Muon flux limits for Majorana dark matter from strong coupling theories

Konstantin Belotsky,1 Maxim Khlopov,1,2 and Chris Kouvaris3

1Moscow Engineering Physics Institute, Moscow, Russia and Center for Cosmoparticle Physics, ‘‘Cosmion,’’ Moscow, Russia2APC Laboratory, Paris, France

3The Niels Bohr Institute, Copenhagen, Denmark(Received 1 November 2008; published 16 April 2009)

We analyze the effects of the capture of dark matter (DM) particles, with successive annihilations,

predicted in the minimal walking technicolor model (MWT) by the Sun and the Earth. We show that the

Super-Kamiokande upper limit on excessive muon flux disfavors the mass interval between 100 and

200 GeV for MWT DM with a suppressed standard model interaction (due to a mixing angle), and the

mass interval between 0 and 1500 GeV for MWT DM without such suppression, upon making the

standard assumption about the value of the local DM distribution. In the first case, the exclusion interval is

found to be very sensitive to the DM distribution parameters and can vanish at the extreme of the

acceptable values.

DOI: 10.1103/PhysRevD.79.083520 PACS numbers: 95.35.+d, 12.60.Nz

I. INTRODUCTION

The possibility of breaking the electroweak symmetry ina natural dynamical way in the context of technicolor hasbeen very appealing since it was first introduced in the1970s [1,2]. It was soon realized that in order to overcomethe problems of the first models, and, in particular, thenecessity to give mass even to the heaviest standard modelparticles like the top quark, a walking coupling is required.However, such a quasiconformal behavior was associatedwith a large number of extra flavors coupled to the elec-troweak sector, making it impossible to evade the strictconstraints from the electroweak precision measurements.Recently, in a series of papers [3–6], it has been demon-strated that the above problem can be avoided as soon asthe techniquarks transform under higher dimensional rep-resentations of the gauge group. More specifically, theorieswith fermions in the two-index symmetric representationof the gauge group can accommodate such a quasiconfor-mal behavior with a very small number of flavors, incontrast to the case where the fermions transform underthe fundamental representation. This set of theories doesnot violate the experimental constraints, thus being aneligible candidate for the upcoming search of the LHC.

Since, in principle, such theories are strongly coupled, aperturbative treatment can offer very little.Nonperturbative techniques and tools should be imple-mented for the exploration and study of these theories. Inthis context, the first lattice simulations have been devel-oped [7–9]. Similarly, low energy effective theories, validat the scale where the LHC will operate, can offer analternative approach to the problem, as well as distinctsignatures that can rule in or rule out walking technicolor(WTC) models [10–12]. In addition, holographic methodsinspired by the AdS/CFT correspondence can reduce theparameter space of the arbitrary couplings of the effectivetheories and make more definite phenomenological pre-

dictions [13,14]. This last idea is quite promising, espe-cially if one takes into consideration the fact thatquasiconformal theories resemble the exact conformalN ¼ 4 theory more than QCD, where this method hasgiven results close to the known experimental values.The simplest of the WTC models is the one with only

two flavors in the technicolor sector, i.e. the techniquarksU and D, where they transform under the two-index sym-metric representation of an SUð2Þ gauge group.1 There isan extra family of heavy leptons, i.e. �0 and � , that coupleto the electroweak sector, in order to cancel the Wittenglobal anomaly [10]. Such a theory, although simple interms of particle content, has a very rich structure. Thetechniquarks possess an enhanced SUð4Þ global symmetrythat includes SUð2ÞL � SUð2ÞR as a subgroup. This is dueto the fact that the adjoint representation is real. Afterchiral symmetry breaking, the vacuum is invariant underan SOð4Þ symmetry that includes the SUð2ÞV as a sub-group. Nine Goldstone bosons emerge from the breaking,three of which are eaten by the W and Z bosons. Theremaining six bosons come in three particle-antiparticlepairs, and are UU, UD, and DD, where we have sup-pressed color and Dirac indices. Their main feature isthat, although they are Goldstone bosons, they are notregular mesons, like the pions, since they are composedof two techniquarks, rather than a quark-antiquark pair.Therefore these Goldstone bosons carry technibaryonnumber that can protect the lightest particle from decaying.This fact opens interesting possibilities for dark matter(DM) candidates.There is an anomalous-free hypercharge assignment for

the techniquarks, which makes one of them electricallyneutral. For the sake of our study, we choose D to be theone, although the results are identical if we makeU neutralinstead of D. For the above hypercharge assignment, the

1The two-index representation of the SUð2Þ is the adjoint one.

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corresponding electric charges are U ¼ 1, D ¼ 0, � ¼�1, and � ¼ �2. The first possibility of having a darkmatter candidate is the case where the DD (which iselectrically neutral) is also the lightest technibaryon [15].The technibaryon number ofDD can protect it from decay-ing to standard model particles, as long as there are noprocesses that violate the technibaryon number belowsome scale. The sphaleron processes violate the techni-baryon number; however, they become ineffective once thetemperature of the Universe drops below the electroweakscale. Such a particle can account for the whole dark matterdensity, as long as its mass is of order TeV. However, such ascenario is excluded by direct dark matter search experi-ments, since the cross section of DD scattering off nucleitargets is sufficiently large and should be detected in theseexperiments. This problem can be avoided in a slightlydifferent version of the MWT [16]. Another possibility isto assume thatUU is the lightest technibaryon. In this case,it is possible to form electrically bound neutral statesbetween 4Heþþ and �U �U [17–19]. Alternatively, 4Heþþcan be bound to ���. This scenario cannot be excluded byunderground detectors, and therefore, it is a viablecandidate.

