Pis'ma v ZhETF, vol. 92, iss. 4, pp. 219 - 225 © 2010 August 25
Renormalization group improved black hole space-time in large extra
dimensions
T. Bnrschil ■ *, B.Koch
+ Institut für Theoretische Physik, Johann Wolfgang Goethe - Universität, D-60438 Frankfurt am Main, Germany
* Departamento de Física, Pontificia Universidad Católica de Chile, 4860 Santiago, Chile
Submitted 31 May 2010
By taking into account a running of the gravitational coupling constant with an ultra violet fixed point, an improvement of classical black hole space-times in extra dimensions is studied. It is found that the thermodynamic properties in this framework allow for an effective description of the black hole evaporation process. Phenomenological consequences of this approach are discussed and the LHC discovery potential is estimated.
1. Introduction. Models with extra spatial dimensions offer an elegant solution to the hierarchy problem [1, 2]. In the case of [1] this is achieved by banning all standard model particles and forces onto a 4-dimensional subspace, while gravity can propagate also into d additional spatial dimensions. In order to keep the model consistent with todays gravity experiments the additional dimensions are assumed to be compacti-fied in a small volume Vd. By this construction the measured gravitational coupling (or equivalently the Planck mass Mpi) can be explained by a fundamental mass Mf which might be as low as a few TeV. Since this would be much closer to the electro-weak scale, such models give a possible solution of the hierarchy problem. The relation
where
M2Pl = VdMf+2
(1)
connects these two couplings via the volume V,d which is spanned by the extra dimensions. In a world with extra dimensions and a gravitational coupling in a TeV range colliders like the LHC could create tiny black holes (BH) [3-8]. The line element of a higher dimensional spherically symmetric black hole is given by [9]
with
da2 = f(r)dt2 - f-'^dr2 - r2dÜd+2, (2)
f{r) = l-Kg-1/rd+1. (3)
The event horizon Rh depends on the black hole mass M and the universal gravitational coupling G
d+1 _ 167rGM Uh - (d+ 2)Ad+2 '
e-mail: bkoch0fis.puc.cl
Ad+2 =
27r
d+3
rm
marks the surface of a d + 3 unit sphere. Please note, that there are different definitions of the higher dimensional coupling constant G. We use the definition of [6]. For more discussion and other definitions see [5]. By redefining the coupling in terms of the fundamental mass Md+2 = 1/G, the radius of the event horizon is given by
d+1 = 167T M H (d + 2)Ad+2 Md+2 '
(4)
This form of the black hole horizon holds as long as Rh -C R which is for TeV masses true since R n exceeds R typically by fifteen orders of magnitude. A black hole emits thermal radiation [10]. The temperature of this radiation is given by the radial derivative of the metric coefficient f(r) at the horizon. For the case of d extra dimensions this temperature is given by
T„
4w Rh
(5)
However, this prediction is limited to large black hole masses M Mf. For masses close to the fundamental mass one expects modifications of the Hawking temperature and it was conjectured that the thermal radiation could be suppressed, leading to the formation of a stable final state [11-13].
Although, some extra dimensional models like in eq. (1) can solve the hierarchy problem, they are not the desired unified description of all forces yet. The reason for this is that gravity (with or without extra dimensions) can not be generalized in the usual loop expansion to a renormalizable quantum field theory. It was
ÜHCbMa B ?K3T<J> Tom 92 Bbin.3-4 2010
219
conjectured that this problem results from expanding the theory in the gravitational coupling instead of solving the complete theory that might even contain higher powers in the curvature R. In [14-17] it was shown that by using special truncation methods an exact renormalization group (RG) equation for the gravitational coupling can be derived. In a first order truncation those studies have been generalized to extra dimensions [18, 19] leading to a fundamental mass that depends on the energy scale k: Mj+2 Mdf+2{k). It was shown that the running gravitational coupling has the form
Md+2(k) = Md+2
k
tMf
d+2"
(6)
which also depends on a parameter, t. Going beyond one loop, this transition behavior between the infrared and ultraviolet regime was found to be even more pronounced with increasing number of extra dimensions [20]. For the case of d = 0 the effect of this running coupling on the structure of the Schwarzschild metric was derived in [21]. The aim of this paper is to repeat the construction for d ^ 0 and to study its phenomeno-logical implications.
2. RG improved black holes in extra dimensions and black hole remnants. In flat space-time the de Broglie relation connects energies k and distances d by k = 1/d. In curved space-time more care is needed since distances are determined locally by the metric. For the case of a spherically symmetric Schwarzschild spacetime, modifications of the de Broglie relation can only depend on the radial coordinate r. This leads to the ansatz
k(r) =
d(r)
(7)
where £ is a parameter of order one. Before calculating the distance function d(r) it is essential to remember its behavior for large distances r oo. In this limit the metric should approach the flat Minkowski metric
lim
d(r)
= 1 .
