Spontaneous scalarization of dyonic black hole in Einstein–Maxwell-scalar theory

In this paper, we study the scalarization of the static and spherically symmetric dyonic Reissner–Nordstrom (RN) black holes in the Einstein–Maxwell-scalar theory where the scalar field is coupled to an electromagnetic Chern–Simons term. When both electric and magnetic charges are present, there exists an unstable region of parametric space for the dyonic RN black holes where the scalarization of black holes should occur. That is to say, mixing electric and magnetic charges can reduce the scalarization in this theory. Firstly, we calculate the perturbation field equations under the dyonic RN black hole background and obtain the corresponding asymptotic-flat perturbation solutions, which are the bifurcation points at the dyonic RN branch. The results show that the perturbation scalarization demands a lower bound of the coupling constant. Then, we calculate the scalarized black hole solutions bifurcating from the dyonic RN solutions. We find that there exist a lot of discrete branches of the scalarized solutions. Contract to the dyonic RN solutions, these scalarized solutions can be overcharged and their mass could even approach zero. After illustrating the behavior of the entropy for the scalarized black holes, we demonstrate that the scalarized configurations might be thermodynamically more stable than GR configurations. Moreover, we also show that for each scalarized branch, the black hole cannot reach the extremal limit with vanishing temperature.


Introduction
The no-hair theorem is very important for understanding the properties of black holes. It states that a black hole is uniquely a e-mail: jiejiang@mail.bnu.edu.cn b e-mail: jiatansust@163.com (corresponding author) determined only by three parameters: mass, electric charge, and angular momentum [1][2][3]. Although the no-hair theory has been proven in Einstein-Maxwell theory, various counterexamples can be found when the gravitational theory possesses an extra field degree of freedom, such as Einstein-Yang-Mills theory [4][5][6], Dilatonic black holes [7], and a black hole with Skyrme hairs [8][9][10]. These hairy black hole solutions prompt people to further study the no-hair theorem. The recent developments in the gravitational wave detection [11] and black hole shadow [12] provide us a chance to detect the features of the black hole and then test alternative theories. A key difference between General Relativity (GR) and the alternative theory of gravity is the phenomena of scalarization (a counterexample of the no-hair theorem).
The phenomena of spontaneous scalarization were first investigated for neutron stars in the scalar-tensor theory where the scalar field non-minimally couples to the Ricci curvature [13]. It was illustrated that there is a certain region in parametric space where the scalar-free and scalarized solutions coexist and the scalarized one is energetically favored. After that, the spontaneous scalarization in scalar-tensor theories was extended to the black hole cases [14,15]. Recently, the spontaneous scalarization was widely studied in extended scalar-tensor-Gauss-Bonnet gravity, in which the scalar field is non-minimally coupled to a Gauss-Bonnet gravity correction term [16][17][18][19][20][21][22][23][24][25][26][27]. The dynamics of spontaneous black hole scalarization in Einstein-Scalar-Gauss-Bonnet gravity is also investigated in Ref. [28][29][30]. Moreover, this mechanism is also studied in Einstein-Chern-Simons-Scalar theory with a scalar field non-minimally coupled to a Chern-Simons gravity correction term [31][32][33][34][35]. In these models, spontaneous scalarization is induced by the geometry of spacetime. Alternatively, this mechanism can also be found in the Einstein-Maxwell-scalar models where the scalar field is non-minimally coupled to some Maxwell invariant terms.
For the Einstein-Maxwell-scalar theories mentioned above, the Maxwell invariant term is chosen as or their function. Apart from these Maxwell invariant quantities, the electromagnetic Chern-Simons term I CS = abcd F ab F cd is also a candidate as a Maxwell invariant coupled to the scalar (pseudoscalar) field and this term will violate the parity invariance. Usually the pseudoscalar field is also called axion and it is widely applied in the cosmology and astronomy to study the dark matter [62][63][64][65][66][67]. When the coupling function is a constant, this term is just a pure topological term and does not contribute to the equation of motion. Moreover, different from F ab F ab , the electromagnetic Chern-Simons term is nonvanishing only for the case where both electric and magnetic charges are present. For nonvanishing electric and magnetic charges, I CS shows a similar feature as F ab F ab . Moreover, there are a lot of similarities between this theory and the geometric Einstein-Chern-Simons-Scalar theory, which can also induce the spontaneous scalarization and violates the parity invariance. That is to say, this flavor of Einstein-Maxwell-scalar theory can help us build intuition about the scalarization in Chern-Simons gravity in a much easier way because of its relatively simple field equations. Then, it is natural to ask whether spontaneous scalarization can occur in this model. Therefore, in this paper, we would like to consider the dyonic Reissner-Nordstrom (RN) black holes in the Einstein-Maxwell-scalar theory where the scalar field is coupled to an electromagnetic Chern-Simons term and investigate whether the mixing of electric and magnetic charges can reduce the spontaneous scalarization.
