Holographic anisotropic background with confinement-deconfinement phase transition

We present new anisotropic black brane solutions in 5D Einstein-dilaton-two-Maxwell system. The anisotropic background is specified by an arbitrary dynamical exponent ν, a nontrivial warp factor, a non-zero dilaton field, a non-zero time component of the first Maxwell field and a non-zero longitudinal magnetic component of the second Maxwell field. The blackening function supports the Van der Waals-like phase transition between small and large black holes for a suitable first Maxwell field charge. The isotropic case corresponding to ν = 1 and zero magnetic field reproduces previously known solutions. We investigate the anisotropy influence on the thermodynamic properties of our background, in particular, on the small/large black holes phase transition diagram. We discuss applications of the model to the bottom-up holographic QCD. The RG flow interpolates between the UV section with two suppressed transversal coordinates and the IR section with the suppressed time and longitudinal coordinates due to anisotropic character of our solution. We study the temporal Wilson loops, extended in longitudinal and transversal directions, by calculating the minimal surfaces of the corresponding probing open string world-sheet in anisotropic backgrounds with various temperatures and chemical potentials. We find that dynamical wall locations depend on the orientation of the quark pairs, that gives a crossover transition line between confinement/deconfinement phases in the dual gauge theory. Instability of the background leads to the appearance of the critical points (μϑ,b, Tϑ,b) depending on the orientation ϑ of quark-antiquark pairs in respect to the heavy ions collision line.

by anisotropy, it is useful to deal with an explicit analytical solution. For this purpose we take the particular case of the simplest warp factor b(z) = e cz 2 2 . We find the dilaton potential by the potential reconstruction method similar to the isotropic case [46] (and refs therein). We show that only c ≤ 0 guarantees real solutions for the dilaton (compare with [19] and refs therein). We construct the blackening function that supports the Van der Waals-like phase transition between small and large black holes for a suitable first Maxwell field charge. The isotropic case corresponding to ν = 1 and zero magnetic field reproduces previously known solutions [45,46]. We investigate the anisotropy influence on the thermodynamic properties of our background, in particular, on the small/large black holes phase transition diagram. We find that the anisotropy changes the location of the domain of instability.
We also discuss applications of the model to the bottom-up holographic QCD. We note that the RG flow interpolates between the UV section with two transversal suppressed coordinates and the IR section with the suppressed time and longitudinal coordinates due to anisotropic character of our solution. We study the temporal Wilson loops extended in longitudinal and transversal directions by calculating the minimal surfaces of the corresponding probing open string world-sheet for various values of temperature and chemical potential. We find that for particular sets of the model parameters the dynamical wall appears. The appearance of dynamical walls also depends on the orientation of the temporal Wilson loop, that gives a crossover transition line between confinement-deconfinement phases in the dual gauge theory. This effect has been also observed in the anisotropic model considered in [35]. In the background, investigated in the present paper, there are two more anisotropic effects. Namely, the instability of the background restricts leads to the appearance of critical points (µ ϑ,b , T ϑ,b ). Each critical point is located at intersection of the confinement/deconfinement open string phase transition line and the small/large black holes phase transition line of our background. The lines of the first type depend on orientation of the quark-antiquark pairs and the lines of the second type are fixed for the given anisotropy parameter ν. In other words, positions of the critical points (µ ϑ,b , T ϑ,b ) depend on the orientation of quark-antiquark pair in respect to the heavy ions collision line. Averaging on all possible orientations on the quark-antiquark pairs, one gets a family of the critical points. In our model it also happens, that the confinement/deconfinement transition line, oriented along the transversal direction, is below the small/large black holes phase transition line. This means that near the small chemical potential the small/large black hole transition line is hidden by the confinement/deconfinement transition line for the pair of quarks oriented in the transverse direction. Recall that a small/large black holes transition line near the top of the holographic phase diagram is treated as a problem, since this behavior is not supported neither by experimental data nor by calculations performed in the framework of effective theories. Let us also remind, that most of the effective models suggest the existence of a QCD critical point (µ CEP , T CEP ) somewhere in the middle of the phase diagram, where the crossover line becomes a first order transition line. There were attempts to relate (µ CEP , T CEP ) with the small/large black holes background transition [51,52], but here there is a problem with the first order phase transition in the top of the QCD phase diagram, that is not present at the QCD phase diagram [1,2]. The isotropic holographic model improving this has been proposed in [49] just by removing the small/large black holes background transition, see also [53,54]. We note, that the presence of the small/large black holes background transition endows our anisotropic model by a rich phase structure.
The paper is organized as follows. In Sect. 2 we construct the anisotropic 5dimensional solution with an arbitrary dynamical exponent ν, a nontrivial warp factor, a non-zero time component of the first Maxwell field and a non-zero longitudinal magnetic component of the second Maxwell field. In Sect. 2.3 we consider exponential warp factors with quadratic exponent and show that only negative definite quadratic form guarantees the real solutions for the dilaton. In Sect. 3 we discuss the thermodynamics of the constructed background and find out the small/large black holes transition line in the (µ, T )-plane. Sections 2.5 and 4 are devoted to applications to QCD. In Sect. 2.5 we shortly discuss the RG flows corresponding to constructed solutions. In Sect. 4.1 we find dynamical walls corresponding to the temporal Wilson loops extended in the longitudinal and transversal directions. In Sect. 4.2.5 we determine the relative position of the background and the confinement-deconfinement phase transition lines and discuss the corresponding critical points. In Appendix A we derive E.O.M. and in Appendix B we present simplest solutions, the black hole solutions for c = 0, with zero and non-zero chemical potential, and the vacuum solution for c < 0 for completeness.

