Dynamical spectral structure of density fluctuation near the QCD critical point

Abstract

The expression for the dynamical spectral structure of the density fluctuation near the QCD critical point has been derived using linear response theory within the purview of Israel–Stewart relativistic viscous hydrodynamics. The change in the spectral structure of the system as it approaches the critical point has been studied. The effects of the critical point have been introduced in the system through a realistic equation of state and the scaling behaviour of various transport coefficients and thermodynamic response functions. We have found that the Rayleigh and Brillouin peaks are distinctly visible when the system is away from the critical point but the peaks tend to merge near the critical point. The sensitivity of the structure of the spectral function on wave vector (k) of the sound wave has been demonstrated. It has been shown that the Brillouin peaks get merged with the Rayleigh peak because of the absorption of sound waves in the vicinity of the critical point.

Appendix A

In this appendix, the expression for the dynamical spectral structure, $\mathcal {S}_{nn}(\mathbf {k},\omega )$ derived by considering contributions up to second order in transport coefficients (i.e $\eta ^{2},\zeta ^{2},\kappa ^{2},\eta \zeta , \eta \kappa ,\zeta \kappa$) has been provided. The coupling and relaxation coefficients ($\tilde{\alpha _{0}},\tilde{\alpha _{1}},\beta _{0},\tilde{\beta _{1}},\beta _{2}$) have been taken non-zero in obtaining the results displayed in the text, but have been taken as zero in the following to avoid a more lengthy and complex expressions.

