Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://theory.sinp.msu.ru/~qfthep04/2004/Proceedings2004/Alexeyev_QFTHEP04_220_226.ps Äàòà èçìåíåíèÿ: Thu Sep 17 21:19:08 2009 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 08:28:36 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 87 jet

Gauss­Bonnet black holes at new colliders: beyond the
dimensionality of space
S.Alexeyev 1,2 , A.Barrau 2 , J.Grain 2
1 Sternberg Astronomical Institute,
Universitetsky Prospect, 13, Moscow 119992, Russia
2 CNRS­IN2P3 Laboratory for Subatomic Physics and Cosmology
53, avenue des Martyrs, 38026 Grenoble cedex, France
Abstract
The Gauss ­ Bonnet invariant is one of the most promising candidates for a quadratic curvature
correction to the Einstein action in expansions of supersymmetric string theory. We study the evap­
oration of such Schwarzschild ­ Gauss ­ Bonnet black holes which could be formed at future colliders
if the Planck scale is of order a TeV, as predicted by some modern brane world models. We show
that, beyond the dimensionality of space, the corresponding coupling constant could be measured
by the LHC. This opens new windows for physics investigation in spite of the possible screening of
microphysics due to the event horizon.
1 Introduction
It has recently been pointed out that black holes could be formed at future colliders if the Planck scale is
of order a TeV, as is the case in some extra­dimension scenarios [1, 2]. This idea has driven a considerable
amount of interest (see e.g. [3]). The same phenomenon could also occur due to ultrahigh energy neutrino
interactions in the atmosphere [4]. Most works consider that those black holes could be described by the
D­dimensional (D # 5) generalized Schwarzschild or Kerr metrics [5]. The aim of this work is to study
the experimental consequences of the existence of the Gauss­Bonnet term (as a step toward quantum
gravity) if it is included in the D­dimensional action. This approach should be more general and relies
on a real expansion of supersymmetric string theory.
2 Black hole formation at colliders
The ''large extra dimensions'' scenario [6] is a very exciting way to address geometrically the hierarchy
problem (among others), allowing only the gravity to propagate in the bulk. The Gauss law relates the
Planck scale of the e#ective 4D low­energy theory MP l with the fundamental Planck scale MD through
the volume of the compactified dimensions, VD-4 , via: MD = # M 2
P l /VD-4 # 1/(D-2)
. It is thus possible
to set MD # TeV without being in contradiction with any currently available experimental data. This
translates into radii values between a fraction of a millimeter and a few Fermi for the compactification
radius of the extra dimensions (assumed to be of same size and flat, i.e. of toroidal shape). Furthermore,
such a small value for the Planck energy can be naturally expected to minimize the di#erence between
the weak and Planck scales, as motivated by the construction of this approach. In such a scenario, at
sub­weak energies, the Standard Model (SM) fields must be localized to a 4­dimensional manifold of
weak scale ''thickness'' in the extra dimensions. As shown in [6], as an example based on a dynamical
assumption with D=6, it is possible to build such a SM field localization. This is however the non­trivial
task of those models.
Another important way for realizing TeV scale gravity arises from properties of warped extra­dimensional
geometries used in Randall­Sundrum scenarios [7]. If the warp factor is small in the vicinity of the stan­
dard model brane, particle masses can take TeV values, thereby giving rise to a large hierarchy between
the TeV and conventional Planck scales [2, 8]. Strong gravitational e#ects are therefore also expected in
high energy scattering processes on the brane.
In those frameworks, black holes could be formed by the Large Hadron Collider (LHC). Two partons
with a center­of­mass energy # s moving in opposite directions with an impact parameter less than the
horizon radius r + should form a black hole of mass M # # s with a cross section expected to be of order
# # #r 2
+ . Thoses values are in fact approximations as suppression e#ects should be considered [9, 10] and
220

are taken into account in the section 5 of this paper. Although the accurate corss section values are not
yet known, a semiclassical analysis of quantum black hole formation is now being constructed and the
existence of a closed trapped surface in the collision geometry of relativistic particles in demonstrated.
