure of the gluon and ghost propagatorsrequires the knowledge of the latter in the whole complex plane. However, thelattice simulations only provide the propagators in the real positive range of theEuclidean momenta.In this chapter, we will look in detail at rational functions, particularly at thePA, and explore their use to identify the singularities and branch cuts for arbitrarycomplex momenta. A series of tests will be made, to examine the reliability of PAsin the reproduction of analytic structures.
The numerical analytic continuation, i.e. , the task of extending the domain of afunction beyond the regime where the information is available, for example from afinite set of data points, is a known problem in Physics. The reconstruction of real-time correlations of spectral functions [34], the calculation of scattering amplitudes[29], and situations, e.g. [5, 21, 27, 35, 36], where the analytic continuation allowsus to access information in different regions of momenta, while our knowledge isrestricted to the physical one, are good examples of this problem. A graphicalrepresentation of the analytic continuation is shown in Figure 2.1.By adjusting a function to a set of data points, we can use the former tocalculate its values in the whole domain of the function, and not only where thedata is available. Yet, if we do not know the form of the function represented by5 igure 2.1: Graphical representation of an example of analytic continuation, from [29]. the finite set of data, which one do we choose? Power series may seem a goodsolution. However, we will see that rational functions, particularly PAs, offer amore general and faithful approximation, making it more useful to the numericalcontinuation of the data to the complex plane.
In order to convince ourselves that rational functions are indeed richer structuresby capturing the analytic properties of the approximated functions, let us considerthe following example, taken from [21, 39].Consider the function F ( x ), given by F ( x ) = s x x , (2.1)represented graphically in Figure 2.2, that has the following expansion in powerseries, around x = 0 F ( x ) = ∞ X n =0 a n x n = 1 + x − x x − x
128 + O ( x ) , (2.2) Without loss of generality, the expansion is made around the origin. The problem ahead isindependent of the expansion point. ( x ) F [ ] ( x ) F [ ] ( x )- - x - F ( x ) F ˜ [ ] ( w ( x )) F [ ] ( x )- - x - Figure 2.2:
Left:
Comparison between the original function F ( x ) and the truncatedTaylor expansions F [4] ( x ) and F [100] ( x ). Right:
Comparison between the originalfunction F ( x ), the rational function e F [4] ( w ( x )), and the PA F [2 | ( x ). For x >
0, thethree curves overlap. where a n are the Taylor coefficients of F ( x ).By denoting the Taylor expansion truncated at order N as the partial sum F [ N ] ( x ) ≡ N X n =1 a n x n , (2.3)we have, for N = 4, F [4] ( x ) = 1 + x − x x − x . (2.4)Let us suppose that we only have access to the partial sum F [4] and that wehave no idea of the exact function that originated it. If we want to compute thevalue of F ( x ) at the origin, we have F [4] (0) = 1, which concurs perfectly with theexact result of F (0). On the other hand, if we try to calculate the value of F ( x ) at x = 0 . F [4] ( x ), the approximation is less accurate. In fact, F [4] (0 . ≈ . F (0 .
5) = 2 √ / ≈ . r = 1 /
2, the approximationfails drastically, even using expansions truncated at higher orders, as it can be seenin Figure 2.2.In fact, computing F ( ∞ ) = lim x →∞ F ( x ), with only the Taylor expansion, isa hopeless task. In this limit, F [ N ] ( x ) diverges for any N ∈ N . Nonetheless, theoriginal function F ( x ) does not diverge when x → ∞ , lim x →∞ F ( x ) = √
2. Howcould we achieve this value with only F [4] ( x )? A cunning trick to transform the7 · · · F [ N ] (1 /
2) 1 1.25 1.34375 1.38281 1.39990 · · ·
Table 2.1: Results for the lowest partial sums of Eq. (2.6) with w → / expansion in one which will let us estimate the value of F ( ∞ ) is: to perform thechange of variables x ≡ w/ (1 − w ); to define e F ( w ) ≡ F ( x ( w )) = (1 − w ) − ; (2.5)to re-expand it in w , e F ( w ) = ∞ X n =0 b n w n = 1 + w w w
16 + 35 w
128 + O ( w ); (2.6)and, in a similar way to F ( x ), to define the truncated Taylor expansion of e F ( w )as the partial sum e F [ N ] ( w ) ≡ N X n =0 b n w n . (2.7)By doing this, the limit x → ∞ is translated into w → /
2. For this value of w , the Taylor expansion of (2.6) converges, and we are able to approximate thevalue of e F (1 /
2) and, therefore, of F ( ∞ ). Let us, then, do so for the first lowest N of e F [ N ] (1 / √ ≈ . e F [4] ( w ) in terms of x , e F [4] ( w ( x )) = 1 + (17 / x + (219 / x + (637 / x + (2867 / x (1 + 2 x ) . (2.8)Clearly, this is not a power expansion, but a rational function. We already sawthat an approximation like (2.8) allows us to estimate not only the value of F ( x )near the origin, but also in the limit x → ∞ . By graphically comparing e F [4] ( w ( x ))with the original function (Figure 2.2) an improvement can be seen, which was Note that e F [4] ( w ( x )) and F [4] ( x ) are not the same. The first comes from the expansion of e F ( w ) in w , while the latter comes from the expansion of the original function before the changeof variables. e F [4] ( w ( x )) isdefined in x ∈ [ − , − / F ( x ) is undefined (visible in Figure 2.2), theoverall behaviour of the original function can be reproduced.If the use of rational functions can considerably improve the approximatedescription of a function, how do we build one? Surely, it is not of our interest tofind the right change of variables for every function we come across. A particulartype of rational functions is the PA. The idea of PAs is to use the first Taylorcoefficients of a given function to build a ratio of polynomials, i.e. , a rationalfunction. A simple PA is P ( x ) = a + a x b x . (2.9)In this case, the goal is to fix the unknowns a , a and b in such a way that thefirst three coefficients of the Taylor expansion of P ( x ) match the first three Taylorcoefficients of the function to be approximated. For the function F ( x ), defined in(2.1), we find P ( x ) = 1 + (7 / x / x = 1 + x − x x − x
128 + O ( x ) . (2.10)When comparing it with (2.2), it can be seen that the first three coefficients areexactly the same (but not the remaining ones). If we now use the limit of P ( x )to estimate the value of F ( ∞ ), we get lim x →∞ P ( x ) = 1 .
4, which is a betterdetermination than any in Table 2.1. By requiring the matching of the first fiveTaylor coefficients, we obtain the PA P ( x ) = 1 + (13 / x + (41 / x / x + (29 / x , (2.11)and lim x →∞ P ( x ) = 41 / ≈ . F ( ∞ ) is increased. Indeed,for eleven matching coefficients we reach a precision of ∼ − .Graphically (Figure 2.2), the precision of the approximation via PA is evident.The reproduction of the divergence at x = − .3 The Padé Approximant In order to use the PA as a tool, a rigorous definition must be made, as in [6,21, 39]. For a more formal and complete definition see, e.g. , [12]. Thus, let usconsider a function f ( z ) that has a series expansion in the complex plane f ( z ) = ∞ X n =0 c n z n , (2.12)where c n are its Taylor coefficients. Let us also denote f [ N ] ( z ) as the respectivetruncated Taylor expansion of order N , f [ N ] ( z ) ≡ N X n =0 c n z n . (2.13)A Padé Approximant of order [ L | M ] is defined as the ratio of two polynomials Q L ( z ) and R M ( z ), of orders L and M respectively, P [ L | M ] ( z ) ≡ Q L ( z ) R M ( z ) = q + q z + q z + ... + q L z L r z + r z + ... + r M z M . (2.14)As it is usually done, the normalisation r = 1 is considered. The coefficients q , ..., q L and r , ..., r M will be called Padé coefficients .The PA of the function f ( z ) is denoted by f [ L | M ] ( z ), and is built such thatthe Taylor expansion of f [ L | M ] ( z ) reproduces exactly the first L + M + 1 Taylorcoefficients of f ( z ). In this sense, we say that the PA has a contact of order L + M with the expansion of f ( z ), and the difference between the PA and the originalfunction satisfies f ( z ) − f [ L | M ] ( z ) = O ( z L + M +1 ) . (2.15)When it exists, the PA is unique for any L and M .As we will see later, sequences of PAs are extremely important (and fundamentalin the scope of the present work), for it is their stability that gives us the confidenceon the outcome of the approximations. Sequences with L = M + J are called near-diagonal when J = 0, and diagonal when J = 0.Despite the already seen advantages of using PAs, there is a downside: unlike Without loss of generality, an expansion around the origin is considered.
