Spectral representation of lattice gluon and ghost propagators at zero temperature. (English) Zbl 1472.81309
Summary: We consider the analytic continuation of Euclidean propagator data obtained from 4D simulations to Minkowski space. In order to perform this continuation, the common approach is to first extract the Källén-Lehmann spectral density of the field. Once this is known, it can be extended to Minkowski space to yield the Minkowski propagator. However, obtaining the Källén-Lehmann spectral density from propagator data is a well known ill-posed numerical problem. To regularise this problem we implement an appropriate version of Tikhonov regularisation supplemented with the Morozov discrepancy principle. We will then apply this to various toy model data to demonstrate the conditions of validity for this method, and finally to zero temperature gluon and ghost lattice QCD data. We carefully explain how to deal with the IR singularity of the massless ghost propagator. We also uncover the numerically different performance when using two – mathematically equivalent – versions of the Källén-Lehmann spectral integral.
MSC:
| 81V35 | Nuclear physics |
| 81V05 | Strong interaction, including quantum chromodynamics |
| 47A10 | Spectrum, resolvent |
| 51B20 | Minkowski geometries in nonlinear incidence geometry |
| 58J47 | Propagation of singularities; initial value problems on manifolds |
References:
| [1] | Sauli, V., Minkowski solution of Dyson-Schwinger equations in momentum subtraction scheme, JHEP, 02, Article 001 pp. (2003) |
| [2] | Frederico, T.; Duarte, D. C.; de Paula, W.; Ydrefors, E.; Jia, S.; Maris, P., Towards Minkowski space solutions of Dyson-Schwinger equations through un-Wick rotation (2019) |
| [3] | Solis, E. L.; Costa, C. S.R.; Luiz, V. V.; Krein, G., Quark propagator in Minkowski space (2019) |
| [4] | Meyer, H. B., Transport properties of the quark-gluon plasma: a lattice QCD perspective, Eur. Phys. J. A, 47, 86 (2011) |
| [5] | Dudal, D.; Oliveira, O.; Silva, P. J., Källén-Lehmann spectroscopy for (un)physical degrees of freedom, Phys. Rev. D, 89, 1, Article 014010 pp. (2014) |
| [6] | Alkofer, R.; von Smekal, L., The infrared behavior of QCD Green’s functions: confinement dynamical symmetry breaking, and hadrons as relativistic bound states, Phys. Rep., 353, 281 (2001) · Zbl 0988.81551 |
| [7] | Roberts, C. D.; Williams, A. G., Dyson-Schwinger equations and their application to hadronic physics, Prog. Part. Nucl. Phys., 33, 477-575 (1994) |
| [8] | Maris, P.; Roberts, C. D., Dyson-Schwinger equations: a tool for hadron physics, Int. J. Mod. Phys. E, 12, 297-365 (2003) |
| [9] | Bhagwat, M.; Pichowsky, M. A.; Tandy, P. C., Confinement phenomenology in the Bethe-Salpeter equation, Phys. Rev. D, 67, Article 054019 pp. (2003) |
| [10] | Windisch, A.; Huber, M. Q.; Alkofer, R., On the analytic structure of scalar glueball operators at the Born level, Phys. Rev. D, 87, 6, Article 065005 pp. (2013) |
| [11] | Eichmann, G.; Sanchis-Alepuz, H.; Williams, R.; Alkofer, R.; Fischer, C. S., Baryons as relativistic three-quark bound states, Prog. Part. Nucl. Phys., 91, 1-100 (2016) |
| [12] | Sanchis-Alepuz, H.; Fischer, C. S.; Kellermann, C.; von Smekal, L., Glueballs from the Bethe-Salpeter equation, Phys. Rev. D, 92, Article 034001 pp. (2015) |
| [13] | Strauss, S.; Fischer, C. S.; Kellermann, C., Analytic structure of the Landau gauge gluon propagator, Phys. Rev. Lett., 109, Article 252001 pp. (2012) |