To compute the real probability to form black holes at the LHC, it is necessary to take into account
that only a fraction of the total center­of­mass energy is carried out by each parton and to convolve the
previous estimate with the parton luminosity [1]. Many clear experimental signatures are expected [2], in
particular very high multiplicity events with a large fraction of the beam energy converted into transverse
energy with a growing cross section. Depending on the value of the Planck scale, up to approximately a
billion black holes could be produced at the LHC.
3 Schwarzschild ­ Gauss ­ Bonnet black holes
The classical Einstein theory can be considered as the weak field and low energy limit of a some quantum
gravity model which is not yet built. The curvature expansion of string gravity therefore provides an
interesting step in the modelling of a quasiclassical approximation of quantum gravity. As pointed out
in [11], among higher order curvature corrections to the general relativity action, the quadratic term is
especially important as it is the leading one and as it can a#ect the graviton excitation spectrum near
flat space. If, like the string itself, its slope expansion is to be ghost free, the quadratic term must be the
Gauss ­ Bonnet combination: LGB = R µ### R µ###
- 4R##R ## +R 2 . Furthermore, this term is naturally
generated in heterotic string theories [12] and makes possible the localization of the graviton zero­mode
on the brane [13]. It has been successfully used in cosmology, especially to address the cosmological
constant problem (see e.g. [14] and references therein) and in black hole physics, especially to address
the endpoint of the Hawking evaporation problem (see e.g. [15] and references therein). We consider here
black holes described by such an action:
S = 1
16#G # d D x # -g # R + #(R µ### R µ###
- 4R##R ## +R 2 ) + . . . # ,
where # is the Gauss ­ Bonnet coupling constant. The measurement of this # term would allow an
important step forward in the understanding of the ultimate gravity theory. Following [16], we assume
the metric to be of the following form :
ds 2 = -e 2# dt 2 + e 2# dr 2 + r 2 h ij dx i dx j
where # and # are functions of r only and h ij dx i dx j represents the line element of a (D- 2)­dimensional
hypersurface with constant curvature (D- 2)(D - 3). The substitution of this metric into the action [11]
leads to the following solutions :
e 2# = e -2# = 1 + r 2
2#(D - 3)(D - 4) â
# # 1 ± # 1 + 32# 3-D
2 G#(D - 3)(D - 4)M#( D-1
2
)
(D - 2)r D-1
# # .
The mass of the black hole can then be expressed [11, 16] in terms of the horizon radius r + ,
M = (D - 2)# D-1
2 r D-3
+
8#G# # D-1
2 # # 1 + #(D - 3)(D - 4)
r 2
+
#
where # stands for the Gamma function. The temperature is obtained by the usual requirement that no
conical singularity appears at the horizon in the euclidean sector of the hole solution,
TBH = 1
4# (e -2# ) # | r=r+ = (D - 3)r 2
+ + (D - 5)(D - 4)(D - 3)#
4#r+ # r 2
+ + 2#(D - 4)(D - 3) # .
In the case D = 5, those black holes have a singular behavior [16] and, depending on the value of #,
can become thermodynamically unstable or form stable relics. For D > 5, which is the only relevant
hypothesis for this study (as D = 5 would alter the solar system dynamics if the Planck scale is expected
to lie #TeV), a quantitatively di#erent evaporation scenario is expected.
221

4 Flux computation
Using the high­energy limit of multi­dimensional grey­body factors [17], the spectrum per unit of time t
and of energy Q can be written, for each degree of freedom, for particles of type i and spin s as:
d 2 N i
dQdt =
4# 2
# D-1
2 # 2
D-3
# D-1
D-3 # r 2
+ Q 2
e
Q
T BH - (-1) 2s
.
