The generalized uncertainty principle, motivated by string theory and
non-commutative quantum mechanics, suggests significant modifications to the
Hawking temperature and evaporation process of black holes. For
extra-dimensional gravity with Planck scale O(TeV), this leads to important
changes in the formation and detection of black holes at the the Large Hadron
Collider. The number of particles produced in Hawking evaporation decreases
substantially. The evaporation ends when the black hole mass is Planck scale,
leaving a remnant and a consequent missing energy of order TeV. Furthermore,
the minimum energy for black hole formation in collisions is increased, and
could even be increased to such an extent that no black holes are formed at LHC
energies.Comment: 5 pages, 2 figures. Minor changes to match version to appear in
  Class. Quant. Gra

Adelberger E G (EOT-WASH Group Collaboration)

Ahn E J

Amati D

Banks

Barvinsky A

Casadio R

Cavaglià M

Chamblin A

Chen P

Emparan R

Garay L J

Giudice G F

Ida D

Kanti P

Maggiore M

Marco Cavagli

Randall L

Roy Maartens

Saurya Das

English

arXiv

Cavagli, M.

Das, S.

Maartens, Roy

Portsmouth University Research Portal (Pure)

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Will we observe black holes at LHC?
Marco Cavaglia`∗
Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth P01 2EG, U.K.
Saurya Das†
Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick E3B 5A3, Canada
Roy Maartens‡
Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth P01 2EG, U.K.
(Dated: February 1, 2008)
The generalized uncertainty principle, motivated by string theory and non-commutative quantum
mechanics, suggests significant modifications to the Hawking temperature and evaporation process
of black holes. For extra-dimensional gravity with Planck scale O(TeV), this leads to important
changes in the formation and detection of black holes at the the Large Hadron Collider. The
number of particles produced in Hawking evaporation decreases substantially. The evaporation
ends when the black hole mass is Planck scale, leaving a remnant and a consequent missing energy
of order TeV. Furthermore, the minimum energy for black hole formation in collisions is increased,
and could even be increased to such an extent that no black holes are formed at LHC energies.
The possible existence of large extra dimensions
(LEDs) [1] has opened up new and exciting direc-
tions of research in quantum gravity. In this scenario,
standard-model particles are confined to the observable
4-dimensional “brane” universe, whereas gravitons can
access the whole d-dimensional “bulk” spacetime, being
localized at the brane at low energies. The effects of
LEDs show up in new particle states at energies below the
fundamental gravitational scale [2] and in nonperturba-
tive effects in the trans-Planckian energy regime (for a re-
view see [3].) The fundamental Planck scale in LED mod-
els may be as small as a TeV, within reach of near-future
experiments. This leads to many interesting possibilities
in “quantum gravity phenomenology”. One of the most
exciting quantum gravity signatures would be production
and decay of black holes (BHs) and other extended ob-
jects in cosmic ray air-showers and at particle colliders
such as the Large Hadron Collider (LHC) [4, 5, 6].
Hawking evaporation provides the observable signature
of BH formation at the TeV energy scale. After forma-
tion, BHs are expected to lose the hair associated with
multipole and angular momenta, decay via Hawking radi-
ation, and eventually either disappear completely or leave
a Planck-sized remnant. The final state of BH decay is
presently not understood, being in the realm of quantum
gravity theory. Here we investigate the implications for
BH decay of the generalized uncertainty principle (GUP).
It is commonly believed that quantum gravity implies
the existence of a minimum length [7]. This in turn leads
to a modification of the quantum mechanical uncertainty
∗Email: marco.cavaglia@port.ac.uk
†Email: saurya@math.unb.ca
‡Email: roy.maartens@port.ac.uk
principle:
∆xi &
~
∆pi
[
1 +
(
α′ℓPl
∆pi
~
)2]
, (1)
where ℓPl is the Planck length and α
′ is a dimension-
less constant of order one which depends on the details
of the quantum gravity theory. Equation (1) can be de-
rived in the context of non-commutative quantum me-
chanics [8], string theory [9] or from minimum length
