Physics Letters B 634 (2006) 130–136

www.elsevier.com/locate/physletb

Bounds on low scale gravity from RICE dataand cosmogenic neutrino flux models

Shahid Hussain a,∗, Douglas W. McKay b

a Bartol Research Institute, University of Delaware, Newark, DE 19716, USAb University of Kansas, Department of Physics and Astronomy, Lawrence, KS 66045-2151, USA

Received 10 October 2005; received in revised form 6 January 2006; accepted 8 January 2006

Available online 31 January 2006

Editor: H. Georgi

Abstract

We explore limits on low scale gravity models set by results from the Radio Ice Cherenkov Experiment’s (RICE) ongoing search for cosmicray neutrinos in the cosmogenic, or GZK, energy range. The bound on MD , the fundamental scale of gravity, depends upon cosmogenic fluxmodel, black hole formation and decay treatments, inclusion of graviton mediated elastic neutrino processes, and the number of large extradimensions, d. Assuming proton-based cosmogenic flux models that cover a broad range of flux possibilities, we find bounds in the interval0.9 TeV < MD < 10 TeV. Heavy nucleus-based models generally lead to smaller fluxes and correspondingly weaker bounds. Values d = 5, 6 and7, for which laboratory and astrophysical bounds on LSG models are less restrictive, lead to essentially the same limits on MD .© 2006 Elsevier B.V. All rights reserved.

PACS: 96.40.Tv; 04.50.Th; 13.15.+g; 14.60.Pq

1. Introduction

The highest energy cosmic ray events range to values above105 PeV [1], with hundreds of TeV equivalent center-of-massenergies. Prospects of achieving comparable energies at an ac-celerator laboratory are remote, at best. The opportunities forprobing fundamental physics at these ultrahigh energies will beconfined to cosmic rays for the foreseeable future. The weak-ness of neutrino interactions at currently available energies al-lows neutrinos to carry information to us line of sight from deepwithin highly energetic astrophysical sources. For the same rea-son, enhancements of neutrino interaction rates by new physicsat ultra-high energies can be dramatic and distinctive. On theother hand, similar sized enhancements in the electromagneticand strong interaction rates can easily be obscured. These two,unique features of neutrinos—direct propagation from sourceto observer and sensitivity to new physics—drive the effort to

* Corresponding author.E-mail address: shahid@bartol.udel.edu (S. Hussain).

0370-2693/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2006.01.029

build neutrino telescopes to cover energies from fractions ofan MeV to detect p–p neutrinos from the sun to thousandsof PeV to detect cosmogenic neutrinos, those originating fromthe decays of pions created by collisions of the highest en-ergy cosmic rays with the cosmic microwave radiation photons[2,3].

In this Letter we present a detailed study of bounds onnew physics, specifically on the scale MD in models of lowscale gravity [4], that follow from predictions by ultrahigh-energy (UHE) proton-based models of cosmogenic neutrinofluxes [5–8] and the absence of UHE neutrino events in the Ra-dio Ice Cherenkov Experiment (RICE) data from 2000 through2004 [9]. We briefly compare these limits to limits obtainedby using predictions by recently proposed heavy nuclei-basedmodels for the cosmogenic neutrino flux [10].

Before proceeding to a description of the calculation, the re-sults and the summary and conclusions, we sketch the essentialfeatures of the RICE detection system and the features of lowscale gravity that lead to prediction of event rates in RICE thatexceed those expected from neutrino interactions in the Stan-dard Model (SM).

S. Hussain, D.W. McKay / Physics Letters B 634 (2006) 130–136 131

1.1. The RICE experiment

The advantages of weakly interacting neutrinos raise the ob-vious question of detection. Neutrino detection is possible withsome combination of large fluxes and large detector volumes.Accelerator neutrino facilities strive for high fluxes and de-tectors the size of large fixed target experiments. But in theenergy range above a PeV, the predicted cosmogenic fluxes areextremely small and decrease rapidly with energy. The high en-ergy end of the expected cosmogenic flux range, 104–105 PeV,can be characterized by the observed cosmic ray figure of 1event per km2 per century. Many cubic kilometers is the effec-tive detector volume that must be achieved.