This is an approximation as modifications might arise when the exact values of the greybody factors are
taken into account due to their dependence, in the low energy regime, on both the dimensionality of the
spacetime and on the spin of the emitted particle. Fortunately, as demonstrated in the 4­dimensional
case [18], the pseudo­oscillating behaviour induces compensations that makes the di#erences probably
quantitatively quite small. The mean number of emitted particle can then be written as
N tot = 15(D - 2)# D-9
2 #(3)
#( D-1
2
)G
3
4 N f +N b
7
8 N f +N b
# r D-2
init+
D - 2 + 2(D - 3)#r D-4
init+
#
where N f and N b being the total fermionic and bosonic degrees of freedom, r init+ is the initial horizon
radius of a black hole with mass M init and, interestingly, the ratio of a given species i to the total emission
is given by :
N i
N tot
= # s g i
3
4 N f +N tot
where # s is 1 for bosons and is 3/4 for fermions and g i is the number of internal degrees of freedom for the
considered particles. The mean number of particles emitted by a Schwarzschild ­ Gauss ­ Bonnet black
hole ranges from 25 to 4.7 depending on the values of # and D, for MD # 1 TeV and M init # 10 TeV.
Those values are decreased to 5 and 1.05 if M init is set at 2 TeV. Figure 1 shows the flux for di#erent
values of # and D.
5 String coupling constant measurement
To investigate the LHC capability to reconstruct the fundamental parameter #, we have fixed the Planck
scale at 1 TeV. Although a small excursion range around this value would not change dramatically our
conclusions, it cannot be taken much above, due to the very fast decrease of the number of formed black
holes with increasing MD . Following [1], we consider the number of black holes produced between 1
TeV and 10 TeV with a bin width of 500 GeV (much larger than the energy resolution of the detector),
rescaled with the value of r + modified by the Gauss ­ Bonnet term. For each black hole event, the emitted
particles are randomly chosen by a Monte­Carlo simulation according to the spectra given in the previous
section, weighted by the appropriate number of degrees of freedom. The Hawking radiation takes place
predominantly in the S­wave channel [19], so bulk modes can be neglected and the evaporation can be
considered as occurring within the brane. As the intrinsic spectrum dN i /dQ is very strongly modified by
fragmentation process, only the direct emission of electrons and photons above 100 GeV is considered.
We have checked with the Pythia [20] hadronization program that only a small fraction of directly
emitted #­rays and electrons fall within an hadronic jet, making them impossible to distinguish from
the background of decay products. Furthermore, the background from standard model Z(ee)+jets and
#+jets remains much lower than the expected signal. The value of the Planck scale is assumed to be
known as a clear threshold e#ect should appear in the data and a negligible uncertainty is expected on
this measurement. For each event, the initial mass of the black hole is also assumed to be known as it
can be easily determined with the full spectrum of decay products (only 5% of missing energy is expected
due to the small number of degrees of freedom of neutrinos and gravitons). The energy resolution of
the detector is taken into account and parametrized [21] as #/E = # a 2 /E + b 2 with a # 10% # GeV
and b # 0.5%. Unlike [1], we also take into account the time evolution of the black holes and perform
a full fit for each event. Once all the particles have been generated, spectra are reconstructed for all
the mass bins and compared with theoretical computations. The values of D and # compatible with the
222

Energy (TeV)
10 ­3
10 ­2
10 ­1
1 10
Integrated
flux
(1/TeV)
10 ­5
10 ­4
10 ­3
10 ­2
dim=6
dim=7
dim=8
dim=9
dim=10
dim=11
Energy (TeV)
10 ­3
10 ­2
10 ­1
1 10
Integrated
flux
(1/TeV)
10 ­5
10 ­4
10 ­3
10 ­2
dim=6
dim=7
dim=8
dim=9
dim=10
dim=11
Energy (TeV)
10 ­3
10 ­2
10 ­1
1 10
Integrated
flux
(1/TeV)
10 ­7
10 ­6
10 ­5
10 ­4
10 ­3
10 ­2
10 ­1
=0.1
l
=0.5
l
=1
l
=5
l
=10
l
Energy (TeV)
10 ­3
10 ­2
10 ­1
1 10
Integrated
flux
(1/TeV)
10 ­6
10 ­5
10 ­4
10 ­3
10 ­2
=0.1
l
=0.5
l
=1
l
=5
l
=10
l
Figure 1: Integrated flux as a function of the total energy of the emitted quanta for an initial black hole mass
M = 10 TeV. Upper left: # = 0, D = 6, 7, 8, 9, 10, 11. Upper right : # = 0, 5 TeV -2 , D = 6, 7, 8, 9, 10, 11. Lower
left : D = 6, # = 0.1, 0.5, 1, 5, 10 TeV -2 . Lower right : D = 11, # = 0.1, 0.5, 1, 5, 10 TeV -2 .
