PHYSICAL REVIEW D, VOLUME 65, 064023
Cosmic rays as probes of large extra dimensions and TeV gravity
Roberto Emparan* Theory Division, CERN, CH-1211 Geneva 23, Switzerland
Manuel Masip† Centro Andaluz de Fı´sica de Partı´culas Elementales (CAFPE) and Departmento de Fı´sica Teo´rica y del Cosmos, Universidad de Granada, E-18071, Granada, Spain
Riccardo Rattazzi‡ Theory Division, CERN, CH-1211 Geneva 23, Switzerland ͑Received 18 October 2001; published 27 February 2002͒ If there are large extra dimensions and the fundamental Planck scale is at the TeV scale, then the question arises of whether ultrahigh energy cosmic rays might probe them. We study the neutrino-nucleon cross section in these models. The elastic forward scattering is analyzed in some detail, hoping to clarify earlier discussions. We also estimate the black hole production rate. We study energy loss from graviton mediated interactions and conclude that they cannot explain the cosmic ray events above the GZK energy limit. However, these inter- actions could start horizontal air showers with characteristic profile and at a rate higher than in the standard model.
DOI: 10.1103/PhysRevD.65.064023 PACS number͑s͒: 04.50.ϩh, 04.70.Ϫs, 98.70.Sa
I. INTRODUCTION whether the first signatures from low-scale unification and large extra dimensions might come from the study of In this paper we explore the possibility that the primary ultrahigh-energy cosmic rays ͑UHECR͒. particles for ultrahigh-energy cosmic rays are neutrini inter- While this work was in progress, detailed analyses of the acting gravitationally with atmospheric nucleons. An obvious possibility of black holes forming at the CERN Large Had- objection to this idea is that the gravitational interaction is ron Collider ͑LHC͒ have appeared ͓7–9͔. During the final ͓ ͔ too weak to produce any sizable cross section for this pro- stage of this work, a paper 10 has appeared which studies cess. However, this point needs to be fully reconsidered in the production of black holes in cosmic rays, and also inves- theories where the fundamental scale is around the TeV, as tigates their detection in horizontal air showers. Our work ͓ ͔ postulated in models involving large extra dimensions and a complements that of Ref. 10 : the latter focuses on the phe- four-dimensional brane-world ͓1,2͔. In these scenarios not nomenology of the detection of black holes, whereas we ad- only does the cross section for elastic gravitational scattering dress in more detail the theoretical aspects of neutrino- increase at cosmic ray energies, but there is also the possi- nucleon scattering in TeV-gravity theories. A paper bility that the collision results into the formation of micro- discussing further aspects of the detection of these showers ͓ ͔ scopic black holes. Both effects can dramatically increase the has appeared when this work was ready for submission 11 . cross section for scattering between neutrini and atmospheric nucleons, and hence they may play a role in explaining the II. ULTRAHIGH ENERGY SCATTERING ON THE BRANE most energetic cosmic ray events. In the recent past, this possibility has been entertained in An essential feature of the gravitational interaction is that a number of papers, with differing conclusions ͓3–5͔. There at center-of-mass ͑c.o.m.