Volume 180, number 4 PHYSICS LETTERS B 20 November 1986

A H I G H - E N E R G Y N E U T R I N O S I G N A T U R E F R O M S U P E R S Y M M E T R I C R E L I C S

John S. H A G E L I N

Department of Physics, Maharishi International University, Fairfield, IA 52556, USA

K.W. N G and Kei th A. O L I V E

School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA

Received 21 July 1986

We compute the energy spectrum of high-energy (0.1-10 GeV) neutrinos produced by the annihilation of supersym- metric (SUSY) cold dark matter trapped in the sun. We compare this spectrum to the spectrum of atmospheric neutrinos and find that in the direction of the sun the solar flux of neutrinos can exceed the atmospheric background for neutrino energies E v >~ 1 GeV, and are as much as a factor ~ 30 above background for energies E v ~ few GeV. We discuss these sig- natures for standard SUSY relics as well as for superstring relics.

Although observations of the overall mass-density of the universe are consistent [1 ] with ~2 ~- 0.1 (~2 = P/Pc, Pc = 1.88 X 10 -29 h 2 g c m -3 a n d h 0 =H0/100 km Mpc-1 s - 1 ) inflation as well as theoretical prejudice demand that ~2 = 1. This discrepancy is accounted for by postulating the existence of dark matter . By utilizing big-bang nucleosynthesis constraints [2] on the abundances o f the light elements, one finds that the cosmological mass density o f baryons is limited to ~2 B <~ 0.2. Thus, if I2 = 1, the majori ty of the dark matter must be non-baryonic. On the scale of galaxies, large spherical mass distributions (halos) surrounding spiral galaxies have been argued as necessary for disk stability [3] and to account for the flatness of observed rotat ion curves [4]. Observations [5] on the scale of galactic halos indicate ~2 ~- 0 .05 -0 .15 and are consistent with big-bang nucleosynthesis limits on the baryon density. It has been argued [6], however, that unless the stellar mass distribution has a very special form or the halo is filled with massive ~ 100Mo) black holes, the halo is also largely non-baryonic. In this paper we will assume that the galactic halo is filled with non-baryonic cold dark matter * 1. We will assume that in the solar neighborhood the dark matter is present at a density n = 0.3(1 GeV/m) cm -3 and velocity ~ = 300 km s -1 where m is the mass of the dark matter candidate.

At the above density, dark matter would be gravitationally trapped by the sun [8,9] and annihilations in the sun would provide a source o f high-energy neutrinos [10]. The neutrino energy spectrum will depend on the dark mat ter candidate [11,12]. For Dirac neutrinos or sneutrinos, the spectrum will be monochromat ic [11 ] with E v = m, while for Majorana particles the spectrum is very broad [11 ] , with the bulk o f the neutrinos arriving with rela- tively low energies o f tens o f MeV. In this paper, we compute in detail the high-energy tail o f this distribution and compare it with the background of atmospheric neutrinos produced in cosmic ray collisions (for a review, see ref. [13]).

Specifically, we will be interested in the neutrino spectrum produced by the annllfilation of supersymmetric (SUSY) dark matter candidates [ 14,15]. In SUSY theories, R-par i ty [ 16] implies that the lightest SUSY particle (LSP) must be stable and provide us with an excellent dark mat ter candidate. In the minimal SUSY standard

,1 For a review of the pros and cons of various types of dark matter see ref. [7]. Here it sufficies to say that cold matter has a mass typically in excess of O(1 GeV).

