Why is "a phase for the QCD vacuum" considered a fundamental constant and what does it even mean?

Question

We can read about fundamental physical constants in Wikipedia:

There is, however, no single "correct" way of enumerating them, as it is a matter of arbitrary choice which quantities are considered "fundamental" and which as "derived". Uzan (2011) lists 22 "unknown constants" in the fundamental theories, which give rise to 19 "unknown dimensionless parameters", as follows:

the gravitational constant G,

the speed of light c,

the Planck constant h,

the 9 Yukawa couplings for the quarks and leptons (equivalent to specifying the rest mass of these elementary particles),

2 parameters of the Higgs field potential,

4 parameters for the quark mixing matrix,

3 coupling constants for the gauge groups SU(3) × SU(2) × U(1) (or equivalently, two coupling constants and the Weinberg angle),

a phase for the QCD vacuum.

The number of 19 independent fundamental physical constants is subject to change under possible extensions of the Standard Model, notably by the introduction of neutrino mass (equivalent to seven additional constants, i.e. 3 Yukawa couplings and 4 lepton mixing parameters).

Besides the rather obscure and hidden couplings of the photon, gluons, and W and Z to matter, it's the the last constant I'm interested in: a phase for the QCD vacuum. What does it mean and why is it considered a fundamental constant?

Answer 1

They're referring to the $\theta$-angle of the $\theta$-vacuum. See this question and my answer there for a rough overview of why there's a $\theta$-vacuum for an unbroken non-Abelian gauge theory like QCD.

In practice, the $\theta$-angle measures how strongly its corresponding gauge interaction violates CP symmetry. That we observe no CP violation by the strong force even though we don't really know a reason for why $\theta$ should be close to zero is an observation often called the strong CP problem.