͒ energies ͱs well above the funda- has been a controversy as to the right way to perform the mental scale, the coupling to gravitational coupling grows so calculations, and how to implement unitarity at high ener- large that graviton exchange dominates over all other inter- gies. We hope to shed some light on these issues, and show actions. This is actually the case for atmospheric nucleons 11 that, actually, the situation is quite simple once the appropri- being hit by neutrini of energy Eϳ10 GeV (ͱs ate point of view is taken ͓6͔. In addition, the possibility of ϳ106 GeV) if the fundamental scale is around 1 TeV. In producing black holes in cosmic ray collisions needs to be particular, if the impact parameter b is sufficiently smaller addressed in detail. Once we have, hopefully, settled the than the radius of compactification, the extra dimensions can terms for the analysis, we will turn to the actual discussion of be treated as non-compact. In this regime one would be prob- ing the extra dimensions purely by means of the gravitational interactions. *Also at Departamento de Fı´sica Teo´rica, Universidad del Paı´s Another consequence of ultra high energies in gravita- Vasco, E-48080, Bilbao, Spain. Email address: tional scattering is that, to leading order, its description in- roberto.emparan@cern.ch volves only classical gravitational dynamics. In particular, †Email address: [email protected] this means that we do not need any detailed knowledge of ‡On leave from INFN, Pisa, Italy. Email address: quantum gravity to perform the calculations: any theory that [email protected] has general relativity as its classical limit should yield the
0556-2821/2002/65͑6͒/064023͑7͒/$20.0065 064023-1 ©2002 The American Physical Society ROBERTO EMPARAN, MANUEL MASIP, AND RICCARDO RATTAZZI PHYSICAL REVIEW D 65 064023 same results.1 One can distinguish different regimes in the Fourier-transform to impact parameter space of the Born am- scattering ͑and we will do so below͒, but perhaps the most plitude. Alternatively, it can be obtained from the deflection spectacular effect at such energies is the expected formation of a particle at rest when crossing the gravitational shock- of black holes, via classical collapse, when the impact pa- wave created by a second particle ͓12͔. rameter is of the order of the horizon radius of the ͑higher Note that the transforms in impact parameter space are dimensional͒ black hole ͓15͔,2 two-dimensional, since the particles scatter in three spatial dimensions. Nevertheless, the exchanged gravitons propa- ϩ 1/(nϩ1) Ϫ n 3 gate in the (4ϩn)-dimensional space. Moreover, we are 2n(n 3)/2⌫ͩ ͪ ͩ 2 ͪ working at a scale where the spectrum of Kaluza-Klein R ϭ S nϩ2 modes is essentially continuous. In this case the Born ampli- tude comes out easily as ͓16͔ 1/2(nϩ1) s ϫͩ ͪ . ͑1͒ 2 ϩ n s M 2n 4 iM ϭin/2⌫ͩ 1Ϫ ͪ ͑ϪtϪi⑀͒n/2Ϫ1. ͑3͒ D Born 2 2ϩn M D Ͻ This implies that any dynamics at b RS is completely 4 shrouded by the appearance of trapped surfaces: Ultrashort Hence the eikonal phase, distances are directly probed only for energies around the 1 d2q fundamental energy scale. ͑ ͒ϭ ͵ iq•b M ͑ ͒ s,b 2 e i Born , 4 In the following we will assume for simplicity that no 2is ͑2 ͒ scale for new physics arises before reaching the scale for the which is finite for bϭ0, is (s,b)ϭ(b /b)n, where we have fundamental energy M . In particular, we assume that scales ” c D defined such as the string tension, or the tension and thickness of the brane, do not appear before that scale. This prevents the pos- Ϫ ͑4͒n/2 1 n s sibility of additional effects arising at impact parameters bnϭ ⌫ͩ ͪ . ͑5͒ c 2 2 2ϩn larger than the ones that give rise to black hole formation. If M D this is the case, then the picture for ultrahigh-energy scatter- ing that we describe here should be largely universal. Nev- Having the phase (s,b) is sufficient for numerical evalu- ertheless, stringy effects below the regime where general ation of the eikonalized amplitude ͑2͒. The result in Eq. ͑2͒ relativity can be trusted may be readily accommodated ͓9͔ can be written in terms of Meijer functions. However, it is and should not introduce large changes in our results. easy to get simple analytical expressions for the amplitude in ӷ Ӷ ӷ Ϫ1 Ultrahigh energy scattering in the