Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://nuclphys.sinp.msu.ru/conf/lpp14/210809/Galtsov.pdf Äàòà èçìåíåíèÿ: Wed Sep 16 16:30:20 2009 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 00:40:58 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: gravitational radiation

No black holes LHC?
XIV Lomonosov Conference, Moscow 2009
Dmitri Gal'tsov
Department of Theoretical Physics Moscow State University Russia

No black holes LHC? ­ p. 1/2


ADD model
The simplest model with large extra dimensions was suggested by Arkani-Hamed, Dimopoulos and Dvali (ADD) as solution to the hierarchy problem. The weakness of gravity in four dimensions is explained by the presence of n extra dimensions compactified on a torus large compared to the electroweak scale, while the fundamental mass parameter M (higher-dimensional Planck mass) is taken of the order of 1 TeV. Assuming that the extra space is flat, the D-dimensional Einstein action can be related to the 4-dimensional one by 1 GD V D RD - g D d x = GD 1 4 R4 - g 4 d x = G4 R4 -g4 dD x,

where V is the volume of the n-torus, so GD = V G4 , or in terms of masses, 2 MPl = M (n+2) V .

No black holes LHC? ­ p. 2/2


Denoting V = (2 R)n one finds the compactification radius R for fixed value of M = 1 T eV :
l = V n
1/n

103 R

0/n-17

cm

1 2 4 6

1.5 â 1013 cm 0.5 mm = 1/(10-4 eV) 3 â 10-8 mm = 3/(20KeV) 10-10 mm = 1/(1MeV)

The case n = 1 is excluded, but n = 2 gives l 1mm. This is enormous with respect not only to electroweak scale, but also to atomic scale, so such large extra dimensions have to be seen in the low energy processes not to say about direct mechanical tests of the Newton law.

No black holes LHC? ­ p. 3/2


Static force
Two test masses m1 , m2 at a distance small with respect to the compactification radius r R feel a gravitational potential following from Gauss's law in (4 + n) dimensions: m1 m2 1 V ( r ) n+2 n+1 , (r R ) M r with the Tev-scale gravitational constant. Gravity is strong in this case. But being placed at large distances r R, they feel four-dimensional potential, since the gravitational flux lines joining them do not penetrate into the extra dimensions: m1 m2 1 (r R ) V ( r ) n+2 n , M R r Gravity is weakened by the ratio (R/r)n , so our effective 4 dimensional gravitational coupling is reproduced in view of 2 2 the ratio MP l M +n Rn . Therefore an observed weakness of gravity is explained by the Gauss law: only a small fraction of the gravitational flux lines propagates along the brane.

No black holes LHC? ­ p. 4/2


Linearized theory
Expanding the D-dimensional metric gM N = M N + D hM N we get Fierz-Pauli lagrangian in D-dimensional Minkowski space, with n D - 4 of spatial dimensions forming a torus T n with equal radii R 1 MN 1 1 MN 1 MN L=-h hM N + h h - h M N h + h M P hP N 4 4 2 2 D M N - h TM N , 2 where M , N , .. = 0, 1, 2, ..., D - 1. The Minkowski metric is M N = diag(1, -1, -1, ...), h M N hM N hM and M M N M N . The gravitational field hM N is coupled in a universal way to a conserved matter stress-tensor TM N (N T M N = 0). The latter is fur ther assumed to be localized on the brane by some confinement mechanism

No black holes LHC? ­ p. 5/2


Momentum quantization
Consider first vacuum case TM N = 0. Metric functions depend on xM = (xµ , y i ), µ = 0, . . . , 3, i = 1, . . . n, and must be periodic under translations yj yj + 2 R, which leads to quantization of the momentum in compact directions
+ +

hM N (xP ) =
n1 = -

···

nn = -

hn N (x) ni y M ex p i R V

i

ii , µ, = 0, 1, 2, 3 and i, j = 5, 6, · · · , 4 + n

Graviton polarizations can be split in the Kaluza-Klein spirit h + µ Aµj -1/2 µ , hM N = Vn A i 2ij

No black holes LHC? ­ p. 6/2


Massless modes
Different polarization states for zero modes (n = 0) give rise to massless fields: 4D graviton , n massless graviphotons A0 i , massless radion 0 and n(n + 1)/2 massless moduli 0j . µ i These fields which have no momentum in the compactified dimensions are confined to the brane. The trace of the scalar matrix, the radion 0 describes fluctuations of the torus. It is supposed that there must exist a mechanism giving it the mass, which stabilize the volume of the compact space. The lagrangian for the massless modes reads 1 µ L ni =0 = h µ h - µ hµ h - 2hµ hµ + 2hµ µ h 4
n

