De-singularizing the extremal GMGHS black hole via higher derivatives corrections

The Gibbons-Maeda-Garfinkle-Horowitz-Strominger (GMGHS) black hole is an influential solution of the low energy heterotic string theory. As it is well known, it presents a singular extremal limit. We construct a regular extension of the GMGHS extremal black hole in a model with $\mathcal{O}(\alpha')$ corrections in the action, by solving the fully non-linear equations of motion. The de-singularization is supported by the $\mathcal{O}(\alpha')$-terms. The regularised extremal GMGHS BHs are asymptotically flat, possess a regular (non-zero size) horizon of spherical topology, with an $AdS_2\times S^2$ near horizon geometry, and their entropy is proportional to the electric charge. The near horizon solution is obtained analytically and some illustrative bulk solutions are constructed numerically.


Introduction
Low energy string theory compactified to four spacetime dimensions admits a famous black hole (BH) solution, found by Gibbons and Maeda [1] and, independently, by Garfinkle, Horowitz and Strominger [2] from now on dubbed the GMGHS BH. This solution can be described either in the Einstein frame or in the conformally related string frame. In the former case (which is the one considered in this work), the model's action is the sum of an Einstein term, a kinetic term for a scalar field (the dilaton) and a Maxwell term with an exponential coupling to the dilaton -see Eq. (2.3) below. The GMGHS BH is the simplest solution of this model, representing a charged, static and spherically symmetric horizon surrounded by scalar hair. This stringy extension of the Reissner-Nordström (RN) BH has attracted an enormous attention, finding a variety of interesting applications, e.g. .
An awkward property of the GMGHS solution is that its extremal limit is singular: when taking this limit the area of the spatial sections of the horizon shrinks to zero and the Kretschmann scalar blows up at the (would be) horizon. Since neither Ref. [1] nor [2] take into account the possible stringy α ′ -corrections to the Einstein-Maxwell-dilaton action one may ask if such corrections could de-singularize the extremal solution.
A perturbative extension of the extremal magnetic GMGHS BH has been constructed by Natsuume in Ref. [9] (to first order order in α ′ ). As found therein [15,20,21,[26][27][28], the corrected solution inherits all basic properties of the extremal GMGHS BH; in particular, the horizon area still vanishes. On the other hand, to the best of our knowledge, the task of constructing the fully non-linear BH solutions of the O(α ′ ) corrected action has not yet been considered in the literature. This is presumably due to the complexity of the field equations. Yet, such construction can reply to the key question whether such corrections can desingularise the extremal GMGHS solution. The main purpose of this work is to report results in this direction. Starting with a general model for the O(α ′ ) corrections to the Einstein-Maxwell-dilaton action (which is essentially the one in Ref. [9]), we find that the field equations of the full model possess an exact solution describing a Robinson-Bertotti-type vacuum, with an AdS 2 × S 2 metric, an electric field and a constant dilaton. On the other hand, we also find there is no counterpart of this solution with a magnetic charge.
The supergravity action includes, in general, a tower of corrections of all powers in α ′ because of the recursive definition of the Kalb-Ramond 3-form field strength, which breaks the supersymmetry in the supergravity theory. The term of quadratic order in curvature is obtained imposing supersymmetry of the theory at first order in α ′ if we consider the Chern-Simons term in the field strength. There are further higher power corrections in the curvature of the torsionful spin connection, which are required to preserve supersymmetry order by order [31]. Additional higher-curvature corrections unrelated to the supersymmetrization of the Kalb-Ramond kinetic term are also present, although those appear first at cubic order in α ′ . These additional higher-order (like α ′ 2 , α ′ 3 · · ·) corrections may drastically modify the non-perturbative result obtained from our setup. The properties of the solutions with higher curvature corrections, within the heterotic string theory, is still largely an uncharted territory. Although the conditions imposed by supersymmetry have been studied in detail, the solution of the Einstein equations are not well known yet, especially for the interesting case of the heterotic string with fluxes.