0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (Nor th -Hol land Physics Publishing Div is ion)

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model, the LSP is in general a mixture of four neutral f~ermions [15] : the photino ~; the zino Z (or equivalently the wino ~, and the bino B); and two higgsinos H and ft. The present mass density of the LSP, X, can be deter- mined from the annihilation cross section which is derived from the effective low-energy lagrangian for the inter- action between X and ordinary fermions (quarks or leptons) f:

£ = ~ (XT#75X) [ fTu(AfP L + B f P R ) f ] , (1) f

where PL,R are the left and right projection operators (1 + 75)/2 and expressions for Af and Bf can be found in ref. [15]. In terms of Af and Bf the annihilation cross section is given by

<OOrel> = ~ + b x , (2)

a = ~ O(mx - mr) -- m f / m x ) m f ( A f - Bf) , (3a) f 27r (1 2 2 1/2 2 2

"~ ~ j O ( m × - m f ) ( 1 - 2 2 1/2 2 + B 2 ) ( 4 m 2 x3 2- ~ A f B f m 2 + 7 ~ - m r / m × ) ] = m f / m x ) [(Af - ~ - m f ) + 3 m 2 ( A f _ B f ) 2 ( 1 2 2 - 1 f 2rr (3b)

where (O2el > = 6 T im x = 6x . The resulting value of ~2 x will o f course depend on the squark and slepton masses contained in A f and Bf.

We are interested in the high energy tail o f the neutrino spectrum resulting from "pr imary" neutrinos XX ~ ff -~ X + v + ~, and in the case where f is a b-quark, from "secondary" neutrinos due to b -~ c ~ X + v. These primary and secondary neutrinos are directly calculable, since the intermediate heavy fermions b, c and r have lifetimes that are short compared to the mean free path in the strongly interacting solar interior. In addition to these primary and secondary neutrinos there are expected to be a significant number of soft neutrinos due to pion and kaon decays, with energy spectra that are more difficult to estimate. These soft neutrinos have been the basis o f previous analyses [ 10,11 ] and are not our principle focus here.

To compute the primary and secondary neutrinos, we first need the annihilation rate of X's in the sun. De- pending on the mass of the LSP, the sun will loose X'S either through evaporation [8] or annihilation. It was found [17] that for m x >~ 6 GeV, annihilation dominates over evaporation. This domain, m× >~ 6 GeV, is actually favored by terrestrial and cosmological arguments. For example, an LSP higgsino must be more massive than the b-quark in order to annihilate efficiently enough to present over-closing the universe [15]. With regard to the photino, experimental searches for anomalous single photons (ASPs) in e+e - annihilations appear to rule out m~ <~ 6 GeV for selectron masses m g ~< 50 GeV [18]. However, for m~- ~< 6 GeV cosmology requires m-~ ~< 60 GeV to avoid overclosure [15], leaving only a restricted domain for m~-~< 6 GeV, i.e., i f5 GeV ~< rn~- ~< 6 GeV with 50 GeV ~< m g ~< 60 GeV. It may be that this window is already excluded by the expanded data sample now available to the leading ASP experiment [19].

Assuming m× t> 6 GeV, the annihilation rate at equilibrium will be equal to 1/2 the trapping rate, which is given by [9,11 ]

F T -~ (6.8 × 1028 s - 1 ) a p , 3 6 f E m x l , (4)

2 h e e 1 er ross f r ' s o n r on d mm 1 where Op,36 = o /10 -36 cm is t last'c s c a t t i n g c section o X p ot s an fE = " [ , 4 3 m p m × ( m 2 + ~ 2 ) - 1 ]. In terms ofAu, d and Bu, d (the Af and Bf for the up and down quarks), the elastic scat- tering cross section is [11,20]

Op = (A d - Bd)2(~j -- ¼)2(3 / r r )m2m2(mx + rap) -2 , (5)

where ~'--- (A u - Bu) / (A d - Bd). One can identify several sources of high energy neutrinos: X)( ~ r + r - followed by r ~ £ + u r + ~ (with a

branching ratio of 0.175); XX -+ c~ followed by c ~ s + £ + u~ (0.126); XX -~bb followed by b ~ c + £ + 9~ (0.117)

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and the subsequent c decay; and finally i fx is massive enough XX -+ tt- followed by t ~ b + ~ + v~ (0.11) and the subsequent b decay. The branching ratios for XX ~ t t , bb, c~ or r+r - are obtained from eqs. (2) and (3). The average (in a strong sense, since the spectrum turns out to be very flat) neutrino energy will be E v ~- (1/3)m x.

