Neutrino coannihilation on dark-matter relics?

Gabriela Barenboim,1,* Olga Mena Requejo,2,† and Chris Quigg2,‡

1Departament de Fısica Teorica, Universitat de Valencia, Carrer Dr. Moliner 50, E-46100 Burjassot (Valencia), Spain2Theoretical Physics Department, Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, Illinois 60510, USA

(Received 16 April 2006; published 27 July 2006)

High-energy neutrinos may resonate with relic background neutralinos to form short-lived sneutrinos.In some circumstances, the decay chain that leads back to the lightest supersymmetric particle would yieldfew-GeV gamma rays or charged-particle signals. Although resonant coannihilation would occur at anappreciable rate in our galaxy, the signal in any foreseeable detector is unobservably small.

DOI: 10.1103/PhysRevD.74.023006 PACS numbers: 96.50.Zc, 14.80.Ly, 95.35.+d

The possibility of detecting relic neutrinos by observingthe resonant annihilation of extremely energetic neutrinoson the background neutrinos through the reaction � ��! Z0

has been the object of extensive studies [1–3]. Given thesmall neutrino masses ( & 1 eV) indicated by current ex-perimental constraints, suitably intense sources of ex-tremely energetic (1021–1025 eV) cosmic neutrinos arerequired. The positions and shapes of the absorption linesin the extremely high-energy neutrinos spectrum are influ-enced by Fermi motion of the relics and by the thermalhistory of the Universe.

Relic neutrinos have a special standing: According to thestandard cosmology, neutrinos should be the most abun-dant particles in the Universe, after the photons of thecosmic microwave background, provided that they arestable over cosmological times. But the weight of cosmo-logical observations argues that neutrinos are not the onlyundetected relics. According to the Wilkinson MicrowaveAnisotropy Probe (WMAP) three-year analysis [4], thematter fraction of the present Universe is �mh

2 �0:127�0:007

�0:014, where h � 0:73� 0:03 is the reducedHubble parameter [5] and �m is the ratio of the matterdensity to the critical density %c � 3H2=8�GN. (HereH isthe Hubble parameter and GN is Newton’s constant.) Thebaryonic fraction is �bh2 � 0:0223�0:0007

�0:0009, and the neu-trino fraction ��h

2 � �Pim�i�=94 eV � 0:0072 & �bh

2.Accordingly, we have reason to believe that the matterfraction is dominated by an unseen ‘‘dark-matter’’ compo-nent. A popular hypothesis, realized in many extensions tothe standard model including supersymmetry, holds thatthe dark matter is dominated by a single species of weaklyinteracting massive particle (WIMP).

There is no confirmed observation of the passage ofWIMPs through a detector [6,7]. A positive signal reportedby the dark-matter experiment DAMA [8] seems in conflictwith upper limits set by the Cryogenic Dark Matter Search[9] and the EDELWEISS experiment [10]. Acceleratorexperiments have so far not yielded evidence for the pro-

duction of a superpartner candidate for the dark-matterparticle [6].

In this note, we ask whether resonant coannihilation ofneutrinos with dark-matter superpartners might be observ-able. We study a particular representative case of neutra-lino dark matter, in which the absolutely stable lightestsupersymmetric particle (LSP) is ~�0

1, a superposition ofneutral wino, bino, and Higgsino. In circumstances apt fordetection, resonant formation of a sneutrino in the reaction�~�0

1 ! ~�, shown in Fig. 1, might induce absorption lines ordirect signatures. A sneutrino heavier than the neutralino isimplied by the assumption that ~�0

1 be the LSP. A directsignature of sneutrino formation and decay requires that ~�decay into channels beyond the �~�0

1 formation channel.Informative examples include the parameter sets (I0 and L0)presented among the post-WMAP benchmarks for the con-strained minimal supersymmetric standard model inRef. [11]. (The same two models have been consideredby Datta and collaborators [12] in their recent study of theLorentz-boosted situation of ultrahigh-energy neutralinosscattering on the relic neutrino background.)

We summarize the relevant information in Table I.The first point to notice is that the neutrino energy at

resonance is approximately 300 and 500 GeV for the twosuperparticle spectra under study—squarely in the atmos-pheric neutrino range. Accordingly, there is no need toinvoke unknown mechanisms to produce the required neu-

ν

ν

ν χ

χ

0

FIG. 1. Resonant sneutrino formation in neutrino-neutralinocollisions. Double lines denote superpartners.

*Electronic address: Gabriela.Barenboim@uv.es†Electronic address: omena@fnal.gov‡Electronic address: quigg@fnal.gov

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trino ‘‘beam,’’ as we must for the � ��! Z0 resonance. Themodest resonant energies follow from the fact that in thesemodels the neutralino mass is more than 11 orders ofmagnitude larger than the (relic) neutrino mass. In bothcases, inelastic processes—sneutrino decays that do notreturn to the entrance channel—are prominent.

