QCD sum rules for ρ mesons in vacuum and in-medium, re-examined

An updated investigation of QCD sum rules for the first two moments of ρ-meson spectral functions, both in vacuum and in-medium, is performed with emphasis on the role of the scale related to spontaneous chiral symmetry breaking in QCD. It is demonstrated that these lowest moments of vector current spectral distributions do permit an accurate sum-rule analysis with controlled input including QCD condensates of the lowest dimensions, whereas higher moments are subject to uncertainties from higher dimensional condensates. Possible connections with Brown-Rho scaling are discussed. The factorization approximation for four-quark condensates is shown not to be applicable in any of the cases studied.

Figure 1

Vector-isovector spectral function in vacuum showing the ρ resonance and continuum parts as described in the text and compared to $e^{+}{e}^{-}\to {\pi }^{+}\hspace{0.5em}{\pi }^{-}$ (ρ resonance region) and $e^{+}{e}^{-}\to n\hspace{0.5em}\pi$ data with $n$ even [26, 27].

Figure 2

QCD sum-rule analysis of the ρ-meson spectral function in vacuum. First moment (solid line, left-hand side of Eq.(24)) is plotted versus right-hand side (grey band including uncertainties) as function of the gap scale $\sqrt{s_0}$ delineating low-mass resonance region from high-mass continuum.

Figure 3

Dependence of $\sqrt{\hspace{0.5em}{s}_0}$ [determined from Eqs. (30, 31, 32)] on the slope $(s_2-s_1){}^{-1}$ of the ramp function $W(s)$ describing the onset of the continuum in the vacuum sum rule. The gray band indicates the uncertainty range of the result obtained with step-function parametrization of the continuum.

Figure 4

In-medium isovector vector spectral functions at nuclear matter density, ${\rho }_0=0.16$ ${\mathrm{fm}}^{-3}$, taken from Refs. [4] (KKW) and [16] (RW). The ρ-meson spectrum in vacuum is also shown for comparison.

Figure 5

QCD sum-rule analysis of the KKW in-medium spectral function [4]. First moment [solid line, left-hand side of Eq. (33)] is plotted versus right-hand side (gray band including uncertainties) as function of the in-medium gap scale $\sqrt{s_0^{*}}$.

Figure 6

QCD sum-rule analysis of the RW in-medium spectral function [16]. First moment [solid line, left-hand side of Eq. (33)] is plotted versus right-hand side (gray band including uncertainties) as function of the in-medium gap scale $\sqrt{s_0^{*}}$.

Figure 7

Dependence of $\sqrt{\hspace{0.5em}{s}_0^{*}}$, as in Fig. 3, on the slope $(s_2-s_1){}^{-1}$ of the ramp function $W(s)$, now describing the onset of the continuum in the in-medium sum rules. (Upper panel) Result for the KKW spectral function. (Lower panel) Result for the RW spectral function. The gray bands indicate the uncertainty ranges of the results obtained with step-function parametrizations of the continuum.