Black brane anisotropic solutions 2.1 The equations of motion and boundary conditions
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.
We search the black brane solution in the anisotropic background. For this purpose we use the metric ansatz in the following form: 3) where b(z) is the warp factor and g(z) is the blackening function; we set the AdS radius L = 1 and all the quantities in formulas and figures are presented in dimensionless units. The variation of the action (2.1) over metric components g µν gives 4 independent equations, corresponding to 00-, 11-, 22-and 44-components of the Einstein tensor, that are presented in the Appendix A. These equations can be transformed to the following ones: Here and below = d/dz. The variation of the action (2.1) over the scalar field φ and the components A (1) µ of the first Maxwell field leads to the following EOM: The EOM for the second Maxwell field doesn't give any contribution into system (2.5)-(2.10) as its left-hand side is identically zero: We also impose the boundary conditions in the form: g(0) = 1 and g(z h ) = 0, (2.12) A t (0) = µ and A t (z h ) = 0, (2.13) where z h is the horizon. As to the scalar field, it is natural to require that φ(z) is real for 0 < z ≤ z h and that φ(z h ) = 0. (2.14) 2.2 Solutions with factor b(z) = exp P (z) and spatial anisotropy One can use the following strategy to find particular solutions of the system of equations (2.5)-(2.10).
• Choose the form of functions b(z) and f 1 (z).
• Using these b(z) and f 1 (z), find the time component of the electric field A t (z) from (2.10).
• Using b(z), find the derivative of the scalar field φ (z) from (2.6). To have a solution one has to be sure that • Using g(z) and b(z), get f 2 (z) from (2.7).
Let us express the warp factor b(z) via a polynomial P (z) 1 : and take the coupling factor f 1 (z): 2.3 Solutions with factor b(z) = exp(cz 2 /2) and spatial anisotropy As we are interested in effects that can be caused by the anisotropy of the chosen metric ansatz, it is needed to find some particular solution of the system (2.5)-(2.10) and investigate it's properties explicitly. For this purpose we preferred to start from the simplest form of the warp-factor, the same as in [37]: and take the factor f 1 (z): In this case the equation (2.5) becomes and together with the boundary conditions (2.35)
The blackening function used in [4] g [2] (z) = 1 − z 2+ 2 is different by the factor ρ 2 in the second coefficient in front of z 2+ 2 ν . Near the horizon this factor is approximately equal to 1: The behavior of the blackening function from the holographic coordinate z till horizon is depicted on Fig.1. The main feature is that the blackening function values decrease faster for larger chemical potential µ (Fig.1.A) and for smaller warp factor coefficient c (Fig.1.B). The difference between the approximations (2.45) and (2.46) and the exact expression (2.37) is irregular and depends on the model parameters ( Fig.1.C). In the isotropic case the blackening function values are larger than in the anisotropic ones ( Fig.1.C and D). For µ close to zero it is the desreasing function of z till the horizon, but for growing µ the local minimums and the second horizons small than the original ones appear. Changing the values of c almost does not influence on the horizon position. Using the first two terms of the expansion (2.40) on z and z h we have the following expansion of coupling factor f 2 : • c > 0 For c > 0 expression (2.56) can be parametrized as Note, that we can get a real solution only for Integrating (2.60) we obtain We see that the solution (B.21) becomes complex for α < z < β. It leads to an instability region for the scalar field.
• c = 0 For c = 0 we get