As for the extremal RN solution in the Einstein-Maxwell model, we expect this "attractor" to describe the near-horizon geometry of an extremal BH with a regular horizon. That is, the O(α ′ )-corrections desingularize the extremal limit of the GMGHS BH, leading to a non-zero size, regular horizon 1 . It is well known that such near horizon geometry is a key feature of static supersymmetric BHs, providing the attractor mechanism by which horizon scalar fields values are determined by the charges carried by the BH and insensitive to the asymptotic values of the scalar fields [32][33][34][35]. The entropy of these BHs is consistent with the microscopic states counting of the associated D-brane system. These are described by the inclusion of higher derivative corrections in the generalized prepotential together with the supergravity low energy description [36][37][38].
In this work we shall provide numerical evidence for the existence of non-perturbative extensions of the GMGHS BHs which are extremal, asymptotically flat and regular, on and outside the horizon. These solutions represent global (bulk) extensions of the aforementioned attractors found analytically and possess a variety of interesting properties. For example, as for the attractors, the entropy of the extremal BHs is proportional to the electric charge. Also, their charge to mass ratio is always greater than one. This paper is organised as follows. In Section 2 we describe the full model, including α ′ corrections and describe briefly the GMGHS solution. In Section 3 we consider the α ′ corrected solutions; first we construct (analytically) the attractors and then we give (numerically) examples of global bulk solutions. Section 4 gives some concluding remarks. In the appendix, we summarize some formulas, including the equations of motion and our static spherically symmetric ansatz, which are needed for the results in section 3.

The model 2.1 The action
The starting point is the string-frame action including order α ′ terms Finding the local solutions in the vicinity of the (extremal) horizon (i.e. the attractor) does not guarantee the existence of global solutions with the right asymptotics. Progress in this direction requires an explicit construction of the bulk extremal BHs.
whereg ,R ,R µν ,R µνρσ ,∇ denote determinant, Ricci scalar, Ricci tensor, Riemann curvature tensor, covariant derivative constructed from the string frame metricg µν , respectively. Also, F = dA is the U (1) field strength tensor and φ is a real scalar field: the dilaton. a, b, c, h are constant coefficients.
The corresponding expression of the action in the Einstein frame (which is the case of interest in this work) is found via a conformal transformation, with g µν = e 2φ g µν . (2.2) This results in (g µν being the Einstein frame metric): with the leading order corrections in α ′ The variation of (2.3) with respect to g µν , φ and A µ leads to the equations of motion of the model. However, their expression is too complicated to include here. Let us briefly comment on the string theory origin of the above action. The U (1) gauge field arises as a subgroup of the E 8 × E 8 or SO(32) gauge group in the low-energy effective theory of the heterotic string [3]. We set the remaining gauge fields and the antisymmetric field strength H µνρ to zero. Then the four-dimensional action emerges from heterotic string theory compactified on a six-dimensional torus. This includes corrections due to the next order terms such as R 2 , F 4 , F 2 (∇φ) 2 [9,15] . After eliminating terms in the effective action by field redefinitions, the corrections to the leading order in α ′ can be expressed as (2.4).
The higher-order (α ′ ) terms in the four-dimensional effective action can be considered corrections due to quantum gravity [15,21,39]. Let us estimate the mass scaling of these terms, for extremal BHs, near the horizon r ∼ M . The GMGHS BH solution implies that any derivative contributes a factor of order M −1 ; then, the curvature tensors are R ∼ M −2 , R 2 ∼ M −4 and the gauge field strength gives F ∼ M −1 . Hence we have, e.g.
We assume the classical mass-charge relation for extremal BHs approximately holds and they are sufficiently large, Q ≃ M ≫ 1, in units of the Planck mass M Pl . We conclude that the α ′ corrections are suppressed by powers of (α ′ /M ) 2 . Thus, we consider the leading-order α ′ corrections throughout only, assuming the BHs are sufficiently macroscopic α ′ /M ≪ 1.
Moreover, the leading order terms in the Lagrangian L have M −2 while the leading order correction in the Einstein frame are given by (2.4) [9,15]; all terms involving ∇ ρ F µν are dropped, without loss of generality, since they are equivalent, via the Bianchi identities, to terms already accounted for or terms involving ∇ µ F µν [21,39] . Such terms vanish in the absence of charged matter sources, which is the case considered in this paper.
