Effective Lagrangians for Qcd, Duality and Exact Results

I briefly discuss effective Lagrangians for strong interactions while concentrating on two specific lagrangians for QCD at large matter density. I then introduce spectral duality in QCD a la Montonen and Olive. The latter is already present in QCD in the hadronic phase. However it becomes transparent at large chemical potential. Finally I show the relevance of having exact non perturbative constraints such as t’Hooft anomaly conditions at zero and nonzero quark chemical potential on the possible phases of strongly interacting matter. An important outcome is that for three massless quarks at any chemical potential the only non trivial solution of the constraints is chiral symmetry breaking. This also shows that for three massless flavors at large quark chemical potential CFL is the ground state. 1. Effective Lagrangians for QCD In the non perturbative regime of strongly interacting theories effective Lagrangians play a dominant role since they efficiently describe the non perturbative dynamics in terms of the relevant degrees of freedom. Symmetries, anomalous and exact, are used to constrain the effective Lagrangians. An important point is that the effective Lagrangian approach is applicable to any region of the QCD or QCD-like phase diagram whenever the relevant degrees of freedom and the associated symmetries are defined. 1.1 Zero temperature and quark chemical potential At zero temperature and quark chemical potential the simplest effective Lagrangian describing a relevant part of the nonperturbative physics of the YangMills (YM) theory is the glueball Lagrangian whose potential is [1–4]: V = H 2 ln [ H Λ4 ] . (1)