{'fraction_non_alphanumeric': 0.03524229074889868, 'fraction_numerical': 0.007757134648534764, 'mean_word_length': 4.626616379310345, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Dropping Sigma-Meson Mass and In-Medium Pion-Pion Correlations

The influence of a dropping sigma-meson mass on previously calculated in-medium pion-pion correlations in the J=I=0 (sigma-meson) channel is investigated. It is found that the invariant-mass distribution around the vacuum threshold experiences a further strong enhancement over and above the standard many-body effects. The relevance of this result for the theoretical interpretation of this result for the theoretical interpretation of recent A(pi-2 pi)X data is pointed out.


Introduction
It is about ten years now that the first theoretical work suggesting a strong enhancement around the 2m π threshold of the in-medium 'sigma'-meson mass distribution was published [1]. The threshold enhancement originates from in-matter p-wave renormalization of the two pions to which the 'bare sigma' meson of about 1 GeV decays. The existence of such a bare sigma meson has in the meanwhile been confirmed by quenched lattice calculations [2]. Its decay into two pions leads to a strong hybridization of the sigma meson and finally only a broad bump of about 600 MeV width survives at considerably lower energy. The ππ interaction constants in the completely phenomenological model of ref. [1] had been adjusted to reproduce the experimental ππ phase shift in the scalar -isoscalar channel. Since the latter raises right from threshold with considerable positive slope, it signals substantial attraction among low-energy pions. Pions are, however, too light to be bound by the interaction. On the other hand, the p-wave renormalization of the pions in the nuclear medium (nucleon-and delta-hole couplings) increases their kinetic mass and thus suppresses the kinetic energy. As a consequence binding of the two pions is favored at high density. Before this occurs, a strong accumulation of strength close to the 2m π threshold takes place. Even if, with increasing density, the mass distribution receives contributions from below the 2m π threshold, this does not lead to a sharp 2-pion bound state. Various decay channels such as πN-π∆-or NN render it broad [4]. To describe the subthreshold behavior of the scalar-isoscalar strength function in accordance with chiral symmetry requirements it turned out later that the ππ scattering needs to be implemented by using chiral models such as the linear or non-linear sigma model, rather than an empirical parametrization of the s-wave phase shift. In addition a formalism is called for which respects chiral symmetry beyond the tree level, [10]. Otherwise the in-medium effects become so strong that they lead to unreasonable pion-pair condensation [11]. In the meantime s-wave pion-pion correlations have attracted a significant amount of attention both on the theoretical [6,7,8] and the experimental [9,12] sides. This is primarily related to the fact that these studies are of relevance for the evolution of the chiral condensate and its fluctuations with increasing density. The enhancement of the in-medium ππ correlations close to and below the vacuum threshold has recently been confirmed by an independent calculation within the non-linear sigma model. Also on the experimental side important progress has been made. The CHAOS collaboration has studied the A(π, 2π) knock-out reaction [9,12] and the invariant-mass measurements of the outgoing pions reveal a strong low-mass enhancement which seems to corroborate the theoretical predictions. Performing elaborate reaction calculations, including initial-and final-state absorption, Vincente-Vacas and Oset [7] have claimed that the theory outlined above underestimates the measured ππ mass enhancement. This claim may be partly questioned, since the reaction theory calls for the inclusion with a finite total three momentum of the in-medium pion pairs which sofar are only propagated in back-to-back kinematics ( q = 0). It was shown in [3] that, allowing for finite three momenta of the pion pair, leads to further increase the π + π − invariant-mass distribution near threshold. On the other hand, Hatsuda et al. [8] have argued that the partial restoration of chiral symmetry in nuclear matter, which leads to a dropping of the σ-meson mass [13], induces similar effects as the standard many-body correlation mentioned above. It is therefore natural to study the combination of both effects.
As a model for ππ scattering we consider the linear sigma model treated in leading order of the 1/N-expansion [10]. The scattering matrix can then be cast in the following form where s is the Mandelstam variable. In Eq. (1) D π (s) and D σ (s) denote respectively the full pion and sigma propagators, while σ is the sigma condensate. The expression in Eq. (1) reduces, in the soft-pion limit, to a Ward identity which links the ππ four-point function to the π and σ two-point functions as well as to the σ one-point function. To this order, the pion propagator and the sigma-condensate are obtained from a Hartree-Bogoliubov approximation [10]. In terms of the pion mass m π and the decay constant f π , they are given as The sigma meson, on the other hand, is obtained from the Random-Phase Approximation (RPA) involving ππ scattering [10] and reads where Σ ππ (s) is the ππ self energy, regularized by means of a form factor which is used as a fit function [3] and allows to reproduce the experimental ππ phase shifts. The coupling constant λ 2 denotes the bare quartic coupling of the linear σ-model, related to the mean-field pion mass, m π , the sigma mass, m σ , and the condensate, σ , via the mean-field saturated Ward identity It is clear from the above that the sigma-meson propagator in this approach is correctly defined, since it satisfies a hierarchy of Ward identities. In cold nuclear matter the pion is dominantly coupled to ∆−h, p−h, as well as to 2p − 2h excitations which, on the other hand, are renormalized by means of repulsive nuclear short-range correlations, (see [4] for details). Since the pion is a (near) Goldstone mode, its in-medium s-wave renormalization does not induce appreciable changes. The sigma meson, on the other hand, is not protected by chiral symmetry from large s-wave renormalization, resulting in a density-dependent mass modification [2] . We extract an approximate density dependence at mean-field level from the (leading-order) density dependence of the condensate. Indeed from Eq. (4) it is clear that the density dependence of the sigma meson mass is essentially dictated by the density dependence of the condensate. Thus, for densities below and around nuclear saturation density, ρ 0 , we take for the in-medium sigma-meson mass the simple ansatz where ρ is the nuclear matter density and m σ is the vacuum σ-meson mass. Such a density dependence very naturally arises in the linear sigma model from the tad-pole graph where the sigma meson directly couples to the nuclear density. In [8] this density dependence was investigated quantitatively and a value of α in the range of 0.2 to 0.3 was found. These are the values which we also will use. It should be mentioned at this point that between the (s-wave) density renormalization via the tad-pole of the bare sigma-meson mass in the linear sigma model (as we mentioned already, such a bare sigma exists in quenched QCD lattice calculations [2]) and the in-medium p-wave renormalization of the pion there exists no double counting. Besides the different partial waves involved, a simple inspection of the corresponding Feynman graphs shows that both density effects are of totally different microscopic origin.

