Parity andCTRealization in QCD

Abstract

We show that an essential assumption in the Vafa and Witten's theorem on P and CT realization in vector-like theories, the existence of a free energy density in Euclidean space in the presence of any external hermitian symmetry breaking source, does not apply if the symmetry is spontaneously broken. The assumption that the free energy density is well defined requires the previous assumption that the symmetry is realized in the vacuum. Even if Vafa and Witten's conjecture is plausible, actually a theorem is still lacking.

Vafa-Witten theorem on spontaneous breaking of parity

“…We conclude that the concerns raised in [2,3] are valid in that the partition function would not be positive definite in a system exhibiting spontaneous parity violation. However, this situation will not arise for systems with a Hamiltonian bounded from below such that the density of states does not grow faster than the exponential of the energy.…”

“…The results summarized in the previous section were criticized in [2] based on the following observation. Suppose we construct a Landau-Ginzburg functional of the order parameter defined as X E /V , where V denotes the spatial volume of the system.…”

“…In section II we review the original VW argument. In section III we examine the large volume limit of the theory and show that, as claimed in [2], it is well defined only for the case where spontaneous symmetry breaking does not occur; yet we will argue that this deficiency does not invalidate the results of VW. We revisit the lattice arguments in section IV where we argue that the Aoki phases (at least for a large number of fermions) do not represent absolute minima of the effective potential but only metastable states 1 .…”

“…It was argued in [2] and later in [3] that the expression for the free energy on which the VW argument is based has an ill-defined infinite-volume limit and is therefore insufficient to determine the absence of spontaneous parity violation. Previously, Aoki [4] had noted the possibility that lattice QCD has a parity-violating phase which disappears in the continuum limit; this apparently presents a problem for the VW result since the original proof can be applied to the lattice-regulated theory as well as to the continuum.…”

Vacuum energy and θ-vacuum

“…However, some doubts have been recently raised about the validity of the Vafa-Witten proof of the theorem [7][8][9][10]. In fact, E θ is not always smooth at θ = π: there are cases where CP is spontaneously broken at that point.…”