Digamma diagnostics for the mixed-phase generation at NICA

A novel type of diagnostics for dense and/or hot nuclear matter produced in heavy-ion collisions at NICA and similar future colliders (FAIR, etc.) is suggested. The diagnostics is based on an assumption (confirmed in many experiments worldwide) about intensive generation of light scalar mesons (σ) the consequent decay of which produces γγ pairs with the mass and width dependent upon density and temperature of the fireball produced in the collision process. Thus, measurements of the absolute yield, mass and width of the γγ signal carry valuable information about the state of fireball generated during the high-energy nuclear collision.

The phase transition in hot matter and the temperature behavior of pseudoscalar as well as scalar meson parameters were investigated in, e.g., [3,6]. Figure 1 illustrates the temperature dependence of M π , M σ and Γ σ obtained in [6] in the framework of the effective nonlocal quark model. It is quite evident from figure 1 that the σ-meson mass and width drop strongly when the temperature approaches its critical value T c . Figure 1. Temperature dependence of M π , M σ and Γ σ [6]. M 0 σ corresponds to M σ in the chiral limit. The quark deconfinement temperature T d and the critical temperature T c of the chiral phase transition are also shown.
The density dependence of the σ-meson parameters has been studied in, e.g., [4,5]. The density dependence of M σ is essentially dictated by that of the condensate σ . When the nuclear (quark) matter density becomes sufficiently high, the Pauli principle or temperature destroys the qq condensate, the valence quark motion is liberated from the condensate, and thus the CSR happens. This effect is rather similar to disappearance of superconductivity in electron gas in metals when the temperature is rising, since Cooper pairs are destroyed.
The in-medium σ-mass ansatz for the nuclear matter densities ρ < ρ 0 (ρ 0 being the nuclear saturated density) is M σ (ρ) = M σ (1 − αρ/ρ 0 ), where M σ is the vacuum σ-meson mass and the parameter α was found from QCD sum rules or model calculations to be α 0.2-0.3 [4,5]. From this ansatz, it is evident that the σ mass goes down when the matter density increases, and its width gets much reduced (due to the phase-space reduction). Thus, the σ-meson propagator in nuclear medium becomes much sharper than in vacuum due to CSR effects [5]. The overall importance of CSR can be concluded from the citation [3]: "Observing the restoration of chiral symmetry at finite temperature is one of the central aims in the future relativistic heavy-ion experiments at the BNL RHIC and CERN LHC".

Sources for σ-meson production in hadronic collisions
Below we will review shortly some possible sources for the light scalar σ-meson production in hadronic collisions.
(i) Dibaryon production in pn, pd and dd collisions at E ∼ 1 GeV/A. Recent experiments of the WASA-at-COSY Collaboration [9] on the basic 2π-production reaction pn → d + π 0 π 0 have shown a direct correlation of the old ABC-puzzle [10] with 2π emission from the I(J P ) = 0(3 + ) dibaryon state D 03 (2380). The same isoscalar resonance along with ABC enhancement have been observed also in pd → 3 He + π 0 π 0 and dd → 4 He + π 0 π 0 [11]. So, the authors [11] concluded that "the ABC effect in the double-pionic fusion to nuclei is traced back to a pn resonance, which obviously is strong enough to survive even in the nuclear medium." However, the detailed mechanism of the D 03 resonance decay which leads to the ABC enhancement is still not firmly established. According to the dibaryon model for short-range NN interaction [12,13], production of intermediate dibaryons in hadronic collisions is often associated with the scalar σ-meson production, with σ mass and width being renormalized strongly compared to those observed for σ mesons produced in vacuum (e.g., in ππ scattering). This renormalization happens due to CSR effects in the dibaryon state. Indeed, the D 03 (2380) dibaryon is a strongly excited hadron (with the excitation energy E * 500 MeV), and the partial CSR was predicted for such states rather reliably [1]. Besides that, the D 03 (2380) dibaryon is a very dense object, since it has six quarks in a volume of one nucleon (r(D 03 ) 0.7-0.9 fm). So, the ABC effect can be interpreted as a consequence of the near-threshold σ-meson emission from the dibaryon, with the σ-meson parameters [14] 100 MeV (cf. the σ-meson parameters found from ππ scattering in free space [8,15]: The π 0 π 0 invariant mass spectrum in the pn → dπ 0 π 0 reaction at √ s = 2.38 GeV calculated in the dibaryon model with the above (reduced) parameters of σ-meson is shown in figure 2 (a). (a) Invariant π 0 π 0 mass spectrum in pn → dπ 0 π 0 at T p = 1.14 GeV ( √ s = 2.38 GeV) [9] in comparison with the dibaryon model calculations [14] (solid line). Contributions of the near-threshold σ-meson production and the "background" process D 03 (2380) → D 12 (2150)π 0 → dπ 0 π 0 are shown by dashed and dash-dotted lines, respectively. (b) Invariant π + π − mass spectrum in pp → ppπ + π − at p = 6.6 GeV/c in comparison with a resonance fit (upper solid line). The 2π background without and with the contribution of the P 11 at 1400 MeV is given by the dotted and lower solid line, respectively (for details, see [16]).
(ii) Generation of the Roper resonance N * (1440) in NN, Nα, etc., collisions. The authors [16] have analyzed carefully the N * (1440) generation in pp collisions at beam momenta 6-30 GeV/c and low momentum transfers −t ≤ 0.06 GeV/c 2 . The Roper resonance produced in such a way decays immediately (due to its very large width Γ N * 350 MeV) into Nπ or Nππ channel, the latter decay going with a high probability via generation of an intermediate σ meson [7]. The π + π − invariant mass distribution in the pp → ppπ + π − process at 6.6 GeV/c is shown in figure 2 (b). It is clearly seen that the π + π − production is dominated by the Roper resonance decay and is concentrated near M π + π − 400 MeV corresponding to the σ-meson mass. So, the degree of CSR in the Roper resonance is less than in the dibaryon, which is likely due to the lower density of the Roper (three quarks instead of six).
(iii) Single and double σ production from Pomeron. It is well known that the high-energy pp scattering has a diffraction character and is governed by a Pomeron exchange. Furthermore, the Pomeron, being a scalar object, can produce σ mesons in high-energy pp collisions (see figure 3). The single and double σ emissions from Pomeron have been studied by Kisslinger et al. [17] who demonstrated that ca. 6-7% of the whole pp elastic cross section at 50 GeV goes to σ emission.  The solid line represents the fit by an exponential-type function (for details, see [18]).
Thus, we demonstrated that in many hadronic collisions at intermediate and high energies there appears a rather high yield of light scalar σ mesons which mass and width values should carry information about the stage of CSR inside the fireball, or, in other words, about its temperature and density. In turn, this important information can be found by measuring the invariant-mass distributions of digamma signals from the σ-meson decays. Since the in-medium CSR effects lead to reduction of the σ mass and thus to suppression of the σ → ππ decay mode, one should observe an increasing branching ratio for the σ → γγ decay in hot and/or dense matter.

