Clearly, this U(1)a problem that Weinberg discovered needed to be solved, and so Gerard ‘t Hooft, a brilliant Dutch physicist, looked into it, and found the solution. There was no U(1)a problem, he said, because the vacuum of QCD is so complicated that there is no true U(1) a symmetry, even though the QCD Lagrangian makes it look like there is. Unfortunately, the newly-discovered complex QCD vacuum came with an even larger problem than Weinberg’s U(1)a problem.
Discovering the complicated nature of the QCD vacuum added a new term to the QCD Lagrangian – one that contained something called “the quark mass mixing phase”, or “theta”. Due to the kind of gauge theory that QCD is (nonabelian), such a parameter is required. Theta comes with a lot of baggage: if it is not zero or less than zero, something called “CP symmetry” is violated.
The “C” in CP stands for charge conjugation: for this to be preserved, the physical system in question must be invariant under swapping a particle for its antiparticle. Parity (the “P” in CP), on the other hand, requires that a physical system be invariant under inversion of its spatial coordinates. Therefore, CP symmetry tells us that if we were to swap out particles for antiparticles, and invert the spatial coordinates of our system, the new system with the switched particles would be physically equivalent to the old system that we had before we messed around with the particles and the coordinates!
Now CP symmetry is violated in weak interactions, but not in strong or electromagnetic interactions. It makes sense why it’s not violated in electromagnetic interactions, but since it’s preserved in QCD, this means that the theta term in the QCD Lagrangian must be zero or, well, basically zero. There isn’t any reason why this should be the case, though, because of all the values that theta can lie between (from -pi to pi), it is absurd that some how, miraculously, it found itself a value in which we get CP symmetry. This is a so-called “fine tuning problem” in the standard model, and we call it “The Strong CP Problem”.
The Strong CP Problem is one of nature’s great unsolved mysteries, and it’s up there on the list with quantum gravity and dark energy . We know so much about the standard model, and we’ve even now found the Higgs boson, but we have no clue why on earth this theta in quantum chromodynamics is zero (or very close to zero).