considerations [10]. In the classical limit, ∆xi ≫ ℓPl,
we recover the standard Heisenberg uncertainty princi-
ple ∆x∆p & ~. The correction term in Eq. (1) becomes
effective when momentum and length scales are near the
Planck scale. Equation (1) implies a minimum length
scale:
∆xi & ∆xmin ≡ 2α′ℓPl . (2)
The Hawking thermodynamical quantities can be derived
heuristically by applying the uncertainty principle to 4-
dimensional BHs, and the GUP leads to modified ther-
modynamical quantities [11]. Here we generalize this to
d-dimensional BHs, which are relevant for BH production
at TeV scales.
A d-dimensional Schwarzschild BH has gravitational
potential Φ ∝ (rs/r)d−3, where the horizon radius is
given by
rs = ωdℓPlm
1/(d−3) , m =
M
MPl
. (3)
Here M is the mass and the dimensionless area factor is
ωd = {16π/[(d− 2)Ωd−2]}1/(d−3), where Ωd−2 is the area
of Sd−2. Since Hawking radiation is a quantum process,
the “emitted” quanta must satisfy the uncertainty prin-
ciple. By modelling a black hole as a (d−1)-dimensional
cube of size 2rs, the uncertainty in the coordinate of a
massless Hawking particle at emission is estimated as
2∆x ∼ 2rs. The true value is ∆x = 2Krs, where K
is a correction factor of order one that can in principle
be calculated for the spherical geometry of the horizon.
The Hawking temperature T can be identified up to nor-
malization with the energy uncertainty of the emitted
particles, T ∼ ∆E ∼ c∆p (the Boltzmann constant is
set to one). This leads to the formula for the BH tem-
perature [11]. If one uses the standard uncertainty rela-
tion, one deduces the standard formula [11], and this is
readily generalized to recover the d-dimensional Hawking
temperature [12]
T0 =
(
d− 3
4πωd
)
MPlc
2 m1/(3−d) . (4)
For the GUP case, ∆p = (2~/∆x)[1 +√
1− 4ℓ2Plα′2/∆x2]−1 and ∆x = 2Krs. This leads
to
T = 2T0
(
1 +
√
1− α
2
ω2dm
2/(d−3)
)−1
, (5)
where α = α′/K. Equation (5) generalizes the 4-
dimensional result of Ref. [11] to d dimensions, and also
to incorporate the GUP parameter α′ and the geometri-
cal correction factor K. In Ref. [11], α = 1.
From Eq. (5) it is evident that GUP-quantum gravity
effects increase the characteristic temperature of the BH.
Therefore, we expect quantum BHs to be hotter, shorter-
lived and with a smaller entropy than semiclassical BHs
of the same mass. The BH temperature is undefined for
M <Mmin, where
Mmin =
d− 2
8Γ
(
d−1
2
) (α√π)d−3 MPl . (6)
BHs with mass less than Mmin do not exist, since their
horizon radius would fall below the minimum allowed
length. Hence Hawking evaporation must stop once the
BH mass reachesMmin. This can be shown by calculating
the mass loss during evaporation. The energy radiated
per unit time is governed by the Stefan-Boltzmann law.
Following Ref. [13], we assume that the radiation takes
place mainly along the 4-dimensional brane. (We neglect
the energy losses due to emission of gravitons into the
bulk.) Since the surface gravity is constant over the hori-
zon, the Hawking temperature of the higher-dimensional
BH and that of the induced BH on the brane are identical.
Thus Eq. (5) can be used to calculate the emission rate.
The mass loss for a BH emitting on an n-dimensional
brane is given by [14]
dm
dt
= − 1
cMPl
σ¯nA(n)T
n , (7)
where A(n) is the area of the induced BH on the brane
and σ¯n is the effective n-dimensional Stefan-Boltzmann
constant,
σ¯n =
Ωn−3Γ(n)ζ(n)
(2π~c)n−1(n− 2)
∑
i
ci(n)Γsi(n)fi(n) . (8)
Here the sum is over all particle flavours, ci are the n-
dimensional degrees of freedom of the individual species,
the Γsi ’s are the n-dimensional greybody factors [15], and
fi(n) = 1 or 1 − 21−n for bosons or fermions. The area
of the induced BH on the brane is taken as the optical
area in n− 2 dimensions and is given by
A(n) = Ωn−2r
n−2
c , (9)
where rc = [(d− 1)/2]1/(d−3) [(d− 1)/(d− 3)]1/2 rs is the
optical radius [13]. Using Eq. (5) the BH rate of mass
loss on a 4-dimensional brane can be written as
dm
dt
= 16
(
dm
dt
)
0
(
1 +
√
1− α
2
ω2dm
2/(d−3)
)−4
, (10)
where the standard rate is(
dm
dt
)
0
= − µ
tPl
m−2/(d−3) , (11)
with
µ =
π2
120
(d− 1)(d−1)/(d−3)
22/(d−3)ω2d
(
d− 3
4π
)3∑
i
ci(4)Γsi(4)fi(4) .
(12)
As expected, the GUP-corrected mass loss is larger than
the standard Hawking mass loss. Equation (10) can be
integrated to give the decay time,
τ =
(
d− 3
16µ
)(
α
ωd
)d−1
I(4, d− 6, ωdm1/(d−3)/α) tPl ,
(13)
where
I(m,n, x) =
∫ x
1
dz zn
(
z +
√
z2 − 1
)m
. (14)
