Photo-emission rate of sQGP at finite density
We calculate the thermal spectral function of SYM plasma with finite density using holographic technique. The gravity dual of the finite temperature and density is taken as the RN-AdS black hole. In the presence of charge, linearized vector modes of gravitational and electromagnetic perturbation are coupled with each other. By introducing master variables for these modes, we solve the coupled system and calculate spectral function. We also calculate photo-emission rate of our gauge theory plasma from spectral function for light like momentum. AC, dc conductivity and their density dependence is also computed.
The gauge/gravity duality [1, 2, 3] opened a new possibility to quantitative study for strongly interacting system. Although it is not developed enough to describe realistic QCD, we expect to learn some features of QCD from it based on the universality of the hydrodynamics: in longwavelenth limit, the details of the theory does not matter. For example, the shear viscosity/entropy ratio [4, 5] is universal if we neglect the higher derivative terms. We also expect, due to analytic structure of the theories, that there are similarities of supersymmetric and non-supersymmetric theories can continue to the finite wavelenth/frequency regime.
The quarks and gluons are liberated at high enough temperature. However, over , the experimental data shows that quarks and gluons are not free but are strongly interacting: the small viscosity and the presence of the coherent flow show that the interactions should be very strong. Such strongness of the interaction is the motivation why one has to abandon perturbative QCD in such energy/temperature regime. One way to avoid that difficulty is to rely on hQCD for the quark gluon plasma in RHIC(Relativistic Heavy Ion Collider). The hydrodynamic calculations of hQCD were shown to be useful to discuss the transport phenomena [6, 7]. It is interesting to see what happens in LHC(Large Hadron Collider) where the energy scale is much higher .
The finite baryon density effect is very essential ingredient to understand how the core of neutron star and the early universe behave. It maybe uncover some significant features of the evolution of our universe, galaxies and stars. At RHIC experiment, the temperature reached is above but the density is almost zero. In near future J-PARC(Japan Proton Accelerator Research Complex), LHC and especially FAIR (Facility For Antiproton and Ion Research) which probes the regime of a few times of normal nuclear density  will tell us much about the density effect of quarks and gluons. The holographic study for the system with finite density in hydrodynamic regime was made in [8, 9, 10, 11, 12].
The dual gravity background for the finite density and temperature is taken to be Reissner-Nordström AdS blackhole. In the phase diagram, we know much about high temperature, low density regime but not low temperature high density regime. Previously we studied some issues ie. meson mass shifted by density effect for zero temperature, finite density sector . Now we will study the effect of both finite temperature and density. The bulk U(1) charge is identified with the particle number density of the boundary field theory. To see the finite frequency/momentum dependence of the response of the system, the spectral function is a good tool. It gives us ac conductivity and its trace is related to the photo-emission and di-lepton production rate [13, 14, 15].
The spectral function with and without baryon density in the probe brane approach was already calculated [16, 17, 18, 19, 21, 22, 23]. However in that approach the gravity back reaction to the presence of the charge is neglected. In this paper, we take bottom up approach where the back reaction is taked into account. We will compute the spectral function of tensor and vector modes which describe the fluctuation of energy momentum tensor and currents of hot plasma. After that the finite temperature and density effects of photo-emission rate are calculated and discussed.
2 A recipe for Greens function
In this section, we will briefly review how to calculate thermal spectral function. To describe thermalized plasma holographically we need the black hole background. The general equations of motions for the linearized fluctuations in this background are
where denote the fluctuating fields in given background and runs 1 to n, the number of independent fields and are dimensionless frequency and momentum. Near the boundary, (, there are two local Frobenius solutions
is the solution of indicial equation near the boundary, where is the conformal dimension of an operator and is the dimension of the dual source field. Near the horizon, u=1, there are also two local solutions
The two different local solutions of eq.(1) should be matched:
However not all solutions are allowed physically because this system contains the blackhole: no outgoing-wave can propagate from the horizon, therefore we should impose , , which is so called the infalling boundary condition. Taking the normalization using the linearity of differential equation, we have
Note that the coefficient is the source of the boundary theory operator , so by differentiating the generating functional twice with respect to we get the retarded Green function. And another coefficient corresponds to the condensate or vacuum expectation value of the operator which couples to the source . The retarded Green function is given by the ratio between . We will give a sketch of the on-shell quadratic action, with eq. (2), (5)
obviously the first term in last line is divergent () so it should be renormalized holographically  or we can ignore it because the imaginary part of the Greens function do not care about the real number which comes the first term. For the issues how to regulate the on-shell action, see the appendix D. The spectral function is its imaginary part . Notice that
Here is real because the equation of motion and initial conditions are real for . Above expression is independent of since it is a kind of the conserved flux 
where , with the normalization constant. The retarded Green function is defined by the recipe 
From the eq. (2), the only thing we should know is the value of . The spectral function, for example, is given when =0
Given the normalization constant we can calculate the spectral function numerically.
