Cosmic ray constraints on the annihilations of relic particles in the galactic halo

Volume 214, number 3

PHYSICS LETTERS B

24 November 1988

COSMIC RAY CONSTRAINTS ON THE ANNIHILATIONS OF RELIC PARTICLES IN THE GALACTIC HALO J o h n ELLIS, R.A. F L O R E S CERN, CH-1211 Geneva 23, Switzerland K. F R E E S E Department of Physics, MIT, Cambridge, MA 02138, USA S. R I T Z 2 UniversiO~ of Wisconsin, Madison, W15 3 706, USA D. S E C K E L 3 University of California, Santa Cruz, CA 95064, USA and Joseph SILK University of California, Berkeley, CA 94720, USA Received 23 August 1988

We re-evaluate the fluxes of cosmic ray antiprotons, positrons and gamma rays to be expected from the annihilations of relic particles in the galactic halo. We stress the importance of observational constraints on the possible halo density of relic particles, and specify their annihilation cross sections by the requirement that their cosmological density close the Universe. We use a Monte Carlo programme adapted to fit e+e - data to calculate the 15, e + and y spectra for some supersymmetric relic candidates. We find significantly smaller ~ fluxes than previously estimated, and conclude that present upper limits on cosmic ray ~ and e + do not exclude any range ofsparticle masses. We discuss the prospects for possible future constraints on sparticles from cosmic y rays.

A s t r o p h y s i c a l o b s e r v a t i o n s ~1 strongly suggest t h a t galactic haloes c o n t a i n d a r k m a t t e r w h i c h has n o t d i s s i p a t e d like t h e c o n v e n t i o n a l m a t t e r in the galactic disc, a n d is likely to be n o n - b a r y o n i c (see, e.g. ref. [ 2 ] ). T h e r e is n o s h o r t a g e o f p a n i c l e physics c a n d i dates for this d a r k m a t t e r , w i t h t h r e e o f t h e m o s t fat Address after September 1st, 1988: School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA. 2 Address after September 1st, 1988: Department of Physics, Columbia University, New York, NY 10027, USA. 3 Address after September 1st, 1988: Bartol Research Foundation, University of Delaware, Newark, DE 19716, USA. ~' Fora recent review, see e.g. ref. [ 1 ].

v o u r e d b e i n g m a s s i v e n e u t r i n o s , a x i o n s a n d relic s u p e r s y m m e t r i c particles [ 3,4 ]. E x p e r i m e n t a l physicists are n o w d e v i s i n g strategies for d e t e c t i n g these d i f f e r e n t species o f relic p a n i c l e s . T h e y h a v e b e e n off e r e d t h r e e possible signatures o f h e a v y d a r k m a t t e r p a n i c l e s ~ such as m a s s i v e n e u t r i n o s or s u p e r s y m m e t r i c p a n i c l e s . T h e s e are Z scattering o f f nuclei in the laboratory which may produce detectable nuclear recoil energy a n d / o r inelastic n u c l e a r e x c i t a t i o n [ 5 ], a n n i h i l a t i o n s o f relic p a n i c l e s ~ t r a p p e d in the Sun w h i c h m a y p r o d u c e an o b s e r v a b l e flux o f h i g h energy solar n e u t r i n o s [ 6 ], a n d a n n i h i l a t i o n s o f relics ~ in the h a l o w h i c h m a y p r o d u c e d e t e c t a b l e fluxes o f ant i p r o t o n s [7 ], p o s i t r o n s a n d g a m m a rays [7,8 ]. N e w 403

