In-medium effects on hidden strangeness production in heavy-ion collisions

. We study φ meson production in heavy-ion collisions from sub-threshold energies of 1.23 A GeV up to RHIC energies within the microscopic Parton-Hadron-String Dynamics (PHSD) transport approach where novel production channels for φ mesons based on a coupled channel T-matrix approach are implemented along with the collisional broadening of the φ meson spectral width in medium. Since φ meson production is closely related to the production of kaons and antikaons, antikaon properties are described via the self-consistent coupled-channel unitarized scheme within a SU(3) chiral Lagrangian (G-matrix) which incorporates explicitly the s − and p − waves of the kaon-nucleon interaction, while the in-medium modiﬁcations of kaons are accounted for via a kaon-nuclear potential, which is assumed to be proportional to the local baryon density.


Introduction
Recently it has been reported by the HADES collaboration that φ mesons are produced in a relatively large quantity in Au+Au collisions at subthreshold energies and the ratio of hidden strangeness to open strangeness reaches values of ≈ 0.5 [1].The same tendency of an enhanced φ production in Ni+Ni and Al+Al collisions at 1.93 A GeV has been reported earlier by the FOPI collaboration [2,3].With increasing beam energy this ratio decreases to 0.2 as has been measured recently by the STAR collaboration [4] and at high energies the dependence on the collision energy is mild [5][6][7][8][9].
The goal of this study is to show that the observed 'enhanced' φ multiplicity and φ/K − ratio close to threshold can be understood by considering a collisional broadening of the φ meson spectral function and accounting for additional multi-step meson-baryon and mesonhyperon reactions for φ meson production as predicted by the SU(6) extension of the mesonbaryon chiral Lagrangian within a unitary coupled channel T-matrix approach.

In-medium modification of φ meson properties
In order to explore the influence of in-medium effects on the vector-meson spectral function we introduce the collisional broadening by where M is the mass, Γ φ (M) the total width of the vacuum spectral function of the φ meson, and Γ coll the collisional width approximated as Here v is the velocity of the φ meson in the rest frame of the nucleon current, γ 2 = 1/(1−v 2 ), ρ the nuclear density scaled by ρ 0 = 0.168 fm −3 (normal nuclear density) and σ tot VN the mesonnucleon total cross section in vacuum.In order to simplify the calculations of Γ coll (ρ) we use the linear density approximation [10] with a coefficient α coll which is taken to be 25 MeV [11].

φ production/absorption within a SU(6) based T-matrix approach
The s-wave scattering amplitude of meson and baryon from the SU(6) chiral effective Lagrangian is written as [12], where i( j) indicates the initial (final) meson-baryon scattering states, M i( j) and E i( j) are, respectively, mass and center-of-mass energy of the baryon, f i( j) the decay constant of the meson in the i( j) state, and ε S IJ i j the degeneracy coefficient, corresponding to the scattering channel with S , I, and J being total strangeness, isospin and angular momentum of the collision, respectively [13,14].The T-matrix approach can be formulated on the basis of the Born scattering amplitude V S IJ ik , where k is the intermediate meson-baryon state and the sum is performed over all possible states.G S IJ kk is the product of the meson and baryon propagators of the state k [15], which is renormalized such that G S IJ  kk (s = m 2 N + m 2 π ) = 0 with m N and m π being nucleon and pion masses, respectively.The channels considered in this study for φ meson production are ηN, KΛ, KΣ, ρN, KΣ * , ρ∆, K * Λ, K * Σ, K * Σ * → φN for I = 1/2 and KΣ, ρN, η∆, KΣ * , ρ∆, K * Σ, K * Σ * → φ∆ for I = 3/2, including their inverse reactions by detailed balance.

φ meson production in heavy-ion collisions
Fig. 1 shows the PHSD results for the rapidity distribution of reconstructed φ mesons from the decay into K + K − pairs, compared with the experimental data from the HADES and STAR collaborations.The short dashed orange lines show the PHSD results without including the novel mB channels from the T-matrix approach and without any in-medium modifications of φ and K, K mesons.The dash-dotted green lines show the results without in-medium modifications of φ and K, K mesons [17]  with φ collisional broadening, but without in-medium effects for K, K mesons.The solid blue lines show the results with collisional broadening for φ mesons and with in-medium modifications of K, K mesons.The number of φ mesons, reconstructed from K + K − pairs, is divided by the branching ratio Br(φ → K + K − ).We note that the rescattering of the K + or K − in the medium reduces the reconstructed φ meson to 60-70% in Au+Au reactions at energies between E kin = 1.23A GeV and √ s NN = 3 GeV.In addition, φ yield rapidly increases from E kin = 1.23A GeV to E kin = 1.58A GeV, because both are sub-threshold energies for φ production.Fig. 2 shows the φ/K − ratio as a function of the collision energy from E kin = 1.23A GeV to √ s NN = 200 GeV.The PHSD results are presented for four different scenarios as in Fig. 1.
An inclusion of the in-medium effects for K, K, which leads to a strong enhancement of the K − yield [17] and, as a result, to a reduction of the φ/K − ratio.

Summary
In this study we have investigated the hidden strangeness (φ meson) production in heavy-ion collisions from subthreshold to relativistic energies within the microscopic off-shell PHSD transport approach.We have found that a collisional broadening of the φ meson spectral function lead to an enhancement of φ meson production, especially at subthreshold energies  and that the novel mB → φB channels from the SU(6) chiral Lagrangian in the T-matrix approach also enhance considerably the φ production in heavy-ion collisions.

Figure 2 .
Figure2.The PHSD results for the ratio φ/K − at midrapidity (|y| ≤0.3) as a function of the collision energy for four different scenarios: with and without novel mB channels for the φ meson production from the T-matrix approach and with and without the collisional broadening of the φ meson width and in-medium effects on (anti-)kaons (cf. the legend).The solid symbols show the compilation of the experimental data from Refs.[3][4][5][6][7][8][9].
. The dashed red lines indicate the φ rapidity distributions