In this paper we focus on yet a third possibility. Becauseof the fact that in WTC the techniquarks transform underthe adjoint representation of the gauge group, it is possibleto form bound states between a D and technigluons [20].The object DaGa, where G represents the gluons of thetheory and a runs over the three color states (since it is theadjoint representation), is electrically neutral and colorless.If DLG has a Majorana mass, a seesaw mechanism isimplemented and the mass eigenstates are two Majoranaparticles, namely, a heavy N1 and a light N2. Although theMajorana mass term breaks the technibaryon symmetry, aZ2 symmetry, like the R parity in neutralinos, protects N2

from decaying. BecauseN2 is a linear combination ofDLGand DRG (which has no electroweak coupling), N2 has asuppressed coupling to the Z boson of the form

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig2 þ g02

p2

Z�sin2� �N2�

5��N2; (1)

where g and g0 are the electroweak and hypercharge cou-plings, respectively. We have omitted terms that couple N1

with N2, as N1 is very heavy and decays to N2 very fast.The mixing angle � is defined through the relation tan2� ¼2mD=M, where mD and M are the Dirac and Majoranamasses of theDLG particle. Although the Z2 symmetry canprotect N2 from decaying, two N2 can coannihilate tostandard model particles through Z boson mediation. Forsmall masses of N2, the main annihilation channel in theearly Universe is to light particle-antiparticle pairs, likequarks and leptons. For larger masses, the dominant chan-nel is annihilation to pairs of Wþ �W�. It was shown in[20] that because of the sin� dependence of the annihila-tion cross section, sin� can be chosen accordingly in such a

way that the relic density of N2 matches the dark matterdensity of the Universe. This is depicted in Fig. 1, wheresin� is given as a function of the mass ofN2, in order forN2

to account for the dark matter density. The fact that N2 is aMajorana particle and consequently does not have coherentenhancement in scattering off the nuclei targets in theunderground detectors, along with the suppression of thecross section that scales as sin4�, makes N2 evasive fromexperiments like the Cryogenic Dark Matter Search(CDMS) for almost any mass of interest. For the samereason, this form of dark matter cannot explain the positiveresults of DAMA/NaI and DAMA/Libra experiments [21].Under specific conditions, N2 can be excluded as a maindark matter particle only for a small window of masses,roughly between 100 and 150 GeV [22]. N2 is also sus-ceptible to indirect signatures such as those suggested in[23].In a similar fashion, for a hypercharge assignment where

�0 is neutral, this heavy neutrino can be a dark matterparticle if the evolution in the early Universe is dominatedby quintessence-like dark energy [24]. However, it waspointed out in [22] that such a candidate, even in the casewhere �0 is a Majorana particle, is excluded for masses upto 1 TeV (depending on the value of the local dark matterdensity of the Earth).We should emphasize here that the investigation and the

results of [20] can also apply to �0 when it is electricallyneutral, if we assume that the left-handed �0 has aMajorana mass and a Dirac mass. Because of the factthat DG (for the hypercharge value that makes it neutral)interacts with the electroweak sector as �0 does (for adifferent hypercharge that makes �0 neutral), N2 canequally represent the lightest Majorana particle made ofDG or �0.We would like to stress a few points regarding the

models we use in this paper. By now, it has been estab-lished in a solid way that only quasiconformal technicolortheories can pass the electroweak precision tests and si-multaneously avoid the problems of old technicolor theo-ries (like the large effect of flavor changing neutralcurrent). To our knowledge, the technicolor models weexamine are the only viable technicolor models that arenot excluded and also have desired features (light Higgs,stable DM, etc.). The model that we have focused on, the

50 100 500 1000

N2 mass, GeV

0.150.2

0.3

0.5

0.7

1

sin

θ

FIG. 1. Dependence of the mixing parameter sin� on the massof N2.

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MWTmodel, is the simplest of all and is comprised of twotechniquarks in the adjoint of SUð2Þ. Our analysis is validfor the general case of technicolor dark matter models thatare not based on the stability of technibaryon number or onthe excess of particles over antiparticles.

The paper is organized as follows: In Sec. II, we calcu-late the relevant annihilation cross sections for the earlyUniverse and the Sun. In Sec. III, we calculate the capturerate of the relic density by the Sun and the Earth. In Sec. IV,we provide the muon flux from the captured dark matterparticles. We present our conclusions, and a discussion ofthe uncertainties that might enter in our results, in Sec. V.

