Black Hole's Life at colliders
arXiv:0704.1056v1  [hep-ph]  9 Apr 2007
Black hole’s life at colliders
Seong Chan Park∗†
FRDP, School of astronomy and physics, Seoul National University, Seoul 151-742, Korea
E-mail: spark@phya.snu.ac.kr
In the series of papers by Ida, Oda and Park, the complete description of Hawking radiation to the
brane localized Standard Model fields from mini black holes in the low energy gravity scenarios
are obtained. Here we briefly review what we have learned in those papers.
From Strings to LHC
January 2-10 2007
International Centre Dona Paula, Goa India
∗Speaker.
†We thanks to Daisuke Ida and Kin-ya Oda for valuable collaborations.
c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/
http://arxiv.org/abs/0704.1056v1
mailto:spark@phya.snu.ac.kr
Black hole’s Life at colliders Seong Chan Park
1. Introduction
We briefly review recent developments in the mini black hole production and evaporation
mainly based on the series of works done by Ida, Oda and Park [1, 2, 3, 4]. In the low energy
gravity scenarios such as ADD and RS-I, the CERN Large Hadronic collider (LHC) will become
a black hole factory [5, 6]. Above the TeV Planck scale, the classical production cross section of
the (4+n)-dimensional black hole grows geometrically σ ∼ ŝ1/(n+1), with
ŝ being the center of
mass energy of the parton scattering.
Once produced, black hole loses its energy or mass primarily via Hawking (thermal) radia-
tion. The Hawking radiation goes mainly into the standard model quarks and leptons (spinors) and
gauge bosons (vectors) localized on the brane, except for a few gravitons and Higgs boson(s). The
quanta of Hawking radiation will have characteristic energy spectrum determined by the Hawking
temperature and the greybody factor. The process of Hawking radiation in four dimensional rotat-
ing black hole has been treated in detail by Teukolsky, Press, Page and others in 1970s’. In higher
dimensions, however, it is shown that the process has quite different features.
• Hawking temperaure TH ∝ (Mbh)/M∗)1/(n+1) of a (4+ n) dimensional black hole is much
higher than 4 dimensional one with the small fundamental scale M∗ ∼ TeV ≪ MPlanck. With
this high temperature, the number of available degrees of freedom for Hawking radiation
are much bigger in (4+n) dimensions with all the standard model particles localized on the
brane [8].
• The near horizon geometry of (4+ n) dimensional black hole is quite complicated. Its ge-
ometry is different from that of a four dimensional Kerr black hole. With the highly mod-
ified geometry in the vicinity of the event horizon, frequency dependent correction factor
of Hawking radiation, i.e., greybody factor, is also largely modified (also see the references
[9, 10, 11] as independent studies on the same topic).
To understand the physics of those black holes, we have to understand the greybody factor of
higher dimensional, rotating black hole [12, 13].
2. Generalized Teukolsky equation and greybody factor
The induced metric on the three-brane in the (4+ n)-dimensional Myers-Perry solution [7]
with a single nonzero angular momentum is given by
∆−a2 sin2 ϑ
dt2 +
2a(r2 +a2 −∆)sin2 ϑ
2 +a2)2 −∆a2 sin2 ϑ
sin2 ϑdϕ2 − Σ
dr2 −Σdϑ2, (2.1)
where
Σ = r2 +a2 cos2 ϑ , ∆ = r2 +a2 −µr1−n. (2.2)
Black hole’s Life at colliders Seong Chan Park
The parameters µ and a are equivalent to the total mass M and the angular momentum J
(2+n)A2+nµ
16πG4+n
, J =
A2+nµa
8πG4+n
(2.3)
evaluated at the spatial infinity of the (4+n)-dimensional space-time, respectively, where A2+n =
2π(3+n)/2/Γ((3+ n)/2) is the area of a unit (2+ n)-sphere and G4+n is the (4+ n)-dimensional
Newton constant of gravitation.
1.Subtracting outgoing wave contamination at NH and separating ingoing and outgoing wave
at FF are described. 2.Here we answer the question :what fraction of energy would be radiated into
Hawking radiation in spin-down phase.
2.1 Asymptotic solutions in Kerr-Newman frame
We are given a linear, second-order equation, say
+ τR = 0, (2.4)
where η and τ are determined in Kerr-Newman frame as:
η = −
(s−1)∆′+2iK
, (2.5)
2iωr(2s−1)−λ
. (2.6)
∆ = r2 +a2 − (r2H +a2)
K = (r2 +a2)w−ma. (2.7)
The asymptotic solutions are given at NH and FF limits:
RNH ∼ Win +Woute2ikr∗∆s, (2.8)
RFF ∼ Yinr2s−1 +Youte2ikr∗/r. (2.9)
2.2 BC: Subtracting outgoing contamination at NH
The solutions near the horizon r → rH are
RNHin = 1+a1(r− rH)+
(r− rH)2 + · · · ,
RNHout = e
2ikr∗(r− rH)s (1+b1(r− rH)+ · · ·) , (2.10)
where the coefficients ai’s and bi’s are straightforward to compute:
a1 = −
, (2.11)
a2 = −
(η0 + τ−1)a1 + τ0
1+η−1
, (2.12)
· · · , (2.13)
Black hole’s Life at colliders Seong Chan Park
where η j and τ j are j-th order coefficients of Taylor expansion of η and τ , respectively.