The RICE detector concept relies on characteristics of UHEshowers, radio wavelength emission from these showers, andthe transmission of radio wavelengths in cold, pure ice [11]. Theshowers in dense media (e.g., ice), are compact, travel fasterthan light in the medium and are smaller in transverse size thansub-GHz frequency wavelengths. The showers develop a net ex-cess charge at shower maximum that is about 106 electrons ata PeV, and is proportional to energy. This net charge emits co-herent Cherenkov radiation at frequencies up to a GHz. Thisradiation has an attenuation length of more than a kilometer inAntarctic ice. A single radio antenna can be sensitive to neu-trino induced showers within more than a km3 of ice at thehighest energies; even a modest, pilot array has an effective vol-ume, Veff, of many cubic kilometers. Discussion of the recentRICE analysis can be found in Ref. [9], while more details ofthe full experiment are given in Refs. [12,13].

1.2. Low scale gravity and UHE cosmic ray neutrinos

The phenomenology of low scale gravity (LSG) [4] predictsenhanced cross sections mediated by gravity, including produc-tion of black holes in high energy particle reactions [14,15].Under quite general assumptions, the gravity enhanced crosssections, such as that for black hole production, are large, andgrow with a power of the center of mass energy. After forma-tion, black holes are thought to evaporate to a statistical mixtureof many particles [16], producing a characteristic signature.There is great interest in black hole production at current andfuture colliders as a “low energy” probe of extra dimensions[17,18]. Yet even the LHC energies may not be large enough toreach the black hole regime [18].

A number of ultra-high energy neutrino studies take advan-tage of the fact that the highest energies accessible in particlephysics occur not at colliders, but in cosmic rays; this is thencoupled to the prediction that low scale gravity cross sectionsin extra dimension models, while much smaller than hadronic,can be orders of magnitude larger than Standard Model (SM)neutrino cross sections at ultra-high energies [19]. With the as-sumption that the collision energy dumped into a black hole isreleased as visible energy in air shower detectors or neutrinotelescope detectors, predictions of event rates and consequentbounds on MD , the “Planck mass” of extra-dimension gravity,have been presented for various model estimates of neutrinoflux [20].

A low scale gravity effect that does not depend on the blackhole dynamics is graviton exchange, producing an hadronicshower from the graviton–parton interaction in the nucleon.This is a variety of deep-inelastic, neutral-current process, andwe include this in our study as well. This class of events iscomplementary to black hole production and decay. Thoughthe cross sections are large, the elasticity is high, greater than∼90%, so relatively little of the neutrino energy is depositedin the showers. Nonetheless, in part of the parameter space this“deep inelastic” gravity effect competes with the black hole for-mation.

2. Summary of direct black hole production and gravitonmediated neutrino deep inelastic scattering

We present the essential components of our calculation ofthe number of events expected in RICE data [9] from SM, blackhole, and graviton-mediated interactions. For impact parame-ters less than the Schwarzschild radius, rS , the black disk, blackhole cross section formula has often been adopted, which reads

(1)σBH ≈ πr2S,

without including gray body [21], or impact parameter [22] andform factor [23], effects. Later we include a model for the im-pact parameter dependence for comparison. In Eq. (1), rS is the(4 + d)-dimensional Schwarzschild radius of a black hole ofmass MBH [24]:

rS = 1

MD

[MBH

MD

] 11+d

κd,

where κd = 2.1, 2.44, or 2.76 for cases d = 5, 6, or 7, respec-tively. Here MD is the (4 + d)-dimensional Planck mass scaleof extra-dimensional physics, which is the scale we aim to ex-plore with UHE RICE data in this study. Eq. (1) is a parton levelcross section, and the effective black hole mass is MBH = √

s,with s = xs; s is the square of the center-of-mass energy and x

is the momentum fraction of the struck parton. The correspond-ing cross section is given by

(2)σνN→BH(Eν) =1∫

xmin

dx σBH(xs)∑

i

fi(x,Q),

where the fi(x,Q) are the parton distribution functions, xmin =M2

BH0/s or 1/r2s s, whichever is larger; the sum over index i

accounts for different parton flavors. Here s = 2MNEν and Q ischosen to be

√xs. Choosing Q = 1/rs [17] makes insignificant

difference. Eν is the primary neutrino energy. The choices ofxmin and Q are not unique, as discussed in the papers listed inRefs. [17,18,20]. Here MBH0 is a free parameter correspondingto the minimum energy needed to form a black hole.

