Analytic structure of QCD propagators in Minkowski space

Analytical functions for the propagators of QCD, including a set of chiral quarks, are derived by a one-loop massive expansion in the Landau gauge, deep in the infrared. By analytic continuation, the spectral functions are studied in Minkowski space, yielding a direct proof of positivity violation and confinement from first principles. The dynamical breaking of chiral symmetry is described on the same footing of gluon mass generation, providing a unified picture. While dealing with the exact Lagrangian, the expansion is based on massive free-particle propagators, is safe in the infrared and is equivalent to the standard perturbation theory in the UV. By dimensional regularization, all diverging mass terms cancel exactly without including mass counterterms that would spoil the gauge and chiral symmetry of the Lagrangian. Universal scaling properties are predicted for the inverse dressing functions and shown to be satisfied by the lattice data. Complex conjugated poles are found for the gluon propagator, in agreement with the i-particle scenario.

Figure 1

Two-point graphs with no more than three vertices and no more than one loop. The crosses are the counterterms $\delta {\mathrm{\Gamma }}_g=m^2$, $\delta {\mathrm{\Gamma }}_q=-M$. In this paper, the quark and ghost self-energy and the gluon polarization are obtained by the sum of all the graphs in the figure.

Figure 2

The real part (dashed line) and the imaginary part (solid line) of the gluon propagator are displayed together with the lattice data of Ref. [2] ($N=3$, $\beta =5.7$, $L=96$). The propagator is normalized by its finite value at $p^2=0$ and is evaluated by Eq. (28) with $F_0=-1.05$ and $m=0.73\mathrm{GeV}$.

Figure 3

Real part of the gluon propagator according to Eq. (28) for $m=0.73\mathrm{GeV}$ and several values of $F_0$ in the range $-1.6<F_0<-0.6$. The points are the lattice data of Ref. [2] ($N=3$, $\beta =5.7$, $L=96$). The solid line is obtained for $F_0=-1.05$.

Figure 4

The real part (dashed line) and the imaginary part (solid line) of the gluon propagator (enlargement of Fig. 2).

Figure 5

Real part $\mathrm{Re}\chi$ and imaginary part $-\mathrm{Im}\chi =\pi p^2\rho$ of the ghost dressing function according to Eq. (28) for $m=0.73\mathrm{GeV}$ and several values of $G_0$ in the range $0.2<G_0<0.3$. The points are the lattice data of Ref. [2] ($N=3$, $\beta =5.7$, $L=80$). The best agreement with the data points is obtained for $G_0=0.24$ (solid line). The dressing function is scaled by a finite renormalization constant $Z_G$.

Figure 6

The ghost dressing function $\chi (s)$ of Eq. (28) is shown as a function of the Euclidean momentum $p_E^2=s{m}^2$ for $m=0.7\mathrm{GeV}$ and $G_0=0.195$. The points are lattice data extracted from Fig. 1 of Ref. [5] and renormalized by the factors $Z_G=0.86$ ($N_f=0$, open squares, originally taken from Ref. [2]); $Z_G=0.78$ ($N_f=2$, two light quarks, circles); and $Z_G=0.78$ ($N_f=2+1+1$, two light and two heavy quarks, filled squares). The energy units of the quenched data ($N_f=0$) have been multiplied by a factor of 1.12 with respect to the original data of Ref. [2] (a point at $p=1\mathrm{GeV}$ in the figure was at $p=1.12\mathrm{GeV}$ in the original figure).

Figure 7

The inverse gluon dressing function $1/J(s)=F(s)+\mathrm{\Delta }F(s)+F_0$, as arising from Eqs. (28), (44), is plotted as a function of the Euclidean momentum $p_E^2=s{m}^2$ for $N_f=2$, $m=0.8\mathrm{GeV}$, $M=0.65\mathrm{GeV}$ (optimal set, solid line); for $N_f=2$, $m=0.73\mathrm{GeV}$, $M=0.65\mathrm{GeV}$ (suboptimal, dotted line); and for $N_f=0$, $m=0.73\mathrm{GeV}$ (pure Yang-Mills, $\mathrm{\Delta }F=0$, dashed line). The constant $F_0$ is $-0.65$ for $N_f=2$ and $-1.05$ for $N_f=0$ [47]. The circles are lattice data extracted from Fig. 1 of Ref. [5] and scaled as described in the text. The triangles are lattice data extracted from Fig. 2 of Ref. [2].

Figure 8

The real part of the gluon propagator is evaluated by setting $s=-p^2/m^2-iϵ$ in Eqs. (28), (44), for $m=0.80\mathrm{GeV}$ and several values of $M=0.48$, 0.52, 0.56, 0.65 GeV. The constant $F_0$ varies in the range $-0.65<F_0<-0.6$ in order to keep all curves on the lattice data in the Euclidean space, for $p^2<0$. The data points are extracted from Fig. 1 of Ref. [5] for $N_f=2$. The propagator is normalized by its finite value at $p^2=0$.

