Rotating BHs at future colliders:Greybody factors for brane fields

Kin-ya OdaKin-ya Oda, Tech. Univ. Munich

• Why Study BHs at Collider?• BH at Collider (Basic Facts)

• Production• BHs areare produced with large angular momentalarge angular momenta.• (Black ring formation)

• Evaporation• Brane-field eqs. are separable separable in any dim.• Greybody factorsGreybody factors for 5-d BH• Power spectraPower spectra from 5-d BH

• Summary

with S. Park and D. Ida hep-th/0212108hep-th/0212108, to appear in PRD

Why study BHs at collider?In ADD/RS1 scenario, hierarchy problem is solvedto result in the fundamental gravitational scale ~ O(TeV).

Experimentally accessible quantum gravity!!

There is no complete description at this non-perturbative region. There is no complete description at this non-perturbative region.

E, M

Correspondence principlein string theory

S, T, σ

g -2Ms

Another point of view

Truly QG effects will be observed as the deviationfrom the asymptotic behavior (in BH picture).

It is essential to predict BH behavior as precisely as possible!as precisely as possible!

BH pictureBH picture

String pictureString picture

BH production is NOTNOT mere a model but the general consequencegeneral consequence of the theoryif nature realizes ADD/RS1 scenario.

(Voloshin’s criticisms are already answered.)• Giddings 01• Eardley, Giddings 02; Yoshino, Naumbu 02• Dimopoulos, Emparan 02• …

BH at collider (basic facts)1.1. Balding PhaseBalding Phase

• Dynamical production phase• BH loses its “hair”.

2.2. Spin Down PhaseSpin Down Phase• BH loses its mass and angular momentum..

3.3. Schwartzschild PhaseSchwartzschild Phase• Angular momentum is small.• BH loses its mass.

4.4. Planck PhasePlanck Phase• Truly QG, highly unpredictable• A few quanta would be emitted.

Temperature gets higher and higher.Temperature gets higher and higher.

We consider the region where the produced BH is:• largelarge enough to be treated semi-classciallysemi-classcially and• smallsmall enough to be “spherical”“spherical” in the bulk.

BH radiates mainly on the brane.on the brane.

(Both are typically satisfied in LHC energy range.)

Rotating BH formation

It must must be producedwith finite angular momentum. finite angular momentum.

€

rh =rS(M )

(1+a*2)1/(n+1)

€

J =bM/2Initial angular momentum:

When we neglect balding phaseneglect balding phase,

€

b<2rh(M,J )

gives condition for BH formation.

€

bmax=21+n+2

2

⎛ ⎝ ⎜

⎞ ⎠ ⎟

2⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

−1n+1

rS(M )

There is maximum b allowed for BH formation.

The BH formation process is non-perturbative but classical.

bparton

parton

€

M /2

€

M /2

Banks, Fischler 99;Giddings, Thomas 01; Dimopoulos, Landsberg 01

Naively, BH forms when b <rS .~

RHS is decreasing function of J (or b).

rescaled angular momentum

This is obtained by This is obtained by neglecting balding neglecting balding phase.phase.

How good is this How good is this approximation?approximation?

€

J max~

2.9 (n=1)

4.5 (n=2)

M

11.5 (n=7)

⎧

⎨ ⎪ ⎪

⎩

⎪ ⎪

for M /M p =5

(typical LHC energy)

Our formula nicelynicely fits numerical result with full GR

Closed trapped surfaceClosed trapped surface formswhen b < bmax.

cf) Numerical result utilizesthe Aichelburg-Sexl solution(Eardley, Giddings 02) Yoshino, Nambu 02

Setup: two Schwarzschild BHs with• boost→∞,• mass→0,• energy: fixed.

t

z

b

€

R(n)=bmax

rS

A few % for A few % for n n =1 and ~1% for =1 and ~1% for n n >1.>1.

There are two direct There are two direct consequences.consequences.

So what?So what?

1. Production cross section becomes largerlarger

€

F =σ

πRS2

€

σ =πbmax2 =41+

n+22

⎛ ⎝ ⎜

⎞ ⎠ ⎟

2⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

−2n+1

πrS2

Production cross section becomes largerlarger when one take angular momentumangular momentum into account.

2. BHs are produced with large large angular momentaangular momenta

€

(J max=bmaxM /2)

€

dσdJ

=8πJ /M 2 (J <J max)

0 (J >J max)

⎧ ⎨ ⎩ ⎪

€

dσ =2πbdb

€

db

Differential cross sectionincreasesincreases linearly with J.

BHs are really produced with larlarge angular momentage angular momenta!!

€

J =bM/2Initial angular momentum:

Radiation from rotating BH

Once we have established that BHs are produced with large angular molarge angular momentamenta,we want to find out which sigwhich signal would resultnal would result.

Radiations from rotating BH

€

dEs,l,m

dt dω dΩ=

ω2π

Γs,l,m(ω)

eω−mΩ m1Ss,l,m(ω,Ω)

2

BH radiates mainly into the brane fieldsbrane fields via Hawking radiationHawking radiation.

Greybody factorsGreybody factors are obtained by solving the brane field equationsbrane field equations.

What we have found:• Brane field equationBrane field equation is separableseparable into angular and radial parts for any spin any spin ss and in any dimensions any dimensions nn.• We have analytically solved this equation for any spin any spin ss in 5 dim.5 dim. and found greybody factorsgreybody factors in low frequency expansionlow frequency expansion.• We show that radiations are highly anisotropicanisotropic (initially).

