High Temperature Dimensional Reduction and Parity Violation

TL;DR

At high temperature the electroweak sector reduces to a parity-preserving 3d effective theory, with parity violation arising only from higher-dimensional operators such as $O^{--}_4$. The leading P-odd, CP-odd operator is generated by top-quark loops with coefficient $c_4=\frac{2}{3} g^2 g_Y^2 T D_6$ where $D_6=\frac{7\zeta(3)}{128\pi^4 T^2}$, and $O^{--}_4$ is purely imaginary in Euclidean space, affecting lattice simulations. The work clarifies that P-violating effects are typically subleading for thermodynamics but can modify certain correlators and screening properties, and it establishes the parametric accuracy ${\cal O}(g^3)$ for dimensional reduction when including dim=6 operators.

Abstract

The effective super-renormalizable 3-dimensional Lagrangian, describing the high temperature limit of chiral gauge theories, has more symmetry than the original 4d Lagrangian: parity violation is absent. Parity violation appears in the 3d theory only through higher-dimensional operators. We compute the coefficients of dominant P-odd operators in the Standard Electroweak theory and discuss their implications. We also clarify the parametric accuracy obtained with dimensional reduction.

Figures (1)

• Figure 1: The graphs giving the dim=6 operator $O_4^{--}$ in eq. (\ref{['o4']}). The solid line is a (top) quark propagator, the dashed line a Higgs field and the wiggly line a gauge field. The symbols L and R indicate the handednesses of the quarks. For the dim=6 operator one needs the momentum dependence of the 4-leg diagram, which supplies two more indices $j,k$. An obvious 5-leg diagram is not shown.