Photon polarization tensor in the light front field theory at zero and finite temperatures Silvana Perez * Faculdade de Fı ´sica, Universidade Federal do Para ´, 66075-110, Bele ´m, Para ´, Brazil Charles R. Silva † Instituto Federal do Para ´, 66093-020, Bele ´m, Para ´, Brazil Stefan Strauss ‡ Institut fu ¨r theoretische Physik, Justus-Liebig-Universita ¨t, 35392 Gießen, Germany (Received 11 February 2012; published 21 May 2012) In this work, we consider the light front quantum electrodynamics in (3 þ 1) dimensions and evaluate the photon polarization tensor at one loop for both zero and finite temperatures. In the first case, we apply the dimensional regularization method to extract the finite contribution and find the transverse structure for the amplitude in terms of the light front coordinates. The result agrees with one-loop covariant calculation. For the thermal corrections, we generalize the hard thermal loop approximation to the light front and calculate the dominant temperature contribution to the polarization tensor, consistent with the Ward identity. In both zero as well as finite temperature calculations, we use the oblique light front coordinates. DOI: 10.1103/PhysRevD.85.105021 PACS numbers: 11.10.Wx, 11.15.Bt, 11.30.Ly, 12.20.Ds I. INTRODUCTION In recent years, light front (LF) quantized field theories have been successfully generalized to finite temperature. The light front frame was introduced by Dirac [1], and the quantization of field theories on the null-plane has found applications in many branches of physics [2,3]; see also Ref. [4] for a review and a guide to the extensive literature. The proper thermal description of LF quantized field theories was pointed out in a number of publications, including Refs. [5–10]. It was shown that the thermal contributions to the self-energy in scalar theories at one loop [11] coincide with the results from conventional calculations. Furthermore, the anomaly term and fermion condensate at zero and finite temperature in the LF Schwinger model match their conventional counterparts [7]. Thermodynamical properties were computed nonper- turbatively using discrete light cone quantization [12] in the massive Schwinger model [13], two-dimensional supersymmetric theories [14,15], and, in four dimensions, for SUðN c Þ pure gauge theory in the large N c approxima- tion using the transverse lattice approach [16]. Moreover, the formalism was applied investigating the in-medium properties of quark bound states [17,18] and the nontrivial vacuum structure of the Unruh effect [19]. Particularly in Refs. [8,11], it has been shown that there is a convenient coordinate system: the oblique one, in which the study of thermal effects is straightforward. One can collect both the usual light front coordinates as well as the oblique one in the general light cone coordinate frame,  t ¼ t þ z;  z ¼ At þ Bz;  x ¼ x;  y ¼ y; (1) where A and B are arbitrary real constants with the restric- tion that A  B Þ 0 and x  ¼ðt; x; y; zÞ are the usual Minkowski coordinates. In particular, for A ¼ 0, B ¼ 1, Eq. (1) represents the oblique light front coordinates (OLFC) proposed in Ref. [5] and used in Refs. [7,11,13] to carry out the discussions of statistical mechanics within the LF. One of the distinct features of LF dynamics is the energy-momentum dispersion relation which is linear in the LF energy. This property is also present in the OLFC coordinates. Accordingly, the propagators in the OLFC momentum space behave differently at  k 0 !1. For in- stance, the propagator of a scalar particle reads iGðkÞ¼ i k 2  m 2 þ i ¼ i 2k 0 k 3  k 2 1  k 2 2  k 2 3  m 2 þ i ¼ i 2k 0 k 3  k 2 i  m 2 þ i ; i ¼ 1; 2; 3: (2) Because of the 1=k 0 dependence of the propagator, the computation of loop integrals is more demanding. One has to properly take into account contributions from the arc contours, used to close the complex integration at infinity, and singular point contributions from moving poles to recover the correct covariant result [20,21]. Particularly, in Ref. [21], various techniques were used to demonstrate the equivalence between equal time and light front dynamics for certain one-loop computations at zero temperature. * silperez@ufpa.br † charles.rocha@ifpa.edu.br ‡ stefan.strauss@theo.physik.uni-giessen.de PHYSICAL REVIEW D 85, 105021 (2012) 1550-7998= 2012=85(10)=105021(8) 105021-1 Ó 2012 American Physical Society