PHYSICAL REVIEW D 82, 103012 (2010)

Constraints on neutrino-nucleon interactions at energiesof 1 EeV with the IceCube Neutrino Observatory
Shigeru Yoshida*
Department of Physics, Graduate School of Science, Chiba University, Chiba 263-8522, Japan(Received 8 September 2010; published 22 November 2010)
A search for extremely high energy cosmic neutrinos has been carried out with the IceCube Neutrino
Observatory. The main signals in the search are neutrino-induced energetic charged leptons and their ratedepends on the neutrino-nucleon cross section. The upper limit on the neutrino flux has implications forpossible new physics beyond the standard model such as the extra space-time dimension scenarios whichlead to a cross section much higher than the standard particle physics prediction. In this study we constrain
the neutrino-nucleon cross section at energies beyond 109 GeV with the IceCube observation. The
constraints are obtained as a function of the extraterrestrial neutrino flux in the relevant energy range,
which accounts for the astrophysical uncertainty of neutrino production models.
DOI: 10.1103/PhysRevD.82.103012

PACS numbers: 98.70.Sa, 13.15.+g, 95.85.Ry

I. INTRODUCTION
High energy cosmic neutrino observations provide a rareopportunity to explore the neutrino-nucleon (N) interaction behavior beyond energies accessible by the present
accelerators. These neutrinos interact during their propagation in the Earth and produce energetic muons and taus.
These secondary leptons reach underground neutrino detectors and leave detectable signals. The detection rate is,therefore, sensitive to neutrino-nucleon interaction probapﬃﬃﬃ
bility. The center-of-mass energy of the collision, s, iswell above 10 TeV for cosmic neutrino energies on theorder of 1 EeV ( ¼ 109 GeV). This is a representative
energy range for the bulk of the GZK (Greisen-ZatsepinKuzmin) cosmogenic neutrinos, generated by the interactions between the highest energy cosmic ray nucleons andthe cosmic microwave background photons [1].
The N collision cross section can vary greatly if nonstandard particle physics beyond the standard
pﬃﬃﬃ model (SM)is considered in the high energy regime of s  TeV. The
extra-dimension scenarios, for example, have predicted
such effects [2,3]. In these scenarios, the virtual exchangeof the Kaluza-Klein graviton [2] or microscopic black hole
production [4] leads to a substantial increase of theneutrino-nucleon cross section by more than 2 orders ofmagnitude above the SM prediction. The effect would besizable enough to affect the expected annual event rate[Oð0:1  1Þ] of the GZK neutrinos in the km3 instrumentation volume of an underground neutrino telescope
such as the IceCube observatory. Thereby, the search for
extremely high energy (EHE) cosmic neutrinos leads toconstraints on nonstandard particle physics [5].
The IceCube neutrino observatory has already begun
EHE neutrino hunting with the partially deployed underground optical sensor array [6]. The 2007 partial IceCube*syoshida@hepburn.s.chiba-u.ac.jp
URL: http://www.ppl.phys.chiba-u.jp/~syoshida1550-7998= 2010=82(10)=103012(6)detector realized a 0:7 km2 effective area for muons with
109 GeV and recently placed a limit on the flux of EHE
neutrinos approximately an order of magnitude higher thanthe expected GZK cosmogenic neutrino intensities with
242 days of observation [7]. Since new particle physics
may vary the cross section by more than an order ofmagnitude as we noted above, this result should already
imply a meaningful bound on the N cross section. In thispaper, we study the constraint on the N cross section(N ) by the null detection of EHE neutrinos with the2007 IceCube observation. A model-independent boundis derived by estimating the lepton intensity at the
IceCube depth with the SM cross section scaled by aconstant. The constraint is displayed in the form of the
excluded region on the plane of the cosmic neutrino fluxand N . It is equivalent to an upper bound on N for agiven flux of astrophysical EHE neutrinos. We also study
the model-dependent constraint on the microscopic blackhole creation by neutrino-nucleon collision predicted in theextra-dimension scenario [5]. We calculate the fluxes of
leptons propagating in the Earth including the black holecross section and the final states to estimate expected event
rate in an equivalent IceCube 2007 measurement as afunction of extraterrestrial neutrino intensity. The null
detection of signal candidates leads to a constraint on
this particular scenario.
There are several works on model-independent upperbounds of N using the observational limit of EHE neutrino flux in the literature. References [3,4,8] derived thebound using the results of horizontal air shower search by
AGASA [9] and Fly’s Eye [10]. References [8,11] set thelimit based upon the flux bound by the RICE experiment[12]. Our approach in the present study is different mainly
in two respects. The previous works assumed the GZK
cosmogenic neutrino bulk as the guaranteed beam anddeduced the cross section limit using the GZK neutrino
intensity. Here extraterrestrial neutrino intensity is considered a free parameter. This method is an application of the103012-1

