Rotating BHs at future colliders:
Greybody factors for brane fields
Kin-ya Oda, Tech. Univ. Munich
with S. Park and D. Ida
hep-th/0212108, to appear in PRD
• Why Study BHs at Collider?
• BH at Collider (Basic Facts)
• Production
• BHs are produced with large angular momenta.
• (Black ring formation)
• Evaporation
• Brane-field eqs. are separable in any dim.
• Greybody factors for 5-d BH
• Power spectra from 5-d BH
• Summary
Why study BHs at collider?
In ADD/RS1 scenario, hierarchy problem is solved
to result in the fundamental gravitational scale ~ O(TeV).
criticisms
BH production is NOT mere a model but (Voloshin’s
are already answered.)
•
Giddings
01
the general consequence of the theory • Eardley, Giddings
02; Yoshino, Naumbu 02
• Dimopoulos, Emparan 02
if nature realizes ADD/RS1 scenario.
•…
Experimentally accessible quantum gravity!!
Another point of view
S, T, σ
Correspondence principle
in string theory
BH picture
String picture
There is no complete description at this non-perturbative region.
g -2Ms
E, M
Truly QG effects will be observed as the deviation
from the asymptotic behavior (in BH picture).
It is essential to predict BH behavior as precisely as possible!
BH at collider (basic facts)
1. Balding Phase
•
•
Dynamical production phase
BH loses its “hair”.
2. Spin Down Phase
•
BH loses its mass and angular momentum.
3. Schwartzschild Phase
•
•
Angular momentum is small.
BH loses its mass.
4. Planck Phase
•
•
Truly QG, highly unpredictable
A few quanta would be emitted.
BH radiates mainly
on the brane.
We consider the region where the produced BH is:
• large enough to be treated semi-classcially and
• small enough to be “spherical” in the bulk.
(Both are typically satisfied
in LHC energy range.)
Temperature gets higher and higher.
Rotating BH formation
The BH formation process is nonperturbative but classical.
There is maximum b allowed for
BH formation.
Naively, BH forms when b <r
~ S.
1
2 n1
Banks, Fischler 99;
Giddings, Thomas 01; Dimopoulos, Landsberg 01
M /2
parton
b
bmax
parton
M /2
It must be produced
with finite angular momentum.
J max
 n  2  
 21 
  rS (M )

  2  

2.9 (n  1)

4.5 (n  2)
~ 
for M / M p  5

11.5 (n  7) (typical

J  bM / 2
Initial angular momentum:
When we neglect balding phase,
b  2rh ( M,J )
rS ( M )
rh 
2 1/(n1)
(1 a* )
gives condition for
BH formation.
rescaled angular momentum
RHS is decreasing function of J (or b).
LHC energy)
This is obtained by
neglecting balding
phase.
How good is this
approximation?
Our formula nicely fits
numerical result with full GR
bmax
R(n) 
rS
A few % for n =1 and ~1% for n >1.
cf) Numerical result utilizes
the Aichelburg-Sexl solution
b
(Eardley, Giddings 02) Yoshino, Nambu 02
t
z
Closed trapped surface forms
when b < bmax.
Setup: two Schwarzschild BHs with
• boost→∞,
• mass→0,
So what?
• energy: fixed.
There are two direct
consequences.
1. Production cross section
becomes larger

F 2
RS
  b
2
max
2
2 n1
 n  2  
2


 4 1 
rS


  2  

Production cross section becomes larger when
one take angular momentum into account.
2. BHs are produced with large
angular momenta
Initial angular momentum:
db
J  bM / 2
2

8

J
/
M
(J  J max )
d
 
dJ 
(J  J max )
 0
d  2bdb
(J max  bmax M /2)
Differential cross section
increases linearly with J.
BHs are really produced with
large angular momenta!!
Radiation from rotating BH
Once we have established that BHs
are produced with large angular
momenta,we want to find out which
signal would result.
Radiations from rotating BH
BH radiates mainly into the brane fields via Hawking radiation.
dEs,l,m
2
 s,l,m ( )

