1 Introduction

One of the primary objectives of heavy-ion collision (HIC) experiments at Relativistic Heavy Ion Collider (RHIC) and at Large Hadron Collider (LHC) is to create and characterize a deconfined state of thermal quarks and gluons, called quark gluon plasma (QGP). It is expected that after a proper time \(\tau _0 \sim 1~\text {fm}/c\) of the collision, the system attains a state of local thermal equilibrium. The evolution of the QGP in spacetime can be studied by using relativistic hydrodynamics [1,2,3,4], which is a low frequency or long wavelength effective theory of many body interacting systems. In the non-central collisions of nuclei at RHIC and LHC energies large electric and magnetic (EM) fields will be produced due to the electric current generated by the accelerated motion of charged spectators, i.e. protons. In non-central collisions of heavy nuclei (Au + Au or Pb + Pb) at RHIC and LHC energies, the transient magnetic field (B) can be as high as \(\sim 10^{17}{-}10^{18}\) Gauss) [5,6,7,8,9]. Therefore, in such a situation, it is imperative to consider the effect of time varying magnetic and electric fields on the characterization of the QGP. The survival time of the fields crucially depends on the electrical conductivity of the QGP [10,11,12]. We theoretically evaluate the dynamic structure factor [13] of the system in the present work in an idealistic scenario of non-expanding QGP with critical point under a static magnetic field. The inclusion of magnetic field in relativistic hydrodynamics is studied within the ambit of relativistic magnetohydrodynamics (MHD) which is a self-consistent macroscopic framework that deals with the evolution of mutually interacting charged fluid along with the EM fields. Recently, several authors have studied the effect of the EM fields on QGP fluid in the context of special relativistic systems [14,15,16,17,18,19,20].

In the relativistic viscous hydrodynamics and relativistic magnetohydrodynamics, the transport coefficients, such as the shear viscosity, bulk viscosity, thermal conductivity, etc. are taken as inputs which can be estimated from an underlying microscopic theory [21,22,23,24]. A straightforward extension of the non-relativistic viscous fluid dynamics (Navier–Stokes equation) to relativistic regime (without magnetic field) [25, 26] leads to acausal and unstable solutions [27,28,29]. These issues were addressed and resolved by Müller, and Israel and Stewart (MIS) [30, 31] who developed a causal and stable second-order relativistic hydrodynamics. Here the order of the theory is dictated by different orders of gradients in the expansion hydrodynamic quantities i.e., energy–momentum tensor (EMT).

The dynamic density correlator or the structure factor, \({\mathcal {S}}_{nn}({\varvec{k}},\omega ),\) has been studied extensively in condensed matter physics by using linear response theory [13] within the ambit of fluid dynamics. The correlation of density fluctuations can be examined through the dynamic spectral structure \({\mathcal {S}}_{nn}({\varvec{k}},\omega )\) in wave vector (k),  frequency \((\omega )\) space. Ordinarily, for a fluid modelled by relativistic hydrodynamics, the \({\mathcal {S}}_{nn}({\varvec{k}},\omega )\) consists of three distinct peaks, two of them are Brillouin (B) peaks, created due to pressure fluctuation at constant entropy and the other is the Rayleigh (R) peak, originates due to thermal or entropy fluctuation at constant pressure [32, 33].

The spectral structure has been extensively investigated experimentally in condensed matter physics to determine the speed of sound by using the scattering of photons and neutrons. It is observed that the position of the B-Peaks in \({\mathcal {S}}_{nn}({\varvec{k}},\omega )\) depends on the speed of sound and the width of both B and R peaks enable us to evaluate various transport coefficients (shear and bulk viscosities, thermal conductivity), and thermodynamic response functions (specific heats). For QCD (quantum chromodynamics) matter, no such external probes are available making the direct detection of the peaks extremely difficult, if not impossible. However, such investigation may shed light on the speed of sound waves, and consequently on the equation of state (EoS) of the QCD matter. In the previous studies [32, 33], the relativistic hydrodynamics is used to find out the correlation in density fluctuation in QCD matter without including magnetic field. In the present work, we have considered the relativistic MHD to study the dynamical density correlator in a baryon-rich fluid. We derive \({\mathcal {S}}_{nn}({\varvec{k}},\omega ),\) which allows us to determine the speed of the perturbation propagating as sound waves through the QCD medium immersed in a constant magnetic field.

The manuscript is organized as follows: in Sect. 2, we discuss briefly the form of energy–momentum tensor of fluid in the presence of EM fields. In the next section i.e. in Sect. 3, we find the linearized equations of MIS hydrodynamics and derive the density-density correlation function. In Sect. 4, we discuss results that stem from our analysis, and the summary and conclusion is presented in Sect. 5.

Throughout the article, we use the natural units, \(c=k_{B}=\epsilon _{0}=\mu _{0}=1\) and the metric tensor is \(g^{\mu \nu }=\) diag\(\left( +1,-1,\right. \)\(\left. -1,-1\right) .\) The time-like fluid four velocity \(u^{\mu }=\frac{1}{\sqrt{1-v^2}}\,(1,{\varvec{v}})\) satisfies the relation, \(u_{\mu }u^{\mu }=1.\) Also, we use the following decomposition \(\partial _{\mu }\equiv u_{\mu }u_{\nu }\partial ^{\nu }+(g_{\mu \nu }-u_{\mu }u_{\nu }) \partial ^{\nu }=u_{\mu }D+\nabla _{\mu }.\) The fourth-rank projection tensor is defined as \(\Delta ^{\mu \nu }_{\alpha \beta }=\frac{1}{2}\left( \Delta ^{\mu }_{\alpha } \Delta ^{\nu }_{\beta }+\Delta ^{\mu }_{\beta }\Delta ^{\nu }_{\alpha }\right) -\frac{1}{3}\Delta ^{\mu \nu }\Delta _{\alpha \beta },\) where \(\Delta ^{\mu \nu }\equiv g^{\mu \nu }-u^\mu u^\nu \) is the projection operator, such that \(\Delta ^{\mu \nu }u_\nu =0.\)

2 Relativistic magnetohydrodynamics

In this section, first we discuss the equation of motion of electromagnetic (EM) field and magnetohydrodynamics.

2.1 Equation of motion of the EM field

We start by discussing the relativistically covariant formulation of classical electrodynamics. The second rank anti-symmetric EM field tensor \(F^{\mu \nu },\) can be defined as, in terms of electric field four-vector \(E^{\mu }(=F^{\mu \nu }u_\nu ),\) magnetic field four-vectors \(B^{\mu }=(\frac{1}{2}\epsilon ^{\mu \nu \alpha \beta }u_\nu F_{\alpha \beta }=\tilde{F}^{\mu \nu }u_\nu ),\) and the fluid four-velocity \(u^{\mu }\) [34,35,36],

$$\begin{aligned} F^{\mu \nu }=E^{\mu }u^{\nu }-E^{\nu }u^{\mu }+\epsilon ^{\mu \nu \alpha \beta }u_{\alpha }B_{\beta }, \end{aligned}$$
(1)

and its dual counter part is represented by,

$$\begin{aligned} \tilde{F}^{\mu \nu }=B^{\mu }u^{\nu }-B^{\nu }u^{\mu }- \epsilon ^{\mu \nu \alpha \beta }u_{\alpha }E_{\beta }, \end{aligned}$$
(2)