Ó 2010 The American Physical Society

SHIGERU YOSHIDAPHYSICAL REVIEW D 82, 103012 (2010)

technique to derive the flux limit based upon the quasidifferential event rate [7,12,13], which is independent ofspecific neutrino flux models. It is valid when the cosmic
neutrino flux and the cross section N do not rapidlychange over a decade of a given neutrino energy. As
EHE cosmic ray composition and their origin are still quite
uncertain, this approach provides more appropriate conservative limits on N . It also allows estimation of theminimum intensity of neutrino flux required to constrain
the cross section. Another difference is that the previous
works introduced the simplification that event rate solely
depends on rates of electromagnetic or hadronic cascades
directly initiated by neutrinos inside the effective volumeof the detector. This is in fact a good approximation for the
RICE experiment which is sensitive to radio emission fromshower events. However, underground neutrino telescopes
such as IceCube have larger effective areas for throughgoing muons and taus in EHE neutrino search [14–16].
This study of the model-independent limit includes calculation of not just intensities of neutrinos but also thesecondary muon and tau fluxes reaching the detection
volume for a given N and includes their contributionsin the overall event rate.
The paper is outlined as follows: First we discuss themodel-independent constraint in Sec. II. The method to
calculate the neutrino and the secondary lepton propagation from the Earth’s surface to the IceCube detector depth
is described. The fluxes for different strengths of N are
calculated and the resultant constraint is shown for bothN and the cosmic neutrino flux at neutrino energies of 1
and 10 EeV, respectively. Section III describes the constraint on the microscopic black hole production by
neutrino-nucleon interaction as an example of the modeldependent bound on N . Fluxes of muons and taus from
evaporation of black holes produced in the neutrinonucleon collision in the Earth are calculated. Their contributions, as well as those from contained hadronic showers
induced directly by the evaporation, would give an observable event rate in the IceCube 2007 measurement, andthereby put constraints on the black hole scenario. We
summarize our conclusions in Sec. IV.
II. MODEL-INDEPENDENT CONSTRAINT ON THE
NEUTRINO-NUCLEON CROSS SECTION
The flux limit obtained by the present IceCube observations allows us to place an upper bound on the neutrinonucleon cross section in a model-independent manner; newphysics cannot increase N too much, otherwise EHE
neutrinos would have produced observable events. As anunderground neutrino telescope is sensitive to not justshower events induced from neutrinos, but also tothrough-going muons and taus generated by the neutrinonucleon scattering, one must understand how much fluxesof these leptons reaching an underground detection volume
is increased with N . In this section, we first discuss our

method to calculate intensities of neutrinos, muons, andtaus at the underground depth of the IceCube observatory
for a wide range of N strength, followed by a descriptionof how they would contribute to the event rate. Finallythe constraint on both N and cosmic neutrino flux is
described together with the relevant discussions.
A. The method
Given a neutrino flux at the surface of the Earth, theneutrino and charged lepton fluxes at the IceCube depth arecalculated by the coupled transportation equations [16]:dJml Z
1 dndl¼ NA N;CCþNC J þ
dEJ ðE Þ
l
El dE l l
dX
cdl
Z
d
þ NA dE0 N;NC J ðE0 Þ
dE
Z
d
(1)
þ NA dE0l lN;CC Jl ðE0l Þ;
dE
dJl
ml
¼ NA lN Jl JldX
cdl ElZZddþ NA dE0 N;CC J ðE0 Þ þ NA dE0l lN Jl ðE0l Þ
dEl
dEl
d
Z
m
1 dn
þ l d dE0l 0 l Jl ðE0l Þ;(2)
El dElclwhere Jl ¼ dNl =dEl and J ¼ dN =dE are differential
fluxes of charged leptons (muons and taus) and neutrinos,
respectively. X is the column density, NA is the Avogadro’s
number,  is the local density of the medium (rock/ice) in
the propagation path,  is the relevant interaction crosssection, dndl =dE is the energy distribution of the decay
products which is derived from the decay rate per unit
energy, c is the speed of light, and ml and dl are themass and the decay life time of the lepton l, respectively.
CC(NC) denotes the charged (neutral) current interaction.
In this study we scale N to that of the SM prediction withthe factor Nscale , i.e., N  Nscale SM
N . It is an extremely
intensive computational task to resolve the coupled questions above for every possible value of N . To avoid thisdifficulty, we introduce two assumptions to decouple thecalculation of J from the charged lepton transportation
equation. The first is that distortion of the neutrino spectrum by the neutral current reaction is small and the other isthat regeneration of neutrinos due to muon and tau decay
and their weak interactions is negligible. These are very
good approximations in the energy region above 108 GeV
where even tau is unlikely to decay before reaching the
IceCube instrumentation volume. Then the neutrino flux issimply given by the beam dumping factor as