Let us now focus on the analytic structure of functions and on how well thePA can reproduce it. In [39], a series of general examples with test functions arecarried out with this objective. Here, we will aim our attention to some of thembefore we move further into more specific tests.We begin to consider the following complex test functions: f ( z ) = e − z , (2.16) f ( z ) = (cid:18) z − z + 2 (cid:19) e − z , (2.17) f ( z ) = e − z/ (1+ z ) , (2.18) f ( z ) = s z z . (2.19)The analytic structures of each of the functions above are represented, in thecomplex plane, in Figures 2.4 to 2.7. These representations, made with the use ofthe software Mathematica [38], are built in such a way that the poles, zeros, andbranch cuts are enhanced. To do so, the argument of f i ( z ) is represented, insteadof its value. This allows us to use the following key to read figures made in thisway. Figure 2.3: Key for the identification of poles, zeros and essential singularities in represen-tations of complex functions, where it is used a cyclic colour function over the argumentof the represented function. Image from [24]. π - π / π / π ∞ ○○○○○ ○○ ○○○○○○○○ ○○ ○○ ○○○○○○○○○○○○○○○○ - -
20 0 20 40 - - z I m z f [ | ] Poles ○ f [ | ] Zeros f [ | ] Poles ○ f [ | ] Zeros f [ | ] Poles ○ f [ | ] Zeros
Figure 2.4:
Left:
Representation of the test functions’ analytic structure f ( z ). Thekey for the structure identification is in Figure 2.3. Right:
Distribution, in the complexplane, of poles and zeros for the sequence of diagonal PAs of orders [5 | |
10] and[20 | f ( z ). The branch cut, it is identified by a black dashed line.By looking at expressions (2.16) to (2.19) and Figures 2.4 to 2.7, we can concludethat:• f ( z ) has no singularities for | z | < ∞ ;• f ( z ) has a zero at z = 2 and a simple pole at z = − f ( z ) has an essential singularity at z = − f ( z ) has a branch cut along a line on [-1,-1/2].After computing the PA for a given function, its analytic structure can beextrapolated from the PA’s own analytic structure. As a fraction between polyno-mials, the zeros of a PA correspond to the roots of its numerator - Q L ( z ) in (2.14)-, and its poles correspond to the roots of its denominator - R M ( z ) in (2.14). Notethat these are the only structures present in the analytic structure of a PA. As aconsequence, the reconstruction of the original function’s analytic structure onlyrelies on the distribution of poles and zeros of the associated PA.For some functions, an analytic expression of the respective PA can be found.For example, for f ( z ) we have Q L ( z ) = L X k =0 (2 L − k )! L !(2 L )! k !( L − k )! ( − z ) k , (2.20)12 π - π / π / π ∞ ○○○○○ ○ ○○○○○○○○○ ○ ○○○ ○○ ○○○○○○○○○○○○○○ - -
20 0 20 40 - - z I m z f [ | ] Poles ○ f [ | ] Zeros f [ | ] Poles ○ f [ | ] Zeros f [ | ] Poles ○ f [ | ] Zeros
Figure 2.5:
Left:
Representation of the test functions’ analytic structure f ( z ). Thekey for the structure identification is in Figure 2.3. Right:
Distribution, in the complexplane, of poles and zeros for the sequence of diagonal PAs of orders [5 | |
10] and[20 | f ( z ). The pole at z = − z = 2 appear atthe same position for the three represented orders, and are, therefore, overlapped. - π - π / π / π ∞ ○ ○○ ○○ ○○○○○○○○ ○○ ○○○○○○○○○○○○○○○○○○○○ - - - - - - - z I m z f [ | ] Poles ○ f [ | ] Zeros f [ | ] Poles ○ f [ | ] Zeros f [ | ] Poles ○ f [ | ] Zeros
Figure 2.6:
Left:
Representation of the test functions’ analytic structure f ( z ). Thekey for the structure identification is in Figure 2.3. Right:
Distribution, in the complexplane, of poles and zeros for the sequence of diagonal PAs of orders [5 | |
10] and[20 | f ( z ). π - π / π / π ∞ ○ ○ ○ ○ ○ - - - - - - z I m z f [ | ] Poles ○ f [ | ] Zeros ○○ ○ ○○ ○ ○ ○ ○○ - - - - - - z I m z f [ | ] Poles ○ f [ | ] Zeros ○○○○ ○ ○ ○ ○ ○○○○○○○○○○○○ - - - - - - z I m z f [ | ] Poles ○ f [ | ] Zeros
Figure 2.7:
Left:
Representation of the test functions’ analytic structure f ( z ). Thekey for the structure identification is in Figure 2.3. Right:
Distribution, in the complexplane, of poles and zeros for the sequence of diagonal PAs of orders [5 | |
10] and[20 | f ( z ). R M ( z ) = M X k =0 (2 M − k )! M !(2 M )! k !( M − k )! z k . (2.21)However, in general, a PA can be found numerically for a given order [ L | M ]. Onceit is done, the poles and zeros can be obtained and represented graphically in thecomplex plane, and the analytic structures can be compared. For our test functions,some orders of diagonal PAs were calculated. The corresponding distributions ofpoles and zeros are presented in Figures 2.4 to 2.7.For f ( z ), the distribution of poles and zeros of the obtained PAs is symmetricaround the origin, as seen in Figure 2.4. However, their position strongly dependson the order of the PA used. Indeed, by increasing the PA’s order, its distributionseems to spread and move towards infinity, leaving no structure behind.The same happens with f ( z ) (Figure 2.5), except that, in this case, a pole anda zero appear in the expected positions, z = − z = 2 respectively. Thesepole and zero are stable, and their positions, in the complex plane, seem to beindependent of the order of approximation. This behaviour indicates that theoriginal poles and zeros are identified by stable ones, in the complex plane, in aPA sequence. On the other hand, unstable poles and zeros do not correspond toany characteristic of the analytic structure of f ( z ). For all the numerical calculations in the present work, the software
Mathematica [38] wasused.
14 similar, but opposite, behaviour to the one seen for f ( z ) and f ( z ) canbe observed in the distribution of poles and zeros for f ( z ). However, insteadof spreading, the poles and zeros gather around the position where the essentialsingularity should be (see Figure 2.6).Lastly, the branch cut of f ( z ) is reproduced by the PA in the form of analternating sequence of poles and zeros, between z = − z = − /
2. In thiscase, the distance between nearby zeros and poles decreases when the order of thePA is increased.Additional numerical tests were performed in [39]. It was found that “spuriouspoles” may appear with the increase of the order of the approximation, due toinsufficient numerical accuracy. Some of the poles have an associated zero thatcancels its contribution to the analytic structure. These pole-zero pairings, oftencalled
Froissart doublets , are artefacts of the approximation, and accumulate instructures around the origin. These have to be identified as such, in order toproperly identify the correct analytic structure. Later on, we will introduce a wayto remove these unwanted poles by performing a residue analysis (Section 3.6).We can, now, draw some conclusions regarding the reproduction of the analyticstructure of a function:• For a function with or without singularities, distributions of poles and zerosmay appear. As the PA’s order is increased, if they spread to infinity or areoverall unstable, they are not associated with the analytic structure of theoriginal function, but are artefacts of the method;• For a function with an essential singularity, the structures of poles andzeros tighten around the position of the singularity for increasing orders ofapproximation;• Poles and zeros that are stable throughout the PA sequence may be correctlyidentified as being part of the original function’s analytic structure;• A branch cut can be identified by a PA as a sequence of alternating polesand zeros, for which the distance between nearby poles and zeros decreaseswhen the order of the PA is increased;• The increase of the order of approximation may cause the emergence of15roissart doublets (pole-zero pairings), which do not contribute to the analyticstructure. 16 hapter 3Preliminary tests
In the previous chapter, we explored the use of PAs to identify the analyticstructure of a function based on the distribution of poles and zeros. Notwithstand-ing, the functions that we are dealing with in our study are not so simple as theones covered there, and more specific tests have to be performed. These will allowto identify a set of tools for a proper identification of poles and branch cuts.
The results from the renormalisation group improved perturbation theory forthe gluon propagator and its dressing function are given, respectively, by D gl ( p ) = 1 p " N f α s π ln p Λ ! + 1 − γ , (3.1) d gl ( p ) ≡ p D gl ( p ) = " N f α s π ln p Λ ! + 1 − γ . (3.2)Following [11], in the numerical tests we will use α s = 0 . .
425 GeV and γ = 13 /
22. These results offer us a valuable opportunity to study the reliability ofusing PAs to study the QCD propagators, since we expect an equivalent behaviourin the UV limit. Throughout this chapter we look at the Equations (3.1) and (3.2)and study them as test functions to understand the behaviour and validity of the17 p [ GeV ] D gl ( p ) [ GeV - ] p [ GeV ] d gl ( p ) Figure 3.1: Graphical representation of the gluon propagator D gl ( p ) (left), and of itsdressing function d gl ( p ) (right). - π - π / π / π ∞ - π - π / π / π ∞ Figure 3.2: Analytic structures of the gluon propagator D gl ( p ) (left), and of its dressingfunction d gl ( p ) (right). The key for the structure identification is in Figure 2.3. PA approach.To give us a visual idea of the behaviour of these test functions, a graphicalrepresentation of (3.1) and (3.2) is shown in Figure 3.1. As for the respectiveanalytic structures, they are represented in the complex p -plane in Figure 3.2. Forthe propagator, we see a simple pole at the origin, created by the factor 1 /p , aswell as a branch cut on the whole real negative p -axis, from the logarithm. Onthe other hand, only the branch cut in the real negative p -axis appears in theanalytic structure of the dressing function. These are the structures we want toreproduce using PA sequences. 18 .2 Relation between L and M The first step in the construction of a PA sequence is to establish the bestrelation between the orders L and M of the polynomials, in (2.14), since thisrelation is what dictates the limit behaviour of a PA.We know that the propagator, as well as the dressing function, have a de-pendence only on p . For this reason, we may impose that only the coefficientsassociated with even powers of momentum have nonzero values. Hence, for sim-plicity, we will build our PAs in order of p and not p , i.e. , d gl ( p ) → d [ L | M ] gl ( p ) = Q L ( p ) R M ( p ) = q + q p + q ( p ) + ... + q L ( p ) L r p + r ( p ) + ... + r M ( p ) M = q + q p + q p + ... + q L p L r p + r p + ... + r M p M . (3.3)The same happens to the propagator, D gl ( p ) → D [ L | M ] gl ( p ) . (3.4)By looking at the representation of the dressing function (Figure 3.1), we seethat it slowly goes to zero for high values of p . It does so as [ln p ] − / , and,thus, the right choice seems to be a relation that reproduces a similar behaviourat large momenta. Unfortunately, a ratio of polynomials cannot describe exactly alogarithmic function over a wide range of its arguments and, so, we have to lookfor the best approach.In Figure 3.3, the functions [ln p ] − / and (1 /p )[ln p ] − / are shown to-gether with some simple PAs. For relatively high values of momentum ( p ∼
