Quenched Chiral Logarithms

TL;DR

This work develops a diagrammatic framework to compute quenched chiral logarithms, incorporating eta' loops and finite-volume effects while highlighting which quantities are robust to quenching (safe quantities) and which exhibit enhanced logs. It shows that leading-order logs vanish for quenched $f_pi$ with degenerate light quarks, while $B_K$ shares the same logarithmic behavior as full QCD for degenerate masses; non-degenerate cases introduce eta'–driven logs, producing non-analytic mass dependences in several observables. Finite-volume corrections, predicted via Gasser-Leutwyler methods, are tested against lattice data, with reasonable agreement: most volume effects are small, but $B_V$ and $B_A$ display the expected volume dependence and signs. The findings offer practical guidance for interpreting quenched lattice results and for planning future simulations at smaller pion masses and varied volumes to scrutinize chiral-log predictions.

Abstract

I develop a diagrammatic method for calculating chiral logarithms in the quenched approximation. While not rigorous, the method is based on physically reasonable assumptions, which can be tested by numerical simulations. The main results are that, at leading order in the chiral expansion, (a) there are no chiral logarithms in quenched $f_π$, for $m_u=m_d$; (b) the chiral logarithms in $B_K$ and related kaon B-parameters are, for $m_d=m_s$, the same in the quenched approximation as in the full theory; (c) for $m_π$ and the condensate, there are extra chiral logarithms due to loops containing the $η'$, which lead to a peculiar non-analytic dependence of these quantities on the bare quark mass. Following the work of Gasser and Leutwyler, I discuss how there is a predictable finite volume dependence associated with each chiral logarithm. I compare the resulting predictions with numerical results: for most quantities the expected volume dependence is smaller than the errors, but for $B_V$ and $B_A$ there is an observed dependence which is consistent with the predictions.

Figures (15)

• Figure 1: Diagrams contributing to $f_\pi$: (a) leading order; (b) wavefunction renormalization; (c) vertex renormalization. The lines represent pseudo-Goldstone bosons, the box is the axial current, and the circle is the vertex coming from ${\cal L}$.
• Figure 2: Quark diagram corresponding to Fig. 1(a). The lines represent quark propagators, as described in the text. The box is the axial current. The apex where the propagators meet is the operator which is used to create the pion.
• Figure 3: Quark diagrams corresponding to Fig. 1(b).
• Figure 4: Quark diagrams contributing to pion scattering.
• Figure 5: Quark diagrams corresponding to Fig. 1(c).