Randall-Sundrum both regimes of qbc 1 and qbc 1. When q bc the model has been addressed in ͓6͔, and the different regimes phase (s,b) yields a sharp peak for the eikonal amplitude for the scattering in the present case are qualitatively the in Eq. ͑2͒, which allows for an evaluation near the saddle ϭ Ϫ1/(nϩ1)Ӷ same as described there. At large impact parameters one does point bs bc(qbc /n) bc : not expect formation of black holes, but in this case, the ϩ ϩ leading contribution to the scattering amplitude is exactly 4iei n s n 2 1/(n 1) M ϭ ͫ͑ ͒n/2Ϫ1⌫ͩ ϩ ͪͩ ͪ ͬ ͑non-perturbatively͒ calculable within an eikonal approach saddle 4 1 ͱnϩ1 2 qMD ͓12,13,17͔. This is known to work particularly well for high energy gravitational scattering at large impact parameter s (nϩ2)/(nϩ1) 3 ϵ ͩ ͪ ͑ ͒ ͓13,18͔. Zn . 6 qMD The eikonal resummation of ladder and crossed-ladder ϭ ϩ n Ͻ diagrams is achieved by computing the scattering amplitude The phase (n 1)(bc /bs) is real for t 0. Observe that as the amplitude is non-perturbative in the gravitational cou- 2ϩn → pling 1/M D . In the limit q 0 one gets instead 2s M͑s,t͒ϭ ͵ d2beiq•b͑ei (s,b)Ϫ1͒ i 2 M͑qϭ0͒ϭ2isb2⌫ͩ 1Ϫ ͪ eϪi/n, ͑7͒ c n 4s ϭ ͵ db bJ ͑bq͒͑ei(s,b)Ϫ1͒. ͑2͒ i 0 which is finite for nϾ2. For nϭ2 the real part of M has a logarithmic singularity This amplitude is well defined for any values of the ex- changed momentum qϭͱϪt (tϽ0 since the scattering is q→0 ͒ M ϭ Ϫ 2 ͑ ͒ ͑ ͒ elastic . The eikonal phase (s,b) is obtained from the 4 sbc ln qbc . 8 Notice that also at small q the amplitude is non-analytic in 1This has been noted often earlier, e.g., in ͓12–14,6–8͔. the gravitational coupling. Indeed the amplitude at q→0is 2 ͓ ͔ We define M D as in 16 . 3Loops involving only momenta of internal gravitons are sup- 2 4 ͑ ͒ pressed by factors of 1/(M Db) . This corresponds to the linearized approximation to Eq. 2 .
064023-2 COSMIC RAYS AS PROBES OF LARGE EXTRA . . . PHYSICAL REVIEW D 65 064023 effectively described by the ͑Born͒ operator T defined in Ref. At present, the cross section for black hole production can ͓ ͔ ϳ Ϫ1 only be estimated as the geometric cross section, 16 but with an effective UV cutoff bc on the mass of the exchanged Kaluza-Klein ͑KK͒ modes. This cut-off origi- ϳ 2 ͑ ͒ nates from the interference with the multigraviton exchange bh RS , 12 diagrams in the eikonal series. ͑ ͒ ϳ 1/(nϩ1) The eikonal amplitude will be used in the next section to with RS as in Eq. 1 . In this case bh s , again slower compute the differential cross section for neutrino-nucleon than linear. scattering. At the partonic level we have Clearly this result cannot be very accurate. Radiation is expected to be emitted during the collapse, and the amount d 1 of energy that is expected to be radiated in the process can be ϭ ͉M͉2. ͑9͒ a sizable fraction of the total energy ͑perhaps around 15– dq2 16s2 30 %, from four-dimensional estimates ͓21͔͒, but at large We can also derive the total elastic cross section from the enough energies it will not be able to prevent the collapse. optical theorem: This effect will tend to reduce the above value for the cross section. However, there are also factors which increase it, Im M͑qϭ0͒ 2 such as the fact that a black hole acts as a somewhat larger ϭ ϭ 2⌫ͩ Ϫ ͪ ͑ ͒ ͑ ͓ ͔͒ el 2 bc 1 cos , 10 scatterer 40–75 % larger radius 22 . It seems reasonable to s n n expect that the above expression is not off by any large fac- tors. Finally, note that these black holes form through classi- i.e., it is essentially given by the area of a disk of radius cal collapse. In ͓23͔ the semiclassical instanton contribution ϳb . c to the nucleation of black holes was considered. Being a Observe that tunneling process, it is exponentially suppressed. Hence, it can be neglected relative to the ͑real time͒ classical collapse