For s = 1/2 and 1,
τ−1 =
2iωδs,1 −λ
, (2.14)
τ0 = λ
+δs,1
(∆1 −∆2), (2.15)
η−1 =
δs,1/2 −2i
, (2.16)
η0 = −
(K1∆1 −K0∆2), (2.17)
where
∆1 = 2+(n−1)(1+a2), (2.18)
∆2 = 1−n(n−1)(1+a2)/2, (2.19)
K0 = (1+a
2)ω −am, (2.20)
K1 = 2ω . (2.21)
The problem is to integrate Eq.2.4 from purely ingoing initial conditions at r = r0 out to r → ∞.
Choosing the positive choice for s makes Yin stable and easily determined by an outward integration.
However, for such an integration RNHout is unstable against contaminating the purely ingoing solution.
We can avoid the difficulty in mid-integration, relying on a mathematical transformation of the
equation to stabilize the solutions in the two asymptotic regimes r → rH and r → ∞ as follows. To
counteract the above contamination, let
R̃ = R− (1+a1(r− rH)). (2.22)
Then f satisfies the equation
L R̃ = g, (2.23)
where L = d2/dr2+ηd/dr+τ and g=−L (1+a1(r− rH)) =−ηa1−τ (1+a1(r− rH)). Equa-
tion 2.23 is now stably integrated through both asymptotic regimes, i.e., from the near horizon to
far field regimes.
Near the horizon, R̃ becomes
R̃(r → rH) =
(r− rH)2 + · · · . (2.24)
2.3 Separating solution at FF
For s = 1/2,
RFFin ∼ 1+
+ · · · ,
RFFout ∼ e2ikr∗
· · ·), (2.25)
Black hole’s Life at colliders Seong Chan Park
where
1 = −i
, (2.26)
· · · (2.27)
Then, R̃1/2 becomes
R̃1/2(r → ∞)≃ (Yin +1−a1)+a1r+
+Yout
e2iωr∗
. (2.28)
Using this expression, we can easily read out Y ’s without numerical difficulties.
Finally, the greybody factor could be written as
Γs=1/2 = 1−
|c f1 |
|Yout|2
|Yin|2
. (2.29)
For s = 1,
RFFin ∼ r(1+
+ · · ·),
RFFout ∼ e2ikr∗
+ · · ·), (2.30)
where
cv1 = −i
, (2.31)
cv2 = −
λ 2 −4aω(aω −m)
, (2.32)
· · · .
Then, R̃1 becomes
R̃1(r → ∞)≃ (Yin −a1) r+(Yincv1 −1+a1)+
+Yout
e2iωr∗
. (2.33)
Using this expression, we can easily read out Y ’s without numerical difficulties.
Finally, the greybody factor could be written as
Γs=1 = 1−
|cv2|
|Yout|2
|Yin|2
. (2.34)
3. Hawking radiations of mini-black hole
Schematically black hole evolution follows five successive steps as is depicted in Fig.1: the
production phase, the balding phase, the spin-down phase, the Schwarzschild phase and the Planck
phase. When a black hole is produced in high energy collision (production phase), the geometry
is highly irregular, and could even be topologically non-trivial. By emitting (bulk) gravitons and
other particles, the black hole will be settled down to a rotating black hole which is supposed to be
well described by Myers-Perry solution in (4+n) dimensional spacetime (balding phase).
Black hole’s Life at colliders Seong Chan Park
The decay in spin-down and Schwarzschild phases are calculable in terms of Hawking radi-
ation. We are interested in those phases (spin-down phase and Schwarzschild phase) and would
answer what fraction of energy will be lost in each of these phases.
The rate of energy (and angular momentum) loss by Hawking radiation is given as follows:
2π ∑s,l,m
dω〈Ns,l,m〉
, (3.1)
where gs is the number of “massless" degrees of freedom at temperature T , namely, the number of
degrees of freedom whose masses are smaller than T , with spin s. The expected number of particles
of the species of spin s emitted in the mode with spheroidal harmonics l, axial angular momentum
〈Ns,l,m〉=
sΓl,m(a,ω)
e(ω−mΩ)/T − (−1)2s
. (3.2)
1 2 3 4 5
Ω r_h
D=10, s=1, a*=0.01
1 2 3 4 5 6
Ω r_h
D=10, s=1, a*=0.3
2 4 6 8
Ω r_h
D=10, s=1, a*=0.6
2 4 6 8 10 12
Ω r_h
D=10, s=1, a*=0.9
2.5 5 7.5 10 12.5 15
Ω r_h
D=10, s=1, a*=1.2
5 10 15 20
Ω r_h
D=10, s=1, a*=1.5
Figure 1: Hawking radiation from D = 10 black hole. s = 1,a = 0.01,0.3,0.6,0.9,1.2,1.5.