10 GeV), the dressing function seems to tend to 0 between 1 and 1 /p . A similaranalysis for the propagator can be made. This time, for high values of momentum,the propagator goes to zero with (1 /p )[ln p ] − / , between 1 /p and 1 / ( p ) , as Here, the value 1 is an example of a constant value. In fact, for any constant c >
0, there isa value p min such that [ln p ] − / < c, ∀ p>p min . Furthermore, it is verified that ∃ p min > : c > [ln p ] − / > p , ∀ p>p min . n - ( p ) / p p ( / p ) ln - ( p ) / p /( p ) p Figure 3.3: Graphical representation of the asymptotic behaviour of D gl ( p ) (left), and d gl ( p ) (right), together with PAs of orders [ N | N ] (constant function 1), [ N − | N ](function 1 /p ), and [ N − | N ] (function 1 / ( p ) ). seen in Figure 3.3 .A criterion to determine the best sequence for both cases, by choosing thesuitable relation between L and M to use, will be discussed in Section 3.4. By definition, the PA of a function is built using its Taylor expansion, and so,it depends on the point around which it is made. Throughout the examples inChapter 2, we showed little concern for this matter, making all the expansionsaround the origin. However, we are now confronted with functions that are notdefined at the origin, e.g. , the logarithm. For this reason, we have to find theexpansion point that enables us to make the best approximation.Since we are interested in values of p between ∼ ∼
10 GeV, werequire a good precision in the reproduction of the original function in this range ofmomentum. In this sense, we choose the central point p = 5 . p . Figure 3.4 shows d gl ( p ), together withthe respective PA of order [1 | i.e. , p = 1 GeV and p = 10 GeV. The PA of order [1 | Formally, it is verified that ∃ p min > : 1 p > p [ln p ] − / > p ) , ∀ p>p min . gl ( p ) d gl [ | ] ( p ) , p = p [ GeV ] d gl ( p ) ; d gl [ | ] ( p ) d gl [ | ] ( p ) , p = p [ GeV ] (%) Figure 3.4: Representation of d gl ( p ), together with the respective PA of order [1 |
1] using,as expansion point, p = 5 . d gl ( p ) d gl [ | ] ( p ) , p = d gl [ | ] ( p ) , p =
10 GeV2 4 6 8 10 p [ GeV ] d gl ( p ) , d gl [ | ] ( p ) d gl [ | ] ( p ) , p = d gl [ | ] ( p ) , p =
10 GeV2 4 6 8 10 p [ GeV ] (%) Figure 3.5: Representation of d gl ( p ), together with the respective PAs of order [1 | p = 1 GeV and p = 10 GeV (left), and the approximationerror (right). these expansion points are shown in Figure 3.5.By comparing the obtained errors, we see that if we choose the expansion pointto be near an endpoint we gain precision around it, but loose it at the oppositeside. Furthermore, by analysing the error curves, we conclude that the error isat its lowest at p = p , reaching its maximum values at the endpoints. Thus, agood expansion point should be one that lowers the error on both endpoints. Forthis reason, we just need to analyse the error at the endpoints, since we know thatthere will be no higher values in between.Let us, now, examine how the errors at p = 1 GeV and p = 10 GeV evolvewhen we increase the order of approximation [ N | N ] in a diagonal sequence with p = 5 . N , the error at p = 1 GeV is the highest of the two, making it the maximum errorreached in the interval [1 ,
10] GeV. The decrease of the error for both values ofmomentum is more pronounced for lower orders of approximation, up to N ∼ = p = p = p = d gl [ N | N ] ( p = ) d gl [ N | N ] ( p =
10 GeV ) N - - - (%) Figure 3.6: Evolution of the approximation error at the endpoints p = 1 GeV and p = 10 GeV with N , using p = 5 . , , . From then on, the decrease in the maximum error is slower. Nonetheless, it isassured that the approximation should not present errors higher than ∼ .
02% fororders of approximation with N greater than 11.Nonetheless, we notice a discrepancy between the errors at p = 1 GeV and p = 10 GeV. We can lower the error at p = 1 GeV, without letting the error on theopposite side grow too much, by changing the expansion point. The evolution of theerrors at p = 1 GeV and p = 10 GeV, is represented in Figure 3.6, for four differentvalues of the expansion point, p = 5 . , , . p = 4 . N . Thus, during the next tests,PAs made around p = 4 . . Although this value may not be globally the best one - a deeper analysis could be made -, itdoes not have to be very precise. Variations up to 0 . p translate on minimal magnitudevariations of the errors, as Figure 3.6 suggests. q i r i . × − − . × − . × − . × − . × − . × − . × − (a) i q i r i . × − − . × − . × − . × − . × − . × − . × − − . × − (b)Table 3.1: Padé coefficients obtained for the orders of approximation [3 |
3] (a) and [3 | d gl ( p ). With the expansion point already chosen, we are finally able to calculate PAs.Let us go back and continue the discussion of Section 3.2, on the relations between L and M , and calculate, e.g. , the PAs of orders [3 |
3] (with J = 0) and [3 |
4] (with J = −
1) for the dressing function. The Padé coefficients are displayed, respectively,in Tables 3.1a and 3.1b.Notice the last coefficients of each polynomial in the tables mentioned above,from which a careful analysis grants us an important result. In Table 3.1a, the lastcoefficients, q and r , are both of the same order of magnitude. On the other hand,if we compare the last coefficients in Table 3.1b, we see that r is three orders ofmagnitude smaller than q . However, still in Table 3.1b, r is just one order ofmagnitude higher than q . In general, for any PA of order [ L | M ] of the dressingfunction, the following relations between the Padé coefficients can be verified : q L ∼ r M , L = Mq L (cid:28) r M , L > Mq L (cid:29) r M , L < M . (3.5)Regarding the coefficients for the propagator, we observe that, for the order[3 | Here, the last coefficients are understood as the coefficients of the terms of highest orders. The first relation in (3.5), and later in (3.6), are considered to be true for differences with amaximum of two orders of magnitude, i.e. , | log( q L /r M ) | (cid:46) q i r i . × − − . × − . × − . × − . × − . × − . × − − . × − (a) i q i r i . × − − . × − . × − . × − . × − . × − . × − − . × − − . × − (b)Table 3.2: Padé coefficients obtained for the orders of approximation [3 |
4] (a) and [3 | D gl ( p ). for the order [3 | q and r . Even if we had not discussed earlier, in Section 3.2, thatthe relation between L and M should be M = L − M = L −
2, we could arrive tothe same conclusion with a diagonal PA. A quick calculation for a PA of order [3 | q = 1 . × − , q = 3 . × − and r = 2 . × − . In thissense, similar relations to (3.5) can be established for the PAs of the propagator, q L ∼ r M , L = M − q L (cid:28) r M , L > M − q L (cid:29) r M , L < M − . (3.6)The relations (3.5) and (3.6) seem to indicate that, in a certain way, the PAis sensitive to the relation between L and M that better reproduces the originalfunction. For the dressing function, the PA “hints” that the most faithful approxi-mation is achieved for approximants where M = L , while for the propagator thePA “advises” us to use L = M −
1. This ability of the PA to tell us the properrelation between L and M , and to “correct it” if a different relation is considered,will be of great importance later on. We are now in a position to study the distributions of poles and zeros withinPA sequences, and to compare them with the expected analytic structures (Figure24 ○ - -
50 50 100 Re p [ GeV ]- - p [ GeV ] Poles ○ Zeros d gl [ ] ( p ) ○ ○ - -
50 50 100 Re p [ GeV ]- - p [ GeV ] d gl [ ] ( p ) ○ ○ ○ ○○○○○ - -
50 50 100 Re p [ GeV ]- - p [ GeV ] d gl [
11 11 ] ( p ) ○ ○ ○ ○○○○○ ○ - -
50 50 100 Re p [ GeV ]- - p [ GeV ] d gl [
12 12 ] ( p ) ○ ○ ○ ○○○○○○○○ ○○○ ○○○ - -
50 50 100 Re p [ GeV ]- - p [ GeV ] d gl [
20 20 ] ( p ) ○ ○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ - -
50 50 100 Re p [ GeV ]- - p [ GeV ] d gl [
50 50 ] ( p ) Figure 3.7: Distribution of poles and zeros for some values of N within the diagonal PAsequence of order [ N | N ], for the perturbative gluon dressing function. p -plane, for some values of N within the diagonal PA sequence of order [ N | N ],for the dressing function.Form N = 1 to N = 11 we see an accumulation of alternating poles and zeros onthe real negative p -axis. Following the conclusions of Section 2.4, this representsthe original branch cut, associated with a branch point at p = 0. Beginning at N = 12, poles and zeros start to emerge in the rest of the complex plane, mostly onits right side. These new poles and zeros come in pairs, i.e. , in the same position,25 ○ - -
50 50 100 Re p [ GeV ]- - p [ GeV ] Poles ○ Zeros D gl [ ] ( p ) ○ ○ - -
50 50 100 Re p [ GeV ]- - p [ GeV ] D gl [ ] ( p ) ○ ○ ○ ○○○○○ ○ - -
50 50 100 Re p [ GeV ]- - p [ GeV ] D gl [
11 12 ] ( p ) ○ ○ ○ ○○○○○ ○○ - -
50 50 100 Re p [ GeV ]- - p [ GeV ] D gl [
12 13 ] ( p ) ○ ○ ○○○○○○○ ○○ ○○○○○○ - -
50 50 100 Re p [ GeV ]- - p [ GeV ] D gl [
20 21 ] ( p ) ○ ○ ○○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○ - -
50 50 100 Re p [ GeV ]- - p [ GeV ] D gl [
50 51 ] ( p ) Figure 3.8: Distribution of poles and zeros for some values of N within the near-diagonalPA sequence of order [ N − | N ], for the perturbative gluon propagator. and, therefore, the zeros cancel the pole’s contributions to the total function.An analogous behaviour is seen for the near-diagonal PA sequence of order[ N − | N ] for the propagator, in Figure 3.8. However, this time we face anadditional problem: the pole from the factor 1 /p is in the same position as thebranch point, so we cannot distinguish them by simply representing the distributionof poles and zeros.In the next section we will try to solve this problem, by carrying out a residueanalysis. 26 .6 Residue analysis Consider, for example, the distribution of poles and zeros of the PA of order[50 | | | A k | for each pole k . The existence of two distinct levels is clear. The first one, inred tones, with residue values above 10 − , corresponds to the poles lying on thereal negative p -axis. The second one, in green/blue tones, corresponds to thepoles belonging to the pole-zero pairings. We can clear these last poles out of therepresentation by performing a cut in the residues at | A k | = 10 − , i.e. , by onlyrepresenting the poles k for which | A k | (cid:38) − .For the PA of order [50 | | A k | = 10 − is represented in Figure 3.10. We see that only the poles on the realnegative p -axis remain after the cut. In this way, the branch cut of the dressingfunction is faithfully reproduced by the poles that remain after performing the cutsin the residues.Let us, now, compare the distribution of poles and zeros for the PA of order[50 |