where, \(\epsilon ^{\mu \nu \alpha \beta }\) is the Levi-Civita tensor. It is quite straight forward to see by using the anti-symmetric property of \(F^{\mu \nu }\) that both \(E^{\mu }\) and \(B^{\mu }\) are orthogonal to \(u^{\mu }\) i.e., \(E^{\mu }u_{\mu }=B^{\mu }u_{\mu }=0.\) It may be noted that in the rest frame \(u^{\mu }=(1,\textbf{0}),\) \(E^{\mu }=(0,\textbf{E}),\) and \(B^{\mu }=(0,{\textbf{B}}),\) where \(\textbf{E}\) and \({\textbf{B}}\) correspond to the electric and the magnetic three vectors fields with \({\textrm{E}}^{i}=F^{i 0}\) and \({\textrm{B}}^{i}=-\frac{1}{2} \epsilon ^{i j k} F_{j k}.\) The indices i, j, k run over \(1, \,2,\, 3.\) The \(E^\mu \) and \(B^\mu \) can be interpreted as the electric and magnetic fields respectively measured in a frame in which the fluid moves with velocity \(u^\mu .\)

Using the EM field tensor and its dual, we can express the Maxwell’s equations in a covariant way as,

$$\begin{aligned} \partial _{\mu }F^{\mu \nu }= & {} J^{\nu }, \end{aligned}$$
(3)
$$\begin{aligned} \partial _{\mu }\tilde{F}^{\mu \nu }= & {} 0, \end{aligned}$$
(4)

where, \(J^{\nu }\) represents the electric charge four-current and it acts as the source of EM field. The electric charge four-current \((J^\mu )\) can be decomposed as follows:

$$\begin{aligned} J^{\mu }=j^{\mu }+d^{\mu }, \end{aligned}$$
(5)

where, \(j^{\mu }\) is the conduction current and \(d^{\mu }=\Delta ^{\mu }_{\nu }J^{\nu },\) is the charge diffusion current, \(n_{q}=u_{\mu }J^{\mu } \) the proper net charge density. If we consider a linear relation between \(j^{\mu }\) and \(E^{\mu }\) (Ohm’s law) then we can write \( j^{\mu }=\sigma ^{\mu \nu }E_{\nu },\) where \(\sigma ^{\mu \nu }\) is the second rank conductivity tensor. The construction of \(j^{\mu }\) follows \(u_{\mu }j^{\mu }=0.\) It implies that the conduction current exists even in the absence of any net charge. The solutions of Eqs. (3) and (4) for given electric charge four current \(J^{\mu }\) in Eq. (5) completely determine the evolution of electromagnetic field. It also acts as a coupling between the fluid and EM fields as it contains the fluid information e.g. fluid conductivity \(\sigma ^{\mu \nu },\) net charge density \(n_{q}\) etc. We consider a system consists of quarks, anti-quarks and gluons. All the relevant thermodynamic quantities and the EoS used in this work have been calculated by taking these degrees of freedom into account (we do not repeat the discussions on EoS here but refer to Ref. [37] for details). Therefore, the net (quark–antiquark) charge density is equivalent to net (quark–antiquark) number density and the relation \(n_{q}=q n_{f}\) holds, where \(n_{f}\) corresponds to net number density.

At first, we assume that the fluid does not possess any polarization or magnetization. In that case the EM field stress-energy tensor (EMT) can be written as,

$$\begin{aligned} T^{\mu \nu }_{EM}=-F^{\mu \lambda }F^{\nu }_{\lambda }+ \frac{1}{4}g^{\mu \nu }F^{\alpha \beta }F_{\alpha \beta }. \end{aligned}$$
(6)

If we take the partial derivative of the field stress-energy tensor, we obtain the equation of motions,

$$\begin{aligned} \partial _{\mu }T^{\mu \nu }_{EM}=-F^{\nu \lambda }J_{\lambda }. \end{aligned}$$
(7)

In the above equation the current density due to external source is ignored. In presence of an external source \((J^{\mu }_{ext})\), the total current is given by,

$$\begin{aligned} J^{\mu }=J^{\mu }_f +J^{\mu }_{ext}. \end{aligned}$$
(8)

In such case, the external current acts as a source term in energy–momentum conservation equation.

In this article, we consider an ideal magneto-hydrodynamic limit which resembles very large magnetic Reynolds number \(R_{m}=Lv\sigma \mu \gg 1,\) where L is the characteristic macroscopic length or time scale, v is the characteristic velocity of the flow, \(\sigma \) is the isotropic electrical conductivity \((\sigma ^{\mu \nu }=\sigma g^{\mu \nu })\), and \(\mu \) is the magnetic permeability of the QGP. It is evident that \(R_m\) increases with \(\sigma .\) From the induced current density equation, \(J^{\mu }_{ind}=j^{\mu }=\sigma E^{\mu },\) it is clear that when \(\sigma \) attains a very large value \((\rightarrow \infty )\) then the electric field will approach zero \((E^\mu \rightarrow 0)\) to keep \(J^{\mu }_{ind}\) but finite. This simplifies the EM tensor \(F^{\mu \nu }\) to the following form,

$$\begin{aligned} F^{\mu \nu }\rightarrow B^{\mu \nu }=\epsilon ^{\mu \nu \alpha \beta }u_{\alpha }B_{\beta }. \end{aligned}$$
(9)

Using Eqs. (8) and (9) in the Maxwell’s equations Eq. (3) we obtain,

$$\begin{aligned} \epsilon ^{\mu \nu \alpha \beta }\left( u_{\alpha }\partial _{\mu }B_{\beta } +B_{\beta }\partial _{\mu }u_{\alpha }\right) =J^{\nu }_{f} +J^{\nu }_{ext}. \end{aligned}$$
(10)

The EMT in absence of electric field can be obtained from Eqs. (6) and (9) as,

$$\begin{aligned} T^{\mu \nu }_{EM}\rightarrow T^{\mu \nu }_B=\frac{B^2}{2} \left( u^{\mu }u^{\nu }-\Delta ^{\mu \nu }-2b^{\mu }b^{\nu }\right) , \end{aligned}$$
(11)

where \(B^{\mu }B_{\mu }=-B^{2}\) and \(b^{\mu }=\frac{B^{\mu }}{B}\) with the constraints \(b^{\mu }u_{\mu }=0\) and \(b^{\mu }b_{\mu }=-1.\) Using Eq. (9), one can show that \(B^{\mu \nu }B_{\mu \nu }=2B^{2}.\) We can define another anti-symmetric tensor as: \(b^{\mu \nu }=-{B^{\mu \nu }}/{B}.\)

2.2 Equation of motion of magnetohydrodynamics

In this section, we derive the equation of motion for relativistic fluid under the influence of external EM field.

