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

Abstract

We give a simple argument for the cancellation of the log(−k 2 ) terms (k is the gluon momentum) between the zero-temperature and the temperature-dependent parts of the thermal self-energy.There have been many studies of thermal Green functions in gauge field theories [1][2][3][4][5][6][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(−k 2 ) terms cancel in the sum of the T = 0 and the T = 0 contributions to the gluon self-energy. As we shall see, this happens because the log(−k 2 ) terms appear in the thermal part of the self-energy only in the combination log(−k 2 /T 2 ). But one can show that the log(T 2 ) contributions have the same structure as the ultraviolet divergent terms which occur at zero temperature [9]. Consequently, the log(−k 2 /T 2 ) terms combine directly with the log(−k 2 /µ 2 ) contributions which occur at T = 0 (µ is the renormalization scale), so that the log(−k 2 ) terms cancel in a simple way in the thermal self-energy of the gluon. The branch cut in the log(−k 2 ) 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 = 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 T , Π L andwhere the projection operators P T,L µνare transverse with respect to the external four-momentum k µ and satisfy: 6,7]. The non-trnsverse projection operator P C µν can be written in the plasma rest frame as follows [10]Although Π C vanishes at T = 0 because of the Slavnov-Taylor identity, it is in general a non-vanishing function of the temperature, so that k µ Π µν = 0 for the exact self-energy. We will discuss here, for definiteness, the retarded thermal self-energy of the gluon, which is obtained by the analytic continuation k 0 → k 0 + iǫ.(A rather similar analysis can be made in the case of the time-ordered self-energy, following the approach presented in reference [11]). In order to illustrate in a simple way the mechanism of the cancellation of the log(−k 2 ) contributions, let us first consider the special case of...