One-loop Chiral Perturbation Theory with two fermion representations
  We develop Chiral Perturbation Theory for chirally broken theories with fermions in two different representations of the gauge group. Any such theory has a non-anomalous singlet $U(1)_A$ symmetry, yielding an additional Nambu-Goldstone boson when spontaneously broken. We calculate the next-to-leading order corrections for the pseudoscalar masses and decay constants, which include the singlet Nambu-Goldstone boson, as well as for the two condensates. The results can be generalized to more than two representations. 
Thomas DeGrand, Maarten Golterman, Ethan T. Neil and Yigal Shamir
hep-ph hep-lat