103012-2

SM;CCJ ðE ; XIC Þ ¼ J ðE ; 0ÞeNscale N

XIC

;

(3)

CONSTRAINTS ON NEUTRINO-NUCLEON INTERACTIONS . . .

where XIC is column density of the propagation path fromthe Earth’s surface to the IceCube depth. The charged
lepton fluxes, Jl¼; ðEl ; XIC Þ, are obtained asJ; ðE; ; XIC Þ ¼ NA


Z XIC

Z

0

dX

Z

dE0;

dE NscaledN; 0ðE ! E; ÞdE; ;dSM;CCN
dE0;SM;CC

 J ðE ; 0ÞeNscale N

X

:

(4)

Here dN; =dE; ðE0; ! E; Þ represents distributionsof muons and taus with energy of E; at XIC created byN collisions at depth X with an energy E0; . This iscalculated in the transportation equation, Eq. (2), with areplacement of J ðE0 Þ by Eq. (3).
Calculation of the neutrino and the charged lepton fluxes
with this method is feasible for a wide range of Nscale
without any intensive computation. A comparison of thecalculated fluxes with those obtained without the introduced simplification for a limited range of Nscale indicatesthat the relative difference we found in the resultant
J;; ðXIC Þ is within 40%. Since this analysis involves anorder of magnitude of increase in N , the introduced
approximations provide sufficient accuracy for the present
study.
Figure 1 shows the calculated intensities of the secondary muons and taus for various Nscale factors. Here theprimary neutrino spectrum is assumed to follow the GZK
cosmogenic spectrum and flux calculated in Ref. [17] assuming an all-proton cosmic ray composition with a moderately strong source evolution, ð1 þ zÞm with m ¼ 4
4

GZK ν

Flux [10 −18 cm−2 sec −1sr−1]

E > 10PeV

3

x10
2

1

x3
x1

0
−1

−0.5

0

0.5

1

cos (Zenith Angle)

FIG. 1 (color online). Integral fluxes of the muon and taus
above 10 PeV ( ¼ 107 GeV) at IceCube depth (  1450 m) for
GZK cosmogenic neutrinos [17]. The solid lines representmuons while the dashed lines represent taus. Numbers on eachof the curves are the multiplication factors (Nscale ) that enhance
the standard N cross section [19] in the relevant calculations.

PHYSICAL REVIEW D 82, 103012 (2010)

extending to z ¼ 4. One can see that the intensity is nearly
proportional to Nscale as expected since the interaction
probability to generate muons and taus linearly depends
on N . It should be pointed out, however, that the dependence starts to deviate from the complete linearity when
the propagation distance is comparable to the mean freepath of neutrinos, as one can find in the case of Nscale ¼ 10
in the figure. This is because the neutrino beam dumping
factor in Eq. (3) becomes significant under thesecircumstances.
The flux yield of leptons at the IceCube depth, Yl (l ¼0
 s, , ), originating from neutrinos with a given energy
at the Earth’s surface, Es , is given by Eq. (4) for muons andtaus and by Eq. (3) for neutrinos, with an insertion ofJ ðE ; 0Þ ¼ ðE  Es Þ. Here Es denotes a given incoming neutrino energy at the Earth’s surface. The resultant
event rate per neutrino energy decade is then obtained by[7,12,13],N ðEs Þ ¼

