The order $p^8$ mesonic chiral Lagrangian

TL;DR

This paper derives the mesonic chiral Lagrangian at order p^8 for various numbers of light quark flavors, providing a comprehensive enumeration of contact terms and special cases.

Contribution

It presents the first derivation of the NNNLO mesonic chiral Lagrangian for general and specific flavor numbers, including contact terms and external field considerations.

Findings

Derived the NNNLO chiral Lagrangian for arbitrary N_f

Enumerated contact terms separately

Discussed cases with missing external fields

Abstract

We derive the chiral Lagrangian at next-to-next-to-next-to-leading order (NNNLO) for a general number $N_f$ of light quark flavours as well as for $N_f=2,3$. We enumerate the contact terms separately. We also discuss the cases where some of the external fields are not included. An example of a choice of Lagrangian is given in the supplementary material.