We do not consider terms that are proportional to the Kalb-Ramond three-form field strength H, where B is 2-form gauge potential, ω L , ω Y denote the Lorentz and Yang-Mills Chern-Simons forms, respectively. The presence of H in the heterotic string is required due to α ′ corrections in the Bianchi identity which are needed for anomaly cancellation. Since the Chern-Simons forms act as sources for H , we are in general not able to simply set H = 0 . However, the Lorentz Chern-Simons form can be written by the exterior derivative of a three form for the spherically symmetric metrics. Hence, we can absorb it into the definition of 2-form gauge potential B [40]. Moreover, the Yang-Mills Chern-Simons form vanishes for the purely magnetic case [9].
In the expression (2.3), a, b, c and h are constants which are not fixed a priori. The coefficient a is arbitrary (and nonzero), since it can be changed by scaling α ′ . The choice in Ref. [9] (employed also here) is a = 1/8. That reference gives also an argument that h = 0.
As for b and c, to the best of our knowledge, the only concrete computation is in Ref. [9]. The idea there was to construct the (leading order) α ′ -correction to the magnetic GMGHS BH, in the extremal limit. The corrected solution is supposed to possess the same near-throat behaviour and the same far field asymptotics as the original solution (with α ′ = 0). This argument implies (for the conventions here), while the value of b 0 is arbitrary 2 .
Generalizing the discussion, α ′ − (or quantum gravity) corrected Einstein-Maxwell-dilaton theories have attracted much attention [15,[26][27][28] even beyond the stringy models. Within string theory and its low-energy effective field theories, the coefficients of each terms in the string effective action are invariant under field redefinitions [9,15], which were then taken from the heterotic string calculations [3]: a = 1/8 and h = 0 . The parameter c can be fixed by a requirement of consistency with exact results that were obtained for the GMGHS BH [7] while the value of b does not affect the correction to the BH mass. On the other hand, there are studies of BHs based on the most general collection of four-derivative terms for 4-dimensional Einstein-Maxwell-dilaton theories [26]. We will here construct the α ′ corrected GMGHS solution within the framework of the low-energy effective theory of the heterotic string. But our choice here is to work with some slight more generality. Thus we take:

Ansatz and entropy
In this work we are interested in static spherically symmetric solutions with a purely electric U (1) potential. An Ansatz suitable to address both the (generic) BH solutions and the Robinson-Bertotti ones (with an AdS 2 × S 2 near horizon geometry) reads with (r, t) the radial and time coordinates, respectively, while dΩ 2 = dθ 2 + sin 2 θdϕ 2 is the usual metric on S 2 . The entropy of generic solutions (extremal or not) is fully accounted for by the Wald formula [41] where A H is the event horizon area, ε µν is the binormal to the horizon of the BH, normalized so that ε µν ε µν = −2 and relation (2.9) is evaluated at the horizon.

The α ′ = 0 limit: the GMGHS solution
The GMGHS solution can be written in the form (2.8), with the metric functions (2.10) and the Maxwell potential and dilaton field The two free parameters r + , r − (with r − < r + ), corresponding to outer and inner horizon radius, respectively They are related to the ADM mass, M , and (total) electric charge, Q, by Other quantities of interest are the horizon area A H and the Hawking temperature T H (2.13) The extremal limit, which corresponds to the coincidence limit r − = r + , results in a singular solution (as can be seen e.g. by evaluating the Kretschmann scalar). In this limit, the area of the event horizon goes to zero. The Hawking temperature, however, approaches a constant. For completeness, let us mention the existence of a magnetic version of the GMGHS solution, which possesses the same metric, while the U (1) potential and the dilaton are (2.14) However, this solution also becomes singular as r − → r + . A full duality orbit of solutions that interpolate between these two in Einstein-Maxwell-scalar models was recently discussed in [42] (See also [43,44]).
3 Non-perturbative electrically charged solutions

The attractors
Taking into account α ′ corrections, the possible existence of non-perturbative generalizations of the extremal GMGHS BHs with a nonzero horizon size is suggested by the presence of a Robinson-Bertotti-type exact solution, which we shall now discuss. The existence of such near-horizon geometry is, moreover, closely connected with the attractor mechanism [32][33][34][35], as discussed above. We consider the following Ansatz, which is a particular case of (2.8), where v 0 , v 1 , φ 0 and q are parameters which satisfy a set of algebraic relations which result from the field equations. The ansatz (3.15) was discussed by [34,45,46] as the most general near horizon field configuration consistent with the SO(2, 1)×SO(3) symmetry of AdS 2 ×S 2 . Instead of following this route, however, in what follows we choose to determine the unknown parameters by using the formalism proposed in Refs. [34,45,46], thus by extremizing an entropy function 3 . This alternative approach allows us to also compute the entropy of these BHs, and to show that the solutions exhibit attractor behaviour.