X
1 dJe þ þ s ZðE Þ d¼e ; ; 3 d logEl¼e ; ; ;;X

Z

dEl Al ðEl ÞYl ðEs ; El ; XIC ðÞ; Nscale Þ;(5)where Al is the effective area of the IceCube to detect the
lepton l. In the equation above, the l ¼ ,  terms represent the through-going track events while the contributionof events directly induced by neutrinos inside the detection
volume is represented by the terms l ¼ e ,  ,  . The
effective area for 0 s, A , is proportional to N i.e., Nscale
so the rate of contained shower events is linearly dependent
on the neutrino-nucleon scattering probability. Note thatthe differential limit of the neutrino flux is given by Eq. (5)for Nscale ¼ 1 with N ¼ 
 90 which corresponds to the
90% confidence level average upper limit. It calculates anupper bound of the number of events observed with bin
width of a decade of energy with the condition that energy
dependence of neutrino flux multiplied by the effective
area behaves as 1=E [4,12]. Limiting N in the present
analysis corresponds to an extraction of the relation between Nscale and the (unknown) cosmic neutrino fluxJe þ þ yielding N ¼ 
 90 . The obtained constraintson N is represented as a function of Je þ þ for agiven energy of Es . It consequently accounts for astrophysical uncertainties on the cosmic neutrino flux.
In scenarios with extra dimensions and strong gravity,
Kaluza-Klein gravitons can change only the neutral current(NC) cross section because gravitons are electrically neutral. Any scenarios belonging to this category can beinvestigated by scaling only NC
N in the present analysis.
The event rate calculation by Eq. (5) is then performed forYl ðNscale ¼ 1Þ with the effective area for ’s, A , enhanced
þ Nscale SM;NC
Þ=ðSM;CCþ SM;NCÞ since theby ðSM;CC
N
N
N
N
rate of detectable events via the NC reaction by IceCube

103012-3

SHIGERU YOSHIDAPHYSICAL REVIEW D 82, 103012 (2010)is proportional to NC
N . We also show the constraint in

10−6

IceCube 2007

this case.

10EeV

In this analysis we use the IceCube observation resultswith 242 days data in 2007 to limit N using Eq. (5). No
detection of signal candidates in the measurement has led
to an upper limit of the neutrino flux of 1:4 
106 GeV cm2 sec1 sr1 [7] in the energy range from3  107 to 3  109 GeV. The effective area Al is0:7 km2 for , 0:4 km2 for , and 3  104 km2 for0 s [7]. Constraints on N are then derived with Eq. (5).
The results for Es ¼ 109 and 1010 GeV are shown in
Fig. 2. Enhancing the charged current cross section bymore than a factor of 30 for E ¼ 1 EeV (109 GeV) is
disfavored if the astrophysical neutrino intensities arearound 107 GeV cm2 sec1 sr1 , near the upperbound of the GZK cosmogenic neutrino bulk. Note thatneutrino-nucleon
collision with E ¼ 1 EeV corresponds
pﬃﬃﬃ
to s  40 TeV and the present limit on N would placea rather strong constraint on scenarios with extra dimensions and strong gravity, although more accurate estimation requires studies with a model-dependent approach
which implements the cross section and the final-stateparticles from the collision predicted by a given particle
physics model. Taking into account uncertainty on theastrophysical neutrino fluxes, any model that increases
10−6

IceCube 20072

j (E) E [GeV cm−2 sec −1 sr −1]

10EeV

10−7

1EeV

10−8

10−9
10−33

10−32

10−31

10−30

10−29

σνN [cm ]2

FIG. 2 (color online). Constraints on the all-flavor sum of
cosmic neutrino flux and the charged current N cross section
based on the null detection of neutrino signals by the IceCube
2007 observation. The right upper region is excluded by thepresent analysis. The horizontal lines provide references of the
expected GZK cosmogenic neutrino fluxes [20] and the verticallines correspond to the SM cross section [19].

j(E) E 2 [GeV cm −2 sec−1 sr−1]