2.2.1 Conservation of energy and momentum of fluid and electromagnetic field

In the absence of any magnetic field, the EMT and the particle currents are conserved separately according to the following conservation laws,

$$\begin{aligned} \partial _{\mu }N^{\mu }= & {} 0, \end{aligned}$$
(12)
$$\begin{aligned} \partial _{\mu }T_f^{\mu \nu }= & {} 0. \end{aligned}$$
(13)

In presence of external EM field the total EMT tensor (fluid+field), \(T^{\mu \nu }\) is given by,

$$\begin{aligned} T^{\mu \nu }=T^{\mu \nu }_f+T^{\mu \nu }_{EM}, \end{aligned}$$
(14)

where \(T^{\mu \nu }_f \) and \( T^{\mu \nu }_{EM}\) are the contributions from the fluid and EM field respectively. The total EMT in Ref. [35] contains additional terms which can not be unambiguously attributed to the fluid or to the field. But for constant susceptibility and vanishing \(E^{\mu }\) such contributions vanish and then Eq. (14) becomes a good approximation. As the electric charge is conserved, the charge current of the fluid is individually conserved too,

$$\begin{aligned} \partial _{\mu }J^{\mu }_f=0. \end{aligned}$$
(15)

If we have an external charge current, then it will act as a source term of the EMT:

$$\begin{aligned} \partial _{\mu }T^{\mu \nu }=-F^{\nu \lambda }J_{ext,\lambda }. \end{aligned}$$
(16)

The conservation equation for the electromagnetic field Eq. (7) with external source can be written as,

$$\begin{aligned} \partial _{\mu }T^{\mu \nu }_{EM}= -F^{\nu \lambda }\left( J_{f,\lambda }+J_{ext,\lambda }\right) . \end{aligned}$$
(17)

Using Eqs. (14), (16) and (17) we get,

$$\begin{aligned} \partial _{\mu }T^{\mu \nu }_f= F^{\nu \lambda }J_{f,\lambda }. \end{aligned}$$
(18)

Usually, the total EMT of an isolated system remains conserved but in the presence of an external charge current an appropriate source term should be taken into account. In this case, the fluid evolution depends on the fluid charge current density through Eq. (18).

It is also useful to express the conservation equations by taking projection along and perpendicular to fluid four velocity in an alternative way. If we take the parallel projection of Eqs. (17) and (18), then we obtain,

$$\begin{aligned} u_{\nu }\partial _{\mu }T^{\mu \nu }_{EM}= & {} 0, \end{aligned}$$
(19)
$$\begin{aligned} u_{\nu }\partial _{\mu }T^{\mu \nu }_f= & {} 0. \end{aligned}$$
(20)

In presence of current density given by Eq. (8), the perpendicular projection of Eqs. (17) and (18) can be written as:

$$\begin{aligned} \Delta ^{\alpha }_{\nu }\partial _{\mu }T^{\mu \nu }_{EM}= & {} Bb^{\alpha \lambda }\left( J_{f,\lambda }+J_{ext,\lambda }\right) , \end{aligned}$$
(21)
$$\begin{aligned} \Delta ^{\alpha }_{\nu }\partial _{\mu }T^{\mu \nu }_f= & {} -Bb^{\alpha \lambda }J_{f,\lambda }. \end{aligned}$$
(22)

It implies that the momentum density of fluid depends on diffusion current/magnetic field, momentum density of the field, external current and fluid diffusion current.

2.2.2 Ideal and dissipative non-resistive magnetohydrodynamics

The EMT for non resistive magnetohydrodynamics is given by,

$$\begin{aligned} T^{\mu \nu }_{EM}\equiv T^{\mu \nu }_{B}=\frac{B^2}{2} \left( u^{\mu }u^{\nu }-\Delta ^{\mu \nu }-2b^{\mu }b^{\nu }\right) . \end{aligned}$$
(23)

In case the ideal fluid, the total EMT takes the form-

$$\begin{aligned} T^{\mu \nu }_{(0)}=\left( \epsilon +\frac{B^2}{2}\right) u^{\mu }u^{\nu }- \left( P+\frac{B^2}{2}\right) \Delta ^{\mu \nu }-B^2b^{\mu }b^{\nu }, \nonumber \\ \end{aligned}$$
(24)

where, \(\epsilon \) and P denote energy density and the thermodynamic pressure respectively. For dissipative fluid with non-zero shear and bulk viscosities, and non-zero thermal conductivity, the EMT becomes-

$$\begin{aligned} T^{\mu \nu }= & {} \left( \epsilon +\frac{B^2}{2}\right) u^{\mu }u^{\nu }- \left( P+\Pi +\frac{B^2}{2}\right) \Delta ^{\mu \nu }\nonumber \\{} & {} -B^{\mu }B^{\nu }+q^\mu n^\nu +q^\nu n^\mu + \pi ^{\mu \nu }, \end{aligned}$$
(25)

where, \(\Pi , q^\mu \) and \(\pi ^{\mu \nu }\) denote bulk pressure, heat flux and shear stress respectively, and are expressed in Eckart’s frame of reference [25, 31]. The system of equations can be closed with the help of constitutive relation of charged-current and with an Equation of State (EoS), which relates the thermodynamic pressure to energy and number density \(P=P\,(\epsilon ,\,n_{f}).\)

The inclusion of the polarization modifies the energy–momentum tensor as follows [14, 38, 39]:

$$\begin{aligned} T^{\mu \nu }_0= & {} T_\textrm{F0}^{\mu \nu }+T_{\textrm{EM}}^{\mu \nu }, \end{aligned}$$
(26)
$$\begin{aligned} T^{\mu \nu }_\textrm{F0}= & {} \epsilon \, u^\mu u^\nu -P\Delta ^{\mu \nu }-\frac{1}{2}\left( M^{\mu \lambda }F^{\;\;\nu }_{ \lambda }+M^{\nu \lambda }F^{\;\;\mu }_{\lambda }\right) , \end{aligned}$$
(27)
$$\begin{aligned} T_{\textrm{EM}}^{\mu \nu }= & {} -F^{\mu l}F_{\;\;\lambda }^\nu +\frac{g^{\mu \nu }}{4}F^{\rho \sigma }F_{\rho \sigma }, \end{aligned}$$
(28)

where, \(M^{\mu \nu }\) is the polarization tensor. In the non-dissipative limit, the entropy is also conserved along with the charge. The charge and entropy currents can be expressed in the non-dissipative hydrodynamics as,

$$\begin{aligned} n_{0}^\mu= & {} n\, u^\mu , \end{aligned}$$
(29)
$$\begin{aligned} s_0^\mu= & {} s\, u^\mu , \end{aligned}$$
(30)

where, n and s are the electric charge density and entropy density measured in the local rest frame. It is convenient to decompose the tensor \(F^{\mu \nu }\) into components parallel and perpendicular to \(u^\mu \) as:

$$\begin{aligned} F^{\mu \nu }= & {} F^{\mu \lambda }u_\lambda u^\nu -F^{\nu \lambda }u_\lambda u^\mu +\Delta ^\mu _{\;\;\alpha }F^{\alpha \beta }\Delta _\beta ^{\;\;\nu }\nonumber \\\equiv & {} E^\mu u^\nu -E^\nu u^\mu +\frac{1}{2}\epsilon ^{\mu \nu \alpha \beta }\left( u_\alpha B_\beta -u_\beta B_\alpha \right) . \end{aligned}$$
(31)

