
DOI: 10.22661/AAPPSBL.2019.29.4.09
Fluctuations of Conserved Charges,
a Signature of QCD Phase Transition : A Review
ABHIJIT BHATTACHARYYA^{1}
DEPARTMENT OF PHYSICS, UNIVERSITY OF CALCUTTA, 92, A. P. C. ROAD, KOLKATA  700009, INDIA
^{1 }Email address: abphy@caluniv.ac.in
In this work I have reviewed the fluctuations and the higher order susceptibilities of quark number, electric charge and strangeness at vanishing chemical potential. I have argued that this can be used as a signature for QCD phase transitions. This review originates from the series of work that our group has done in this area
INTRODUCTION
It is believed that strongly interacting matter undergoes a phase transition at high temperatures and/or high densities. To find experimental evidence of such a phase transition one needs to search for its signatures. Fluctuations of different conserved charges are one such signature of phase transitions, which can be verified by experiments. Fluctuations can be of several classes. Here, we are concerned about the dynamical fluctuations that reflect the underlying dynamics of the system. Different types of phase transitions give rise to different signatures of fluctuations. For example, a second order phase transition gives rise to critical opalescence [1, 2] and this corresponds to order parameter fluctuation. On the other hand, the first order phase transition gives rise to large multiplicity fluctuations [3].
In principle, fluctuations in strongly interacting matter should be calculated using quantum chromodynamics (QCD) which is the theory of strong interaction. However, due to our limited knowledge of non perturbative physics, we have to use either lattice QCD (LQCD) or effective models. Here we discuss our results using one such effective model of QCD, namely the PolyakovNambuJonaLasinio (PNJL) model. In the next section we discuss the PNJL model very briefly. In section three, the methodology of the calculation is discussed and in section four we present our results.
PNJL MODEL
The PNJL model provides an excellent description of the chiral properties and the confinement physics of the QCD phase transition at finite temperature and density. In this model quark dynamics is studied with a background gauge field having only the temporal component. For a detailed review of the 2 flavor and 2+1 flavor PNJL model, see Ref. [414]. The PNJL model successfully reproduces the thermodynamic properties of QCD, calculated on a lattice, at zero baryon density. The thermodynamic potential for the multifermion interaction in the mean field approximation (MFA) of the PNJL model can be written as [9],
 (1)

In the above expression, g_{S} and g_{D} are the fourquark and sixquark coupling constants respectively and g_{1} and g_{2} are the eightquark coupling constants. Here σ_{f} = <𝜓̅_{f} 𝜓_{f}> denotes the chiral condensate of the quark with flavor f and the single quasiparticle energy. The constituent mass M_{f} for flavor f can be written as
 (2)

where f, f + 1 and f + 2 take the labels of flavor u, d and s in a cyclic order. Therefore, when f = u then f + 1 = d and f + 2 = s and so on. In the above integrals, the vacuum integral has a cutoff Λ, whereas the medium dependent integrals are extended to infinity. The coupling constants may be obtained by reproducing different physical observables.
Where U(Φ,, T) is the LandauGinzburg type potential given by [15],
 (3)

The Polyakov loop Φ and its charge conjugate are defined as,
Further explanation of the different terms and values of different constants may be obtained in references [811].
FLUCTUATIONS AND TAYLOR EXPANSION OF PRESSURE
In the case of a phase transition, fluctuations play an important role. The fluctuations are related to the second cumulants of the partition function or the susceptibilities of the system via the fluctuationdissipation theorem. These susceptibilities can be expressed in terms of the integrals of equaltime correlation functions. In the context of heavy ion collision experiments, if the fluctuations are Gaussian then they can be related to the two point correlators of the system and thus are related to the susceptibilities that measure the response of a system to external perturbations. A system in thermal equilibrium is characterized by its partition function
 (4)

where H is the Hamiltonian of the system, and Q_{i} and Âµ_{i} denote the conserved charges and the corresponding chemical potentials respectively. These conserved charges can be baryon number, charge, strangeness, etc. The mean and the variances are then expressed in terms of the derivatives of the partition function with respect to different chemical potentials
 (5)

with Î´Q_{i} = Q_{i } <Q_{i}>. The susceptibilities can be defined as
 (6)

The susceptibilities are the measures for the fluctuations of the system. The higher order susceptibilities are obtained as
 (7)

