Seong Chan Park SNU Rotating Black Holes at future colliders I: Greybody factors for brane fields D. Ida, K.-y. Oda, SPARK, Phys.Rev.D67:06405,003, Erratum-ibid.D69:049901,004. II: Anisotropic scalar field emission D. Ida, K.-y. Oda, SPARK, Phys.Rev.D71:14039,005. III: Determination of black hole evolution D. Ida, K.-y. Oda, SPARK, Phys.Rev.D73:140,006.
Suppose that you want to make a black hole for your wife.. How can you make it? Hoop Conjecture (Kip Thorne 197) states An imploding object forms a Black Hole when, and only when, a circular hoop with a specific critical circumference could be placed around the object and rotated. The critical circumference is given by times Pi times the Schwarzschild Radius corresponding to the object s mass. So, build enough energy in a small space!!
It s like putting an elephant into a freezer..
Particle Physicists idea Collide two particles with a big CM energy. G. 't Hooft Phys.Lett.B198, 61 (1987) ( E,0,0, E) E 6 0.510 MPl ( E,0,0, E) b 6 ECM E 10 MPl R GE M 6 S CM 10 / Pl ( 1/ ) G M Pl If the impact parameter is smaller than Million times Planck length, a black hole forms.
Current technology (and budget) allows us just limited amount of CM energy. LEP-II (CERN) : 00 GeV (closed) Tevatron (Fermi Lab.) : GeV (running) The LHC (CERN): 14 TeV(from 008-) Require enormous adjustment: b E 1 1 M M M CM 1TeV=1000 GeV M Pl Pl Pl Pl 19 1.1 10 GeV It seems impossible to make a BH.
Wait! If there are extra dimensions, gravity becomes stronger at small distances: Mm G4, ( r r ) C r Mm G4 n, ( r r ) n C r Also notice that the Planck scale has to be renormalized: " Vol( n)" 1 G G n M4 n " Vol ( n)" M4 4n 4
Current experimental data suggest Microscopic torsion-pendulum Experiment rc 197m Hoyle et.al. Phys.Rev.D70:04004,004 M4 n O(1)TeV TeV scale gravity is an open possibility.
TeV gravity Is Realized in many string theory models. Arkani-Hamed, Dimopoulos, Dvali, Phys.Lett.B49:63-7,1998. L. Randall, R. Sundrum, Phys.Rev.Lett.83:3370-3373,1999 Is able to address Hierarchy problem. ( = why is Higgs so light? = why gravity is so week? ) Got a lot of attention from physics community. (String theory community, GR community, Particle physics community, SF community )
Black hole by high energy collision The LHC as a black hole factory T. Banks /W. Fischler, ``A model for high energy scattering in quantum gravity,'' hep-th/9906038. S. B. Giddings / S. D. Thomas, ``High energy colliders as black hole factories: The end of short distance physics,'' Phys. Rev. D65, 056010 (00) S. Dimopoulos / G. L. Landsberg, ``Black holes at the LHC,'' Phys. Rev. Lett.87, 16160 (001) Black holes from the sky J. L. Feng / A. D. Shapere, ``Black hole production by cosmic rays,'' PRL88, 01303 (00) L. Anchordoqui / H. Goldberg, ``Experimental signature for black hole production in neutrino air showers,'' PRD 65, 04750 (00) R. Emparan/ M. Masip /R. Rattazzi, ``Cosmic rays as probes of large extra dimensions and TeV gravity,'' PRD65, 06403 (00)
Now we have obvious and urgent questions Q1) How many black holes will be produced? Determination of production cross section. Q)What will be their signals? Determination of black hole s life. Decay pattern of black hole. Need to calculate Greybody factors.
Set-up(1) Our black hole is best described by (4+n)Dimensional, rotating black hole solution. R. C. Myers and M. J. Perry, ``Black Holes In Higher Dimensional Space-Times,'' Annals Phys. 17, 304 (1986). ds g (4) ( r, ) r cos dn g (4) (r,) a sin ( r a )asin 0 0 [(r a ) a sin ]sin * 0 0 0 0 0 0 0 0 r a cos r a r r 1 n1
Set-up() We live on 3-brane world. (as was suggested by ADD, RS) y i e x Electron, photon, gluon, quarks, Me, Don. N. Page, everything except graviton is confined on this hyper surface (D3).