B. Results
10−7

1EeV

10−8

10−9
10−33

10−32

10−31

10−30

10−29

σνN [cm2 ]FIG. 3 (color online). Constraints on the all-flavor sum of
cosmic neutrino flux and the neutral current N cross section
for the scenario that only the neutral current reaction is enhanced
by a new physics beyond the standard model. The right upper
region is excluded by the present analysis. The horizontal lines
provide references of the expected GZK cosmogenic neutrino
fluxes [20] and the vertical lines correspond to the SM crosssection [19].

the neutrino-nucleon cross section to produce charged
pﬃﬃﬃ
leptons by more than 2 orders of magnitude at s 
40 TeV is disfavored by the IceCube observation.
However, we should point out that the IceCube 2007 data
could not constrain the charged current cross section if theintensity of cosmic neutrinos in the relevant energy regionis fewer than 108 GeV cm2 sec1 sr1 , within thelower range of prediction for the cosmogenic neutrino
fluxes [18]. Absorption effects in the Earth becomes sizable in this case, resulting in less sensitivity to the crosssection. This limitation will be improved for larger detection areas of the full IceCube detector.
Figure 3 shows the constraints when only the NC crosssection p
isﬃﬃﬃvaried. Enhancement of NC
N by a factor beyond
100 at s  40 TeV is disfavored, but this strongly depends on the cosmic neutrino flux one assumes. Because
the NC interaction does not absorb neutrinos during theirpropagation though the Earth, the cross section could bebounded even in the case when the neutrino flux is small,but the limit becomes rather weak; the allowed maximum
enhancement factor is on the order of 103 .
III. CONSTRAINT ON THE MICROSCOPIC
BLACK HOLE PRODUCTION
A constraint on a specific physics model that enhances
the neutrino-nucleon cross section is obtained by the same

103012-4

CONSTRAINTS ON NEUTRINO-NUCLEON INTERACTIONS . . .
6

GZK ν

PHYSICAL REVIEW D 82, 103012 (2010)

10

−7

10

−8

10

−9

10

−10

10

−11

10

−12

4

µ

2

τ
2

0
−1

−0.5

0

0.5

Downward

τ

−2

−1

dN/dE E [GeV cm sec ]Flux [10−18 cm−2 sec−1 sr−1]

E > 10PeV

1

106

Atm µ

µ

108

1010

12

10

Energy [GeV]

cos (Zenith Angle)

FIG. 4 (color online). (Left) integral fluxes of the muon and taus above 10 PeV ( ¼ 107 GeV) at IceCube depth (  1450 m) of the
GZK cosmogenic neutrinos [17] in case of the microscopic black hole creation scenario [5] for ðMD ; xmin Þ ¼ ð1 TeV; 1Þ (solid line),(1 TeV,3) (dotted-dashed line). The dashed line corresponds to the intensities obtained by the SM N cross section [19]. (Right) energy
spectra of the GZK  induced muons and taus at IceCube depth with downgoing [i.e. cosðzenithÞ  0] geometry expected by the blackhole model for ðMD ; xmin Þ ¼ ð1 TeV; 1Þ. The spectra produced by the SM cross section are also shown as dashed lines for comparison.
The curve labeled ‘‘Atm ’’ represents the atmospheric muon intensity estimated by the IceCube observation with its uncertainty
expressed by the two dashed lines [21].

procedure for the model-independent bound, except thetransport equations, Eqs. (1) and (2), would have total
and differential neutrino cross section provided by both
SM and the new model. Here we study the model of blackhole creation as a possible consequence of low-scale gravity that may occur if space-time has more than four dimensions. We use the predicted cross section of lack hole
production via the neutrino-nucleon scattering describedby Ref. [5], parametrized by the Planck scale MD , the ratioof the minimal black hole mass to the Planck scale xmin ,and the space-time dimension D ¼ 4 þ n. In this papern ¼ 6 and MD ¼ 1 TeV are assumed as representative
numbers. The resultant cross section may exceed SMinteraction rates by 2 orders of magnitude or even greater.
Therefore, the model-independent bound shown in theprevious section indicated that the 2007 IceCube observation should be already sensitive to some of the parameter
space in the black hole creation model.
The final states in the neutrino-nucleon scattering in thismodel are quite different from the SM case. Black holes
evaporate and generate multiple particles of all kinds,
like leptons, quarks, gluons, and bosons. These products
are distributed according to the number of degrees of
freedom. Consequently, the average number of muonsand taus, Nþ , are 1=30 of all particle average multi which is also determined by the specific model.
plicity N,
As N  10 at neutrino energy of E ¼ 1 EeV, multiple
muon or tau production would very rarely occur. Then theeffective differential cross section dN =dE; in the transport equations (1) and (2) in the black hole model isrepresented by

