The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

3.2. Effective Lagrangians 55
would observe parity doublets Ih) and Q'Alh) in the hadron speetrum. No sueh parity doubling
is observed in the real world. Another argument in favor of spontaneously breaking of the chiral
symmetry is the presenee ofvery light pseudoscalar mesons (pions in the case of Nf = 2), which ean
be associated with the Goldstone bosons. In fact, spontaneous breaking of the chiral symmetry is
also evident for other reasons like, for instance, large differenees in vector and axial-vector spectral
functions, results of lattice ealculations and so on [158]. Clearly, only SU(Nf)A is spontaneously
broken.7 Otherwise one would also observe scalar Goldstone bosons. A more evident reason is the
presenee of SU(Nf)V multiplets in the hadron spectrum. For instanee, SU(Nf = 2)v corresponds
to an ordinary isospin transformation group. The high quality of the isospin symmetry of the
strong interaction may be demonstrated by comparing the proton and neutron masses mp = 938.27
MeV and mn = 939.57 MeV, which are indeed very elose to eaeh other.
In faet, the ehiral symmetry is broken not only spontaneously but also explicitly due to the presenee
of the quark mass term .c�CD in the QCD Lagrangian (3.1). Let us now be more eonerete and
regard Nf = 2.
Furthermore, we will assume that the matrix M is already in diagonal form:
M ( � u �d )
�(mu + md) (� �) + �(mu -md) ( � �1 )
1 1
2(mu + md)I + 2(mu -md)T3 , (3.50)
where I is the unit matrix in 2 dimensions and T3 a Pauli matrix. Whereas the term proportional to
the quark mass difference breaks both SU(2)v and SU(2)A symmetries (or, equivalently, SU(2)L
and SU(2)R), the one proportional to mu + md remains SU(2)v invariant. It turns out that the
quark masses mu rv 3 --;-7 MeV and md rv 7 --;- 15 MeV are mueh smaller than the typieal hadronie
mass seale of the order of rv
as a small perturbation.
3.2 Effective Lagrangians
1
Ge V. Thus, the quark mass term .c�CD indeed ean be considered
In the last seetion we have discussed the general symmetry aspeets of QCD. Now we would like to
eoneentrate on the two fiavor case. Let us furt her switeh from quark and gluon to hadron degrees
of freedom. At present, one is not able to direetly determine the strueture and dynamics ofhadrons
from QCD. Therefore, it is useful to apply the powerful method of effective field theory (EFT)
technique to analyze the interactions between the hadrons at low energies. Some examples of
such a teehnique were already given in the previous ehapter. Clearly, one would like to extraet all
possible information from QCD in order to minimize the number of free parameters in the effective
Lagrangian. One expects that the ehiral symmetry of the QCD Lagrangian is very important for
EFT eonsiderations, sinee the lightest hadronic degrees of freedom, the pions, are elosely related
to its spontaneous breaking. We will see below that the spontaneously broken ehiral symmetry
indeed provides strong eonstraints on the strueture of pion-pion and pion-baryon interaetions.
The starting point in the EFT program is the construction of the effective Lagrangian. Apart
from the usual requirements like loeality, invarianee und er Lorentz transformations, parity, timereversal
invarianee and hermeticity the effective Lagrangian should also be chirally symmetrie.
7 In fact, Vafa and Witten have proven that the global SU(Nf)v symmetry of QCD can not be spontaneously
broken [159). This holds more generally for any vector-like (gauge) theory.