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, $i \int d^4x e^{ip\cdot x} < ρ| T[{\bar d}(x) i γ_5 u (x) {\bar u}(0) γ_μγ_5 d (0)] | ρ>$ is revisited. In particular, we argue that the dimension 5 condensate, $<{\bar q} (i D_0)^2 q >_N + {1\over 8} < {\bar q} g_s σ\cdot {\cal G} q >_N$, 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 $-0.025 {\rm GeV}^2 \sim -0.019$ GeV$^2$. The role of the $N$ and $Δ$ intermediate states in the correlation function is also investigated. The $N$ intermediate state is found not to contribute to the sum rules. For the $Δ$ 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 $Δ$ subtraction while allowing the in-medium pion mass to vary within 139 MeV $ \le m_π^* \le$ 159 MeV, we obtain $f_s/f_π= 0.37 \sim 0.78$ and $f_t / f_π= 0.63 \sim 0.79$.