Description
There is no empirical evidence for CP conservation in the strong
interactions. As there generally is a renormalizable, parity-odd
coupling between the field strength and its dual, this requires an
explanation from theory.
I will therefore first review what interactions are present when
constructing an effective theory for hadrons from QCD. But I will also
point out, that from such considerations alone, it cannot be decided
whether the effective interactions (that, e.g., give mass to eta-prime)
are misaligned (CP violation) or aligned (no CP violation) with the
quark mass phase.
To see whether or not there is a material effect of the parity-odd
operator in QCD requires therefore an understanding of how field
configurations from different topological sectors contribute to the path
integral or, in canonical quantization, whether topology implies
different ground states that are in general not parity eigenstates. To
that end, I will review the pertinent homeomorphisms between the SU(2)
subgroups of the strong interactions and the boundaries of spacetime or
spatial hypersurfaces.
As for the Euclidean path integral approach, I will note that pure gauge
configurations on the boundary only follow when the latter is placed at
infinity. Picard-Lefschetz theory then implies that steepest-descent
integration contours cover all field configurations within a topological
sector that one can find in the infinite spacetime volume. Consequently,
the limit of infinite spacetime volume must be taken before summing over
sectors, and it turns out that parity violation then vanishes. Commuting
these limits, as tacitly done in standard approaches, corresponds to a
singular deformation of the original Cauchy contour, falsely suggesting
parity-violating results.
Regarding canonical quantization, I will note that the usually
considered theta-vacua are not properly normalizable, which is at odds
with the probability interpretation from the axioms of quantum
mechanics. The root of this problem is the summation over
gauge-redundant configurations in the orthonormality relations among
theta-vacua. Imposing that (in temporal gauge) the wave functionals and
Hilbert-space operators are well-defined when the inner product covers
each physical field configuration one time and one time only, I recover
that the consistent states satisfy Gauß' law and are moreover
eigenstates of parity.
References:
2001.07152 [hep-th]
2403.00747 [hep-th]
2404.16026 [hep-ph]