(This integral can be solved analytically.) Setting α = 0
we obtain the standard Hawking decay time in d dimen-
sions,
τ0 =
1
µ
(
d− 3
d− 1
)
m(d−1)/(d−3) tPl . (15)
The BH entropy is
S = 2πωd
(
α
ωd
)d−2
I(1, d− 4, ωdm1/(d−3)/α) , (16)
and is always smaller than the standard (α = 0) Hawking
value,
S0 =
(
4πωd
d− 2
)
m(d−2)/(d−3) . (17)
From Eq. (11) and Eq. (15), it is evident that the final
stage of standard Hawking evaporation is catastrophic.
The BH reaches in a finite time a stage with zero mass,
3infinite radiation rate and infinite temperature. By con-
trast, in the GUP scenario the existence of a minimum
length prevents the mass becoming smaller than Mmin,
Eq. (6). At this point the emission rate, Eq. (10), be-
comes imaginary. Since the emission rate is finite at
the end, it could be argued that the BH decays non-
thermally by emitting a hard Planck-mass quantum in a
finite time O(tPl), once the final stage of evaporation has
been reached. However, the specific heat, C = T∂S/∂T ,
is
C = −2πωdm(d−2)/(d−3)
√
1− α
2
ω2dm
2/(d−3)
×
×
(
1 +
√
1− α
2
ω2dm
2/(d−3)
)
, (18)
and vanishes at the endpoint, so that the BH cannot ex-
change heat with the surrounding space. Thus the end-
point of Hawking evaporation in the GUP scenario is
characterized by a Planck-sized remnant with maximum
temperature,
Tmax = 2T0
∣∣∣
M=Mmin
. (19)
The way that the GUP prevents BHs from evaporat-
ing completely is similar to the way that the standard
uncertainty principle prevents the hydrogen atom from
collapsing. The existence of a remnant as a consequence
of the GUP was pointed in Ref. [11], in the context of
primordial BHs in cosmology. Primordial BH remnants
are possible candidates for dark matter. Remnants have
also been predicted in certain models of quantum black
holes [16].
The multiplicity of a particle species i produced in BH
decay is [14]
Ni = N
ciΓsifi(3)∑
j cjΓsjfj(3)
, (20)
where the total multiplicity N is
N =
30ζ(3)
π4
S
∑
i ciΓsifi(3)∑
j cjΓsjfj(4)
. (21)
The observable effects of the GUP at particle collid-
ers are related to the existence of a minimum mass and
to the multiplicity of the decay. We assume that the
parton-parton center-of-mass (CM) energy Ecm is larger
than Mmin. Since the GUP-corrected entropy is smaller
than the standard one, we expect a significantly smaller
multiplicity, i.e., a smaller number of emitted particles in
the evaporation process, and a larger average energy of
the produced quanta. Moreover, the BH stops evaporat-
ing when it reaches the minimal mass. This should lead
to the detection of a missing energy of the order of the
minimal mass plus the missing energy due to invisible
decay products.
FIG. 1: Minimal mass for BH formation vs the parameter α,
for d = 6, . . . , 10 dimensions (from below). The yellow and
red regions are the exclusion zones for d = 10 and d = 6
respectively.
What happens if Ecm < Mmin? In this case, the col-
lision will produce no BH. Either a different nonpertur-
bative gravitational object (perhaps a brane [5] or string
ball [6]) forms in the collision, or the scattering is dom-
inated by nongravitational effects. In the former case,
depending on the details of the quantum gravity theory,
we could have formation of a gravitational object that
is either stable or does not produce any quanta during
the decay phase. Therefore, the formed object could be
totally invisible, with the only observable effect being a
missing energy of about the minimal mass in the final
state of the collision.
The minimal BH mass depends sensitively on the (un-
known) O(1) parameter α and on the spacetime dimen-
sion d. Figure 1 gives the parton-parton Ecm in Planck
units vs. the parameter α, for d = 6, . . . , 10 spacetime di-
mensions. A BH in d dimensions at fixed α can form only
if Ecm/MPl is above the curve corresponding to the space-
time dimension. For instance, if α = 1 and d = 10, BHs
with mass smaller than ∼ 4.72MPl do not form. If α = 2,
the minimum BH mass will be ∼ 600MPl! In this case,
LHC will not see any BH. The situation is somewhat bet-
ter in atmospheric ultra high-energy cosmic ray events,
because the collisions are expected to have a larger Ecm
than in particle colliders, and BHs are expected to form
with a higher average mass. However, the bulk of the
BHs which are formed in cosmic ray collisions have mass
of the order of few MPl, so that the rate of BH formation
in the GUP scenario could be dramatically suppressed.
