Goldstone Realization of Lorentz-symmetry and Supersymmetry at Finite Temperature*

The Ward-Takahashi identities of Lorentz symmetry are examined at finite temperature. It is shown that the Lorentz symmetry is broken with the Goldstone realization. The analogy with finite-temperature supersymmetry-breaking in an R-invariant model is explained. Submitted to Physics Letters B * Work supported by the Department of Energy, contract DE AC03 76SF00515. The supersymmetry has been known to be broken at finite temperature for some time.““’ Recently,““’ it was found that the supersymmetry is realized in the Goldstone fashion. Namely, at finite temperature the supersymmetry is broken so that the Ward-Takahashi identities are saturated by zero-momentum singularities of the field operators, which couple to the supercurrent. The purpose of this letter is to illustrate that an analogous phenomena occurs for Lorentz symmetry in a usual field theory, and thereby advance the understanding of the nature of the symmetry breaking at finite temperature. The formalism to treat field theories at finite temperature is well established. In particular, the real-time formalism (Therm0 Field Dynamics) Is1 gives a sound basis for dealing with real-time Green’s functions which are physical correlation functions in real space. In ref. 3, the real-time fermion propagator was calculated to the one-loop order in a (R-noninvariant) supersymmetric model, and it was shown to acquire a zero-momentum singularity, implying the Goldstone realization of supersymmetry. A similar calculation”1 in the l/IV-expansion yielded the same conclusion. These works motivated the systematic investigation of the Ward-Takahashi identities for the real-time Green’s function for the supersymmetric theories at the tree and one-loop levels.‘4’s1 Ref. 4 has also investigated the R-invariant model, in which the above phenomena does not occur. We have found that both at the tree and one-loop levels at finite temperature there always are some field-operators (even for the R-invariant model) that couple to the supercharge at zero momentum (even for a massive theory), thus confirming the Goldstone realization of the supersymmetry. Correspondingly, the Green’s functions of those operators have been demonstrated to have singularities at zero momentum. + In the following, we shall see that essentially the same phenomena is found for the Lorentz symmetry in field theories. For the purpose of the illustration, we take a real scalar field 4 of mass m and a cubic coupling X@ and observe t These singularity were called the ‘Goldstone mode” “’ or the ‘thermal superpair”. IS1