3 RN AdS
The dual geometry for the finite temperature and density is chosen as charged AdS black hole . The action is
where the cosmological constant is , the last term is the Gibbons-Hawking term. And K is the extrinsic curvature on the boundary, l is the AdS radius. The metric of RN AdS is
where the gauge charge Q is related to the black hole charge q
the five dimensional gauge theory coupling constant and the gravitational constant 111Usually five dimensional gravitational constant is used as but here we will use as itself. can be chosen as 
but we will not use these parameters explicitly. The metric function is rewritten as
where and . The charge is expressed by and m,
the range of outer horizon is . Finally, black hole temperature is given as
From the horizon regularity, the black hole charge q is related with the chemical potential
Notice that there is maximum value of q, which corresponds to . Horizon radius should be real, so 1+2 cos() must be positive.
It is very useful to express the frequency and momentum as a dimesionless quantities . This choice is good enough to see the finite temperature behavior of system but not good in zero temperature limit. Alternative way is to rescale w and k by the black hole radius , that is by : let . At the extremal limit, by eq. (16), and
and the chemical potential is written
If we rescale w and k with b, in the extremal limit we rescale w and k with chemical potential, .
The origin of charged black hole in string theory can be understood by STU model : the diagonally charged STU black hole is RN AdS. The diagonal is the subgroup of the SU(4) R-symmetry originally but here we assume that this is a part of flavor U(1) group which is relevant if we assume that the bulk filling branes are embedded in our AdS5 space time 222Please note that this is no more than a conceptual introduction of bottom up approach in string theory.. The merit of doing this is that we can have a back-reacted gravitational background which is a solution of glue-quark coupled system in terms of gauge theory it means our approach is beyond quenched approximation.
4 Tensor mode
The gravitational and gauge field perturbation is classified by the boundary SO(2) rotational symmetry. This classification is summarized in the appendix A. Tensor mode perturbation is easy to treat because it is completely decoupled from other fields. The equation of motion for component is
where prime denotes derivative with respect to r and . Introducing new coordinate ,
The equation of motion is simplified
This differential equation has two independent solutions near boundary,
In eq. (5), we choose . By dividing both side, we get the normalized solution for tensor mode
The on-shell action is given in eq. (5.7) of ,
Our normalization is such that at the boundary. 333More properly, we should express , where hatted variable is the value at the boundary or the external source of the boundary theory. By taking the imaginary part of the Greens function and renormalizing divergent terms, the thermal spectral function is
Here the ratio is
This ratio is independent of the evaluation point. As explained before, imposing the infalling condition at the horizon and Dirichlete boundary condition at the UV boundary, we get the numerical solution for and ,
Using , one can show that the zero temperature spectral function is
See appendix B for detail. Figure 1 shows the difference between normalized thermal spectral function at non-zero and zero temperatures for . The thick line is zero chemical potential case =0 which is the AdS Schwarzschild case of ref. . The dashed and solid line correspond to the = 0.5, 1 case respectively. When the chemical potential increases, the spectral difference grows up and the position of the small peak is shifted to the larger .
The position of peak in the spectral difference is the pole position of retarded Greens function . The shift of the peak is the shift of the quasi normal mode. When chemical potential grows in the unit of T, the pole position grows faster than T.
The right side of the fig. 1 shows the spectral function as a function of spatial momentum . When =0, the peak position can be identified with the inverse screening length. However, for the tensor mode, there is no peak for =0. As we know SYM has conformal symmetry, hence it can not have any scale. For the finite , there is a broad peak and the peak is more sharpened when chemical potential grows. This shows that in a dense system the thermal particle collides more often so that the particles propagates shorter distance. For the light like momentum, this screening is maximized. For spacelike momentum, the thermal fluctuation of spectral function rapidly vanishes leaving only zero temperature piece.
5 Vector mode
Vector type perturbation consist of three independent fields, and equation of motion for these modes are coupled with each other. In hydrodynamic limit, this mode has a diffusion pole so that it is also named as diffusive mode. Here we are interested in general energy/momentum regime. The equations of motion for vector modes are
This equation is simplified by introducing gauge invariant combination ,
where . It is not easy to solve these 2nd order coupled differential equations, but the authors of  have decoupled these equations by introducing master variables. Let us define first as
The equation of motion is rewritten as
In order to use our recipe for spectral function, we need the on-shell action for vector modes:
From the equations for master field we can get the spectral function of master fields. We however need the spectral function of original variables not the master fields itself. In ref. , authors showed a systemetic way to compute the spectral function of original variables in terms of master variables. Let us first find the series solution of ,
which defines the conjugate momentums . The boundary action can be written in terms of the boundary values of original variables and their conjugate momentums:
where are the boundary values of the fields and are their conjugate momentum which will be identified with one point function and dots denote contact terms which does not have any derivatives with respect to u. Note that these conjugate momentums depends on the boundary source terms implictly. The master variables and have series solutions near the boundary,
Define the transformation matrix R,
the boundary value of the master fields is simply related to the boundary value of the original fields by R,
Then the conjugate momentum are written as
The boundary action is now written only by boundary values () and conjugate momentum of master field . The two point function for and is related to the gauge invariant variable as
therefore the correlation functions for each components are
Note that when spatial momentum or density ”a” vanishes two point function vanishes. It means that the holographic operator mixing between and comes from the density effects. By following the standard recipe described in section 2, , the conjugate momentum of master fields, are computed as the ratio of two connection coefficients
By imposing infalling IR boundary condition for , the spectral functions are computed.