So far we have described the simplest versions of BH for-mation to estimate cross sections and UHE shower rates. Manypossible modifications show up when the horizon forming col-lision is studied in detail. A proposal by Yoshino and Nambu

132 S. Hussain, D.W. McKay / Physics Letters B 634 (2006) 130–136

[22] to sharpen the horizon formation criterion is straightfor-ward to implement [25], and we include it here to illustratethe effect on the event rates for BH production. The impactparameter dependence of the apparent horizon formation is con-veniently presented as a plot of mass contained within an ap-parent horizon (MA.H.) vs. impact parameter. We fit the graphof MA.H. vs. b with a function MA.H.(b/bmax = z), the masscontained within an horizon as a function of impact parame-ter, scaled by the impact parameter beyond which no horizonforms, bmax, as in Ref. [22]. We set the lower limit of x, xmin =(MBH0)

2/M2A.H.(z). The form of the cross section is still taken

to be σ(s) = πr2s (s), but the threshold for BH production is

raised, which accounts for c.m. energy that does not contributeto BH formation and subsequent decay to observable particles.In addition to the x-integral, the cross section now includes anaverage over impact parameter, taken to be weighted geometri-cally by the area element d(πb2)/πb2

max [25]. The cross sectionfor this modeling of the impact-parameter effects reads

(3)σνN→BH(Eν) =1∫

0

2z dz

1∫xmin

dx σBH(xs)∑

i

fi(x,Q).

Again, s = 2MNEν and Eν is the primary neutrino energy. Allof the cross sections relevant in this Letter are shown in Fig. 1for representative values of MD and MBH0, with d = 6. Theblack hole production cross sections are sensitive to the valuesof MD and MBH0 and to the inclusion of impact parameter ef-fects. The bounds we obtain for d = 5 or 7 differ by only fiveto ten percent from those obtained for d = 6, so we give detailsonly for d = 6 throughout. Laboratory and astrophysical dataalready place strong lower bounds on the cases d < 5.

We treat the graviton exchange in the higher dimension, lowscale gravity picture in the eikonal approximation [26]. Formore than 3 spatial dimensions, an impact parameter scale bc

Fig. 1. Standard model and low scale gravity (MD = 1 TeV, d = 6) cross sec-tions. Pair of dotted curves gives standard model NC (lower curve) and CC(upper curve) interactions. Pair of solid (dashed) curves gives black hole for-mation cross section, without (with) inelasticity, for MBH0 = MD (upper solid(dashed) curve) and MBH0 = 3MD (lower solid (dashed) curve). The dash-dotcurve is for graviton exchange case.

enters the problem, and the dominant contribution to the eikonalamplitude when

√s � MD comes from momentum transfer

q = √−(p − p′)2 in the range 1/rS > q > 1/bc, where thestationary phase evaluation of the amplitude is a good approx-imation. The four momenta p and p′ refer to the incident andscattered neutrinos, respectively. The eikonal amplitude can bewritten in this approximation

(4)|Md | = Bd(bcMD)d+2[bcq]− d+2d+1 ,

where Bd = 0.23, 0.039 or 0.0061 when d = 5, 6, or 7. Thesaddle point impact parameter reads

bc = 1

MD

(s

M2D

)1/d

βd .

The factor βd = 1.97, 2.32, or 2.66 for d = 5, 6, or 7. The elasticparton level cross section then reads

σEK(x, q) = 1

16πxs

∑i

fi(x, q)|Md |2,

which enters into the rate calculations below. The cross sectionsare displayed in Fig. 1 for the case MD = 1 TeV.

At this point we can write the shower production rates for theeikonal and direct black hole production cases, specifying therestrictions on the range of integrations from impact parameterand threshold considerations. For the black hole production wehave the folding of the neutrino flux energy distributions withthe effective volume of the detector and the scattering cross sec-tions on the nucleon target integrated over x. The eikonal set-uphas an additional integration over the inelasticity parameter y.These expressions are presented below. These results have ap-peared in various forms elsewhere [27].

