Effect of in-medium mass-shift on transverse-momentum spectrum and elliptic anisotropy of $\phi$ meson

TL;DR

This study investigates how in-medium mass-shift impacts the transverse-momentum spectrum and elliptic flow of $\, ext{phi}\, ext{mesons}$, revealing that mass-shift enhances yields and suppresses elliptic flow, with effects depending on source and probability.

Contribution

It provides the first detailed analysis of how in-medium mass-shift influences $\, ext{phi}\, ext{meson}$ observables in heavy-ion collisions, offering a method to constrain mass-shift ranges.

Findings

Mass-shift increases $\, ext{phi}\, ext{meson}$ yields.

Mass-shift suppresses elliptic flow $v_2$ at high momentum.

Effects vary with source and probability of mass-shift.

Abstract

We study the effect of in-medium mass-shift on transverse-momentum spectrum and elliptic anisotropy of $\phi$ meson. It is found that the mass-shift enhances the $\phi$ yields and suppresses the elliptic flow $v_2$ in large momentum region, and the effects increase with the increasing mass-shift. The effects are various for different sources and decrease with the increasing expanding velocity. We further study the effects for parts of all $\phi$ meson with mass-shift, and the effects decrease with the decreasing probability of $\phi$ meson with mass-shift. Since the different mass-shift lead to different transverse-momentum spectrum and $v_2$, our study may provide a way to restrict the ranges of in-medium mass-shift of $\phi$ meson in high-energy heavy-ion collisions.

Equations

$$N(\mathbf{k})={\omega }_{\mathbf{k}}\frac{d^{3}N}{d\mathbf{k}}={\omega }_{\mathbf{k}}⟨a_{\mathbf{k}}^{†}{a}_{\mathbf{k}}⟩,$$

$$N(\mathbf{k})=\frac{V}{2{\pi }^3}{\omega }_{\mathbf{k}}{n}_B(\mathbf{k}),$$

$$n_B(\mathbf{k})=\frac{1}{\exp ({\omega }_{\mathbf{k}}/T)-1}.$$

$$a_{\mathbf{k}}=c_{\mathbf{k}}{b}_{\mathbf{k}}+s_{-\mathbf{k}}^{\ast }{b}_{-\mathbf{k}}^{†}.\text{vspace}\ast -2mm$$

$$N(\mathbf{k})=\frac{V}{2{\pi }^3}{\omega }_{\mathbf{k}}{n}_1(\mathbf{k}),$$

$$n_1(\mathbf{k})=|c_{\mathbf{k}}{|}^2{n}_{\mathbf{k}}+|s_{-\mathbf{k}}{|}^2(n_{-\mathbf{k}}+1),$$

$$n_{\mathbf{k}}=\frac{1}{\exp ({\Omega }_{\mathbf{k}}/T)-1},$$

$$N_0(\mathbf{k})=\int \frac{g_i}{(2\pi {)}^3}{d}^4{\sigma }_{\mu }(r)k^{\mu }{n}_{{\mathbf{k}}^{\prime }}^0,$$

$$n_{{\mathbf{k}}^{\prime }}^0=\frac{1}{\exp ({\omega }_{{\mathbf{k}}^{\prime }}(r)/T(r))-1},$$

$${\omega }_{{\mathbf{k}}^{\prime }}^{\prime }(r)=\sqrt{{\mathbf{k}}^{\prime 2}(r)+m^2}=k^{\mu }{u}_{\mu }(r),$$

$$N_{\ast }(\mathbf{k})$$

$$c_{\pm {\mathbf{k}}^{\prime }}^{\prime }=\cosh [f_{{\mathbf{k}}^{\prime }}^{\prime }],s_{\pm {\mathbf{k}}^{\prime }}^{\prime }=\sinh [f_{{\mathbf{k}}^{\prime }}^{\prime }],$$

$$f_{{\mathbf{k}}^{\prime }}^{\prime }=\frac{1}{2}\log \left[{\omega }_{{\mathbf{k}}^{\prime }}^{\prime }/{\Omega }_{{\mathbf{k}}^{\prime }}^{\prime }\right]=\frac{1}{2}\log \left[k^{\mu }{u}_{\mu }(r)/k^{\ast \nu }{u}_{\nu }(r)\right],$$

$${\Omega }_{{\mathbf{k}}^{\prime }}^{\prime }(r)=\sqrt{{\mathbf{k}}^{\prime 2}(r)+m_{\ast }^2}$$

