Nonperturbative QCD thermodynamics in the external magnetic field

The thermodynamics of quarks and gluons strongly depends on the vacuum colormagnetic field, which grows with the temperature $T$, so that spatial string tension $\sigma_s ={\rm const}~ g^4 (T) T^2$. We investigate below what happens when one imposes in addition constant magnetic field and discover remarkable structure of the resulting thermodynamic potential.


Introduction
The problem of quark gluon thermodynamics in magnetic field is of the high interest in the modern physics, since heavy-ion experiments produce important data on the properties of resulting hadron yields and hadron interactions, which might be influenced by the strong magnetic fields (MF) created during the collision process [1,2,3]. For a recent review on the effects of MF see [4]. On the theoretical side the problem of MF in the quark gluon plasma (qgp) was studied in different aspects, e.g. in the NJL-type models [5] and in the holographic approach [6,7]. Within the nonperturbative QCD the theory of qgp in MF was developed in [8,9,10], where the general form of the thermodynamic potentials was found in MF with zero or nonzero baryon density, summing over all Landau levels including LLL.
In this approach the only nonperturbative interaction, which was taken into account, reduced to the inclusion of Polyakov lines in the resulting expression for the pressure.
The resulting expressions for magnetic susceptibilitiesχ q (T ), obtained in [8], were used in [9,10] to compare with the lattice data from [11,12], and a reasonable agreement was found forχ q (T )with different q = u, d, s, in [9,10] as well as for the sum [10], however somewhat renormalised values of effective quark masses were used.
Recently in [13,14] a new step in the development of the np QCD thermodynamics was made, where the colormagnetic confinement (CC) was included in the dynamics of the qgp. This interaction with the spatial string tension σ s grows with temperature,σ s ∼ g 4 T 2 and is important in the whole region T c < T < 10 GeV. A concise form of the final expression was found in [13,14] in the case of an oscillatory type CC and an approximate one in the realistic case of the linear CC. The resulting behavior both in the SU(3) case, found in [15,16], and in the qgp case [13,14] agrees well with the corresponding lattice data.
It is the purpose of the present paper to extend our previous analysis of the qgp thermodynamics in MF, done in [8,9,10], including the dynamics of CC with the explicit form of the magnetic screening mass m D , generated by CC.
As will bee seen, we propose a simple generalization of the results [8,9,10], where the CC produces the mass M D , entering the final expressions in the combination m 2 q + m 2 D ≡M instead of m q . We check the limiting cases and compare the result with lattice data.
The paper is organized as follows. In the next section the general analysis of the MF effects in thermodynamics is explained, in section 3 the magnetic susceptibility is defined, in section 4 the results are compared to the lattice data, and in the section 5 the summary and prospects are given.
2 General structure of the pressure with and without magnetic field We start with the quark pressure of a given flavor as expressed via the 3d quark Green's function S 3 (s) in the stochastic field of the colormagnetic confinement (CMC). From the path integral representation [14,15] one obtains is the quark Polyakov line, and S 3 (s) can be expanded in a series over eigenstates in the CMC on the 2d minimal area surface in 3d space.
As it was argued in [14,15], in the case of linear CMC one obtains for S 3 (s) an approximate form and pressure can be written as On another hand, using the relation from [8] s = nβ 2ω , β = 1/T and the representation one obtains as in [8] the pressure of the given quark flavor Let us now introduce the magnetic field B along the z axis, so that our system of quarks undergoes the influence of both CMC field and (electro)magnetic field (MF) at the same time.
We consider the influence of the MF only, and write the corresponding equations from [8] It is interesting, that in the case µ = 0 one can replace in (7) the exponent (as was suggested in [8]) and using the phase space in MF and the relation one obtains the same Eqs. (8), (9), but with the replacement where c ≃ 1 for T → ∞. Therefore in what follows we shall be using the Eq.(8) with the replacement in (9), m 2 q →M 2 = m 2 q + cσ s .
As was shown in [8], the form (8) can be summed up over n ⊥ , σ to obtain the following result (we consider below for simplicity only the case µ = 0) (14) Note, that the first term in (14) appears from the lowest Landau levels (LLL). For these levels it is known from analysis in [17], that the asymptotic quark energy values do not depend on eB and are equal to the √ σ for small quark mass. This agrees with our valuesM = m 2 q + σ s ≈ √ cσ s and supports our expression (14) at least in the high e q B limit, e q B ≫M , when the second and the third term in (14) tend to zero. Hence one obtains in the limit |e q B| ≫M, T Note, that the factor |e q B| appears due to the phase space relation in MF, Eq. (11). Now we turn to the limit of small MF, |e q B| ≪M , T . One obtains from (14) the contribution of the second term only One can compare (16) with (4), obtained in the case of zero MF, and insertion of (5) in (4) yields the same answer as in (16).

Magnetic susceptibility of the quark matter
Using general expression for the quark pressure (14), one can define a more convenient quantity, the magnetic susceptibilitiesχ (n) q ,χ (2) q ≡χ q , To this end one expands the Mc Donald functions K n ( √ n 2 + b 2 ) entering in (14)in powers of b, following [8], and one obtains As a consequence, one has forχ q It is possible to sum up the series over n in (18), when one exploits the representation As a result one obtains for the quadratic magnetic susceptibility (ms) The ms in (22) is defined for a given quark flavor q, and the total m.s. for the quark ensemble, e.g. for 2 + 1 species of quarks can be written aŝ

Results and discussions
We have presented above the thermodynamic theory of quarks in the magnetic field, when quarks are affected also by Polyakov line interaction and the CC interaction, generalizing in this way our old results of [8]- [10], where the CC part was absent.
We have included the CC interaction in the energy eigenvalues ε σ n ⊥ , Eq.(9) tentatively via the replacement (13). This substitute can be corroborated in the case of lowest Landau levels withσ = 1, n ⊥ = 0, where magnetic field does not eneter, and the effective quark mass in subject to the CC interaction only. In the general case one can expect possible interference of eB and CC terms, which can spoil the suggested replacement.
To make this first analysis more realistic, we have checked the limits of small and large values of eB. In the first case we have shown the correct correspondence with the eB = 0 result of [13,14], and in the second case of large eB, the leading linear in eB term is just LLL term, which is not influenced by magnetic fields, except for the phase space redefinition. These results enable us to proceed with the analysis and comparisons of obtained equations with lattice data.
We present below an analysis of the MF influence on the quark thermodynamics. It is interesting, that the basic expression for the pressure P (f ) q in (14) has the property, that at small eB < (eB) crit , the dependence of ∆P (B, T ) on eB is quadratic with a good accuracy, according to Eq. (17). At larger eB, eB > (eB) crit , one has the linear dependence of ∆P (B, T ) on eB, given in (15). Fig.1 illustrates this behaviour for thee fixed temperatures  (14) and Fig.1 that (eB) crit >M , and actually is around 0.5 GeV 2 for T ≃ 0.2 GeV . We have computed analytically the difference ∆P (T ) = P (T, eB) − P (T, 0) using Eq. (14) and compare with the lattice data from [18] for averaged u, d, s quark ensemble.
One can see in Fig (14) are compared with the lattice data from [18]. We have used in (14) the Polyakov line L(T ) obtained in [19]. One can see a reasonable agreement within the accuracy of the lattice data, which supports the main structure of the theory used in the paper. Detailed analysis for larger intervals of eB and T and for specific quark flavours is possible within the approach and is planned for next publications.