$$\begin{aligned} \mathcal {S^\prime }_{nn}(\mathbf {k},\omega )= & {} k^2 n_0 \Bigg [ \omega ^2 n_0 T^{2}\Big (\frac{{\partial s}}{{\partial T}}\Big )_n \Big \{h_0 \kappa T \omega + \Big (\frac{{\partial p}}{{\partial T}}\Big )_n k^2 \nonumber \\&\times \Big ( \zeta +\frac{4}{3} \eta \Big )-k^{2}T^{2} \kappa \omega ^2+h_{0}k T\kappa \nonumber \\&\times \Big (\frac{{\partial s}}{{\partial n}}\Big )_T +k^2T \kappa ( \zeta +\frac{4}{3}\eta )\nonumber \\&- T^{2} \Big (\frac{{\partial p}}{{\partial T}}\Big )_n^2\Big \}+k^2 T \kappa h_{0} \Big \{- h_0T \Big (\frac{{\partial p}}{{\partial T}}\Big )_n \nonumber \\&+k^2T \kappa (\zeta +\frac{4}{3} \eta )+T^{2} \Big (\frac{{\partial p}}{{\partial T}}\Big )_n^2\Big \}+n_0^{2}T^{2} \Big (\frac{{\partial s}}{{\partial T}}\Big )_n \nonumber \\&\times \Big (\frac{{\partial s}}{{\partial n}}\Big )_T \Big \{\Big (\frac{{\partial p}}{{\partial T}}\Big )_n k ( \zeta +\frac{4}{3} \eta )+T^{3} \kappa \omega ^2 \nonumber \\&- h_0T \kappa \Big \}+k^4 n_{0} h_{0} (\zeta +\frac{4}{3} \eta )^{2}h_{0}\Big (\frac{{\partial p}}{{\partial n}}\Big )_T \nonumber \\&\times \Big \{n_0 T^2 \omega ^{2} \Big (k^2 (\zeta +\frac{4}{3} \eta )+ \kappa T \omega ^2- k^2 T\kappa \Big ) \Big \}\nonumber \\&+ h_{0}^{2} \kappa T^{2} k^{2} \Big (\frac{{\partial p}}{{\partial T}}\Big )_n- n_0h_{0} \kappa T^{3} k^{2}\omega ^2 \Big (\frac{{\partial s}}{{\partial n}}\Big )_T\Bigg ] \nonumber \\&\Bigg / \Bigg [ h_0^2 \Big \{ k^4 T^{2}\kappa ^2 \omega ^4+n_0^2 T^2 \omega ^4\Big (\frac{{\partial s}}{{\partial T}}\Big )_n\Big \}\nonumber \\&+h_0 k^2 \omega ^2 \kappa ^2T^{4} \Big (\frac{{\partial s}}{{\partial T}}\Big )_n \times \Big \{ \omega ^4 -2n_{0} \Big (\frac{{\partial p}}{{\partial n}}\Big )_T\Big \}\nonumber \\&-2 k^2 n_0^2 T^{4} \kappa ^2 \omega ^2\Big ( \frac{{\partial s}}{{\partial T}}\Big )_n^{2}-2n_{0}h_{0}k^2 T^{4} \kappa ^2 \omega ^2 \nonumber \\&\Big (\frac{{\partial p}}{{\partial T}}\Big )_n-2n_0^3 T^2k^{4} \omega ^2 \Big (\frac{{\partial s}}{{\partial T}}\Big )_n\times \Big \{\Big (\frac{{\partial p}}{{\partial T}}\Big )_n\nonumber \\&\times \Big (\frac{{\partial s}}{{\partial n}}\Big )_T+ \Big (\frac{{\partial s}}{{\partial T}}\Big )_n \Big (\frac{{\partial p}}{{\partial n}}\Big )_T\Big \}+k^4 n_0^4 T^2 \omega ^2 \nonumber \\&\times \Big (\frac{{\partial s}}{{\partial n}}\Big )_T^2 \Big (\frac{{\partial p}}{{\partial n}}\Big )_T^2+2k^2 n_0 T^{2} \kappa \omega ^2 \Big (\frac{{\partial p}}{{\partial T}}\Big )_n\nonumber \\&\times \Big \{ k^4 T\kappa \Big (\frac{{\partial p}}{{\partial n}}\Big )_T+T \omega ^2 \Big (\frac{{\partial s}}{{\partial n}}\Big )_T k^2( \zeta +\frac{4}{3} \eta ) \nonumber \\&-T^{4} \kappa \omega ^2\Big \}+n_0^2 T^{2}\Big \{ h_{0}k^{2} \kappa ^2 \Big (\frac{{\partial p}}{{\partial n}}\Big )_T^2\nonumber \\&+k^{2}T^4 \kappa ^2 \omega ^4-2 k^4 T^{2} \kappa \omega ^4 \Big (\frac{{\partial p}}{{\partial n}}\Big )_T^{2}\nonumber \\&+2k^2 h_{0}T \kappa \omega ^2 ( \zeta +\frac{4}{3} \eta )+\Big (\frac{{\partial s}}{{\partial T}}\Big )_n^2 k^4 ( \zeta \nonumber \\&+\frac{4}{3}\eta )^{2}+2k^4T^{2} \kappa \Big (\frac{{\partial p}}{{\partial T}}\Big )_n \nonumber \\&\times \Big (\frac{{\partial s}}{{\partial n}}\Big )_T (\zeta +\frac{4}{3} \eta )\Big \}\Bigg ] \Big < \delta n(\mathbf {k},0)\delta n(\mathbf {k},0)\Big > \end{aligned}$$

(50)

where

$$\begin{aligned} \mathcal {S}_{nn}(\mathbf {k},\omega )=\frac{\mathcal {S^\prime }_{nn}(\mathbf {k},\omega )}{\Big < \delta n(\mathbf {k},0)\delta n(\mathbf {k},0)\Big >} \end{aligned}$$

(51)

The expression for $\mathcal {S}_{nn}(\mathbf {k},\omega )$ contain derivatives of several thermodynamics quantities. In this appendix we recast these derivatives in terms of response functions like: isothermal and adiabatic compressibilities ($\kappa _T$ and $\kappa _s$), specific heats ($C_P$ and $C_V$), baryon number susceptibility ($\chi _B$) and velocity of sound ($c_s$), etc. The baryon number density (n) and the entropy density (s) can be written as:

$$\begin{aligned} n=\Big (\frac{\partial p}{\partial \mu }\Big )_{T};\,\,\,\,\, s=\Big (\frac{\partial p}{\partial T}\Big )_{\mu } \end{aligned}$$