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data have recently become available from non-dedicated detectors which may constrain some relic particle candidates. These include upper limits on dark matter scattering rates in 76Ge detectors [ 9 ] - which exclude Dirac neutrinos vD or sneutrinos 9 with masses between 12 and 103 GeV if they have the expected halo density, upper limits on high-energy solar neutrinos [10] - which exclude 9e or 9, with masses between ~ 2.5 GeV and 25 GeV but do not exclude any range of photino ~ or higgsino h masses [ 11 ], and new upper limits on the low-energy cosmic ray ~ flux [ 12 ] ~2 whose implications we analyze in this paper. The observation of a surprisingly high flux of lowenergy O's in the cosmic rays a few years ago [14] was very difficult to understand in terms of secondary 0 production by matter cosmic ray primaries [ 15 ], and its origin was the subject of many speculations. A particularly interesting suggestion [ 7 ] was the idea that these O's could be due to relic particle annihilations in the galactic halo: Zg--'0 + X. By making favourable choices of uncertain astrophysical parameters such as the relic halo density ph and the confinement time zp of ~'s in the galactic magnetic field, it appeared possible [8] to reproduce the claimed flux of O's with the annihilations o f Dirac neutrinos vD or higgsinos h, but this was more difficult with photinos ~, and impossible with sneutrinos 9. Recently, new measurements [ 12 ] ~2 have failed to reproduce the low-energy 0 flux claimed previously, and have instead established upper limits on the flux which are more than an order of magnitude below the previous claim. We study the question whether these new observations impose significant constraints on relic particle models, and also examine whether upper limits on the fluxes of positrons or gamma rays could constrain such models. For each relic particle ~ that we consider, we fix its total annihilation cross section axx by requiring that its present cosmological density give closure [16]: t2x=px/p~= 1. These values o f axz are significantly smaller than some used in the literature [ 8 ]. We calculate the differential 0, e + and'/fluxes using a Monte Carlo programme with parameters adjusted to fit e÷e - annihilation data [ 17 ] ~3. We find 0 spectra that ~2 See also ref. [ 13 ]. "~ For an up-to-date review and comparison with experimental data, seee ref. [ 18 ]. 404

24 November 1988

are smaller than those used previously, especially in the case o f the bb final states that are important in hh (and to a lesser extent ~?) annihilations. Moreover, the spectra have different shapes and do not scale as functions o f Ep/m x. We combine these particle physics calculations with the observational upper bounds [ 12,13 ] on the 0 flux to give upper limits on the combination p h x ~ ~ of astrophysical quantities - p~ being the local halo dark matter density and zo the time of 0 confinement in the galactic magnetic field. For all values o f the relic ~ or h masses, we find upper limits on p~ x ~ which are much larger than the values expected from astrophysics [17]. Thus no range o f relic ~ or K masses is excluded by present 0 data, and the same conclusion holds for present e ÷ data [20]. We also discuss the prospects for relic detection by possible future 0 experiments, and possible future constraints on sparticles from cosmic 7 rays, presenting both the expected shapes of ), ray spectra and best guesses for their normalizations. We central formula for the flux observable at the Earth of O's produced by relic XZ annihilation in the halo is [7,8] @0(Ep) = ~

AVpfp(Eo)

ZO,

(1)

where < azzvz> A is the velocity-averaged )~)~annihilation cross section, Vp is the interstellar O velocity, f~(E0) is the differential cross section for inclusive 0 production dN

1 da(x~-~0+X)

A(E~) = dE---~ ~

dE~

(2)

suitably modulated by the solar wind, and zp is the 0 confinement time. We will now discuss each of these factors in turn, and then the analogous formulae for the e + and 7 fluxes. The annihilation cross section for Z's in the halo is controlled by model parameters: mostly by sparticle masses i f ~ = ~ , and by the ratio v,/v2 of Higgs VEVs giving masses to the up- and down-type quarks if Z = h. The values of these parameters often used in the literature [8 ] give large annihilation cross sections and thus relatively large 0 fluxes @p. However, these same model parameters also control the annihilation rate o f Z's in the early Universe, and hence the cosmological relic density Px of X's, which we ex-

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PHYSICS LETTERS B

press in units I2x=pz/pc of the critical density pc. The annihilation cross section in the early Universe, averaged at a temperature T<< rex, is given by [ 3 ]

(axzVx) TA= a + b ( T / m z ) + O ( T / m z )

2,

(3a)

where a and b are determined by the model parameters, whereas the halo annihilation cross section

(~rz~Vx)A = a [ 1 + O ( v ~ ) ] ,

(3b)

where the relevant velocities vz ~ 10- 3c are very small. For 5 GeV < mz< 30 GeV, the cosmological relic density is related to the annihilation cross sections (3) by [3] 1 0 - 2 6 cm 3 s - i g2zh2= 20xf(a+ ½bxf)

(10-26 cm3 s - ' ~

~- \

-~xx--~x)-A I {20xf[l+½(b/a)xf]}-',

(4) where h is the present-day Hubble constant in units of 50 kms- ~ Mpc- ~, and xr is the ratio of the freezeout temperature 7",-in the early Universe to mx. Eq. (4) gives a close connection between I2~ and (crxzvz) a, because xr~ 1/20 and b/a are fairly modelindependent. For example, if~=)~ we find

½(b/a)xr= ( ~ 4~ ) (ms,/GeV) 2.