There ends the good news, at least for the Universe atlarge. Whereas we expect that stable neutrinos should bethe most abundant particles in the Universe after the pho-tons of the cosmic microwave background, the neutralinogas is on average very tenuous. The neutralino fraction ofthe Universe (identified with the dark-matter fraction) is

�~�h2 �%~�h2

h%c; (1)

where the numerical value of the critical density is

%c � 1:05 10�5h2 GeV cm�3: (2)

Consequently, the mass density of relic neutralinos is

%~� � 1:05 10�5�~�h2 GeV cm�3 � �N ~�M�~�

01�; (3)

where �N ~� is the mean number density of relic neutralinosthroughout the Universe. In the two models we considerhere, �N ~� & 10�8 cm�3, some 10 orders of magnitudesmaller than the relic neutrino density and 31 orders ofmagnitude smaller than the density of electrons in water.

The peak cross sections for inelastic sneutrino formation

�peak~�: inel �

8�m2~�

�m2~� �M

2~�0

1�2B�~�! ~�0

1��1�B�~�! ~�01���

(4)

are about an order of magnitude smaller than that [3] for Z0

formation, ��� ��! Z�peakvisible � 0:80��� ��! Z�peak �

365 nb. The cross sections integrated over the peak aresome 50 to 70 times smaller than �Z��� ��! Z�peak

visible �912 nb GeV, where �Z � 2:4952 GeV. At the resonancepeak, the interaction length for neutrinos on neutralinos inthe present Universe is

L ~�int � 1=�peak

~��N ~�: (5)

For the two spectra under consideration, we find L~�int �

�1:2; 2:4� 1015 Mpc for model (I0, L0), for �e;� interac-tions, with similar values for �� interactions. These valuesare roughly 11 orders of magnitude longer than the inter-action length at resonance for � ��! Z0, which is L� ��

int �1:2 104 Mpc in the current Universe [3]. A distance of1015 Mpc corresponds to approximately 1021:5 light years,so exceeds the 1:3 1010-year age of the Universe by aprodigious factor. If ��0

1 ! ~� coannihilation were to beobservable, absorption lines would not be the signature.We should have to rely on the direct detection of few-GeV� rays or charged particles from the inelastic decay chainsthat lead back to the �~�0

1 ground state.Our location in the Universe may not be privileged, but it

is not average. Cold dark-matter clusters in galaxies. Mostof the analytic expressions proposed in the literature fordark-matter halo profiles can be cast in the form

%~��r� �%0

�r=a��1� �r=a�������=�; (6)

where %0 is a characteristic density, a sets the radius of thehalo, � is the asymptotic logarithmic slope at the center, is the slope as r! 1, and � controls the detailed shape ofthe profile in the intermediate regions around a. A usefulcatalog of dark-matter halo profiles for the Milky Waygalaxy is given in Ref. [17]. The parameters of four profilesexamined there are collected in Table II. In each case, thedensity parameter %0 is adjusted to reproduce the presumeddark-matter density (neglecting local influences) at ourdistance (r� � 8:5 kpc) from the galactic center, %~��r�� �0:3 GeV cm�3 [21], which represents 5 orders of magni-tude enhancement over the relic density in the Universe at

TABLE I. Superpartner parameters in two minimal supersym-metric standard model scenarios [11], evaluated using theSPHENO [13] code for particle properties and the MICROMEGAS

[14] code for relic densities, as implemented at the Comparisonof SUSY spectrum generators web site [15] with mt �172:5 GeV [16] and the default values mb�mb� � 4:214 GeVand �s�M2

Z� � 0:1172.

Model I0 Model L0

M�~�01� 141 GeV 185 GeV

�~�h2 0.105 0.111

�N ~� 7:7 10�9 cm�3 6:4 10�9 cm�3

N ~��r�� 2:2 10�3 cm�3 1:6 10�3 cm�3

N ~� <400 cm�3 <300 cm�3

M�~�02� 265 GeV 349 GeV

M�~��� 265 GeV 349 GeVm~�e;� 290 GeV 422 GeVEres� 299 GeV 481 GeV

�~� 324 MeV 857 MeV�peak

~�: inel 40.2 nb 20.6 nb�~��

peak~�: inel 13.0 nb GeV 17.7 nb GeV

B�~�! ~�01�� 0.720 0.434

B�~�! ~�02�� 0.086 0.179

B�~�! ~��‘�� 0.194 0.386m~�� 277 GeV 386 GeVEres� 272 GeV 403 GeV

�~� 612 MeV 2.011 GeV�peak

~�: inel 52.3 nb 14.3 nb�~��

peak~�: inel 32.0 nb GeV 28.8 nb GeV

B�~�! ~�01��� 0.342 0.153

B�~�! ~�02��� 0.012 0.023

B�~�! ~����� 0.026 0.052B�~�! ~��W��a 0.620 0.772

aThe prominence of this decay mode is owed to the presence of alight stau (m~�1

� 150, 201 GeV) and appreciable ~�L-~�R mixingin these models.