ϳs2/n. ͑11͒ el we are considering. This growth of the cross section at high energy is slower than the perturbative result ϳs2, and also slower ͑for nϾ2) III. NEUTRINO-NUCLEON SCATTERING AND than the linear dependence ϳs postulated ͑apparently for BLACK HOLE PRODUCTION all n)in͓4͔. Unitarity in impact parameter space is manifest ͑ ͒ 5 These results can now be readily applied to neutrino- in the eikonal amplitude 2 . For large impact parameter this nucleon scattering at ultrahigh energies. At impact param- implies as well unitarity for high partial waves. Partial wave eters bϽ1 GeVϪ1 the neutrino interacts essentially with the unitarity at shorter impact parameter is a harder problem, and Ͼ partons, and if b RS the eikonal approximation gives a indeed, corrections to the eikonal amplitude are expected to good description of the scattering. At smaller distances, become crucial. As t grows, graviton self-interactions, which trapped surfaces are expected to form and the neutrino and carry factors of t associated to the vertices, increase the at- the parton will collapse to form a black hole. traction among the scattered particles, and it is expected that, In order to numerically evaluate the amplitude ͑2͒,we eventually, gravitational collapse to a black hole will take proceed as follows. First, we write it as place. Hence the initial state is expected to be completely absorbed, but in such a way that any short distance effects Mϭ 2 ͵ ͑ ͒͑ i/xnϪ ͒ϭ 2Mˆ ͑ ͒ will be screened by the appearance of a horizon. Indeed as i 4 sbc xdxJ xqbc e 1 4 sbc qbc . shown in Ref. ͓13͔ the effects of the non-linearity of gravity ͑13͒ are suppressed by a power of RS /b, so our eikonal approxi- mation should be valid for bӷR and its breakdown be as- S At large values of qbc we know this is well described by the sociated to the formation of black holes. This relation be- simple result ͑6͒. It is convenient to extract this behavior, and tween eikonal breakdown and black-hole formation can also ͉Mˆ ͉2 Ӷ write the squared amplitude as be established as follows. In the region b bc , there is a one to one correspondence between the transferred momentum q n2/(nϩ1) and the saddle point impact parameter b . The case qϳͱs, ͉Mˆ ͉2ϭ ϩ͑ ͒2 Ϫ(nϩ2)/(nϩ1) ͑ ͒ ͑ ͒ s „1 qbc … ϩ F qbc . 14 where the ͑small angle͒ eikonal approximation breaks down, n 1 ϳ corresponds precisely to bs RS . Notice in passing that we ͑ ͒ ϭ The prefactors have been chosen in such a way that for can also write Eq. 9 as d 2 bsdbs , as expected for a →ϱ ϭ qbc the function F goes to 1. Apart for the case n 2 classical trajectory with impact parameter bs . → ͓ where it has a mild logarithmic singularity at qbc 0 see ͑ ͔͒ Eq. 8 , F is O(1) over the full range of qbc . For our applications it is useful to study the cross section 5Notice that in order to achieve unitarity we have needed to per- as a funtion of the fraction y of energy transferred to the form an all-order loop resummation. As argued in ͓13͔, this is es- nucleon: sential when considering energies above M D . This point is missed ͓ ͔ in some of the earlier work, such as the last reference in 5 . Note as 2 ͓ ͔ EϪEЈ q well that the Froissart bound 19 , generalized to higher dimensions yϭ ϭ , ͑15͒ in ͓20͔, does not apply since the exchanged particle is massless. E xs
064023-3 ROBERTO EMPARAN, MANUEL MASIP, AND RICCARDO RATTAZZI PHYSICAL REVIEW D 65 064023
FIG. 2. As in Fig. 1 but for M ϭ5 TeV. FIG. 1. Elastic cross section vs minimum fraction of energy lost D ϭ ϭ by the neutrino for M D 1 TeV and n 2,3,6 large extra dimen- sions. Solid, long-dashed and short-dashed lines correspond respec- Finally, to estimate the total cross section to produce a 14 12 10 black hole in a neutrino-nucleon scattering we compute tively to Eϭ10 GeV, Eϭ10 GeV, and Eϭ10 GeV.