In the worst case of large parameter α, no BHs will be
observed, either at particle colliders or in cosmic ray air-
showers.
The region of BH detection at LHC for d = 6 (yellow)
and d = 10 (red) dimensions is shown in Fig. 2. The up-
per line correspond to the CM energy of LHC (MPl = 14
TeV). The two lower lines correspond to the current ex-
4TABLE I: Thermodynamical quantities for two typical 10-dimensional BHs produced at LHC (masses M = 9 and 12 TeV),
assuming MPl = 1 TeV. The values in brackets give the percentage deviation from standard Hawking quantities.
M = 9 TeV
Min. mass (TeV) Temperature (TeV) Decay time (TeV−1) Entropy Multiplicity
α = 0 – .508 .689 15.5 5
α = 0.5 .037 .538 (+6%) .511 (-26%) 13.3 (-8%) 5 (0%)
α = 1.0 4.72 .721 (+42%) .071 (-90%) 5.24 (-66%) 2 (-60%)
M = 12 TeV
Min. mass (TeV) Temperature (TeV) Decay time (TeV−1) Entropy Multiplicity
α = 0 – .488 .997 21.5 7
α = 0.5 .037 .513 (+5%) .760 (-24%) 20.0 (-7%) 7 (0%)
α = 1.0 4.72 .657 (+35%) .157 (-84%) 9.62 (-55%) 3 (-57%)
FIG. 2: (α,MPl) moduli space for BH observation at LHC.
The yellow (red) region is for d = 6 (d = 10) spacetime di-
mensions.
perimental limits onMPl for d = 6 (MPl = 1.6 TeV, from
submillimeter tests of the gravitational inverse-square
law [17]) and d = 10 (MPl = 0.25 TeV, from collider
experiments [18]). The maximum value of the parame-
ter α that allows observation of BH formation at LHC is
α ∼ 1.6 (d = 6) and α ∼ 1.4 (d = 10).
If the parameter α is sufficiently small, BHs can still be
observed at LHC. If this is the case, what are the observ-
able signatures of the GUP? We have mentioned that the
multiplicity of the decay process is significantly smaller
than in the standard case. Table 1 shows the parameters
for two typical 10−dimensional BHs produced at LHC
with masses M = 9 and 12 TeV, and parameters α = 0,
0.5, 1, assuming MPl = 1 TeV. The most significant dif-
ference between the standard BH and the GUP-corrected
BH is given by the multiplicity of the decay products.
As α increases, the total number of particles produced
during the evaporation phase becomes smaller. The sup-
pression in the multiplicity is most severe for α = 1 and
M = 9 TeV. Indeed, as α increases and M decreases,
the minimum BH mass Mmin becomes a significant frac-
tion of the BH mass M . Therefore, a smaller amount of
energy can be converted into Hawking thermal radiation.
The decrease in multiplicity can lead to important ob-
servational signatures. The hadron to lepton ratio of the
decay products is roughly expected to be 5:1. In the case
M = 12 TeV, α = 0, for instance, the BH evaporates into
five quarks, one charged lepton, and one gluon, with the
charged leptons accounting for about 15% of the emis-
sion. As the parameter α increases the leptonic compo-
nent becomes negligible. When α = 1, for example, the
charged leptonic component is reduced to ∼ 6% of the
emission.
What happens after Hawking evaporation has
stopped? The BH remnant becomes invisible to the de-
tector. This leads to a large missing energy. (For α = 1
the missing energy of the 9 TeV BH is more than 50% of
the total CM energy.) Therefore, an event with charac-
teristics of a standard BH event (large visible transverse
energy and high sphericity), but with smaller multiplic-
ity and large missing energy, would be the GUP smoking
gun.
In summary, the generalized uncertainty principle of
string theory or non-commutative quantum mechanics
brings important qualitative and quantitative changes to
BH thermodynamics. The BH is hotter, evaporates more
rapidly, and has smaller entropy, relative to the stan-
dard case. BH evaporation involves a smaller number of
particles with higher average energy. And the endpoint
of evaporation is a Planck-scale remnant with zero heat
capacity. We have shown, in the context of TeV-scale
higher-dimensional gravity, that these changes could have
significant implications for the possible formation and de-
tection of BHs at LHC (and also in cosmic ray showers).
In particular, the minimum energy for formation is in-
creased, and could be pushed beyond the reach of LHC
even if MPl ∼ 1 TeV.
5Acknowledgements
We thank R. Bhaduri, P. Chen, J. Gegenberg, V. Hu-
sain, M. Maggiore and J. Waddington for useful com-
ments and discussions. M.C. and R.M. are supported by
PPARC (UK), and S.D. by NSERC (Canada).
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Will we observe black holes at the LHC?