Results
The result for the invariant-mass distribution ImD σ (E ππ ), as calculated from Eq. (3) by using the in-medium mass (5), is shown in Fig. 1 at saturation density. One observes a dramatic downward shift of the mass distribution as compared to the vacuum. The low-energy enhancement, already present without sigma-mass modification (α = 0) and induced by the density-dependence of the pion loop, is strongly reinforced as the in-medium σ-meson mass is included. For α = 0.2 and α = 0.3 the peak height is increased by a factor 2 and 4 respectively. Similarly for the T-matrix, a sizable effect can be noticed in its imaginary part, as shown in Fig. 2, which might be sufficient to explain the findings of the CHAOS collaboration [9]. Further work in this direction is in progress.

Discussions and Conclusion
Let us further comment and discuss the above results. They clearly indicate that there exists a large conspiracy between s-and p-wave in-medium renormalisations of the hybrid 'sigma meson'. Both effects induce a strong shift of the strength towards and even below the 2m π threshold. However, one has to worry about opposing effects. For example it may happen that vertex corrections, usually a source of repulsion and not taken into account in this work, weaken the effects. They seem, however, to be of minor importance as was recently shown by Chanfray et al. [14]. More care should also be taken in properly incorporating Pauli-blocking when renormalizing the pion pairs in matter, although preliminary investigations [15] have shown that this effect is weak.
In conclusion we have shown that a dropping sigma-meson mass, linked to the partial restoration of chiral symmetry in nuclear matter, further enhances the build up of previously found ππ strength in the I = J = 0 channel. This scenario holds in the linear sigma model but is likely to hold true in the nonlinear sigma model as well, since the tad-pole graph which renormalizes the bare sigma meson mass in the linear sigma model has its equivalent in the non- linear version. Whether our findings are linked to similar ones found recently in the experiment by Bonutti et al [9] is an exciting possibility which, however, must still be consolidated by refined reaction calculations. Last but not least there remains the challenge that the results of Bonutti et al. are confirmed by other experiments. A promising reaction will be the A(γ, 2π) reaction off nuclei with varying mass number, since the average density increases from low to high mass numbers and photons, in contrast to pions, probe the nuclear interior. Existing data from MAMI Mainz [16] are waiting to be analyzed! We hope to have shown in this contribution [17] that the scalar-isoscalar channel, in what concerns chiral symmetry, is at least as interesting and promising as the vector-isovector channel (rho-meson).