Ghost spectral function from the spectral Dyson-Schwinger equation

We compute the ghost spectral function in Yang-Mills theory by solving the corresponding Dyson-Schwinger equation for a given input gluon spectral function. The results encompass both scaling and decoupling solutions for the gluon propagator input. The resulting ghost spectral function displays a particle peak at vanishing momentum and a negative scattering spectrum, whose infrared and ultraviolet tails are obtained analytically. The ghost dressing function is computed in the entire complex plane, and its salient features are identified and discussed.

Figure 1

Diagrammatic representation of the Dyson-Schwinger equation for of the inverse ghost propagator. Blue dots represent full 1-PI vertices. Internal lines stand for full propagators.

Figure 2

Diagrammatic notation used throughout this work: Lines stand for full propagators, small black dots stand for classical vertices, and larger blue dots stand for full vertices.

Figure 3

Ghost-gluon vertex dressing ${\lambda }_{Ac\overline{c}}^{(\mathrm{cl})}(p,q)$, see (13), data taken from [4]. The dressing is shown at the symmetric point $p^2=q^2=(p+q{)}^2$ for scaling and lattice-type decoupling solution, more details can be found in [4].

Figure 4

Reconstructed gluon spectral function (left) based on [4] and (28) and the respective gluon propagators (right). The spectral functions and propagators differ in the infrared and are labeled by $G_A^{(\mathrm{lat})}$ (blue) for our lattice-type input, and $G_A(0)$ $[{\mathrm{GeV}}^{-2}]=4.4$ (green), 1.9 (yellow). (a) Gluon spectral functions (scaling and decoupling), see (27), based on the reconstruction of the scaling spectral function in [4] (red-dashed line). The spectral functions dier only in the infrared, shown in the inset and (b) Euclidean gluon propagators obtained from the KL- representation (6) with the gluon spectral functions in Fig. 4. The small IR-difference for $\omega \underset{\sim }{<}0.7$ GeV shown in the inset in Fig. 4 translate into the IR-differences for $p\underset{\sim }{<}1$ GeV. The lattice data is taken from [69].

Figure 5

Ghost spectral functions (left) and respective Euclidean dressings (right), obtained with the decoupling gluon spectral functions in Fig. 4, with the same color coding by $G_A^{(\mathrm{lat})}$ (blue), $G_A(0)[{\mathrm{GeV}}^{-2}]=4.4$ (green), 1.9 (yellow). (a) Ghost spectral function: direct computation by iteration with the spectral DSE (18) and the gluon spectral function from Fig. 4 for ${G_A}^{(lat)}$ . The inlay also indictaes the -function contribution in the origin, indicated by an arrow. Its amplitude is given by the value of corresponding Euclidean dressing function $1/Z_c(p)$ at $p=0$. The squares show our best t, comp. Sec. 4 and (b) Euclidean ghost dressings: $1/Z_c(p)$ from KL-representation via the $\rho c$’s in Fig. 5 (squares), $1/Z_c(p)$ from the direct solution with the Euclidean DSE (straight lines), $1/Z_c(p)$ from the Euclidean fRG computations in [33]: we have taken the solutions with matching values of G0 A (dashed lines).

Figure 6

Ghost spectral functions: direct computation by iteration with the spectral DSE (18) and the gluon spectral functions from Fig. 4 with the same color coding, i.e., $G_A(0)[{\mathrm{GeV}}^{-2}]=4.4$ (green), 1.9 (yellow). The inlays also show the $\delta$-function pole in the origin, indicated by an arrow. The residue is given by the value of corresponding Euclidean dressing function $1/Z_c(p)$ at $p=0$. The squares show our best fits, comp. Sec. 4b.

Figure 7

Comparison of the ghost dressing function obtained via the KL representation (10) from the spectral function (solid line) and its fit (squares) for the different input gluon propagators.

Figure 8

Real (left) and imaginary (right) part of the ghost dressing function $1/Z_c(\omega )$ for the lattice-like gluon input (comp. Fig. 5) as a function of complex frequencies. Purely real $\omega$ correspond to Minkowski frequencies, purely imaginary $\omega$ to Euclidean frequencies. The color coding serves to guide the eye. The branch cut along the real frequency axis is clearly visible.

Figure 9

One of the two BSEs comprising the system that controls the scalar glueball formation. The blue (red) ellipses denote the glueball-gluon (ghost) BS amplitudes, and $k_{\pm }=k\pm P/2$, $q_{\pm }=q\pm P/2$, where $q$ denotes the loop momentum.

Figure 10

Convergence of an exemplary spectral function through iteration of the DSE. The color coding indicates the iteration number $n_{\mathrm{iter}}$. After about 10 iterations, the curves become visually indistinguishable, i.e., the iteration converges.

Figure 11

Ghost spectral functions (solid lines) compared to their fits via ansatz (32). The best fit parameters are listed in Table 1. The change of sign around 1.2 GeV is an imprint of the oscillations of the input reconstructed gluon spectral function from [4] and is discussed in Appendix pp3-s4.