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large. The resulting neutralino number densities,N ~��r�� � %~��r��=M�~�

01�, are given in Table I; the den-

sity profiles themselves are shown in Fig. 2, along with thedark mass enclosed within a sphere of specified galacto-centric radius.

What little we know about the distribution of dark matterwithin the solar system has been deduced by investigatingpossible dark-matter perturbations on the orbits of planetsand asteroids. In a recent contribution, Khriplovich andPitjeva [22] have deduced bounds on the amount of darkmatter that lies within the orbits of Mercury, Earth, andMars. On the assumption that the dark-matter density isconstant within the spherical volumes delimited by thesethree planetary orbits, the Earth and Mars orbits constrain

%�~� < 10�19 g cm�3 � 5:6 104 GeV cm�3; (7)

which would imply upper bounds on the number density ofneutralinos at Earth, N

~� & 350 cm�3.What are the prospects for observing the coannihilation

of neutrinos on the relic neutralino background? Considerfirst the interaction of neutrinos created in the collisions ofcosmic rays with Earth’s atmosphere. According to thecalculations of the atmospheric neutrino flux reviewed byGaisser and Honda [23], the muon-neutrino flux at E� �300 GeV is

dN�dE�

�10�5

300cm�2 s�1 GeV�1: (8)

The quantity �~��peak~�: inel provides a rough measure of the

coannihilation cross section integrated over the resonancepeak. Consequently, the product

R ~� � �~��peak~�: inel �

dN�dE�

(9)

characterizes the ~� coannihilation rate of a single neutra-lino in the rain of atmospheric neutrinos.

For model I0, we compute the coannihilation rateR�I0�

~� � 4:3 10�40 s�1. If we consider the neutralinodensity for the Universe at large, then the rate per unittarget volume is

R�I0�~� �

�N ~�R�I0�~� � 3:3 10�48 cm�3 s�1

� 10�40 cm�3 y�1; (10)

which is laughably small, and smaller still for model L0.The rates (10) are multiplied by �2:9 105; 5:2 1010� ifwe consider the mean density at Earth’s distance from thegalactic center or the orbital-dynamics upper bound (7) onthe density of dark matter in the solar system.

To pursue this line to a logical conclusion, we ask howmany interactions of atmospheric neutrinos with relic neu-tralinos occur in the atmosphere per unit time. Taking theEarth to be a sphere with radius R � 6371 km, we com-pute the volume of a 10-km-thick shell above the Earth’ssurface to be Vatm � 5:1 1024 cm3. The number of sneu-trino events induced in this volume is therefore &

3 10�5 y�1, taking the largest conceivable neutralinonumber density. Such an infinitesimal rate renders moot adiscussion of signatures and detection efficiencies. Settingaside questions of detection, the total number of eventswithin the Earth’s volume, V � 1:08 1027 cm3, is nomore than 6 10�3 y�1.

Extraterrestrial sources may interact with relic neutrinosover a larger volume. A representative estimate [24] of thediffuse neutrino flux from active galactic nuclei suggeststhat, at E� � 300 GeV, the flux of cosmic neutrinos arriv-ing from all directions is

TABLE II. Parameters of four radial density profiles of theMilky Way dark halo considered in Ref. [17], according to theparametrization given by expression (6), plus the mass enclosedin a sphere of radius r�.

Model � � a [kpc] %0GeV cm�3� ��1067 GeV�

ISO 2 2 0 4.0 1.655 3.8847BE [18] 1 3 0.3 10.2 1.459 3.8351NFW [19] 1 3 1 16.7 0.347 4.4205M99 [20] 1.5 3 1.5 29.5 0.0536 4.8675

M99M99

NFWNFW

BEBE

ISOISO

FIG. 2 (color online). Top panel: Dark-matter density profilesgiven by the form (6) with parameters specified in Table II. Thesolid (blue) curve shows the canonical Navarro-Frenk-Whiteuniversal profile [19]. The dot-dashed (purple) curve representsthe cuspier profile of Moore and collaborators [20]. Also shownare the Binney-Evans profile [18] (dotted green line) and thenonsingular profile corresponding to an isothermal sphere(dashed red line). Bottom panel: The corresponding enclosed-mass profiles.