(5)

We shall assume that the relic X's are the dominant form of dark matter in the Universe, and shall fix the numerical values of the model parameters by requiring I2z = 1, as suggested by naturalness and by inflation [16]. We note that model parameters used in the past [ 8 ] give relatively low values of the cosmological relic density, t'2z~ 0.2 to 0.05, so eq. (4) tells us that our ( a = v x )A and hence our fluxes will be reduced by a factor ~ 5 to 20. The O's, e+'s and "f's are produced indirectly via the annihilation reactions ~z-~ff: f=z, c or b for the candidates we consider. When f = z, e +'s and 7's may be produced either directly or indirectly in the x decay chain, but no O's are produced. When f = c or b, O's may be produced either directly in the fragmentation process, or indirectly in the decay chains of heavier particles, just as e+'s and "/'s are produced in the decays of fragmentation particles. It is therefore impossible to calculate these spectra analytically. Since the Z~-+t'f reaction is similar to e+e--,elq, a

24 November 1988

natural possibility is to use ere - data [8 ] to determine the spectrum f0(Ep) of eq. (1) [and analogouslyfe+ y(Ee+ v) ]. However, the e % - data require three important corrections to meet our purpose. (1) The ;~Z~t'f annihilation fractions are in general quite different from the mix of (tq final states in e r e - annihilation. This can be an important effect, because the c ~ D and b ~ B meson fragmentation functions are known to be substantially harder than those of the lighter quarks, leaving on the average less energy available for the production of other fragmentation particles in ec and fib final states. Therefore, one would expect, for example, the O spectra from the )0~ annihilations considered here to be somewhat softer, and the overall yield of O's to be less, than the e+e - data. (2) The second difference from e r e - annihilation is that particles which are usually considered to be stable in an accelerator experiment (e.g., n -+, K -+, la-+, neutrons and antineutrons) have time to decay in flight, making additional contributions to the inclusive spectra (dN / dx); x - E i/m z. We obtain the ( d N / d x ) i using the Lund Monte Carlo programme [ 16 ], which reproduces the available e+e - data fairly accurately, making the modifications needed to include the decays of all unstable particles. We have used the Peterson et al. [ 21 ] form of fragmentation function of the c and b quarks, with ec and/~b adjusted to give =0.65 and =0.81 as inferred from ere - data. The inclusive distributions (dN/dx)~ are determined for the ~, c and b final states separately, and fit to the functional form

(~)

2 or3

dN

=fliCi ~ i

A,Sexp(-fl,Sx).

(6)

j=l

The values of the fit parameters A~, B~ and Ci are given in table 1. The shapes of the ( d N / d x ) i distributions are relatively insensitive to the centre-of-mass energy. For the y's and e+'s, there is some increase at low x ( x < 0 . 0 5 ) with increasing centre-of-mass energy. However, we will show that the most favourable signal-to-noise region for the e r ' s and )"s is most likely to be in the range x > 0.1 where this effect is considerably smaller. By choosing a 20 GeV centre-of-mass energy (corresponding to the central m x value of 10 GeV) to fit the parameters in table 1, we limit the errors in the spectrum due to these scaling violations 405

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24 November 1988

Table 1 Fit parameters describing the spectra obtained from the Monte Carlo, with mx=10 GeV using the functional form dN/dx= C#~e4, exp ( - B/c).

0 from b e + from c e + from b e + from c e + from z 7 from b 7 from c 7 from z

C

A,

( 1 - 2 5 / m z2) ,/2 × (10'2/mz+0.151) 9.67/mx+0.033 1.0 1.0 1.0 1.0 1.0 1.0

181

A2

B,

378 678 537 3.49 989 607 15.8

to less t h a n ~ 30% o v e r t h e i n t e r e s t i n g r a n g e o f m z. In light o f t h e o t h e r u n c e r t a i n t i e s , a m o r e e x a c t t r e a t m e n t o f t h e s p e c t r a is h a r d l y j u s t i f i e d . F o r t h e ~ ' s t h e