223

simulated data are then investigated. Figure 2 shows the # 2 /d.o.f. for the reconstructed spectra for 2
di#erent couples (# [ TeV -2 ], D)=(1,10) and (# [ TeV -2 ], D)=(5,8). The statistical significance of this
# 2 should be taken with care since a real statistical analysis would require a full Monte­Carlo simulation
of the detector. Nevertheless, the ''input'' values can clearly be extracted from the data. Furthermore,
it is important to notice that for reasonable values of # (around the order of the quantum gravity scale,
i.e. around a TeV -2 in our case) it can unambiguously be distinguished between the case with and the
case without a Gauss ­ Bonnet term.
6 Discussion
In case the Planck scale lies in the TeV range due to extra dimensions, this study shows that, beyond
the dimensionality of space, the next generation of colliders should be able to measure the coe#cient of
a possible Gauss ­ Bonnet term in the gravitational action. This would allow an important step forward
in the construction of a full quantum theory of gravity. It is also interesting to notice that this would be
a nice example of the convergence between astrophysics and particle physics in the final understanding
of black holes and gravity in the Planckian region.
Then, as studied in [16, 23], a cosmological constant could also be included in the action. On the
theoretical side, this would be strongly motivated by the great deal of attention paid to the Anti­de
Sitter and, recently, de Sitter / Conformal Field Theory (AdS and dS /CFT) correspondences. On the
experimental side, this would open an interesting window as there is no unambiguous relation between
the D­dimensional and the 4­dimensional cosmological constants.
Acknowledgments
S.A. would like to thank the AMS Group in the ``Laboratoire de Physique Subatomique et de Cosmolo­
gie (CNRS/UJF) de Grenoble'' for kind hospitality during the first part of this work. This work was
also partially supported (S.A.) by ``Universities of Russia: Fundamental Investigations'' via grant No.
UR.02.01.026.
References
[1] S. Dimopoulos & G. Landsberg, Phys. Rev. Lett. 87 (2001) 161602
[2] S.B. Giddings & S. Thomas, Phys. Rev. D 65 (2002) 056010
[3] K. Cheung, Phys. Rev. Lett. 88 (2002) 221602
P. Kanti & J. March­Russell, Phys. Rev. D 66 (2002) 024023
A.V. Kotwal & C.Hays, Phys. Rev. D 66 (2002) 116005
S. Hossenfelder, S. Hofmann, M. Bleicher & H. Stocker, Phys. Rev. D 66 (2002) 101502
A. Chamblin & G.C. Nayak, Phys. Rev. D 66 (2002) 091901
V. Frolov & D. Stojkovic, Phys. Rev. D 66 (2002) 084002
M. Cavaglia, Phys. Lett.B 569 (2003) 7
D. Ida, K.­Y. Oda & S.C. Park, Phys. Rev. D 67 (2003) 064025
. Cavaglia, S. Das & R. Maartens, Class. Quantum Grav. 20 (2003) 205
R. Casadio & B. Harms, Int. J. Mod. Phys. A 17 (2002) 4635
P. Kanti & J. March­Russell, Phys. Rev. D 67 (2003) 104019
I.P. Neupane, Phys. Rev. D 67 (2003) 061501
[4] A. Ringwald & H. Tu, Phys. Lett. B 525 (2002) 135
R. Emparan, M. Masip & R. Rattazzi, Phys. rev. D 65 (2002) 064023
J.L. Feng & A.D. Shapere, Phys. Rev. Lett. 88 (2002) 021303
L.A. Anchordoqui, J.L. Feng, H. Goldberg & A.D. Shapere, Phys. Rev. D 65 (2002) 124027
E.­J. Ahn, M. Ave, M. Cavaglia & A.V. Olinto, Phys. Rev. D 68 (2003) 043004
224

Figure 2: Upper part: values of the # 2 /d.o.f. for the reconstructed spectra as a function of D and # for ''input''
values # = 1 TeV -2 and D = 10; the right side shows rectangles proportional to the logarithm of the # 2 /d.o.f.