Bulk vs Brane emission Boltzmann law 1 1 de / dt A T r ( ),(brane) 4 4 4 s rs rs 1 1 de / dt A T r ( ),(bulk) 4n n 4n 4n s rs rs de de ( brane) ( bulk), (for each d.o.f) dt dt d.o.f. (brane)=all the SM particles d.o.f.(bulk) = graviton R. Emparan, G. T. Horowitz and R. C. Myers, ``Black holes radiate mainly on the brane,'' Phys. Rev. Lett.85, 499 (000)
Production Cross section M/ M, J Mb / b M/ r a 1 ( ) 0 r r s 1n r r ( M, J) r ( M)(1 a ) 1/ n1 H S 1/ n1 S ( ) n( 4n ), n (1) r M C G M C O Hoop Conjecture: n1 n 41 rs bmax D. Ida, K.-y. Oda, S.PARK, Phys.Rev.D67:06405,003, Erratum-ibid.D69:049901, 004
Related numerical studies b t cf) Numerical result utilizes the Aichelburg-Sexl solution z Setup: two particles (BHs) with boost, mass 0, energy: fixed. D. M. Eardley and S. B. Giddings, ``Classical black hole production in high-energy collisions,'' Phys. Rev. D66, 044011 (00) H. Yoshino and Y. Nambu, ``Black hole formation in the grazing collision of high-energy particles,'' Phys. Rev.D 67, 04009 (003) H. Yoshino, A. Zelnikov and V. P. Frolov, ``Apparent horizon formation in the head-on collision of gyratons,'' arxiv:gr-qc/070317. Closed trapped surface forms when b < b max.
Excellent Agreements in the Results: R( n) b max r S Yoshino-Nambu (0 ), Yoshino et.al.(04,05 ) n 1 3 4 5 6 7 R YN 1.056 1.158 1.8 1.76 1.314 1.344 1.368 R IO P 1.110 1.170 1.18 1.6 1.300 1.334 1.364 Ida, Oda, Park PRD03 Error ~ a few %
BH Differential production cross section Geometrical Cross section Form factor F r S n1 n 41 rs bmax n 1 3 4 5 6 7 F YN 1.084 1.341 1.515 1.64 1.741 1.819 1.883 F IO P 1.31 1.368 1.486 1.59 1.690 1.780 1.863 db d bdb d 8 J / M dj 0 ( Jmax bmax ( J ( J M / ) J J max max Most of BHs are produced with large angular momentum! ) )
Decay of Black Hole Most of black holes are produced with large angular momentum. Geometry is not spherically symmetric. Hawking radiation is anisotropic, not equally probable. s,. l m d dt M J 1 g d s, l, m s m/ T s, l, m e 1m :The probability is not equal to every particle but crucially depends on spin and angular mode. Anisotropic and nontrivial Hawking radiation is expected. We have to know this greybody factor to understand Hawking Radiation.
A Good News Rotating Black Holes at future colliders I: Greybody factors for brane fields D. Ida, K.-y. Oda, SPARK, Phys.Rev.D67:06405,003, Erratum-ibid. D69:049901,004. II: Anisotropic scalar field emission D. Ida, K.-y. Oda, SPARK, Phys.Rev.D71:14039,005. III: Determination of black hole evolution D. Ida, K.-y. Oda, SPARK, Phys.Rev.D73:140,006. Greybody factors for all the SM particles (s=0,1/,1) are obtained in general (4+n)dimensional cases.
In this series of papers We developed analytic and numerical method to understand Mini Black holes. Cross section for BH production is estimated. Hawking radiation in (4+n)D is fully calculated after taking greybody factors precisely. Mini BH s Life is (almost) completely described.
Greybody factor d dt M J 1 g d s, l, m s m/ T s, l, m e 1m = Absorption Probability of wave mode (s, l, m) by BH. = Modification factor to take the curved geometry NH into account. Looks not black to me. It looks Grey! T
Caution: Our calculation is valid only if 1/ MBH rh rc Higher Dimensional Classical 1 M BH 1/ 1 ( ) n rbh GM BH (1/ ) n or BH M G M Pl Trans-Planckian Domain. Large (or Warped) extra dimensions: Compactification radius is BIG NOTE) if Mp~ 1 TeV, E=14 TeV at the LHC, these relations are fine.
Brief History of Greybody Factor for Rotating BHs. Teukolsky equation (Kerr) =Wave equation for general (s,l,m) wave for 4D Kerr BH (197,1973) Generalized to (4+n) for brane fields. Ida, Oda, Park-1 (PRD 04 ) Solution to Teukolsky equation/ Greybody Factors : Analytic and Numerical methods was developed by Teukolsky-Press, Starobinsky, Unruh, Page in 1973-1976. Analytic (5D),low energy: Ida,Oda,Park-1 Numerical ((4+n)D) : s=0 Ida, Oda, Park-II (PRD 05 ), Harris, Kanti (PLB 06 ), Duffy, Harris, Kanti, Winstanley (JHEP 05 ) s=1/,1 Ida, Oda, Park-III (PRD 06 ) Casals,Kanti,Winstanley (for s=1 only) (JHEP 06 ) Hawking radiation and its evolution : Hawking 1975, Page 1976 (4D) Ida, Oda, Park-III (PRD 06 )((4+n)D) including all the SM fields.