NEUTRINO COANNIHILATION ON DARK-MATTER RELICS? PHYSICAL REVIEW D 74, 023006 (2006)

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dN�dE�

� 6:3 10�17 cm�2 s�1 GeV�1; (11)

approximately 2 10�9 the vertical atmospheric neu-trino flux we have just considered. The upper bound on theamount of dark matter in the sphere defined by Earth’sorbit,��1 au�< 7:85 1044 GeV [22], could therefore bethe site of up to (144 130) �~�0

1 ! ~� inelastic interactionsper year for model (I0, L0).

Let us compute the contribution of each event, at its ownlocation, to the signal recorded by a detector of unit area atEarth. It will be sufficient to assume that the detectorrecords signals from all relevant directions with perfectefficiency. We define the vector from the Sun to the detec-tor as ~s � �0; 0; s�, and denote the location of the target as~r � r�sin cos�; sin sin�; cos�. Then the vector thatpoints from the target to the detector is ~d � ~s� ~r, sothat d2 � r2 � s2 � 2rs cos. The decay products of thesneutrino produced at ~r are distributed isotropically.Accordingly, the fraction incident on a detector of areaA will be f �A=4�d2, the fraction of the solid anglethat the detector subtends.

The effective number of targets seen by a detector of unitarea at ~s is given by

neff�s� �1

4�M~�

Zd3r

%~��r�

d2

�1

2M~�

Z 1

�1d�cos�

Z s

0

r2dr%~��r�

r2 � s2 � 2rs cos: (12)

For the special case of a constant density %�~� �5:6 104 GeV cm�3, and s � 1 au � 1:496 1013 cm,the effective number of targets is

neff�s� �%~�

�s2M~� � 3 1015 cm�2: (13)

Using the cosmic-neutrino flux (11), we estimate that adetector in the vicinity of Earth would be sensitive to 7:710�26 events cm�2 y�1. With such a small number ofevents, there is no hope of detecting the few-GeV gammarays that would be created in the cascade back to ~�0

1.The halo of our galaxy contains a great quantity of dark

matter: for the Navarro, Frenk, and White (NFW) profile,4:4 1067 GeV lies within the galactocentric radius of oursolar system. The number of neutralino targets with whichneutrinos might coannihilate to form sneutrinos is thus�3:1; 2:4� 1065 for model (I0, L0). Focusing again onmodel I0 and taking the estimate (11) for the cosmic-neutrino flux, we find that the sneutrino formation ratethroughout the galaxy is 2:61017 s�1�8:11024 y�1.

ISOISO BEBE

NFWNFW M99M99

n eff

[cm

–2]

n eff

[cm

–2]

n eff

[cm

–2]

n eff

[cm

–2]

FIG. 3 (color online). Effective number of neutralino targets per unit of detector area viewed from a location at radius s from thegalactic center, according to four dark-matter density profiles for the Milky Way galaxy. Our location corresponds to s � 8:5 kpc.

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These rates are prodigious in absolute terms, but representan insignificant disturbance to the galaxy’s neutralinopopulation.

Can we hope to observe the sneutrino formation thatmight be bubbling away in our neighborhood? In this case,we define ~s � �0; 0; s� to be the vector from the galacticcenter to the detector, and use (12) to compute the effectivenumber of targets seen from Earth. For the NFW profile,we find neff�r�� � 4:9 1019 cm�2. For a detector of anyplausible area, only a tiny fraction of the 3:1 1065 neu-tralinos in the halo produce a visible signal. Indeed, usingthe neutrino flux (11), we expect that a detector in thevicinity of Earth would be sensitive to 1:310�21 events cm�2 y�1.

It is not obvious that Earth’s location should happen tobe optimal for viewing coannihilation events in the galaxy.In Fig. 3 we show how the effective number of targetsviewed depends on s, the galactocentric radius of theobservation point. For the smooth isothermal sphere(ISO) and Binney and Evans (BE) profiles, our solar sys-tem lies near the optimal distance, while for the NFW

profile the effective number of targets viewed is insensitiveto the detector position. The most singular profile weconsider, M99, favors an observatory close to the galacticcenter—not that we could contemplate one—but even inthis case, the sensitivity is enhanced by less than 2 orders ofmagnitude, which is far too little to enable detection.Singular profiles have more effect on the rates forneutralino-neutralino annihilations, which are proportionalto the square of the neutralino density.

If neutralinos account for much of the cold dark matterof the Universe, then it is possible that neutrino-neutralinocoannihilation into sneutrinos is happening all around us,but at a rate that will forever elude detection.

We thank Luis Ibanez for posing a question that stimu-lated this investigation. We are grateful to Celine Boehm,Dan Hooper, and Peter Skands for helpful conversations.Fermilab is operated by Universities Research AssociationInc. under Contract No. DE-AC02-76CH03000 with theU.S. Department of Energy. G. B. and C. Q. acknowledgethe hospitality of the CERN Particle Theory Group.

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