(52)

Baryon number susceptibility, isothermal compressibility and adiabatic compressibility are given by,

$$\begin{aligned} \chi _{B}=\Big (\frac{\partial n}{\partial \mu }\Big )_{T}; \kappa _{T}=\frac{1}{n_{0}}\Big (\frac{\partial n}{\partial p}\Big )_{T}; \kappa _{s}=\frac{1}{n_{0}}\Big (\frac{\partial n}{\partial p}\Big )_{s} \end{aligned}$$

(53)

Specific heats can be expressed as:

$$\begin{aligned} C_{P}=T\Big (\frac{\partial s}{\partial T}\Big )_{p}; C_{V}= & {} T\Big (\frac{\partial s}{\partial T}\Big )_{V}=T\Big (\frac{\partial s}{\partial T}\Big )_{n}\nonumber \\= & {} \Big (\frac{\partial \epsilon }{\partial T}\Big )_{V}=\Big (\frac{\partial \epsilon }{\partial T}\Big )_{n} \end{aligned}$$

(54)

Now we write down the expression for partial derivatives, $(\frac{\partial p}{\partial T})_{n}, (\frac{\partial p}{\partial n})_{T}, (\frac{\partial \epsilon }{\partial T})_{n}$ and $\Big (\frac{\partial \epsilon }{\partial n}\Big )_{T}$ below. $(\frac{\partial p}{\partial T})_{n}$ can be evaluated as:

$$\begin{aligned} \Big (\frac{\partial p}{\partial T}\Big )_{n}= & {} \frac{\partial (p, n)}{\partial (T,n)}\nonumber \\= & {} \frac{\partial (p, n)}{\partial (T, p)}\frac{\partial (T, p)}{\partial (s, p)}\frac{\partial (s, p)}{\partial (s, \epsilon )}\frac{\partial (s, \epsilon )}{\partial (s, n)}\frac{\partial (s, n)}{\partial (T, n)} \nonumber \\= & {} \Big [-\Big (\frac{\partial n}{\partial T}\Big )_{p}\Big ]\Big (\frac{\partial T}{\partial s}\Big )_{p} \Big (\frac{\partial p}{\partial \epsilon }\Big )_{s/n}\Big (\frac{\partial \epsilon }{\partial n}\Big )_{s}\Big (\frac{\partial s}{\partial T}\Big )_{n} \nonumber \\= & {} n_{0}\alpha _{P}\frac{T}{C_{P}}c^{2}_{s}\Big (\frac{\partial \epsilon }{\partial n}\Big )_{s}\frac{C_{V}}{T}\nonumber \\= & {} n_{0}c^{2}_{s}\alpha _{p} \frac{C_{V}}{C_{P}}\Big (\frac{\partial \epsilon }{\partial n}\Big )_{s} \end{aligned}$$

(55)

Using the relation,

$$\begin{aligned} d\epsilon =Tds+\mu dn \,\,\,\,\, \text {and } \mu =\Big (\frac{\partial \epsilon }{\partial n}\Big )_{s} \end{aligned}$$

(56)

we write:

$$\begin{aligned} \Big (\frac{\partial p}{\partial T}\Big )_{n}=\mu n_{0}c^{2}_{s}\alpha _{p} \frac{C_{V}}{C_{P}} \end{aligned}$$

(57)

Next we consider $\Big (\frac{\partial p}{\partial n}\Big )_{T}$:

$$\begin{aligned} {\Big (\frac{\partial p}{\partial n}\Big )_{T}= \frac{1}{n_{0}\kappa _{T}}} \end{aligned}$$