The entropy function is defined as where Q is the electric charge of the solutions, while f (v 0 , v 1 , q, φ 0 ) is the Lagrangian density of the model (2.3) evaluated for the Ansatz (3.15) and integrated over the angular coordinates, The attractor equations are: and The unique solution of the eqs. and Replacing the above expression and back in the entropy function we obtain the remarkable simple expression of the entropy of an extremal BH: which can also be expressed in terms of the electric charge, As a check, we note that the result agrees with Wald's form (3.11) evaluated for the near horizon geometry. Let us also remark that the constant c which enters (2.3) does not enter the above expressions 4 . Moreover, one can easily see that (3.22) solves indeed the field equations, being a consistent solution.
As a final remark, we mention that there is no magnetic counterpart of the above solution. That is, when employing the same metric ansatz as in (3.15), a constant scalar and a magnetic U (1) form A = Q cos θdϕ, the field equations imply the following relation which does not possess any physical solution.

The bulk extremal BHs
On general grounds, we expect that the above attractor solution describes the neighbourhood of the event horizon of a bulk extremal BH. In what follows, we give numerical evidence for the existence of these configurations.
In the numerical study of the solutions, it is convenient to work in Schwarzschild-like coordinates, with the following choice in (2.8): b(r) = r, a(r) 2 = e −2δ(r) N (r) and c(r) 2 = 1/N (r), which results in the line element To construct RN-like extremal BH solutions, we assume the existence of a horizon located at r = r H > 0. In its exterior neighbourhood, one finds the following approximate solution (which holds for an extremal BH only): as imposed by the field equations. Thus it turns out that only the parameters φ 0 and δ 0 in the above near-horizon expansion are essential, the coefficients φ 1 and δ 1 being determined in terms of these. However, their expression is too complicated to include here. For large r, one finds the following asymptotic form of the solution: The essential parameters in the above expansion are the mass M , electric charges Q, electrostatic potential at infinity V 0 and scalar 'charge' Q s . These extremal BHs have finite global charges M, Q as well as a finite scalar 'charge' Q s . Their Hawking temperature vanishes, while the entropy takes a very simple form, as resulting from (2.9) with R the Ricci scalar of the induced horizon metric: After replacing the near horizon expression of the solution (e.g. A H = 4πr 2 H , R = 2/r 2 H ), the relation (3.24) is recovered 5 .
We also note that the equations of the model are invariant under the transformation  The metric functions N (r), δ(r), the scalar field φ(r) and the electric potential V (r) are found by solving a set of four ordinary differential equations. However, these equations are too long to display here, with hundreds of independent terms. However, we mention that we have been able to find a suitable combination of the field equations such that the functions N, δ still solve first order equations, while the functions φ and V satisfy second order equations. Moreover, the 2nd order equation for the electric potential V (r) possesses a 1st integral which is used in practice to check the accuracy of the numerical results.
The solutions which smoothly connect the asymptotics (3.28) and (3.30) are constructed numerically. The only input parameters are while the constants φ 0 , δ 0 and M , V 0 , Q, Q s result from the numerical output, with the electric charge Q given by (3.23), a relation which provides an extra-test of the numerical accuracy. We follow the usual approach in such problems and, by using a standard ordinary differential equation solver, we evaluate the initial conditions at r = r H (1 + 10 −5 ) for global tolerance 10 −14 , adjusting for 'shooting' parameters φ 0 and δ 0 , and integrating towards r → ∞, looking for solutions with proper asymptotics. The profile of a typical BH solution is shown in Figure 1. There, apart from the metric and matter functions, we display also the Ricci and Kretschmann scalars, which show that the geometry is regular on and outside the horizon.
The basic properties of the extremal solutions can be summarized as follows. First, given the input constants (α ′ , b, c), regular BH solution appears to exist for a single set of the 'shooting' parameters (φ 0 , δ 0 ), while the profile of the scalar field φ(r) is nodeless. Thus it is natural to conjecture that, similar to the α ′ = 0 case in [1,2], only nodeless solutions exist also for the model in this work.
