Notes on strongly coupled QCD in the continuum

By continuum here we mean using methods different from lattice QCD, which is currently our main instrument for quantitative understanding of QCD physics at strong coupling.

What can we actually do apart from lattice simulations to study properties of QCD in this regime? Not much really. As recent minireview paper by M. Pennington explains, one approach to the problem would be solving Schwinger-Dyson equations at strong coupling.

While coupling grows, we have more and more equations to deal with – which is a kind of apparent. Indeed, at very small coupling we have roughly three equations (not taking color/flavor index structure into account) – one for bosonic propagator (gluon), one for fermionic propagator (quark) and one for the boson-fermion vertex, see the fig. below.

Note that solid lines here mean exact propagators, so if we need to unleash the power of perturbation theory (pardon my French), we need to write diagrammatic expansion for them as well. What we will find is that Schwinger-Dyson equation relates 3-point function (vertex) to 2,3 and 4-point functions – therefore, we need additional equation for the 4-point function which in turn has terms
containing 2,3,4 and 5-point functions etc. etc. ad infinitum.

If coupling is small, we might expect that the chain of these equations might be effectively closed (by, say, neglecting at some point contribution of 10-point function into Schwinger-Dyson equation for 9-point function) – after that we are turning on numerics and start getting results like large effective mass of quarks nearly massless in zero approximation (from experiment we know that current u and d quarks have masses < 10 MeV, while constituent quarks, partons, have masses as large as several hundred MeV, essentially, ).

Will we get large parton mass this way? Not really – one can explicitly check that solutions of the corresponding Schwinger-Dyson (SD) equation give you as long as the (bare) coupling is small. On the other hand, if the bare coupling is larger than 1, there exist solutions of truncated SD equations that do feature parton mass , althogh bare mass is taken to be zero.

This shows that generation of large mass of constituent quark is a non-perturbative phenomenon. Is it the end of the story – i.e., is the theory absolutely untreatable in this regime? Yes and no. As it turns out, one can use renormalization group and rewrite Swinger-Dyson equations in terms of renormalized coupling, not the bare one. Second, gauge invariance (or Ward-Takahashi identities at the practical level) help to strongly reduce the number of Schwinger-Dyson equations one has to solve. On the other hand, taking properly gauge invariance into account also means further complications – for example, appearance of ghosts in diagrammatic expansions (using axial gauge you can get rid of ghosts but in a sense it reappears since gluon propagator now depends on two functions of momentum, not one).