The differential flux of neutrinos from the sun is now easily computed: 1

d ~ _ ~ F T B f 1 dI ' f

dE v f (4~rd 2) I 'f dE v ' (6)

where d is 1 au, Ff and Bf are the decay rates and branching ratios for f-+ f ' + £ + v~. For Ff, we use the standard weak decay matrix element ]M} 2 = 64G2(pf • Pv~)(P~" Pf ') , which yields the following differential rates:

2 2 m4,pf dFf GF Ev~ I mf, mf lq 2Efp f (m2_ 2m2,) 4 2 3 4 2 [ m 2 - 2Ev~(Ef + Pf) dEv~---47r3Efpf - 3 E v ~ ( 3 E f p f + P f ) Ev~ 4E2~ l n ~ m ~ - - - - - ~ - ~ ] d ' t .~. ~ .1. (7a)

for Ev~ <~ (m 2 - m2,)/2 (El + pf);

dFf G2 F_ 6 4 2 2 4 - [.24 (mf - 6 m f m f , + 3 m f m f , + 2m6,) + ~1 m4,(Ef_pf)Ev~

dEv~ 41r3Efpf

( m 2 - 2 E v ~ ( E f - P f ) ' )] (m 2 _ 2 2 2 + 2 3 3 1 4, 2111 , ( 7 b ) _1 2 m f , ) ( E f - p f ) Ev~ ~ ( E f - p f ) E v ~ + ~ m f m f m 2,

for Ev~ >1 (m 2 - m2,)/2 (El + pf); and

dFf G 2 pf, [EfEf'(m 2 + m2,) 4L.2L. 2 2 2 2 4p2p2,] (8a) . . . . ~L~f~f, - - j m f m f , - - ~

dEf, 4rr3Ef

for mf,Ef/mf <~ Ef, <~ m2/2 (El +pf) + m~./2 (El - pf);

dFf G 2 . . . . dEf, ig mfmf , (mf + 1 2 m2,) 4rr3Efpf [~_~(m6+m6,) a 2 2. 2 m2,)+~EfEf, p fpf , (mf +

(8b) a 2 2 + 2 2 2 2 2 3 3 -- ~EfEf,pfpf,(EfEf, - pfpf,) - ~ (PfPf' - EfEf') m2m2' - "4 (EfEf, pfpf , ) (mf + mf,) + ~ (EfE~,- p3fp3,)] ,

for Ef, /> m2/2(Ef + p f ) + m2,/2(Ef-pf ) . (Eqs. (8a), (8b) are needed for the computation of secondary neutrinos b -+ c -+ X + v decays.) The total rate Ff for a relativistic particle f is given by

4 _ 1 6 1 4 2_I 6 1 m8,/m2+l 2 4 , Pf [~ mf -- g mfmf , 1" g mf, - - ~ ~ mf mf, In (mf/mf )]. (9)

47r3Ef

We have assumed that m~ = mv~ = 0. In fig. 1 we have plotted d'I'/dEve + d~/dE~- e = 2dcb/dEve. There is an equal contribution from the muon neu-

trino flux. Also included in the figures is the differential flux of atmospheric neutrinos taken from ref. [13]. We have plotted only the flux o f v e +-re. Although the solar flux of v u + -~u equals the flux o f v e + re, the atmospheric background flux o f v u + ~ ranges from -~ 2 - 7 times larger than the v e + ve flux. We have also multiplied the iso-

• . . / . t o

tropic flux given mref . [13] by a factor 2rr(1 - cos 30 ) ~- 0.84 sr for the flux coming from a cone in the direc- tion o f the sun with a half opening angle o f 30 °, which is meant to represent the capabilities o f proton-decay de- tectors to determine the directionality of neutrino induced events.