II. ANNIHILATION OF N2 IN THE EARLYUNIVERSE AND IN THE SUN

A pair of N2 annihilates mainly into pairs of fermions,f �f, and into pairs of W bosons, WþW� (with longitudinalpolarization), provided that the energy is sufficient to openthe corresponding channel. In our calculation we adopt thefollowing formulas for the annihilation cross sections mul-tiplied by the relative velocity and averaged over thethermal velocity distribution at temperature T,

h�viff ¼ 2G2Fm

2�f

�PZ

�C2A

2

m2f

m2þ

�ðC2

V þ C2AÞ

þ�C2V

2� 17C2

A

8

�m2f

m2

�T

m

�sin4�; (2)

h�viWW ¼ 2G2Fm

2ð2m2 �m2WÞ2�3

W

�m4Z

PZ

T

msin4�; (3)

which are deduced from [20,24–26]. Here

PZ ¼ m4Z

ð4m2 �m2ZÞ2 þ �2

Zm2Z

; �f;W ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�m2

f;W=m2

q;

where mf, mZ, mW , and m are the masses of the final

fermion f, Z and W bosons, and N2, respectively; GF isthe Fermi constant; CV ¼ T3L � 2Qsin2�W ; and CA ¼ T3L

are standard model parameters of f. T3L and Q are theweak isospin and the electric charge of the correspondingparticle. Equations (2) and (3) represent the cross sectionsin a nonrelativistic approximation in the form of

h�vi ¼ �0 þ �1

T

m: (4)

In the standard big bang scenario, the modern relic densityof N2 is given by [20]

�N2h2 ¼ 1:76� 10�10 ffiffiffiffiffi

g�p

g�s�1½GeV�2��m

T�

�2; (5)

where the freeze-out temperature is given by

m

T�¼ L� 3

2ln½L�; L ¼ ln

�mPlm�1

6:5ffiffiffiffiffig�

p�: (6)

mPl is the Planck mass, and g� and g�s are effective spindegrees of freedom contributing to the energy and entropydensities of the plasma at T ¼ T�, respectively. For the T�values of interest, g� ¼ g�s ¼ 80� 100.By inspection of Eqs. (2) and (3), we see that at the

freeze-out, the cross section is dominated by h�vi ¼ �1 �T=m for all the annihilation channels (N2N2 ! WW,N2N2 ! f �f, with f being �e;�;, e, �, , u, d, s, c, b).

For the annihilation of N2 into top-quark-antiquark pairs,the first term of Eq. (4) can be important (for the mass

20 50 100 200 500 1000 2000

N2 mass, GeV

10 11

10 10

10 9

10 8

10 7

10 6

10 5

10 4

v,G

eV2

light fermions WW

tt

at T T

20 50 100 200 500 1000 2000

N2 mass, GeV

10 14

10 12

10 10

10 8

10 6

v,G

eV2 light fermions

WW

tt

at T T

FIG. 2 (color online). The cross sections of N2N2 ! light fermions, t�t, WW are shown for the models from [20] (solid lines) and[24] (dot-dashed lines) at freeze-out (left panel) and for the Sun (right panel). ‘‘Light fermions’’ include all fermions except the topquark; channels with off-mass shell W or f were not taken into account.

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interval mt < m & 500 GeV); however, for m>mt, theWW channel strongly prevails during the freeze-outperiod.

The condition that N2 saturates all the cold dark matter(CDM),

�N2h2 ¼ �CDMh

2 ¼ 0:112; (7)

fixes the free parameter of the model sin� [or �1 inEqs. (5) and (6)]. It is given in Fig. 1 as a function of themass of N2. In the model of pure Majorana neutrinos (i.e.�0) [24], where sin� ¼ 1, such a condition is provided byan extension of the big bang scenario due to quintessence[27]. Because of this, there is a difference between theannihilation cross sections predicted in these two variantsof MWT DM particles, as shown on Fig. 2 (left panel). Inthe model from [20], the total cross section at the freeze-out is virtually independent of the mass due to the fact thatit is always adjusted via Eq. (7) in order to give the properrelic density. However, at the temperatures of the solarcore, T� � 1:3 keV T�, the first term in Eq. (4) [orEq. (2)] dominates for many types of final fermions.Therefore, annihilation rates inside the Sun are differentfrom those in the early Universe (see Fig. 2, right panel).Channels with light final fermions in the Sun becomesuppressed with respect to those in the early Universe,while the t�t channel becomes of special importance (seeSec. IV).

III. CAPTURE OF RELIC N2 BY THE SUN ANDTHE EARTH

RelicN2 with density in the vicinity of the Solar System,assumed to be loc ¼ 0:3 GeV=cm3, may scatter off nucleiinside the Sun and the Earth and can be trapped by theircorresponding gravitational potential wells. The interac-tion of N2 with nuclei A is spin dependent and the respec-tive cross section can be represented as [20]

�N2A ¼ 2G2F�

2

�Issin

4�; (8)

where � is the reduced mass of N2 and A, and

Is ¼ C2 � �2JðJ þ 1Þ: (9)

The coefficient C takes into account quark contributions tothe spin of the nucleon, and for weak interactions it is[28,29]

C ¼ Xq¼u;d;s

T3q�q

�� 12 0:78� 1

2 ð�0:48Þ � 12 ð�0:15Þ ¼ 0:705 for p

12 ð�0:48Þ � 1

2 0:78� 12 ð�0:15Þ ¼ �0:555 for n:

(10)

The other coefficient in Eq. (9) relates the nucleon contri-bution (with spin s and orbital momentum l) to the spin J ofthe nucleus, and within the single unpaired nucleon model,

it is

�2JðJ þ 1Þ ¼ ½JðJ þ 1Þ þ sðsþ 1Þ � lðlþ 1Þ�24JðJ þ 1Þ : (11)

We assume that only hydrogen contributes to the capture ofrelicN2 in the Sun. In this case, from Eqs. (9)–(11) we haveIs ¼ 0:705 � 34 � 0:37. This estimate agrees with that for

