Quark mass function in Minkowski space

We investigate the properties of quark mass functions in quantum chromodynamics calculated by the Schwinger-Dyson equation in the strong coupling region, in which the loop integration is performed in Minkowski space. The calculated results are compared with those obtained by integration in Euclidean space.


Introduction
The Schwinger-Dyson equation (SDE) [1,2] is one of the methods for evaluating nonperturbative phenomena, such as chiral phase transition. So far, much work on chiral symmetry breaking has been done with the SDE in momentum representation, in which a one-loop contribution is integrated over Euclidean space.
Some calculations of fermion mass functions with the SDE have been done in Minkowski space. In Ref. [3], a spectral representation for Green functions is assumed, in which the mass functions are calculated in Lorentz-invariant form. In Ref. [4], explicit one-loop contributions of the mass function have been calculated. However, the mass function is evaluated for only one iteration from a constant initial mass as an input.
It has been pointed out that the spectral functions for the gluon calculated by the lattice simulation in Euclidean space, which is numerically continued to Minkowski space, becomes negative for some range [5]. Similar behavior has been found for generalized perturbative calculations [6].
Analytic continuation from Euclidean space to Minkowski space is valid in perturbative calculation if the pole positions in the complex plane of the energy are known. However, it is not trivial in the nonperturbative region.
So far, the structure of the fermion mass function in the strong coupling region in the entire range of energy and momentum space has not been fully studied in Minkowski space, even at zero temperature.
In the previous paper [7], we formulated the SDE for quantum electrodynamics (QED) in which the momentum integration is performed in Minkowski space without the instantaneous exchange approximation (IEA) [8,9].
In this paper, we apply our previous method to calculate the quark mass function in quantum chromodynamics (QCD) with the SDE in Minkowski space at zero temperature.
In Sect. 2, we present the SDE for QCD in Minkowski space. In Sect. 3, some numerical results are shown and calculated results are compared with those obtained by the SDE in Euclidean space. Section 4 is devoted to a summary and some comments.

The SDE for the quark mass function
We calculate a quark self-energy (P) in QCD in four dimensions, which is given by where S(Q) and D μν (K) are propagators of a quark with momentum Q = (q 0 , q) and a gluon with momentum K = P − Q = (k 0 , k), respectively. Here, P = (p 0 , p) is an external momentum of the quark. The strong coupling constant and the color factor are denoted by g s and C F = 4/3, respectively. The quark propagator is given by where m 0 is a bare quark mass.
In this paper, we calculate a mass function in the Landau gauge, in which the wave-function renormalization constant is Z = 1 for a one-loop order of perturbation. Therefore, we calculate the self-energy given in Eq. (1) with A = 1, M M = B = m 0 + Tr[ ]/4, and the quark-gluon vertex with ν = γ ν . Here, the gluon propagator is given as in the Landau gauge. In this paper, we neglect an effective gluon mass. Integrating over the azimuthal angle of the momentum q, the mass function is given by with p = |p|, q = |q|, and α s = g 2 s /(4π). The energy q 0 and the momentum q are integrated over the ranges − 0 ≤ q 0 ≤ 0 and δ ≤ q ≤ , respectively, with cutoff parameters 0 , , and δ. Here, I and J are given by and respectively, with η ± = |p ± q| and k = |k|. Here, (M 2 ) R and (M 2 ) I denote the real part and the imaginary part of the mass function M 2 M , respectively. In Minkowski space, the propagator I in Eq. (5) rapidly varies near Q 2 (M 2 ) R if the imaginary part of the mass function (M 2 ) I is small, which is one of the difficulties for numerical calculation in Minkowski space. Therefore, it is necessary to perform integration efficiently. As implemented in the previous work for QED [7], we divide the q 0 integration into small ranges and integrate the quark propagator over q where the remaining contributions of the integrand are averaged for the range q Convergence of the calculations is significantly improved by our method. The explicit expressions are summarized in Appendix A, in which the improved ladder approximation is implemented.

Numerical results
In this section, some numerical results are presented. We solve the SDE presented in Eq. (1) by a recursion method starting from a constant mass. 1 For each iteration, we calculate the quark mass function normalized as where n denotes the number of iterations. Here, the mass function is normalized by a current quark mass at large μ 2 , in which perturbative calculations are reliable. In the iteration, the mass function M M (p 0 , p) in the integrand of Eq. (4) is replaced by the renormalized one obtained by the previous iteration. Here, m(μ 2 ) is a renormalized mass at a renormalization scale μ. 2 Here, M M (P 2 ) is defined as for P 2 > 0, and for P 2 < 0, respectively, where P 2 = p 2 0 − p 2 . The mass function can be written as an absolute value |M M (P 2 )| and a phase factor exp(i (P 2 )) as In Fig. 1, the convergence property of the mass function |M M | integrated over |P 2 | as is presented. The solid line and the + symbols represent |M M | integrated over P 2 > 0 and P 2 < 0, respectively. The horizontal axis denotes the number of iterations. The dash-dotted line represents the calculated results for M E , where the mass function in Euclidean space M E (P 2 E ) is given by [10,11] 1 The initial input parameters are M R = QCD and M I = 0, with 0 = = 20 QCD and δ = 0.2 QCD , where we define M M = M R + iM I . We set QCD = 0.5 GeV with ε = 10 −6 . 2 We take m(μ 2 ) = 3 MeV at μ = 20 QCD .

