# A pedagogical exposition of the hadron physics?

I am looking for a textbook/lecture notes/etc. on the basics of hadron physics.

I wish to understand how to construct the effective Lagrangian for pions and nucleons starting from the QCD Lagrangian. In other words, from $$\mathcal{L}_{QCD}=i\bar{Q}\hat{D}Q+(\text{mass terms})+ (\text{gauge fields})$$

derive

$$\mathcal{L}_{hadrons}=-\frac{f_{\pi}^2}4Tr(\partial^\mu U\partial_\mu U^{\dagger})+i\bar{N}\hat{D}N+\text{(gauge interactions)+(higher-order terms)}$$

I've read Peskin and Schroeder and Srednicki's textbooks and sort of understand the general idea but still missing a lot of important points. Among the unclear things are the following

• What is the exact quantitative relation between the quark fields and the hadron fields? To my understanding, this is not a direct one. However, in order to make computations something more then "hadrons are composed out of appropriate combinations of quarks" is needed.

• How do we construct terms in the hadron Lagrangian? It is relatively clear to me how pion terms appear, but the nucleon ones are confusing. For example, in Srednicki's book there are two different bases for nucleons non-trivially related to each other and it is not clear to me why do we interpret one of them and not the other as "real" protons and neutrons.

In brief, I would like a text on the subject which is i) as pedagogical and ii) as self-contained as possible. I'm not concerned too much with generality, detailed treatment of the lowest-terms only would suffice. Also, I would strongly favour more modern expositions since some old-fashioned terms and approaches are by themselves a great source of confusion for me.

What you are looking for are explanations of effective field theory (for example see this review by Burgess http://arxiv.org/abs/hep-th/0701053) and chiral perturbation theory in particular (for example see this review by Scherer http://arxiv.org/abs/hep-ph/0210398, and here are some slides by Tiburzi that look good at first glance: http://www.int.washington.edu/PROGRAMS/12-2c/week2/tiburzi_01.pdf).

To briefly answer your questions--I think you will end up being disappointed! The exact quantitative relationship between the quarks and hadrons is extremely complicated and not fully understood. Indeed fully understanding this is more or less what the Millenium Prize Problem on "Yang Mills Existence and Mass Gap" is about.

The relationship between quarks and hadrons is mainly that the lagrangian for the hadrons should reflect the symmetries of the underlying QCD lagrangian (and also including the effects of chiral symmetry breaking). This may seem like an uncomfortably indirect mapping, and it is true that it would be better to have a firm, direct map that showed in detail how to go from one picture to the other. Unfortunately, we have to make do with what we have: the spirit of effective field theory is that we don't have to know exactly how the low energy behavior emerges from the high energy one, it is enough to know what degrees of freedom we want to describe in the low energy theory and what symmetries the lagrangian should have. This effective lagrangian, with all possible operators consistent with all the symmetries, will produce the most general S-matrix consistent with the symmetries, and combined with power counting we can turn this into a useful scheme where we can determine a finite number of parameters from experiment and from those parameters predict the results of other experiments.

The method that is usually taken for Chiral Perturbation Theory is:

1. Write down the fields corresponding to the degrees of freedom you want to describe--for example the pions.

2. Write down all operators consistent with the symmetries of the theory (gauge invariance, lorentz invariance, spontaneously broken chiral symmetry, etc). At this stage you have an infinite number of operators.

3. Order these by they scaling dimension, which tells you at what order in an expansion in energy that they will appear. Decide what order in energy you want to work to. This will leave you with a finite number of operators.

4. Fix the coefficients of these operators by experiment.

In principle one could derive the coefficients from QCD. However there is much too hard to be done analytically. People also compute these coefficients using lattice QCD.

As to your question about Srednicki, I'm not exactly sure what you mean but typically the notion of an asymptotic state / particle that propagates to infinity is clearest when the terms in the lagrangian are quadratic in fields (the free part) has the mass and kinetic terms diagonal. So it might be that the field redefinitions you are talking about are needed to put the lagrangian in that form.