The relation between the susceptibility and the fluctuation allows us to verify the theoretical predictions with the experimental data. In the presence of phase transitions, various susceptibilities and thermodynamic quantities, calculated in the framework of LQCD and effective phenomenological models, have indicated different degrees of fluctuations in different phases. The transverse momentum fluctuations, on the other hand, are related to energy/temperature fluctuations that provide a measure of the specific heat of the system [1]. Presence of a phase transition corresponds to the maximum of specific heat and if the system passes through critical end point then these fluctuations are expected to show diverging behavior.
In 2+1 flavor, we have three independent chemical potentials. We can take those to be the quark chemical potential (Âµ_{q}), the charge chemical potential (Âµ_{Q}) and the strange chemical potential (Âµ_{S}). The pressure of the strongly interacting matter can then be expressed as
 (8)

From the usual thermodynamic relations we can show that the first derivative of pressure with respect to Âµ_{q} gives the quark number density and the second derivative is the quark number susceptibility (QNS). Our first job is to minimize the thermodynamic potential numerically with respect to the fields σ_{u}, σ_{d}, σ_{s}, Φ and for a particular temperature. The values of the fields can then be used to evaluate the pressure. Then we can expand the scaled pressure at a given temperature in a Taylor series for the chemical potentials Âµ_{q}, Âµ_{Q} and Âµ_{S}, as
 (9)

where,
 (10)

The flavor chemical potentials Âµ_{u}, Âµ_{d}, Âµ_{s} are related to Âµ_{q}, Âµ_{Q}, Âµ_{S} by,
 (11)

To study the diagonal terms of the expansion, we can write
 (12)

where,
 (13)

