〈 Berger | Dillon 〉

But for weak interactions such linear combinations (also for the charged pions, e.g. π^- = u'd) are still good, right? (Since they produce correct decay rates, cross ratios, for the pions.) What do you think?

well the thing is the thing is that QCD becomes strongly coupled at distances ~ 10 femtometer ~ 1/Λ_QCD s.. so we've never actually seen such a state at those energies .. indeed a bound state makes perfect sense there, but that's just not what we call a π^0 w. mass ~135MeV

i think about it in terms of distances cause it's easier for me to visualize .. at distances<1/Λ QCD is well defined in terms of quarks and gluons .. as E->Λ the coupling becomes strong and our perturbative expansion breaks down .. (cont)

So how do we predict the spectrum? .. well for the lighter mesons we can .. QCD has an approximate global SU(3)_L x SU(3)_R family symmetry which is also spontaneously broken to SU(3)_V by the quark condensate <qq> .. (cont)

So by Goldstone's theorem we will have 3^2 - 1 = 8 goldstone bosons .. these are the light pseudoscalar mesons K^\pm π^\pm etc Since the symmetry is Global this means that they should be massless, though which is not the case .. the small masses are treated as a pertbtn .. cont

However, Gell-Man showed with the "eightfold way" if we operate under the assumption that all asymptotic states are color singlets of SU(3)_c we get a spectrum that identically matches the ones we observe, in terms of quantum numbers .. cont

Now comes the fun part: Chiral perturbation theory Weinberg showed that we can still write down a meaningful lagrangian for the low energy DOFs in the following way: Write down all terms of the new DOFs, not forbidden by the global symmetries of the 'parent' theory (cont)

This is the good part.