Meson Screening Masses in Thermal QCD

The meson screening masses in the scalar and pseudoscalar channels are calculated from the momentum dependent meson spectral functions in the Hard Thermal Loop Approximation (HTL) of the QCD[1]. A new subtraction procedure is invented to get an UV finite result. It is a numerical approach, which requires the analytical form of the mesonic spatial correlation functions in the free case[2] in order to calculate the divergent integrals. Results are compared to the recent QCD lattice results[3,4]. This analysis leads to a better understanding of the excitations of Quark Gluon Plasma in finite temperatures and is of relevance for interpreting lattice results.


Spatial correlation function in HTL approximation
The spatial correlations are exponentially suppressed at large values of z G(z) ∼ z→+∞ e −m scr z and the suppression parameter, which is governing the large distance behavior of the correlator is the screening mass m scr .It gives information on the nature of the excitations characterizing the QGP phase.
One can express the spatial correlation function through the finite momentum spectral function of the quark-antiquark excitations in QGP which in the scalar channel is represented by σ S (ω, p).
Scalar meson correlation function in HTL approximation is calculated in the way similar to the pseudoscalar one [1]. with and a e-mail: piotr.czerski@ifj.edu.plwhere ω ∓ (k) are dispersion relations for plasmino and quasi-particle excitations.For high energy ω → ∞ the spectral function σ S (ω, p z ) → ω 2 and (1) diverge.
To get a finite results for G(z) one has to regularize integrals.For the non interacting QGP the problem was solved for all mesonic channels in [2] adopting adopting Pauli-Villars regularization scheme.In the interacting case the problem is much more complicated since the meson spectral function is obtained only numerically and one has to adopt a numerical regularization.
For numerical regularization of the integrals we follow the method proposed in [3].We calculate numerically the difference of two functions which is converging to zero for high energies and allows to perform all integrations numerically for a given p z momentum.σ di f f S (ω, p z ) = σ f ree S (ω, p z )−σ S (ω, p z ), with the asymptotic mass m = √ 2m q used in the free case.The final result is: z .The screening mass m scr is a smallest between all m i masses for a given T.

Results
In Fig 1 we show the temperature dependence of the screening masses with comparison with the recent lattice results.In Table 1 we summarize screening masses for all temperatures and channels, where m f ree scr = 2 √ π 2 T 2 + m 2 is the free screening mass, m = √ 2m q is the asymptotic mass and m q = g(T )T/ √ 6 is the thermal gap mass of the quark.m scr PS and m scr S are the pseudo-scalar and scalar screening mass of the interacting QGP.
The numbers in the Table 1 are calculated assuming N f = 2 and the transition temperature, following [6], T c = 202 MeV.The ratio T c /Λ MS = 0.7721 for the N f = 2 case is taken from [7].The running of the gauge coupling is given by the two-loop perturbative beta-function, leading to the expression: 3 .We adopt µ = 1.142πT suggested in [6].Finally the thermal gap mass of the quark m q = g(T )T/ √ 6.

Conclusions
The main goal of this work is to calculate the asymptotic mass of the scalar and pseudo scalar meson in the interacting quark gluon plasma for different temperatures.
The meson screening masses in the scalar and pseudoscalar channels are calculated from the momentum dependent meson spectral functions in the Hard Thermal Loop Approximation (HTL) of the QCD [1,3].A new subtraction procedure is invented to get an UV finite result.It is a numerical approach, which requires the analytical form of the mesonic spatial correlation functions in the free case [2] in order to calculate the divergent integrals.Results are compared to the recent QCD lattice results [4,5].
This analysis leads to a better understanding of the excitations of Quark Gluon Plasma in finite temperatures and is of relevance for interpreting lattice results.[5] and compared to the m f ree scr .

Fig. 1 .
Fig. 1.Temperature dependence of the scalar and pseudoscalar screening masses compared with the lattice results extracted from Ref. [4] (upper left Fig.) and from Ref. [5] (upper right Fig.).Lower Fig. shows the temperature dependence of the scalar and pseudoscalar screening masses compared with the lattice results extracted from Ref.[5] and compared to the m f ree scr .

Table 1 .
The pseudo-scalar and scalar screening mass of the interacting QGP.