Electric-magnetic asymmetry of the $A^2$ condensate and the phases of Yang-Mills theory

We study the finite-temperature behavior of the $A^2$ condensate in the Landau gauge of $SU(2)$ Yang-Mills theory on the lattice in a wide range of temperatures. The asymmetry between the electric (temporal) and magnetic (spatial) components of this unconventional dimension-2 condensate is a convenient ultraviolet-finite quantity which possesses, as we demonstrate, unexpected properties. The low-temperature behavior of the condensate asymmetry suggests that the mass of the lowest thermal excitation in the condensate is unexpectedly low, about 200 MeV, which is much smaller than the glueball mass. The asymmetry is peaking at the phase transition, becoming a monotonically decreasing function in the deconfinement phase. A symmetric point is reached in the deconfinement phase at a temperature approximately equal to twice the critical temperature. The behavior of the electric-magnetic asymmetry of the condensate separates the phase diagram of Yang-Mills theory into three regions. We suggest that these regions are associated with the condensed, liquid, and gaseous states of the confining gluonic objects, the Abelian monopoles.

Figure 1

The normalized electric-magnetic asymmetry of the $A^2$ condensate, ${\Delta }_{A^2}(T,m)/T^2$, as a function of the normalized temperature, $T/m$, in the Abelian Higgs model. The horizontal dashed line in the plot shows the high-temperature limit (27), for $T\gg m$, which recovers the case of photodynamics. The inset illustrates the low-temperature behavior of the condensate asymmetry (28).

Figure 2

The electric-magnetic asymmetry of the $A^2$ condensate (9) for $SU(2)$ gauge theory normalized by the critical temperature squared as a function of $T/T_c$. The high-temperature fit (39) with the best fit parameters (40) is shown by the solid line. The horizontal dashed line corresponds to ${\Delta }_{A^2}=0$, while the vertical dashed line marks the critical temperature. The symmetric point is explicitly indicated.

Figure 3

The same as in Fig. 2 but for electric-magnetic asymmetry of the $A^2$ condensate (9) normalized by the temperature $T^2$. The inset zooms in on the high-temperature region of the fit. The peaked value at $T=T_c$, Eq. (40), and the asymptotic value at $T\to \infty$, Eq. (36), are indicated by vertical dotted and horizontal dash-dotted lines, respectively.

Figure 4

The behavior of the electric-magnetic asymmetry in the low-temperature region. The lines represent the fits by the function (41), which are discussed in the text. Notice the different normalization of the expression under the logarithm: in (a) the normalization factor is $1/T_c^2$, while in (b) this factor is $1/T^2$.