Partially quenched chiral perturbation theory without $Φ_0$

TL;DR

This work resolves the long-standing barrier to using partially quenched lattice QCD as a quantitative probe of real QCD by showing that the PQ chiral Lagrangian can be formulated without the flavor-singlet field $\Phi_0$. It demonstrates that $\Phi_0$ can be treated as a heavy auxiliary field with mass $m_0$, whose decoupling (via $m_0\to\infty$) yields a theory whose low-energy constants match those of unquenched QCD, thereby legitimizing PQ simulations for physical parameter extraction. The paper also clarifies the pole structure of neutral flavor-singlet correlators, predicting double poles as a robust signature of PQ dynamics, and provides a consistent framework that maps earlier $\Phi_0$-inclusive results to the $\Phi_0$-free theory. Collectively, these results justify PQ chiral perturbation theory as a sound bridge between lattice PQQCD and real QCD, enhancing the reliability of parameter extraction from PQ simulations. The analysis combines symmetry arguments, effective field theory construction, and a functional-integral decoupling approach to establish a solid foundation for PQ chiral perturbation theory and its connection to QCD phenomenology.

Abstract

This paper completes the argument that lattice simulations of partially quenched QCD can provide quantitative information about QCD itself, with the aid of partially quenched chiral perturbation theory. A barrier to doing this has been the inclusion of $Φ_0$, the partially quenched generalization of the $η'$, in previous calculations in the partially quenched effective theory. This invalidates the low energy perturbative expansion, gives rise to many new unknown parameters, and makes it impossible to reliably calculate the relation between the partially quenched theory and low energy QCD. We show that it is straightforward and natural to formulate partially quenched chiral perturbation theory without $Φ_0$, and that the resulting theory contains the effective theory for QCD without the $η'$. We also show that previous results, obtained including $Φ_0$, can be reinterpreted as applying to the theory without $Φ_0$. We contrast the situation with that in the quenched effective theory, where we explain why it is necessary to include $Φ_0$. We also compare the derivation of chiral perturbation theory in partially quenched QCD with the standard derivation in unquenched QCD. We find that the former cannot be justified as rigorously as the latter, because of the absence of a physical Hilbert space. Finally, we present an encouraging result: unphysical double poles in certain correlation functions in partially quenched chiral perturbation theory can be shown to be a property of the underlying theory, given only the symmetries and some plausible assumptions.