Shower rates from BH production and decay, with or withoutincluding the impact parameter effects (black hole inelasticity),are determined by folding the appropriate cross section with theeffective volume and differential flux:

RBHshower = 2πρNA

∑i

Eν max∫Eth

dEν

dFi(Eν)

dEν

(5)× Veff(Eν)σνN→BH(Eν),

where xmin = M2BH0/s or 1/r2

s s, whichever is larger, for thesimple black disk case (Eq. (2)); xmin = (MBH0)

2/M2A.H.(z)

when the impact-parameter-dependent inelasticity estimate isincluded (Eq. (3)). The mass enclosed in the apparent hori-zon, MA.H., and the scaled impact parameter, z, are discussedabove. Eth is the experimental threshold for detection of show-ers, taken to be 100 PeV for RICE. Above the cutoff Eν max,the integral gives negligible contribution to R. The rate shownis for down-going neutrinos, assuming an isotropic, diffuse fluxof neutrinos, appropriate to those of cosmogenic origin. Fluxesare given per steradian, and the 2π factor accounts for the inte-gration of isotropic flux times angular average of the effectivevolume over the dome of the sky.

S. Hussain, D.W. McKay / Physics Letters B 634 (2006) 130–136 133

For the eikonal scattering case, the rate expression reads

REKshower = 2πρNA

∑i

Eν max∫Eth

dEν

dFi(Eν)

dEν

(6)×1∫

Eth/Eν

dy Veff(yEν)

1/r2s ys∫

M2D/ys

dx σEK(x, q),

with q = √xys. The graviton exchange is affected only by the

value of MD , though there is an implicit dependence on thechoice of minimum impact parameter where the eikonal, semi-classical approximation is expected to be valid. We have chosenthis value as 1/rs . The fluxes we used to explore the range ofevent rates are shown in Fig. 2. Recent work that estimates cos-mogenic neutrino fluxes based on UHE nuclei is reported inRef. [10].

3. Numerical work and results

To produce representative event rate predictions, we fix num-ber of extra dimensions d at 6, consider both MD = 1 TeVand 2 TeV and the minimum BH mass values MBH0 = MD

and 3MD . As we noted in our outline of cross sections, chang-ing d to 5 or 7 affects our bounds by only 5–10 percent. We useCTEQ6 [28] for the nucleon parton distributions. The cross sec-tions for MD = 1 TeV are plotted from 106 GeV to 1012 GeVin Fig. 1.

When MD = 2 TeV, each cross section is reduced byroughly a factor 10. To assess flux model dependence on re-sults, we chose large, medium, and small fluxes for comparison,as described in the caption of Fig. 2. We are primarily interestedin cosmogenic flux models, generated by the GZK effect [5–8].These produce neutrinos at the very highest energies, and are ar-guably less model-dependent than those generated from modelsof specific types of astrophysical sources. In Fig. 2 we show thecomparison between the three proton-based flux models used,which show factors of three or more difference over most of theenergy range of interest, and more than an order of magnitudein some regions [5–8]. Though this and previous studies, forexample [20,27], focus on the proton-based cosmogenic fluxmodel, we also comment below on the consequence if heavynuclei [10] are the UHE cosmic rays rather than protons. Theselatter models themselves vary widely in their neutrino flux pre-dictions, depending upon assumed energy cutoff of the sourceand on nuclear species. Moreover, a mixture of several species,including protons, is consistent with cosmic ray data.

The qualitative features of our calculations in all channelscan be anticipated from the power law growth of cross sections,times stronger power law fall of flux, times (slower) power lawgrowth of effective volume to detect shower energy. The rapiddecrease in flux with energy wins, but the growing cross sec-tions and effective volume delay the loss of events as one scanshigher and higher energies. In Table 1 we present the numberof events expected in RICE over the 5 year period 2000–2004using the three different cosmogenic flux models. We have inte-

Fig. 2. Cosmogenic νe fluxes by ESS [5], PJ [7], and KKSS [6]. We show theKKSS maximal and a relatively smaller flux for the KKSS case.