(58)

$$\begin{aligned} \Big (\frac{\partial p}{\partial n}\Big )_{T}= & {} \frac{\partial (p, T)}{\partial (n, T)} =\frac{\partial (p, T)}{\partial (p, s)}\frac{\partial (p, s)}{\partial (\epsilon , s)}\frac{\partial (\epsilon , s)}{\partial (n,s)}\frac{\partial (n, s)}{\partial (n, T)}\nonumber \\= & {} \Big (\frac{\partial T}{\partial s}\Big )_{p} \Big (\frac{\partial p}{\partial \epsilon }\Big )_{s/n} \Big (\frac{\partial \epsilon }{\partial n}\Big )_{s}\Big (\frac{\partial s}{\partial T}\Big )_{n}\nonumber \\= & {} \frac{T}{C_{P}}c^{2}_{s}\Big (\frac{\partial \epsilon }{\partial n}\Big )_{s}\frac{C_{V}}{T} \end{aligned}$$

(59)

$$\begin{aligned}= & {} \mu c^{2}_{s} \frac{C_{V}}{C_{P}} \end{aligned}$$

(60)

The factor, $\Big (\frac{\partial s}{\partial T}\Big )_{n}$ can be written as:

$$\begin{aligned} \Big (\frac{\partial s}{\partial T}\Big )_{n}=\frac{1}{T}\Big (\frac{T_{0}\partial s}{\partial T}\Big )_{n}=\frac{C_{V}}{T} \end{aligned}$$

(61)

For fixed net baryon number, $c_n$ can be written as $c_{n}=C_{V}$. Therefore,

$$\begin{aligned} {\Big (\frac{\partial \epsilon }{\partial T}\Big )_{n}=C_{V}} \end{aligned}$$

(62)

We evaluate the derivative $\Big (\frac{\partial \epsilon }{\partial n}\Big )_{T}$ as

$$\begin{aligned} \Big (\frac{\partial s}{\partial n}\Big )_{T}= & {} -\Big (\frac{\partial s}{\partial T}\Big )_{n}\Big (\frac{\partial T}{\partial n}\Big )_{s}=-\frac{1}{T}\Big (\frac{T\partial s}{\partial T}\Big )_{n}\Big (\frac{\partial T}{\partial n}\Big )_{s}\nonumber \\= & {} -\frac{C_{V}}{T}\frac{1}{n_{0}}\Big [n_{0}\Big (\frac{\partial T}{\partial n}\Big )_{s}\Big ]= \frac{C_{V}}{n_{0}T\alpha _{s}} \end{aligned}$$

(63)

The velocity of sound is given by:

$$\begin{aligned} c_s^2=\Big (\frac{\partial p}{\partial \epsilon }\Big )_{s/n}= \frac{nd\mu +sdT}{\mu dn+Tds}= \frac{nF dT+sdT}{\mu (\frac{\partial n}{\partial T})_\mu dT+ \mu (\frac{\partial n}{\partial \mu })_T d\mu +T(\frac{\partial s}{\partial T})_\mu dT+ T(\frac{\partial s}{\partial \mu })_T d\mu }\nonumber \\ =\frac{nF dT+sdT}{\mu (\frac{\partial n}{\partial T})_\mu dT+ \mu F(\frac{\partial n}{\partial \mu })_T dT+T(\frac{\partial s}{\partial T})_\mu dT+ TF(\frac{\partial s}{\partial \mu })_T dT} = \frac{nF+s}{\mu (\frac{\partial n}{\partial T})_\mu + \mu F(\frac{\partial n}{\partial \mu })_T+T(\frac{\partial s}{\partial T})_\mu + TF(\frac{\partial s}{\partial \mu })_T} \end{aligned}$$

(64)

where,

$$\begin{aligned} F= & {} \frac{(\frac{\partial s}{\partial T})_\mu - \frac{s}{n}(\frac{\partial n}{\partial T})_{\mu }}{\frac{s}{n}(\frac{\partial n}{\partial \mu })_{T} -(\frac{\partial s}{\partial \mu })_T} \end{aligned}$$

(65)