The anti-symmetric polarization tensor \(M^{\mu \nu }\) represents the response of matter to \(F^{\mu \nu }.\) It is given by \(M^{\mu \nu }\equiv -\partial \Phi /\partial F_{\mu \nu },\) where \(\Phi \) is the thermodynamic potential function. It is also convenient to define the in-medium field strength tensor \(H^{\mu \nu }\equiv F^{\mu \nu }-M^{\mu \nu }.\) Similar to \(F^{\mu \nu },\) we decompose \(M^{\mu \nu }\) and \(H^{\mu \nu }\) as follows:

$$\begin{aligned} M^{\mu \nu }= & {} P^\nu u^\mu -P^\mu u^\nu +\frac{1}{2}\epsilon ^{\mu \nu \alpha \beta }\left( M_\beta u_\alpha -M_\alpha u_\beta \right) , \end{aligned}$$
(32)
$$\begin{aligned} H^{\mu \nu }= & {} D^\mu u^\nu -D^\nu u^\mu + \frac{1}{2}\epsilon ^{\mu \nu \alpha \beta }\left( H_\beta u_\alpha -H_\alpha u_\beta \right) , \end{aligned}$$
(33)

with \(P^\mu \equiv -M^{\mu \nu }u_\nu ,\) \(M^\mu \equiv \epsilon ^{\mu \nu \alpha \beta }M_{\nu \alpha }u_\beta /2,\) \(H^\mu \equiv \epsilon ^{\mu \nu \alpha \beta } H_{\nu \alpha }u_\beta /2,\) and \(D^\mu \equiv H^{\mu \nu }u_\nu .\)

In the local rest frame of the fluid, the non-trivial components of these tensors are \((F^{10}, F^{20}, F^{30})=\textbf{E},\) \((F^{32}, F^{13}, F^{21})=\textbf{B},\) \((M^{10}, M^{20}, M^{30})=-\textbf{P},\) \((M^{32}, M^{13}, M^{21})=\textbf{M},\) \((H^{10}, H^{20}, H^{30})=\textbf{D},\) and \((H^{32}, H^{13}, H^{21})=\textbf{H},\) where \(\textbf{P}\) and \(\textbf{M}\) are the electric polarization and magnetization vector respectively. In linear domain, they are related to the fields \(\textbf{E}\) and \(\textbf{B}\) by the expressions \(\textbf{P}=\chi _e\textbf{E}\) and \(\textbf{M}=\chi _m{{\textbf{B}}},\) where \(\chi _e\) and \(\chi _m\) represents the electric and magnetic susceptibilities respectively. Here \(E^\mu , B^\mu \) are space-like, i.e. \(E^\mu E_\mu =-E^2\) and \(B^\mu B_\mu =-B^2\) and orthogonal to \(u^\mu \) i.e. \(E^\mu u_\mu =0, B^\mu u_\mu =0\) with \(E\equiv |\textbf{E}|\) and \(B\equiv |{\textbf{B}}|.\)

There are several physical system where the electric field is much weaker than the magnetic field. The interior of a neutron star is one such example. In the following discussions, we will omit the contribution from the electric field. We introduced the four-vector \(b^\mu \equiv B^\mu /B,\) which is normalized as \(b^\mu b_\mu =-1\) along with the anti-symmetric rank-2 tensor \(b^{\mu \nu }\equiv \epsilon ^{\mu \nu \alpha \beta } b_\alpha u_\beta .\) In the absence of electric field, we have,

$$\begin{aligned} F^{\mu \nu }= & {} -B b^{\mu \nu }, \end{aligned}$$
(34)
$$\begin{aligned} M^{\mu \nu }= & {} -M b^{\mu \nu }, \end{aligned}$$
(35)
$$\begin{aligned} H^{\mu \nu }= & {} -H b^{\mu \nu }, \end{aligned}$$
(36)

where \(M\equiv |\textbf{M}|\) and \(H\equiv |\textbf{H}|.\) In absence of electric fields, the matter and field contributions to the EMT (26) can now be written in terms of \(b^{\mu }\) and \(b^{\mu \nu }\) as (see, e.g., Refs. [14, 40])

$$\begin{aligned} T^{\mu \nu }_\textrm{F0}= & {} \varepsilon u^\mu u^\nu -{P_\perp }\Xi ^{\mu \nu }+{P_\parallel } b^\mu b^\nu , \end{aligned}$$
(37)
$$\begin{aligned} T^{\mu \nu }_{\textrm{EM}}= & {} \frac{1}{2}B^2\left( u^\mu u^\nu -\Xi ^{\mu \nu }-b^\mu b^\nu \right) , \end{aligned}$$
(38)

where \(\Xi ^{\mu \nu }\equiv \Delta ^{\mu \nu }+b^\mu b^\nu \) is a new projection tensor with \(\Xi ^{\mu \nu }u_\mu =\Xi ^{\mu \nu }b_\mu =0.\) The transverse and longitudinal pressures relative to \(b^\mu \) can be defined as \({P_\perp }=P-MB\) and \({P_\parallel }=P\) relative to the vector \(b^\mu .\) In the absence of magnetic field, the fluid is isotropic and \({P_\perp }={P_\parallel }=P,\) where P is the thermodynamic pressure defined in Eq. (28). In the local rest frame of the fluid, the direction of the magnetic field is chosen as the z-axis without loss of generality, so we have \(b^\mu =(0,0,0,1).\) Then the EMT takes the form, \(T^{\mu \nu }_\textrm{F0}=\mathrm{{diag}}(\epsilon ,{P_\perp }, {P_\perp }, {P_\parallel }).\)

In the presence of polarization (or non-zero magnetization), the full energy momentum tensor (EMT) for relativistic MHD can be expressed as,

$$\begin{aligned} T^{\mu \nu }= & {} (\epsilon +P-MB)u^\mu u^\nu -(P-MB+\frac{1}{2}B^2)g^{\mu \nu }\nonumber \\{} & {} +(MB-B^2)b^\mu b^\nu \nonumber \\{} & {} - \Pi \Delta ^{\mu \nu }+q^{\mu }u^{\nu }+q^{\nu }u^{\mu }+\pi ^{\mu \nu }. \end{aligned}$$
(39)

The form of \(q^{\mu }, \Pi , \pi ^{\mu \nu }\) used in MIS hydrodynamics contains additional coupling and relaxation coefficients arising due to inclusion of second order gradients [27, 41, 42]:

$$\begin{aligned} \Pi= & {} -\zeta \Big [\partial _{\mu }u^{\mu }+\beta _{0}D\Pi -\alpha _{0}\partial _{\mu }q^{\mu }\Big ] ,\nonumber \\ \pi ^{\lambda \mu }= & {} 2\eta \Delta ^{\lambda \mu \alpha \beta } \Big [\partial _{\alpha }u_{\beta }-\beta _{2}D\pi _{\alpha \beta }- \alpha _{1}\partial _{\alpha }q_{\beta }\Big ],\nonumber \\ q^{\lambda }= & {} \kappa T\Delta ^{\lambda \mu }\Big [ \frac{1}{T}\partial _\mu T -Du_{\mu }-\beta _1 D{q_\mu }-\alpha _0\partial _\mu \Pi \nonumber \\{} & {} +\alpha _1\partial _{\nu }\pi ^{\nu }_{\mu }\Big ], \end{aligned}$$
(40)

where, \(\zeta ,\) \(\eta ,\) and \(\kappa \) are the coefficient of bulk viscosity, shear viscosity, and thermal conductivity respectively, and \(\Delta ^{\mu \nu \alpha \beta }= \frac{1}{2}\big [\Delta ^{\mu \alpha }\Delta ^{\nu \beta }+\Delta ^{\mu \beta }\Delta ^{\nu \alpha }- \frac{2}{3}\Delta ^{\mu \nu }\Delta ^{\alpha \beta }\big ].\) Here, \(\beta _0,\beta _1,\beta _2\) are relaxation coefficients, \(\alpha _0\) and \(\alpha _1\) are coupling coefficients. The relaxation times for the bulk pressure \((\tau _{\Pi })\), the heat flux \((\tau _q)\) and the shear tensor \((\tau _{\pi })\) are defined as [43]

$$\begin{aligned} \tau _{\Pi }=\zeta \beta _0,\quad \tau _q=k_BT\beta _1,\quad \tau _{\pi }=2\eta \beta _2. \end{aligned}$$
(41)