We are now in a position to discuss the results o f the standard SUSY candidates. In fig. la, we show the spec- trum of v e + ~e for a "generic" higgsino. A generic higgsino includes a H, a H, or in fact any weakly interacting majorana dark matter candidate with canonical couplings to the Z 0. In this case, we cannot require ~ = 1, since ~2 is determined by m~ through eqs. (2) and (3). Taking h 0 = 1/2, we find for m~ = 6, 10, 20 and 40 GeV, ~2 "~ 1,0.37,0.12 and 0.036 respectively. From fig. 1 a, we see that for Eu > 1 GeV, the solar flux o f neutrinos exceeds the atmospheric background (in the direction o f the sun) for m~ = 6 GeV. Even for m~ = 40 GeV, R - (dqb/dE~)/

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V o l u m e 180, number 4 PHYSICS LETTERS B 20 November 1986

10-2

~ 104 0

10 .4

I : 10-5

lO-°

1°~.1 o12' o.5 1.o 2:0 lO

Eve(GeM) 1o .2 ~ - - r ~ - ~ r - - ~ - ' r T ~ - -

\ \

\ c) Photino ,> \

10 -3 \ O \ \ '7, 6 \

E \ \ 0 10 -4 10

I~ lO-5 + 20

&

40

10-7 ~ 0.1

' ' ' ~ . . . . I . . . . . . . . I ~] 102 ~ ~ ~ " k - ~ - \

\ \ \ a) Higgsino 1 \ \ b) Higgsino \ (generic) [ ~ \ (closure)

= 6 \ , '~ 10 -3

'E 1 o - ' ~ ~ - a ~ . \

L- /"l\ e l j10-° .o °'°I

L , , ~ , 1 , , ~ l ~ , , t x I_1 1 0 - 7 ~ 5.0 20 0.1 0.2 0.5 1.0 2.0 5.0 10

I=ve(GeV) 20

10215 ~ " ' ~,' ' ' ' I

"7, \

10_3 \ \ \

' I \ \

¢dl 4 I \ ~ 10- \ \ \

\

1>~ 10_s ~E- \

e l j lO5L-2,0 ,,o °'° I f

10,~ ~o . . . . . . . , 2:0

I I I I ~ i=[

d) Superstring

\

\ \ \

\ \ 50

~o~-~ I I ~ i i i l l

5.0 10 20 0.2 0.5 1.0 2.0 5.0 10 20 0.1 0.2 0.5 1.0

Evo(GeV ) Eve(GeV)

Fig. 1. The differential flux of v e + ~e from the sun due to the annihilation of cold dark matter as compared to the atmospheric background flux in the direction of the sun (dashed line) for (a) a generic higgsino, X = H or H for rn x = 6, 10, 20 and 40 GeV. In this case, S2 x = 1, 0.37, 0.12 and 0.036 respectively; (b) a symmetric higgsino, x = (~H + o~l) (o 2 + ~ 2 ) - 1 ~ with o/-~ adjusted so that S2 x = 1 : u/E = 3, 1.8, 1.4 and 1.2 for the same set of LSP masses as in (a) ; (e) a photino with degenerate squark and slepton masses, m ~ , adjusted so that $Z x = 1: m~, = 71, 8 8 , 1 1 7 and 166 GeV for the same set of LSP masses as in (a); (d) superstring LSP (photino) with masses 20, 40 and 50 GeV and ffermion masses taken from the relations of refs. [ 20,21 ].

( d ¢ / d E a t m ) > 1 when E v t> 3 GeV. At its m a x i m u m , R ~- 10 for m ~ = 6 GeV at E v TM 3 G e V ; R -~ 15 for m ~ = 10 G e V at E v -~ 4 G e V ; R ~- 40 for m ~ = 20 GeV at E v -~ 7 GeV a n d R ~- 120 for mfi = 40 GeV at E v = 20 GeV.

