GUP black hole remnants in quadratic gravity

The Hawking radiation of static, spherically symmetric, asymptotically flat solutions in quadratic gravity, is here explored. The emitted fermions are governed by the Dirac equation in the generalized uncertainty principle (GUP) context, that probes the spacetime (quantum) nature. The Hawking thermal spectrum is scrutinized out of the tunnelling method and the WKB procedure. The Hawking flux is shown to vanish for small black holes, for a precise combination of the GUP parameter and the coupling constants that drive the gravitational interaction in quadratic gravity, yielding absolutely stable black hole remnants.


I. INTRODUCTION
The Hawking evaporation consists of a quantum effect involving black holes, irrespectively of their masses. It is usually described via the tunnelling procedure [1][2][3][4] in the WKB semiclassical approximation. In the fermionic sector, the Hawking radiation spectrum was studied as the tunnelling of fermions satisfying the Dirac equation through an event horizon. The tunnelling method has also been used to calculate the Hawking flux of dark fermions [5,6] across the event horizon of black hole geometries. This method also encompasses small black holes, whose masses are of the order of the Planck scale [7][8][9][10].
In this paper, we will scrutinize the Hawking flux of fermions across the event horizon of black holes that are solutions of static, spherically symmetric, asymptotically flat solutions in higher-derivative gravity with quadratic curvature terms, including quantum effects on the fermion dynamics predicted by the generalized uncertainty principle (GUP) [11][12][13][14][15]. One may argue that if the evaporation process has an end, it will give rise to a remnant black hole. Similarly, if the Hawking flux is extinguished, leaving a black hole with vanishing quantum luminosity, then a remnant black hole may be produced. A black hole remnant consists of a black hole phase that evaporates under the Hawking radiation, which is either (absolutely) stable or long lived. The latter is also known as a metastable remnant [16]. The central concept involving black hole remnants consists of black holes whose size decreases during the Hawking evaporation process, reaching a minimal length, possibly near the Planck scale l p at which point the black hole ceases to evaporate.
We show, in particular, that the absolutely stable remnant case is attained for black hole solutions of quadratic gravity. We will compute higher-derivative corrections to the Hawking flux using the tunnelling method in a GUP context, governed by the GUP parameter β.
The paper is organized as follows: in Sect. II, we will briefly review and discuss the black hole metric solution arising in higher-derivative gravity. Using the semiclassical approach of the WKB approximation, the tunnelling rate and the black hole luminosity will be calculated in Sect. III.
Sect. is then devoted to the concluding remarks IV.

II. STATIC SPHERICALLY SYMMETRIC SOLUTIONS IN HIGHER-DERIVATIVE GRAVITY
Motivated by the divergence structure appearing in the quantization of general relativity at oneloop, Stelle came up with a gravitational theory containing quadratic curvature invariants which turned out to be renormalizable [22], but saddly suffers from a ghost in its spectrum. Several solutions to the ghost issue have been proposed [23][24][25][26], but no consensus has been reached so far.
The ghost seems to be harmless at energies below the Planck scale [27], which is the regime we are mostly interested in this paper. In any case, one can always project the ghost out by a suitable choice of boundary conditions [28,29].
The action of quadratic gravity where G denotes the 4D Newton constant, is both renormalizable and asymptotically free. In fact, the coefficients of the quadratic curvature terms vanish asymptotically in the ultraviolet regime of the theory. The action (1) yields the EOMs [30] B µν = T µν , where for G µν being the Einstein tensor. The tensor (3), whose trace reads satisfies the effective field equations ∇ ν B µν = 0. When b = 0, corresponding to the Einstein-Weyl theory, the sign of a can be derived when one linearizes the Minkowski metric, namely, The range a > 0 implies a stable theory, in the sense that no tachyonic instabilities sets in. In addition, there are massive spin-2 and spin-0 excitations, respectively with masses m 2 2 = 1 32πGa and m 2 0 = 1 96πGb . The former corresponds to the aforementioned ghost. Hence, one can write Solutions of the EOM (2) were scrutinized in Ref. [30], using the Frobenius procedure, with respect to the radial coordinate, r, to implement indicial equations for the leading asymptotic behaviour as r → 0. Ref. [31] derived the leading asymptotic profiles of the temporal and radial metric coefficients, respectively as where α 1 , α 2 , α 3 , ζ are the only free parameters on the solution. As it is an expansion around r = 0, the metric (7, 8) is trustworthy for the computation of the Hawking radiation spectrum, as terms in order beyond O(r 6 ) are totally negligible. The family (7,8) includes the standard Schwarzschild solution [31], as a solution of the higher-derivative EOMs. At the origin, the family of solutions (7,8) presents a physical singularity, as lim r→0 R µνρσ R µνρσ ∼ r −6 [30,31]. This family also includes non-Schwarzschild black holes.