Newman-Penrose Formalism Null Tetrad Get equations for scalar, Weyl Spinor and Vector( nd Rank symmetric spinor) on the background geometry of Myers-Perry by perturbation.
Generalized Teukolsky equation Turned out to be separable.(petrov type-d) 0 1) ( ) csc cot ( ) cos ( sin sin 1 A as s s m s a s d ds d d 0 ) ( 4,, 1 R A a ma K i r i s K dr dr dr d rr r s s ma a r K r a r r a r n ) ( 1 cos 1 Spin-weighted-spheroidal harmonics We have to solve this radial equation.
Analytic method Low energy approximation (D=5) NH limit: FF limit: Matching here! Overlapping region
Greybody factors: D=5 Low energy approximation. D. Ida, K.-y. Oda, SPARK, Phys.Rev.D67:06405,003, Erratum-ibid.D69:049901,004. Cf) For s=0, V. P. Frolov and D. Stojkovic, Phys. Rev. D67, 084004 (003)
Numerical method. Numerical Integration Of Teukolsky equation. 1.Near Horizon BC: Imposing Purely Incoming FF 3. Read out Ingoing& outgoing wave D=4, D. N. Page, ``Particle Emission Rates From A Black Hole. :Massless Particles From A Rotating Hole,'' Phys. Rev. D14, 360 (1976). D=4+n, D. Ida, K.-y. Oda, SPARK, Phys.Rev.D71:14039,005. Phys.Rev.D73:140,006
Two main difficulties: 1. Imposing purely incoming BC at NH R Outgoing wave contamination is growing faster than the value we want to calculate! 1 incoming ² Outgoing r r H :Error/value grows fast.
Idea: Get rid of this part! :Error/value decreases. Now, we don t worry about the outgoing wave contamination. Numerical integration is done for 1 ²
. Separation of Ingoing and Outgoing parts at FF s=0: (In) ~ (Out) s=1/: (In) > (Out) s=1: (In) >> (Out) Numerically difficult to separate since there is a big hierarchy!
Idea: Analytically expand the solution We can always find the same order terms. By comparing them, we can safely separate In and Out parts.
Higgs-I : S-wave, Various Dimensions l=m=0, a=0 D=4 D=6 D=8 D=10 After angle integration and scaling w^{-} Ida, Oda, Park PRD 05 Area of Black hole event horizon
Higgs-II : S-wave (l=0), Rotating hole (a=0,.3,.6,.9) a=.9.6.3 D=5, l=m=0 a=0 Ida, Oda, Park PRD 05 D=4, l=m=0 D=10, l=m=0 Ida, Oda, Park PRD 05 Ida, Oda, Park PRD 05 NOTE: low energy enhancement appears in S-wave mode when BH Is highly rotating.
S = 1=; D = 5 Ida, Oda, Park PRD 06
D = 10; s = 1= Ida, Oda, Park PRD 06
Super-radiance modes: negative probability modes Ida, Oda, Park PRD 06 Ida, Oda, Park PRD 06 Ida, Oda, Park PRD 06
D = 5; s = 1 Ida, Oda, Park PRD 06
D = 11; s = 1 Ida, Oda, Park PRD 06
Time Evolution of Mini Black Hole (5D, 10D) 1 0.8 1 0.8 s 5d M 0.6 0.4 0. 5D J 0.6 0.4 v 0. f SM 0 0 0. 0.4 0.6 0.8 1 1. J Mass vs Angular momentum 0 0 0. 0.4 0.6 0.8 time Angular momentum vs t/t(.01) M 1 0.8 0.6 0.4 0. 0 v s SM f 10d Ida, Oda, Park PRD 06 0 0.5 1 1.5.5 J v 1 0.8 0.6 f 0.4 0. 0 s 0 0.5 1 1.5.5 Time 10d J Ida, Oda, Park PRD 06
Black Hole s Life Balding Phase (Production of BHs) Time Spin Down Phase (Losing energy and angular momentum)? Schwarzschild Phase (Losing Mass) Planck Phase (Remnant???) Today s topic.
Summary BH production cross section is estimated using Hoop conjecture and taking angular momentum into account. Greybody factors of D=4+n, rotating black holes are calculated for all the SM particles. (s=0,1/,1) BH decay by Hawking radiation is understood in the spin-down and Schwarzschild phases.
Open questions Still bulk gravitational radiation is missing. (For non-rotating black hole) A. S. Cornell, W. Naylor and M. Sasaki, ``Graviton emission from a higher-dimensional black hole,'' JHEP060, 01 (006) Numerical simulation of scattering process (Q. Can we use the BH-BH merging code?) Table for Black hole hunters. M. Cavaglia, R. Godang, L. Cremaldi and D. Summers, ``Catfish: A Monte Carlo simulator for black holes at the LHC,'' arxiv:hep-ph/0609001. Planck Phase. String theory? Quantum Information? Boltzmann -Brain???