Vector meson spectral function in a dynamical AdS/QCD model

A new matter state, quark-gluon plasma, has been generated at the Relativistic Heavy-Ion Collider (RHIC) in Brookhaven. These substances are interconnected by strong interactions which can be elaborated by quantum chromodynamics (QCD). So this new state of matter is also extensively called the strongly-correlated quark-gluon plasma (sQGP) Gyulassy and McLerran (2005). The strength of the coupling between them is related to the energy. At high energy, the coupling is very small, and quarks have the property of asymptotic freedom, which can be studied by perturbation theory. At low energy, the coupling is very large, and quarks have the property of color confinement, so the perturbation theory is no longer applicable and a non-perturbation method needs to be employed. Fortunately, gauge/gravity correspondence or the bulk/boundary correspondence  Maldacena (1998); Witten (1998); de Haro et al. (2001); Rey and Yee (2001); Klebanov and Witten (1999); Karch and Katz (2002); Erlich et al. (2005); Freedman et al. (1999); Policastro et al. (2002); Casalderrey-Solana et al. (2014); Gopakumar and Vafa (1999); Emparan et al. (1999); Caldarelli et al. (2000); Hawking et al. (1999), a new approach, provides a powerful tool to study the non-perturbative regime of the physic system. The most attractive achievement of the gauge/gravity correspondence is that one can also perform an analytical calculation by this technique.

The suppression of $J/\Psi$ and $\Upsilon(1S)$ peaks in the dilepton spectrum plays an important role in many signals of phase transition. Heavy quarkonium may survive, due to Coulomb attraction between quark and antiquark, as bound states when the quark matter encounters phase transitions. The seminal paper of quarkonium suppression is from Matsui and Satz Matsui and Satz (1986) where the estimate for the dissociation temperature was first made, $T_{dis}\sim 1.2T_{c}$. By analyzing the spectral function in the Nambu-Jona-Lasinio model, Ref. Hatsuda and Kunihiro (1985) points out that quark-antiquark excitations, small mass and narrow width in the $\pi-\sigma$ channels, still exist at $T>T_{c}$ where $T_{c}$ is the phase transition temperature. However, Ref. Asakawa and Hatsuda (2004) studies $J/\Psi$ and $\eta_{c}$, by discussing the correlation function of $J/\Psi$ at finite temperature on $32^{3}*(32-96)$ anisotropic lattices using the maximum entropy method (MEM) , in the deconfined plasma from lattice QCD, the results show that $J/\Psi$ and $\eta_{c}$ can survive in the plasma with obvious resonance state up to $T\simeq 1.6T_{c}$ and dissociate at $1.6T_{c}\leq T\leq 1.9T_{c}$. Next, Ref. Datta et al. (2004) considers the $Q\overline{Q}$ with quark masses close to the charm mass on very fine isotropic lattices, the results show that the $J/\Psi$ and $\eta_{c}$ can survive up to $1.5T_{c}$ and vanish at $3T_{c}$.