1 ϭ ͵ dxͩ ͚ f ͑x,͒ͪ R2 , ͑17͒ where x is the fraction of proton momentum carried by the i S M 2 /s i parton. Summing over partons we have D ͑ ͒ ϭ Ϫ1 where RS is given in Eq. 1 and RS . Again for the d 1 1 choice of scale in the PDF’s the previous discussion applies: ϭ ͵ dx ͩ ͚ f ͑x,͒ͪ ͉M͑x,y,ͱs/M ͉͒2. i D dy 0 16xs i the Schwarzschild radius rather than the black-hole mass sets ͑16͒ the time scale of gravitational collapse. Notice that in a more standard case of, say, neutrino-quark fusion into an elemen- ͑ ͒ tary leptoquark the right choice would be of the order of Here f i(x, ) are the parton distribution functions PDFs Ϫ the lepto-quark mass. The crucial difference is that the black- ͑we use the CTEQ5 set extended to xϽ10 5 with the meth- hole is not an elementary object: its physical size is much ods in ͓24͔͒. Notice that quarks and gluons interact in the bigger than its Compton wavelength. same way. The scale should be chosen in order to mini- We plot in Fig. 3 this cross section versus the energy of mize the higher order QCD corrections to our process. A ϭ ϭ ϭ the incoming neutrino for n (2,3,6) and M (1,5) TeV. simple, but naive, choice would be q. However 1/q does Ͼ 2 ͑ ͒ Ͼ 2 We include plots with xs M D solid and xs (10M D) not really represent the typical time or length scale of the ͑ ͒ interaction. As we have seen, in the stationary phase regime, dots . These correspond to the cross sections for producing the neutrino is truly probing a distance b ӷ1/q from the black holes with a mass larger than M D or 10M D , respec- s tively. The sizeable difference between the two choices of a parton. Heuristically: the total exchanged momentum can be ͑ ͒ large, but through the exchange of many soft gravitons. So minimum x, indicates that the production of light small ϭ Ϫ1 black holes dominates: the fast decrease with x of the PDFs we believe that a better normalization is to take bs ϰ 1/(nϩ1) Ͼ Ϫ1 ϭ Ͻ Ϫ1 wins over the growth x of the partonic BH cross when q bc and q if q bc . The latter choice is ef- ϭ Ϫ1 section. Notice, on the other hand, that the total elastic cross fectively equivalent to choosing bc as at small q the section is less dependent on the small x region and is domi- eikonal corresponds to a pointlike interaction. Our choice of nated by xϳ1 for the case nр3. This is because of the faster is consistent with the fact that gravity at ultra-Planckian growth ϰx2/n of the partonic elastic cross section. energies is dominated by long distance classical physics. Choosing ϭq would also make little sense. q can be as big ϳͱ ӷ 2Ͼ 2 IV. DISCUSSION as s M D , but the evolution of the PDF’s at Q M D cannot be simply performed withing QCD, as truly quantum We are now ready to discuss the implications of our re- gravitational effects ͑string theory͒ would come into play. sults on the phenomenology of ultra-high energy cosmic ͱ Instead as s grows above M D , and t/s is kept fixed but rays. The first question is whether neutrino nucleon scatter- small, the impact parameter bs grows and we are less sensi- ing at super-Planckian energies can explain the observed cos- Ͼ ϭ ϫ 10 tive to short distance physics. As a matter of fact, for large mic ray events with energy E EGZK 5 10 GeV. It is enough s the total N will be bigger than the proton area known since long ago that cosmic protons with energy above ϳ 2 (GeV) : at higher energies the parton picture breaks down, EGZK are damped by inelastic scattering with the microwave the proton interacts gravitationally as a pointlike particle, and background photons. The relevant reaction is pϩ␥→pϩ, the neutrino scatters elastically on it. and EGZK is the threshold proton energy given the photon A useful quantity to study is the cross section integrated temperature. Because of this reaction, ultraenergetic cosmic Ͼ Ӎ for y y 0. In Fig. 1 and Fig. 2 we plot this quantity for protons are brought down to E EGZK within a few Mpc. ϭ ϭ M D 1 TeV and M D 5 TeV, respectively. We include the Since there are good reasons to believe that the cosmic pro- 10 12 14 cases with nϭ(2,3,6) and Eϭ(10 ,10 ,10 ) GeV. tons have extra-galactic origin, we should observe a sharp
064023-4 COSMIC RAYS AS PROBES OF LARGE EXTRA . . . PHYSICAL REVIEW D 65 064023
FIG. 3. Cross section for black hole produc- ϭ tion as a function of E , for M D 1,5 TeV and nϭ2,3,6. Solid and dotted lines correspond to Ͼ 2 Ͼ 2 xs M D and xs (10M D) , respectively.