Ss ,l,m ( ,)
 m
dt d d 2  e
1
Greybody factors for Brane fields determine the evolution of BH (up to
Planck phase).
M 
 

d
1
s,l,m
  
g
d


 
s 
 m
dt  J  2  s ,l,m
e
1 m 
Greybody factors are obtained by solving the brane field equations.
Conventionally one has simply assumed g.o. limit:
 
2
What we have found:
• Brane field equation is separable into angular and radial parts
for any spin s and in any dimensions n.
• We have analytically solved this equation for any spin s in 5 dim.
and found greybody factors in low frequency expansion.
• We show that radiations are highly anisotropic (initially).
Higher dim. Kerr metric
(just to show how it looks)
ds 2  g (4) (r,  )  r 2 cos2  d2n
g (4 ) (r,  ) 
   a 2 sin 2 




*



0

0

vanishes on the brane
(  r 2  a 2 )asin 2 

[(r 2  a 2 )2  a 2 sin 2  ]sin 2 

0
0

0 0

0 0



0


0 
  r 2  a 2 cos
2 


a
  r 2 1 n1  2 
r 
 r
Newman Penrose formalism
(just to show how it looks)
We set null tetrad as follows:
 r
t
2

n    asin   

 t
1 r
2

n' 


asin






2
2
i sin 
r  ia cos 
t
2
2

m 
a


(r

a
)






2(r  ia cos )
2
m'  m
Scalar:
Spinor:
Vector:
g(4 )    0
  r 2  a 2 cos
  a 2 
2
  r 1 n1  2 
r 
 r
Brane field equations
Decomposition: , 0 , 0 ~ R(r)S( )e
angular part:
1 d 
dS 
sin 

sin  d 
d 
i t  im 
  r 2  a 2 cos
  a 2 
2
  r 1 n1  2 
r 
 r
 (s  a cos  )2  (s cot   m csc  )2  s(s 1)  AS  0
radial part:
Same as 4-dim. Can be treated in a standard manner.
2
2
d  s 1 dR 
K

(r

a
)  ma



dr 
dr 
K 2 

,r K 
2
   s4ir  i
 ,rr  2 2ma  (a )  AR  0

 
 

n1
Gives the greybody factor.
n(n 1)r
s
Note: This term is absent in Kanti, MarchRussell 02 (appeared after our work)
because they utilized Cvetic-Larsen
equation which essentially relies on the
fact that this term vanishes (in 4-d).
This term vanishes for n=1
(5-d). For this case, we can
find greybody factors
analytically.
Greybody factors in 5-d
(just to show how it looks)
Analytic solution in 5-d
matching NH ( 
˜  1) and FF (   1 Q˜ ) solutions
1
˜
at the overlapping region ( 1 Q    ), we obtain

˜
1
2s 1






R  Yin ei˜    Yout ei˜   .
2 
2 
Greybody factors in low frequency expansion.
r  rh
rh

˜  rh
  m
Q˜ 
 
˜  O(a* )
2 T

Scalar power spectrum
dE
rh
dtd
rh
Spinor power spectra
dE
rh
dtd
rh
Vector power spectrum
dE
rh
dtd
rh
dE
dtd
rh
rh
Scalar ang. power spectrum
a* 1.5
dE
rh
dt d d cos 
cos 
rh
Spinor ang. power spectrum
a* 1.5
dE
rh
dt d d cos 
cos 
rh
Vector ang. power spectrum
a* 1.5
dE
rh
dt d d cos 
cos 
rh
Summary

What we have done
– Production of rotating BH


BHs are produced with large aungular momenta.
Production cross section of BH becomes larger
when one takes angular momentum into account.
– Evaporation of rotating BH





Brane field equation is separable for any spin and in any
dimensions.
Analytic expression of greybody factors for n=1
Power spectrum is substantially different from
g.o. limit.
Especially spinor and vector are highly anisotropic.
Works in progress
– Greybody factor in any dimensions without low frequency limit
– Complete determination of time integrated power and angular
spectrum which can be observed in real experiment
Usable by experimentalists!!