RHIC experiments show that sQGP not only has the properties of high temperature and density, but also a very intense magnetic field engendered in noncentral relativistic heavy-ion collision and exists long enough to also affect the produced quark-gluon plasma  Skokov et al. (2009); Bzdak and Skokov (2012); Zhu et al. (2020, 2021); Voronyuk et al. (2011); Roy et al. (2017); Tuchin (2013); Seiberg (1995); Gusynin et al. (1996). In recent years, a lot of work has revealed the dependence of thermal QGP properties on the magnetic field and chemical potential Kharzeev and Yee (2011); Cao et al. (2020); Preis et al. (2011); Albash et al. (2008); Kalaydzhyan and Kirsch (2011); Cao et al. (2021); Finazzo et al. (2016); Preis et al. (2013); Rougemont et al. (2016); Bu et al. (2013); Hoyos et al. (2011); Gorsky et al. (2011); Cai et al. (2013); Mamo (2015); Jensen et al. (2010); Dudal et al. (2016); Jokela et al. (2012); Callebaut et al. (2013); Filev and Raskov (2010); Gürsoy et al. (2017); Evans et al. (2016); Braga and Ferreira (2018); Colangelo et al. (2013); Gursoy et al. (2018); Braga and Ferreira (2019); Zhu et al. (2019); Li et al. (2017); Zhou et al. (2020); Feng et al. (2020). Ref. Fujita et al. (2009) deliberates the finite-temperature influences on the spectral function in the vector channel and pointed out that the dissociation of the heavy vector meson tower onto the AdS black hole results in the in-medium mass shift and the width broadening. The lattice QCD studies, for the $J/\psi$ spectral function at finite temperature, show that mesons can survive beyond $2T_{c}$ where $T_{c}$ devotes to the phase transition temperature Umeda et al. (2001); Asakawa and Hatsuda (2004); Datta et al. (2004). Ref.Ghoroku et al. (2006) calculates low-lying (axial-)vector meson mass spectra in a five-dimensional deformed $AdS_{5}$ holographic QCD by a bulk scalar field, which is a kind of soft-wall model. They pointed out that such deformation effects are important to reproduce the experimental data for vector mesons and the axial-vector mesons. The paper Ávila and Patiño (2020), by introducing the probe D7-branes to the 10d gravitational backgrounds to add flavor degrees of freedom, indicates that the magnetic field decreases the mass gap of the meson spectrum along with their masses. Ref. Braga and Ferreira (2018) displays the heavy meson dissociation in a plasma with magnetic fields and states clearly the dissociation effect increases with the magnetic field for magnetic fields parallel and perpendicular, respectively, to the polarization. The case of finite density is studied in  Braga et al. (2017), which gives that increasing temperature and chemical potential decrease the peak of spectral function showing the thermal dissociation process. Ref. Hou and Ren (2008) calculates the dissociation temperature of $J/\Psi$ and $\Upsilon(1S)$ state by solving the Schrödinger equation of the potential model. They find the ratios of the dissociation temperatures for quarkonium with the $U$-ansatz of the potential to the deconfinement temperature to agree with the lattice results within a factor of two.

Inspired by  Braga et al. (2017); Braga and Ferreira (2018), the main objective of this manuscript is to investigate the spectral function for $J/\psi$ and $\Upsilon(1S)$ state by considering a more general situation, including chemical potential and magnetic field in gravity background, which could better simulate the sQGP environment produced by RHIC experiments. The introduced magnetic field breaks the $SO(3)$ invariance into $SO(2)$ invariance, that is, the magnetic field only changes the geometry perpendicular to it which is different from other models. The manuscript is organized as follows. In section II, we review the magnetized holographic QCD model with running dilaton and chemical potential. In section III, we give the specific derivation process for calculating the spectral function of $J/\psi$ and $\Upsilon(1S)$ states. In section IV, we show and discuss the figure results.

We consider a 5d EMD gravity system with two Maxwell fields and a neutral dilatonic scalar field as a thermal background for the corresponding hot, dense and magnetized QCD, where the finite temperature effects are introduced by black hole thermodynamics, the finite density effects are related to the charge of the black hole, and the real physical magnetic fields are related to the five-dimensional magnetic field. The 5-d Lagrangian is given by