$$=\sqrt{[k^{\mu }{u}_{\mu }(r){]}^2-m^2+m_{\ast }^2}$$

$$=k^{\ast \mu }{u}_{\mu }(r),$$

$$n_{\pm {\mathbf{k}}^{\prime }}^{\prime }=\frac{1}{\exp ({\Omega }_{{\mathbf{k}}^{\prime }}(r)/T(r))-1},$$

$$N_{\ast }(\mathbf{k})=\int \frac{g_i}{(2\pi {)}^3}{d}^4{\sigma }_{\mu }(r)k^{\mu }[F_1{n}_{{\mathbf{k}}^{\prime }}^{\prime }+F_2],$$

$$F_1=1+2F_2,F_2=|s_{{\mathbf{k}}^{\prime }}^{\prime }{|}^2,$$

$$N_{\ast }(\mathbf{k})=\int \frac{g_i}{(2\pi {)}^3}{d}^4{\sigma }_{\mu }(r)k^{\mu }[n_{{\mathbf{k}}^{\prime }}^{\prime }+F_2].$$

$$N_{\ast }(\mathbf{k})=N_1(\mathbf{k})+N_2(\mathbf{k}),$$

$$N_1(\mathbf{k})=\int \frac{g_i}{(2\pi {)}^3}{d}^4{\sigma }_{\mu }(r)k^{\mu }{n}_{{\mathbf{k}}^{\prime }}^{\prime },$$

$$N_2(\mathbf{k})=\int \frac{g_i}{(2\pi {)}^3}{d}^4{\sigma }_{\mu }(r)k^{\mu }{F}_2.$$

$$\cos \beta =k_x/|{\mathbf{k}}_T|,\left(|{\mathbf{k}}_T|=\sqrt{k_x^2+k_y^2}\right).$$

$$N_{\ast }^p(\mathbf{k})$$

Full text

I Introduction

The main goal of high-energy heavy-ion collisions is to create the matter of quark-gluon plasma (QGP), and study its properties by analyzing the observables of particles in final state BRAHMS_NPA2005 ; PHOBOS_NPA2005 ; STAR_NPA2005 ; PHENIX_NPA2005 . $\phi$(s$\bar{s}$) meson is an excellent probe for studying QGP because it is sensitive to several aspects of the collision Shor_PRL1985 ; PHENIX_PRC2005 ; STAR_PRL2007 ; PHENIX_PRL2007 ; STAR_PLB2009 ; PHENIX_PRC2011 ; PHENIX_PRC2015 and it is expected to have a small interaction with the hadronic medium Shor_PRL1985 ; STAR_PRL2007 ; PHENIX_PRL2007 ; JHChen_PRC06 ; Hirano_PRC08 ; Nasim_PRC13 ; STAR_PRL16 . However, the recent experimental data of the elliptic flow of identified hadrons in the Pb-Pb collisions at $\sqrt{s_{NN}}=2.76$ GeV indicate that the $\phi$ meson may have a larger hadronic cross section than its current theoretical estimate ALICE_JHEP15 . It is still an open issue to determine the interaction between $\phi$ and the hadronic medium.

The transverse-momentum spectrum and elliptic flow of $\phi$ meson are important observables in high-energy heavy-ion collisions PHENIX_PRC2005 ; STAR_PRL2007 ; PHENIX_PRL2007 ; STAR_PLB2009 ; PHENIX_PRC2011 ; PHENIX_PRC2015 ; STAR_PRL16 ; ALICE_JHEP15 . The transverse-momentum spectrum can reveal information about the thermalization and expansion of the $\phi$ emission source PHENIX_PRC2005 ; STAR_PLB2009 ; PHENIX_PRC2011 ; PHENIX_PRC2015 , and the elliptic flow $v_{2}$ of $\phi$ meson was an important evidence for the formation of hot and dense matter with partonic collectivity STAR_PRL2007 ; PHENIX_PRL2007 ; STAR_PRL16 ; ALICE_JHEP15 . The $\phi$ meson is expected to have a mass-shift in the hadronic medium, and the mass-shift are various for several theoretical calculations Hatsuda ; ask_npa94 ; Song_PLB ; Smith_PRC98 ; Lin_npa02 ; Fuch_epja ; Cab_PRC03 ; Metag . In this work, we will study the effect of in-medium mass-shift on transverse-momentum spectrum and elliptic flow $v_{2}$ of $\phi$ meson, by using the ideal relativistic hydrodynamics in $2+1$ dimensions to describe the transverse expansion of sources with zero net baryon density and combine the Bjorken boost-invariant hypothesis Bjorken_PRD83 for the source longitudinal evolution. In the calculations, we use the equation of state of s95p-PCE Shen_PRC10 and take the initial energy density distribution in the transverse plane as the Gaussian distribution: $\epsilon=\epsilon_{0}\exp[-x^{2}/(2R_{x}^{2})-y^{2}/(2R_{y}^{2})]$, where $\epsilon_{0}$ and $R_{i}~{}(i=x,y)$ are the parameters of the initial source energy density and radii YZHANG_PRC15 . And the freeze-out temperature of $\phi$ is taken as $140$ MeV AsaCsoGyu99 ; Padula06 ; YZHANG16 ; YZHANG_PRC15 .