where X is q, Q, and S. Here we will use the expansion around Âµ_{X} = 0, where the odd terms vanish due to CP symmetry. For diagonal susceptibilities we evaluate the expansion coefficients up to eighth order. To extract the Taylor coefficients, first the pressure is obtained as a function of different combinations of chemical potentials for each value of T and fitted to a polynomial about zero chemical potential using the gnuplot fit program [16]. Stability of the fit has been checked by varying the ranges of fit and by simultaneously keeping the values of least squares to 10^{10} or even less.
RESULTS
We now present the coefficients of the Taylor expansion of pressure for the 2+1 flavor PNJL model with terms up to sixquark (6q) and eightquark (8q) interactions, and we study the nature of the quark number, charge, strangeness and isospin susceptibilities and their higher order derivatives [10]. Here, we consider a maximum eighth order term in the polynomial in Âµ_{X}. We restrict our expansion range to Âµ_{q} ~ 300 MeV, above which diquark physics is expected to become important. In addition, kaon condensation takes place in the NJL model for Âµ_{S} > 240 MeV. Therefore, we restrict our r ange within Âµ_{S} < 200 MeV below T_{c}. However, above T_{c}, approximate restoration of chiral symmetry implies that the chiral condensates become almost zero. Therefore, above T_{c}, we have extended the range of Âµ_{S} for better fit of the coefficients. Near T_{c}, the χ^{2} (which is same as the least square here) of the fit varies rapidly with the variation in range of Âµ_{X}, over which the fit was done. Consequently, near T_{c} we have fitted the pressure for a 1 MeV gap of temperature and the data points are spaced by 0.1 MeV of chemical potential for all temperature values [10, 11].
Fig. 1: Variation of c_{2}, c_{4}, c_{6}, c_{8} with T/T_{c}, for Âµ_{X} = Âµ_{q}, for 6q and 8q interactions. The arrows on the right show the corresponding SB limit. The lattice data are taken from Ref. [17].
Now we present the behavior of the coefficients c_{2}, c_{4}, c_{6 }and c_{8} for three sets of chemical potentials for 6q and 8q interactions [10]. In figure (1) we show the variation of c_{2}, c_{4}, c_{6} and c_{8} with T/T_{c} for Âµ_{X} = Âµ_{q} for both models and for lattice data. It can be seen that QNS () shows an order parameterlike behavior. At low temperatures the differences in data between 6q and 8q interactions are small and PNJL model with 6q interaction is closer to the lattice data [17, 18]. At high temperature for 6q interaction reaches almost 98% of its ideal gas value whereas for 8q interaction it reaches almost 99% of its ideal gas value. However, lattice data for N_{t} = 6 at high temperatures reaches almost the StefanBoltzmann (SB) limit. The fourth order derivative can be thought of as the susceptibility of . The figure, for this quantity, shows a peak near T_{c}. Near T_{c} the for the 8q interaction shows a much higher peak than the 6q interaction and the peak for the eightquark interaction is closer to the lattice data. At higher temperatures, both cases match very well with the lattice data. But, the lattice result is little bit closer to the SB limit, at high temperature, compared to the model results. Since, the coupling strength is large enough for T < 2.5T_{c}, a sufficient amount of interaction is present in the system. So, it is expected that will not converge exactly to the SB limit within T < 2.5T_{c}. The higher order coefficients, and , show interesting behavior near T_{c}. Although at very low and at very high temperatures both of them converge to zero, shows sharp peaks, near T_{c}, for both cases. However, for 8q interaction the peak is much sharper. Similar behavior can be observed for , which shows more peaks near T_{c}. The reason behind the peaks near the transition temperature is due to the increase in fluctuation near T_{c}. The sharper peaks of 8q interaction are probably due to the introduction of enhanced attractive interaction through the eightquark term. The number of peaks increases near T_{c} for higher order coefficients.
Fig. 2: Variation of c_{2}, c_{4}, c_{6}, c_{8} with T/T_{c}, for Âµ_{X} = Âµ_{Q}, for 6q and 8q interactions. The arrows on the right show the corresponding SB limit. The lattice are data taken from ref. [17].
Fig. 3: Variation of c_{2}, c_{4}, c_{6}, c_{8} with T/T_{c}, for Âµ_{X} = Âµ_{S}, for 6q and 8q interactions. The arrows on the right show the corresponding SB limit. The lattice data are taken from ref. [17].
In Figure 2 the variation of susceptibility and the higher order coefficients for the charge chemical potential is shown. The nature of all the coefficients are similar as the quark chemical potential. At high temperature the fluctuation for 8q interaction is closer to the SB limit compared to the model with 6q interaction.
However, lattice data is slightly above the SB limit for . For the case of , our data (for both cases) show a better convergence towards SB limit, unlike . At low temperature, the behavior of the model with 6q interaction is closer to the lattice data compared to the model with 8q interaction. The quartic fluctuations show a peak near T_{c}. The peak for 8q interaction is sharper than 6q interaction and the plot for 8q interaction matches well with the lattice result. The higher order coefficients show similar behavior as the quark chemical potential case [10].
Figure 3 shows the variation of susceptibility and the higher order coefficients for the strangeness chemical potential with T/T_{c}. The for both the models is slightly different from the lattice data. Both of the plots are almost 98% of the SB limit at high temperatures, however the lattice data coincides with the SB limit. The has similar behavior as to the . However, in the present case, the peak is not near T_{c}. Near T_{c} we can see a small bump for both of the models but the peaks in both cases are at higher temperatures, as compared to lattice data. This is due to the fact that during the chiral crossover, the strange quark (the only element which carries strangeness) is sufficiently heavy and the corresponding condensate σ_{s} melts at much higher temperatures than T_{c}. The maxima of dσ_{s}/dT and that of coincide. However, in case of a lattice, there is only one peak at T_{c}. Therefore, one cannot really pin down the cause of the double peak structure in our model study. It may be a model artifact. At high temperatures both the curves are above the SB limit. But the lattice data is slightly below the SB limit. The higher order derivatives also show a sharp peak at the transition temperature followed by a broader peak at higher temperatures
Fig. 4: Variation of c_{4}/c_{2} with T/T_{c}, for Âµ = q, Q or S for both model with 6q and 8q interactions. The arrows on the right show the corresponding SB limit. The lattice data are taken from ref. [17]. The upper left panel corresponds to the quark chemical potential, the upper right panel corresponds to the charge chemical potential and the lower panel corresponds to the strangeness chemical potential.
Let us now try to look at the ratio of these coefficients. These ratios are sensitive probes of deconfinement. For an ideal gas consisting of particles with baryon number b, we have [3]
 (14)

where χ_{B}, χ_{N} and χ_{N̅} are the baryon number, particle number and antiparticle number cumulants respectively. Then we can write [3]
 (15)