-

i=1

1 µ 1µ Fi Fµ i + µ + 4 2
µi

n(n+1)/2

µ ij µ ij ,
( i j )= 1

where Fµ i = µ A i - A

No black holes LHC? ­ p. 7/2


Massive modes
Fourier components with n = 0 can be suitably rearranged to form an infinite tower of massive 4D spin two fields hn , n - 1 µ towers of massive vector fields Ani and n(n - 1) towers of µ massive scalars nj , with the remaining 'could be' massive i vectors and scalars eliminated by D-dimensional coordinate transformations. This rearrangement is similar to spontaneous symmetry breaking. Like in the Higgs mechanism, the massless spin-2 graviton fields absorb the spin-1 and spin-0 fields and become massive. The massive fields obey the equations 1 2 n ( + mn ) (hµ - µ hn ) = 0, 2 2 2 ( + mn ) Ani = 0, ( + mn ) nj = 0, µ i where is 4D d'Alember t operator and m
2 n

=

4 2 n R2

2

.

No black holes LHC? ­ p. 8/2


Coupling to matter
D-dimensional gravity strongly couples to bulk fields in a universal way. However, for the matter occupying only the brane one gets the four-dimensional Newton coupling. In 4d terms the lowest order interaction lagrangian reads 4 - d4 x(hµ Tµ + T µ ) , µ 2 where Tµ is the 4d stress-tensor. Rewriting this in terms of physical massive modes one finds 4 - d4 x(hµ,n Tµ + n T µ ) . µ 2
n

Note that the vector KK modes Ani fully decouple and the µ scalar KK modes nj only couple through their trace n , the i dilaton mode. Although each individual graviton couples very weakly to ordinary matter, their large number may enhance to an observable scale the effects due to both the vir tual graviton exchange and emission of real KK gravitons.

No black holes LHC? ­ p. 9/2


Emission of KK gravitons
Radiation is the most efficient tool to probe extra dimensions. Emission of light KK gravitons may lead to drastic and inadmissible modification of cer tain astrophysical and cosmological patterns thus providing restrictions to LED scenarios. First are the effects of KK graviton emission in hot stars such as the Sun, red giants and supernova SN1987A. Excessive energy losses in the stars can alter the stellar evolution. Emission of the KK gravitons (gK K ) is due to: + g
KK

, photon-photon annihilation; , electron-positron annihilation;
KK

e- + e+ g

KK

e- + Z e e- + Z e + gK K , gravi-bremsstrahlung in a static electric field of the nuclei; N + N N + N + gK K , nucleon-nucleon bremsstrahlung.

+ e- ( Z e) e- ( Z e) + g

, gravi-Compton scattering;

No black holes LHC? ­ p. 10/2


SN cooling and related effects
The dominant graviton emission process from a SN core is nucleon-nucleon bremsstrahlung. The requirement that KK gravitons do not carry away more than half of the energy emitted by the supernova SN1987A gives the bounds M > 13 T eV for n = 2 and M > 1 T eV for n = 3. Fur ther evolution of these gravitons leads to more stringent bounds. After being created, KK gravitons are quasi-stable except for their slow, gravitational-strength, decay into photons, neutrinos, and other standard par ticles. Therefore, the decays of KK gravitons produced in all cosmic SNe will contribute to the measured diffuse cosmic -ray background, providing more restrictive limits than the SN 1987A energy-loss argument. Measurements by the EGRET satellite imply M > 34 T eV for n = 2 and M > 3 T eV for n = 3. Limits on gamma-rays from all the neutron-star sources imply M > 180 T eV for n = 2 and M > 10 T eV for n = 3.

No black holes LHC? ­ p. 11/2


Even more stringent restriction follow from the secondary effects due to KK gravitons. The decay products of the gravitons forming the halo can hit the surface of the neutron star, providing a heat source. The low measured luminosities of some pulsars imply M > 670 T eV for n = 2 and M > 22 T eV for n = 3. These are the most restrictive bounds which probably make n = 2 case uninteresting as a solution of the hierarchy problem. Astrophysical constraints set very strong bounds on M for n < 4 in some cases even ruling out the possibility to observe any signature of KK gravitons at the LHC. But it has to be kept in mind, however, that these constraints refer to soft KK gravitons lighter than 100 MeV. They disappear in more sophisticated models in which the graviton spectrum is bounded from below at this value.