Scalar potential
From equation (2.8) we get the expression for the scalar potential V as a function of z: . (2.69) The dependence V (φ) can't be expressed explicitly due to nontrivial behavior of φ(z) (2.57), but it can be displayed graphically (Fig.6). For c = −1 it can be approximated by a sum of two exponents and a negative constant: (2.70) The best fit is given by Note, that in [57] an explicit isotropic solution for the dilaton potential as a sum of two exponents and zero chemical potential has been constructed. It would be interesting to generalize this construction to the anisotropic and non-zero chemical potential cases.
The behavior of V (φ) for positive warp factor coefficient is quite different. Let us recall that for c > 0 the scalar field φ becomes complex under horizon (Fig.7). The function V (Re(φ)) doesn't display visible dependence on chemical potential (Fig.8.A) and the function V (|φ|) stops to depend on µ rather soon ( Fig.8.B).

Scalar invariants
For completeness we present here the dependence of the scalar invariants R, R 2 = R µν R µν and K = R µνρσ R µνρσ on the parameter µ for the unit horizon and negative warp factor coefficient c. All the invariants are smooth inside the black hole and start to diverge for z > z h (Fig.9, 10). In isotropic case it happens earlier, means for smaller z, than is anisotropic one. Thus the horizons of the blackening function, depicted on

RG flow
Our background is an anisotropic analog of the background used in the improved holographic QCD model [22]. The holographic coordinate z corresponds to the 4D RG scale. According to holographic dictionary one identifies the 4D energy scale E with the metric scalar factor, i.e.
The running 't Hooft coupling λ t is identified with the string coupling λ = e φ up to a factor, λ = κλ t . In Fig.11.A we show the dependence of coupling constant on the energy parameter for isotropic and anisotpopic cases. We see that the running coupling constant decreases from the IR region to the UV region. This behavior reproduces our expectations of the running coupling view in a nonperturbative QCD. Note, that the anisotropic case does not differ much from the isotropic one. The difference becomes more essential for small z h . The β-function in terms of the background is defined as [5,58] Introducing the function X, related with the β-function as the function Y , related with the blackening function as and the function H, related with the vector field provided by non-zero chemical potential as one can check that in the isotropic case due to E.O.M. these quantities satisfy the first order differential equations where ∂ φ = ∂/∂φ. The anisotropic case is more subtle and will be the subject of a forthcoming paper. Fig.12 shows the X-flow in the anisotropic case for ν = 4.5 and different z h in the (φ, X)-and in the (λ, X)-planes. The function X(φ) decreases starting from a constant value up to a local minimum as the argument grows and is shifted to the right for larger horizon. The function X(λ) for z h < 4 smoothly decreases with increasing λ and for z h > 4 the dependence is more complicated. In Fig.13 the behavior of functions β(λ) in anisotropic, ν = 4.5, (Fig.13.A) and isotropic ( Fig.13.C) cases and the corresponding potentials V (φ) (Fig.13.B and D) for the same horizon values are shown. We see, that for both cases β(λ) < 0 in an agreement with the asymptotical freedom. For larger z h both functions display more non-linearity and decrease faster with the argument grow. This tendency is peculiar  either for the isotropic or the anisotropic case. The only special difference is the nonzero value of β(0) for ν = 1. We expect that changing the form of P (z) we can, as in the isotropic case [5,40,58], to recover the first orders expression of the perturbative β-function.
In Fig.14 we show the dependence of the β-function on H for the isotropic and anisotropic (ν = 4.5) cases. We see that β is the increasing function of H, approxi-mately linear for large negative argument values and displaying its non-linearity near zero. The function values are visibly larger for larger chemical potential, while the anisotropy does not change this picture much.
In Fig.15 the Y -flow is shown for anisotropic case with zero and non-zero chemical potential. The function grows rapidly and this growth does not essentially depend on the size of the horizon.
In Fig.16 we display the RG flows in the (X, Y )-plane for anisotropic case ν = 4.5 and zero and non-zero chemical potential. The X and Y have the inverse ratio dependence and does not change much for different z h . Fig.17 shows the RG flows in the (X, Y, H)-space. We see that our anisotropy essentially changes the character of the flow. 3 Thermodynamics of the background 3

.1 Temperature
Calculating the derivative of the blackening function (2.48) at the horizon we get the temperature Here the dependence on ν is caused by the function G (2.39) in the right-hand side of (3.1). In particular, for the zero chemical potential, µ = 0, .

