3.3 Meson spectra at finite baryon density
3.3.2 Spectra
Here we introduced the dimensionless frequency w= ω
2πT . (3.61)
In order to numerically integrate the equations of motion(3.60), we deter-mine local solutions of that equation near the horizon atρH = 1, which obey the infalling wave boundary condition. This condition ensures causality by demanding that the excitations can propagate in inward direction, but nothing can exit the horizon. The local solutions can be used to compute initial values in order to integrate(3.60)forward towards the boundary. The equation of motion(3.60)has coefficients which are singular at the horizon. According to mathematical standard methods, the local solution of this equation behaves as(ρ−ρH)β, whereβis a so-called ‘index’ of the differential equation[75]. We compute the possible indices to be
β =±iw. (3.62)
Only the negative sign will be retained in the following, since it casts the solutions into the physically relevant incoming waves at the horizon and therefore satisfies the incoming wave boundary condition. The solutionE can be split into two factors, which are(ρ−1)−iwand some functionF(ρ), which is regular at the horizon. The first coefficients of a series expansion of F(ρ)can be found recursively as described in[64, 65]. At the horizon the local solution then reads
E(ρ) = (ρ−1)−iwF(ρ)
= (ρ−1)−iw
1 +iw
2 (ρ−1) +· · ·
. (3.63)
So,F(ρ)asymptotically assumes values F(ρ= 1) = 1, ∂ρF(ρ)
ρ=1= iw
2 . (3.64)
To calculate numeric values for E(ρ), we have to specify the baryon densityd˜and the initial valueχ0, which determines the mass parameterm.
These parameters determine the embeddingsχappearing in(3.60). We can then obtain a solutionEfor a given frequencywby numerical integration of the equation of motion (3.60), using the initial values (3.63) and (3.64).
Spectral functions are finally obtained by combining (3.59) and (3.8), R(ω,0) =−NfNcT2
4 Im lim
ρ→∞
ρ3∂ρE(ρ) E(ρ)
. (3.65)
FIGURE3.5: An example for a spectral function at finite baryon density, compared to the zero temperature result.
0 5 10 15 20 25 30 35
0 5000 10 000 15 000
w
R
m= 5 d˜= 0.25
R(w,0)
R0
baryon density. In figure 3.5 an example for the spectral function at fixed baryon density nB ∝ d˜is shown. In the limit of large w, corresponding to asymptotically small temperatures, the spectral function can be derived analytically. This zero temperature result is given by
R0=NfNcT2πw2. (3.66)
Figure 3.5 shows this function as well.
All graphs shown here are obtained for a value ofd˜aboved˜∗, given by (3.38), such that we investigate the regime in which there is no fundamental phase transition of first order. Recall that the parameters of our theory are given byd˜∝ nB/T3 andm ∝ mq/T. Therefore variations in the quark density at fixed temperature and quark mass are introduced by tuningd˜only.
The effects of different quark masses can be seen by tuningm alone. The effect of changes in temperature involves changes in bothmandd.˜
It is interesting to compare the spectra we obtain at finite temperature and density to the vector meson spectrum obtained at zero temperature and vanishing quark density. It is given by the same relation as the mass spectrum (2.81)which we encountered in the example of scalar mesons[43]. In our case, where the mesons do not carry spatial momentum, we can translate the mass Mnof thenth excitation into an energyωn=Mn. At this energy we would see a resonance in a supersymmetric setup. In terms of the dimensionless quantities we use here, these resonance energies are given by
wn= Mn
2πT =m
r(n+ 1)(n+ 2)
2 , n= 0,1,2, . . . , (3.67) wherenlabels the Kaluza-Klein modes arising from the D7-brane wrapping theS3.
Finite temperature effects
We analyze finite temperature effects by choosing two values ofmandd, which˜ correspond to a given values of quark mass, quark density and a temperature
FIGURE 3.6: The effect of variations in temperature on the meson spectrum.
low T m= 5, d˜= 1 med. T m=5/2, d˜=1/8 high T m=5/3, d˜=1/27
0 2 4 6 8 10
0 500 1000 1500 2000
w
R
T. A change in temperature amounts to
T 7→αT (3.68)
and thereby leads to d˜7→ d˜
α3 , m7→ m
α . (3.69)
An example is shown in figure 3.6. There we plot spectra for three different temperatures, which we call low (m= 5,d˜= 1), medium (m=5/2,d˜=1/8) and high (m=5/3, ,d˜=1/27) temperature. We can see that at high temperature there is hardly any structure visible in the spectral function. However, we have chosen a temperature at which already a slight excitation is visible at low energiesw. Decreasing temperature leads to more and more pronounced peaks in the spectral function. Moreover, at decreasing temperature these peaks move closer to the resonance energies(3.67), corresponding to zero temperature and density (drawn as the corresponding dashed lines in the figure).