Table 195% C.L. upper limit shower rates, for RICE 2000–2004 operation, in low scalegravity (MD = 1 TeV, d = 6); rates are given for different cosmogenic neutrinoflux models; BH and BHinel give black hole formation rates (and hence showerrates) without and with inelasticity included, respectively; EK gives the showerrates due to graviton exchange. KKSS corresponds to the maximal flux caseshown in Fig. 2

Flux SM EK BH1 (MBH0 = MD) BH2 (MBH0 = 3MD)

BH BHinel BH BHinel

ESS 0.1 0.66 38.4 4.0 15 1.4KKSS 3.0 43 1509 238 697 98PJ 0.27 1.7 106 11 41 3.7

grated using an updated effective volume of the RICE detector[9] for our visible energy signal. In all cases Eth (our detec-tor threshold) is 107 GeV. All rates are integrated over primaryneutrino energy from threshold to the maximum energy wherethe flux cuts off. As shown in Table 1, since at 95% C.L., withSM cross sections, the maximum number of events compati-ble with no background and zero observed events is 3.0, RICEnearly rules out the maximal KKSS cosmogenic flux model.The MD = 1 and d = 6 choices made for Table 1 are irrelevantfor the SM result, of course.

For the SM, EK and the direct BH production cases, thehadronic component of showers is assigned pulse strength equalto an electromagnetic shower of the same energy [29]. An im-portant difference between electromagnetic and hadronic show-ers is that the hadronic showers do not suffer the loss of ef-fective volume that is estimated to affect the electromagneticshowers due to the LPM [30] effect when computing eventrates.

In the EK, BH1, and BH2 columns of Tables 1 and 2, weshow expected numbers of events in the RICE 2000–2004 data,with the acceptance (effective volume) and efficiencies in can-didate event analysis included, coming individually from theeikonalized graviton exchange (EK), and black hole produc-tion processes. Representative scales MD = 1,2 TeV are used,

134 S. Hussain, D.W. McKay / Physics Letters B 634 (2006) 130–136

Table 2Same as above with MD = 2 TeV

Flux EK BH1 (MBH0 = MD) BH2 (MBH0 = 3MD)

BH BHinel BH BHinel

ESS 0.10 4.5 0.30 1.5 0.12KKSS 7.2 195 29 78 10PJ 0.26 12 1.2 4.0 0.32

and for each the black hole formation thresholds MBH0 = MD ,3MD are assumed and the black hole rate calculated. It is usu-ally argued [14,17] that in the range MD < MBH0 < 2MD , thequantum gravity effects may dominate [31,32]; our point ofview is that the new physics leading to shower formation willoccur, and the EK and BH shower rates act as reasonable esti-mates of the enhancement in the transition region. Certainly oneexpects that the different energy regions tie together smoothlyparametrically, and they do [31]. In addition we show two val-ues of black hole events for each black hole MD , MBH0 para-meter pair, corresponding to the simple “black disk” calculationand the calculation corrected for loss of efficiency for blackhole formation when the impact parameter offset between neu-trino and parton increases. The second number in the columnslabeled “BHinel” is that obtained when the inelasticity effectsare modeled as described above. The black hole (BH) ratesdrop by an order of magnitude. The EK rate, unaffected bythe BH inelasticity effects, is now about 20% of the BHinel forMBH = MD case and 50% of BHinel for the MBH = 3MD case.Again this is true for all the flux models we apply, which covera wide range of possibilities.

Pumping up the cross sections above SM extrapolationspushes harder on the flux bounds. Given that UHE neutrinosare not yet observed [9], choosing a flux model then allows oneto limit cross sections, which predict at some point more eventsthan current observational limits allow. For RICE, this numberis 3.0 at 95% C.L., as noted earlier. Clearly most of the en-tries in Tables 1 and 2 are already disallowed at better than 95%C.L. To quantify this situation, we allow the mass scale of grav-ity to decrease (increase) and ask what value allows the eventrate for a given flux to rise above (fall below) the 95% limit setby RICE. This value is what we quote in Table 3 and shown asa function of MBH0/MBH in Fig. 3.