(3.2)
For c = 0 it reproduces the result from [33]: From (3.1) we get the dependence of temperature on z h , µ, c and ν. In Fig.18.A and Fig.18.B we present the dependence of T on z h for different µ and fixed c = −1 for isotropic (Fig.18.A) and anisotropic ( Fig.18.B) cases, respectively. In Fig.18.C we compare the plots, presented in Fig.18.A and Fig.18.B. Plot in Fig.18.D is a zoom of Fig.18.B. In Fig.19 we present the dependence of temperature on z h for different µ = 0 and c < 0, and in Fig.20 the dependence of temperature on µ keeping z h = 1 for different c < 0 is shown. Fig.21 displays contour plots for the temperature dependence on the horizon position and chemical potential for the isotropic (A) and anisotropic (B) cases at fixed c = −1.
These plots show the following behavior of the temperature: • For µ = 0 (dashed lines) there is one extremal point (minimum) for the temperature as a function of the horizon position; we denote the corresponding horizon z h as z h,min (0) = z h,min (0, c, ν), and we get the following picture: for 0 < z h < z h,min (0) (large black holes) the temperature drops as z h grows and for z h,min (0) < z h (small black holes) the temperature increases with the growth of z h ; the minimal isotropic horizon z min (c) and the corresponding critical temperature T min (0) is higher for the isotropic case, i.e. T (z min (c), c, ν); one can read these inequalities from the plot in Fig.18.C and D; for negative c with decreasing |c| the temperature T (iso) min (the brown dashed line) is below the green one and its minimum is shifted to the right from the minimum of the green one ( Fig.19.A); T  • For 0 < µ < µ cr there are two extremal points z h,min (µ) = z h,min (µ, c, ν) and z h,max = z h,max (µ, c, ν); corresponding T min (µ) and T max (µ) are shown in Fig.18: for 0 < z < z h,min the temperature drops with the growth of z h ; for z h,min < z < z h,max the temperature increases with the growth of z h ; for z h,max < z < z h 0 the temperature decreases again with the growth of z h ; here z h 0 is the position of the new horizon, µ cr , z h,min and z h,max depend on the warp factor coefficient c and the anisotropic parameter ν; the anisotropy increases the size of the new horizon, z h 0 (µ, c, 1) < z h 0 (µ, c, ν).
• For µ = µ cr (dotted lines in Fig.18) there is no extremal point, but there is an inflection point (z h,cr , T cr ), z h,cr = z h,cr (µ cr ), T cr = T cr (µ cr ), therefore for all values of z h with its growth the temperature decreases; the temperature becomes equal to zero at a new horizon.
• For µ > µ cr increasing z h we decrease the temperature and there is a point

a new horizon appears;
for negative c with the growth of |c| the values of for |c 1 | > |c 2 | and all ν ≥ 1 (Fig.19).   We can also investigate the behavior of T (µ) using expression (3.1) and taking some fixed values of the horizon. In Fig.20 we plot the curves for different negative values of the warp factor coefficient c in isotropic (A) and anisotropic (B) cases for z h = 1. In Fig.20.C we compare these cases plotting them together. Fig.20.D and E display T (µ) for large black holes with z h = 0.5 and small black holes with z h = 1.5. The function T (µ) decreases faster for smaller c. The isotropic case curves lie higher than the anisotropic ones and reach zero temperature at larger chemical potential values (Fig.20.C). For smaller horizons we have the same picture (Fig.20.D and E).
To summarize, note that the Van der Waals type of the temperature-horizon dependence T (z h ) observed in [45,46] for isotropic case, also takes place in the anisotropic one (see Fig.21). In both cases this behavior becomes more pronounced with decreasing of negative c, see Fig.19. The approximate solution considered in [35] does not inherit this property.