This paper is organized as follows. In Sec.II, we present the formulas of the single-particle momentum distribution for boson with in-medium mass-shift. In Sec. III, the effect of mass-shift on the transverse momentum spectrum and the elliptic flow $v_{2}$ of $\phi$ meson will be shown. Finally, a summary of this paper are given in Sec. IV.

II Formulism

Denote $a_{\mathbf{k}}\,(a^{\dagger}_{\mathbf{k}})$ the annihilation (creation) operator of the free boson with momentum ${\mathbf{k}}$ and mass $m$, and the invariant single-particle momentum distribution can be expressed by

[TABLE]

where $\langle\cdots\rangle$ means the thermal average, and $\omega_{\mathbf{k}}=\sqrt{{\mathbf{k}}^{2}+m^{2}}$ is the energy of the free particle. For a homogeneous source with volume $V$ and temperature $T$, the single particle momentum distribution becomes

[TABLE]

[TABLE]

Denote $b_{\mathbf{k}}\,(b^{\dagger}_{\mathbf{k}})$ the annihilation (creation) operator of the boson with momentum ${\mathbf{k}}$ and modified mass $m_{\!*}$ ($m_{\!*}=m-\delta\!m$) in hadronic medium. The operators $(a_{\mathbf{k}}\,,a^{\dagger}_{\mathbf{k}})$ and $(b_{\mathbf{k}}\,,b^{\dagger}_{\mathbf{k}})$ were related by the Bogoliubov transformation AsaCso96 ; AsaCsoGyu99

[TABLE]

For a homogeneous source, $c_{{\mathbf{k}}}=\cosh f_{{\mathbf{k}}}$, $s_{{\mathbf{k}}}=\sinh f_{{\mathbf{k}}}$ and $f_{{\mathbf{k}}}=\frac{1}{2}\log(\omega_{{\mathbf{k}}}/\Omega_{{\mathbf{k}}})$. Where $\Omega_{\mathbf{k}}=\sqrt{{\mathbf{k}}^{2}+m_{\!*}^{2}}$ is the energy of the particle in medium, and the single particle momentum distribution becomes AsaCsoGyu99 ; Padula06

[TABLE]

[TABLE]

[TABLE]

as can be seen from equation $(5)-(7)$, the single particle momentum distribution $N({\mathbf{k}})$ changes with the particle mass in medium.

For hydrodynamic sources, the single particle momentum distribution of the boson without in-medium mass-shift was expressed as Cooper ; KolHei03 ; Kol00

[TABLE]

[TABLE]

[TABLE]

and the single particle momentum distribution of the boson with in-medium mass-shift was expressed as AsaCsoGyu99 ; Padula06 ; YZHANG_PRC15 ; YZHANG_CPC15

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $g_{i}$ is the degeneracy factor for hadron species $i$. The quantity $d^{4}\sigma_{\mu}(r)$ is the four-dimension element of freeze-out hypersurface, and $u_{\mu}(r)$, $T(r)$ are the source four-velocity and temperature at freeze-out, respectively. $k^{\mu}=(\omega_{{\mathbf{k}}},{{\mathbf{k}}})$ is the four-momentum of the particle, and ${{\mathbf{k}}}^{\prime}$ is the local-frame momentum corresponding to ${\mathbf{k}}$. If there is no mass-shift, the coefficients $|c^{\prime}_{{\mathbf{k}}^{\prime}}|^{2}$ and $|s^{\prime}_{{\mathbf{k}}^{\prime}}|^{2}$ will be 1 and 0, respectively. And $N_{*}({\mathbf{k}})$ will be equal to $N_{0}({\mathbf{k}})$.

III Results

In this section, we will present the results in three subsections. In section A, we will show the effect of in-medium mass-shift on the transverse momentum spectrum of $\phi$. In section B, the effect of in-medium mass-shift on the elliptic flow $v_{2}$ will be shown. We further study the effect for parts of all $\phi$ mesons with mass-shift in section C.

