Matching of UL(3)⊗UR(3) and SUL(3)⊗SUR(3) chiral perturbation theories

Abstract

The heavy singlet field is integrated out from the U L (3) ⊗ U R (3) Chiral Perturbation Theory and it is shown how its effects on the low-energy dynamics are reduced to effective vertices for the light mesons. The results are matched against the standard SU L (3) ⊗ SU R (3) Chiral Perturbation Theory in order to establish the relations between the coupling constants from both theories to one-loop level accuracy.

Large $N_ c$ in chiral perturbation theory

“…As explicitly demonstrated in ref. [21], the path integral for the effective U(3) Lagrangian indeed reduces to the one of SU (3) if the singlet meson field is integrated out. The calculation is analogous to the one described in [4], where the SU(3) effective theory is matched with the one relevant for SU (2).…”

“…Like in that case, loops involving the propagation of light as well as heavy mesons require special attention, because the momentum scale of these is set by the heavy masses, while in the light sector, that scale does not occur. We briefly sketch the essential steps, referring to [21] for a more detailed discussion.…”

“…, L 5 , L 9 , L 10 and H 1 are the same in the two versions of the theory. For B, L 6 , L 7 , L 8 and H 2 , the matching conditions 4 read [21] …”

“…[4], where the Green functions of QCD were studied by means of a simultaneous expansion in powers of momenta, quark masses and 1/N c . A considerable amount of work concerning the structure of QCD at large N c has been carried out since then [6]- [21] and quite a few phenomenological applications of the 1/N c expansion have appeared in the literature. For a review of these, in particular also for a discussion of the η-η ′ mixing pattern, we refer to [22].…”

$\eta$-$\eta\prime$ mixing in Large-Nc ChPT: discussion, phenomenology, and prospects

“…In other words, in the combined chiral and LN c limits, QCD at low energies is expected to generate (π, K, η, η ) as the Goldstone bosons. This scenario is the starting point for Large-N c chiral perturbation theory (LN c ChPT) as the EFT of QCD at low energies including the singlet field [14,15,16,17,18,19,20,21], which we will also refer to as U(3) effective theory.…”