Entropy
The entropy is given by formula and is plotted in Fig.22. Fig.22.A shows that the entropy is a monotonously decreasing function of the horizon z h both for the isotropic and the anisotropic cases, in other words the entropy values are bigger for larger black holes, whose horizons are smaller. As we see from Fig.22.B, the velocity of the entropy decreasing depends on parameters c and ν. It is interesting to note that absolute value of this velocity for the same c is bigger in the isotropic case for large black holes and is smaller for small black holes. More precisely, for c = −1 and for c = −0.5 Here s z h (z h , c, ν) = ∂s(z h , c, ν)/∂z h . In Fig.23 we present the entropy dependence on temperature T for µ = 0 and different c for isotropic (green lines) and anisotropic (blue lines) cases. The plots in    23 show that in both cases there are minimal temperatures for which the black holes exist. The minimal temperature in the isotropic case is higher then in the anisotropic one for the same value of c < 0, T min (c, 4.5) < T min (c, 1), that agrees with the plots in Fig.18. In both cases the entropy is a double-valued function and has a large black holes branches and a small black holes one. For the small black holes the entropy increases with decreasing T thus leading to the negativity of the specific heat c v = T ds/dT . Therefore small black holes are thermodynamically unstable, whereas entropy of the large black holes grows while temperature increases and therefore large black holes are thermodynamically stable.
In Fig.24 we present the entropy dependence on temperature T for c = −1 and different µ ≥ 0 for isotropic (green lines) and anisotropic (blue lines) cases. The plots in Fig.24 show that in both cases for fixed µ, 0 ≤ µ ≤ µ cr (c, ν), there are minimal T min (µ, ν) and maximal T max (µ, ν) temperatures, between which the entropy is a multivalued function of T with three branches. We see well only two braches in Fig.24.A, to see the third one has to draw the picture Fig.24.B for small values of s(T ). The schematic picture of three branches is presented in Fig.24.C. When we decrease the temperature, the entropy decreases along the first branch (T min (µ, ν) < T < ∞). Then the entropy decreases along the second branch with an increase of temperature from T min (µ, ν) to T max (µ, ν), i.e. here the black holes are unstable. Finally the entropy increases along the third branch with an increase of temperature for 0 ≤ T < T max (µ, ν), see also at µ = µ cr (−1, ν) (dotted green and blue lines for the isotropic and anisotropic cases). The entropy dependence on the temperature at unified branches corresponding to µ ≥ µ cr (c, ν) is presented in plots

Free energy
To study transitions between different branches in more detail it is reasonable to consider the free energy behavior of the corresponding solutions. The free energy for a given chemical potential and fixed volume is related to the entropy as dF = −s dT (3.9) and can be found by integration of (3.9) that gives The dependence of the free energy on the horizon position z h is presented in Fig.26, the dependence on T is presented in Fig.27 and Fig.28. In Fig.26.A, which corresponds to µ = 0, we can see that the free energy as the function of z h is equal to zero at z h = z h,HP (0, c, ν), and at this point the Hawking-Page phase transition takes place. The value z h,HP (0, c, ν) depends on c and ν, and z h,HP (0, c 1 , ν) < z h,HP (0, 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. In particular,  As we can see from the plots in Fig.26.B, for 0 < µ < µ cr,HP (c, ν) the free energy as the function of the horizon position keeps the same behavior as for µ = 0. At µ = µ cr,HP (c, ν) the free energy becomes non-positive and for µ > µ cr,HP (c, ν) the Hawking-Page horizon disappears. But for the chemical potential values in the interval µ cr,HP (c, ν) < µ < µ cr (c, ν), the free energy still is double-valued what causes the black hole to black hole phase transition (see below).
In Fig.27 and Fig.28 we show the behavior of the free energy as function of the temperature. At µ = 0, as we can see in From Fig.26 we see that for zero chemical potential the free energy increases with z h growth for large black holes, i.e. for z h < z hcr (c, ν), and decreases for small black holes.
For 0 < µ < µ cr (c, ν) the dependence of the free energy from the temperature looks like the swallow-tailed shape both in isotropic and anisotropic cases. When we decrease the temperature from very large values up to T min (µ) (T min (µ) = T min (µ, c, ν), see Fig.18), the free energy riches its maximum value, then goes down to its local minimum at T max (c, ν) and turns back to increase. It intersects itself at T = T BB (µ), where a large black hole transits to a small one. Since both free energy values at T = T BB are equal and negative, meanwhile the free energy of the thermal gas is zero, the system undergoes the phase transition not to a thermal gaz, but to small black hole background. When we increase the chemical potential µ from zero to µ cr , the loop of the swallow-tailed shape shrinks to disappear at µ = µ cr (c, ν). For µ > µ cr (c, ν), the curve of the free energy increases smoothly from higher to lower values of temperature. It is interesting to compare the phase diagrams corresponding to isotropic and anisotropic backgrounds, see Fig.29. We see that the first order phase transitions start at (0, T HP (0, 1)) and (0, T HP (0, 4.5)), so that T HP (0, 4.5) < T HP (0, 1) and the transition lines describing transitions from large black holes to small ones stop at points (µ It is also important for us to know the position of the large black holes to small black holes transition points at (z h , T )-plane, see Fig.30. In these plots the horizontal arrows show transitions from the large black holes to small black holes for the anisotropic ν = 4.5 and isotropic cases. The shaded by these arrow areas define the instability zones. Fig.31 summarizes our discussion of the phase transitions of our black hole isotropic and anisotropic backgrounds. In the next section we put probe strings in these backgrounds to find out information about the confinement/deconfinement phase transition.  4 Confinement-deconfinement phase transition 4