50] for the dressing function with the distribution of poles and zeros for thePA of order [50 |
51] of the propagator, represented in Figure 3.11, already with thecut in the residues. With an attentive look, we observe that, while for the dressingfunction the absolute value of the residues decreases as the poles get nearer thebranch point, the opposite happens to the propagator, thus suggesting the presenceof a pole at the origin, as expected. Recalling the definition of residue of a complex function f ( z ) at z = z , it is defined as thecoefficient of ( z − z ) − in the respective Laurent expansion [13]. Throughout this work, the residues were numerically computed using the software
Mathe-matica [38]. ○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ - -
50 50 100 Re p [ GeV ]- - p [ GeV ] Poles ○ Zeros d gl [
50 50 ] ( p )
10 20 30 40 50 k - - - A k Figure 3.9: Distribution of poles and zeros for the PA of order [50 |
50] of d gl ( p ) (left),and the absolute value of the residues | A k | for each pole k (right). The values of | A k | are arranged in descending order. The colour code used in the left graphic correspondsto the one used in the right graphic, for the residues’ absolute value. ○ ○ ○ ○○○○○○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○ ○○ ○○○○○○○○ - - - - -
20 20 40 Re p [ GeV ]- - p [ GeV ] Poles ○ Zeros d gl [
50 50 ] p cut at 10 - log | A k | - - - - Figure 3.10: Distribution of poles and zeros for the PA of order [50 |
50] of d gl ( p ), with acut in the residues at | A k | = 10 − . The colour scheme codes the residue’s absolute valueof each pole. ○ ○ ○○○○○○○○○○○○○○ ○○○○○○○○ ○○○○○○○○○○○○○○○○○ ○○○○○ - - - - -
20 20 40 Re p [ GeV ]- - p [ GeV ] Poles ○ Zeros D gl [
50 51 ] p cut at 10 - log | A k | - - - - Figure 3.11: Distribution of poles and zeros for the PA of order [50 |
51] of D gl ( p ), with acut in the residues at | A k | = 10 − . The colour scheme codes the residue’s absolute valueof each pole. In order to investigate the reproduction of poles and branch cuts in otherlocations of the complex p -plane, we can change their positions by adding massterms, m and m , to D gl ( p ) and d gl ( p ), D gl ( p ) = 1 p + m " N f α s π ln p + m Λ ! + 1 − γ , (3.7) d gl ( p ) = " N f α s π ln p + m Λ ! + 1 − γ . (3.8)Thereby, the branch points are moved from the origin to p = − m , both in D gl ( p )and d gl ( p ), and the pole of D gl ( p ) will now appear at p = − m .As an example, the mass term m in d gl ( p ) is set to four different values: − − i
10 and i
10 GeV . These should cause a translation of the branch point inthe complex p -plane to p = 5 , − , i
10 and − i
10 GeV , respectively. However,these changes should not alter the direction of the branch cut, which is, accordingto Figure 3.2, parallel to the real p -axis and goes from the branch point to the leftside of the plane. In Figure 3.12, the poles and zeros distributions obtained for the29 ○ ○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ - - - - -
20 20 40 Re p [ GeV ]- - p [ GeV ] Poles ○ Zeros d gl [
50 50 ] ( p ) m =- ○ ○ ○ ○○○○○○○○○○○○○○○○○○○○○○○ ○○○○ ○○○○○○○○○○○○○○○○ - - - - -
20 20 40 Re p [ GeV ]- - p [ GeV ] d gl [
50 50 ] ( p ) m = ○ ○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ XX - - - - -
20 20 40 Re p [ GeV ]- - p [ GeV ] d gl [
50 50 ] ( p ) m =- i10 GeV ○ ○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ XX - - - - -
20 20 40 Re p [ GeV ]- - p [ GeV ] d gl [
50 50 ] ( p ) m = i10 GeV log | A k | - - - - Figure 3.12: Distribution of poles and zeros for the elements of the PA sequences of order[ N | N ] with N = 50 of d gl ( p ), obtained for m = − , , − i
10 and i
10 GeV . The cut inthe residues has been done at | A k | = 10 − , by following the residue analysis of Section3.6. The colour scheme codes the residue’s absolute value of each pole. approximants in the PA sequences of order [ N | N ] with N = 50 are representedfor the four values of m .In Figure 3.12, we see that, for m = − , the branch cut andthe branch point are in the expected positions and, thus, correctly identified.Notwithstanding, for m = − i
10 and i
10 GeV , the branch cut is not so wellreproduced . Despite the correct identification of the branch point, the identifiedbranch cut is not parallel to the real p -axis, as anticipated. In fact, the grey dashedlines (Figure 3.12) reveal that the branch cut has the direction that connects the Despite the fact that only one element from each PA sequence is presented here, the positionsof the important poles in the distribution of poles and zeros, according to the residue analysis,are very stable throughout the sequences, and, therefore, they are not shown. This is made based on the convention that the branch cut from the logarithm makes an angleof π with the real axis in the complex plane. ○ ○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○○○○ - - - - -
20 20 40 Re p [ GeV ]- - p [ GeV ] Poles ○ Zeros D gl [
49 50 ] ( p ) m = m =- ○ ○ ○ ○ ○○○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○ - - - - -
20 20 40 Re p [ GeV ]- - p [ GeV ] D gl [
49 50 ] ( p ) m = m = ○ ○ ○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ - - - - -
20 20 40 Re p [ GeV ]- - p [ GeV ] D gl [
49 50 ] ( p ) m = m =- - i
10 GeV ○ ○ ○ ○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ - - - - -
20 20 40 Re p [ GeV ]- - p [ GeV ] D gl [
49 50 ] ( p ) m = m = + i
10 GeV log | A k | - - - - - - Figure 3.13: Distribution of poles and zeros for the elements of the PA sequences of order[ N − | N ] of D gl ( p ) with N = 50, obtained for m = − , , − − i
10 and 5 + 10 GeV and m = 0. The cut in the residues has been done at | A k | = 10 − , by following theresidues analysis of Section 3.6. The colour scheme codes the residue’s absolute value ofeach pole. branch point to the expansion point .Lastly, we need to examine the pole’s and branch cut’s behaviour for thepropagator when the mass term m is introduced. In Figure 3.13, the distributionof poles and zeros obtained from the approximants within the PA sequences oforder [ N − | N ] with N = 50 are represented for four different values of m : − , , − − i
10 and 5 + 10 GeV .A quick analysis of Figure 3.13 shows that, for the four different values of m ,the pole and the branch point are in the expected positions. Thus, the position of The same result can be obtained by moving the expansion point through the complex p -plane. Again, only one element from each PA sequence is presented here, due to the stability of theimportant poles and structures that emerge in the poles and zeros distributions throughout thesequences.
Regarding the ghost, the expression for its propagator and for its dressingfunction, obtained perturbatively, are very similar to the ones of the gluon. Indeed,the only difference between them is the anomalous dimension value, which is δ = 9 /
44 for the ghost.The tests made throughout the sections above were repeated for the ghostpropagator and identical results were obtained. For this reason, the conclusionsdrawn in this chapter are valid for both particles.32 hapter 4Approximating a discrete set ofpoints
The tests performed in the last chapter, move us a step closer in our journeyto find the analytic structure of the gluon and ghost non-perturbative propagators.To do so, we have to approximate the lattice data to PAs. This is exactly whenwe reach a cul-de-sac . Until now, the tests were all done with full knowledge ofthe original function’s analytic expression. Now, we are confronted with a discreteset of data points, with associated statistical errors.In this chapter, we introduce a methodology to approximate a discrete set ofdata points to a PA. The method will be tested against data sets generated froma collection of simple functions, before applying it to the lattice data for the gluonand ghost propagators.
The problem of building a PA from a discrete set of points, and not based onthe Taylor expansion of a function, is not new. In fact, several methods used toapproximate a set of data points by a rational function were already explored andemployed in Physics, in situations of numerical analytic continuation. The
NormMethod and the
Moment Method - both require solving a system of equations -, andthe
Point Method (SPM) - in which the coefficients can be determined recursively-, all introduced by Schlessinger [29], are examples of the usage of PAs, e.g. , in [5,337, 34]. Some of these methods were tested in the context of this work, but withvery poor results concerning the quality of approximation. Moreover, none of thesemethods take into account the statistical errors of the data. For these reasons, adifferent approach must be considered.Essentially, we want to reproduce a function f ( x ) based on a number K of datapoints { ( x , y ) , ..., ( x K , y K ) } , with a statistical error σ i associated with each value y i . To do so, these data points will be approximated by a ratio of polynomials, i.e. ,a PA. The Padé coefficients can be computed by minimising an objective functionthat measures the deviation of the data points to the approximated function.Throughout the current work, we will use the chi squared χ as the objectivefunction, which is defined as χ ≡ K X i =1 y i − f ( x i ) σ i ! , (4.1)where σ i is the statistical error associated with the value y i . The quality ofapproximation can be measured from the reduced chi squared, given by e χ ≡ χ degrees of freedom , (4.2)where the number of degrees of freedom is given by the difference between thenumber K of points to be approximated and the number of Padé coefficients tobe calculated. A good approximation to the data translates into a reduced chisquared close to unit, i.e. , e χ ∼ χ . Global optimisationproblems are non-trivial, and there is no available method that solves such class ofproblems for any function. Herein, the minimisation was done numerically using Mathematica [38]. The minimisation methods used in this work for this globaloptimisation problem were the
Differential Evolution (DE) method [23], and the
Simulated Annealing (SA) method [7] . Both are stochastic optimisation methods.The first one, the DE, relies on the maintenance of a population of points, fromwhich a new one is generated based on random processes. This population evolves The
Levenberg Marquardt method [37] was also used, but with very unstable results, comparedwith the other two methods.