The formation of sharp resonances at low temperature indicates the in-tuitively expected behavior of long living mesons in a cold medium, which melt,i. e.decay faster, at high temperatures. However, we did not perform an analysis of the quality factor of the resonance peaks,i. e.we did not calcu-late the lifetimes of the vector mesons. From figure 3.6 we can see that the height-to-width ratio of the peaks seems not to improve to a great extent at low temperatures.
Finite density effects
To investigate the effects of finite baryon densitynB, we tuned˜while keeping mconstant. This amounts to varying the quark density at constant temperature and quark mass. The effect is shown in figure 3.7. We observe that the peak width is considerably influenced by baryon density. At low baryon density the resonances are close to line-like excitations, while they are broadened with increasing particle density. Additionally, increasing the particle density also causes a slight shift of the resonances tohigherenergies.
FIGURE 3.7: The depen-dence of the spectra on baryon density. The dashed lines again mark the supersymmetric spectrum.
0 2 4 6 8 10
0 500 1000 1500 2000 2500
w
R
m= 3.5 d˜= 0.1 d˜= 0.5
These observations are interesting from a phenomenologically inclined point of view. The in-medium effects on mesonic bound states are important to interpret the results of heavy ion collision experiments. Estimations from the early 1990sbased on effective models predicted decreasing vector meson masses at increasing densities[76], known as Brown-Rho scaling. Experimental data from experiments at the SPS facility at CERN, however, is better described by models like the one found in refs. 77, 78. There the in-medium effects also are reflected in peak broadening and shifts to higher energies.
For information on the spectral functions at vanishing particle density we refer the reader to ref. 48. Where the low temperature regime ford˜= 0was investigated.
Dependence on quark mass
To observe the dependence on the mass of the quarks, we plotted spectra for differentmat constant values ofd˜in figure 3.8. We observe more and more pronounced resonances as we increase the meson mass. These mesons eventu-ally nearly resemble the line spectrum(3.67)known from the supersymmetric case of zero temperature and vanishing quark density. This observation reflects the decreasing effect of finite temperature and chemical potential with increas-ing quark mass. In a regime where the scale of the quark mass outweighs both additional scalesT andd˜their effects seem to be negligible. This is the case when we observe a configuration which is located close to the Minkowski phase in the phase diagram, cf. figure 3.1.
In ref. 4 we elaborate on the spectral functions behavior at low quark masses. There we observed that the position of the vector meson excitations in the regime of very low quark massesdecreasedwith increasing quark mass.
Further increasing the quark mass lead to increasing quark masses as described in this section. We omit this discussion here, but resume on the topic when we discuss the pole structure of the spectral functions. The reason is that the peaks referred to in ref. 4 are only visible after subtraction of the zero temperature part R0 from the spectral function. To interpret the spectral function as a probability density for the detection of a quasiparticle, we cannot subtractR0,
0 2 4 6 8 10 0
1000 2000 3000 4000 5000
0 2 4 6 8 10
0 1000 2000 3000 4000 5000
w w
R
R
m= 2 m= 4
d˜= 0.1 d˜= 0.1
FIGURE3.8: The dependence of the spectra on quark mass. The dashed lines again mark the supersymmetric spectrum.
as we would otherwise produce negative probability densities, which are not well defined.
Our setup is a modification of the one used in ref. 48. There, the authors considered vector meson spectra at vanishing baryon density. These spectra only show peaks moving to smaller frequency as the quark mass is increased.
There is no contradiction to the results presented this work. Note that the authors of ref. 48 by construction are restricted to the regime of high temper-ature/small quark mass. Nevertheless, they continue to consider black hole embeddings below the temperature of the fundamental phase transition where these embeddings are only metastable, theMinkowski embeddingsbeing ther-modynamically favored. At small baryon density and smallmour spectra are virtually coincident with those of[48]. However, in our case, at finite baryon density, black hole embeddings are favored for all values of the mass over temperature ratio.