III.1 Effect of in-medium mass-shift on the transverse momentum spectrum

In Fig. 1 (a) and (b), we show the normalized transverse momentum spectrum of $\phi$ meson with various mass-shift $\delta$m for the initial conditions $\epsilon_{0}=$ 20 and 40 GeV/fm3 and $R_{x}=3$ fm, $R_{y}=4$ fm. The ratio of the normalized transverse momentum spectrum with mass-shift to which without mass-shift is also shown in Fig. 1 (c) and (d). A small mass-shift leads to an increase in the yield of $\phi$ meson in large transverse momentum regions and thus reduce the slope of the transverse momentum spectrum, and the effect increases with the increasing mass-shift. For a fixed mass-shift, the transverse momentum spectrum for the initial conditions $\epsilon_{0}=$ 20 GeV/fm3 is more affected than for $\epsilon_{0}=$ 40 GeV/fm3.

With the Eq. (12) and (15), the Eq. (11) can be rewritten as

[TABLE]

[TABLE]

the quantity $n^{\prime}_{{\mathbf{k}}^{\prime}}$ in Eq. (16) is approximately equal to the quantity $n^{0}_{{\mathbf{k}}^{\prime}}$ in Eq. (8) for a small mass-shift, so the coefficient $F_{2}$ is the main factor for the particle spectrum.

In Fig. 2, we show the average coefficients $\overline{F_{2}}$ with various mass-shift and the ratio of $\overline{F_{2}}$ to $\overline{n^{\prime}_{{\mathbf{k}}^{\prime}}}$, where ”—” means the average over all freeze-out points. The average coefficient $\overline{F_{2}}$ is very small (see Fig. 2 (a) and (b)) and the coefficient $F_{2}$ is positive, so the coefficient $F_{1}$ is approximately equal to 1 and the Eq. (16) can be rewritten as

[TABLE]

The ratio of $\overline{F_{2}}$ to $\overline{n^{\prime}_{{\mathbf{k}}^{\prime}}}$ increases with the increasing transverse momentum, so the effect of in-medium mass-shift on spectrum increases with the increasing transverse momentum (see Fig. 1). With the same mass-shift, the average coefficients $\overline{F_{2}}$ are almost the same for the initial conditions $\epsilon_{0}=$ 20 and 40 GeV/fm3. The high initial energy density leads to a large expanding velocity and thus increases the yield ${n^{\prime}_{{\mathbf{k}}^{\prime}}}$ for large transverse momentum, so the ratio of $\overline{F_{2}}$ to $\overline{n^{\prime}_{{\mathbf{k}}^{\prime}}}$ for $\epsilon_{0}=$ 40 GeV/fm3 is smaller than for $\epsilon_{0}=$ 20 GeV/fm3 for a fixed mass-shift. This is the reason for the transverse momentum spectrum for $\epsilon_{0}=$ 20 GeV/fm3 is more affected than for $\epsilon_{0}=$ 40 GeV/fm3.

III.2 Effect of in-medium mass-shift on the elliptic flow

In Fig. 3, we show the $v_{2}$ of $\phi$ meson as a function of $k_{T}$ with various mass-shift and the ratio of $v_{2}$ with mass-shift to which without mass-shift for the initial conditions $\epsilon_{0}=$ 20 and 40 GeV/fm3 and $R_{x}=3$ fm, $R_{y}=4$ fm. The $v_{2}$ of $\phi$ meson for $\epsilon_{0}=$ 40 GeV/fm3 is a little greater than for $\epsilon_{0}=$ 20 GeV/fm3 when there is no mass-shift (\delta$$m = 0). The $v_{2}$ is suppressed by a small mass-shift in large transverse momentum regions, and this effect increases with the increasing mass-shift. The suppression effect for $\epsilon_{0}=$ 20 GeV/fm3 is a little stronger than for $\epsilon_{0}=$ 40 GeV/fm3 (see Fig. 3 (c) and (d)).

To analyse the effect of mass-shift on $v_{2}$, we rewrite the Eq. (18) as

[TABLE]

In Fig. 4, we show the ratio of $N_{1}$($\beta$) to $N_{1}$($\pi$/2) and the ratio of $N_{2}$($\beta$) to $N_{2}$($\pi$/2) for the initial condition $\epsilon_{0}=$ 20 GeV/fm3 and $R_{x}=3$ fm, $R_{y}=4$ fm. Where $\beta$ is the azimuthal angle of $\phi$ meson

[TABLE]