Second, for a given b, the existence of solutions depends on the value of input constant c, which enters the α ′ -term in the action (2.3) (we recall that c is absent in the attractor solution). Taking α ′ = 1/2, we have considered various values of the constant b and varied the value of the parameter c. Then no upper bound on c was found. However, for a given b, the solutions stop to exist for c < c 0 , with c 0 increasing as b increases (see Figure 2, both panels). As c → c 0 , the value of the scalar field at the horizon appears to increase (in modulus) without bound, and the solution becomes singular, a feature which cannot be predicted by the attractor analysis in Section 3.1. A detailed study of the critical behaviour in this limit may be of interest, but it is outside the scope of this work.  If we compare it with the figure 2, we observe that the charge-to-mass ratio Q/M decreases as M decreases.
Given the above remark, the existence of regular solutions with c = 2 (as implied by the perturbative results in Ref. [9]) seems to require very small values of the parameter b, of the order 10 −5 or smaller. Unfortunately, the numerical accuracy deteriorates in that region of the parameter space and we could not explore this case.
Finally, perhaps the most interesting feature of the solutions found so far is that the ratio Q/M is always greater than one -see Figure 2, left panel. But the charge-to-mass ratio Q/M decreases when c increases for fixed b. On the other hand, from Figure 3, we find that as the parameter c grows, the BH mass M decreases for fixed value of b . This feature seems to be at tension with the rationale of the weak gravity conjecture.

Discussion
In this work we have confirmed that α ′ corrections can de-singularize the extremal GMGHS solution, an influential stringy BH whose extremal limit is long known to be singular. This gives an illustrative example of how higher order corrections motivated by quantum gravity can be key (and non-negligible) to understand the BH geometry on and outside a horizon. These higher-curvature terms also contribute to the global charges and energy of the system [20,47,48].
The BH solutions with α ′ corrections in string theory, moreover, shows the importance of corrections in the Riemann tensor (rather than, say, just in the Ricci scalar as in f (R) models); taking them into account is fundamental because the curvature scalars do not capture all the possible terms in the equations of motion at higher orders in α ′ corrections [49]. Here we have focused on static BHs but there are also studies with rotating BHs with first order correction in α ′ [40,50,51] -see also the related studies in [52][53][54][55][56].
We observe that the charge-to-mass ratio Q/M decreases when the BH mass M decreases for fixed value of b , which is different from previous examples [21,24]. Whether the charge-to-mass ratio increases or decreases depends on the particular structure of the higher derivative terms. If it increases then BHs can decay to smaller BHs, while such decays are forbidden in the other case. Although we get Q/M > 1, the charge-to-mass ratio decreases as the mass decreases. Hence, our results do not assure that an extremal BH is always able to decay to smaller extremal BHs of marginally higher charge-to-mass ratio.
Let also remark that the results reported here are exploratory, but establish a proof of concept that such higher order corrections lead to BH solution that are non-singular on the horizon and can be extended throughout the whole spacetime. We have also confirmed the existence of non-extremal BH solutions of the same model. These BHs possess very similar far field and near horizon expressions, the main difference being that the function N (r) possesses a single zero as r → r H , (also the value r H is not fixed in this case). Moreover, we have found clear numerical evidence that the non-extremal BH solutions exist for a large part of the (b, c)-plane. We hope to return elsewhere with a detailed study of these configurations. As a final intriguing remark, it would be interesting to clarify the absence of a magnetic counterpart to the electric attractor in Section 3.1, and its implication in the contex electric/magnetic duality for the α ′ corrected theory.
δ ′ + rφ ′2 + α ′ e −2φ 4P δ a + ce 2(δ−φ) rφ ′2 V ′2 = 0, (.37) while the equations of motion for scalar and gauge fields are where a prime " ′ " states a derivative with respect to the radial coordinate r . The equation (.40) for the electric potential V (r) consists of a first integral. Also, in order to simplify the relations, we have defined the following quantities − N (1 + rφ ′ (4 + 3rφ ′ )) + 2φ ′′ (N − 1 + rN φ ′ (4 + 3rφ ′ )), The component of Einstein equations displayed in (.38) is the constraint which will be used when searching for black hole solutions, including the extremal configuration. Here the functions N, δ solve first order equations, while the functions φ and V satisfy second order equations, which was used in our numerical treatment of the problem.