In a SUSY t h e o r y , it is poss ib le to arrange for a d e c o u p l i n g o f the LSP h iggs ino f r o m the Z 0 , w h i c h occurs

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when the LSP higgsino is o f the form ~0 _+ ~0. This happens in a theor~ when the ratio o f VEVs o/~ (= (01HI0)/ (01ill0)) -~ 1 in which case the LSP higgsino is of the form S = (-Off + off) (02 + ~2)-1/2. In fig. lb, we plot the neutrino fluxes resulting from such an LSP S, adjusting the ratio 0/-6 so that ~2 = 1 (for h 0 = 1/2). Thus for m~- = 6, 10, 20 and 40 GeV we choose o/-6 = 3, 1.8, 1.4 and 1.2 respectively. The curves in fig. lb are independent o f the choice o f rn~ for scalar masses >~ 20 GeV; both scattering and annihilations are controlled by Z ° exchange. Although for mff = 6 GeV, the spectrum is quite similar to that in the generic higgsino case, for larger masses, the fluxes decrease quite rapidly. This is because for larger values o fmff we have chosen v ~ ~- 1 in order to maintain ~2 = 1. This results in a partial decoupling from the Z 0 which boosts the cosmological relic density o f the S but also suppresses the trapping rate. We nonetheless find that R > 1 for E v < 10 GeV provided mff < 40 GeV. Also R ~ 5 for mff = 6 GeV at E v = 3 GeV. For both higgsino cases (figs. 1 a, b) secondary neutrinos from b ~ c ~ x + v are important since XX ~ bb is the dominant annihilation channel.

The photino [14,15] is perhaps the most plausible LSP candidate in many models and is the most frequently discussed SUSY dark matter candidate. The relic photino density can be maintained at ~2 = 1 by adjusting the squark and slepton masses. For now we will assume that these are all degenerate. As in the symmetric higgsino case, S, demanding ~2 = 1 results in reduced neutrino fluxes as m~ is increased (fig. lc) . For m~ = 6, 10, 20 and 40 GeV we have taken mi~ "- 71 ,88 , 117 and 166 GeV to ensure ~2 = 1. The flux falls off with increasing m x be- cause the elastic scattering cross section depends on m ? , and because PT also depends on m x l . Quantitatively, the photino results are similar to the symmetric higgsino results.

For completeness, we note that other gaugino LSPs (a W, B or Z) would produce fluxes comparable to the cases, though none of these other gauginos represent favored dark matter candidates in most SUSY models. For any type o f gaugino, the neutrino fluxes are reduced if m ff > m fi-: heavy squarks suppress the XP elastic cross sec- tion, Op, while fight sleptons maintain an acceptably low relic mass density as is the case in the superstring candi- dates we are about to discuss.

The t~mal dark matter candidate we wish to discuss is the LSP found [20] in a minimal phenomenological model [21] inspired by the superstring [22]. In these models E 6 can be broken by Wilson loops via the Hosotani mechanism [23] to one o f several low-energy gauge groups [24] the smallest being SU(3)o X SU(2)L × U(1)y × U(1)E. The matter fields are contained in a 27 of E 6 and have the following SU(10) and SU(5) decompositions: 27 = 16 + 10 + 1 = (10 + 5 + 1) + (5 + 5) + 1. The most general superpotential one can write down is invariant under R-parity with the assignments to the fermion partners R(16) = + 1, R (10) = - 1 and R ( I ) = - 1 . Thus in ad- dition to the gauginos ~ and Z there is at least one extra ZE due to the U(1)E factor, and in addition to the ~I and H (contained in the 10) there is an SO(10) singlet N which must be included. The LSP may now be a linear combination o f these six fields [20]. Even in this general case, depending on the scale of supersymmetry breaking m 1/2, the LSP is still predominantly a ~ with some admixture of S.