The ratio of $N_{1}$($\beta$) to $N_{1}$($\pi$/2) and the ratio of $N_{2}$($\beta$) to $N_{2}$($\pi$/2) can express the levels of the elliptical anisotropy of $N_{1}({\mathbf{k}})$ and $N_{2}({\mathbf{k}})$ in Eq. (III.2), respectively. The ratio of $N_{1}$($\beta$) to $N_{1}$($\pi$/2) is almost equal to the ratio of $N_{2}$($\beta$) to $N_{2}$($\pi$/2) in low transverse momentum region, so the mass-shift does not affect the $v_{2}$ in this region. However, the elliptical anisotropy of $N_{1}({\mathbf{k}})$ is much greater than $N_{2}({\mathbf{k}})$ in large transverse momentum region, so the $v_{2}$ is suppressed by mass-shift in large transverse momentum region. Although the elliptical anisotropy of $N_{2}({\mathbf{k}})$ is almost the same for different \delta$$m, but the ratio of $\overline{F_{2}}$ to $\overline{n^{\prime}_{{\mathbf{k}}^{\prime}}}$ for large \delta$$m is greater than for small \delta$$m, so the $v_{2}$ is more suppressed by a large mass-shift.

In Fig. 5, we show the ratio of $N_{1}$($\beta$) to $N_{1}$($\pi$/2) and the ratio of $N_{2}$($\beta$) to $N_{2}$($\pi$/2) for the initial condition $\epsilon_{0}=$ 40 GeV/fm3. The elliptical anisotropy of $N_{1}({\mathbf{k}})$ for $\epsilon_{0}=$ 40 GeV/fm3 is a little greater than for $\epsilon_{0}=$ 20 GeV/fm3 in large transverse momentum region, and thus leads to a more greater $v_{2}$ of $\phi$ meson without mass-shift for $\epsilon_{0}=$ 40 GeV/fm3 (see Fig. 3). The elliptical anisotropy of $N_{2}({\mathbf{k}})$ for $\epsilon_{0}=$ 20 GeV/fm3 is almost the same as for $\epsilon_{0}=$ 40 GeV/fm3, but the ratio of $\overline{F_{2}}$ to $\overline{n^{\prime}_{{\mathbf{k}}^{\prime}}}$ for $\epsilon_{0}=$ 40 GeV/fm3 is smaller than for $\epsilon_{0}=$ 20 GeV/fm3. And thus the $v_{2}$ for $\epsilon_{0}=$ 40 GeV/fm3 is less suppressed than for $\epsilon_{0}=$ 20 GeV/fm3 for a fixed mass-shift.

III.3 Parts of all $\phi$ meson have a mass-shift

When parts of all $\phi$ meson have a mass-shift, the single particle momentum distribution becomes

[TABLE]

where $p$ is the probability of $\phi$ meson with mass-shift.

In Fig. 6 and 7, we show the normalized transverse momentum spectrum and $v_{2}$ for parts of all $\phi$ meson with mass-shift. The effect of mass-shift on the transverse momentum spectrum and $v_{2}$ decreases with the decreasing probability of $\phi$ meson with mass-shift.

The above results indicate that the different mass-shift leads to different transverse momentum spectrum and $v_{2}$. The ranges of mass-shift of $\phi$ meson predicted by theory can be restricted by comparing the simulated transverse momentum spectrum and $v_{2}$ to the experimental data in high-energy heavy-ion collisions.

IV Summary

In this paper we studied the effect of in-medium mass-shift on the transverse-momentum spectrum and $v_{2}$ of $\phi$ meson. The mass-shift leads to an increase in the yield of $\phi$ meson in large transverse momentum region and thus reduce the slope of the transverse-momentum spectrum, and the elliptic flow $v_{2}$ of $\phi$ meson is suppressed by the mass-shift in large transverse momentum region. The effects of mass-shift on the transverse-momentum spectrum and $v_{2}$ increase with the increasing mass-shift. The effects also decrease with the increasing expanding velocity of the source. We further studied the effect of mass-shift on the transverse-momentum spectrum and $v_{2}$ for parts of all $\phi$ meson with mass-shift, and the effect of mass-shift on the transverse-momentum spectrum and $v_{2}$ decreases with the decreasing probability of $\phi$ meson with mass-shift. Our study may provide a way to restrict the ranges of mass-shift of $\phi$ meson by comparing the simulated transverse-momentum spectrum and $v_{2}$ to the experimental data in high-energy heavy-ion collisions.

Acknowledgements.

This research was supported by the National Natural Science Foundation of China under Grant No. 11647166, 11747155, the Natural Science Foundation of Inner Mongolia under Grant No. 2017BS0104, Changzhou Science and Technology Bureau CJ20180054, and the Foundation of Jiangsu University of Technology under Grant No. KYY17028, KYY18048.