Analytic properties of the quark propagator from an effective infrared interaction model

In this paper, I investigate the analytic properties of the quark propagator Dyson-Schwinger equation (DSE) in the Landau gauge. In the quark self-energy, the combined gluon propagator and quark-gluon vertex is modeled by an effective interaction (the so-called Maris-Tandy interaction), where the ultraviolet term is neglected. This renders the loop integrand of the quark self-energy analytic on the cut plane $-\pi <arg\left(x\right)<\pi$ of the square of the external momentum. Exploiting the simplicity of the truncation, I study solutions of the quark propagator in the domain $x\in [-5.1,0]{\text{GeV}}^2\times i[0,10.2]{\text{GeV}}^2$. Because of a complex conjugation symmetry, this region fully covers the parabolic integration domain for Bethe-Salpeter equations (BSEs) for bound state masses of up to 4.5 GeV. Employing a novel numerical technique that is based on highly parallel computation on graphics processing units (GPUs), I extract more than 6500 poles in this region, which arise as the bare quark mass is varied over a wide range of closely spaced values. The poles are grouped in 23 individual trajectories that capture the movement of the poles in the complex region as the bare mass is varied. The raw data of the pole locations and residues is provided as Supplemental Material, which can be used to parametrize solutions of the complex quark propagator for a wide range of bare mass values and for large bound-state masses. This study is a first step towards an extension of previous work on the analytic continuation of perturbative one-loop integrals, with the long-term goal of establishing a framework that allows for the numerical extraction of the analytic properties of the quark propagator with a truncation that extends beyond the rainbow by making adequate adjustments in the contour of the radial integration of the quark self-energy.

Figure 1

A diagrammatic representation of the Landau gauge quark propagator Dyson-Schwinger equation. The momentum routing is shown explicitly.

Figure 2

Plots of the quark propagator dressing functions (a) $A\left(p^2\right)$ and (b) $B\left(p^2\right)$, as well as the alternative functions (c) ${\sigma }_S\left(p^2\right)$ and (d) ${\sigma }_V\left(p^2\right)$. The plots have been generated to provide a comparison with the solutions presented in Ref. [13], thereby validating the graphics-processing unit (GPU) parallelized code used in this study.

Figure 3

This figure shows the real part of the solution for ${\sigma }_V\left(p^2\right)$ for $m_0=225$ MeV, obtained in the region $p^2\in [-5.1,0]\times i[0,10.2]$ of the square of the external momentum.

Figure 4

The real part of the solution for ${\sigma }_V\left(p^2\right)$ for $m_0=225$ MeV features one pole on the real axis and eight further poles at locations with nontrivial imaginary part. Various steps that are used for the automated identification of the poles are highlighted and are referred to in the discussion of the pole search procedure in the main text.

Figure 5

All poles that have been detected as the bare mass were varied from 5 to 2500 MeV in increments of 5 MeV. While the following analysis has been restricted to the second quadrant, the third quadrant has also been calculated in order to verify the complex conjugate nature of the poles. Overall, for all mass parameters and in both quadrants together, more than 13 000 poles, as well as the corresponding residues in ${\sigma }_S$ and ${\sigma }_V$ have been extracted.

Figure 6

This figure shows the main result of this study, obtained in the region $p^2\in [-5.1,0]\times i[0,10.2]$ of the square of the external momentum. For the sake of completeness, the behavior of the poles on the real axis is included. A detailed analysis of the trajectories of the real poles is presented in Sec. 5c. The plot is to be read as follows. Trajectories which do not possess any real valued poles are self-explanatory. The bare mass value of the occurrence of the first pole is explicitly provided (in MeV), and also the last mass value for which a particular trajectory has been tracked. The highest mass value in the computation is 2500 MeV, and, apart from trajectory J, all trajectories which have an endpoint within the domain specified above end at this value. The scale that emerges around $p^2=-4 {\mathrm{GeV}}^2$ is discussed in the main text.

Figure 7

These plots show the real and imaginary part of the trajectories traced out by the poles as the bare mass is varied from 5 to 2500 MeV in increments of 5 MeV. The various trajectories have been grouped together in four sets. Each subfigure is dedicated to such a set. Since this computation has been performed in the second quadrant, all real parts are negative and all imaginary parts are positive. Consequently, all graphs below zero correspond to the real part of the respective trajectory, while graphs greater than zero represent the imaginary part of the trajectory. In panel (a), the imaginary part of trajectory H left the region of computation, but re-entered after a short while, which is why this graph appears to be cut off. Panels (b) and (c) contain the trajectories B1 and B2, which originate on the negative real axis. As the poles meet, they leave the real axis, as indicated by them developing a nonzero imaginary part; see Sec. 5c for a detailed analysis.

Figure 8

This figure shows the behavior of the poles in the interval $p^2\in [-5.1,0]$, which collide, and scatter off into the imaginary direction. In panel (a), two trajectories, B1a (approaching from the left) and B1b (approaching from the right), come together and form a new (complex conjugate) trajectory B1c. The collision occurs between a bare mass value of 105 and 110 MeV. In panel (b), again two trajectories collide. At low mass values, trajectory B2a1 approaches from the left and reverses its direction of movement between a mass value of 230 and 235 MeV, after which it is labeled as B2a2. Because the trajectories B2a1 and B2a2 would overlap in the figure, they have been slightly shifted away from the real axis, even though their imaginary parts are zero. Finally, a pole moving to the right along trajectory B2a1 collides with a pole moving to the left along trajectory B2b, creating a new complex (conjugate) trajectory B2c between mass values of 450 and 455 MeV. Above 455 MeV, there are no real poles in the interval $[-5.1,0]$ for all bare mass values of up to 2500 MeV.

Figure 9

Residues of ${\sigma }_S$ and ${\sigma }_V$ as a function of $m_0$ for trajectories A, B2, K [(a)–(d)], and B1, L and M [(e)–(h)].

Figure 10

Residues of ${\sigma }_S$ and ${\sigma }_V$ as a function of $m_0$ for trajectories E, G, H, I [panels (a)–(d)] and N, O, P, Q [panels (e)–(h)].

Figure 11

Residues of ${\sigma }_S$ and ${\sigma }_V$ as a function of $m_0$ for trajectories C, D, F, and J [panels (a)–(d)].