The relaxation lengths which couple to heat flux and bulk pressure \((l_{\Pi q}, l_{q\Pi } )\), the heat flux and shear tensor \((l_{q\pi }, l_{\pi q})\) are defined as follows:

$$\begin{aligned} l_{\Pi q}=\zeta \alpha _0,\quad l_{q\Pi }=k_B T \alpha _0,\quad l_{q\pi }=k_BT\alpha _1,\quad l_{\pi q}=2\eta \alpha _1 . \nonumber \\ \end{aligned}$$
(42)

In the ultra-relativistic limit, \(\beta (=m/T)\rightarrow 0,\) where m is the mass of the particle. We also have [31],

$$\begin{aligned} \alpha _0\approx & {} 6\beta ^{-2}P^{-1},\quad \alpha _1\approx -\frac{1}{4}P^{-1},\quad \beta _0 \approx 216 \beta ^{-4} P^{-1},\nonumber \\ \beta _1\approx & {} \frac{5}{4}P^{-1},\quad \beta _2 \approx \frac{3}{4}P^{-1} . \end{aligned}$$
(43)

To check the consistency of the terms in \(T_{\textrm{F0}}^{\mu \nu }\) involving electromagnetic fields, we use the thermodynamic relation,

$$\begin{aligned} \epsilon =Ts+\mu n-P, \end{aligned}$$
(44)

where, \(\mu \) is the chemical potential introduced to constrain the conservation of net (baryon) number. By using the conservation equations for \(n_{0}^\mu \) and \(s_0^\mu \) in the ideal hydrodynamics, it is straight forward to show that the hydrodynamic equation \(u_\nu \partial _\mu T^{\mu \nu }_0=0\) along with the Maxwell equation Eq. (3) implies,

$$\begin{aligned} D\epsilon =TDs+\mu D n-MDB, \end{aligned}$$
(45)

which is in accordance with the standard thermodynamic relation,

$$\begin{aligned} d\epsilon =Tds+\mu dn-MdB. \end{aligned}$$
(46)

From Eqs. (44) and (46), we obtain the Gibbs–Duhem relation,

$$\begin{aligned} dP=sdT+nd\mu +MdB. \end{aligned}$$
(47)

The complete set of non-dissipative hydrodynamic equations in presence of an external magnetic field is formed by Eqs. (39) and (40) along with the following two equations:

$$\begin{aligned} \partial _\nu \left( B^\mu u^\nu -B^\nu u^\mu \right)= & {} 0, \end{aligned}$$
(48)
$$\begin{aligned} \partial _\mu H^{\mu \nu }= & {} n^\nu . \end{aligned}$$
(49)

Contracting Eq. (48) with \(b_\mu ,\) one obtains the induction equation,

$$\begin{aligned} \theta +D\ln B-u^\nu b^\mu \partial _\mu b_\nu =0, \end{aligned}$$
(50)

where, \(\theta \equiv \partial _\mu u^\mu \) is the velocity divergence and \(D\equiv u^\mu \partial _\mu \) is the co-moving derivative.

All of these facts confirm that the form of EMT, \(T_\textrm{F0}^{\mu \nu }\) is consistent with thermodynamic formulas for matter in the presence of electromagnetic fields [14].

3 Hydrodynamic equations in linearised form to estimate the dynamic structure factor

Presently, we aim to evaluate the dynamic structure factor [\({\mathcal {S}}_{nn}({\varvec{k}},\omega )\)] by taking the correlation of dynamical density fluctuations in \(\omega -k\) space. A small deviation from the equilibrium state of a thermodynamic variable can be accompanied by linearized form of the hydrodynamic equations. The density fluctuation can be obtained from the linearised equation to estimate the structure factor. Let the equilibrium state and the away from the equilibrium state of a thermodynamic quantity \((n, \,\epsilon ,\, u^{\alpha },\, q^{\alpha },\, s,\, \Pi ,\, \pi ^{\alpha \beta }\) etc.) are generically denoted by \(Q_{0}\) and Q respectively. Therefore, any state (away from equilibrium) can be expressed as \(Q=Q_{0}+\delta Q,\) where \(\delta Q\) is some tiny perturbation to the equilibrium value, \(Q_{0}.\) We can thus express the hydrodynamic equations around the equilibrium in the linearized form as:

$$\begin{aligned} 0= & {} \frac{\partial \delta n}{\partial t}+n_{0} \varvec{\nabla }.\delta {\varvec{v}}, \end{aligned}$$
(51a)
$$\begin{aligned} 0= & {} n_{0}\frac{\partial \delta s}{\partial t}+\frac{1}{T_{0}}\varvec{\nabla }.\delta \varvec{q}- \frac{1}{n_0 \, T_0}(B^2-M \, B)\frac{\partial \delta n}{\partial t}, \end{aligned}$$
(51b)
$$\begin{aligned} 0= & {} (h_{0}+B^2-M \, B) \frac{\partial \delta v}{\partial t}+\nabla (\delta P+ \delta \Pi )\nonumber \\{} & {} +\frac{\partial \delta q}{\partial t}+\varvec{\nabla }.\delta \varvec{\pi }, \end{aligned}$$
(51c)
$$\begin{aligned} 0= & {} \delta \Pi +\zeta \Big [\varvec{\nabla }.\delta {\varvec{v}}+\beta _{0}\frac{\partial \delta \Pi }{\partial t}-\alpha _{0}\varvec{\nabla }.\delta \varvec{q}\Big ] , \end{aligned}$$
(51d)
$$\begin{aligned} 0= & {} \delta \pi ^{ij}-\eta \Big [\partial ^{i}\delta v^{j}+\partial ^{j}\delta v^{i}-\frac{2}{3}g^{ij}\varvec{\nabla }.\delta {\varvec{v}}-2\beta _{2}\frac{\partial \delta \pi ^{ij}}{\partial t}\nonumber \\{} & {} -\alpha _{1}(\partial ^{i}\delta q^{j}+\partial ^{j}\delta q^{i}-\frac{2}{3}g^{ij}\varvec{\nabla }.\delta \varvec{q})\Big ], \end{aligned}$$
(51e)
$$\begin{aligned} 0= & {} \delta q-\kappa T_{0}\Big [-\frac{\nabla \delta T}{T_{0}}- \frac{\partial \delta v}{\partial t}-\beta _{1}\frac{\partial \delta q}{\partial t}+\alpha _{0}\nabla \delta \Pi \nonumber \\{} & {} +\alpha _{1}\varvec{\nabla }.\delta \varvec{\pi }\Big ] . \end{aligned}$$
(51f)