The spectral function is plotted in terms of . The figure 2 shows that the imaginary part of the divided by w, which is AC conductivity of thermalized plasma ( with normalization constant, ). The peak position becomes larger as the charge increases. The strength of that peak also increases, when charge grows. In ref., they calculate only zero density case which is denoted by the thick line in fig. 2. The right figure in fig. 2 is the density dependence of DC conductivity. From the spectral function, DC conductivity can be computed by taking zero frequency limit, . As density increases, it decreases and in sufficiently large density regime, DC conductivity is negligible. This is rather surprizing since Drude formula in Maxwell theory says the conductivity is proportional to the density of the charge carrier. It seems that interaction between the charge carriers dominates the abundancy effect. Such drastic reduction of the DC conductivity can be an another explanation of the jet quenching phenomena which are different from the explanation in ref. [29, 30]. In highly dense system, the strongly interacting plasma can not carry charge over long distance because of density effect. If this is the relevant mechanism, raising the temperature supresses the Jet quecnching in LHC since it reduces .
In fig. 5, we plot the spectral function in terms of spatial momentum with fixed frequency. The left one shows thick, dashed, thin line corresponds to = 0.1, 0.3, 0.5 with =1 and the right of fig. 5 also shows thick, dashed, thin line corresponds to =1,2,3 with =0.1. These results can be interpreted as a inverse thermal screening length of the super Yang-Mills plasma. It is interesting that for the tensor and vector mode the peak position is different. For the , is zero so screening mass is zero. But for the finite there is the peak and the position is a function of both and . Because the diffusive nature affects the interactions inside the medium, the screening effect are more complicated.
The hydrodynamic pole in is appeared in fig 3 at
The left figure shows that the hydrodynamic pole position is shifted from 0.0225 (=0.3) to 0.01 (=0.2). This comes from the density effect, when goes to zero the hydrodynamic pole in disappears, see appendix C. This is the operator mixing result. The diffusion pole is only appeared in or not . The right figure shows that reaches very rapidly to the zero temperature spectral function.
Fig. 4 shows real part of ac conductivity, with the normalization unit . By definition, . The ac registivity is defined as . For large , the system has zero registivity, it means that at any density charge carrying is almost perfect in high frequency.
The (xt,xt) component of spectral function has the diffusion pole at . The dispersion relation for diffusive channel gives us the diffusion constant from the hydrodynamic analysis. The left figure in fig. 6 is with =0.3 for various values of : 0.5(thick), 1(dashed), 1.5(thin). When the chemical potential grows , the strength of peak also grows but the position itself does not. The reason of this increase comes from the factor in front of the Greens function. When increases the parameter goes up so the overall factor increases rapidly. When the system reaches the extremal limit, , that factor diverges and our analysis is broken down. It should be computed separately for the zero temperature or for the extremal RN spectral function from finite temperature or non extremal RN AdS blackhole.
In the right figure of the fig. 6, the peak position is shifted when is moved. Again, the position is at .
6 Photo-emission rate
In the heavy ion collision, the emitted photons are a good measure to see the medium effect. The photo emission rate of SYM plasma was calculated holographically for AdS Schwarzschild , for D4/D8/ with finite chemical potential  and for D3/D7 with finite baryon density . We will focus on the photo-emission rate for our gauge theory dual to the RN AdS background here. Let be the number of photons emitted per unit volume. To leading order in electromagnetic coupling e,
where is the Fourier transformed Wightman function which is related to the spectral function multiplied by Bose-Einstein distribution function . Convert the differential photo-emission rate into the emission rate per unit volume as a function of ,
In order to calculate photo-emission rate we need to know the longitudinal part of the spectral function. But for the case of light like momentum , the trace of spectral function is obtained only by . From the appendix A,
where d is the dimension of the boundary field theory. For the light like momentum, the longitudinal correlator should be vanished because the projection operator diverges. The trace of spectral function is only given by . The frequency dependent spectral measure for the light like momenta is in fig. 7. The left one in fig. 7 has three lines = 0(thick), 5(dashed), 10(thin).
In figure 7, the photo-emission rate is compared with , the peak position is the maximum value is 0.01567 with unit