 Þ
dN
Nþ ðE ÞNðEN ðE Þ
¼
dE;
2E
2

(6)

 We take N from Ref. [5] in thewith 0  E;  2E =N.
present calculation. In this specific scenario, a muon or atau carries a small fraction (1=N  0:1) of incoming neutrino energy E in average, in contrast to the SM collisionthat takes away 1  y  0:8 of neutrino energy by a generated charged lepton.
Solving the transport equations gives the intensities ofsecondary muons and taus, which are shown in Fig. 4. One
can find in the zenith angle distribution (the left panel) thatthe intensities are increased by more than 2 orders ofmagnitude above the SM case. The large increase of Nenhances downgoing event rates while the upgoing muonand tau rates are more suppressed. The zenith angle distribution is consistent with the original work in Ref. [5]. It
should also be noted that the energy spectra is substantiallymodified from those in the SM case (the right panel). The
peak energy is around 1 EeV, an order of magnitude lower
than the SM spectrum, reflecting the fact that a smaller
fraction of neutrino energy is channeled into muons andtaus via the black hole evaporation. The peak happens tomatch the most sensitive energy region in the IceCube EHE
neutrino search [7].
Because N is solely predicted by the specific model,
the model-dependent constraints on N interactions isrepresented in the plane of extraterrestrial neutrino flux
and the number of events the IceCube 2007 run would
have detected. Figure 5 shows the number of events as afunction of the neutrino intensity at energy of 1 EeV, if the103012-5

SHIGERU YOSHIDAPHYSICAL REVIEW D 82, 103012 (2010)j(E) E 2 [GeV cm−2 sec−1 sr−1]

10 −6

IV. CONCLUSIONS

10 −7

10 −8

10 −9
10 −2

10 −1

1

10

Number Of Events

FIG. 5 (color online). Number of events as a function of
extraterrestrial neutrino flux at 1 EeV for ðMD ; xmin Þ ¼
ð1 TeV; 1Þ (solid line), (1 TeV,3) (dashed line). The 90% C.L.
line determined by the Poisson statistics is also shown as avertical line for reference.

The IceCube 2007 observation indicated that any
scenario to enhance either the NC or both the NCpﬃﬃandﬃ
CC equivalent cross section by more than 100 at s 
40 TeV is unlikely if the sum of the all three flavors ofastrophysical neutrino fluxes are greater than 3 
108 GeV cm2 sec1 sr1 in the EeV region. Many models of the GZK cosmogenic neutrinos exist to predict thisflux range, thus the present constraints limit new particle
physics beyond the SM, unless the extraterrestrial neutrino
intensity is smaller than the expectation. The example ofthe model-dependent bound on N has been also shownfor the microscopic black hole evaporation scenario. A
high cosmic neutrino intensity constrains the parameter
space of the black hole creation. Future observation by
the rapidly growing IceCube detectors will strongly limit
particle physics models which predict an increase ofneutrino-nucleon interaction probability.
ACKNOWLEDGMENTS

microscopic black hole evaporation occurs as in Ref. [5].
The Poisson statistics then determine the upper limit ofneutrino flux that can be still consistent with the null
observation by IceCube. It is indicated that the neutrino
intensity of 107 GeV cm2 sec1 sr1 is disfavored in this
scenario. More parameter space of MD and xmin will befurther constrained by near future observation with
IceCube whose detection volume is rapidly growing withan increase of the number of detectors in operation.

We wish to acknowledge the IceCube Collaboration for
useful discussions and suggestions. We thank Jonathan
Feng for helpful discussions on the physics models beyond
the standard model and providing numerical data of thecross section and multiplicity distribution predicted by the
micro black hole creation scenario. We also wish to thankLisa Gerhardt for helpful comments on the manuscript.
This work was supported in part by the Grants-in-Aid in
Scientific Research from the JSPS (Japan Society for the
Promotion of Science) in Japan.

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