We can further simplify the set of Eqs. (51a)–(51f) by decomposing the fluid four velocity along the directions parallel and perpendicular to the direction of wave vector, \({\varvec{k}},\) and are termed as longitudinal component \((\delta \varvec{v_{||}})\) and transverse component \((\delta \varvec{v_{\perp }})\) respectively. The longitudinal and the transverse components of the fluid four velocity completely decouple the linearized hydrodynamic equations and we can get two sets linearly independent solutions for the two components. But the density perturbations appears only in the longitudinal component, which is our main focus here. Therefore, we consider the longitudinal component only for the present analysis. The hydrodynamic equations can be solved for a given set of initial condition, \(n(0),\, v_{||}(0),\, T(0),\, q(0),\, \Pi (0)\) and \(\pi (0),\) by using the Fourier-Laplace transformation as:

$$\begin{aligned} \delta Q( \varvec{k}, \omega )= \int ^{\infty }_{-\infty } d^3\varvec{r} \int ^{\infty }_{0}dt e^{-i(\varvec{k}.\varvec{r}-\omega t)} \delta Q(\varvec{r}, t). \end{aligned}$$
(52)

The \(\delta P\) and \(\delta s\) can be written in terms of the independent variables n and T as follows by using the thermodynamic relations:

$$\begin{aligned} \delta P= & {} \Bigg (\frac{\partial P}{\partial n}\Bigg )_{T}\delta n+ \Bigg (\frac{\partial P}{\partial T}\Bigg )_{n}\delta T,\nonumber \\ \delta s= & {} \Bigg (\frac{\partial s}{\partial n}\Bigg )_{T}\delta n+ \Bigg (\frac{\partial s}{\partial T}\Bigg )_{n}\delta T. \end{aligned}$$
(53)

We use Eqs. (52) and (53) to write down the longitudinal linearized hydrodynamic equation as:

$$\begin{aligned} {\mathbb {M}}\delta Q(\varvec{k}, \omega )= \delta Q(\varvec{k},0), \end{aligned}$$
(54)

where,

$$\begin{aligned} {\mathbb {M}} = \begin{bmatrix} -i\omega &{} ikn_{0} &{} 0 &{} 0 &{} 0 &{} 0 \\ -i\omega \big \{n_{0}\big (\frac{\partial s}{\partial n}\big )_{T}- \frac{{\mathcal {A}}}{n_0\,T_0} \big \} &{} 0 &{} -i\omega n_{0}\big (\frac{\partial s}{\partial T}\big )_{n} &{} 0 &{} \frac{k}{T_0} &{}0 \\ {ik} \big (\frac{\partial P}{\partial n}\big )_{T}&{} -i\omega (h_0+{\mathcal {A}}) &{} {ik} \big (\frac{\partial P}{\partial T}\big )_{n}&{} {ik} &{} -{i\omega } &{} i k \\ 0 &{} ik\zeta &{} 0 &{} (1-i\omega \beta _{0} \zeta ) &{} - ik\alpha _{0} \zeta &{} 0 \\ 0 &{} -i\frac{4}{3} \eta k &{} 0 &{} 0 &{} \frac{4}{3}\alpha _{1}k \eta &{} (1-2i\omega \beta _{2}\eta )\\ 0 &{} -i\omega \kappa T_{0} &{} ik\kappa &{} -ik\alpha _{0} \kappa T_{0}&{} (1-i\omega \beta _{1} \kappa T_{0}) &{}- ik \alpha _{1} \kappa T_{0} \\ \end{bmatrix},\nonumber \\ \end{aligned}$$
(55)

and

$$\begin{aligned} \delta Q(\varvec{k},\omega )&{=} \begin{bmatrix} \delta n(\varvec{k},\omega ) \\ \delta v_{||}(\varvec{k},\omega )\\ \delta T(\varvec{k},\omega )\\ \delta \Pi (\varvec{k},\omega )\\ \delta q_{||}(\varvec{k},\omega )\\ \delta \pi _{||}(\varvec{k},\omega ) \end{bmatrix};\nonumber \\ \delta Q(\varvec{k},0)&{=} \begin{bmatrix} \delta n(\varvec{k},0) \\ \Big \{n_{0}\Big (\frac{\partial s}{\partial n}\Big )_{T}- \frac{{\mathcal {A}}}{n_0\,T_0} \Big \} \delta n ( \varvec{k}, 0)+n_{0}\Big (\frac{\partial s}{\partial T}\Big )_{n}\delta T(\varvec{k},0)\\ (h_0+{\mathcal {A}})\delta v_{||}(\varvec{k},0)+ \delta q_{||}(\varvec{k},0)\\ \beta _{0}\zeta \delta \Pi (\varvec{k},0)\\ 2\beta _2 \eta \delta \pi _{||}(\varvec{k},0)\\ \kappa T_{0}\delta v_{||}(\varvec{k},0)+\kappa T_{0}\beta _{1}\delta q_{||}(\varvec{k},0)\\ \end{bmatrix},\nonumber \\ \end{aligned}$$
(56)

here, \({\mathcal {A}}=\,(B^2-MB).\)

The solution for density fluctuation is obtained by solving the set of above algebraic equations which gives,

$$\begin{aligned} \delta n(\varvec{k},\omega )= & {} \Bigg [ {\mathbb {M}}^{-1}_{11}+\Bigg \{n_{0}\Bigg (\frac{\partial s}{\partial n}\Bigg )_{T}- \frac{{\mathcal {A}}}{n_0\,T_0} \Bigg \}\, {\mathbb {M}}^{-1}_{12}\Bigg ]\, \delta n(\varvec{k},0)\nonumber \\{} & {} +\Bigg [n_{0}\Bigg (\frac{\partial s}{\partial T}\Bigg )_{n}\,{\mathbb {M}}^{-1}_{12} \Bigg ] \,{\delta T}(\varvec{k},0) \nonumber \\{} & {} +\Bigg [(h_0+{\mathcal {A}})\,{\mathbb {M}}^{-1}_{13} +\kappa T_{0} {\mathbb {M}}^{-1}_{16} \Bigg ]\,\delta v_{||}(\varvec{k},0)\nonumber \\{} & {} - \Bigg [{\mathbb {M}}^{-1}_{13}+\kappa T_{0}\beta _{1} \,{\mathbb {M}}^{-1}_{16}\Bigg ]\,\delta q_{||}(\varvec{k},0)+\Bigg [\beta _{0}\zeta \,{\mathbb {M}}^{-1}_{14}\Bigg ] \nonumber \\{} & {} \times \delta \Pi (\varvec{k},0) {+} \Bigg [ 2\beta _{2}\eta {\mathbb {M}}^{-1}_{15} \Bigg ]\delta \pi _{||}(\varvec{k},0). \end{aligned}$$
(57)

The appearance of \({{\mathcal {A}}}\) in the density fluctuation indicates the presence of magnetic field in the system. The expression of density fluctuation is a function of the other fluctuating hydrodynamic variables such as \(\delta T,\, \delta v_{||},\,\delta q_{||},\,\delta \Pi ,\) and \(\delta \pi _{||}.\) The independence among hydrodynamic variables does not necessarily imply the absence of correlations [44]. However, in this work, we have specifically assumed that there are no correlations between density fluctuations and other fluctuating hydrodynamic variables [32, 37]. Therefore, the correlation between two independent thermodynamic variables, say, \(Q_i\) and \(Q_j\) vanishes i.e.,

$$\begin{aligned} \left\langle \delta Q_{i}(\varvec{k},\omega )\delta Q_{j}(\varvec{k},0)\right\rangle =0, \quad i\ne j. \end{aligned}$$
(58)