Experiments on digamma production in hadronic collisions
The initial γγ-production experiments in intermediate-energy pp collisions were performed by the CELSIUS/WASA Collaboration [19]. They found a clear γγ signal at M γγ 300-400 MeV (see figure 5 (a)), which turned out to be very stable against cuts. Furthermore, the model which incorporated very well the γγ events from π 0 , η and π + π − production gave practically no events in the area M γγ 300-400 MeV, while the π 0 π 0 production at T p = 1.36 GeV gave about 15% of the observed counts between π 0 and η peaks. So, the authors [19] have come to the conclusion: "Since none of these simulated processes is able to account for the structure observed near the ππ threshold . . ., we are led to consider seriously the possibility that the observed structure (at M γγ 300-400 MeV) is real and might be due to the process pp → ppσ → ppγγ, in particular also since pp → ppπ + π − and pp → ppπ 0 π 0 reactions are dominated by σ production".
Quite recently, there appeared new very encouraging γγ data with high statistics. In experiments done at the Dubna Nuclotron [20,21], the authors analyzed the γγ spectra from pC and dC collisions at beam momenta 5.5 GeV/c (for protons) and 1.7-3.8 GeV/c per nucleon (for deuterons). The γγ   [19], (b) dC → γγ + X at p = 2.75 GeV/cA [20] and (c) pC → γγ + X at p = 5.5 GeV/c [20]. The spectra in figures (b) and (c) were obtained after the background subtraction [20].
It is very interesting that the γγ signal at M γγ 360 MeV with the width Γ γγ 60 MeV clearly observed in dC collisions is absent in pC collisions (while the η meson is clearly seen in both cases). This enhancement at M γγ 2 − 3M π is quite similar to the ABC effect observed for near-threshold dipion production from the nucleon-nucleon and lightest nuclei collisions. So, it can be interpreted in terms of σ-meson production from the meson cloud surrounding the dibaryon component of the incident deuteron [13], see figure 6. In the kinematics of the Dubna experiment [20,21], the possible interpretation of the above γγ signal can also be related to the well-known Landau-Pomeranchuk mechanism for bremsstrahlung in QED. According to Weldon [22], in hot and dense matter, due to removal of some important constraints valid in vacuum (Lorentz invariance, etc.), some processes forbidden in vacuum become possible, like the one-photon σ decay with a final dilepton production σ → π + π − → γ → e + e − . Since the standard two-photon decay σ → γγ is also valid in medium, one can relate the dilepton and digamma signals to each other. Thus, one gets two types of signals from the hot fireball, i.e., dilepton and digamma production, which carry valuable information about the state of hot matter produced in heavy-ion collisions. Recent results of the HADES Collaboration [23] seem to confirm this tight interrelation between dilepton and digamma production in heavy-ion collisions.

Conclusion
We have demonstrated above that in the intermediate-and high-energy collisions of nucleons and light or heavy ions a lot of light scalar σ mesons are emitted from different sources (excited hadrons, high-energy NN collisions, etc.). Masses and widths of these σ mesons will be dependent upon the hadron excitation energy (for excited hadrons) and the temperature and density of fireball arising in heavy-ion collisions. All these σ mesons have in general a very small branching ratio for the γγ decay (≤ 10 −5 ). So, it would be highly uneasy to observe these digamma signals at modest temperature and density of fireball.
When, however, the temperature and/or density become sufficiently high for restoration of chiral symmetry, M σ and Γ σ→ππ drop down while the γγ signal gets much higher at M σ 2M π . When M σ → 2M π and Γ σ→ππ → 0, all numerous σ mesons produced in fireball will emit the γγ pairs. So, one will observe the large γγ burst from fireball with M γγ 2M π 280 MeV. Simultaneously, observation of such a γγ burst will be a good signal for onset of the chiral phase transition in hot and/or dense nuclear matter. Thus, measuring the invariant mass and width of the γγ signal (together with the dilepton signal) will give unambiguous information about the chiral phase transition in hot and/or dense nuclear matter. Moreover, just the combination of the dilepton and digamma signals might be very advantageous in reducing the γγ background coming from π 0 → γγ decay [24].