Holographic study of Wilson loop in the anisotropic background with conﬁnement/deconﬁnement phase transition

. Within the bottom-up holographic QCD using anisotropic black brane solutions in 5D Einstein-dilaton-two-Maxwell system constructed in [1, 2], we study the temporal Wilson loops with arbitrary orientation in respect to the anisotropy direction. We calculate the minimal surfaces of the corresponding probing open string world-sheet in anisotropic backgrounds with various temperatures and chemical potentials. The dynamical wall locations, providing the quark conﬁnement, depend on the orientation of the quark pairs, that gives a crossover transition line between conﬁnement / deconﬁnement phases in the dual gauge theory.


Introduction
Study of the QCD phase diagram, as a function of temperature T and chemical potential µ, is one of the most important problems. The AdS/CFT duality provides an alternative tool for dynamics understanding of quark-gluon plasma (QGP) produced in the heavy-ions-collisions (HIC) [3][4][5][6].
It is believed that there are indications that QGP is anisotropic right after the collision of ions. Isotropisation occures at times ∼ 2 fm/s. It is natural to consider an anisotropic metric within the holographic approach. Anisotropy is usually provided by adding magnetic ansatz of Maxwell field to dilaton gravity action. Non-zero chemical potential is introduced via electic ansatz for the second Maxwell field. Thereby the 5-dimensional dilaton gravity with two Maxwell fields turns out to be the most suitable model. Such model was considered in [1,7]. In the simplest case anisotropic Lifshitz-like models, characterized by anisotropic parameter ν, have been investigated [8].
We consider a 5-dimensional Einstein-dilaton-two-Maxwell system. In the Einstein frame the action of the system is specified as where F 2 (1) and F 2 (2) are the squares of the Maxwell fields F (1) µν = ∂ µ A ν − ∂ ν A µ and F (2) µν = q dy 1 ∧ dy 2 , f 1 (φ) and f 2 (φ) are the gauge kinetic functions associated with the corresponding Maxwell fields, V(φ) is the potential of the scalar field φ.
To find the black brane solution in the anisotropic background, we used the metric ansatz in the following form: where b(z) is the warp factor and g(z) is the blackening function; we set the AdS radius L = 1. All the quantities in formulas and figures are presented in dimensionless units. Note that in [1,2] the following strategy to study holographic model is used. Firstly, one takes b suitable for phenomenological application, in particular one can take b = e cz 2 2 . Secondly, the anisotropic multiplier z 2−2/ν is also fixed by phenomenological reasons [8]. Thirdly, one takes a specific function f 1 by reasons of simplicity. And finally, using E.O.M. following from (2), one finds coupling function f 2 , potential V, Maxwell field potential A µ and blackening function g. The last one has the form: where the function G(x) has the following expansion:

The Wilson loop
The purpose of our consideration is to calculate the expectation value of the temporal Wilson loop oriented along vector n, such that n x = cos ϑ, n y = sin ϑ.
Following the holographic approach [9][10][11] we have to calculate the value of the Nambu-Goto action for test string in our background: where and G µν is given by (2). The world sheet presented in Fig.1 is parameterized as  We can use the following notation ξ ≡ ξ 1 . In our case the action can be rewritten: where τ = dξ 0 , The right-hand side of (12) defines the action of 1-dim dynamical model. This system possesses the first integral of motion: If we introduce z min at which z (ξ) = 0, then the first integral can be expressed as: Let us introduce the effective potential: In order to calculate the length of the curve, at first we have to find the character length of string: where V(z) ≡ M(z) F (z).

Confinement/deconfinement phase transition
To find the conditions of the phase transition [7] we make an expansion of V 2 (z)/V 2 (z min ) in the Taylor series at the point z min : For the confinement/deconfinement phase transition the character length must be infinite for z → z min . So we have two different cases: V (z) = 0 and V (z) 0. 1) If V (z) 0 we can consider the first order only: so that → 0 as z → z min − 0.
2) If V (z) = 0 we have to use the second order: so that → ∞ as z → z min − 0. Note that z min really is the minimum of function V(z) as V (z min ) ≡ 0 and thus V (z min ) > 0. So to find stationary points of V(z) we should solve the following equation: The potential with the field of dilaton is: In our case the effective potential depends on the warp factor, the scalar field and the angle. To find stationary points of V(z) we solve the equation (26) for the potential (27) with arbitrary angle: It is possible to obtain particular cases for θ = 0 0 , 90 0 [1,2] from the expression (28): The expression for the temperature T (z h , µ, c, ν) is Let us remind that in [1,2] the thermodynamical properties of the constructed black hole background were studied and the large/small black hole phase transitions (BB-transition) were found at the temperature T BB (µ). Hawking-Page phase transition takes place at z h,HP , where the free energy equals zero. The particular value of z h,HP depends on c and ν and is larger for larger negative c, i.e. z h,HP (c 1 , ν) < z h,HP (c 2 , ν) for c 1 < c 2 < 0. For the anisotropic background the Hawking-Page horizon is less than for the isotropic one with the same c < 0.
At µ = 0 and for T < T HP (0) the black hole dissolves to thermodynamically stable thermal gas. If the system cools down with the non-zero chemical potential less than some critical value µ cr , the background undergoes the phase transition from a large to a small black hole. This is a generalization of the corresponding effect in the isotropic case [12][13][14][15][16]. The temperature of the large/small black hole phase transition in anisotropic case is less than in the isotropic one, i.e. T (aniso) BB (µ). The value of the critical chemical potential, up to which this phase transition exists, is bigger in the anisotropic case compared to the isotropic one, µ (ν) cr > µ (iso) cr . Also, the point (µ (ν) cr , T (ν) cr ) for ν → 1 goes smoothly to (µ (iso) cr , T (iso) cr ). In Fig.2-4 we can see the angle dependence of the confinement/deconfinment phase transition on the Wilson loop orientation. We choose θ = 0 0 , 10 0 , 45 0 , 60 0 , 90 0 . In the boundary cases the graphs coincide with the graphs for W x (blue solid line) and W y (magenta solid line) from [1,2].
In our consideration we take into account the Hawking-Page phase transition (dashed pink line).

Conclusion
We have studied the dependence of behavior of the temporal Wilson loops on the orientation specified by arbitrary angle θ in the background (2). This result is the generalization of the two particular cases of orientation, considered in [1,2], that can be associated with boundary values θ = 0 0 , 90 0 . We demonstrated that the phase diagram depends on the orientation [7]. Taking into account the instability zones of the anisotropic background, we have found more complicated confinement/deconfinement phase diagrams for differently oriented temporal Wilson loops. For this purpose we have studied the behavior of the temporal Wilson loops in the particular 5-dimensional anisotropic background supported by dilaton and two-Maxwell field. The diagram is defined in (µ, T )-plane for arbitrary angles.
In this model we have determined the critical angles θ cr1 = 22 0 , θ cr2 = 54 0 , θ cr3 = 78 0 . For the critical angle θ cr1 = 22 0 the part of confinement/deconfinement phase transition line is determined by the Hawking-Page phase transition. For the angle θ cr2 = 54 0 the top point of the Hawking-Page phase transition coincides with the top point of the confinement/deconfinement phase transition. For the θ cr3 = 78 0 the whole confinement/deconfinement phase transition line is determined by the Wilson loop.
More detailed consideration for the arbitrary orientation is presented in [17]. In all likelihood these calculations are relevant in the context of the future NICA and FAIR projects.