Pseudo-Hermitian approach to Goldstone’s theorem in non-Abelian non-Hermitian quantum field theories

We generalize previous studies on the extension of Goldstone’s theorem from Hermitian to non-Hermitian quantum field theories with Abelian symmetries to theories possessing a glocal non-Abelian symmetry. We present a detailed analysis for a non-Hermitian field theory with two complex two component scalar fields possessing an $SU(2)$ symmetry and indicate how our findings extend to the general case. In the PT-symmetric regime (parity and time-reversal) and at the standard exceptional point the Goldstone theorem is shown to apply, although different identification procedures need to be employed. At the zero exceptional points the Goldstone boson can not be identified. Comparing our approach, based on the pseudo-Hermiticity of the model, to an alternative approach that utilizes surface terms to achieve compatibility for the non-Hermitian system, we find that the explicit forms of the Goldstone boson fields are different.

Figure 1

Physical regions in parameter space bounded by exceptional and zero exceptional points as a function of (${\mu }^4/m_1^4$, $m_2^2/m_1^2$) for the theory expanded around the $SU(2)$-symmetry breaking vacuum. Left panel for $c_1=-c_2=1$ and right panel for $c_1=-c_2=-1$.

Figure 2

Physical regions in parameter space bounded by exceptional and zero exceptional points as function of (${\mu }^4/m_1^4$, $m_2^2/m_1^2$) for the theory expanded around the $SU(2)$-symmetry invariant vacuum.