F_(i)μν$($i=1,2$)isthefieldstrengthtensorforU(1)gaugefield,$g_i(ϕ_0)$($i=1,2$)isthegaugecouplingkineticfunction,$ϕ_0$isthedilatonfield,$V(ϕ_0)$isthepotentialofthe$ϕ_0$(see\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Bohra:2019ebj}{\@@citephrase{(}}{\@@citephrase{)}}}forexactexpression).Byintroducingexternalmagneticfieldin$x_1$directiondamagesthe$SO(3)$invarianceto$SO(2)$invarianceinboundaryspatialcoordinates,soweconsiderthefollowinganisotropicansatzforthebackgroundblackeningmetric$f_μν$and$(ϕ_0,A_μ)$fieldsinEinsteinframe~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Bohra:2019ebj}{\@@citephrase{(}}{\@@citephrase{)}}}\begin{aligned} ds^{2}&=\frac{R^{2}S(z)}{z^{2}}(-f(z)dt^{2}+dx_{1}^{2}+e^{B^{2}z^{2}}(dx_{2}^{2}+dx_{3}^{2})+\frac{dz^{2}}{f(z)}),\\ \phi_{0}&=\phi_{0}(z),\quad A_{(1)\mu}=A_{t}(z)\delta_{\mu}^{t},\quad F_{(2)}=Bdx_{2}\wedge dx_{3},\end{aligned}with\begin{aligned} f(z)&=1+\int_{0}^{z}d\xi\xi^{3}e^{-B^{2}\xi^{2}-3A(\xi)}[K+\frac{\widetilde{\mu}^{2}}{2R_{gg}R^{2}}e^{R_{gg}\xi^{2}}],\\ K&=-\frac{1+\frac{\widetilde{\mu}^{2}}{2R_{gg}R^{2}}\int_{0}^{z_{h}}d\xi\xi^{3}e^{-B^{2}\xi^{2}-3A(\xi)+R_{gg}\xi^{2}}}{\int_{0}^{z_{h}}d\xi\xi^{3}e^{-B^{2}\xi^{2}-3A(\xi)}},\\ \widetilde{\mu}&=\frac{\mu}{\int_{0}^{z_{h}}d\xi\frac{\xi e^{-B^{2}\xi^{2}}}{g_{1}(\xi)\sqrt{S(\xi)}}},\end{aligned}where$R$istheradiusoftheasymptoticAdSspaces,$S(z)$isthescalefactor,$f(z)$representstheblackeningfunctionand$μ$isthechemicalpotential.Westipulatethattheasymptoticboundaryisat$z=0$and$z=z_h$labelsthelocationofhorizonwhere$f(z_h)=0$.Theconcreteformofgaugecouplingfunction$g_1$canbedeterminedbyfittingthevectormesonmassspectrum.ThelinearReggetrajectoriesfor$B=0$canberestoredwhen\begin{equation}g_{1}(z)=\frac{e^{-R_{gg}z^{2}-B^{2}z^{2}}}{\sqrt{S(z)}}.\end{equation}Here,this$B$,inunits$GeV$,isthe5dmagneticfield.The4dphysicalmagneticfieldis$eB∼constR×B$where$R=1GeV^-1$istheAdSlengthand$const=1.6$(SeedetailsinRef.\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Dudal:2015wfn}{\@@citephrase{(}}{\@@citephrase{)}}}).Forconvenience,wewillusethe5-Dimensionalmagneticfield$B$toparticipateinthecalculationinthesubsequentparts.Using$S(z)=e^2A(z)$,onecanobtain$R_gg=1.16GeV^2$forheavymesonstate$J/ψ$.Inthefollowingcalculation,wetake$A(z)=-az^2$where$a=0.15GeV^2$matchingwiththelatticeQCDdeconfinementtemperatureat$B=0GeV$~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Dudal:2017max}{\@@citephrase{(}}{\@@citephrase{)}}}.\par\par TheHawkingtemperaturewithmagneticfieldandchemicalpotentialcanbecomputedbysurfacegravity\begin{equation}T(z_{h},\mu,B)=\frac{-z_{h}^{3}e^{-3A(z_{h})-B^{2}z_{h}^{2}}}{4\pi}(K+\frac{\widetilde{\mu}^{2}}{2R_{gg}R^{2}}e^{R_{gg}z_{h}^{2}}).\end{equation}Theoriginalpaper~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Bohra:2019ebj}{\@@citephrase{(}}{\@@citephrase{)}}}assumesthatthedilatonfield$ϕ_0$remainsrealeverywhereinthebulk,whichleadstomagneticfield$B≤B_c≃0.61GeV$.Inthisholographicinversemagneticcatalysismodel,thedeconfinementtemperatureis$T_c=0.268GeV$atzerochemicalpotentialandmagneticfield.\par\par$

In this section, we will calculate the spectral function for the $J/\psi$ state and $\Upsilon(1S)$ state by a phenomenological model proposed in Ref. Braga et al. (2017). The vector field $A_{m}=(A_{\mu},A_{z})(\mu=0,1,2,3)$ is used to represent the heavy quarkonium, which is dual to the gauge theory current $J^{\mu}=\overline{\Psi}\gamma^{\mu}\Psi$. The standard Maxwell action takes the following form

where $F_{mn}=\partial_{m}A_{n}-\partial_{n}A_{m}$. The scalar field $\phi(z)$, including three energy parameters, is used to parameterize different mesons,

where $\kappa$ labels the quark mass, $\Gamma$ is the string tension of the quark pair and $M$ denotes a large mass related to the heavy quarkonium non-hadronic decay. There is a matrix element $\langle 0|J_{\mu}(0)|X(1S)\rangle=\epsilon_{\mu}f_{n}m_{n}$ ($X$ shows the heavy mesons $\Upsilon$ or $\Psi$, $\langle 0|$ is the hadronic vacuum, $J_{\mu}$ is the hadronic current and $f_{n}$ indicates the decay constant)when the heavy vector mesons decay into leptons. The value of three energy parameters for charmonium and bottomonium in the scalar field, determined by fitting the spectrum of masses Braga and Ferreira (2018), are respectively:

