Enhanced Global Symmetries and The Chiral Phase Transition

TL;DR

Problem: Can parity doubling near a chiral phase transition reveal an enhanced global symmetry in non-supersymmetric gauge theories? Approach: Use a linear effective Lagrangian with $SU_L(N_f)\times SU_R(N_f)$ symmetry (and, for pseudoreal representations, $SU(2N_f)$) to study how the spectrum can split into two sectors and under what parameter choices vector and axial-vector become degenerate. Findings: a specific set of couplings (e.g., $s=\tilde{g}$, $r=\tilde{g}^2/2$, $h=-\tilde{g}^2$) yields an enlarged symmetry and parity degeneracy, suppressing the parity-doubled sector’s contribution to the $S$ parameter; similar constructions apply to pseudoreal cases, with the enhanced symmetry further reducing $S$ and providing custodial protection upon electroweak gauging. Significance: provides an EFT framework linking near-critical dynamics, parity doubling, and electroweak precision constraints, motivating lattice tests and deeper dynamical understanding of whether such enhanced symmetries emerge in underlying gauge theories.

Abstract

We examine the possibility that the physical spectrum of a vector-like gauge field theory exhibits an enhanced global symmetry near a chiral phase transition. A transition from the Goldstone phase to the symmetric phase is expected as the number of fermions N_f is increased to some critical value. Various investigations have suggested that a parity-doubled spectrum develops as the critical value is approached. Using an effective Lagrangian as a guide, we note that parity doubling is associated with the appearance of an enhanced global symmetry in the spectrum of the theory. The enhanced symmetry would develop as the spectrum splits into two sectors, with the first exhibiting the usual pattern of a spontaneously broken chiral symmetry, and the second exhibiting an additional, unbroken symmetry and parity doubling. The first sector includes the Goldstone bosons and other states such as massive scalar partners. The second includes a parity-degenerate vector and axial vector along with other possible parity partners. We note that if such a near-critical theory describes symmetry breaking in the electroweak theory, the additional symmetry suppresses the contribution of the parity doubled sector to the S parameter.