III. HAWKING RADIATION SPECTRUM, FLUX AND BLACK HOLE HAWKING LUMINOSITY
The GUP asserts that ∆x ∆p 2 1 + β ∆p 2 , for β = β 0 /m 2 p , being β 0 a dimensionless parameter that accounts for effects of quantum gravity, having the bound |β 0 | 10 21 [32,33]. In the GUP apparatus, x j = X j and p j = P j (1 + β p 2 ) are respectively position and momentum operators, where [X j , P k ] = i δ jk . It implies that The Dirac equation, governing fermions, with electromagnetic field A µ , reads where Ω µ = − i 2 ω ρσ µ Σ ρσ , and ω ρ µ σ = e ρ ν e α σ γ ν µα is the spin connection. Eq. (9) can be substituted into Eq. (10), together with the energy of a particle of mass m and electric charge e on the mass shell, i ∂ 0 1 + β p 2 + m 2 [20]. The Dirac equation then becomes The black hole Hawking radiation for fermions can be computed with the aid of the tunnelling procedure, where the fermion is assumed to have the following form, without loss of generality [5]: for an action J and wavefunctions ψ 1 and ψ 2 . The metric (6) yields the tetrads Since the method for computing the tunnelling rate is representation-independent, the one used hereon is more appropriate, for γ 5 = iγ 0 γ 1 γ 2 γ 3 , where the σ i denote the Pauli matrices: Eqs. (12,14) replaced in the GUP-corrected Dirac equation, (11), yields the following EOMs, using the WKB regime to order in : with J = ∂J ∂r ,J = ∂J ∂t , and Expressing the action as with ω denoting the energy of the emitted fermionic spectrum. The tunnelling rate will be then computed [1,2,18]. Substituting Eq. (19) in Eq. (17) yields As the part of the equation, that is in the inner side of the square brackets, will be not identically null, one must have This means that the Θ function will not contribute for the tunnelling process. Now, replacing Eqs. (19,21) into Eqs. (15,16), yields where In what follows the scaling ζ → ζ mp , α 1 → α 1 m p , α 2 → α 2 m 3 p and α 4 → α 4 m 4 p is more illustrative and will be adopted. Solving Eq. (22) on the event horizon yields the imaginary part of the action, where and Eq. (27) is displayed for α 1 = 0, where f α 2 ,α 3 = 3 36α 2 2 m 6 p + 1296α 4 2 m 8 p − 3α 3 3 m 12 p . For α 1 = 0, the general solution having dozens of pages is opted not to be displayed here.
Thus, the tunnelling rate of fermions reads As ζ = M/2 in Eq. (7)  In what follows, for the sake of simplicity, one takes A t = 0 and express The tunnelling rate of evaporation, Γ , is plotted in Figs. 1-5 for various cases.
Taking into account that ω ∼ m 2 p /ζ, the tunnelling rate of evaporation (28) can be written as the Boltzmann term Γ = exp (−ω/T ), where It is worth emphasizing that is the Hawking temperature of the black hole (7,8), obtained with the tunnelling method [3]. The tunnelling rate (28), thus, coincides to the Hawking standard one for black holes with a sufficiently large mass, M = ζ/2, such that the GUP correction is insignificant.
When the black hole mass is near the Planck scale, the Λ function (25b) depends on ω ∼ M = ζ/2 ∼ m p . In this regime, the fermion mass is clearly negligible. Besides, also the temperature is dependent on ω ∼ m p . To carry out this dependence, let us consider Hawking fermions of energy ω in some given mode . Their emission probability can be described by the rate Γ (ω) = e −ω/T (ω) , up to a factor that encodes the absorption probability of the fermions by the black hole. To quantify this reasoning, one denotes the average number of fermions carried by each mode, n (ω) = (1 + exp(ω/T )) −1 = Γ (ω) 1+Γ (ω) . It is then possible to consider the emission rate 2 π ṅ (ω) = n (ω) dω [34]. The black hole luminosity reads where G (ω) denotes the gray-body factors. For small black holes, when m 2 p /M ω, in the continuum limit, the luminosity reads Modelling the black hole by a sphere introduces an upper bound on the absorbed modes, given by ( + 1) m 4 p 27 4 ζ 2 ω 2 [17], as the modes beyond this range will not constitute the absorption spectrum of the black hole.
The action (1) was expressed in Ref. [22] as for C µνρσ being the Weyl tensor components. The action (39) can be expressed as (1) due to the Gauss-Bonnet invariant. As the Frobenius expansion was obtained near the origin, it is now opportune to analyze the possibility of black hole remnants for an expansion around a nonzero radius r h . Ref. [22] showed that in static and asymptotically flat backgrounds, the existence of an event horizon yields a vanishing Ricci scalar. Hence, the quadratic theory consists of Einstein-Weyl gravity, with action (39) with b = 0. The corresponding EOMs read, for the metric (6), The Schwarzschild standard metric, considering A(r) = 1/B(r) = 1 − 2M r , for r h = 2M in this case, satisfies the EOMs (40, 41).
The quadratic gravity theory does not present an analytical solution, but a Frobenius expansion around the event horizon r h : A(r)= γ 2 (r−r h ) + − 1 128πG aγ 1 + 1 128πG aγ 2 1 r h