Behavior of logarithmic branch cuts in the self-energy of gluons at finite temperature

There have been many studies of thermal Green functions in gauge field theories [1–7], which show that their behavior at finite temperature is rather different from the one at zero temperature. In particular, it was recently pointed out by Weldon [8] that in QED, the logarithmic branch cut singularities cancel to one loop-order, in the thermal self-energy of the electron. The purpose of this note is to show that in the Yang-Mills theory, a somewhat similar behavior occurs in the full gluon self-energy, which includes finite temperature effects. Of course, in this theory, the massless gluons are quite modified by these effects and the gluon propagator requires the Braaten-Pisarski resummation. Nevertheless, it is interesting to remark that, even before such a procedure is carried out, the one-loop log(−k2) terms cancel in the sum of the T = 0 and the T 6= 0 contributions to the gluon self-energy. As we shall see, this happens because the log(−k2) terms appear in the thermal part of the self-energy only in the combination log(−k2/T ). But one can show that the log(T ) contributions have the same structure as the ultraviolet divergent terms which occur at zero temperature [9]. Consequently, the log(−k2/T ) terms combine directly with the log(−k2/μ2) contributions which occur at T = 0 (μ is the renormalization scale), so that the log(−k2) terms cancel in a simple way in the thermal self-energy of the gluon. The branch cut in the log(−k2) contribution at T = 0 is associated with the imaginary part of the self-energy, which gives the rate of decay of a time-like virtual gluon into two real gluons. Although this contribution cancels at T 6= 0, there appear then additional, temperature-dependent logarithmic branch points. These singularities indicate processes not available at zero temperature, where particles decay or are created through scattering in the thermal bath. To one-loop order, the thermal self-energy of gluons generally depends on three structure functions, Π , Π and Π [10]