Parity breaking in 2 + 1 dimensions and finite temperature

Abstract

An expansion in the number of spatial covariant derivatives is carried out to compute the ζ-function regularized effective action of 2+1-dimensional fermions at finite temperature in an arbitrary non-Abelian background. The real and imaginary parts of the Euclidean effective action are computed up to terms which are ultraviolet finite. The expansion used preserves gauge and parity symmetries and the correct multivaluation under large gauge transformations as well as the correct parity anomaly are reproduced. The result is shown to correctly reproduce known limiting cases, such as massless fermions, zero temperature, and weak fields as well as exact results for some Abelian configurations. Its connection with chiral symmetry is discussed.

Thermal heat kernel expansion and the one-loop effective action of QCD at finite temperature

“…It is convenient to choose the "Polyakov gauge", in which A 0 is time independent and diagonal [45]. In SU(2),…”

“…Indeed, after choosing the Polyakov gauge there is still freedom to make further non stationary gauge transformations within this gauge. Such transformations (named discrete transformations in [45]) are of the form U (x 0 ) = exp(x 0 Λ), where Λ is a constant diagonal matrix . Its eigenvalues λ j , j = 1, .…”

“…In order to obtain the heat kernel x|e −τ K |x (a matrix in internal space) we use the symbols method, extended to finite temperature in [45,46]:…”

The induced Chern–Simons term at finite temperature

“…This requires to expand the current in powers of D i retaining only terms with ǫ ij and with precisely one D i . A suitable way to deal with the matrix element at coincident points, which combines well with gradient expansion, is to use the method of symbols [14], adapted to finite temperature [11,12] and improved by Pletnev and Banin [15,16]. This gives…”

“…This excludes ultraviolet finite pieces, which are always regular. The expansion in the number of spatial covariant derivatives [11,12] is appropriate for this kind of calculations, since this expansion preserves gauge in-variance order by order and, in addition, terms beyond second order are ultraviolet finite. A further simplification can be achieved by selecting only those terms which have abnormal parity, i.e., those containing a Levi-Civita pseudo-tensor, since the normal parity component of the effective action can be renormalized preserving all classical symmetries and is one-valued [23].…”