Dirac neutrinos, taking into account only the axial current(spin-dependent) contribution, which gives [30] Is � 1:32 �3=16 � 0:3. The case of the Earth will be commentedseparately.It is not difficult to estimate the capture rate of the Sun or

the Earth. The common expression for that is

_N capt ¼XA

ZnN2

h�0N2A

vinAdV; (12)

where nN2and nA are the number densities ofN2 and A in a

given volume element dV, and �0N2A

is the cross section for

an N2 � A collision times the probability that N2 losesenough energy to be gravitationally trapped by the Sun.Introducing a nuclear form factor FA, one writes

�0N2A

¼ �N2A

Z �Tmax

T1F2Að�TÞ

d�T

�Tmax

¼ �N2A�F2A

v2escðrÞ � �v21

v2; (13)

where �T is the transferred energy in an N2 � A collision,�Tmax ¼ 2�2v2=mA, � ¼ ðm�mAÞ2=ð4mmAÞ with mA

being the nucleus mass, vesc is the escape velocity atdistance r from the center of gravity, and v ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv21 þ v2

esc

pand v1 are the N2 velocities at distances r

and r ! 1, respectively. �F2A is the F2

A averaged over theinterval �T 2 ½T1; �Tmax�, where T1 mv21=2. Thenumber density nN2

of N2 at distance r can be related to

the one outside the potential well, nN2ðr ! 1Þ nN21, as

[30,31] nN2¼ nN21 � v=v1. We average Eq. (12) over the

velocity distribution, by the substitution

nN21 ! nN21 � f1ðv1Þdv1;

where nN21 ¼ loc=m. We use a velocity distribution of the

form

f1ðv1Þ ¼ v1ffiffiffiffi�

pv0v�

�exp

��ðv1 � v�Þ2

v20

�

� exp

��ðv1 þ v�Þ2

v20

��; (14)

where v� ¼ v0 ¼ 220 km=s. The captured N2 accumulatein the solar core and annihilate. Their number density isgoverned by the equation

_N ¼ _Ncapt � _Nann: (15)

Here, _Nann is the number of N2 disappearing due to anni-hilation per second,

_N ann ¼Z

n2N2h�vidV:

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It is 2 times larger than the rate of annihilation acts. Theeffect of evaporation of the captured and thermalized N2 isneglected, something that is valid for all m * 3 GeV [26].Thermalization of the captured N2 happens, due to succes-sion of collisions with nuclei, well before N2 has time toannihilate (the ratio of respective characteristic times,within N2 mass range of interest, is �10�5��10 for theSun). Resolving Eq. (15) for _Nann, one finds

_N ann ¼ _Ncapttanh2

� ffiffiffiffiffiffiffiffiffiffiffi_Ncapt

_Neq

vuut �:

Here

_N eq ¼ Vtherm

h�vit2age

defines a critical capture rate above which equilibriumbetween capture and annihilation is established duringthe solar lifetime tage. For _Ncapt _Neq, _Nann is suppressed

with respect to _Ncapt as _Ncapt= _Neq. The value

Vtherm ¼�4� �

core

Tcore

Tesc

�3=2

R3 ��TeV

m

�3=2

8>><>>:2:0� 1026 cm3

�Tcore

15�106 K

�3=2

�150 g=cm3

core

�3=2

for the Sun

1:0� 1023 cm3

�Tcore

7000 K

�3=2

�11 g=cm3

core

�3=2

for the Earth(16)

characterizes the effective volume that the captured N2

occupy, after being thermalized, having a Maxwell-Boltzmann velocity distribution. Here, R is the radius ofthe Sun or the Earth; �, core, and Tcore are their mean andcore densities and core temperatures, respectively (for theSun Tcore T�), and Tesc mv2

escðr ¼ RÞ=2. For the deri-vation of Eq. (16) we assume that the density of matter andthe temperature within Vtherm are constant and equal totheir core values. In this case, the potential energy withrespect to the center takes the formUðrÞ ¼ Tescðcore= �Þ�ðr=RÞ2=2, and an integration of the thermalized N2 numberdensity, nN2

ðrÞ ¼ nN2ð0Þ expð�UðrÞ=TcoreÞ, can be done

analytically. Note that the quantity given in Eq. (16) forthe Sun agrees with the one in [32]. For the integration inEq. (12), we assume a matter density distribution in r as in[30]. The effect of the finite size of hydrogen is insignifi-cant in this case. In fact, qa < 0:1 (typically�0:02), whereq ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2mA�Tp

is the transferred three-momentum and a isthe nucleus size, so FAðqaÞ � 1. The capture and annihi-lation rates obtained for the case of the Sun are shown in

Fig. 3. For comparison, the same results obtained withinthe approximation of [33,34]

_N capt � 4:5� 1018 s � loc

0:4 GeV=cm3

�270 km=s

�v

�3

� �H;SD

10�6 pb

�1000 GeV

m

�2

are shown in Fig. 3, too. As seen, the agreement is verygood.As for the capture by the Earth, the essential difference

is that the potential well in this case is very ‘‘shallow,’’ andincident DM particles have a chance of being captured onlyunder special kinematic conditions. This may happen inscattering off nuclei with nonzero spin, if the DM particlehas a mass close to the mass of the nucleus and/or it isinitially very slow. As a rough estimate, we take one of thenuclei with nonzero spin that is quite abundant in the Earth:the isotope 57Fe present in natural Fe with a fraction� 2%,while all iron is assumed to make up� 30%� 40% of the

50 100 200 500 1000 2000N2 mass, GeV

10 20

10 21

10 22

10 23

10 24

10 25

N2

capt

ure

rate

,sec

1

necess

ary fo

r equilib

rium

annihilation

50 100 500 1000 5000

N2 mass, GeV

10 19

10 20

10 21

10 22

10 23

10 24

10 25

N2

capt

ure

rate

,sec

1

necessary for equilibrium

annihilation ratecapture rate

FIG. 3 (color online). The capture and annihilation rates for N2 in the case of the Sun for the models from [20] (left panel) and [24](right panel). Dotted (green) lines show the capture rates obtained in the approximation [33,34].