34o explore the search space, in order to escape possible local minima. The conver-gence is achieved when the best points of two consecutive populations are inside thechosen tolerance. As for the SA method, it is inspired by the physical/metallurgicprocess of annealing. At each iteration, a new point in the research space is ran-domly generated close to the previous one, replacing it if a lower value is reached.However, the algorithm allows it to be exchanged for a point with a higher value,with a probability that follows a Boltzmann distribution, giving the possibility ofescaping local minima. As the number of iterations increases, the probability ofa replacement for a point with a higher value decreases, simulating a decrease intemperature. The process ends when the maximum number of iterations is reached,or the method converges to a point within the chosen tolerance.In order to test the reliability of this methodology, we first apply it to test datasets, generated from given functions.
The functions that will be used to generate the test data sets are: f ( p ) = ln (cid:16) p + m (cid:17) , (4.3) f ( p ) = 1 p + m , (4.4) f ( p ) = D gl ( p + m ) , (4.5)where D gl ( p ) is the perturbative gluon propagator given by (3.1) and, in all cases, p is dimensionless. For the mass terms, the cases with m = 0 and m = 0 . p = − m , for f ( p );• a branch cut parallel to the real axis with the branch point at p = − m , for f ( p );• a pole at p = − m and a branch cut parallel to the real axis with the branchpoint also at p = − m , for f ( p ).35or each function f i ( p ), a set of K points ( p j , f i ( p j )), with j = 1 , ..., K , isgenerated by randomly choosing K values of p j in the chosen interval and calcu-lating f i ( p j ) for each one. The statistical errors can be simulated by replacingthe value of f i ( p j ), at each p j , by a random value near the original one, i.e. , f i ( p j ) → y j = f i ( p j )(1 + ερ ), where ρ is a random number that follows a Gaussiandistribution centred in 0 and with a standard deviation of 1, and ε is the desiredpercent error. Hence, the statistical error associated with y j is σ j = εf i ( p j ). Whenit comes to the minimisation, each point ( p j , y j ) will contribute more or less to itbased on the associated weight, given by 1 /σ j .The lattice data for the gluon and the ghost propagators that will be used has,in all cases, more than a hundred data points in the range p ∈ [0 ,
8] GeV, withstatistical errors between ∼
1% and ∼ . p ∈ [0 , ε = 1%, were generated.Following the conclusions of Chapter 3, near-diagonal PA sequences of order[ N − | N ], with N going from 1 to 20, will be calculated, and the PAs themselveswill take be p as the independent variable. In Figure 4.1, the achieved values of e χ for f ( p ), using m = 0 and m = 0 . f ( p ), the respective generated data, and some PAsobtained within the calculated sequence for, for example, m = 0 . .A summary of the results can be made by representing simultaneously, in thecomplex p -plane, the poles obtained for all values of N . In Figure 4.3, this all-polesrepresentation is made for f ( p ), using m = 0 and m = 0 .
5, and for both methodsof minimisation, DE and SA. A circumference of unit radius, formed by poles withresidues between ∼ − , in purple, and ∼
1, in yellow/orange, is seen around the In general, a very good adjustment can be observed in graphical representations beginningat low orders of approximation, usually N ∼
3. This type of representation will not be presentedfor more cases, to avoid overloading this work with unnecessary figures. The approximations’quality will be evaluated only through the analysis of e χ . ESA0 5 10 15 20 N χ ˜ m = N χ ˜ m = Figure 4.1: Achieved values of e χ for the test data generated from f ( p ), using m = 0and m = 0 .
5, and for both methods of minimisation, DE and SA. f p data f p f [ | ] p f [ | ] p p f p ; f [ L | M ] p Figure 4.2: Representation of the test data points generated from the f ( p ), which isalso represented, using m = 0 . |
4] and [17 | - - - - p - - p DE m = - - - - - p - - p SA m = - - - - - p - - p DE m = - - - - - p - - p SA m = | A k |- - - - Figure 4.3: All-poles representation obtained for the test data generated from f ( p ),using m = 0 and m = 0 .
5, and for both methods of minimisation, DE and SA. Thecolour scheme codes the residue’s absolute value of each pole. origin. In fact, this type of structures are an artefact of the approximation method,created by poles of the Froissart doublets, already studied in [39]. The fact thatthe poles on the right-hand side of the complex p -plane have, in general, smallerresidues, and are much less scattered than the ones on the left-hand side, reflectsthe difficulty on identifying the analytic structure beyond the region where thedata is defined. We also see an accumulation of poles with high residue on the realnegative p -axis, which suggests the presence of a branch cut. However, we cannotbe deceived by this all-poles representation, as it only serves as an overview of thestructures that emerge in the PA sequence. An analysis of the evolution of thepoles and zeros distributions with N is required.The evolution, with N , of the off-axis poles, i.e. , poles that appear at Re( p ) = 0,is represented in Figure 4.4, for f ( p ) with m = 0 and m = 0 . , in orange/red tones, are not stable throughout thePA sequence for any case, since they are mixed with the poles from the Froissartdoublets in the circumference of unit radius. Following the conclusions of Chapter2, these can be considered spurious poles.The evolution of the on-axis poles and zeros, i.e. , poles and zeros that appearon the real p -axis, is shown in Figure 4.5, for f ( p ) with m = 0 and m = 0 . p -axis, in the sameway as the branch cuts were reproduced in Chapters 2 and 3. However, whereasfor the case with m = 0 the branch point is well identified, the alternating polesand zeros begin at the origin, for m = 0 . p ∼ − . In Figure 4.6, the e χ achieved in the minimisation for f ( p ), using m = 0 and m = 0 .
5, is represented for both minimisation methods. Again, the values of e χ show that the PAs provide a good approximation to the data and the function.For f ( p ), the same artefact that was present for f ( p ) (the circumference ofpoles around the origin) also appears in the all-poles representation, in Figure 4.7,for each value of m , and for both minimisation methods.The evolution of the off-axis poles, in Figure 4.8, shows that no important polesappear at complex p .The expected pole at p = − m can be seen in the evolution of the on-axispoles and zeros, in Figure 4.9. For m = 0, both methods successfully identify thepole at the origin. Indeed, a very stable pole is seen at p = 0 throughout thewhole PA sequence. On the contrary, for m = 0 .
5, the pole at p = − . When PAs are used to approximate a discrete set of data points with associated statisticalerrors, a residue analysis is impracticable, as it was presented in Section 3.6, since the levels inthe absolute values of the residues become indistinguishable. Thus, from now on, the residueanalysis is done based on the relative value of the poles’ residues, i.e. , poles with higher residuesare considered to be more relevant. p N DE m =
05 10 15 20 N p - p N SA m =
05 10 15 20 N p - p N DE m = N p - p N SA m = N p log | A k |- - - - Figure 4.4: Evolution of the off-axis poles and zeros with N in the PA sequence, obtainedfrom the test data generated from f ( p ), using m = 0 and m = 0 .
5, and for bothmethods of minimisation, DE and SA. The colour scheme codes the residue’s absolutevalue of each pole. ○○○○○○○ ○○○○ ○○○ ○○○○ ○○○ ○○○○ ○○○○○○○ ○○○ ○○○○ ○○○ ○○○○ ○○○ ○○○○ ○○○○○○○○○ ○○○○ - - -
20 0 20 40 60 Re p N DE m = Poles ○ Zeros ○ ○○ ○○ ○○ ○ ○○ ○○ ○ ○○ ○○ ○ ○○ ○○ ○ ○○ ○○ ○ ○○ ○○ ○ ○○ ○○ ○ ○○ ○ ○ ○○ ○ ○○ ○ ○ ○ - - p N ○○○○○○○○ ○○○○○○○ ○○○○○○○○○ ○○ ○○○○○ ○○○○ ○○○○○○○○○ ○○○ ○○○○ ○○○○○ ○○○○ ○○○○○ ○○○○ ○○○○ - - -
20 0 20 40 60 Re p N SA m = ○ ○○ ○○ ○○ ○ ○○ ○○ ○○ ○ ○ ○○○ ○ ○ ○○ ○○○ ○ ○○○ ○ ○○ ○○ ○ ○○ ○ ○ ○○ ○ ○○ ○ ○ ○○ ○ ○○ ○ ○ ○ - - p N ○○○○○ ○○○ ○○ ○○○ ○○○○ ○○○ ○○○○ ○○○○○○○ ○○○○○○○○○ ○○○○○○○ ○○○ ○○○○ ○○○ ○○○○○○ ○○○○ - - -
20 0 20 40 60 Re p N DE m = ○ ○○○○○ ○○ ○ ○○ ○○ ○ ○○ ○○ ○ ○○ ○ ○ ○○ ○ ○○ ○○ ○ ○○ ○○ ○ ○○ ○○ ○ ○ ○ ○○ ○ ○ ○ - - p N ○○○○○ ○○○ ○○○○○○○○○○○○ ○○○○○○○ ○○○○○○○○○○○○○○ ○○○ ○○○○○○○○○○○○○ ○○○○ - - -
20 0 20 40 60 Re p N SA m = ○ ○○○ ○○ ○ ○○ ○○ ○ ○○○ ○ ○○ ○○○○ ○ ○ ○○ ○ ○○ ○ ○○ ○○ ○ ○○ ○ ○ ○○ ○ ○○ ○ ○ ○ - - p N log | A k |- - - - Figure 4.5: Evolution of the on-axis poles and zeros with N in the PA sequence, obtainedfrom the test data generated from f ( p ), using m = 0 and m = 0 .