Lower part (left and right): values of the # 2 /d.o.f. for the reconstructed spectra as a function of D and # for
''input'' values # = 5 TeV -2 and D = 8; the right side shows rectangles proportional to the logarithm of the
# 2 /d.o.f.
225

[5] R.C. Myers & M.J. Perry, Ann. Phys. (N.Y.) 172 (1986) 304
[6] N. Arkani­Hamed, S. Dimopoulos & G.R. Dvali, Phys. Lett. B 429 (1998) 257
I. Antoniadis et al., Phys. Lett. B 436 (1998) 257
N. Arkani­Hamed, S. Dimopoulos & G.R. Dvali, Phys. Rev. D 59 (1999) 086004
[7] L. Randall & R. Sundrum, Phys. Rev. Lett. 83 (1999) 3370
[8] S.B. Giddings& E. Katz, J. Math. Phys. 42 (2001) 3082
[9] D.M. Eardley & S.B. Giddings, Phys. Rev. D 66 (2002) 044011
[10] H. Yoshino & Y. Nambu, Phys. Rev. D 66 (2002) 065004
[11] D.G. Boulware & S. Deser, Phys. Rev. Lett. 55 (1985) 2656
[12] B. Zwiebach, Phys. Lett. B 156 (1985) 315
N. Deruelle & J. Madore, Mod. Phys. Lett. A 1 (1986) 237
N, Deruelle & L. Farina­Busto, Phys. Rev. D 41 (1990) 3696
[13] S. Nojiri, S.D. Odintsov & S. Ogushi, Phys. Rev. D 65 (2002) 023521
M.E. Mavrotamos & J. Rizos, Phys. Rev. D 62 (2000) 124004
Y.M. Cho, I.P. Neupane and P.S. Wesson, Nucl. Phys. B 621 (2002) 388
[14] B.C. Paul & S. Mukherjee, Phys. Rev. D 42 (1990) 2595
B. Abdesselam & N. Mohammedi, Phys. Rev. D 65 (2002) 084018
C. Charmousis & J.­F. Dufaux, Class. Quantum .Grav. 19 (2002) 4671
J.E. Lidsey & N.J. Nunes, Phys. Rev. D 67 (2003) 103510
[15] S.O. Alexeyev & M.V. Pomazanov, Phys. Rev. D 55 (1997) 2110
S.O. Alexeyev, A. Barrau, G. Boudoul, O. Khovanskaya & M. Sazhin, Class. Quantum Grav. 19
(2002) 4431
M. Banados, C. Teitelboim & J. Zanelli, Phys. Rev. Lett. 72 (1994) 957
T. Torii & K.­I. Maeda, Phys. Rev. D 58 (1998) 084004
[16] R.­G. Cai, Phys. Rev. D 65 (2002) 084014
A. Padilla, Class. Quantum Grav. 20 (2003) 3129
[17] C. M. Harris & P. Kanti, JHEP 010 (2003) 14
[18] A. Barrau et al., Astronom. Astrophys. 388 (2002) 676
[19] R. Emparan, G. T. Horowitz and R. C. Myers, Phys. Rev. Lett 85 (2000) 499
[20] T. Tj˜ostrand, Comput. Phys. Commun., 82 (1994) 74
[21] ATLAS TDR 14, vol 1 CERN/LHCC/99­14 (1999)
[22] S. O. Alexeyev, A. Barrau, G. Boudoul, M. Sazhin & O. S. Khovanskaya, Astronom. Letters 38, 7
(2002) 428
[23] D. Birmingham, Class. Quantum Grav. 16 (1999) 1197
[24] S. Alexeyev, N. Popov, A. Barrau & J. Grain, in preparation (2003)
226