The outline of this paper is as follows. In Sect. 2, we briefly introduce the Einstein-Maxwell-scalar theory in which the scalar field couples to an electromagnetic Chern-Simons term and obtain the equations of motion in the static and spherically symmetric configurations. In Sect. 3, we study the stability of the dyonic RN black hole under the perturbations and evaluate the perturbation scalar field satisfied the asymptotic flatness condition, i.e., the bifurcation points of the dyonic RN solution and scalarized black hole solution. In Sect. 4, based on the shooting method, we calculate the scalarized dyonic RN black hole solutions numerically and discuss their properties. Finally, the conclusion and discussion are presented in Sect. 5.

Einstein-Maxwell-scalar model with an electromagnetic Chern-Simons term
In this paper, we would like to consider the Einstein-Maxwell-scalar theory where the scalar or pseudoscalar field is coupled to an electromagnetic Chern-Simons term. Noting that the Chern-Simons term violates the parity invariance, i.e., it is a pseudoscalar, then the coupling function F (ϕ) should also be a pseudoscalar to ensure the theory is parity invariant. With this in mind, most of the literature only considers the linear coupling between the pseudoscalar field and the electromagnetic Chern-Simons term [62][63][64][65][66][67]. Motivated by the phenomenon of spontaneous scalarization widely discussed in the quadratic Einstein-scalar-Gauss-Bonnet gravity [49][50][51][52][53], we want to consider the quadratic coupling function such that the theory admits both the dyonic Reissner-Nordstrom black hole and scalarized black hole solutions.
In order to ensure that the theory is parity invariant and also has a quadratic coupling function, we introduce a scalar and a pseudoscalar field into the theory, in which the action is given by with the coupling function and the Chern-Simons term of the electromagnetic field where λ is a conpling constant, R is the Ricci scalar of the spacetime metric g ab , ϕ 1 is a scalar field, ϕ 2 is a pseudoscalar field, F ab = ∇ a A b −∇ b A a with the vector potential A a is the electromagnetic strength. In this theory, the parity invariance is restored duce to the pseudoscalar field. For convenience, we define Then, the coupling function becomes By variation of the action, the equations of motion can be obtained and it is given by From the equation of motion, we can see that the Chern-Simons term does not affect the gravitational part of the equations of motion, i.e., it does not contribute to the stress-energy tensor of the scalar field and electromagnetic field. However, it can influence the equation of motion of the scalar field, and the Chern-Simons term can be regarded as the effective potential of the scalar field, which makes the scalarization possible in this theory.
In this paper, we focus on the static and spherically symmetric dyonic black hole solution in which the scalar field and electromagnetic field have the same symmetries. We adopt the following ansatz for the line element For the dyonic solution, considering the symmetries of the electromagnetic field and together with the Maxwell equation in Eq. (7), the electromagnetic strength is obtained and it is given by where are the electric charge and magnetic charge, separately. Using the above results, the Chern-Simons term becomes which means the purely electric solution will reduce to the Einstein-Maxwell-scalar theory without the Chern-Simons corrections and the no-hear theorem will be valid. Therefore, we only consider the dyonic solution with the magnetic charge in this paper. From the field equations for the scalar fields, we can see that the signs of the effective potential for ϕ andφ are opposite. That is to say, one of them has to be stable under the dyonic Reissner-Nordstrom black hole background, and only one scalar field can lead to spontaneous quantization. In this paper, we only consider the scalarized solution induced by ϕ and setφ = 0 (The opposite case can be obtained by changing the sign of Q e Q m ). Then, using the line element (8), the equations of motion reduce to When the scalar field is vanishing, i.e., ϕ = 0, this theory admits the dyonic Reissner-Nordstrom black hole solution in which in which M is the mass of the black hole. The event horizon exists if M 2 ≥ Q 2 e + Q 2 m and the radius of horizon is given by r = r h with

Stability and perturbation of dyonic Reissner-Nordstrom black hole
