# Chiral Perturbation Theory and Weak Matrix Elements

## TL;DR

This paper discusses recent advances in quenched chiral perturbation theory, its application to baryon masses and weak matrix elements, and provides estimates of quenching errors and unquenching effects on quantities like $B_K$.

## Contribution

It introduces new predictions for higher order extrapolations in quenched chiral perturbation theory and provides the first significant evidence of unquenching effects on $B_K$.

## Key findings

- Evidence for $ ext{eta'}$ loops in baryon mass fits.
- A quenched $B_K$ value of approximately 0.598 at 2 GeV.
- Unquenching increases $B_K$ to about 0.66.

## Abstract

I describe recent developments in quenched chiral perturbation theory (QChPT) and the status of weak matrix elements involving light quarks. I illustrate how, with improved statistical errors, and with calculations of the masses of baryons containing non-degenerate quarks, there is now a clear need for extrapolations of higher order than linear in the quark mass. I describe how QChPT makes predictions for the functional forms to use in such extrapolations, and emphasize the distinction between contributions coming from chiral loops which are similar to those present in unquenched theories, and those from $\eta'$ loops which are pure quenched artifacts. I describe a fit to the baryon masses using the predictions of QChPT. I give a status report on the numerical evidence for $\eta'$ loops, concluding that they are likely present, and are characterized by a coupling $\delta=0.1-0.2$. I use the difference between chiral loops in QCD and quenched QCD to estimate the quenching errors in a variety of quantities. I then turn to results for matrix elements, largely from quenched simulations. Results for quenched decay constants cannot yet be reliably extrapolated to the continuum limit. By contrast, new results for $B_K$ suggest a continuum, ``quenched'' value of $B_K(NDR, 2 GeV) = 0.5977 \pm 0.0064 \pm 0.0166$, based on a quadratic extrapolation in $a$. The theoretical basis for using a quadratic extrapolation has been confirmed. For the first time there is significant evidence that unquenching changes $B_K$, and my estimate for the value in QCD is $B_K(NDR, 2 GeV) = 0.66 \pm 0.02 \pm 0.11$. Here the second error is a conservative estimate of the systematic error due to uncertainties in the effect of quenching. A less conservative viewpoint reduces $0.11$ to $0.03$.