Gap equation in scalar field theory at finite temperature

We investigate the two-loop gap equation for the thermal mass of hot massless gφ theory and find that the gap equation itself has an imaginary part which along with the finite terms contains divergent pieces too. This indicates that it is not possible to find the real finite thermal mass as a solution of the gap equation beyond g order of perturbation theory. PACS: 11.10.Wx Electronic address: [email protected] 1 It is well-known [1] that if a theory contains massless bosonic field such as, QCD or massless scalar theory with g2φ4 interaction, then at very high temperature (T ) the perturbative computations beyond certain order of coupling constant are afflicted with infrared (IR) singularities. In the case of massless g2φ4 theory, the one loop contribution to two point function shows that the fields are screened and the screening mass (Debye mass) is found to be of order gT [2, 3]. However, the result of two loop corrections to it is found to be IR divergent. A natural way of avoiding this IR divergences in the two loop computation is to use the dynamically generated one loop thermal mass gT as the lower cut-off of the momentum integration. As a result one finds the appearance of a new g3 order correction to two point function which, although consistent with the spirit of perturbation theory, has not been predicted from the usual perturbative expansion in powers of coupling constant (g) at zero temperature. In addition to that there are infinite number of higher order diagrams that contribute to this particular g3 order which in turn is the signature of the break-down of usual perturbation theory at very high temperature. Moreover, it also suggests that one has to resum this infinite number of diagrams to correctly calculate this g3 order contribution [2, 4, 5] to two-point function. However remaining within the framework of perturbation theory one can in principle be able to calculate the thermal mass to any order of coupling constant by solving the gap equation [6, 7]. In order to obtain the gap equation, the functional integral formulation may be used, and we shall briefly discuss this method following Jackiw et. al. [7]. Consider a massless g2φ4 theory in d dimension (d = 4− 2ǫ) described by the lagrangian L = 1 2 ∂μφ∂ φ− μ g 2 4! φ , (1) and the partition function is given by Z = ∫ Dφe , (2) where S(φ) = ∫ ddxL(φ(x)). We introduce a loop counting parameter l and write down Z as Z = ∫ Dφe i l S( √ lφ) . (3) In the usual perturbation theory we separate the quadratic part of S(φ) and expand the exponential of the remainder in powers of l. To obtain a gap equation for a possible mass m, we add and subtract Sm = − 2 2 ∫ ddxφ2(x), which of course changes nothing (at least in the classical level, the equation of motion remains the same). S = S + Sm − Sm (4) 2 We recognize the loop expansion by expanding S + Sm in the usual way, but taking −Sm as contributing at one loop higher. These can be accounted systematically by replacing Eq. (4) with an effective action Sl. Sl = 1 l [S( √ lφ) + Sm( √ lφ)− Sm( √ lφ)] . (5) Starting from this effective action the self energy Σ of the complete propagator can be calculated to any order in l and set l = 1 at the end of the calculation. The gap equation is obtained by demanding that Σ does not shift the mass m, i.e., Σ(p) |p2=m2= 0 (6) Furthermore, in order to get a real solution of m from Eq. (6) one has to ensure that the imaginary part of Σ(p) at p2 = m2 is zero. We apply the above mentioned ideas to g2φ4 theory (described by Eq. (1)) at finite temperature using real time formalism. Let us recall that in the real time formalism [8, 9] the thermal propagator has a 2 × 2 matrix structure, the 1 − 1 component of which refers to the physical field, the 2− 2 component to the corresponding ghost field, with the off-diagonal 1− 2 and 2− 1 components mixing them. The propagator used here is given by