Update on pion weak decay constants in nuclear matter

Abstract

The QCD sum rule calculation of the in-medium pion decay constants using pseudoscalar-axial vector correlation function, $i\int d^{4}x{e}^{ip\cdot x}<\rho |T[\bar{d}(x)i{\gamma }_{5}u(x)\bar{u}(0){\gamma }_{\mu }{\gamma }_{5}d(0)]|\rho >$ is revisited. In particular, we argue that the dimension 5 condensate, $<\bar{q}(i{D}_0{)}^{2}q{>}_N+\frac{1}{8}<\bar{q}{g}_s\sigma \cdot \mathcal{G}q{>}_N$, which is crucial for splitting the time ($f_t$) and space ($f_s$) components of the decay constant, is not necessarily restricted to be positive. Its positive value is found to yield a tachyonic pion mass. Using the in-medium pion mass as an input, we fix the dimension 5 condensate to be around $-0.025{\mathrm{GeV}}^2\sim -0.019$ GeV${}^2$. The role of the $N$ and $\Delta$ intermediate states in the correlation function is also investigated. The $N$ intermediate state is found not to contribute to the sum rules. For the $\Delta$ intermediate state, we either treat it as a part of the continuum or propose a way to subtract explicitly from the sum rules. With (and without) explicit $\Delta$ subtraction while allowing the in-medium pion mass to vary within 139 MeV $\le m_{\pi }^{\ast }\le$ 159 MeV, we obtain $f_s/f_{\pi }=0.37\sim 0.78$ and $f_t/f_{\pi }=0.63\sim 0.79$.

Comments: 18 pages including 5 postscript figures