The scalarized black hole solution may appear in the regions of the parameteric space in which the corresponding solution becomes unstable. Next, we consider the perturbations of the scalar field, electromagnetic field, and metric for the dyonic Reissner-Nordstrom black hole. From the equations of motion (7), it is not hard to see that the scalar field perturbation δϕ is decoupled to the perturbations δg ab and δ A a of the metric and electromagnetic field, which means that the stability is only determined by the scalar field perturbation. The perturbation equation of the scalar field is given by in which the covariant derivative ∇ a and the Chern-Simons term I CS are evaluated in the dyonic Reissner-Nordstrom geometry (13). Since the background geometry considered is a static and spherically symmetric black hole, we can perform a spherical harmonics decomposition of the scalar field in which t ≡ v − r * with the tortoise coordinate r * (r ) = f (r ) −1 dr, and ω is the frequency of the quasinormal modes (QNM). Then, the perturbation equation becomes with the effective potential in which Q tot denotes the total charge defined by With some physical limitations [69], we require that there is only the ingoing wave near the event horizon and only the outgoing wave at asymptotic infinity, i.e., From the above setups, the unstable QNM frequencies should have negative imaginary part, which means that the boundary conditions for the unstable modes are vanishing at the event horizon and asymptotic infinity. It has been showed in Ref. [68] that if the potential satisfies the condition there will exist an unstable mode for the perturbation scalar field. Since we only consider the spherically symmetric solution, we set l = 0. As mentioned above, the scalarization can only appear for the case with a nonvanishing magnetic charge, also with the nonvanishing total charge. Therefore, in the following, we normalize the electric, magnetic charges and mass to the total charge Q tot , i.e.,Q e = Q e /Q tot , Q m = Q m /Q tot andM = M/Q tot . From the perturbation field equation (15), we can see that the solution is only dependent on the reduced massM as well as the quantityQ eQm λ 2 .
To simplify, we introduce the notation Using the identity Q 2 tot = Q 2 e + Q 2 m ≥ 2Q e Q m , we have 2Q eQm ≤ 1 and therefore λ 2 ≥ ζ . With these setups, the unstability condition (21) gives Note that the right-hand side of this inequality is always positive and the minimum value is one. Then, a necessary condition for the validity of the above inequality is which also givesQ eQm ≥ 0, i.e., the electric charge and magnetic charge have the same signature. Without loss of generality, we set Q e > 0 and Q m > 0. If the inequality (23) is satisfied, the dyonic Reissner-Nordstrom black hole will become unstable. In Fig. 1, we showed the unstable regions of the parametric space. This figure shows that asQ eQm or λ becomes larger and larger, the perturbation scalar field becomes more and more unstable. The coupling parameter λ presents the interaction strength between the scalar field and the electromagnetic field, and the quantityQ eQm gives how large the Chern-Simons term is. It means that stronger interaction from the electromagnetic field can destabilize the perturbation scalar field. Next, we would like to solve the static and spherically symmetric solution of the perturbation equation (15) by demanding that the scalar field is regular on the horizon and vanishing at infinity, which comes from the requirement for asymptotic flatness at infinity. These perturbation solutions are the bifurcation points of dyonic RN solution and scalarized black hole solution, which also represents the onset of scalarization and instability. It is worth mentioning that this boundary condition is not inconsistent with the boundary condition (20) for the unstable mode since the QNMs considered here are the onset of the instability and their frequencies should have vanishing negative imaginary parts.
For the static and spherical scalar field, the field equation (15) becomes in which δϕ is only a function of r and f (r ) is the blackening factor given by Eq. (13). For this field equation, one can find an exact solution where Herer = r/Q tot is the reduced radius coordinate, P y (x) and Q y (x) are the Legendre function of the first and second kinds separately. Considering the regularity of the scalar field on the horizon, i.e., the perturbation scalar field is finite at r = r h , we can fix the undetermined coefficients and the solution is given by in which we can simply set δϕ(r h ) = 1 since the field equation is linear. Then, getting the bifurcation points from a dyonic RN black hole is equivalent to finding the appropriate parameters such that its linear solution satisfies the asymptotic flatness condition δϕ| r →∞ = 0. That is to say, δϕ| r →∞ should be regarded as a function of the parametric space and we need to study the zeros of this function. In This figure also shows that for a given ζ , the first bifurcations give the largest reduced massM. In the bottom two panels of Fig. 2, we illustrate the behaviors near the extremal black holes (i.e.,M 1). From these figures, one can note that as M → 1, each bifurcation gets closer and closer to each other and eventually approaches 1/4. That is to say, the existence of the bifurcation requires ζ above a minimal value 1/4, i.e., ζ ≥ 1/4, which also implies that λ 2 ≥ 1/4. This shows a lower bound than the previous one (24) from the stability analysis, which is also showed. However, it is reasonable since the performed stability analysis provides only a sufficient condition for instability and does not mean that the scalarization cannot occur outside this condition, see Fig. 3, which also implies that the actual unstable region should be larger than that given by Eq. (21).

