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Complex Poles and Spectral Functions of Landau Gauge QCD and QCD-Like Theories

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Complex Poles and Spectral Functions of Landau Gauge QCD and QCD-Like Theories PHYSICAL REVIEW D 101, 074044 (2020) Complex poles and spectral functions of Landau gauge QCD and QCD-like theories † Yui Hayashi 1,* and Kei-Ichi Kondo1,2, 1Department of Physics, Graduate School of Science and Engineering, Chiba University, Chiba 263-8522, Japan 2Department of Physics, Graduate School of Science, Chiba University, Chiba 263-8522, Japan (Received 21 January 2020; accepted 16 March 2020; published 29 April 2020) In view of the expectation that the existence of complex poles is a signal of confinement, we investigate the analytic structure of the gluon, quark, and ghost propagators in the Landau gauge QCD and QCD-like theories by employing an effective model with a gluon mass term of the Yang-Mills theory, which we call the massive Yang-Mills model. In this model, we particularly investigate the number of complex poles in the parameter space of the model consisting of gauge coupling constant, gluon mass, and quark mass for the gauge group SUð3Þ and various numbers of quark flavors NF within the asymptotic free region. Both the gluon and quark propagators at the best-fit parameters for NF ¼ 2 QCD have one pair of complex conjugate poles, while the number of complex poles in the gluon propagator varies between zero and four depending on the number of quark flavors and quark mass. Moreover, as a general feature, we argue that the gluon spectral function of this model with nonzero quark mass is negative in the infrared limit. In sharp contrast to gluons, the quark and ghost propagators are insensitive to the number of quark flavors within the current approximations adopted in this paper. These results suggest that details of the confinement mechanism may depend on the number of quark flavors and quark mass. DOI: 10.1103/PhysRevD.101.074044 I. INTRODUCTION quark, and ghost propagators in the Landau gauge of the Yang-Mills theory and quantum chromodynamics (QCD). Color confinement, absence of color degrees of freedom In the Yang-Mills theory, or the quenched limit of QCD, the from the physical spectrum, is one of the most fundamental so-called decoupling and scaling solutions of the gluon and and significant features of strong interactions. It is a long- ghost propagators are observed based on the continuum standing and challenging problem in particle and nuclear approaches [4]. The recent numerical lattice results support physics to explain color confinement in the framework of the decoupling solution [5]. The decoupling solution has an quantum field theory (QFT). Understanding analytic struc- impressing feature that the running gauge coupling stays tures of the correlation functions will be of crucial finite and nonzero for all nonvanishing momenta and importance to this end because a QFT describing physical eventually goes to zero in the limit of vanishing momen- particles can be reformulated in terms of correlation tum, which cannot be predicted from the standard pertur- functions [1], and there are some proposals of confinement bation theory that is plagued by the Landau pole of the mechanisms whose criteria are expressed by them, e.g., [2]. diverging running gauge coupling. In particular, the analytic structures of propagators encode A low-energy effective model of the Yang-Mills theory is kinematic information as the Käll´en-Lehmann spectral proposed following the decoupling behavior of the gluon representation [3], which will be useful toward under- propagator, which provides the gluon and ghost propaga- standing confinement. tors that show a striking agreement with the numerical In the past decades, numerous studies of both the lattice lattice results by including quantum corrections just in the and continuum approaches have focused on the gluon, one-loop level [6,7]. This effective model is given by the mass-deformed Faddeev-Popov Langrangian of the Yang- *[email protected] Mills theory in the Landau gauge, or the Landau gauge † [email protected] limit of the Curci-Ferarri model [8], which can be shown to be renormalizable due to the modified Becchi-Rouet-Stora- Published by the American Physical Society under the terms of Tyutin (BRST) symmetry, and we call it the massive Yang- the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to Mills model for short. This effective mass term could stem the author(s) and the published article’s title, journal citation, from the dimension-two gluon condensate [9–13] or could and DOI. Funded by SCOAP3. be taken as a (minimal) consequence of avoiding the 2470-0010=2020=101(7)=074044(22) 074044-1 Published by the American Physical Society YUI HAYASHI and KEI-ICHI KONDO PHYS. REV. D 101, 074044 (2020) Gribov ambiguity [14,15]. Moreover, it has been shown complex poles and the propagator on timelike momenta that the massive Yang-Mills model has “infrared safe” from the argument principle [40]. This