Predictions for squeezed back-to-back correlations of $\phi\phi$ and $K^+K^-$ in high-energy heavy-ion collisions by event-by-event hydrodynamics

We calculate the squeezed back-to-back correlation (BBC) functions of $\phi \phi$ and $K^+K^-$ for heavy-ion collisions at RHIC and LHC energies, using ($2+1$)-dimensional hydrodynamics with fluctuating initial conditions. The BBC functions averaged over event-by-event calculations for many events for the hydrodynamic sources are smoothed as a function of the particle momentum. For heavy-ion collisions of Au+Au at $\sqrt{s_{NN}}=200$ GeV, the BBC functions are larger than those for collisions of Pb+Pb at $\sqrt{s_{NN}}=2.76$ TeV. The BBC of $\phi\phi$ may possibly be observed in peripheral collisions at the RHIC and LHC energies. It is large for the smaller sources of Cu+Cu collisions at $\sqrt{s_{NN}}=200$ GeV.


I. INTRODUCTION
In the late 1990s, it was shown [1,2] that the mass modification of the particles in the hot and dense hadronic sources can lead to a squeezed back-to-back correlation (BBC) of bosonantiboson pairs in high-energy heavy-ion collisions. This BBC caused by the interactions in medium is different from the pure quantum statistical correlations between the bosons with different isospins [3], which are negligible in high-energy heavy-ion collisions because the Fourier transformation of source-density, |ρ( 0, m)|, is very small even for the lightest meson [2,4]. Since it is associated with the source medium, the investigations of the BBC may possibly provide another way for people to understand the thermal and dynamical properties of the hadronic sources formed in high-energy heavy-ion collisions, in addition to particle yields and spectra.
The BBC function is defined as [1,2] where G c (k 1 , k 2 ) and G s (k 1 , k 2 ) are the chaotic and squeezed amplitudes, respectively, where a k and a † k are the annihilation and creation operators of the free boson with momentum k and mass m, · · · indicates the ensemble average. For a homogeneous source with volume V and temperature T , the BBC function can be written as [2] C(k, −k) where c k and s k are the coefficients of Bogoliubov transformation between the creation (annihilation) operators of the quasiparticle in medium with modified mass m * and the free observed particle [1,2], n k is boson distribution, and n 1 (k) = |c k | 2 n k + |s −k | 2 (n −k + 1).
For hydrodynamic sources, with the formula derived by Makhlin and Sinyukov [5], the chaotic and squeezed amplitudes can be expressed as [2,[6][7][8] are the coefficients of Bogoliubov transformation and boson distribution associated with the particle pair (see in Ref. [8]).
In Eq. (8), the factor e 2iK 1,2 ·r is equal to e 2iω k t for k 1 = k, k 2 = −k 1 = −k. So the BBC function C(k, −k) is sensitive to the temporal distribution of the freeze-out points [2,[6][7][8][9]. Recent research [8] indicates that the BBC functions for the hydrodynamic sources with Gaussian initial-energy distributions exhibit oscillations as a function of the particle momentum because of the sharp falls of temporal freeze-out distributions at long times.
Investigating the BBC behavior and predicting its effect in high-energy heavy-ion collisions based on more realistic models is of great interest [8]. In this work, we investigate the BBC functions for the hydrodynamic sources with event-by-event fluctuating initial conditions (FIC). We use (2+1)-dimensional hydrodynamics with the HIJING [10] FIC and the equation of state s95p-PCE [11] to describe the source evolution as in Ref. [12] The rest of this paper is organized as follows. In Sec. II, we give a brief review on the relativistic hydrodynamic model used in this work, and discuss the space-time distributions Relativistic hydrodynamics has been extensively applied in high-energy heavy-ion collisions. In this work, we use the ideal relativistic hydrodynamics in (2 + 1) dimensions to describe the transverse expansion of the particle-emitting sources formed in ultrarelativistic heavy-ion collisions, and adopt the Bjorken boost-invariant hypothesis [13] for the longitudinal evolution of the sources. The hydrodynamic equations of motion for the sources with zero net baryon density are from the local conservation of energy-momentum [14,15].
Under the assumption of Bjorken longitudinal boost invariance, we need only to solve the transverse equations of motion in the z = 0 plane [12], and the hydrodynamic solutions at z = 0 (v z = z/t) can be obtained by the longitudinal boost invariance [16,17].
Assuming the local equilibrium of system is reached at time τ 0 , we construct the initial energy density of the hydrodynamic source at z = 0, by using the AMPT code [18] in which the HIJING is used for generating the initial conditions, as [17,19] ǫ(τ 0 , x, y; Here p ⊥α is the transverse momentum of parton α in the fluid element at (x, y), x α (τ 0 ) and y α (τ 0 ) are the transverse coordinates of the parton at τ 0 , σ 0 is a transverse width parameter, and K is a scale factor which can be adjusted to fit the experimental data of produced hadrons [19]. The initial velocity of the fluid element is then determined by the initial energy density and the average transverse momentum of the partons in the element.