To assess the effect of BH inelasticity on MD bounds, wealso use the impact parameter dependent cross sections as mod-eled in [22,25], to place bounds on MD for each flux and MBH0choice. In Table 3 the BH and ALL columns again show twovalues each. The first refers to the case where all parton CMcollision energy is available to the BH, while the second is thevalue after the modeling of inelasticity is included. The reduc-tion in sensitivity to MD is substantial, with reduction rangingfrom factors of 3/4 to 1/2.

Our results are summarized in Fig. 3, where we show theminimum value of the scale of gravity allowed by RICE dataat 95% C.L. For a given case, the region below the curve isexcluded. The points on the curves are obtained by setting thevalue of MBH0, and then varying MD with d = 6 until the pre-dicted number of events falls below 3, consistent with the RICE

Table 3Experimental lower bounds on LSG scale MD , based on 2000–2004 RICE data(0 events, 0 background). Here ALL is the combined bound due to EK and BH;these bounds are due to all flavors. The pairs of numbers under columns BHDand ALL are the bounds without and with black hole inelasticity, respectively.The bounds here include SM interactions. The numbers are in TeV. KKSS cor-responds to the maximal flux case shown in Fig. 2. We fix number of extradimensions d to 6. With d = 5 or 7, we find values 2.0 TeV or 2.3 TeV insteadof 2.15 TeV for the first entry in the ESS, BH slot in the table. These are typicalfractional changes

Flux MBH0 = MD MBH0 = 3MD

EK BH ALL BH ALL

ESS 0.55 2.15, 1.05 2.15, 1.1 1.55, 0.75 1.55, 0.85KKSS 4.3 11, 6.0 11, 6.6 7.5, 4.0 7.9, 5.15PJ 0.8 3.0, 1.45 3.0, 1.55 2.1, 1.05 2.15, 1.15

Fig. 3. Lower bounds on MD as a function of the ratio of the minimum BHformation threshold, MBH, to MD . The upper curve for each flux model is thelower bound when the naive, black disk model is used for the BH cross section.The lower curve is the lower bound when the estimate of impact parametereffects is included. ESS, KKSS (lower of the two KKSS curves in Fig. 2) andPJ refer to Refs. [5–7].

result of zero events on zero background at 95% C.L. The lowerbound on MD is plotted against MBH0/MD , the ratio of a min-imum invariant mass required for black hole formation to thescale of gravity. In setting the limit curves, the SM, LSG unita-rized graviton exchange, EK, and the LSG black hole produc-tion cross sections, BH and BHinel, are all included in determin-ing event rates and, consequently, the bound. The top curve foreach flux model represents the minimum MD when the simple“black disk” model for the black hole formation cross sectionis used; the bottom curve indicates the lower limit when im-pact parameter dependence of the apparent horizon is includedas estimated in [22] and implemented by [25]. The impact ofthe uncertainty in correct cosmogenic flux model and model ofBH cross section on the bound of MD is seen at a glance inFig. 3. Fixing MBH0/MD , the range of possible bounds on MD

follows by comparing the lowest curve to the highest.To contrast the above results to the those from nuclei-

generated cosmogenic neutrino flux estimates, we calculatedbounds for the case MBH0/MD = 1 in “small” (cutoff at

S. Hussain, D.W. McKay / Physics Letters B 634 (2006) 130–136 135

106.5 PeV) and “large”(cutoff at 107.5 PeV) versions of theiron-based flux model from the first reference in [10]. Usingthe most pessimistic black hole production assumptions in the“small” flux case and the most optimistic for the “large”, thelower bound range is 0.5 TeV < MD < 1.0 TeV. Models basedon lighter nuclei such as oxygen or helium produce neutrinofluxes, and therefore MD limits, much closer to those foundfrom the models based on protons as the UHE cosmic rays.