The required correlator, \({{\mathcal {S}}}^{\prime }_{nn}(\varvec{k},\omega )\) is obtained as:

$$\begin{aligned} {{\mathcal {S}}}^{\prime }_{nn}(\varvec{k},\omega )= & {} \left\langle \delta n(\varvec{k},\omega )\delta n(\varvec{k},0)\right\rangle \nonumber \\= & {} \Bigg [ {\mathbb {M}}^{-1}_{11}+\Bigg \{n_{0}\Bigg (\frac{\partial s}{\partial n}\Bigg )_{T}- \frac{{\mathcal {A}}}{n_0\,T_0} \Bigg \}\, {\mathbb {M}}^{-1}_{12}\Bigg ]\nonumber \\{} & {} \left\langle \delta n(\varvec{k},0)\delta n(\varvec{k},0)\right\rangle . \end{aligned}$$
(59)

Finally, the \({\mathcal {S}}_{nn}({\varvec{k}},\omega )\) is defined as:

$$\begin{aligned} {\mathcal {S}}_{nn}(\varvec{k},\omega )=\frac{{{\mathcal {S}}}^{\prime }_{nn}(\varvec{k},\omega )}{\left\langle \delta n(\varvec{k},0)\delta n(\varvec{k},0)\right\rangle }. \end{aligned}$$
(60)

The \({\mathcal {S}}_{nn}({\varvec{k}},\omega )\) contains the transport coefficients such as \(\eta ,\,\zeta ,\,\kappa \) and other thermodynamic response functions, which will be used to study the behaviour of the structure factor. It is well known that the relativistic Navier–Stokes (NS) hydrodynamic equation can be obtained by setting the various coupling \((\alpha _0,\) \(\alpha _1)\) and relaxation \((\beta _0,\) \(\beta _1,\) \(\beta _2)\) coefficients to zero. Therefore, the \({\mathcal {S}}_{nn}({\varvec{k}},\omega )\) for NS hydrodynamics can be obtained under the similar limits as:

$$\begin{aligned} {\mathcal {S}}_{nn}({\varvec{k}},\omega )= & {} \frac{{\mathcal {A}} k^2 \chi \omega -\omega ^{2}h_{0}T) \left( \frac{\partial s}{\partial T}\right) _n -k^{2}\chi \omega h_{0}+k^{2}({\mathcal {A}}-\omega \chi T)\left( \frac{\partial P}{\partial T}\right) _n}{({\mathcal {A}}-2) k^2 \chi \omega ^2+k^2\chi \omega T(4 \eta /3 - \zeta )+\left( \frac{\partial s}{\partial T}\right) _n \{2k^2 \omega ^2 (4 \eta /3 -\zeta )-2T^{2} \chi \omega ^4\}}. \end{aligned}$$
(61)

The appearance of higher order derivatives in MIS hydrodynamics makes the dispersion relation a quintic equation in \(\omega \) and the corresponding dispersion equation in NS hydrodynamics is a cubic equation in \(\omega .\)

The possibility of the existence of the critical end point (CEP) in the QCD phase diagram [45, 46] is considered as one of the most interesting development in the field of relativistic heavy ion research. The location of the CEP in \(T-\mu \) plane is not known from first principle. The model dependent predictions vary widely as indicated in Ref. [47]. In the present work we chose \((T_{c},\,\mu _{c})= (154,\,367)\) MeV to study the effects of the CEP on the spectral function in the presence of external magnetic field, \(\varvec{B}.\) The effects of CEP are included in the calculation of spectral function through the equation of state [48, 49] and scaling behaviour of the transport coefficients [50,51,52]. The details of this part of the calculations are given in Refs. [37, 53]. Therefore, we refer to these references for details to avoid repetition.

4 Results

Before we present the results it is important to mention the following points. In the context nuclear collisions at relativistic energies one needs to solve the relativistic viscous second order hydrodynamical equations in the presence of the CEP and ultra-high magnetic field. Development of such a numerical code is highly time consuming which will be considered in future publications. In the present work we focus on the study of the dynamical density correlation in a non-expanding QGP in the presence of an ultra-high magnetic field with and without the inclusion of the CEP. This study is important to check whether the magnetic field alters the nature of the correlation near the CEP. We find that the nature of the correlation near the CEP marked by the absence of the Brillouin peaks remains unaltered in presence of the magnetic field. This is one of the important observation of the present work as shown below.

It may also be mentioned here that the matter produced after nuclear collisions at LHC and top RHIC energies will be gluon dominated initially, however, quarks–antiquarks pairs will be produced dynamically due to the interactions among the gluons. Therefore, if the magnetic field (B) does not decay (which depends on the electrical conductivity of the QGP) completely before the generation of quark–antiquark pairs then the effects of B will be realized. Moreover, at lower RHIC energies (Beam Energy Scan Program) the quarks and antiquarks will be present from the very early stage and will get affected by the magnetic field.

We aim to study the nature of the dynamic structure factor \([{\mathcal {S}}_{nn}({\varvec{k}},\omega )]\) in the presence of a static magnetic field, B here. The \({\mathcal {S}}_{nn}({\varvec{k}},\omega )\) in the absence of B has been studied in earlier works [37, 53], where it has been shown that the \({\mathcal {S}}_{nn}({\varvec{k}},\omega )\) admits three identifiable peaks. The central Rayleigh peak (R-peak) is positioned at angular frequency \(\omega =0,\) and the other two peaks, the Brillouin peaks (B-peaks) are situated on both sides of the R-peak with even magnitudes. The R-peak and the B-peaks are originated from the entropy (thermal) fluctuation at constant pressure and pressure fluctuation at constant entropy respectively. The position of the B-peaks enables us to evaluate the speed of sound. Also, the width and the integrated intensities of those peaks are associated with various thermodynamic quantities such as the isothermal compressibilities and specific heats of the system [13].

Figure 1 shows the \({\mathcal {S}}_{nn}({\varvec{k}},\omega )\) for \(eB=m_{\pi }^{2}\) at \(r=(T-T_{c})/T_{c}=0.2,\) i.e. when the system is away from the CEP. The transport coefficients are taken as \(\eta /s=\zeta /s=\chi \,T/s=1/4\pi .\) We see three different peaks which are recognized as the R-peak (central), and the other two peaks as the B-peaks. The B-peaks are positioned symmetrically about \(\omega =0,\) but their heights are not identical. The unequal height of the B-peaks may occur due to the local inhomogeneity present in the system. The B-peaks arise from propagating sound modes associated with pressure fluctuations at constant entropy. In condensed matter physics, the asymmetry of the B-peaks is identified from the fact that two sound modes with different \(\omega \) values, \(-c_sk\) and \(+c_sk\) originate from different temperature zones [54, 55]. Furthermore, we see that the widths of the peaks are very narrow, implying the slow relaxation of the thermal as well as the pressure fluctuations, which keeps the system to linger at out-of-equilibrium state for a longer time.