| [14] | Maas, A., Describing gauge bosons at zero and finite temperature, Phys. Rep., 524, 203-300 (2013) |
| [15] | Oliveira, O.; Silva, P. J., Finite Temperature Landau Gauge Lattice Quark Propagator (2019) |
| [16] | Cucchieri, A.; Mendes, T.; Santos, E. M.S., Covariant gauge on the lattice: a new implementation, Phys. Rev. Lett., 103, Article 141602 pp. (2009) |
| [17] | Bicudo, P.; Binosi, D.; Cardoso, N.; Oliveira, O.; Silva, P. J., Lattice gluon propagator in renormalizable ξ gauges, Phys. Rev. D, 92, 11, Article 114514 pp. (2015) |
| [18] | Cucchieri, A.; Dudal, D.; Mendes, T.; Oliveira, O.; Roelfs, M.; Silva, P. J., Faddeev-Popov matrix in linear covariant gauge: first results, Phys. Rev. D, 98, 9, Article 091504 pp. (2018) |
| [19] | Negele, J. W.; Orland, H., Quantum Many Particle Systems, Frontiers in Physics, vol. 68 (1988), Addison-Wesley: Addison-Wesley Redwood City, USA · Zbl 0984.82503 |
| [20] | Peskin, M. E.; Schroeder, D. V., An Introduction to Quantum Field Theory (1995), Addison-Wesley: Addison-Wesley Reading, USA |
| [21] | Laine, M.; Vuorinen, A., Basics of thermal field theory, Lect. Notes Phys., 925, 1-281 (2016) · Zbl 1356.81007 |
| [22] | Kirsch, A., An Introduction to the Mathematical Theory of Inverse Problems (1996), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 0865.35004 |
| [23] | Egonmwan, A. O., The Numerical Inversion of the Laplace Transform (2012), LAP Lambert Academic Publishing |
| [24] | Tripolt, R.-A.; Gubler, P.; Ulybyshev, M.; Von Smekal, L., Numerical analytic continuation of Euclidean data, Comput. Phys. Commun., 237, 129-142 (2019) · Zbl 07683936 |
| [25] | Oehme, R.; Zimmermann, W., Quark and gluon propagators in quantum chromodynamics, Phys. Rev. D, 21, 471 (1980) |
| [26] | Oehme, R., On superconvergence relations in quantum chromodynamics, Phys. Lett. B, 252, 641-646 (1990) |
| [27] | Cornwall, J. M., Positivity violations in QCD, Mod. Phys. Lett. A, 28, Article 1330035 pp. (2013) |
| [28] | Langfeld, K.; Reinhardt, H.; Gattnar, J., Gluon propagators and quark confinement, Nucl. Phys. B, 621, 131-156 (2002) · Zbl 0988.81518 |
| [29] | Qin, S.-x.; Rischke, D. H., Quark spectral function and deconfinement at nonzero temperature, Phys. Rev. D, 88, Article 056007 pp. (2013) |
| [30] | Rothkopf, A., Bayesian inference of nonpositive spectral functions in quantum field theory, Phys. Rev. D, 95, 5, Article 056016 pp. (2017) |
| [31] | Ilgenfritz, E.-M.; Pawlowski, J. M.; Rothkopf, A.; Trunin, A., Finite temperature gluon spectral functions from \(N_f = 2 + 1 + 1\) lattice QCD, Eur. Phys. J. C, 78, 2, 127 (2018) |
| [32] | Cyrol, A. K.; Pawlowski, J. M.; Rothkopf, A.; Wink, N., Reconstructing the gluon, SciPost Phys., 5, Article 065 pp. (2018) |
| [33] | Asakawa, M.; Hatsuda, T.; Nakahara, Y., Maximum entropy analysis of the spectral functions in lattice QCD, Prog. Part. Nucl. Phys., 46, 459-508 (2001) |
| [34] | Aarts, G.; Allton, C.; Oktay, M. B.; Peardon, M.; Skullerud, J.-I., Charmonium at high temperature in two-flavor QCD, Phys. Rev. D, 76, Article 094513 pp. (2007) |
| [35] | Wang, J.; Chakravarty, S., Rational function regression method for numerical analytic continuation |
| [36] | Fournier, R.; Wang, L.; Yazyev, O. V.; Wu, Q. S., An artificial neural network approach to the analytic continuation problem |
| [37] | Ferrari, F., The analytic renormalization group, Nucl. Phys. B, 909, 880-920 (2016) · Zbl 1342.81341 |
| [38] | Pawlowski, J. M.; Strodthoff, N.; Wink, N., Finite temperature spectral functions in the O(N)-model, Phys. Rev. D, 98, 7, Article 074008 pp. (2018) |