5, and for bothmethods of minimisation, DE and SA. The colour scheme codes the residue’s absolutevalue of each pole. ESA0 5 10 15 20 N χ ˜ m = N χ ˜ m = Figure 4.6: Achieved values of e χ for the test data generated from f ( p ), using m = 0and m = 0 .
5, and for both methods of minimisation, DE and SA. - - - - - p - - p DE m = - - - - - p - - p SA m = - - - - - p - - p DE m = - - - - - p - - p SA m = | A k |- - - - Figure 4.7: All-poles representation obtained for the test data generated from f ( p ),using m = 0 and m = 0 .
5, and for both methods of minimisation, DE and SA. Thecolour scheme codes the residue’s absolute value of each pole. p N DE m =
05 10 15 20 N p - p N SA m =
05 10 15 20 N p - p N DE m = N p - p N SA m = N p log | A k |- - - - Figure 4.8: Evolution of the off-axis poles and zeros with N in the PA sequence, obtainedfrom the test data generated from f ( p ), using m = 0 and m = 0 .
5, and for bothmethods of minimisation, DE and SA. The colour scheme codes the residue’s absolutevalue of each pole. N ∈ [3 , N = 9, with some minor deviations. For higher orders of approximation weobserve that the poles obtained with the SA method deviate toward the origin,followed by a decrease in the respective residue. Finally, for the perturbative gluon propagator f ( p ), the generated data was,once more, well adjusted by the PAs within the sequence, as can be seen by thevalues of e χ in Figure 4.10.A quick look to the all-pole representation for f ( p ), in Figure 4.11, showsthat, for the third time, the circumference of poles around the origin is present forall cases, reinforcing the fact that it is an artefact of the approximation. As for f ( p ) and f ( p ), there are no relevant poles in the complex p -plane apart fromthe ones on the real negative p -axis, as can be observed in the representation ofthe off-axis poles, in Figure 4.12.With respect to the expected pole at p = − m , for the case where m = 0, it iswell identified within the PA sequence by a stable pole for both DE and SA, clearlyseen in the on-axis poles and zeros representation, in Figure 4.13. Additionally,the pole at the origin is also identified by poles that appear near the origin in theoff-axis representation obtained for the DE method (Figure 4.12). The case where m = 0 . p = − . p = − . p = − .
3, its exact position cannot beread with high precision. By comparing the two minimisation methods, we noticethat this identification is more difficult for the SA than for the DE, where there aresome intervals of N ( N ∈ [2 ,
5] and [15 , f ( p ).Nonetheless, alternating poles and zeros can be seen in the interval p ∈ [ − , f ( p ), unimportant poles appear44 ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ - - -
20 0 20 40 60 Re p N DE m = Poles ○ Zeros ○ ○ ○○○ ○ ○○ ○○○○○ ○○○ ○○○ ○ ○○ ○ ○○○○ ○ - - p N ○○○○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○ - - -
20 0 20 40 60 Re p N SA m = ○○ ○○○○ ○○○○ ○○ ○○○ ○○○ ○○○ ○○ ○○○○○ - - p N ○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ - - -
20 0 20 40 60 Re p N DE m = ○ ○ ○○○ ○ ○ ○○ ○○○ ○ ○○○ ○○○ ○ ○○○ ○ ○○ ○○○ ○ ○○ ○ ○○ ○○○ ○○ - - p N ○○○○○○ ○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ - - -
20 0 20 40 60 Re p N SA m = ○ ○○ ○○ ○○ ○○○ ○○ ○○○ ○○ ○○ ○ ○○ ○ ○○ ○○ ○ ○○ ○○ ○ ○○ ○○ ○ ○ - - p N log | A k |- - - - Figure 4.9: Evolution of the on-axis poles and zeros with N in the PA sequence, obtainedfrom the test data generated from f ( p ), using m = 0 and m = 0 .
5, and for bothmethods of minimisation, DE and SA. The colour scheme codes the residue’s absolutevalue of each pole. ESA0 5 10 15 20 N χ ˜ m = N χ ˜ m = Figure 4.10: Achieved values of e χ for the test data generated from f ( p ), using m = 0and m = 0 .
5, and for both methods of minimisation, DE and SA. - - - - - p - - p DE m = - - - - - p - - p SA m = - - - - - p - - p DE m = - - - - - p - - p SA m = | A k |- - - - Figure 4.11: All-poles representation obtained for the test data generated from f ( p ),using m = 0 and m = 0 .
5, and for both methods of minimisation, DE and SA. Thecolour scheme codes the residue’s absolute value of each pole. p N DE m =
05 10 15 20 N p - p N SA m =
05 10 15 20 N p - p N DE m = N p - p N SA m = N p log | A k |- - - - Figure 4.12: Evolution of the off-axis poles and zeros with N in the PA sequence, obtainedfrom the test data generated from f ( p ), using m = 0 and m = 0 .
5, and for bothmethods of minimisation, DE and SA. The colour scheme codes the residue’s absolutevalue of each pole. ○○ ○○ ○○○ ○○○○○○ ○○○○ ○○○○○○○ ○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○ - - -
20 0 20 40 60 Re p N DE m = Poles ○ Zeros ○○ ○○ ○○ ○○ ○○ ○ ○○○ ○ ○○○ ○○○ ○○○ ○ ○○○ ○○○ ○ ○○○○ ○ ○ ○○○ ○ ○○ ○ ○ - - p N ○○○ ○○ ○○○ ○○○ ○○○○○ ○○○○ ○○ ○○○○○○○○○○○○ ○○ ○○○ ○○○○○○○○○○○○○○○○○○○○○○○ - - -
20 0 20 40 60 Re p N SA m = ○○ ○○ ○○ ○○ ○ ○○○ ○○○○ ○○ ○ ○○ ○ ○○○○ ○○ ○○ ○○ ○ ○○ ○○ ○ ○○ ○○ ○ ○ - - p N ○○○ ○○ ○○○○ ○○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○ - - -
20 0 20 40 60 Re p N DE m = ○ ○○ ○○○ ○○ ○○ ○○○ ○○ ○○○ ○ ○○ ○○○○ ○ ○ - - p N ○○○ ○○ ○○ ○○ ○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ - - -
20 0 20 40 60 Re p N SA m = ○ ○○ ○ ○○ ○ ○○○ ○ ○ ○ ○○ ○○ ○ ○○○ ○○○ ○ ○ - - p N log | A k |- - - - Figure 4.13: Evolution of the on-axis poles and zeros with N in the PA sequence, obtainedfrom the test data generated from f ( p ), using m = 0 and m = 0 .
5, and for bothmethods of minimisation, DE and SA. The colour scheme codes the residue’s absolutevalue of each pole. p = 0, but somewhere else in the real negative p -axis, asanticipated. Throughout Chapters 2 and 3, we performed numerical tests to investigate theanalytic structure of a function and studied the distributions of poles and zerosobtained form the PA sequences. In this way, we arrive to the following guide-linelist of how the analytic structure of a function is reproduced by a PA and how itcan be identified:• Poles in the complex plane are reproduced by stable poles in the distributionsof poles and zeros throughout a PA sequence. A pole at the origin is wellidentified by poles from the PAs that have relatively high residues. In thiscase, poles with residues of the same order of magnitude can appear close tothe origin in the off-axis representation, for the DE method. On the otherhand, a pole in a position other than the origin is reproduced by poles withhigher residues. However, their position is less stable, specially for higherorders of approximation. In this case, a better stability is usually obtainedwhen using the SA minimisation method.• The branch cuts are reproduced by alternating poles and zeros. These may beeasier to identify at lower orders of approximation. The exact position of thebranch point is very difficult to identify, particularly if it is not at the origin.However, if that is not the case, poles with a low residue in the distributionsof poles and zeros appear near the origin throughout the PA sequence whenthe DE method is used. This does not happen for the sequence obtainedwith the SA minimisation.• As N grows in the PA sequence, spurious poles, as well as Froissart doublets,tend to emerge and gather around the origin, forming what looks like acircumference of unit radius. A quick residue analysis, together with the49tudy on the poles’ stability, seems be enough to discard the undesired polesfrom the analytic structure. 50 hapter 5Results and discussion Now that we have a good idea of what seems to be a good guide to interpretthe evolution of the distribution of poles and zeros within a PA sequence, and howthe analytic structures of functions are reproduced by it, we can proceed and applythe PA analysis to investigate the gluon and ghost non-perturbative propagators.
The data for the non-perturbative ghost propagator investigated here was pub-lished in [9]. It was obtained via lattice simulation in the Landau gauge performedon an hypercubic spacetime lattice of volume 80 with 70 gauge configurations,using the Wilson gauge action for β = 6 . µ = 3 GeV, according to D ( µ ) (cid:12)(cid:12)(cid:12)(cid:12) µ =3 GeV = 1 µ . (5.1)The propagator data can be seen in Figure 5.1.Following the analysis of the previous two chapters, the lattice data for theghost propagator was adjusted by PAs, for various orders of approximation withina PA sequence, using the two minimisation methods: DE and SA. On a first stage,various relations between L and M were tried. Following the conclusions of Section3.4, the use of PAs of order [ N − | N ] showed to be the best choice . This is in accordance with the fact that, in the limit of high momenta, the full propagator .2 0.5 1 2 5 p [ GeV ] D gh p [ GeV - ] Figure 5.1: Landau gauge lattice ghost propagator used in the analysis. The statisticalerrors are smaller than the size of the points.
