I'm trying to set up the problem of deriving the thermodynamics of hadronic matter. I know how to proceed in the case of an effective description such as mean field (Walecka/linear sigma model) but I'm trying to start from QCD. I do not wish to actually obtain the partition function, but I want to at least set up the complete problem.

In the mean field approximation, we usually write baryon and meson fields and baryon/meson couplings as the interactions, meaning that we also bypass the problem of describing the hadrons as QCD bound states. On the extreme oposite, I know how to proceed in the case of pure quark matter (it's just the "QCD Lagrangian").

How do I write the full Lagrangian? How do I represent the hadrons and the hadron-hadron strong (well, and EM) interaction without shifting to an effective approximation? Can it be done with quarks and gluons as the basic degrees of freedom?

I don't have any compromise with the "practical usefulness" of the result and I'm not particularly interested on the peculiarities of the dynamics of the bound states.

• There is presently no way to do so, really because confinement is not 100% understood. The simplest way to get an analytic result is indeed to use some specific low-energy method (e.g. the chiral Lagrangian) which however doesn't make direct contact with QCD. In LHC physics, the calculations are actually more ab-initio: the proton scattering is factored as a problem of pulling partons out of hadrons (using pdfs) and then scattering partons, which is normal QCD. But again, these pdfs cannot be calculated by hand and need to be measured or simulated. Nov 27 '13 at 18:05
• Yeah, while in theory the standard model Lagrangian (in terms of quark and gluon fields etc.) is enough to fully describe hadron scattering, actually seeing that it does is a massively complicated procedure that many hundreds of people are working on. Nov 27 '13 at 18:33
• So I cannot even set up the problem? As I said, I'm not actually trying to solve it.
– user34134
Nov 27 '13 at 18:35
• I'd disagree with that statement. One way to make such a calculation precise would be to actually calculate the pdfs. They are well-defined quantities in field theory, namely a bunch of matrix elements you can write as path integrals. [Edit: as a reference you can look up the "Handbook of perturbative QCD" by Sterman and collaborators. It really explains how to approach a hadron collision from the QCD standpoint.] Nov 27 '13 at 21:24
• @Vibert thank you for you comments. I'll check that reference in detail. I'm interesting in everything they may do prior to the perturbative expansion. In particular, how to set up the hadron fields and what are the form of the couplings in a general and exact way.
– user34134
Nov 27 '13 at 23:32