Figure 9

The imaginary part of the gluon propagator is evaluated by setting $s=-p^2/m^2-iϵ$ in Eqs. (28), (44), for $m=0.80\mathrm{GeV}$, $F_0=-0.6$, and several values of $M=0.4$, 0.5, 0.6, 0.7 GeV. The propagator is normalized by its finite value at $p^2=0$.

Figure 10

The imaginary part of the propagator is evaluated by setting $s=-p^2/m^2-iϵ$ in Eqs. (28), (44), for the optimal set $m=0.80\mathrm{GeV}$, $M=0.65\mathrm{GeV}$, $F_0=-0.65$ (solid line). The dashed line is a detail of the real part. The propagator is normalized by its finite value at $p^2=0$.

Figure 11

The adimensional scalar function ${\sigma }_M(x,s)$ of Eq. (c10) is shown as a function of the Euclidean momentum $s=-p^2/m^2$ for different values of the mass ratio $x=M^2/m^2$ corresponding to $M=2m$ (top), $M=m$ (center), and $M=m/2$ (bottom).

Figure 12

The adimensional scalar function ${\sigma }_p(x,s)$ of Eq. (c10) is shown as a function of the Euclidean momentum $s=-p^2/m^2$ for different values of the mass ratio $x=M^2/m^2$ corresponding to $M=2m$ (dotted line), $M=m$ (solid line), and $M=m/2$ (dashed line).

Figure 13

The scalar function $Z(p)$ as a function of the Euclidean momentum $p_E$ for three different values of the strong coupling ${\alpha }_s=0.3$, 0.6, 0.9.

Figure 14

The mass function $M(p)$ as a function of the Euclidean momentum $p_E$ for several values of the coupling ${\alpha }_s=0.6$, 0.9, 1.2, 1.5, 1.8 and $m=0.7\mathrm{GeV}$. The points are lattice data for unquenched chiral quarks and have been extracted from Fig. 4 of Ref. [8]. For each coupling the mass parameter $M$ is fixed by requiring that $M(0)\approx 0.32\mathrm{GeV}$, yielding $M=0.94$, 0.65, 0.49, 0.39, 0.318 GeV, respectively.

Figure 15

The mass function $M(p)$ as a function of the Euclidean momentum $p_E$ for three values of the mass scale $m=0.7$, 0.5, 0.2 GeV and ${\alpha }_s=0.9$. The mass parameter $M$ takes the values $M=0.65$, 0.62, 0.57 GeV, respectively. The points are the same lattice data of Fig. 14.

Figure 16

The mass function $M(p)$ as a function of the Euclidean momentum $p_E$ for $m=0.7\mathrm{GeV}$ and $M=0.65\mathrm{GeV}$. The coupling ${\alpha }_s$ is taken as a piecewise constant function, decreasing from ${\alpha }_s=0.9$ at $p_E<0.4\mathrm{GeV}$ down to ${\alpha }_s=0.3$ at $p_E\approx 2\mathrm{GeV}$. The points are the same lattice data of Fig. 14.

Figure 17

The real parts $\mathrm{Re}{S}_M(p^2)$, $\mathrm{Re}{S}_p(p^2)$ of the quark propagator and the spectral densities ${\rho }_M(p^2)$, ${\rho }_p(p^2)$ (imaginary parts) are shown as functions of the physical momentum $p^2=-p_E^2$ for $m=0.7\mathrm{GeV}$, ${\alpha }_s=1.2$, and $M=0.49\mathrm{GeV}$.

Figure 18

The same as Fig. 17 but for ${\alpha }_s=0.9$ and $M=0.65\mathrm{GeV}$.

Figure 19

The same as Fig. 17 but for ${\alpha }_s=0.6$ and $M=0.94\mathrm{GeV}$.

Figure 20

Details of the quark spectral functions for ${\alpha }_s=0.9$, $M=0.65\mathrm{GeV}$, $m=0.7\mathrm{GeV}$ (same as Fig. 18). The width of the peak at $p^2\approx 0.1$ depends on the finite value of the imaginary part $ϵ$ in the analytic continuation $p_E^2=-p^2-iϵ$. In the limit $ϵ\to 0$ the peak becomes a genuine $\delta$-function and disappears from the plot.

Figure 21

The quark spectral function $[p{\rho }_p(p^2)-{\rho }_M(p^2)]$ is shown as a function of the physical momentum $p^2=-p_E^2$ for $m=0.7\mathrm{GeV}$, ${\alpha }_s=0.9$, and $M=0.65\mathrm{GeV}$ (same as Fig. 18). The positivity condition of Eq. (64) is violated for $p^2>q_1^2\approx M^2$ below the two-particle threshold $q_2^2\approx (m+M{)}^2$.