One-loop thermodynamic potential of charged massive particles in a constant homogeneous magnetic field at high temperatures

Abstract

The explicit expressions for the high-temperature expansions of the one-loop corrections to the omegapotential coming from charged scalar and Dirac particles and, separately, from antiparticles in a constant homogeneous magnetic field are derived. The explicit expressions for the non-perturbative corrections to the effective action at finite temperature and density are obtained. Thermodynamic properties of a gas of charged scalars in a constant homogeneous magnetic field are analyzed in the one-loop approximation. It turns out that, in this approximation, the system suffers a first-order phase transition from the diamagnetic to the superconducting state at sufficiently high densities. The improvement of the one-loop result by summing the ring diagrams is investigated. This improvement leads to a drastic change in thermodynamic properties of the system. The gas of charged scalars passes to the ferromagnetic state in place of the superconducting one at high densities and sufficiently low temperatures, in the high temperature regime. * Recall that the Lagrangian of the effective action L(1) ef f = −Ω (1) . In particular, it follows from (7), (8) that, in the high-temperature expansion of the total one-loop omega-potential, the vacuum contribution is canceled out by the analogous term in the omega-potential coming from real particles (for fermions in QED see [38] and for the general case see [27,28,32]).The following stability conditions are assumed in formulas (1), (7) and (8):

High-temperature expansion of the one-loop effective action induced by scalar and Dirac particles

“…The term with l = −1 is absent in the expansion since the eta-function does not possess a singularity at z = 1 unlike the zeta-function. The contributions at the even positive l in the second term in (18), (19), and (20) vanish. With the aid of formulas (15), (16) for the singularities of σ s ν (µ), one can also check directly that expressions (19), (20) are finite when ν → 0.…”

“…In this section, we derive formula (9) of [19] for the nonrenormalized one-loop Ω-potential of fermions at zero temperature and nonzero chemical potential with the contribution of the energy of zero-point fluctuations. The formula may be of use when the solution of the Dirac equation reduces to the solution of KG equation with a self-adjoint operator.…”

“…6, and so we do not dwell on it here.As is known [9,10,[26][27][28], the high-temperature expansion of the one-loop Ω-potential without the vacuum contribution can be used to obtain the energy of vacuum fluctuations at zero temperature. Therefore, we shall derive a nonperturbative representation of the vacuum energy for the Dirac fermions in a stationary electromagnetic field of a general configuration without resorting to the Wick rotation prescription (as for bosons, see [17][18][19]). This vacuum energy is real-valued for not too wild fields and corresponds to the standard definition of the vacuum of quantum fields on stationary backgrounds (see, e.g., [29]).…”

“…Besides, we shall derive the high-temperature expansion of the separate contributions of particles and antiparticles to the Ω-potential. This expression can be used to obtain the number of particle-antiparticle pairs in the system [19,33] in the high-temperature regime. The leading order terms coincide with that known in the literature [33].In [17,18], we developed a general procedure of how to obtain the complete high-temperature expansion when the spectral problem is reduced to the solution of the KG type equation with a self-adjoint operator.…”

“…Besides, we shall derive the high-temperature expansion of the separate contributions of particles and antiparticles to the Ω-potential. This expression can be used to obtain the number of particle-antiparticle pairs in the system [19,33] in the high-temperature regime. The leading order terms coincide with that known in the literature [33].…”