Probing unquenching effects in the gluon polarization in light mesons

We introduce an extension to the ladder truncated Bethe-Salpeter equation for mesons and the rainbow truncated quark Schwinger-Dyson equations which includes quark-loop corrections to the gluon propagator. This truncation scheme obeys the axialvector Ward-Takahashi identity relating the quark self-energy and the Bethe-Salpeter kernel. Two different approximations to the Yang-Mills sector are used as input: the first is a sophisticated truncation of the full Yang-Mills Schwinger-Dyson equations, the second is a phenomenologically motivated form. We find that the spectra and decay constants of pseudoscalar and vector mesons are overall described well for either approach. Meson mass results for charge eigenstate vector and pseudoscalar meson masses are compared to lattice data. The effects of unquenching the system are small but not negligible.

Figure 1

A diagrammatical representation of the coupled system of ghost, gluon and quark Schwinger-Dyson equations. Filled blobs denote dressed propagators and empty circles denote dressed proper vertex functions.

Figure 2

Schwinger-Dyson equations for the quark and gluon propagator under the truncation scheme of the phenomenological model. All vertices are tree level. Blobs represent fully dressed propagators. The seed term of the gluon equation is dressed with the Gaussian form Eq. (18) (see text).

Figure 3

Comparison of the quenched and unquenched ghost and gluon dressing functions with recent lattice data [33, 34, 36]. The sea-quark masses are $m_{u/d}\simeq 16\mathrm{MeV}$, $m_s\simeq 79\mathrm{MeV}$ in the lattice simulations and $m_{u/d}\simeq 3.9\mathrm{MeV}$, $m_s\simeq 84\mathrm{MeV}$ in the SDE-approach, cf. Table .

Figure 4

Comparison of the quenched and unquenched interactions used in the two models. In the full-SDE framework the interaction is given by the product $A(p^2)\alpha (p^2)$ whereas for the phenomenological model the equivalent function is $g^2\Delta (p^2)p^2/4\pi$. The unquenching effects are calculated with physical sea quarks, see Table  for the corresponding values of the quark masses and interaction parameters.

Figure 5

Comparison of the quenched and unquenched quark mass and wave functions (left and right panels, respectively) for the full-SDE setup. See text for details of the parameter configurations.

Figure 6

Comparison of the quenched and unquenched quark mass and wave functions (left and right panels, respectively) for the phenomenological model. See text for details of the parameter configurations.

Figure 7

Pseudoscalar (left panels) and vector (right panels) meson masses as functions of the quark mass parameter. We compare the quenched and degenerate unquenched cases for both the full-SDE (upper panels) and phenomenological model (lower panels). Recall that there is no direct comparison between the quark masses of the full-SDE and phenomenological model.

Figure 8

Quark mass function at $p^2=0$ as a function of the quark mass parameter. We compare the quenched and degenerate unquenched cases.

Figure 9

The difference between the pseudoscalar (left panel) and vector (right panel) meson masses and twice the quark mass function $M(p^2=0)$ as functions of the quark mass parameter. We compare the quenched and degenerate unquenched cases.

Figure 10

Vector meson masses as a function of pseudoscalar meson masses. We compare the quenched and degenerate unquenched cases. The lattice results are taken from Refs. 58, 59.

Figure 11

Ghost-gluon vertex DS equation.

Figure 12

Sketch of the closed contour in the complex $z$ plane.

Figure 13

Vector meson masses as a function of pseudoscalar meson masses in the phenomenological model for different values of the renormalization scale $\mu$. We compare the quenched and degenerate unquenched cases.