| [39] | Boguslavski, K.; Kurkela, A.; Lappi, T.; Peuron, J., Spectral function for overoccupied gluodynamics from real-time lattice simulations, Phys. Rev. D, 98, 1, Article 014006 pp. (2018) |
| [40] | Lowdon, P., Spectral density constraints in quantum field theory, Phys. Rev. D, 92, 4, Article 045023 pp. (2015) |
| [41] | Lowdon, P., Nonperturbative structure of the photon and gluon propagators, Phys. Rev. D, 96, 6, Article 065013 pp. (2017) |
| [42] | Roelfs, M.; Kroon, P. C. (October 2018), symfit 0.4.6 |
| [43] | Oliveira, O.; Duarte, A. G.; Dudal, D.; Silva, P. J., Gluon and ghost dynamics from lattice QCD, Few-Body Syst., 58, 2, 99 (2017) |
| [44] | Coleman, S., Notes from Sidney Coleman’s Physics 253a: Quantum Field Theory (2011) |
| [45] | Leinweber, D. B.; Skullerud, J. I.; Williams, A. G.; Parrinello, C., Gluon propagator in the infrared region, Phys. Rev. D, 58, Article 031501 pp. (1998) |
| [46] | Leinweber, D. B.; Skullerud, J. I.; Williams, A. G.; Parrinello, C., Asymptotic scaling and infrared behavior of the gluon propagator, Phys. Rev. D. Phys. Rev. D, Phys. Rev. D, 61, Article 079901 pp. (2000), Erratum |
| [47] | Dudal, D.; Oliveira, O.; Silva, P. J., High precision statistical Landau gauge lattice gluon propagator computation vs. the Gribov-Zwanziger approach, Ann. Phys., 397, 351-364 (2018) |
| [48] | Duarte, A. G.; Oliveira, O.; Silva, P. J., Lattice gluon and ghost propagators and the strong coupling in pure su(3) Yang-Mills theory: finite lattice spacing and volume effects, Phys. Rev. D, 94, Article 014502 pp. (Jul 2016) |
| [49] | Baulieu, L.; Dudal, D.; Guimaraes, M. S.; Huber, M. Q.; Sorella, S. P.; Vandersickel, N.; Zwanziger, D., Gribov horizon and i-particles: about a toy model and the construction of physical operators, Phys. Rev. D, 82, Article 025021 pp. (2010) |
| [50] | Dudal, D.; Guimaraes, M. S.; Sorella, S. P., Glueball masses from an infrared moment problem and nonperturbative Landau gauge, Phys. Rev. Lett., 106, Article 062003 pp. (2011) |
| [51] | Cucchieri, A.; Dudal, D.; Mendes, T.; Vandersickel, N., Modeling the gluon propagator in Landau gauge: lattice estimates of pole masses and dimension-two condensates, Phys. Rev. D, 85, Article 094513 pp. (2012) |
| [52] | Siringo, F., Analytic structure of QCD propagators in Minkowski space, Phys. Rev. D, 94, 11, Article 114036 pp. (2016) |
| [53] | Siringo, F., Quasigluon lifetime and confinement from first principles, Phys. Rev. D, 96, 11, Article 114020 pp. (2017) |
| [54] | Binosi, D.; Tripolt, R.-A., Spectral functions of confined particles (2019) |
| [55] | Hayashi, Y.; Kondo, K.-I., Complex poles and spectral function of Yang-Mills theory (2018) |
| [56] | Edwards, R. G.; Joo, B., The Chroma software system for lattice QCD, Nucl. Phys. B, Proc. Suppl., 140, 832 (2005) |
| [57] | Pippig, M., PFFT: an extension of FFTW to massively parallel architectures, SIAM J. Sci. Comput., 35, 3, C213 (2014) · Zbl 1275.65098 |
| [58] | Chetyrkin, K. G.; Retey, A., Three loop three linear vertices and four loop similar to MOM beta functions in massless QCD (2000) |
| [59] | Gracey, J. A., Renormalization group functions of QCD in the minimal MOM scheme, J. Phys. A, 46, Article 225403 pp. (2013) · Zbl 1269.81196 |
| [60] | Dudal, D.; Guimaraes, M. S., On the computation of the spectral density of two-point functions: complex masses, cut rules and beyond, Phys. Rev. D, 83, Article 045013 pp. (2011) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
