One of the more interesting explanations for the suppression of the pion field, and the corresponding lack of enhancement in the longitudinal response, at short distance scales is the partial restoration of chiral invariance with density [Bro95]. Essentially, this means that the masses of hadrons comprised of up and down quarks, and in particular the vector mesons such as the meson, will decrease with increasing density at about the same rate [BSH91]:
In equation the quantities with the asterisk are the density modified values and those without are free space values. is the pion decay constant, but in this context serves a more basic role; it is the order parameter for chiral symmetry breaking.
The argument for the validity of the relationship in equation has been made in [BrR91]. Essentially the argument states that the structure of free space Lagrangians is constrained by symmetries inherent to QCD and that even as density increases those symmetries, and therefore the form of the Lagrangian, will remain the same. As density increases and the nature of the vacuum changes, if the Lagrangian does not change then the coupling constants and masses must be modified according to QCD constraints to account for the effects of a modified vacuum (i.e. the altered quark condensate ). The details are described in [BrR91] at the mean field level for which the relationship in equation is exact. Higher order loop correction will have some effect which leads to the use of the approximate signs [Bro95]. The pion mass is the exception to this new scaling law because of its special status as a Goldstone boson. The pion mass has actually been shown to increase slightly, but not significantly, with increased density because of a slightly repulsive optical potential [ErW88].
It was the differences in the mass of the pion and meson that caused the differences between the transverse and longitudinal pieces of the particle-hole potential (eqn. ) illustrated in figure . The results of some calculations based on QCD sum rules [HaL92] are shown in figure show that, for nuclear matter density,
is a reasonable ratio for the reduction of the in-medium mass.
Figure: The meson mass as a function of density ; based on QCD sum rule calculations. Functional form is , where . The deviation from linearity is the result of an uncertainty in the calculation [HaL92].
This would account for the difference between the transverse and longitudinal responses and bring close to unity [BrW94]. In fact, because of the surface peaked nature of the reaction the density sampled will actually be which means which could not fully account for the results in previous experiments.
The analysis of several recent heavy-ion experiments lend support to the idea that the meson mass decreases with density [LKB95] [LKB96]. Models based on the medium modification of vector meson properties have been relatively successful at fitting data from heavy-ion collision (S + Au) at the CERN SPS at 200 GeV/nucleon. The dilepton invariant mass spectrum from these experiments show an excess of dileptons between 250 MeV and 1 GeV over that from proton-nucleus collisions, which has been quantitatively explained by the reduction in vector meson masses in the dense medium [LKB95] [LKB96].