3/9
Downloaded from https://academic.oup.com/ptep/article-abstract/2017/12/123B02/4743140 by guest on 30 July 2018  which is renormalized as As shown in Fig. 1, the mass functions integrated over the momentum for the three cases rapidly converge.
In Fig. 2, the dependencies on P 2 for |M M | in Minkowski space and M E in Euclidean space are presented. The three cases give similar |P 2 | dependencies, though the mass function |M M | in time-like momentum has an imaginary part.
In Fig. 3, the P 2 dependencies of M R and M I in Minkowski space are presented. From Eq. (2), we define a spectral function for the quark mass term denoted by ρ M (P 2 ) as 1 4 Tr As shown in Fig. 3, M I for P 2 > 0 becomes positive in some regions of P 2 , such as P 2 > 0.3 GeV 2 , in which ρ M becomes negative. It may be interesting to compare our result with the spectral function for the quark obtained in Ref. [6] in Minkowski space. In Fig. 4, the P 2 dependencies of M R and M I , as well as ρ M (P 2 ) with the analytic coupling α (AN) s defined in Appendix A, which is valid for a time-like momentum region as well as a space-like one, are presented. The results presented in Fig. 4 show similar behaviors to those in Fig. 3, though the mass functions are smaller than those obtained with the coupling constant defined in Euclidian momentum due to the smaller contribution of the coupling constant α (AN) s (P 2 , Q 2 ) than α s (P 2 ,Q 2 ).

Summary and comments
In this paper, we studied a quark mass function solved by the Schwinger-Dyson equation in Minkowski space for QCD. Evaluation of the mass functions in Minkowski space allows us to study the imaginary part of the mass and energy for the massive fermion states.
We examined the properties of the quark mass function in time-like momentum P 2 > 0 as well as that in space-like momentum P 2 < 0, where P 2 denotes the squared four-momentum of the quark.
Furthermore, we also compared our results with the mass function calculated in Euclidean space. We found that the three cases give similar |P 2 | dependencies, though the mass function with time-like momentum has an imaginary part.
We also studied the behavior of the spectral function for the quark mass term. We found that there seems to exist a negative spectral function in some momentum regions for P 2 > 0 as pointed out in Refs. [5,6], in which the imaginary part of the mass function becomes positive.
Theoretically, we should implement a coupling constant with momenta in Minkowski space. However, coupling constants with nonperturbative contributions in Minkowski space have not yet been studied.
Therefore, we implemented a strong coupling constant with Euclidean momenta, which is a simple model to evaluate the quark mass function in Minkowski space. A different definition of the argument of α s is a part of the higher-order contributions to the one-loop approximation. Therefore, it is expected that qualitative features of the mass function in Minkowski space may be preserved in our model. Furthermore, our model gives rather stable numerical results in calculation in Minkowski space. In order to check our results, we calculated the mass functions as well as the spectral function with an analytic coupling α (AN) s , which is the perturbative strong coupling constant analytically continued from a space-like momentum region to a time-like one.
We found similar behaviors as those obtained with the coupling constant defined in Euclidean momentum. The results suggest that the qualitative features of the mass function may not be affected by the choice of the momentum in the coupling constant, such as Euclidian or Minkowski momentum, provided that the coupling constant is large enough to break the chiral symmetry of quarks.
It may be expected that the SDE has multiple solutions in numerical calculations. Further studies are needed for solutions of the quark mass function obtained by the SDE in Minkowski space in the strong coupling region.
In this paper, we examined the qualitative features of the quark mass function in Minkowski space. In order to reproduce physical quantities, such as the pion decay constant, or to fit the results obtained by SDE with the numerical data from lattice simulations, we need to fine-tune the parameter QCD .
In future works, we shall extend our method to finite temperature and density with the real time formalism.

Appendix. Approximation formulas
The mass function is given by where the propagators I and J are defined in Eqs. (5) and (6), respectively. As shown in Eq. (7), we approximate the integration over q 0 as where X l denotes an average of X (q (l+1) 0 ) and X (q Here, the strong coupling constant α s is replaced by the running coupling constant α s (P 2 ,Q 2 ) [12], which is defined as For J in Eq. (6), we can integrate over k as respectively, with η ± = |p ± q| and k = |k|. The real and imaginary parts of the quark propagator I (q