Exploratory study of the temperature dependence of magnetic vertices in SU(2) Landau gauge Yang-Mills theory

Vertices describe the interactions between the fundamental degrees of freedom and are therefore of vital importance in many ab initio descriptions of field theory, especially using functional methods. To this end, we present the first lattice study of the thermal behavior of (minimal) Landau-gauge SU(2) Yang-Mills three-point functions, i.e., three-gluon and ghost-gluon vertices. Focusing on the chromomagnetic sector, we find that the phase transition mainly affects the three-gluon vertex, while the ghost-gluon vertex is relatively inert.

Figure 1

Various kinematic configurations of the ghost-gluon vertex at temperatures below criticality. The lower right graph illustrates the vertex at the critical temperature as a function of ghost and gluon momentum. (a) Ghost-gluon vertex, orthogonal momenta with two equal; (b) ghost-gluon vertex, one momentum vanishing; (c) ghost-gluon vertex, all momenta equal; (d) ghost-gluon vertex, orthogonal momenta.

Figure 2

Ghost-gluon vertex as in Fig. 1, however, for temperatures above the transition. The lower right plot is at the largest temperature.

Figure 3

The ghost-gluon vertex at the phase transition compared to the results from zero and infinite temperature from Ref. [35]. (a) Ghost-gluon vertex, one momentum vanishing; (b) ghost-gluon vertex, all momenta equal.

Figure 4

Various kinematic configurations of the three-gluon vertex at temperatures below criticality. The lower right graph illustrates the vertex at the highest temperature in the figure as a function of two independent momenta. Only data with relative error smaller than 1 are shown. (a) Three-gluon vertex, orthogonal momenta with two equal; (b) three-gluon vertex, one momentum vanishing; (c) three-gluon vertex, all momenta equal; (d) three-gluon vertex, orthogonal momenta.

Figure 5

Three-gluon vertex as in Fig. 4, however, for temperatures above criticality.

Figure 6

The three-gluon vertex at the point of maximum enhancement at $0.96T_c$ and at $1.55T_c$, compared to the results from zero and infinite temperature from Ref. [35]. Only data with relative error smaller than 1 are shown. (a) Three-gluon vertex, one momentum vanishing; (b) three-gluon vertex, all momenta equal.

Figure 7

The three-gluon vertex as a function of temperature for the case of one gluon momentum vanishing at (a) the lowest momentum ($235\pm 35\mathrm{MeV}$) and (b) next-to-lowest momentum ($370\pm 80\mathrm{MeV}$). The variation in momenta stems from the different $\beta$ values. The vertex has been renormalized at 1.3 GeV to 1, using a linear extrapolation in $\beta$ of the zero-temperature data from [35]. The infinite-temperature results are plotted for comparison at $2.5T_c$. However, given the quite different scales and renormalization properties, any comparison should be made with care.

Figure 8

The three-gluon vertex at temperatures around the phase transition for two momentum configurations. The infrared enhancement of the vertex observed for $T<T_c$ quickly disappears in the high-temperature phase. (a) Three-gluon vertex, one momentum vanishing; (b) three-gluon vertex, all momenta equal.

Figure 9

Volume dependence of ghost-gluon vertex. Filled (empty) symbols denote the spatial lattice of size $N_x^3=30^3(20^3)$. (a) Ghost-gluon vertex, one momentum vanishing; (b) ghost-gluon vertex, all momenta equal.

Figure 10

Volume dependence of three-gluon vertex as in Fig. 9. Only data with relative error smaller than 1 are shown.