Fig. 1
figure 1

The variation of the \({\mathcal {S}}_{nn}({\varvec{k}},\omega )\) with \(\omega \) is shown with magnetic field strength, \(eB=m_{\pi }^{2}\) for \(k=0.1\) fm\(^{-1}\) at \(r=0.2.\) It shows one R-peak and two B-peaks, symmetrically located about \(\omega =0\)

Full size image
Fig. 2
figure 2

The factor \(B^{2}-MB\) is plotted with the variation of magnetic field

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Fig. 3
figure 3

The variation of the \({\mathcal {S}}_{nn}({\varvec{k}},\omega )\) with \(\omega \) is shown in the presence of a magnetic field \((B=3\,m^{2}_{\pi })\) for \(k=0.1\) fm\(^{-1}\) at \(r=0.2.\) It shows one R-peak, two away side B-peaks asymmetrically located about \(\omega =0,\) and two near side B-peaks, originated from the coupling of the magnetic field with the other thermodynamic fields

Full size image

The introduction of B results in splitting the pressure into transverse, \(P_\perp =P-MB\) and longitudinal, \(P_L=P\) components. The expression of the magnetization is adapted from Ref. [14]. The value of \({\mathcal {A}}\) determines the effects of the magnetic field on the spectral function as evident from the expression of \(\delta n\) in Eq. (57). Therefore, we show the variation of \({\mathcal {A}}=B^2-MB\) with B in Fig. 2. The small value of \({\mathcal {A}}\) for low B indicates that the effects of B on \({\mathcal {S}}_{nn}({\varvec{k}},\omega )\) will be significant only beyond a certain value of B.

The \({\mathcal {S}}_{nn}({\varvec{k}},\omega )\) in the presence of the magnetic field with \(eB=3\,m_{\pi }^{2}\) is shown in Fig. 3, where we see five distinct peaks. The peak found at \(\omega =0\) is identified as the R-peak. The two distant peaks about the R-peak are recognized as the B-peaks are originated from the thermodynamic pressure fluctuation at constant entropy (longitudinal component). The magnitudes of the B-peaks are not even as well as their position about \(\omega =0\) is not symmetric. The positional asymmetry could be realized due to the presence of a unidirectional magnetic field. The other two nearby peaks around the R-peak are also found to be the B-peaks arises from the pressure fluctuation in the transverse direction. These B-peaks only appear in the presence of the magnetic field beyond a certain strength. The threshold value for the emergence of the B-peaks (near) is found to be at \(B_{th}=2.32\,m_{\pi }^{2}.\) The transverse pressure \((=P-MB)\) in Eq. (39), containing the B field is solely responsible for the appearance of the near side B-peaks.

Fig. 4
figure 4

A 3D plot is shown to understand the role of the magnetic field in emergence of the near side B-peaks. The peaks start appearing after a threshold value of the magnetic field, \(B_{th}\)

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Fig. 5
figure 5

The variation of the \({\mathcal {S}}_{nn}({\varvec{k}},\omega )\) with \(\omega \) is shown in the presence of a magnetic field in Navier–Stokes theory. It shows one R-peak and two away side B-peaks asymmetrically located about \(\omega =0\)

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Fig. 6
figure 6

The variation of the \({\mathcal {S}}_{nn}({\varvec{k}},\omega )\) with \(\omega \) near the critical end point for \(k=0.1\) fm\(^{-1}\) at \(r=0.01\) is shown in the presence of a magnetic field in MIS theory for (a) magnetic field strength \(B=m^{2}_{\pi }<B_{th},\) and (b) magnetic filed strength \(B=3\,m^{2}_{\pi }\ge B_{th}\)

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The importance of the magnetic field in the appearance of those extra B-peaks can be understood from Fig. 4, where the \({\mathcal {S}}_{nn}({\varvec{k}},\omega )\) is plotted as a function of B. The range of \(\omega \) is kept small to focus on the near side B-peaks only. For the smaller values of B,  no extra peak is found, and they start appearing after a threshold value of \(B_{th}.\)

Figure 5 shows the \({\mathcal {S}}_{nn}({\varvec{k}},\omega )\) in the presence of the magnetic field \((eB=3\,m_{\pi }^{2})\) in NS theory, which is recovered from the MIS model by putting the relaxation and coupling coefficients to zero. We see only three peaks, where the extra peaks due to the presence of the magnetic field are not appearing, but the presence of the magnetic field is evident from the magnitude as well as positional asymmetry of the B-peaks. Therefore, we argue that the extra near-side B-peaks are generated due to the coupling of hydrodynamic fields with the magnetic field when we consider second-order hydrodynamics. This is understood because the dispersion relations of MIS and NS hydrodynamics are quintic and cubic equations respectively.

The behaviour of the structure factor near the CEP \((r=0.01)\) is shown in Fig. 6. The transport coefficients and the thermodynamics response functions that appear in Eq. (60) are taken from the scaling laws [37, 50,51,52,53]. As the near side B-peaks appear only after a threshold intensity of the magnetic field, \(B_{th},\) we have plotted the \({\mathcal {S}}_{nn}({\varvec{k}},\omega )\) for two different values of magnetic field: (a) \(B=m^{2}_{\pi },\) and (b) \(B=3\,m^{2}_{\pi }.\) In both cases, we do not observe any B-peaks but only the R-peak with a larger magnitude. This happens due to the absorption of sound near the CEP and hence B-peaks vanish. Comparing both the cases, we also observe that the magnitude of the R-peak in Fig. 6b is larger compared to the R-peak in Fig. 6a. The most important observation here is that the B-peaks disappear near the CEP irrespective of the value of the external fields (zero or non-zero).

5 Summary and conclusion

The dynamic structure factor, \({\mathcal {S}}_{nn}({\varvec{k}},\omega ),\) is evaluated theoretically from the correlation of the density fluctuation, when the system with a CEP is subjected to an external static magnetic field. Without any magnetic field, the \({\mathcal {S}}_{nn}({\varvec{k}},\omega )\) admits a Rayleigh peak and two Brillouin peaks. But in presence of the magnetic field, we find two extra peaks appearing closer to the R-peak which are identified also as the B-peaks due to the adiabatic transverse pressure fluctuation. The asymmetry in magnitudes of the B-peaks (away side) are realized due to the local inhomogeneity of the system, whereas, the positional asymmetry appears due to the presence of the magnetic field. The extra B-peaks at lower \(\omega \) exclusively appear beyond a threshold value of the magnetic field. The magnetic field splits the pressure into transverse, \(P_\perp =P-MB\) and longitudinal, \(P_L = P\) components. The away side and the near side B-peaks are caused by the pressure fluctuations in longitudinal and transverse direction respectively. To understand the role of the relaxation and the coupling coefficients, we have evaluated \({\mathcal {S}}_{nn}({\varvec{k}},\omega )\) for NS theory and find two away side B-peaks, positioned asymmetrically with uneven heights. Therefore, we argue that the near side B-peaks appear due to the coupling of the magnetic field with the hydrodynamic fields in the second-order hydrodynamic theory. It will be interesting to investigate the role of this asymmetry in a realistic scenario with time varying electromagnetic field in an expanding QGP with CEP, which is beyond the scope of the present theoretical case study performed in an idealistic scenario of non-expanding QGP in a static magnetic field.