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Earth’s mass. We assume that it is concentrated in the coreof the Earth. In the core, vescðr � 0Þ � 14:2 km=s. Thecross section of the 57Fe� N2 interaction is defined in Eq.(8). For Is, one can give a maximal estimate in the singleunpaired nucleon approximation of Is � 0:23 [fromEqs. (9)–(11)]. The form factor of iron can be roughlyestimated in the thin sphere approximation [28],

FAðqaÞ ¼ sinðqaÞqa

:

Since the integration interval in Eq. (13) is small (in thecase of the Earth) with respect to the characteristic scale ofthe FA variation, we take �F2

A ¼ F2AðT1Þ. For a simple

(maximal) estimate, we use Eq. (14) as the N2 distributionin the vicinity of the Earth, where the depletion in the vspace for v < 42 km=s caused by solar attraction isignored (moreover, we also neglect the effect of possibleaccumulation of dark matter particles in the Solar System,described in [35]). The capture rates of the Earth’s poten-tial due to 57Fe� N2 collisions as obtained for the twoMWTmodels considered are represented in Fig. 4. As seenin Figs. 3 and 4, in the model from [20], _Ncapt is lower and_Neq is higher (compared to the other model considered),

which is a consequence of the sin4� suppression of bothN2N2 and A� N2 interactions [Eqs. (2), (3), and (8)]. Inthe case of the Earth, the annihilation channel N2N2 !WW is suppressed with respect to the case of the Sun by7000 K=15� 106 K� 5� 10�4, and therefore _Neq in

Fig. 4 does not fall for m * 2 TeV. Similarly, neutrinoyields from annihilation in the Earth and the Sun shouldnot differ form & 2 TeV, except for a difference caused byabsorption effects in the solar matter, being meaningfulonly for high N2 masses.

By comparison of Figs. 3 and 4, we see that for themaxima of _Ncapt in the case of the Earth, which occur for a

mass of N2 close to the mass of 57Fe, the ratio of _Ncapt for

the Earth over the Sun is �10�10 in both models consid-ered [20,24]. The neutrino flux, induced by annihilation ofN2 in the Earth, will differ from that from the Sun as the

aforementioned ratio multiplied by the squared ratio ofdistances to the Sun and the Earth centers 5� 108 and afactor _Ncapt= _Neq � 10�4 for the model from [20]. So, even

for the model from [24], where the _Ncapt= _Neq suppression

is much weaker, the N2-annihilation-induced neutrino fluxfrom the Earth at its maximum is a few tens times less thanthat from the Sun. A greater neutrino flux from the Earthrelative to that from the Sun can hardly be expected fromcollisions of relic N2 with other nuclei present in the Earth,for which the abundance is more uncertain. Since thesensitivity of the Super-Kamiokande (SK) experiment tothe neutrino induced muon flux from both the solar andEarth cores is of the same order of magnitude, we shallneglect N2 annihilation effects in the Earth in the followingconsideration.

IV. MUON FLUX FROM THE CAPTURED N2

Annihilation ofN2 produces e,�, and neutrinos with aflux

�� ¼ _Nann

N�

2 � 4�r2 ;

where N� is the multiplicity of neutrinos produced perN2N2 annihilation, and r is the distance to the center ofthe Sun or the Earth. Neutrinos from annihilation passingthrough the solar matter can reach the Earth, traverse it,and induce at its surface the muon flux

�� ¼ ���

x�

x��

¼ _Nann

2 � 4�r2Z

dN��ðE��

Þ hx�ix��

(17)

where x� and x��are the mean free paths (measured as

matter columns in g=cm2) of � and �� with respect to the

processes of energy loss (mainly ionization) and �� þA ! �þ X. The last equality in Eq. (17) is a general-ization for the case of energy dependence.

50 100 200 500 1000 2000

N2 mass, GeV

10 12

10 14

10 16

10 18

10 20

N2

capt

ure

rate

,sec

1

necessary for equilibrium

50 100 200 500 1000 2000

N2 mass, GeV

10 13

10 14

10 15

10 16

10 17

10 18

N2

capt

ure

rate

,sec

1

necessaryfor equilibrium

FIG. 4. A rough estimate of the capture rates of relic N2 by the Earth for the models from [20] (left panel) and [24] (right panel).