Black hole’s Life at colliders Seong Chan Park
4. Time evolution
From the ratio of energy and angular momentum in eq.3.1, we can define a scale invariant
function γ(as = a/rs) as follows:
γ−1(as) ≡
d lnas
d lnM
(4.1)
. (4.2)
Now we calculate the ratio of final(M f ) and initial(Mi) energy of black hole by integrating the
eq.4.1 with as(ini) for initial angular momentum.
= Exp
∫ as(final)
as(ini)
γ(as)
. (4.3)
The amount of energy which is radiated in spin-down phase (0 ≈ as(final)6 as 6 as(ini)) is (Mi −
M f ) and M f will be also radiated in Schwarzschild phase where the angular momentum of black
hole is zero.
Next, let us consider the evolution of the black hole. Since time scales as rn+3s in (4+ n)
dimensions 1, it is convenient to introduce scale invariant rates for energy and angular momentum
as follows.
α(as) ≡ −rn+3s
d lnM
, (4.4)
β (as) ≡ −rn+3s
d lnJ
, (4.5)
with these new variables γ(as) can be written as γ−1(as) = β/α(as)− (n+ 2)/(n+ 1). We also
introduce dimensionless variables y and z to take angular momentum and mass of the hole:
y ≡ − lnas, (4.6)
z ≡ − ln
, (4.7)
then finally we get the time variation of energy and angular momentum in terms of scale-invariant
time parameter (τ = r−n−3s (ini)t) with initial mass of the hole:
= (β −α
n+1 z. (4.8)
After finding the solutions z(y) and τ(y) of the coupled differential equations 4.8, one can get
y(τ) and z(τ), hence as and M/Mi, as a function of time. From these, one can find how other
quantities evolve, such as the area.
1We can easily understand this by simply looking at the formula −dM/dt ∼ AT 4 where the surface area of horizon
A ∼ r2s for brane fields and the temperature of the hole T ∼ 1/rs and M ∼ rn+1s .
Black hole’s Life at colliders Seong Chan Park
0 0.5 1 1.5 2 2.5
PSfrag replacements
0 0.2 0.4 0.6 0.8 1
PSfrag replacements
Figure 2: Evolution of bh in D = 10.
Up to now we have used a unit where the size of event horizon is fixed as rh = 1 and angular
momentum of the hole is parameterized by (ah = a/rh). For conversion of unit, the following
expressions are useful with as = ah/(1+a
1/(n+1).
α(as) = −ιn+1n (1+a2h)
n+1 r2h
, (4.9)
β (as) = −κn+1n (1+a2h)
n+1 r2h
, (4.10)
where
ιn = rsM−
n+1 =
(n+2)Ωn+2
, (4.11)
κn = ιn(
n+1 . (4.12)
In Fig.2, black hole evolution in units of the initial mass as a function of rotation parameter as
for scalar(s), fermion(f), vector(v), and sum of all the standard model particles(SM) in D = 5(left)
and D = 10 (right) are shown. The initial angular momentum parameter is fixed by as = 0.83
and 2.67 in D = 5 and D = 10 that are the maximal rotations allowed by the initial collision,
respectively. The mass of the hole goes to zero before the rotation parameter goes to zero when
only scalar emission is available. However, when all the standard model fields are turned on, the
evolution is essentially determined by the spinor and vector radiation. It is found that a black hole
spins down quickly at the first stage with large rotation parameter as and the decrease of rotation
parameter slows down as angular momentum of the hole is reduced.
When all the standard model fields are turned on (SM), the evolution is essentially determined
by the spinor and vector radiation. The figures show that a black hole spins down quickly at the first
stage with large rotation parameter and the decrease of rotation parameter slows down as angular
momentum of the hole is reduced.
5. Summary and Discussion
The complete description of Hawking radiation to the brane localized SM fields and the con-
sequent time evolution of mini black hole in the context of low energy gravity scenario has been
made.
Black hole’s Life at colliders Seong Chan Park
We have developed analytic and numerical methods to solve the radial Teukolsky equation
which has been generalized to the higher dimension (D = 4+n). Two main points in our numerical
methods are as follows. First, we have imposed the proper purely-ingoing boundary condition near
the horizon without the growing contamination of the out-going wave by extracting lower order
terms explicitly. Second, we have developed the method to fit the in-going and out-going part from
the numerically integrated wave solution at far field region by explicitly obtaining the next-to-next
order expansion (or next-to-next-to-next order in vector case) of the solution. With these progress
in numerical treatment, we can safely integrate the generalized Teukolsky equation up to very large
r without out-going wave contamination.
Then we have calculated all the possible modes to completely determine the radiation rate of
the mass and angular momentum of the hole. Totally 3407 are computed explicitly, other than the
modes which are confirmed to be negligible. A black hole tends to lose its angular momentum at
the early stage of evolution. However the black hole still have a sizable rotating parameter after
radiating half of its mass. More than 70% or 80% of black hole’s mass is lost during the spin down
phase.
Now that we have completely determined the radiation and evolution of the spin-down and
Schwarzschild phases, only remaining hurdle is the evaluation of the balding phase, which is still
being disputed due to its non-purturbative nature, to extract the quantum gravitational information
at the Planck phase from the experimental data at LHC.
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