Ͼ drop in the observed event rate at E EGZK . However, vari- energy loss is thus dominated by parton scatterings with ͱ ϳ ous experiments do not observe this drop at all. There have xs M D , i.e., in the Planckian regime. been several suggestions to explain that. One idea is that the The observed showers above the GZK cutoff are all con- primary particles for the UHECR are neutrini ͓25,26,4,5͔,as sistent with an incoming particle that loses all its energy to these particles interact negligibly with the microwave back- the shower already in the high atmosphere. From the above ground and are essentially undamped. However, any of these discussion, low scale gravity could explain these events if suggestions has to face the fact that, within the standard the mean free path Lbh for black hole production were some- model ͑SM͒, the neutrini interact too weakly also with the what smaller than the vertical depth of the atmosphere. In nucleons in the atmosphere. In order to explain the ultra- standard units, the vertical depth xv is measured as the num- GZK events by cosmic neutrini one needs new physics en- ϭ ϫ Ϫ2ϭ Ϫ1 ber of nucleons per unit area xv 1033 NA /cm mb hancing their cross section with nucleons at high energy. In ͑ ͒ where NA is the Avogadro number , so the requirement is Ref. ͓4͔ it was suggested that, in models with TeV scale Ͼ Ϫ1ϭ bh xv mb. From Fig. 3 one can see that, at the relevant gravity, the eikonalized cross section could be of the right energies, the black hole cross section, however large, falls ͓ ͔ order of magnitude. However Ref. 4 did not investigate the short of this requirement. In order to satisfy Ͼmb the rate of energy loss in the eikonalized process, and, in par- bh gravity scale M D should be well below a TeV, which would ticular, did not pay attention to its ‘‘softness.’’ The produc- contradict collider limits. ͓ ͔ tion of black holes was also neglected in Ref. 4 . Hence, we conclude that neutrino-nucleon interactions in As a matter of fact, in order to determine the signal it is TeV-gravity models are not sufficient to explain the showers important to establish quantitatively which is the process that above the GZK limit. Also, at present, cosmic rays do not dominates energy loss—whether elastic gravitational scatter- appear to place any significant bounds on such scenarios. ing or black hole production. It turns out that energy loss is Nevertheless, neutrino-nucleon cross sections N in the mostly determined by black hole production and by scatter- range 10Ϫ5 mb to 1 mb, like in our scenario, can still lead to ϳ ͑ ing at y 1. As we already pointed out, and as can be seen interesting new phenomena in cosmic ray physics, which from comparing Figs. 1, 2 with Fig. 3, the gravitational cross may be observed in upcoming experiments. Cosmic prima- ϳ section at y 1 becomes comparable to bh , though its pre- ries with cross section below 1 mb can travel deep into the cise value is not calculable within our linearized gravity ap- atmosphere before starting a shower. In particular they can proximation.͒ To see this, consider a neutrino travelling through a medium of density . The mean free path for black hole production, at which all energy is lost to the shower, is ϭ Ϫ1 Lbh ( bh ) . While travelling through the medium the neutrino also loses energy through the softer, but more fre- quent, eikonalized scattering. After travelling a distance Lbh , Ͻ the energy fraction lost to soft scatterings with y y 0 is con- trolled by the quantity
y0 d 1 y0 d ͑ ͒ϭ ͵ ϭ ͵ ͑ ͒ y 0 y Lbhdy y dy. 18 0 dy bh 0 dy
When is less than 1 the soft scatterings play a negligible role in the transfer of energy to the atmosphere. In Fig. 4 we plot for several cases: they all show that black hole for- FIG. 4. Fraction , defined in Eq. ͑18͒, of neutrino energy lost mation and scattering at large y dominate energy loss. Notice to soft scatterings. Solid, long-dashed, and short-dashed lines cor- 14 12 10 that, by the discussion at the end of the previous section, respond to Eϭ10 ,10 , and 10 GeV, respectively.