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For muon energy losses we use the approximation [36]

� dE�

dx�¼ a � 2:3

MeV

g=cm2; (18)

taking into account ionization effects and ignoring pairproduction and bremsstrahlung effects. This approxima-tion is valid for E� & 800 GeV and seems reasonable

within the energy interval of concern, as will be seenfrom the final result. The muon mean free path is

x�ðE�Þ ¼ E�=a:

For the muon mean energy hE�i in the reaction �� þ A !�þ X, defining hx�i, we use the following [34,37] rela-

tionship:

hE�i ¼ b � E��; b ¼

�0:5 for ��

0:7 for ���:(19)

The neutrino mean free path is governed by the chargedcurrent (CC) interaction with nucleons. The correspondingcross section calculated in [38] can be approximated within�10% accuracy as

�CC �� 0:72� 10�36 E��

E0cm2 for ��

0:37� 10�36 E��

E0cm2 for ���;

(20)

where E0 ¼ 100 GeV. For the respective mean free path,one has

x��¼ 1

NA�CC

� x0E0

E��

;

x0 ��2:3� 1012 g=cm2 for ��

4:5� 1012 g=cm2 for ���;

(21)

where NA ¼ 6� 1023 1=g is the number of nucleons pergram of matter. Separating annihilation neutrinos fromdifferent channels (ch) in Eq. (17), we have

�� ¼ _Nann

8�r2b

ax0E0

Xch

BrchN��ðchÞhE2��ðchÞi; (22)

where Brch, N��ðchÞ, and hE2��ðchÞi are, respectively, the

branching ratio, the neutrino yield, and the mean neutrinoenergy for a given channel. Equation (22) agrees numeri-cally within 20%� 40% with the respective formula of[34].Figure 5 illustrates the difference in the most important

N2 annihilation channels for the cases of freeze-out (leftpanel) and solar core (right panel). Muon neutrinos pro-duced in the channels N2N2 ! �uu, �dd, �ss, as well as indecays of any born muons, are not of interest because theprimary particles have enough time to slow down beforethey decay, preventing neutrinos from producing a signalabove the experimental threshold. Also, in the dense solarcore, c and b quarks with initial energy * 100 GeV par-tially lose their energy. However, we shall neglect thiseffect. It is not expected to cause an essential error, be-cause, as we will show below, the muon signal is predictedto be virtually unobservable when highly energetic c and bquarks are born as a result of N2 annihilation. In fact, forlarge N2 mass, the �tt and WW channels (see Fig. 5) domi-nate. Decaying, they give c and b quarks with degradedenergy. So, a very large mass of N2 is needed to noticeablyproduce highly energetic c and b quarks. However, forsuch masses, the muon signal is predicted to be smallbecause of suppression of _Nann itself (in the model from[20]) and/or because of effects of �� absorption in the solar

matter. Generally, as seen in Eq. (22), in the absence of ��

absorption in matter, a decrease of neutrino energy quad-ratically reduces the intensity of the muon signal. For thisreason, channels giving rise to ��, as a result of a long

cascade chain, can be considered negligible with respect tosimilar channels with shorter cascade chains of �� pro-

duction. Moreover, as a rule of thumb, longer chains have

50 100 500 1000N2 mass, GeV

10 4

10 3

10 2

10 1

1

Bra

nchi

ngra

tio

νµ νµττccbb

WW

tt

at T T

50 100 500 1000N2 mass, GeV

10 7

10 6

10 5

10 4

10 3

10 2

10 1

1

Bra

nchi

ngra

tio

νµ νµ

ττccbb

WW

tt

at T T

FIG. 5 (color online). The branching ratios of the most interesting N2 annihilation channels for the cases of freeze-out (left panel)and in the solar core (right panel). The models from [20,24] do not differ in this plot.

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an additional suppression due to the branching ratio. Forexample, the channelsN2N2 ! WW ! �, bc, cs ! ��X

have a suppression with respect to N2N2 ! WW !��� � 102 times.

We shall take into account the following channels:

N2N2 ! �� ���; (23)

N2N2 ! þ� ! � ����; (24)

N2N2 ! c �c ! � ���X; (25)

N2N2 ! b �b ! � ���X; (26)

N2N2 ! WþW� ! � ���; (27)

N2N2 ! t�t ! ðW� ! � ���Þð �b ! � ���XÞ; (28)

where it is understood that we can have similar channelswith the charge conjugate final states of the reactionsabove. The small branching ratio of the channel ofEq. (23), as Fig. 5 shows in the case of the Sun, is partiallycompensated by its short chain advantage mentionedabove. The distributions of muonic neutrinos for energyE��

E are given, correspondingly, for the channels of

interest by

dN��

dE¼ �ðE�mÞ for Eq: ð23Þ; (29)

dN��

dE¼ 0:18� 2

m

�1� 3

�E

m

�2 þ 2

�E

m

�3� ð0; mÞ

for Eq: ð24Þ;(30)

dN��

dE¼ 0:13

�m

�5

3� 3

�E

�m

�2 þ 4

3

�E

�m

�3� ð0; �mÞ;

�m ¼ 0:58m for Eq: ð25Þ; (31)

dN��

dE¼ 0:103� 2

�m

�1� 3

�E

�m

�2 þ 2

�E

�m

�3� ð0; �mÞ;

�m ¼ 0:73m for Eq: ð26Þ; (32)

dN��

dE¼ 0:107

m�

�m

2ð1� �Þ; m

2ð1þ �Þ

�;

� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�m2

W

m2

sfor Eq: ð27Þ:

(33)

In the channel of Eq. (28), we have contributions from Wand b decays:

dN��

dE¼ dN��ðWÞ

dEþ dN��ðbÞ

dEfor Eq: ð28Þ; (34)

dN��ðWÞdE

¼ 0:107

ð1� m2W

m2tÞm�

ln

� minð Emt

mð1��Þ ;mt

2 Þmaxð Emt

mð1þ�Þ ;m2

W

2mtÞ

�

�mm2

W

2m2t

ð1

� �Þ; m2ð1þ �Þ

�;

