# Scattering amplitude calculation using first order Chiral Perturbation Theory Lagrangian

Consider the Chiral Perturbation Lagrangian to first order(quark masses set to zero): $$L = L^{(2)} = \frac{f^2}{4}tr[\partial_{\mu} U \partial^{\mu} U^{\dagger}] ,$$ where U is a $2 \times 2$ exponential matrix collecting the pion-goldstone fields and the terms of the Lagrangian come from expanding the exponential- this term is similar to the non-linear sigma model for pions. Consider now the relativistic inclusion of nucleons so that: $$L =L_{\pi N}^{(1)} + L_{\pi \pi}^{(2)} ,$$ and where the first term is similar to the non-linear sigma model with the inclusion of nucleon-pion interaction- it's of the form $N D(x) \bar{N}$, where $D(x)$ is a differential operator containing the pion fields and couplings with the nucleon. The nucleon transforms in a highly non-linear way under the chiral group and the same for the pion fieds.

QUESTION: From all I have found there is no clear phrasing of using this Lagrangian for the calculation of NN-scattering amplitudes, instead, higher order terms must be included- the first such term is of the form: $$L_{NN} \simeq \bar{N}N \bar{N}N + (\bar{N}\sigma N)(\bar{N}\sigma N) ,$$ in the heavy baryon formalism.

But what I though I could do, without considering yet the heavy baryon formalism, was to take the $L_{\pi N}$ Lagrangian and use it the usual QFT way, that is in second order perturbation theory: then I wold have something of the form $S =\simeq \int d^4 x H_{\pi N}^2$, a term that contains four N fields and pion fields as exchange terms. I did the same think using a phenomenological Yukawa type Lagrangian for the exchange of pions between nucleons as described by Erkelenz or Machleidt in papers about pion exchange nuclear forces- a Lagrangian with interaction of the type $$L_{phenomenology}= g \bar{N} \tau \cdot \pi N$$ with the inclusion of gamma matrices for pseudo-scalar terms and other kind of couplings.

So, if the $L_{\pi N}$ is of the same form just for chiral symmetry, is there a theoretical reason not to use it in second order for the calculation of the scattering amplitude between two nucleons- or such a use is valid and I just haven' t found any references on this?

1) The basic reference is Weinberg's work on effective lagrangians for the $NN$ interaction, see https://inspirehep.net/record/29549?ln=en .
2) Weinberg shows that in $NN$ scattering the one-pion exchange interaction and the $NN$ contact terms (with no derivatives) appear at the same order.
3) Note that in the $NN$ sector you cannot simply do perturbation theory. This is because there are $NN$ bound states and resonances at very low energy. Weinberg proposes to compute the $NN$ potential in perturbation theory, and then iterate the potential by solving the Schroedinger (Lippmann-Schwinger) equation.