The generalized uncertainty principle, motivated by string theory and non-commutative quantum mechanics, implies significant modifications to the Hawking temperature and evaporation process of black holes. For extra-dimensional gravity with Planck scale O(TeV), this leads to important changes in the formation and detection of black holes at the the Large Hadron Collider. The number of particles produced in Hawking evaporation decreases substantially. The evaporation ends when the black hole mass is Planck scale, leaving a remnant and a consequent missing energy of order TeV. Furthermore, the minimum energy for black hole formation in collisions is increased, and could even be increased to such an extent that no black holes are formed at LHC energies

Cavaglià, M

Das, S

Maartens, R

CERN Document Server

Will we observe black holes at LHC?

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The generalized uncertainty principle, motivated by string theory and non-commutative quantum mechanics, suggests significant modifications to the Hawking temperature and evaporation process of black holes. For extra-dimensional gravity with Planck scale O(TeV), this leads to important changes in the formation and detection of black holes at the large hadron collider. The number of particles produced in Hawking evaporation decreases substantially. The evaporation ends when the black-hole mass is Planck scale, leaving a remnant and a consequent missing energy of order TeV. Furthermore, the minimum energy for black-hole formation in collisions is increased, and could even be increased to such an extent that no black holes are formed at LHC energies

Cavaglia, Marco

Das, Saurya

Missouri University of Science and Technology (Missouri S&T): Scholars' Mine

Will We Observe Black Holes at the LHC?

Will we observe black holes at LHC?

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Missouri University of Science and Technology (Missouri S&T): Scholars' Mine