(35)

dN��ðbÞdE

¼ 0:103 � 2ð1� m2

W

m2tÞ �m�

½FðE; E�; EþÞ ð0; E�Þ

þ FðE;E; EþÞ ðE�; EþÞ�;FðE; E1; E2Þ ¼ 2

3

��E

E1

�3 �

�E

E2

�3�� 3

2

��E

E1

�2 �

�E

E2

�2�

þ lnE2

E1

;

�m ¼ 0:73m; E� ¼ m2t �m2

W

2mt

�mð1� �Þ;

� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�m2

t

m2

s: (36)

The step function is

ðE1; E2Þ �1 for E1 <E< E2;0 otherwise:

Note that most of the formulas are taken from [34,39]. Thevalues �m<m in Eqs. (31), (32), and (36) take into accountthe partial energy losses by c and b quarks, respectively,while they are hadronized. For the WW channel, the effectof W polarization was neglected. Consideration of thiseffect would correct our estimation for �� from this chan-

nel by�20%. However, as we shall see, it is not necessaryto consider this correction because for m * 3 TeV, wherethe WW channel is important, the predicted muon signalbecomes too faint for the existing experimental setups.Also note that the spectrum of Eq. (36) for �� from b

decay in Eq. (28) differs from that given in [34,39]. In thenotation of [34,39], the signs before x2 and y2 in therespective formulas there should be altered in order to becorrect. All spectra predicted by Eqs. (29)–(36) are illus-trated in Fig. 6. The effect of absorption of �� in solar

matter is taken into account as follows [40]:

dN��ðoutside the SunÞ

dE¼ dN��

ðin the solar coreÞdE

� exp

�� E

130 GeV

�; (37)

dN ���ðoutside the SunÞ

dE¼ dN ���

ðin the solar coreÞdE

� exp

�� E

200 GeV

�: (38)

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Note that the spectra in the solar core for neutrinos ��

and antineutrinos ��� do not differ and are given by

Eqs. (29) and (31)–(36).The muon (including both �� and �þ) fluxes predicted

for the models from [20,24] are shown in Fig. 7. Wecompare them with the respective upper limit obtainedby the Super-Kamiokande Collaboration [41],

�� < 6� 10�15 cm�2 s�1:

It relates to an angle of 10 around the solar center, whichis expected to embrace most of the muon flux induced byannihilation in the Sun for the N2 masses of interest. Forour chosen parameters, the SK limit excludes the intervals

100 GeV<m< 200 GeV (39)

for the model from [20] and

m< 1500 GeV (40)

for the model from [24].

V. DISCUSSION

In the present paper we have considered candidatesemerging from the minimal walking technicolor modelwith a suppressed coupling to the Z boson. These candi-dates can account for the dark matter density of theUniverse and can simultaneously avoid any contradictionwith the results of direct dark matter search experiments. Infact, CDMS and Xenon experiments can exclude only atiny window around 100 GeV for the model from [20].Being elusive for direct dark matter searches, we inves-tigated the possibility of indirect effects of N2 dark matterparticles. In particular, we estimated the neutrino fluxes onthe surface of the Earth from N2 annihilations in the Sunand in the Earth. These effects can provide constraints onthe parameters of the considered models.One of the biggest uncertainties entering these con-

straints is that the final results depend on the local densityof DM particles as well as on the velocity distribution. Asone can see in Fig. 7, a decrease of loc down to0:2 GeV=cm3, a value that is currently acceptable, withunchanged velocity distribution, would leave only a tinyinterval ofm around 110 GeVexcluded for the model from[20]. This makes any conclusion for the model from [20]indefinite, based on the searches for muon signals. In anycase, the obtained constraint is more strict than the onebased on direct dark matter search experiments [22].Regarding the model from [24], the existing uncertain-

ties can hardly influence essentially the excluded range ofEq. (40). This result agrees with the analogous result,obtained by the Kamiokande Collaboration for Majorananeutrinos [42], although it differs in some details (whichare most likely related to the estimations of cross sectionsfor different annihilation channels and their neutrinoyields). Moreover, the effects of b and c quarks slowingdown in dense solar matter (which are neglected in ourconsideration) and all the unaccounted for annihilationchains (which produce ��, including chains which go

through b and c quarks) can lead to an increase in the

50 100 500 1000 5000

N2 mass, GeV

10 19

10 18

10 17

10 16

10 15

10 14

10 13

10 12

muo

nfl

ux,

cm2 se

c1

SK upper limit

FIG. 7 (color online). Muon fluxes predicted for the modelsfrom [20] (solid line) and [24] (dot-dashed line) in comparison tothe Super-Kamiokande constraint. Dark and light dots showpredictions based on the results of [46] for the models from[20,24], respectively.

50 100 150 200

0.001

0.002

0.003

0.004

WWbb

ττ

t b

t W

200 400 600 800 1000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

WW

bbττ

t b

t W

FIG. 6 (color online). The energy spectra of �� produced in different channels for m ¼ 200 GeV (left panel) and m ¼ 1000 GeV(right panel).

MUON FLUX LIMITS FOR MAJORANA DARK MATTER . . . PHYSICAL REVIEW D 79, 083520 (2009)

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predicted muon fluxes for large m. This can give neutrinoswith lower energy, which may avoid absorption in solarmatter. For this reason, the channel N2N2 ! �tt ! b !��X, giving softer �� than �� from t ! W and WW

channels (see Fig. 6), has a significant contribution forhigh m.