No black holes LHC? ­ p. 12/2


Emission of gravitons at colliders
KK gravitons may be produced at colliders both in leptonic and hadronic collisions. Since the produced gravitons interact with matter only on 4D gravitational scale, they will remain undetected leaving a "missing energy" signature. Such events were searched for in the processes at LE P and p + p + missing, ¯ e+ + e- + missing, e+ + e- Z + missing p + p jet + missing ¯

at Tevatron. The combined LEP limits are M > 1.4 T eV for n = 2, M > .8 T eV for n = 3, M > .5 T eV for n = 4, M > .3 T eV for n = 5 and M > .2 T eV for n = 6. Experiments at LHC will improve this sensitivity. Theoretical predictions for hadron machines have uncr tainties, and can e be applied only at subplanckian energies s M , where s = (p 1 + p 2 ) 2

No black holes LHC? ­ p. 13/2


Transplanckian regime
Physics at s M can be described only by the full underlying quantum gravity or string theory. However, for the transplanckian energies s M the semiclassical description is possible. Since an effective gravitational coupling grows with energy, gravity becomes dominant. On the other hand, it can be argued that at ultrahigh energies, par ticle scattering not only becomes dominated by gravity, but in addition it involves only classical gravitational dynamics. Indeed, in the usual 4D theory quantum gravity effects should not, by definition, be impor tant in the classical limit 0. This, in terms of the two relevant lengths, i.e. the Planck length lPl = ( G4 /c3 )1/2 = /MPl c and the gravitational radius 4 associated with the energy of the collision rg = G4 s/c , implies that the classicality condition lPl rg , is equivalent to the condition s MPl c2 of transplanckian energies.

No black holes LHC? ­ p. 14/2


In the ADD scenario the D-dimensional Planck length l (marking quantum gravity effects) and the gravitational radius rg (classical) are, correspondingly 1 1 n+1 GD n+2 GD s l = , rg = . 3 4 c M c c The above reasoning remains essentially the same and shows that in the transplanckian regime s M c2 scattering is also classical, at least for some range of momentum transfer. Indeed, from this condition it follows DB l rg where

c D B = s is De Broglie wavelength of collision, marking QFT effects.

No black holes LHC? ­ p. 15/2


The novel feature which is due to extra dimensions is the existence of one more scale parameter GD s bc rg , 5 c D B which does not exists for n = 0. In the limit of vanishing Planck constant 0 this quantity does to infinity, so the classical region is bounded from above by b < bc . For b > bc , the ordinary QFT become impor tant, while quantum gravity effects are still negligible. Another restriction on the feasibility of calculations is the weak gravitational field approximation b rg , otherwise one has to use the fully non-linear Einstein theory. This, however, poses not only technical problems, but also the problem of the overall consistency of the ADD model beyond the linearized level. For tunately, the weak field condition b rg automatically imply the classicality condition b bc since in the transplanckian region rg DB .
1 n

rg

1 n

No black holes LHC? ­ p. 16/2


Black hole production
One of the most amazing predictions of theories with LEDs is that one could actually form black holes from par ticle collisions at the LHC. Black holes are formed when the mass of an object is within the horizon size corresponding to the mass of the object. If the center-mass energy and the impact parameter of the collision are such that the D = 4 + n dimensional gravitational radius is larger than the impact parameter M 2 ds = (1 - 2+n M r
1+n

)dt2 -

dr (1 -
M

2 M

2+n 1+n r

)

+ r2 d2 ,

the horizon size is given by rH M M
1 1+n

1 . M

.

No black holes LHC? ­ p. 17/2


Then the collided par ticles will form a black hole with mass MB H = s, and the cross section as we have seen is roughly the geometric cross section corresponding to the horizon size of a given collision energy 2 1 MB H n +1 2 rH 2 ( ) MP l M The cross section would thus be of order 1/TeV2 400 pb, and the LHC would produce about 107 black holes per year! These black holes would not be stable, but decay via Hawking radiation. This has the features that every par ticle would be produced with an equal probability in a spherical distribution. In the SM there are 60 par ticles, out of which there are 6 leptons, and one photon. Thus about 10 percent of the time the black hole would decay into leptons, 2 percent of time into photons, and 5 percent into neutrinos, which would be observed as missing energy. These would be very specific signatures of black hole production at the LHC.