(8)
In that case the scale approaches asymptotically k(r —¥ oo) « £/r. The distance function is calculated via the definition of distance in general relativity by integrating the line element
m = jc^\ds2class
(9)
along a curve C. The subscript class indicates that the line element is calculated with a fixed coupling Mf. We
parameterize C in Schwarzschild space-time and calculate the distance along the curve
ds
class
d(T) = [
Jo
fclass(r') HP)
dr12
0 VI fclass(r')\
dr1 .
The parameterization along the radial coordinate r' in the range r' € [0,r(P)] is chosen like in [21]. In eq. (9) it is necessary to take the absolute value of ds in order to have always a positive distance. Due to the absolute value the distance function differs inside and outside of the event horizon. Together with definition of the event horizon (4) the distance function is expressed in the two regions by
H
dr<RH(r) = J dr dr>RH(r) = dr
!dr'i^
1 rl id+l
Rr _ r'd+1 '
1 ri rd+l
tyd+l IlH
(10) (11)
It is not possible to find a general analytic solution for those two distance functions. Instead, the functions are interpolated between the two limits r 0 and r oo. For small r the denominator of (10) simplifies and the small r limit can be integrated
d(r) =
Iodr'\iw
nW1
,d+1
,>d+1
2 d+3
-r 2
R
d + 3
(12)
H
For large r one has to integrate in two steps, first form r' = 0 to r' = Rh which just gives a constant summand B, and afterwards from r' = Rh to r' = r. In the second integration the fraction simplifies to a constant and distance behaves like r, again with a summand A
d(r) = dr
fr , I rid+1 JRHdr V r'd+1 ~ Rft1
rr r'd+1
r
= r - A
(13)
r—»oo V
I
This is the dependency as required in (8). Now the distance function is interpolated between eq. (12) and (13) by
d'(r) =
„d+3
rd+! + 7d<+1
7d =
(d + 3)2
(14)
which has the correct asymptotic behavior in the limits r oo: d'(r) r and r 0: d'(r) r<d+3)/2. The parameter is evaluated by the small r limit. Identifying the energy scale k with inverse distance one finds
k(r) =
d'(r)
-d+l
idRdH+1
rd+ 3
(15)
This relation between the energy scale k and the radius r in higher dimensional Schwarzschild space-time allows to express the scale dependent fundamental mass Mf (6) in terms of the radial coordinate r.
Modifying the horizon radius Rh by the radius dependent fundamental mass Mf(r) and defining
gives
RdH+1(r) =
t=m
16?r
d+2
M
(d + 2)Ad+2 Mdf+1 Mf
- Rd+1
— nH
t ( rd+1 + 7dRd+1
<-h
Md+2
„d+3
(17)
(16)
1.0 1.5
r (1/TeV)
Fig.l. Metric coefficients f(r) for different BH masses M at d = 2, t = 0.002, and Mf = 1 TeV. For those parameters a critical mass of Mc = 1.67 TeV is found
sections, it is natural to identify the critical mass with the mass of a black hole remnant
Mr = Mc.
The critical mass can be calculated in dependence of the parameters t and Mf. As it can be seen in fig.2 for d = 2,..., 7, Mc depends strongly on the parame-
1000 800
> 600 S
where t parameterizes the strength of the RG corrections to the classical result. Since this Rh has an explicit independence it can not be interpreted as event horizon. Like usually the event horizon of a spherical symmetric black hole solution is the zero of the radial metric coefficient /(r) = 1 - R^1 (r)/rd+1. As shown in fig.l the metric function does not cross f(r) = 0 for small values of M and so there is no singularity in the line element da2. However, for a larger black hole mass one finds the critical case where the line element gets a zero at one radius. For M > Mc this zero splits up into two zeros, where the outer zero corresponds to the apparent event horizon. As also shown in fig.l this outer horizon grows for large BH masses (r oo) and approaches the classical event horizon of eq. (4). This kind of behavior of the metric function is independent of the number of extra dimensions. As it will be explained in the following
^ 400 200
0_.- ' 1 i ' ' ; i i
-6-5-4 10 10 10 0.001 0.01 0.1 1 10
Fig.2. Remnant masses depending on the quantum gravity parameter t for different numbers of extra dimensions d. Mf = 1TeV
ter t. While Mc tends to zero for small values of t, it grows rapidly beyond the experimentally testable TeV range for larger values of t. Since the strength of the quantum gravity corrections is parameterized by t, it is interesting to note that by construction the continuous limit t —¥ 0 leads to Mc —t 0, whereas in a priory classical calculation (i = 0) no critical mass Mc and therefore no remnant exists.
3. Black hole thermodynamics. The predicted thermal decay of black holes due t
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