Impact of the finite volume effects on the chiral behavior of fK and BK

Abstract

We discuss the finite volume corrections to fK and BK by using the one-loop chiral perturbation theory in full, quenched, and partially quenched QCD. We show that the finite volume corrections to these quantities dominate the physical (infinite volume) chiral logarithms.

Figures (3)

• Figure 1: From top to bottom, we plot the chiral logarithmic corrections as predicted in full, partially quenched ($r_{\rm sea}=m_{\rm sea}/m_s^{\rm phys}=0.5$) and quenched ChPT, respectively, as functions of the light valence quark mass $r=m_q/m_s$, where the strange quark mass is fixed to its physical value. In each plot the thick line corresponds to the physical (infinite volume) chiral logarithm, whereas the other four curves correspond to the logarthmic contributions computed in the finite volume $V=L^3$, where for $L$ we choose the values shown in the legend. The renormalisation scale is chosen to be $\mu=1$ GeV.
• Figure 2: The finite volume corrections to $f_K$ in full, partially quenched and quenched theory, eqs. (\ref{['fK1']}),(\ref{['fK2']}),(\ref{['fK3']}), respectively. The partially quenched case that we consider is the one with $N_{\rm f}=2$ dynamical quarks degenerate in mass for which we take $r_{\rm sea}=m_{\rm sea}/m_s^{\rm phys} = 1, 0.5, 0.2$. Each plot corresponds to a different value of the size of the side of the box $L$, indicated in the plots. We keep the same scale, to better appreciate the reduction of the finite volume effects as $L$ is increased.
• Figure 3: The finite volume corrections to $B_K$ in full, partially quenched and quenched theory, eqs. (\ref{['BK1']}),(\ref{['BK2']}),(\ref{['BK3']}), respectively.