The values of e χ at the minima are represented in Figure 5.2, for N ∈ [1 , p -axis, near the origin, can also beobserved, suggesting the presence of a branch cut.An analysis of the on-axis poles and zeros, which is represented in Figure 5.4,shows a very stable pole with high residue at the origin, for both the DE and theSA methods. This is a clear indication of the presence of a pole at p = 0 in theanalytic structure of the ghost propagator. The remaining poles and zeros seem toindicate the presence of a branch cut on the real negative p -axis, with a branchpoint at p ∼ − . .On the other hand, by looking at the evolution of the off-axis poles, in Figure5.5, we see that no relevant stable poles appear throughout the PA sequence, in has the same behaviour as the one given by the perturbative analysis. We must not forget the numerical tests made in Chapter 4, where we saw that the branchpoint is hard to identify if it is not at the origin. Nonetheless, in this case, the stability of theon-axis poles may allow us to correctly identify the branch point’s position. ESA10 20 30 40 N χ ˜ Figure 5.2: Achieved values of e χ for the Landau gauge lattice ghost propagator, forboth methods of minimisation, DE and SA. - - - p [ GeV ]- - p [ GeV ] DE - - - p [ GeV ]- - p [ GeV ] SAlog | A k |- - - - Figure 5.3: All-poles representation obtained for the ghost propagator data, for bothmethods of minimisation, DE and SA. The colour scheme codes the residue’s absolutevalue of each pole. ○○ ○○○○○○ ○○○○○○○○ ○○ ○○○ ○○○○○○○○○○ ○○ ○○○ ○○ ○○○ ○○ ○○○ ○○ ○○○○○○○○○○ ○○○○○ ○○○ ○○ ○○○ ○○ ○○○ ○○ ○○○ ○○ ○○○ ○○ ○○ - - -
20 0 20 40 60 Re p [ GeV ] N DE Poles ○ Zeros ○○ ○○○ ○○○○○○ ○ ○○ ○○○ ○○○ ○○○ ○○○ ○○○ ○○○ ○○○ ○ ○○○ ○○○ ○○○ ○○○ ○○○ ○○○ ○○○ ○ - - p [ GeV ] N ○○○ ○○ ○○○○○○○○○○○○○ ○○○ ○○ ○○○ ○○○○○ ○○○○○ ○○ ○○○ ○○ ○○○ ○○○○ ○○○ ○○ ○○○ ○○○○○ ○○○○ ○○○ ○○ ○○○ ○○ ○○○ ○○○○ ○○○ ○○ ○○ ○○○ ○○○○ ○○○ ○○ ○ - - -
20 0 20 40 60 Re p [ GeV ] N SA ○○ ○○ ○○ ○○ ○○○○○ ○○○ ○○○ ○○○ ○○○ ○○○ ○○ ○ ○○ ○○○ ○○○ ○○ ○ ○○ ○○○ ○○○ ○○ ○ ○○ ○○○○ ○○ ○ ○○ ○○○ - - p [ GeV ] N log | A k |- - - - Figure 5.4: Evolution of the on-axis poles and zeros within the PA sequence obtained forthe ghost propagator data, for both methods of minimisation, DE and SA. The colourscheme codes the residue’s absolute value of each pole. - p [ GeV ] N DE10 20 30 40 N p [ GeV ] - - p [ GeV ] N SA10 20 30 40 N p [ GeV ] log | A k |- - - - Figure 5.5: Evolution of the off-axis poles within the PA sequence obtained for the ghostpropagator data, for both methods of minimisation, DE and SA. The colour schemecodes the residue’s absolute value of each pole. the remaining complex p -plane.These results are in agreement with [5, 16], where the analytic structure ofthe ghost propagator suggested to be similar to the perturbative result, with nocomplex singularities. The results also support the no-pole condition for the ghostpropagator, as proposed in [15]. For the gluon propagator, we use the results of lattice simulations in the Landaugauge using the Wilson gauge action for β = 6 .
0, renormalised in the MOM-scheme55 lattice80 lattice64 lattice32 lattice0.1 0.5 1 5 p [ GeV ] D gl p [ GeV - ] Figure 5.6: Gluon propagator used in the analysis, obtained with four different latticevolumes. at µ = 3 GeV, similarly to the ghost propagator in the last section. Several datasets are considered, which were obtained using: a 32 lattice, physical volume of(3 .
25 fm) , with 50 gauge configurations, from [3]; a 64 lattice, physical volume of(6 .
50 fm) , with 2000 gauge configurations, from [11]; a 80 lattice, physical volumeof (8 .
13 fm) , with 550 gauge configurations, from [11]; and a 128 lattice, physicalvolume of (13 .
01 fm) , with 35 gauge configurations, from [9]. The mentioneddata sets for the gluon propagator are represented in Figure 5.6 . All of them areessentially compatible with each other at one standard deviation level, and so theydefine a unique curve.As seen in Figure 5.6, the 128 lattice simulation has more information in theIR region of momentum, i.e. , p (cid:46) L and M were tried. According to the conclusion of Section 3.4, the best choice revealed tobe, once more, the use of PAs of order [ N − | N ].In Figure 5.7, the e χ at the minima are shown. The several obtained e χ are The zero momentum propagator is not considered. ESA N χ ˜ lattice64 lattice80 lattice128 lattice Figure 5.7: Achieved values of e χ for the Landau gauge lattice gluon propagator, andfor both methods of minimisation, DE and SA. essentially the same for both minimisation methods. We also observe that thequality of the approximation increases ( e χ decreases) with the increase of thelattice volume.In Figure 5.8, the all-poles representation is shown for the four lattice volumes,and for both minimisation methods. Apart from the usual artefact, a new structureemerges in the complex p -plane: a conjugate pair of complex poles of high residueat Re( p ) <
0. This structure becomes clearer and well defined for higher latticevolumes, which indicates that this pair of poles is associated with the IR structureof the theory. On the other hand, for smaller lattice volumes, a branch cutmay be identified on the real negative p -axis. Also in Figure 5.8, the slightsuggestion of the existence of another conjugate pair of complex poles may be seenat Re( p ) >
0. However, these poles have a lower residue and are not present forall simulations. Additionally, for some cases, the pair is not identified with bothminimisation methods. In this sense, further studies are needed to see if thesepoles are meaningful or artefacts of the method. Herein, we will not consider thepoles at Re( p ) > - - p [ GeV ]- - p [ GeV ] DE32 lattice - - - p [ GeV ]- - p [ GeV ] SA32 lattice - - - p [ GeV ]- - p [ GeV ] DE64 lattice - - - p [ GeV ]- - p [ GeV ] SA64 lattice - - - p [ GeV ]- - p [ GeV ] DE80 lattice - - - p [ GeV ]- - p [ GeV ] SA80 lattice - - - p [ GeV ]- - p [ GeV ] DE128 lattice - - - p [ GeV ]- - p [ GeV ] SA128 latticelog | A k |- - - - Figure 5.8: All-poles representation obtained for the gluon propagator data with 32 ,64 , 80 and 128 lattices, for both methods of minimisation, DE and SA. The colourscheme codes the residue’s absolute value of each pole. .2.1 Complex poles In Figures 5.9 and 5.10, the off-axis poles are represented for the four latticevolumes, and for both minimisation methods. A cut in the residues at | A k | = 1was already performed, and only the relevant poles appear. As seen already in theall-poles representations, the complex poles are more stable throughout the PAsequence for higher lattice volumes, particularly for lower values of N . For thisreason, the results for the simulation with the largest lattice volume, i.e. , the 128 lattice, are those appropriate to read out the position of the poles.In the last chapter, we saw that the position of a pole is identified with precisiononly for lower orders of approximation. In this case, despite the fact that the poleis reproduced throughout the whole PA sequence by poles with high residue, onlythe ones obtained for lower orders of N should be used to estimate the position ofthis singularity in the analytic structure of the propagator. In fact, by looking atthe off-axis poles for the DE method, in Figure 5.10, we observe that the poles aremore stable for N ∈ [2 ,
8] than for the remaining orders. As for the SA method,although in the representation of Re( p ) the pole is very stable in one position for N ∈ [2 , N ∈ [8 , p )shows otherwise: for N ∈ [8 , N ∼ N ∈ [2 ,
8] for the DE method, and the ones obtainedwith N ∈ [2 ,
7] for the SA method, are used to estimate the position of thesesingularities in the analytic structure . An arithmetic average of the respectivepoles’ positions gives the following results for the position of the poles present inthe analytic structure of the gluon propagator, for both minimisation methods:DE: p = − . ± i . ;SA: p = − . ± i . . In Figure 5.11, the above results are represented, together with the followingresults, obtained in previous studies. In [11], the tree level prediction of the RGZ Although the remaining poles are not used, their appearance is important to confirm thepresence of these poles in the analytic structure of the propagator. p [ GeV ] N DE32 latticecut at 15 10 15 20 25 30 N p [ GeV ] - p [ GeV ] N SA32 latticecut at 15 10 15 20 25 30 N p [ GeV ]- p [ GeV ] N DE64 latticecut at 110 20 30 40 N p [ GeV ] - p [ GeV ] N SA64 latticecut at 110 20 30 40 N p [ GeV ] log | A k |- - - - Figure 5.9: Evolution of the off-axis poles within the PA sequence, obtained for the gluonpropagator data with 32 and 64 lattices, for both methods of minimisation, DE andSA. A cut in the residues at log | A k | = 0 has been performed. The colour scheme codesthe residue’s absolute value of each pole. p [ GeV ] N DE80 latticecut at 110 20 30 40 N p [ GeV ] - p [ GeV ] N SA80 latticecut at 110 20 30 40 N p [ GeV ]- p [ GeV ] N DE128 latticecut at 110 20 30 40 N p [ GeV ] - p [ GeV ] N SA128 latticecut at 110 20 30 40 N p [ GeV ] log | A k |- - - - Figure 5.10: Evolution of the off-axis poles within the PA sequence, obtained for thegluon propagator data with 80 and 128 lattices, for both methods of minimisation, DEand SA. A cut in the residues at log | A k | = 0 has been performed. The colour schemecodes the residue’s absolute value of each pole. p ∼ p ) ∈ [ − . , − .
20] GeV and Im( p ) ∈ ± [0 . , .