4. Discussion of results and outlook

The essence of our results is shown in Fig. 3, where the re-gion below a given curve shows the excluded region for whichMD is too small, and the LSG cross section too big, to be con-sistent with the RICE results [9]. As we comment further below,in connection with the information contained in the tables, theconclusions depend heavily on the flux model assumed and thetreatment of the LSG interactions. It is clear that, should eventsbe observed, there are likely to be a number of different para-meter choices in the flux models and the cross section modelsthat will reproduce the observed events. This will still be trueif a component of heavy nuclei is present in the UHE cosmicrays, modifying the cosmogenic flux predictions.

Within the range of cosmogenic flux models we consider,Table 3 reveals that at 95% C.L., RICE and the ESS flux model[5] rule out an LSG model where the naive σBH = πr2

S crosssection is assumed, where MBH = 3MD , and where MD <

1.55 TeV. This can be regarded as a least lower bound onmodels with the naive “black disk” cross section for the blackhole formation within our analysis assumptions. The greatestlower bound corresponds to that obtained with the largest fluxmodel, and the value is greater than 7.9 TeV (11 TeV) whenMBH0 = 3MD (MD). The KKSS maximal flux model [6] isright at the borderline of our 95% C.L. constraint using theSM cross section [9], so the corresponding LSG scale is im-precise. Basically there is no room for LSG if this flux model isassumed.

If the estimates of the effects of non-zero impact parame-ter are included in the manner proposed in [22], and some ofthe collision energy is lost to the BH formation process, thenthe BH formation cross sections decrease and the event ratesand corresponding bounds on the LSG scale weaken. The nom-inal effects on the least lower bound, tied to the flux model [5],and greatest lower bound, tied to KKSS [6] are the reductionsto 0.85 TeV and 5.15, respectively. The greatest lower bound ofall, 11 TeV, occurs if the BH scale is equal to MD and the simpleblack disk cross section is used with the maximum KKSS fluxshown in Fig. 2. Given the lack of precision in these consider-ations, we summarize the most conservative bounds statementas an estimated range on the value of MD as lying in the rangebetween roughly 0.9 TeV and 10 TeV, as quoted in the abstract.

Next, suppose for the sake of argument that a given numberof events were actually observed in the RICE data. There areinteresting degeneracies that occur. For example, there is an ap-proximate degeneracy between the EK MD = 1, 2 TeV entriesand BH (MD = 1, 2 TeV with MBH0 = 3 TeV) entries for everyflux model we used. The enhancements in event rates are more

than an order of magnitude compared to SM in every case, butthe interpretation, just within low scale gravity itself and withthe flux known, will be a challenge. The EK calculation doesnot invoke black hole formation, but for one scale choice EKcan yield the same signal enhancement as BH with a differentscale choice.

When the extra uncertainties in flux are admitted, thereis a sharp increase in the possible degeneracies in parameterchoices, leading to similar event rates. For example, it is no sur-prise to find in Table 3 that the ESS flux with BH1 parameterchoices give the same RICE event rate predictions as the PJflux with the BH2 parameter set for the LSG model, with andwithout inelasticity included. Similarly, the PJ flux with EK in-teractions alone produce, as noted in Table 3, the same boundon MD as the ESS flux with ALL interactions included and in-elasticity estimated for the BH production case.

A number of ideas have been proposed for distinguishingSM from new physics cross sections, and new physics crosssections from one another, independently of flux; see, for exam-ple, Ref. [27] and references cited there. Typically these ideasrequire angular information, at least number of up-going eventsvs. down-going events, to “divide out” the flux effects. To getdiscriminating power takes a sizable sample of events, which isa challenge for all UHE telescope projects.

These remarks remind us that the field of UHE neutrino tele-scope physics and astrophysics is still very young. All-in-all weare at a very complex, fluid and fascinating threshold of discov-ery. Much more experimental and theoretical work remains tobe done.

Acknowledgements

We thank Dave Besson and the RICE Collaboration for dis-cussions and encouragement. Comments on early versions ofthe manuscript from Danny Marfatia are appreciated. John Ral-ston and Prasanta Das participated in the early stages of thiswork, and we thank them for their contributions. We appreci-ate the cooperation of D. Seckel, R. Protheroe, D. Semikoz andA. Taylor, who provided us with data files from their cosmo-genic flux model calculations. This work was supported in partby the Department of Energy High Energy Physics Division andby the NSF Office of Polar Programs.

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