What do the overlines above some terms in the Standard Model represent?

Question

Why do some terms in the Standard Model, like the Theta parameter (among others) sometimes have a little dash or bar, like an overline, above them?

For example the axion article on Wikipedia mentions the symbol $\bar{\Theta}$:

Strong CP problem

As shown by Gerard 't Hooft,[4] strong interactions of the standard model, QCD, possess a non-trivial vacuum structure that in principle permits violation of the combined symmetries of charge conjugation and parity, collectively known as CP. Together with effects generated by weak interactions, the effective periodic strong CP-violating term, $\bar{\Theta}$, appears as a Standard Model input – its value is not predicted by the theory, but must be measured. However, large CP-violating interactions originating from QCD would induce a large electric dipole moment (EDM) for the neutron. Experimental constraints on the currently unobserved EDM implies CP violation from QCD must be extremely tiny and thus $\bar{\Theta}$ must itself be extremely small. Since $\bar{\Theta}$ could have any value between 0 and 2π, this presents a "naturalness" problem for the standard model. Why should this parameter find itself so close to zero?

What does it mean?

P.S.: What about the complex conjugate? Could that be what the overlines are referring to in the middle of the Standard Model? Do the h.c. (hermitian conjugate) and the complex conjugate mean the same thing, mathematically, and refer (physically) to antimatter?

Answer 1

There are two possible sources of strong CP-violation, one from the quark sector and the other in the gluon sector. The QCD Lagrangian is

$$ \mathcal L_\mathrm{QCD}=-\frac14 F_{\mu\nu}^a F^{a\mu\nu}+\frac{g^2\Theta_\mathrm{YM}}{16\pi^2 N_c}F_{\mu\nu}^a\tilde F^{a\mu\nu}+\bar qi\not Dq-\bar qM e^{i\Theta_Q\gamma^5/N_f}q $$

with terms corresponding to gluon field strength, Yang-Mills vacuum parameter, quark kinetic term and quark mass term (coming from the Yukawa terms after Higgs SSB and a field redefinition) respectively. $\Theta_\mathrm{YM}$ and $\Theta_Q$ are in general complex numbers, and their respective terms individually violate CP-symmetry for non-zero values.

However, in the quantum path integral, we can perform a field redefinition on the quarks, eliminating the theta term in the quark mass matrix. This brings with it a chiral anomaly due to the non-invariance of the path integral measure under such a shift, and a change in $\Theta_Q$ induces a corresponding change in $\Theta_\mathrm{YM}$. Thus only their difference is a meaningful contributor to strong-CP violation, and this parameter is denoted by $\bar\Theta\equiv\Theta_\mathrm{YM}-\Theta_\mathrm{Q}$. For unknown reasons, this value is very finely-tuned to zero, causing little to no CP-violation in the strong sector - this is the strong-CP problem. Since $\bar\Theta$ appears in the factor $\exp(in\bar\Theta)$ in the path integral, it is periodic with period $2\pi$

You can also perform this analysis on U(1) theory to show that there is no CP-violation in quantum electrodynamics even without fine-tuning: there are no instantons to prevent the Yang-Mills theta term from decoupling.

tl;dr - there appear to be two CP-violating terms in the strong Lagrangian, only the difference $\bar\Theta$ of their respective parameters has a measurable effect.