investigation renormalization group (RG) flows on which the running extends the previous result [40] obtained for the pure gauge coupling remains finite for all scales [7,15,16]. The Yang-Mills theory with no flavor of quarks NF ¼ 0 that the three-point functions [17] and two-point correlation func- gluon propagator has a pair of complex conjugate poles tions at finite temperature [18] in this model were compared and the negative spectral function while the ghost propa- to the numerical lattice results, showing good agreements. gator has no complex pole. In this article, we will see the Moreover, the two-loop corrections improve the accor- following results. Both the gluon and quark propagators at dance for the gluon and ghost propagators [19]. These the “realistic” parameters for NF ¼ 2 QCD have one pair of works indicate the validity of the massive Yang-Mills complex conjugate poles as well as the gluon propagator in model as an effective model of the Yang-Mills theory. the zero flavor case. By increasing quark flavors, we find a The gluon and ghost propagators for unquenched lattice new region in which the gluon propagator has two pairs of QCD with the number of quark flavors N ¼ 2; 2 þ 1; 2 þ complex conjugate poles for light quarks with the inter- F 4 ≲ 10 1 þ 1 have been studied, for instance [20], and exhibit the mediate number of flavors NF < .However,the decoupling feature as well. The massive Yang-Mills model gluon propagator has no complex poles if very light quarks ≥ 10 with dynamical quarks can reproduce the numerical lattice have many flavors NF or both of the gauge coupling gluon and ghost propagators for QCD as well [21]. and quark mass are small. In the other regions, the gluon However, it is argued that higher-loop corrections are propagator has one pair of complex conjugate poles. On the important for the quark sector in this model [22,23]. other hand, the analytic structures of quark and ghost Despite this shortcoming, it appears that the effective gluon propagators are nearly independent of the number of quarks mass term captures some nonperturbative aspects of QCD. within this one-loop analysis. What is more, QCD phases have been extensively studied This paper is organized as follows. In Sec. II,we in a similar model with the effective gluon mass term of the present machinery to count the number of complex poles Landau-DeWitt gauge [24]. as an application of [40].InSec.III, we review the calculation Apart from the realistic QCD, it is also interesting to study of the massive Yang-Mills model with quarks of [21], gauge theories with many flavors of quarks. For many quark consider the infrared safe trajectories of this model, and flavors, the infrared conformality is predicted [25,26] and argue the infrared negativity of the gluon spectral function. well-studied in line with the walking technicolor for physics Then, in Sec. IV, we analyze the analytic structures of the beyond the standard model [27], for example, [28–30]. gluon, quark, and ghost propagators at the best-fit parameters Some argue that chiral symmetry restores while color of [21], and investigate the number of complex poles for degrees of freedom are “unconfined” in the conformal various parameters and the number of quarks NF.Finally, window. For a better understanding of the confinement Sec. V is devoted to conclusion, and Sec. VI contains further discussion. In Appendix A, we provide a generalization of the mechanism, observing NF dependence will be thus extremely valuable. proposition of Sec. II for various infrared behaviors. All works on the correlation functions described above Appendix B gives complementary one-loop analyses for were implemented in the Euclidean space. Considerable Sec. IVA. efforts have been devoted to reconstructing the spectral functions from the Euclidean data based on the Käll´en- II. COMPLEX POLES IN PROPAGATORS Lehmann spectral representation, e.g., [31,32]. On the other – To elucidate the starting point, we review a generaliza- hand, several models of the Yang-Mills theory [14,33 39], tion of the spectral representation for a propagator so as to including the (pure) massive Yang-Mills model [40,41], and a allow complex poles. We then develop a method for way of the reconstruction from the Euclidean data [42] counting the number of complex poles from the data on predict complex poles in the gluon propagator that invalidate timelike momenta, e.g., the spectral function, as a straight- the Käll´en-Lehmann spectral representation. The existence of forward application of the general relation [40]. complex poles of the propagators of the confined particles is a controversial issue, see, e.g., [43]. The complex singularities invalidate the standard reconstruction from a Euclidean field A. A generalization of the spectral representation theory to a relativistic QFT [1] and might correspond to We introduce some definitions and underlying assump- unphysical degrees of freedom in an indefinite metric state tions on propagators adopted in this article.
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