.1 Equation for the dynamical wall
To guarantee the confinement-deconfinement phase transition one has to check the existence of the dynamical wall (DW). The dynamical wall position z DW is defined by the minimal extremal point of the effective potential, that depends on the orientation [35] and is related to the warp factor power P (z) and the scalar field φ: Here the subscribte indexes show the orientation of the Wilson loop. Therefore the dynamical wall position is given by equations: For zero temperature we have Substituting φ (z) from (2.6) we get the following equations for the positions of the dynamical wall corresponding to x-and y-directions of the quark orientations: We take the simplest case (2.31) again. Therefore we choose the expression with positive sign in (2.56), and equations (4.3) and (4.4) become: Note that in both equations there is also a dependence of the corresponding blackening functions on ν. The phase transition from confinement to deconfinement occurs when the corresponding equations loss solutions. To see the dependence of the dynamical wall position on the parameters of the metric, it is useful to study the details of dependence of the different terms defined DW x , DW y and DW iso on these parameters. Here DW iso denotes the left-hand side of the equation, similar to (4.8), in the isotropic case.
To show that the presence of the dilaton field supports the appearance of the DW, we display parts of the expressions σ x (z, c, ν) and σ y (z, c, ν) without the square roots, that are originated from the dilaton fields, by the dashed lines in Fig.32.B and C. We see that these dashed lines never intersect horizontal axis, therefore in these cases there are no DW solutions.
Solutions to equations (4.9) and (4.10) can also be represented as the boundary between positive and negative values of functions σ x (z, c, ν) and σ y (z, c, ν). Taking ν = 1 provides us with the isotropic case result (Fig.32.D). We see that the critical c = c cr , above which there are no solutions in all cases, is c cr = 0. For c > 0 our consideration is not valued, since the scalar field becomes complex.

Non-zero temperature, zero chemical potential
Let us take the nontrivial blackening function. The blackening function modifies the DW equations. It is convenient to present these equations in the form where Σ(z, z h , c, ν) ≡ σ(z, c, ν) + G(z, z h , c, ν). We find the solutions of equations (4.14) and (4.15) numerically. To visualize the location of these solutions we plot Σ(z, z h , c, ν) as a function of z for different values of parameters z h , ν, c and find its intersection with 2/z for the Wilson loop W x and with (ν + 1)/(νz) for the Wilson loop W y in Fig.33. If there are two intersection points, we take the minimal one and we call it the minimal intersection point. In all cases to get the corresponding DW position we take the minimal intersection point. From Fig.33.A we see that for z h > z h,cr the dynamical wall always appears, as there are intersections of the grey and dark green lines. At the critical horizon (the thick dark green line) there is a touch of these two lines and for z < z h,cr (lighter green lines) there is no intersection at all, therefore confinement disappears.
The light blue curves in Fig.33.B do not cross the grey line and for these cases there are no dynamical dynamical walls. The dark blue lines cross the brown one and for the corresponding temperature there is the quark confinement, meanwhile the thick dark blue line just touches the grey line and at this temperature the phase transition occurs. The similar picture can be seen at Fig.33.C, corresponding to different orientation of quark pairs. Therefore the dynamical wall always appears for z h > z h,cr . The particular values of z h,cr are different for isotropic (A) and anisotropic (B, C) cases and depend on the quark orientations. This appearance/disappearance of dynamical walls corresponds to confinement and deconfinement phases. The phase transition between these two regimes occurs at z = z h,cr . These plots show that all dynamical walls appear below the corresponding horizons, i.e. z DW (z h , ν) < z h and these horizons z h are larger than z h,min (0) and z h,cr for the isotropic case and for W x case, but they can be smaller than z h,min (0) and z h,cr in the anisotropic W y case. The positions of minimal z h admitted the DW, z h,DW are indicated by dark green, dark blue and dark magenta lines.
In Fig.34 solutions to equations (4.14), (4.15) are located on the boundary of the colored and white areas. Since to find the dynamical walls' positions we have to take the minimal solutions, the dynamical walls' positions are located on the left parts of boundaries between the colored and white areas.