where N is the number of particles.
The ratio is called kurtosis and this can be related to experimental data. Now from equation (14) one can see that kurtosis for the baryon number is proportional to b^{2}. In the hadronic phase, all baryons have baryon number B_{hadronic} = 1 and in the QGP phase, the quarks have the baryon number B_{quark} = . Thus, kurtosis for the quark number chemical potential R_{q} = c_{4} / c_{2} = (N_{C}B)^{2} is 9/12 at low temperatures and 1/12 at high temperatures. However, this is a classical estimate that has to be corrected by quantum statistics at high temperatures. Therefore, one has to multiply the ratio by a factor of 6/π^{2} for massless particles at high temperatures [3].
In Figure 4 we plotted the kurtosis for the baryon number along with the lattice data. In this plot, more fluctuations are observed for 8q interaction than 6q interaction. However, lattice data shows higher fluctuation near T_{c} than the model study. At higher temperatures, both models coincide with the lattice data and converge well with the SB limit. The ratio R_{Q} = , at low temperature, is dominated by the charge fluctuation in the pion sector resulting R_{Q} = 1/12. At high temperatures R_{Q} = , which is its SB limit. The model with 8q interaction shows more fluctuation than the 6q interaction and as well as the lattice data. The 8q interaction shows almost 99% convergence with the SB limit at high temperature. For the strangeness fluctuation, we can see two peaks in curve for both the models. First peak occurs at chiral transition for light flavors and the second peak occurs when chiral transition occurs in the strange sector. At intermediate temperatures the PNJL model overestimates the ratio more than the LQCD result.
A reliable way to understand the physics of the phase transitions of strongly interacting matter is to study the fluctuations of conserved charges. Susceptibilities are related to fluctuations via the fluctuationdissipation theorem. A measure of the intrinsic statistical fluctuations in a system close to thermal equilibrium is provided by the corresponding susceptibilities. At finite temperatures, and chemical potentials fluctuations of conserved charges are sensitive indicators of the transition from hadronic matter to quarkgluon plasma (QGP). Moreover, the existence of the CEP can be signaled by the divergent fluctuations. For the small net baryon number, which can be met at different experiments, the transition from hadronic to the QGP phase is continuous and the fluctuations are not expected to lead to any singular behavior. Computations on the lattice have been performed for many of these susceptibilities at zero chemical potentials [1922]. It has been shown that at the vanishing chemical potential the susceptibilities rise rapidly around the continuous crossover transition region.
Acknowledgement: I thank BRNS and Alexander von Humboldt foundation for support.
References
[1] L. Landau and L. M. Liftshitz, "Statistical Physics", Pergamon Press, U.S.A. (1980).
[2] K. Huang, "Statistical Mechanics", Wiley, U.S.A. (1987).
[3] V. Koch, arxiv:0810.2520v1.
[4] K. Fukushima, Phys. Lett. B 591, 277 (2004).
[5] S. K. Ghosh, T. K. Mukherjee, M. G. Mustafa and R. Ray, Phys. Rev. D 73, 114007 (2006).
[6] M. Ciminale, R. Gatto, N. D. Ippolito, G. Nardulli and M. Ruggieri, Phys. Rev. D 77, 054023 (2008).
[7] S. Roessner, C. Ratti and W. Weise, Phys. Rev. D 75, 034007 (2007).
[8] P. Deb, A. Bhattacharyya, S. Datta and S. K. Ghosh, Phys. Rev. C 79, 055208 (2009).
[9] A. Bhattacharyya, P. Deb, S. K. Ghosh and R. Ray, Phys. Rev. D 82, 014021 (2010).
[10] A. Bhattacharyya, P. Deb, A. Lahiri and R. Ray, Phys. Rev. D 82, 114028, (2010).
[11] A. Bhattacharyya, P. Deb, A. Lahiri and R. Ray, Phys. Rev. D 83, 014011 (2010).
[12] C. Sasaki, B. Friman and K. Redlich, Phys. Rev. D 75, 074013 (2007).
[13] C. Ratti, S. Roessner and W. Weise, Phys. Lett. B 649, 57 (2007).
[14] V. Skokov, B. Friman, E. Nakano, K. Redlich and B. J. Schaefer, Phys. Rev. D 82, 034029 (2010).
[15] C. Ratti, M. A. Thaler and W. Weise, Phys. Rev. D 73, 014019 (2006).
[16] http://www.gnuplot.info/.
[17] M. Cheng et al., Phys. Rev. D 79, 074505 (2009).
[18] M. Cheng et al., Phys. Rev. D 77, 014511 (2008).
[19] S. A. Gottlieb et al.,Phys. Rev Lett. 59, 2247 (1987).
[20] R. V. Gavai, S. Gupta and P. Majumdar, Phys. Rev. D 65, 054506 (2002).
[21] C. Bernard et al., Phys. Rev. D 71, 034504 (2005).
[22] C. Bernard et al., Phys. Rev. D 77, 014503 (2008).

Abhijit Bhattacharyya is a professor at the Department of Physics, University of Calcutta, Kolkata, India. His research interests are QCD phase transitions, heavy ion collision and compact stars. Prof. Bhattacharyya did his PhD from the Bose Institute, Kolkata. He was the recipient of the Alexander von Humboldt Fellowship and worked at the Institute for Theoretical Physics and the Frankfurt Institute for Advanced Studies, Frankfurt University, Germany. 