064023-5 ROBERTO EMPARAN, MANUEL MASIP, AND RICCARDO RATTAZZI PHYSICAL REVIEW D 65 064023
cross the atmosphere at a large zenith angle and start char- information will be crucial to secure that the excess is not acteristic horizontal air showers.6 The horizontal depth of the due to an underestimate of the ͑unknown͒ neutrino flux. ͑ atmosphere is xh is about 36 times the vertical one, so that Finally, in order to establish which process gravitational Շ ͒ for N .1 mb a neutrino can travel horizontally down to elastic scattering, or black hole production dominates the the interaction point. In the standard model the charged- signal, one needs some knowledge about the energy depen- ϳ Ϫ5 10 0.363 dence of the incoming neutrino flux. We have already estab- current cross section is N 10 (E/10 GeV) mb. No horizontal air shower has been detected so far. However, lished that BH production dominates energy loss. However, conservative estimates of the flux of ultraenergetic cosmic as the eikonalized cross section grows with E, if the neutrino neutrini ͓26͔ suggest that the next generation of experiments flux J(E) decreases with E slowly enough, the number of should be barely sensitive to neutrino cross sections of the BH events at energy E may be overshadowed by soft scat- tering events due to neutrini with energy ӷE. The signal is order of the SM one. In our scenario can be consider- N the number dN(E) of showers with energy between E and ably bigger, so there is the interesting possibility that gravi- EϩdE. In terms of the neutrino flux J(E) and the differen- tational scattering and black hole production will lead to a tial cross section we can write sizeable event rate, higher than in the SM. The shape of the shower is probably one of the better dN͑E͒ Emax dEЈ d E ways to characterize these processes. In the SM charged- ϰ ͵ J͑EЈ͒ ͩ EЈ,yϵ ͪ . ͑19͒ dE Ј dy Ј current process, a significant fraction of the neutrino energy E E E is released to just one or a few hadrons from the breakdown Emax represents the energy at which bh becomes larger than of the target proton. The shower then builds up from the Ϫ the inverse horizontal depth x 1 . Neutrini with EϾE in- cascading hadronic interactions of these few hadrons. In the h max teract right away and cannot generate horizontal showers. We scenarios we are considering, the production of a black hole have studied the above integrand by assuming J(E)ϳE␣. ϳͱ ϭͱ of mass M bh s 2M m p is followed by its very quick We found that, in the cases of interest, already for ␣ϽϪ2 ͓ ͔ evaporation by emission on the brane 22 of a number of the signal is dominated by events with large y, and then by ϳ ͱ (nϩ2)/(nϩ1) particles of the order of M bh /Tbh ( s/M D) . For black holes. ͑More precisely we find that the critical ␣’s for the energies we are considering this number can be bigger nϭ2, 3 and 6 are respectively equal to Ϫ1.76, Ϫ1.65, and than 100. Then the shower builds more quickly than for SM Ϫ1.48.͒ This condition is satisfied for the cosmogenic neu- processes. It is reasonable to expect that the shapes will dif- trino flux in Fig. 1 of Ref. ͓26͔ for which ␣ӍϪ3. If the fer, very much in the way that a shower formed by a primary cosmogenic neutrini dominate the flux, then black hole pro- iron nucleus differs from the shower formed by a primary duction and gravitational scattering at yϳ1, and not the proton. To investigate the difference in the case at hand re- softer processes, will dominate the signal in horizontal air 7 quires a more detailed study. Note that the BH cross section showers. plotted in Fig. Fig. 3 is inclusive over the mass of the BH. A To conclude, we hope to have convincingly established significant portion of that cross section is due to the produc- that neither higher-dimensional graviton-mediated neutrino- tion of not so heavy BH’s, through scattering with partons nucleon scattering nor black hole production in TeV-gravity with small x. Moreover, as discussed above, the cross section models can explain the observed cosmic ray showers above bh is of the same order as the elastic gravitational scattering the GZK limit. Nevertheless, horizontal air showers may at yϳ1. In the latter processes a significant fraction of the probe these scenarios. In this case, black hole production and neutrino energy is transferred to a few proton fragments. We gravitational deflection by a large angle will be the processes then expect the resulting shower to resemble those induced that dominate the signal. by SM physics. In order to assess how well can one distin- guish gravity induced showers from SM showers requires to ACKNOWLEDGMENTS take into account all these facts. This is an important point: if an excess of horizontal shower is observed, the shower shape We would like to thank Gian Giudice, Michael Kachel- riess, Hallsie Reno and Alessandro Strumia for valuable con- versations. R.E. acknowledges partial support from UPV 6For the observation of horizontal air-showers from neutrino pri- grant 063.310-EB187/98 and CICYT AEN99-0315. M.M. maries, see ͓27͔. acknowledges support from MCYT FPA2000-1558 and 7See ͓11͔. Junta de Andalucı´a FQM-101.
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