The spectral function for $J/\Psi$ state and $\Upsilon(1S)$ state will be calculated with the help of the membrane paradigm Iqbal and Liu (2009). The equation of motion obtained from Eq. (3) are as follows

For the z-foliation, the conjugate momentum of the gauge field $A^{\mu}$ is given by the following formula:

Supposing plane wave solutions for $A^{\mu}$ have nothing to do with the coordinates $x_{2}$ and $x_{3}$-in other words, the plane wave propagates in the $x_{1}$ direction. The equation of motion (6) can be split into two parts: longitudinal-the fluctuations along $(t,x_{1})$; transverse-fluctuations along ($x_{2},x_{3}$). Combined with Eq.(7), the longitudinal components $t,x_{1},z$ of Eq.(6) can be expressed respectively as

By using the Bianchi identity, one can get

The longitudinal ”conductivity” and its derivative are defined as

The transverse channel is decided by following the dynamical equation

The latter two equations from the Bianchi identity are constraint equations. Now, the transverse ”conductivity” and its derivative are defined as

The Kubo’s formula shows that the five-dimensional ”conductivity” at the boundary is related to the retarded Green’s function:

where $\sigma_{L}$ is interpreted as the longitudinal AC conductivity and $\sigma_{T}$ denotes the transverse AC conductivity; Here $G_{R}^{L}(\omega)$ and $G_{R}^{T}(\omega)$ are longitudinal and transverse retarded two-point Green’s function of the currents, which can be calculated from the solutions of equations of motion from AdS/CFT Iqbal and Liu (2009); Son and Starinets (2002). To obtain flow equations  (33) and  (38), we assume $A_{\mu}=A_{n}(p,z)e^{-i\omega t+ipx_{1}}$, where $A_{n}(p,z)$ is the quasinormal modes. Therefore, we have $\partial_{t}F_{x_{1}t}=-i\omega F_{x_{1}t}$, $\partial_{t}j^{x_{1}}=-i\omega j^{x_{1}}$ for Eq.  (33) and $\partial_{t}F_{x_{1}x_{2}}=-i\omega F_{x_{1}x_{2}}$, $\partial_{t}F_{tx_{2}}=-i\omega F_{tx_{2}}$, $\partial_{t}j^{x_{2}}=-i\omega j^{x_{2}}$ for Eq.(38) . By combining Eq.(9), Eq.(10), Eq.(31) for longitudinal case and Eq.(34), Eq.(35), Eq.(36) for the transverse case and the momentum $P=(\omega,0,0,0)$ with metric(II), the Eq.(33) and Eq.(38) can be written as

The initial condition for solving the equation can be obtained by requiring regularity at the horizon $\partial_{z}\sigma_{L(T)}=0$. It is not difficult to find that the metric Eq.(II) restores the SO(3) invariance and the flow equations Eq.(41) and Eq.(42) have the same form when magnetic field $B=0GeV$. The spectral function is defined by the retarded Green’s function

In this paper, By making use of a general magnetic field dependent dynamical AdS/QCD model we investigate the spectral function of heavy mesons($J/\Psi$ and $\Upsilon(1S)$). Here, the heavy vector mesons are represented by a phenomenological model proposed in Ref. Braga et al. (2017). The spectral function are calculated by using the membrane paradigm Iqbal and Liu (2009). First, we show the holographic quarkonium masses in zero temperature case (see Karch et al. (2006) for the detailed calculation process), in tables 1 and 2, for the states $1S,2S,3S,4S$ in two different dilatons, quadratic dilaton(model1)-$\phi(z)=\kappa^{2}z^{2}+Mz+\tanh(\frac{1}{Mz}-\frac{\kappa}{\sqrt{\Gamma}})$, nonquadratic dilaton(model2) Martin Contreras et al. (2021)-$\phi(z)=(\kappa z)^{2-\alpha}+Mz+\tanh(\frac{1}{Mz}-\frac{\kappa}{\sqrt{\Gamma}})$, for the values of parameters in model2, see Ref. Martin Contreras et al. (2021). Further, we compare the theoretical values with the experimental valuesTanabashi et al. (2018) for heavy mesons mass.

This work is supported by the National Natural Science Foundation of China by Grant Nos. 11735007,11890711, and 11890710. In addition, the first author wants to thank Hai-cang Ren and Zhou-Run Zhu for helpful discussions.