Non-zero chemical potential
We can also study how these plots look for non-zero chemical potential. The positions of the dynamical walls for non-zero temperature and non-zero chemical potential in isotropic and anisotropic cases are presented in Fig.35.A and Fig.35.B correspondingly. It is convenient to write equations (4.9) and (4.10) in the form where Σ(z, z h , µ, c, ν) ≡ σ(z, c, ν) + G(z, z h , µ, c, ν), (4.20) G(z, z h , µ, c, ν) ≡ g 2g (4.21) and σ(z, c, ν) is defined by (4.11). The light blue and dark blue curves in Fig.35.A represent the function G(z, z h , µ, c, ν) for ν = 4.5, c = −1, z h = 1.5 and different µ. The light magenta and dark magenta lines correspond to G(z, z h , µ, c, ν) for the same set of parameters and z h = 3. The thick lines, that touch σ x (grey line) and σ y (brown line), depict the critical values of chemical potential µ. Thus the presence and the particular position of the horizon modifies the position of dynamical walls as compare with the zero temperature case (g = 1) presented in Fig.32. Anisotropy also influences on the dynamical walls' position. This can be seen from comparing of Fig.35.A and Fig.35.B, where the isotropic case is pictured.
To find the phase transition line we have to determine µ(z h , c, ν, W i ), here W i indicates the orientation of the Wilson line, W x or W y , for given z h such that for any µ > µ(z h , c, ν, W i ) there is no real solution of equations (4.18) and (4.19). To find these points, it is convenient to draw the contour plots for functions DW x and DW y near zero. They are presented in Fig.36 and Fig.37 correspondingly. For comparison we present in Fig.38 the contour plots for DW iso near 0.

The dynamical wall position
All previous considerations can be summarized in contour plots. Namely, we can draw the contours for the locations of the effective potentials' derivatives V xy = 0 in the (z, z h )-plane, keeping c = −1 and considering two cases ν = 1 (Fig.39) and ν = 4.5

Phase transition lines and critical points
Confinement/deconfinement phase transition for the isotropic case ν = 1 is shown on Fig.42. Note that for zero chemical potential the Hawking-Page temperature is less than the temperature of the confinement/deconfinement transition temperatute, T HP (0) < T CD (0). The temperature of the black hole to black hole transition T BB (µ) is less than the temperature of the confinement/deconfinement transition T CD (µ) for 0 < µ < µ the background transition line stops at (µ (iso) cr , T (iso) cr )). Therefore, the phase transition line for 0 < µ < µ is determined by the background transition line (the orange lines at Fig.42) and for µ ≥ µ by the isotropic confinement/deconfinement transition line (the green lines at Fig.42). This is in agreement with results of the previous studies [46] and refs. therein, where it has been argued that the transition for µ < µ is the first-order phase transition (FOPT) and for µ > µ    W x and W y for ν = 4.5. In this plot we see that for µ = 0 the minimal value of the horizon z h,DW x , for which the DW can appears for W x , corresponds to small black hole, meanwhile the same horizon for W y , z h,DW y , corresponds to large black hole. By this reason we have T DW y (0) < T HP (0) < T DW x (0).  Figure 44. Confinement-deconfinement phase transitions of Wilson lines W iso for ν = 1 (green), W x and W y for ν = 2 (blue and magenta) and ν = 4.5 (dark blue and dark magenta); the background transition lines for ν = 1 (orange), ν = 2 (cyan) and ν = 4.5 (dark cyan).

Explicit numerical calculations show that
T CDy (µ) < T BB (µ) < T CDx (µ) for 0 < µ < µ xb . (4.23) The confinement/deconfinement transition for the anisotropic case ν = 4.5 is shown on Fig.43.A and B. The phase diagram for the longitudinal Wilson line W x is depicted by the blue lines, for transversal lines W y by the magenta lines and for the anisotropic background by the cyan lines. Wilson lines can also have arbitrary orientations, that corresponds to a modification of blue and magenta lines to some intermediate configuration.
The case ν = 2 can be considered as intermediate between the isotropic one and our main case ν = 4.5 (Fig.43.C and D). Fig.44 combines all the three cases ν = 1, 2, 4.5 thus showing isotropization.