As in Ref [12], we solve the hydrodynamic equations numerically by the HLLE scheme and the Sod's operation splitting method [14,[20][21][22][23][24][25]. In the calculations, we use the equation of state s95p-PCE [11], which combines the hadron resonance gas at low temperatures and the lattice QCD results at high temperatures, and take the parameters σ 0 = 0.6 fm, τ 0 = 0.6 and 0.4 fm/c for the heavy-ion collisions at the RHIC and LHC, respectively. With these parameter values the hydrodynamic results of transverse momentum spectra and elliptic flow of identical pion and charged hadrons are consistent with the experimental dada at the RHIC and LHC [12,19].
For hydrodynamic sources with a Bjorken cylinder, the four-dimension element of freezeout hypersurface can be written as where τ , r ⊥ , and η are the proper time, transverse coordinate, and space-time rapidity of the element. The function f µ (τ, r ⊥ , η) is related to the freeze-out mechanism that is considered, and K µ 1,2 f µ (τ, r ⊥ , η) corresponds to the source distributions of proper time and space in the calculations [see Eqs. (7) and (8)]. In this work we assume that φ and K mesons are frozen out at fixed temperatures T f = 140 and 160 MeV, respectively, and use the AZHYDRO technique [15,26,27] to calculate the freeze-out hypersurface element.
We plot in Fig. 1 where N E is the total event number. In Eqs. (7) and (8), the factors e iq 1,2 ·r and e 2iK 1,2 ·r are equal to 1 and e 2iω k t , respectively, for The BBC function C(k, −k) defined with Eq. (1) is proportional to the square of the Fourier transformation of the source temporal distribution with the , and the effects of the source spatial distribution in the numerator and denominator in Eq. (1) may cancel out partially. So, the values of the BBC function are mainly related to the source temporal distribution, particle mass, mass shift, and source temperature. Because we use the same freeze-out temperature for φ meson in the calculations for the collisions with different centralities and energies, the main reason of the dependences of the φ BBC function on the collision centrality and energy (see Fig. 4 for fixed k and m * ) is that the temporal distribution of the source is narrower in peripheral collisions and becomes wider with increasing collision energy (see Fig. 1). For anisotropic sources, the anisotropic source velocity may lead to the dependence of the BBC function on the direction of particle momentum [8,29]. We show in Fig. 6 the dependence of the average BBC functions of φφ, C(k, −k) |k| , on the cosine of the particle azimuthal angle ψ for the collisions at the RHIC and LHC energies. Here, m * is taken as 1.05 GeV, corresponding approximately to the middle between the valley and peak of the BBC function (see Fig. 4), and the momentum region averaged is 0-1 GeV/c. The BBC functions decrease with increasing cos ψ more rapidly for peripheral collisions because the source transverse velocities are more anisotropic in this case [8,29]. We show in Fig. 7 the dependence of the average BBC functions of φφ, C(k, −k) |k| , on the particle pseudorapidity for the collisions at the RHIC and LHC energies. Here, m * is taken as 1.05 GeV and the momentum region averaged is 0-1 GeV/c. One can see that the BBC functions decrease with increasing | y|. This is because the average source longitudinal velocity is higher than the average source transverse velocity for hydrodynamic sources with Bjorken longitudinal boost invariance, and the higher longitudinal velocity leads to larger average values of e −k µ uµ/T f at | y| = 1 than at | y| = 0 [8,29].  GeV. Here, m * is taken as 1.05 GeV and the momentum region averaged is 0-1 GeV/c. C(k, −k) |k| , on the cosine of the particle azimuthal angle ψ and the particle pseudorapidity for Cu+Cu collisions, respectively. Here, m * is taken as 1.05 GeV and the momentum region averaged is 0-1 GeV/c. One can see that the average BBC functions almost independent of cos ψ for central collisions, and decrease slightly with increasing cos ψ for peripheral collisions because of the anisotropic transverse expansion [8,29]. As discussed in section III, the average BBC functions for Cu+Cu collisions also decrease with the increasing | y| like that for the Au+Au and Pb+Pb collisions.

V. SUMMARY AND CONCLUSIONS
In the hot and dense hadronic sources formed in high-energy heavy-ion collisions, the particle interactions in medium might lead to a squeezed BBC of boson-antiboson pairs.
In this paper, we investigate the BBC functions of φφ and K + K − for heavy-ion collisions of Au+Au at √ s N N = 200 GeV at the RHIC and Pb+Pb at √ s N N = 2.76 TeV at the LHC, using (2 + 1)-dimensional hydrodynamics with the FIC generated by HIJING. The investigations indicate that the BBC functions averaged over event-by-event calculations for many events for the hydrodynamic sources with the FIC are smoothed as a function of the particle momentum. This is different from the BBC functions for the hydrodynamic sources with Gaussian initial-energy distributions, which exhibit oscillations with respect to the particle momentum [8].