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Γ ∝ω2Conventionally one has simply assumed g.o. limitg.o. limit:

Greybody factorsGreybody factors for Brane fields determinedetermine the evolution of BH (up to Planck phase).

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−ddt

M

J

⎛

⎝ ⎜

⎞

⎠ ⎟ =

12π

gss,l,m

∑ dωΓs,l,m

eω−mΩ m1

ω

m

⎛

⎝ ⎜

⎞

⎠ ⎟ ∫

Higher dim. Kerr metric(just to show how it looks)

€

ds2 =g(4)(r,ϑ )+r2 cos2ϑ dΩn2

€

g(4)(r,ϑ )=

−Δ−a2sin2ϑ

Σ(Δ−r2 −a2)asin2ϑ

Σ0 0

*[(r2 +a2)2 −Δa2sin2ϑ ]sin2ϑ

Σ0 0

0 0ΔΣ

0

0 0 0 Σ

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

vanishes on the brane

€

Σ=r2 +a2 cosϑ

Δ=r2 1−μ

rn+1 +a2

r2

⎛

⎝ ⎜

⎞

⎠ ⎟

Newman Penrose formalism(just to show how it looks)

€

nμ =δμt −asin2ϑδμ

ϕ −ΣΔ

δμr

n'μ =Δ2Σ

δμt −asin2ϑδμ

ϕ( )+

12δμ

r

mμ =isinϑ

2(r+iacosϑ )aδμ

t −(r2 +a2)δμϕ

[ ]−r −iacosϑ

2δμ

ϑ

m'μ =mμ

We set null tetrad as follows:

€

Σ=r2 +a2 cosϑ

Δ=r2 1−μ

rn+1 +a2

r2

⎛

⎝ ⎜

⎞

⎠ ⎟

Spinor:

Vector:

€

g(4)μν∇μ∂νφ=0Scalar:

Brane field equations

Same as 4-dim. Can be treated in a standard manner.

This term This term vanishes for n=1 vanishes for n=1 (5-d)(5-d). For this case, we can . For this case, we can find greybody factors analytifind greybody factors analytically.cally.

€

1sinϑ

ddϑ

sinϑdSdϑ

⎛ ⎝ ⎜

⎞ ⎠ ⎟

+ (s−aωcosϑ )2 −(scotϑ +mcscϑ )2 −s(s−1)+A[ ]S=0

€

Δ−s ddr

Δs+1 dRdr

⎛ ⎝ ⎜

⎞ ⎠ ⎟

+K 2

Δ+s 4iωr −i

Δ,rKΔ

⎛

⎝ ⎜

⎞

⎠ ⎟ + Δ,rr −2( )+2maω−(aω)2 −A

⎡

⎣ ⎢

⎤

⎦ ⎥ R=0

angular part:

radial part:

€

K =(r2 +a2)ω−ma

Note: This term is absentabsent in Kanti, March-Russell 02 (appeared after our work) because they utilized Cvetic-Larsen equationCvetic-Larsen equation which essentially relies on the fact that this term vanvanishes (in 4-d)ishes (in 4-d).

€

−n(n−1)μr−n−1 Gives the greybody factor.

€

Σ=r2 +a2 cosϑ

Δ=r2 1−μ

rn+1 +a2

r2

⎛

⎝ ⎜

⎞

⎠ ⎟

€

φ,ψ0,φ0 ~R(r)S(ϑ )e−iω t + imϕDecomposition:

Greybody factors in 5-d(just to show how it looks)

€

matching NH (˜ ω ξ <<1) and FF (ξ >>1+ ˜ Q ) solutions

at the overlapping region (1+ ˜ Q <<ξ <<1˜ ω ), we obtain

R∞ =Yine−i ˜ ω ξ ξ

2

⎛ ⎝ ⎜

⎞ ⎠ ⎟

−1

+Youtei ˜ ω ξ ξ

2

⎛ ⎝ ⎜

⎞ ⎠ ⎟

−2s−1

.

Analytic solution in 5-d

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ξ =r −rh

rh˜ ω =rhω

˜ Q =ω−mΩ

2πT=˜ ω +O(a* )

Greybody factors in low frequency expansion.Greybody factors in low frequency expansion.

Scalar power spectrum

€

rhdE

dtdω

€

rhω

Spinor power spectra

€

rhdE

dtdω

€

rhω

Vector power spectrum

€

rhdE

dtdω

€

rhω

€

rhdE

dtdω

€

rhω

Scalar ang. power spectrum

€

a* =1.5

€

cosϑ

€

rhω

€

rhdE

dt dω dcosϑ

Spinor ang. power spectrum

€

cosϑ

€

rhω

€

a* =1.5

€

rhdE

dt dω dcosϑ

Vector ang. power spectrum

€

a* =1.5

€

cosϑ

€

rhω

€

rhdE

dt dω dcosϑ

Summary What we have done

– Production of rotating BH BHs are produced with large aungular momenta.large aungular momenta. Production cross section of BH becomes largerlarger

when one takes angular momentum into account.– Evaporation of rotating BH

Brane field equation is separable separable for any spin and in any dimensions.

Analytic expression of greybody factors for n=1 Power spectrum is substantially different fromdifferent from

g.o. limitg.o. limit. Especially spinor and vector are highly anisotropicanisotropic.

Works in progress– Greybody factor in any dimensions without low frequency limit– Complete determination of time integrated power and angular sptime integrated power and angular sp

ectrumectrum which can be observed in real experiment

Usable by experimentalists!!