One of the features o f the minimal model of ref. [21 ] is that the squark and slepton masses are fixed by m 1/2, with the ratios m~ : ml/2 :m~L : m~R = 1.95 : 1 : 0.7 : 0.4. Thus the LSP mass and relic density are given solely in terms of ml /2 . For ml[ 2 ~ I 0 0 - 1 0 0 0 GeV, it was found [20] that m× ~ 2 0 - 1 0 0 GeV and

~2x(h0/0.5)2 ~- 20±2 . (10)

In fig. ld , we plot the differential flux of v e + ~e for such superstring relics with masses 20, 40 and 50 GeV (we have arbitrarily chosen m t = 40 GeV). For m x = 20 GeV, the LSP is almost a pure photino while for m x = 40 and 50 GeV there is a significant higgsino admixture. Because pho t ino-pro ton elastic scattering takes place exclusively via squark exchange, the large squark masses ( " 200 GeV) in the m x = 20 GeV case greatly suppresses the solar trapping and hence the neutrino flux. At the larger LSP masses, the higgsino admixture allows for a scattering via Z 0 exchange and so the flux in this case is qualitatively similar to previous higgsino examples. This issimilar to the results found in ref. [20], i.e., that as m x is increased the neutrino flux rises due to the enhanced scattering contribution from Z 0 exchange. As one can see however, the flux only rises above the atmospheric background f o r e /> 10 GeV.

l)

Finally, we would like to stress that in order to test these supersymmetric dark matter candidates, we are really

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only interested in neutrino events (in underground detectors) withenergies in excess of I GeV. In addition, be- cause the atmospheric neutrino flux ratio [25] between v e +~e and vu +vu is ~-0.3 f o r e v "" 1 - 2 GeV and the solar flux ratio is uni ty, one is experimentally much more sensitive to v e + ~e" To see how close we are to existing data, we compare our predictions with neutrino observations by the IMB detector when correlated to the direc- t ion of the sun [26]. Data with sufficient information (energy and direction) is available for E v ~- 400 M e V - 2 GeV. When the por t ion of the data between 1 and 2 GeV is examined a neutrino excess ( o f v e + r e + vn + ~ ) is limited [26] b y R ' = (d~/dEv) l o dEv/(dcb/dEv) lam a d E v < 1.3, where (dcb/dEv)lat m is the atmospheric background in the direction o f the sun as in the figures but with v , + ~u included, R ' calculated from the figures must be cor- rected (to include v , + ~u) by a factor 2/(1 + vu/Ve) ~ 0.46 for E v ~- 1 - 2 GeV. Taking m x = 6 GeV, in figs. l a , b, c we find R ' = 0 .9 ,0 .5 and 0.3 respectively, all within a factor of ~-4 of the present experimental upper limit. Clearly this limit can be improved if more data were available which included higher energies (E v > 2 GeV) and/ or could distinguish between v e and vu.

In conclusion, we have calculated the differential flux o f high energy neutrinos due to the annihilation o f supersymmetric relics t rapped in the sun. The fluxes can easily exceed the atmospheric neutrino background flux for E v >~ O(1) GeV. Present experimental data comes close to probing the supersymmetric dark matter candidates but is not ye t in a posit ion to rule them out.

We would like to thank D. Burke, S. Gasiorowicz, J. LoSecco, M. Srednicki and S. Rudaz for valuable discus- sions. The work o f K.-W.N. and K.A.O. was supported in part by DOE grant DE-AC02-83ER-40105.

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L.M. Krauss, Nucl. Phys. B 227 (1983) 556. [15] J. Ellis, J. Hagelin, D.V. Nanopoulos, K.A. Olive and M. Srednicki, Nucl. Phys. B 238 (1984) 453. [ 16 ] P. Fayet, in: Unification of the fundamental particle interactions, eds. S. Ferrara, J. Ellis and P. van Nieuwenhuizen

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