59] GeV . In [5], afixed order PA, computed with the Schlessinger Point Method (SPM) mentioned inSection 4.1, found the singularity at: p = − . ± i . , for the same64 lattice data here; and at p = − . ± i . for the decouplingsolution of the DSE.By comparing the above results (see Figure 5.11), we can conclude that thereis a discrepancy between the results obtained via DSE and the ones obtainedusing lattice data (DE, SA, SPM - lattice, and RGZ), even though the latter wereobtained with different approaches to reproduce the lattice data. Indeed, the polefound with the DSE is at smaller absolute value of Re( p ) and Im( p ), and, thus,closer to the origin.Regarding the results based on the lattice data (DE, SA, SPM - lattice, andRGZ), we see that the identified pole is at a slightly smaller Re( p ), but at a similarIm( p ), when compered to the result from the DSE. Nonetheless, the good overallcompatibility in the position of the conjugate pair of complex poles is reassuring.Furthermore, they match with the analysis inspired by the Gribov-Zwanziger typeof actions [11], for which the solution for the gluon propagator is itself a ratio ofpolynomials and, thus, a type of PA. The evolution of the on-axis poles and zeros within the PA sequence is shownin Figures 5.12 and 5.13, for both minimisation methods, DE and SA, applied tothe four simulation data sets. In contrast to the complex poles identified in thelast subsection, the presence of a branch cut on the real negative p -axis is moreevident for the smaller lattices, where the alternating poles and zeros are morenoticeable.Regarding the branch point, we have concluded, in Chapter 4, that its positionis difficult to identify using the present methodology. Notwithstanding, we caninfer, from Figures 5.12 and 5.13, that the branch point is not at the origin, butsomewhere between p = 0 and p = − . . A possible way to better identifythe branch cut and the branch point might be the use of a much larger ensemble62 - - - - - p [ GeV ] I m p [ G e V ] RGZ
SPM - lattice DE SA
SPM - DSE
Figure 5.11: Position of the poles present in the analytic structure of the gluon prop-agator obtained in the present work - DE and SA -, and in other studies (see text).The confidence region for the positions is defined as an ellipse, for which the semiaxescorrespond to the errors associated with each result. Only one of the complex poles inthe conjugate pair is shown. ○○○○○○○ ○○○ ○○ ○○○ ○○ ○○○○ ○○○○○ ○○○○ ○○○○○ ○○○○ ○○○ ○○○○ ○○○ ○○○○○○○○○○○○○ ○○○○○○○○○○ ○○○ ○○○○ ○○○○○ ○○○○ ○○○ ○○ ○○ - - -
20 0 20 40 60 Re p [ GeV ] N DE32 lattice Poles ○ Zeros ○ ○○○ ○ ○○ ○ ○ ○○ ○ ○ ○○ ○○ ○ ○○○ ○ ○○ ○○ ○ ○○ ○○ ○ ○○ ○ ○○ ○ ○ ○○ ○ ○ ○○ ○ ○ ○○ ○○ ○ ○○ ○ ○○○ ○ ○○ ○○○ ○ - - p [ GeV ] N ○○○○○○○ ○○○○○○○○ ○ ○○○○○○ ○○○○ ○○○○○○ ○○○○○○○○○○○ ○○○○ ○○○○○○○○ ○○○ ○○ ○○○ ○○○○ - - -
20 0 20 40 60 Re p [ GeV ] N SA32 lattice ○○○ ○○○ ○ ○○○○ ○ ○○○○○○ ○ ○ ○○ ○○ ○ ○ ○○ ○○ ○ ○ ○ ○○ ○○○ ○○○ ○○ ○○ - - p [ GeV ] N ○○○○○○ ○○ ○○○ ○○ ○○○ ○○ ○○○ ○○○ ○○ ○ ○○ ○ ○○○ ○○ ○○○ ○○○○○ ○○○ ○○ ○○○ ○○○○ ○○○ ○○○○ ○○○ ○○○ ○○ ○○○ ○○ ○○○ ○○ ○○○ ○○ - - -
20 0 20 40 60 Re p [ GeV ] N DE64 lattice ○ ○○○ ○○ ○○○ ○○○ ○○○ ○ ○○ ○ ○○ ○○ ○○○ ○○ ○○ ○ ○○ ○○○○ ○○○ ○○○ ○○ - - p [ GeV ] N ○○○○○○ ○○○ ○○○ ○○○ ○○ ○○○ ○○○ ○○○ ○○ ○○○ ○○○ ○○○ ○○ ○○○ ○○ ○○○ ○○○ ○○○○ ○○○ ○○ ○○○ ○○ ○○○ ○○ ○○○ ○○○ ○○ ○○ - - -
20 0 20 40 60 Re p [ GeV ] N SA64 lattice ○○ ○○○○○ ○○ ○ ○○ ○○ ○○○ ○○○○ ○○ ○○○ ○○○ ○○○ ○○ ○○ ○ - - p [ GeV ] N log | A k |- - - - Figure 5.12: Evolution of the on-axis poles and zeros within the PA sequence, obtainedfor the gluon propagator data with 32 and 64 lattices, for both methods of minimisation,DE and SA. The colour scheme codes the residue’s absolute value of each pole. ○○○○○ ○○ ○○○ ○○ ○○○ ○○○○ ○○○ ○○ ○○○ ○○○○ ○○○ ○○ ○○○ ○○○○ ○○○ ○○ ○○○ ○○○○ ○○○ ○○○○ ○○○ ○○○○ ○○○ ○○ ○○○ ○○○○ ○○○ ○○○○ ○○○ ○○ ○○○○○ ○○○○ ○○○ ○○○○ ○○○ ○○○○ ○○○○ - - -
20 0 20 40 60 Re p [ GeV ] N DE80 lattice Poles ○ Zeros ○ ○○○○○○ ○○○○ ○○ ○○○○ ○○ ○ ○○○○○ ○○○ ○ ○○ ○○ ○○○ ○○○ ○○ ○ ○○ ○ ○○ ○○○ ○○○ ○○ ○ ○ - - p [ GeV ] N ○○○○○○ ○○ ○○○ ○○○ ○○ ○○○ ○○ ○○○ ○○ ○○○ ○○ ○○○ ○○ ○○○ ○○ ○○○ ○○ ○○○ ○○○○ ○○○○○ ○○○○ ○○○ ○○ ○○○ ○○○○ ○○○○○ ○○ ○○○ ○○ ○○○ ○○ ○○○ ○○ ○○○ ○○○○ ○○○○ - - -
20 0 20 40 60 Re p [ GeV ] N SA80 lattice ○○○○○○ ○○○ ○○ ○○○○ ○○ ○ ○○ ○○○ ○○ ○ ○○ ○ ○○ ○○○○○ ○○○ ○ ○ - - p [ GeV ] N ○○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ - - -
20 0 20 40 60 Re p [ GeV ] N DE128 lattice ○ ○○○○○○ ○○○ ○○ ○○ ○○ ○○○ ○○ ○○ ○ ○○○○ ○ ○○ ○ ○○ ○ ○○○○ ○ ○○○ ○○○ ○ ○ ○○ ○ ○○ ○○○ ○ ○ ○○ ○ ○○ ○○ ○ ○○ ○ ○ ○○ ○ ○ ○○ ○ ○ ○○ ○ ○○○ ○○ ○ ○○ ○ ○ ○○ ○○ - - p [ GeV ] N ○○○○○○○○○○○○○○○○○○○ ○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ - - -
20 0 20 40 60 Re p [ GeV ] N SA128 lattice ○ ○○○ ○○○ ○ ○○ ○○ ○ ○○ ○○ ○○ ○○ ○○ ○ ○○ ○○○ ○○○ ○○ ○ ○○ ○○ ○ ○ ○ ○○ ○○ ○ ○○ ○ ○ ○○○ ○○ ○ ○ ○○○ ○○○ ○○ ○ ○ ○○○ ○○ - - p [ GeV ] N log | A k |- - - - Figure 5.13: Evolution of the on-axis poles and zeros within the PA sequence, obtained forthe gluon propagator data with 80 and 128 lattices, for both methods of minimisation,DE and SA. The colour scheme codes the residue’s absolute value of each pole.
65f gauge configurations.Another approach to this problem could be to use approximants inspired bythe perturbative gluon propagator, for example, D gl ( p ) ≈ Q L ( p ) R M ( p ) " ω ln S F ( p ) T G ( p ) + 1 − γ (5.2)where Q L ( p ) /R M ( p ) and S F ( p ) /T G ( p ) are the usual PAs of orders [ L | M ] and[ F | G ], respectively. In this way, the information about the branch cut andthe branch point could be accessed from the analysis of the poles and zeros of S F ( p ) /T G ( p ). Several tests were made in the context of this work. However, theobtained results were very unstable. Thus, such results are not reported here.Notwithstanding, the interval of momenta identified above for the position ofthe branch point is in agreement with the naive identification of the latter withthe quoted “gluon mass” term found in [30] and [14], which is 0 .
12 GeV and0 .
36 GeV for each, respectively. Additionally, in [11], it was obtained the massscale of 0 .
216 GeV . Thus, we undoubtedly see the connection between the positionof the branch point and the mass scale that regularises the logarithm correctionto the perturbative result. This mass scale prevents the IR logarithmic divergenceof the propagator, making it finite at zero momentum.66 hapter 6Conclusions and future work Throughout this work, we explored the use of Padé Approximants to computethe analytic structure of the Landau gauge gluon and ghost propagators. Theapproximants were build as a global optimisation problem that minimises a chisquared function, resorting to two different numerical minimisation methods. ThePAs showed to faithfully reproduce the original functions, as well as their analyticstructure. This allowed to have a first glimpse of the analytic structure of thepropagators, i.e. , to identify its singularities and branch cuts, without relying ona particular theoretical or empirical model to describe the lattice data.Our methodology revealed the existence of a conjugate pair of poles in the com-plex p -plane, for the gluon propagator, clearly stemming from the IR structureof the theory. The presence of these complex singularities supports their connec-tion with the non-perturbative phenomenon of confinement. Regarding the ghostpropagator, a unique pole was found at p = 0, in agreement with the respectiveperturbative result. A branch cut on the real negative p -axis was identified inthe analytic structure of both propagators, with the branch points at Re( p ) < eferences [1] R Alkofer. “The infrared behaviour of QCD Green’s functions Confinement,dynamical symmetry breaking, and hadrons as relativistic bound states”. In: Physics Reports doi :
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