Uncertainties also come from neutrino propagation ef-fects, i.e. neutrino oscillations and interactions with matter.Oscillation effects in vacuum and matter [43] might changethe flavor content of the neutrino flux going from the solarcore to the detectors on the Earth. The uncertainties in thedescription of oscillations become less ambiguous once weaverage the effect because of the large distance, energy,and time measurement intervals involved. Indeed, neutri-nos of three flavors are generated in the source. The flux ofmuon neutrinos at the Earth is

���¼ P�����

þ Pe���eþ P���

; (41)

where P�� is the probability of transition between flavors

� and �. As we see in the right panel of Fig. 5, the most

important N2 annihilation channels of neutrino production,depending on the mass, are N2N2 ! �bb with b decayinginto cl ��, N2N2 ! �tt with t ! Wb, and N2N2 ! WW withW ! l ��. In order to give a simple estimate of the effect ofneutrino oscillations, we are going to consider the �bbmodeas the dominant one for m< 200 GeV, the �tt mode for0:2<m< 3 TeV, and the WW mode for m> 3 TeV. Inthe �bbmode, the production of � is suppressed because ofthe fact that is much heavier than the muon and theelectron (we ignore here the difference in the neutrinospectra, and we also ignore the production of � from decay), so the produced neutrino flux is roughly �e þ�� þ 0:3�. In the WW mode, one has �e þ �� þ �.

Finally, in the �tt mode both W and b decay, so we haveroughly �e þ �� þ 0:7�. For the transition probabilities

obtained in [44], within the three-flavor scheme, we havethe following: all P�� � 0:3 for E & 10 GeV, and P�� ¼P� ¼ P � 0:4, Pe ¼ P�e � 0:2, and Pee � 0:6 for

E � 10 GeV. Therefore, the following change of the fla-vor content of the neutrino flux going from the solar core tothe detector might take place,

�e þ �� þ 0:3� )� 0:8�e þ 0:8�� þ 0:8�; at E & 10 GeV

0:9�e þ 0:7�� þ 0:7�; at E � 10 GeV;for m & 200 GeV;

�e þ �� þ 0:7� ) 0:9�e þ 0:9�� þ 0:9�; at any E; for 200 GeV & m & 3 TeV;

�e þ �� þ � ) �e þ �� þ �; at any E; for m * 3 TeV:

In our rough estimate we assume that P�� ¼ P�� ¼ðP�� þ P �� ��Þ=2. In this simplified picture we see thatoscillations redistribute the flavor content of the neutrinoflux, making it more homogenous. This would decrease thepredicted �� flux by �20%.

Effects of � interactions lead not only to absorption ofneutrinos in solar matter due to CC interactions, but also toa loss of energy for �e;� due to neutral current (NC)

interactions, and for � due to both NC and CC (in thelatter case � is regenerated from the chain �N ! X, ! �X). Energy loss is a small effect (the ratio ofrespective cross sections is �NC=�CC � 1=3). It shoulddecrease the muon signal a little for small m, but increaseit a bit for large m. This is analogous to the neutrinosproduced from the long cascade chains that we neglected,since a shift to a lower energy partially saves neutrinosfrom absorption.

In the four-flavor oscillation scheme, where a steriletype of neutrino �s is added, there can also be a dampingeffect of the signal for low N2 mass, because of the �� !�s transition. However, an amplification of the signal forhigh mass can occur because of the larger penetratingability of neutrinos oscillating to �s [45].

In [46] (including the website referred to therein), themuon fluxes for separate annihilation channels are ob-tained, taking into account oscillation and interaction ef-fects. For the sake of comparison, we have taken their dataon muon fluxes for the case of ‘‘standard’’ oscillationparameters (case ‘‘B’’ in their notation). For a mass m<175 GeVwe considered the channels b �b, c �c, , for 175<m< 2000 GeV the t�t channel, and for m> 2000 GeV thet�t andWW ones. Since the data were related to muon fluxesper one annihilation act, we multiplied the fluxes by therespective annihilation rates. The points on Fig. 7 show therespective results for a few mass values. As seen, theagreement is extremely good, especially for high N2

mass. Therefore, the upper limits on m in Eqs. (39) and(40) do not change appreciably. At lower mass (which is oflimited interest for the models studied), where the b �bchannel dominates, a slightly larger difference is notedbetween our result and [46], mainly due to ignorance ofthe b-quark energy losses in our calculation and the afore-mentioned oscillation effect. However, the suppression of�� yields from the b �b channel is partially compensated by

the excessive �� yields from the channel scaling as 4%

of the branching ratio at this particular mass range. As it is

BELOTSKY, KHLOPOV, AND KOUVARIS PHYSICAL REVIEW D 79, 083520 (2009)

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shown in Fig. 7, the difference between our calculation andthat of [46] is practically indistinguishable on the plot.

ACKNOWLEDGMENTS

The work of K. B. was supported by Khalatnikov-Starobinsky Leading Scientific School Grant No. N

4899.2008.2 and Russian Leading Scientific School GrantNo. N 3489.2008.2. The work of C. K. was supported by theMarie Curie Fellowship under Contract No. MEIF-CT-2006-039211. We would also like to thank J. Edsjo forhelping with the interpretation of the results published in[46].

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