Scalarization of dyonic RN black hole
In this section, we would like to evaluate the scalarized black hole solutions bifurcating from the dyonic RN solution of the field equations (12) using a shooting method. These black hole solutions are required to be regular on the black hole horizon and satisfy the asymptotic flatness condition.
with α ∞ = α| r →∞ . This condition implies that the asymptotic line element can be expressed as ds 2 | r →∞ −du 2 + 2dudr + r 2 (dθ 2 + sin 2 θ dφ 2 ) (30) with u = α ∞ v, which describes a flat geometry at infinity. Then, we can expand these functions to infinity, i.e., at r → ∞. Here M and D are the mass of the black hole and the charge of the scalar field individually. Considering the regularity condition and together with f (r h ) = 0, we can expand the metric function and scalar field at r = r h , in which we denote ϕ h = ϕ(r h ) and α h = α(r h ). Using the equation of motion (12), we can further obtain For a black hole solution, we need f (r h ) ≥ 0, which leads to The existence of a scalarized black hole solution demands that the right-hand side of the above inequality is positive, i.e., Without loss of generality, we can set α h = 1 by changing the coordinate v → v/α(r h ). For the fixed coupling constant λ, electric charge Q e and magnetic charge Q m , the above discussion implies that all the series coefficients for the function f (r ), ϕ(r ) and ϕ(r ) can be calculated in terms of ϕ h and r h . Then, we can use the series expansions to evaluate values of the functions near the event horizon and perform them as initial values to solve the field equations. Same to the perturbation cases, getting the scalarized dyonic RN black hole reduces to finding the appropriate initial values ϕ h and r h such that its solution satisfies the asymptotic flatness conditions (29). From the field equations (12), we can see that the first condition of Eq. (29) is automatically satisfied at infinity. Therefore, we only need to find the initial parameters which meet the condition ϕ| ∞ = ϕ| r →∞ = 0. In Fig. 4, we illustrate the asymptotic value ϕ ∞ of the scalar field as a function of horizon radius r h for fixed λ = 3,Q m = 0.5 and for fixed λ = 6,Q m = 0.5 with different values ϕ h = 0.03, 0.06, 0.09. These figures show that there exist some discrete zero-points of ϕ ∞ which give the scalarized dyonic RN solutions and these discrete solutions can give the discrete branches of scalarized solution. Then, we present the metric functions f (r ), α(r ) and scalar field ϕ(r ) as a function of r/r h with some fixed λ,Q m and ϕ h for the first three solutions of dyonic RN black holes in Fig. 5, where we rank these solutions by the distance between their horizon radius and the extremal radius which leads to f (r ext h ) = 0. From these figures, we can note that there does not exist a zero-point for the scalar field After giving λ andQ m , a fixed ϕ h can determine the discrete scalarized solutions with some r h . By varying ϕ h , we can obtain the discrete branches of scalarized solution in the parametric space (ϕ h , r h ). For a given scalarized solution, using Eq. (31), we can obtain its corresponding mass M and scalar charge D. In Fig.6, we show the behavior of the reduced scalar chargeD = D/Q tot as a function of the reduced black hole massM with series values of the coupling constant λ and reduced magnetic chargeQ m for the first three branches of scalarized dyonic RN black hole. Wheñ D andφ h approach zero, the solutions (bifurcations) can be described by the results from perturbation analysis. In Table  1, we show the difference between the first three bifurcations calculated by solving the nonlinear equations and the per-turbation equation respectively. As a result, we can see that R =M 1 /M 2 = 1 + O(φ h ) for smallφ h , in whichM 1 is the reduced mass for smallφ h from the nonlinear calculations andM 2 is the reduced mass from the perturbation analysis. That is to say, all the branches of the scalarized solution bifurcate with the GR branch of the dyonic RN black hole and the bifurcation points can be determined by the perturbation scalarized solutions given by Sect. 3. For the GR branch, the existence of the black hole demands thatM ≥ 1, i.e., M 2 ≥ Q 2 tot . However, the reduced massM of the scalarized branches can be less than one or even approach zero, i.e., the scalarized dyonic black hole can be overcharged. This means that the scalarized dyonic RN solution enlarges the range of mass in GR. One can also note from Fig. 6 that the absolute value of the black hole scalar charge smoothly decreases with the mass and the maximal value is reached when the black hole mass approaches zero. Moreover, these figures also illustrate that the absolute value of scalar charge increases as λ or Q e Q m increase when we fixed the black  11), we can see that the larger λ and Q e Q m indicate stronger interaction between the electromagnetic field and scalar field. That is to say, the stronger interaction can lead to larger scalar hair and also enlarge the parametric region of the scalarized solution.