No black holes LHC? ­ p. 18/2


The widely accepted picture consists in the four-stage process of formation and evaporation of BHs in colliders, formation of a closed trapped surface collision of shock waves modeling the collision, the balding phase, during which the B gravitational waves and relaxes to the (CTS) in the head-on par ticle H emits Myers-Perry BH,

Hawking evaporation and superradiance phase in which the experimental signatures are supposed to be produced, and the quantum gravity stage, where more fundamental theory like superstrings is impor tant. This scenario was implemented in computer codes to simulate the BH events in LHC, where they are expected to be produced at a rate of several per second, and in ultra high energy cosmic rays.

No black holes LHC? ­ p. 19/2


Transplankian radiation
Radiation is the main inelastic process which will accompany transplanckian scattering. For b < bc its main features can be understood classically. During recent years radiation in presence of uncompactified LED was extensively studied within the classical theory (Kosyakov '99; D.G. '01; Kazinski et al. '02; Lemos et al. '03; D.G. and Spirin '04-'09, etc.).Bremsstrahlung in transplanckian collisions in the ADD model was recently considered by D.G., Kofinas, Tomaras and Spirin (0908.0675 [hep-ph]).Bremsstrahlung is substantially enhanced due to exchange of KK modes and emission of light massive gravitons. The cross-section exhibits rapid increase with the number of extra-dimensions and may serve an efficient tool to test theories with large n. Radiation emitted in the KK modes is invisible and provides a new channel of the missing energy processes at colliders.

No black holes LHC? ­ p. 20/2


The radiated energy in the rest frame one one of the par ticles r eads 6 m2 m2 D d+3 ~ Erad = CD 3d+3 b with a known dimension-dependent coefficient. Qualitatively the dependence on b and can be understood estimating the number of light KK modes par ticipating in interaction and radiation. To pass to the CM frame, one calculates the relative energy loss (radiation efficiency) Erad /E , and expresses the result in terms of the Lorentz factor in the CM 2 frame via (for m = m ) cm = (1 + )/2: =C
d

rS b

3(d+1)



2d+1 cm

.

No black holes LHC? ­ p. 21/2


The two new features of this expression are: (a) the large 2d factor cm+1 due to the large number of light KK modes involved both in the gravitational force and in the radiation, and (b) the growing with d coefficient: d C b
d

1 10.1 3.45 196

2 184 1.88 7.90

3 3359 1.46 3.15

4 6 · 104 1.29 2.11

5 1.06 · 106 1.21 1.72

6 1.8 · 107 1.17

rS
c

1.53 -1 rS and bc in TeV evaluated for M 1TeV and s 14TeV. The classical description of small angle ultrarelativistic scattering is valid for impact parameters in the region rS < b < bc , where 1/d bc -1/2 [(d/2)GD s/ c5 ] rS (rS /B )1/d is the scale beyond which (for d = 0) classical notion of trajectory is lost

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Another restriction comes from the quantum bound on the radiation frequency cr < m , which is equivalent to b > C /(mc). For d = 0 the two conditions overlap provided C < bc . To estimate set b = C to obtain 3 = Bd (sm/M )d+2 (Bd = 7.4, .8, .6, .9, 1.9, 5.6, 21 for d = 0, 1, . . . , 6). Thus, a simple condition for strong radiation damping is 3 sm M , (1) which may well hold for heavy par ticles and uclei with LHC n energies and cosmic rays. For example, for s = 14 TeV and m = .2 TeV all conditions are met for d = 1, 2, and at the quantum boundary b = C one has 1 5 â 104 , 2 106 . For protons in LHC, C > bc , our formula does not apply for d = 0. For d = 0 and b = C it gives = .25.

No black holes LHC? ­ p. 23/2


Conclusions
Our analysis shows that Kaluza-Klein bremsstrahlung may lead to strong radiation damping in transplanckian collisions. Therefore, One may have to include the reaction force in the study of BH production, which might even exclude the formation of a CTS. Incidentally, there are indications that gravitational collapse of an oscillating string does not take place, once gravitational radiation is taken into account. Our results also imply that bremsstrahlung is a strong process leading to missing energy signatures in transplanckian collisions, which may fur ther constrain the ADD parameters

No black holes LHC? ­ p. 24/2