Nucleon-Pion Lagrangian

We are now ready to consider another set of composite particles,
the nucleons. We know that there are two nucleons in Nature, of almost
equal mass, the neutron and the proton, so they can be combined into the
iso-spinor

Anyhow, the nucleons contain not only the three (valence) quarks, but also plenty of gluons, and plenty of virtual quark pairs, and we are unable to find what exactly this state is. Therefore, here we follow the general strategy of attributing elementary fields to composite particles. Before we arrive at sufficiently high energies, or small distances, at which the internal structure of composite objects becomes apparent, we can safely live without knowing exactly how the composite objects are constructed.

As usual, having defined elementary fields of particles that we want
to describe, we also have to postulate the corresponding Lagrangian
density. And as usual, we do that by writing a local function of fields
that is invariant with respect to all conserved symmetries. When we
have the nucleon and pion fields at our disposal, and we want to construct
the Lorentz and chiral invariant Lagrangian density, the answer is:

There is no magic in this expression - one only has to properly identify generators of the O(4) group with generators and . This is unique, once we fix which components (1,2,3 in our case) transform under the action of . Under the chiral rotation about the same angle , the nucleon fields transform by infinitesimal transformation , within the spinor representation of Eq. (28), i.e.,

It is now a matter of a simple algebra to verify that Lagrangian density (41) remains invariant under chiral rotations of fields (42) and (43). Note that the first term in Eq. (41) is separately chiral invariant, so we could multiply the second term by an arbitrary constant .

We can now proceed with the transformation to
better variables and , given by Eq. (35),
which gives,

There are several fantastic results obtained here. First of all, the nucleon mass term appears out of nowhere, and the nucleon mass,

is given by the chiral-symmetry-breaking value of the field. In principle, we could begin by including the nucleon mass term already in the initial Lagrangian density (41). This is not necessary - the nucleon mass results from the same chiral-symmetry-breaking mechanism that pushes scalar mesons up to high energies. Second, the third term in Eq. (44) gives the coupling of nucleons to mesons, and in the potential approximation it yields the long-distance, low-energy tail of the nucleon-nucleon interaction, i.e., the one-pion-exchange (OPE) Yukawa potential Yuk35. Derivation of this potential from Lagrangian density (44) requires some fluency in the methods of quantum field theory, so we do not reproduce it here. Suffice to say, that the OPE potential appears as naturally from exchanging pions, as the Coulomb potential appears from exchanging photons via the electron-photon coupling term in Eq. (16). Last but not least, the last term in Eq. (44) gives the axial-vector current that defines the weak coupling of nucleons to electrons and neutrinos. From where phenomena like the decay can be derived. [This term is an independent chiral invariant, so again we could put a separate coupling constant there; experiment gives =1.257(3).]