With a similar consideration as Ref. [18], to have an indicator for the stability, it is illustrated in Fig. 7 that the behavior of the reduced black hole entropyÃ h /4 = πr 2 h of the first three scalarized branches as a function of its reduced massM for some given λ andQ m . We see that all scalarized branches start from a bifurcation point at the GR branch and the entropy for all scalarized solutions exceeds the counterpart of dyonic RN solution with the same reduced mass M. This indicates that the scalarized configurations might be thermodynamically more stable compared to the GR configuration. For different scalarized branches, one can see that the entropy of the second and third branches is lower than the first branch, which implies that the first scalarized branch is the most stable one. Moreover, we can also note from Fig. 7 that the entropy of scalarized solutions increases as λ and Q e Q m increase, which means that the stronger interaction can make the scalarized configuration more stable. Together with the result indicated in Fig. 6, these imply that the scalarization might be more likely to appear in situations with stronger interaction between the scalar field and electromagnetic field.
In Fig. 8, we show the behavior of the reduced black hole temperatureT = f (r h )/4π as a function of its reduce mass M with different values of λ andQ e for the first three scalarized branches and GR branches. These figures show that the scalarized dyonic RN black hole solutions cannot reach the extremal limit T = 0 different from the GR branch where the zero-temperature can be obtained forM = 1. This is because the scalarized branches are ended at the point where the spacetime mass vanishes. Moreover, from these figures, we can also see that different from the entropy of the scalarized black holes, the maximal temperature is not always given by the first branch and is determined by the relative position of the scalarized branches for the fixed λ andQ m . Table 1 The table shows the ratio R

Conclusion and discussion
In the current paper, we considered the Einstein-Maxwellscalar theory where the scalar field is coupled to an electromagnetic Chern-Simons term, and studied the scalarization of the static and spherically symmetric dyonic RN black holes in this theory. The Chern-Simons term is nonvanishing only if neither the electric charge nor magnetic charge of black holes is zero. That is to say, the scalarized black hole solution can only appear in cases with nonvanishing electric and magnetic charges. The gravity part of the field equations is not affected by the electromagnetic Chern-Simons term and it only provides an effective potential to the equation of motion for the scalar field. When we pick the coupling constant, electric charge, and magnetic charge appropriately, the dyonic RN black hole will become unstable and there will exist scalarized black hole solutions.
First of all, we investigated the perturbation of the dyonic RN black holes and get the perturbation solutions of the scalar field satisfied the asymptotic flatness condition, which is just the bifurcation points of dyonic RN solution and scalarized solution in nonlinear cases. Our result shows that the bifurcation points only appear if the parameters satisfy ζ = 2Q e Q m λ 2 > Q 2 tot /4, which also leads to λ 2 > 1/4. Then, by employing the shooting method, we calculated the scalarized black hole solutions bifurcating from the dyonic RN black hole solution. As a result, there exist some discrete branches of the scalarized solutions and all scalarized branches bifurcate with the GR branch. Different from the dyonic RN black holes, the scalarized dyonic RN black holes can be overcharged (i.e., M 2 < Q 2 tot ) and their mass could even approach zero. This means that the scalarized solution enlarges the range of mass in GR. By analyzing the scalarized configurations with different coupling constant λ and reduced magnetic chargeQ m , we can conclude that the stronger interaction can lead to larger scalar hair. Moreover, we studied the entropy and temperature of the scalarized dyonic RN black holes as well. The results show that the entropy for all scalarized black holes exceeds the counterpart of dyonic RN black holes and the first scalarized branch captures the largest entropy, which means that the scalarized configurations might be thermodynamically more stable than GR configurations. By comparing the results of different λ andQ m , we also found that the stronger interaction can make the scalarized configuration has a larger entropy. These imply that the scalarization might be more likely to appear in situations with stronger interaction between the scalar field and the electromagnetic field. From the analysis of black hole temperature, we found that the extremal limit T = 0 cannot be reached in the scalarized branches because these branches are ended at the point where the black hole mass vanishes.
In this paper, we only discussed the black hole entropy of the scalarized solutions. However, this does not indicate that the scalarized black holes are dynamically stable. It is also necessary to consider the dynamical stability of these scalarized black holes obtained in this paper and this will be left for our future work. Moreover, it is also interesting to extend the discussion into the dynamical situation to study the scalarization or into the spinning cases to consider the spin-induced scalarization.
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