Conclusion and discussion
We have considered 5-dimensional Einstein-dilaton-two-Maxwell-scalar system. We have found anisotropic solutions for this system by using the potential reconstruction method, i.e. choosing the corresponding dilaton and Maxwell potentials for the given background. This method has been used for isotropic cases in [45,46,49] and refs therein.
Our anisotropic background is the deformed AdS 5 that has the UV boundary with two suppressed transversal coordinates and the IR boundary with the suppressed time and longitudinal coordinates. One can say that in this background two different 3dimensional reductions, one in the UV domain and the other in the IR domain, are realized. The UV reduction is realized by the suppression of the original transversal coordinates, meanwhile the IR reduction is obtained by the suppression of the longitudinal and temporal coordinates. For the corresponding isotropic solution there is no 3-dim reduction neither in UV nor in IR regions.
In our calculations the warped factor is chosen in such a way that the explicit analytical calculations can be performed. This solution can be generalized to provide a more realistic model. In this case, similar to the isotropic case, the solution can be given only in terms of quadratures. We solved the equations of motion to obtain a family of the black hole solutions by modifying the initial potential corresponding to zero temperature. In this construction the special boundary conditions for the dilaton field are chosen, namely we have required that the dilaton field is zero at the horizon.
We have also studied the thermodynamical properties of the constructed black hole background and found the large/small black hole phase transitions at the temperature magenta T BB (µ). This result is presented in Fig.29. At µ = 0 and for T < T HP (0), the black hole dissolves to thermal gas which is thermodynamically stable for T < T HP (0). When the system cools down with the chemical potential less than the 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 [45,46,[50][51][52]. We have found that T (aniso) BB (µ) < T (iso) BB (µ) and the value of the critical chemical potential, value up to which this phase transition exists, is bigger in the anisotropic case as the compare to the isotropic one, µ cr , see Fig.44. Also, we have found that the point (µ The U-shape for large distances between quarks provides the quark confinement and is realized in the presence of the dynamical wall (DW). We have found the domains in the (z h , µ) planes, where the DW can appear for the longitudinal and transversal orientation of the temporal Wilson loops. In these regions the open strings cannot exceed the DW, even the separation of the quark and antiquark goes to infinite and the quark confinement takes place. We have found that the phase diagram depends on the orientation, cf. [35]. Taking into account the instability zones of the anisotropic background, we have found more complicated confinement/deconfinement phase diagrams for different oriented temporal Wilson loops and the details are the following: • In the case of the longitudinal orientation, W T x , parts of regions near zero values of the chemical potential, µ < µ xb , enter to the instability regions of our background, where the small black holes collapse to large ones. Here the horizon suddenly blows up to pass the critical value z h,DW x (µ) (see Fig.18 and Fig.40, at the last plots these jumps are indicated by the arrows), so that the confinement phase transforms to the deconfinement one by a phase transition. While the chemical potential is greater than the critical value µ > µ xb , the black hole horizon grows gradually and continuously passes the critical horizon, corresponding to (µ cr , T cr ), so that the confinement phase transforms to the deconfinement phase smoothly. In other words, the confinement-deconfinement line is determined by the probe string behavior itself. It is worth to notice that the similar situation takes place in the isotropic case.
• In the case of the transversal orientation, W y , situation is more interesting. It happens, that the background phase transition line for small µ, µ < µ yb , is located above the phase transition line for the Wilson line, and for small µ we have a smooth confinement-deconfinement phase transition. For µ yb < µ < µ cr we fall in the zone of instability of the background and the first order phase transition takes place.
As to the future investigations, the following natural questions to static and nonstatic properties of our model are worth noting. As to static properties, it is natural to • investigate the opportunity to fix our holographic QCD model by a suitable choice of the function P (z) in (2.16), so that in the isotropic limit it would fit the Cornell potential known by lattice QCD; it would be interesting to perform calculations on an anisotropic lattice and compare these future results with our model; • consider more general anisotropic backgrounds and derive the corresponding anisotropic RG flows; • study the Regge spectrum for mesons, adding the probe gauge fields to the backgrounds; we expect that similarly to the isotropic case, the gauge potential can be fixed requiring the linear Regge spectrum for mesons; • consider estimations for direct photons; • evaluate transport coefficients and their dependence on the anisotropy.
As to the thermalization processes, which are the main motivations of our considered of the anisotropic background, we suppose to reexamine • the shock wave collisions in the anisotropic background with the warped factor; • thermalization times for 2-point correlators; for the no-dilaton case this question has been addressed in [31,34]; • time dependence of the transport coefficients, for the no-dilaton model see [35].
In this paper we have studied a particular anisotropic model specified by the anisotropy parameter ν and in all plots we take ν = 4.5, since just this case reproduces the total multiplicity dependence on energy, M ∼ s 0.155 . It would be interesting to find isotropization of our solution and it is natural to expect that in this case the both phase transition lines, the large/small black hole transition and the string confinement/deconfinement transition, will smoothly move to their isotropic parters. We leave these matters to future works.
In a similar way we get the expression for f 2 without g and b . Indeed, we differentiate (A.3) and obtain