B2

39.1

11.2

13.2

-

-

42.4 147 160 5.42 63.6 51.3 7.61

16.1 200 187 14.8 94.3 80.9 5.45

11.5 51.8 53.4 22.3 18.5 16.3 21.5

11.6 23.1

13.1 18.1 -

-

a m o u n t r e l a t i v e to t h e m x = 10 G e V fits. Fig. 1 s h o w s a M o n t e C a r l o h i s t o g r a m o f t h e ( d N / dx)i weighted by their fractions in ~ annihilation for m~ = 10 G e V , s u p e r i m p o s e d b y a s i m i l a r l y w e i g h t e d c o m b i n a t i o n o f t h e fits ( 6 ) to t h e ( d N / d x ) i . A l s o s h o w n is a f u n c t i o n u s e d i n p r e v i o u s e s t i m a t e s [ 8 ] o f t h e p flux. W e n o t e t h a t it h a s a s l o w e r f a l l - o f f w i t h x

n i f i c a n t d e p e n d e n c e o n t h e c e n t r e - o f - m a s s energy. T h i s is t a k e n i n t o a c c o u n t b y a n e n e r g y - d e p e n d e n t

i

/

B3

p r e - f a c t o r w h i c h scales t h e C b y t h e a p p r o p r i a t e

s h a p e s o f t h e ( d N / d x ) ~ d i s t r i b u t i o n s are well d e s c r i b e d b y eq. ( 6 ) f o r t h e e n e r g y r a n g e s c o n s i d e r e d h e r e , b u t t h e o v e r a l l n o r m a l i z a t i o n s , C~, h a v e a sig-

5.00

A3

i

\

1.00

0.50

dN dx

~.,,.

[R.f. s3

0.10

0.05

0.01

,

,

,

,

.

n,

0.1

0.5 x -= E~

/m X

: mX

=

n

,

.

1.0

i0 G e V

Fig. 1. Spectra of lb'S produced in relic XXannihilations. The histogram is the Monte Carlo [ 17,18 ] output for a ? weighing 10 GeV, and the solid curve is the result of the corresponding fit (6) for m~ = 10 GeV. The dash-dotted curve is the result of the fit (6) for mn = 10 GeV. Note that it is softer than the ? curve because hh ~ b b has a larger branching fraction than ? ' ~ b b . The dotted curve is that of Stecker and Rudaz [ 8 ] for m~ = 10 GeV: their curve for m~ = 10 GeV would be a factor 1.3 higher. 406

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than our fit ~¢4,and a larger overall normalization. Indeed it gives 1.14 p, 0 per e+e - annihilation event at EcM = 34 GeV, whereas TASSO [22 ] observes 0.76+_0.06 P,0 and our Monte Carlo gives 0.72 P,P per e+e - annihilation event. Also shown in fig. 1 is dN/dx for hh annihilation for m~ = 10 GeV. (3) The produced particle spectra must be corrected for propagation through the interstellar medium and for modulation by the solar wind. The former are expected to be small for O's in the energy range of interest here, but the raw spectra of fig. 1 and table 1 must be corrected for modulation by the solar wind. We make this correction using the simplified approach of Perko [23], which has been shown to give an accurate form for the cosmic ray proton spectrum. This approach approximates the diffusion coefficient by Aflp for P>Pc, and by Aflpc for P
E~=E+AE,

for p>p¢,

p+E

E~=p~.ln~

+Ec+AE,

(7a) forp
(7b)

where AE is an energy shift that depends on the phase in the solar cycle, and E = m o + T = (rn 2 + p 2 ) ~/2. We follow ref. [ 12 ] ,2 by choosing z~d~= 495 MeV to correspond to the phase at which their data were taken. In order to compare these data with the predictions o f different models, the latter would have to be normalized by choosing values for the astrophysical quantities ph and r o in eq. ( 1 ). The local density of halo dark matter is constrained by a variety of astronomical observations. If galaxies form within pre-existing haloes of dark matter, as is widely believed, then dynamical estimates imply [ 19 ] 0.2 GeV c m - 3
(8)

if the fraction F of dissipational matter lies in the range 0 . 0 5 < F < 0 . 2 implied by observations. The range (8) includes most mass-model estimates of p~. For the best-guess value F ~ 0 . 1 , the local halo density p~ "-,0.3 GeV cm -3 [ 19]. It has been noted that p~ would be significantly higher if the dark mat,4 For this reason, there is no sharp cut-off in the lb/p ratio at Ep= rn~ as previously suggested [ 8 ], and higher energy p data give less stringent limits on sparticles.

24 November 1988

ter had the same velocity dispersion as extreme population II stars [24]. However, this does not seem likely if galaxies were formed by dissipational contraction within dark matter haloes. Our best-guess value p ~ ~ 0.3 GeV c m - 3 for the local halo density is considerably smaller than values (0.75 to 1 GeV c m - 3) used previously [ 8 ] in this contect. Thus our fluxes will be reduced by factors ~ 6 to 10. Strictly speaking, the value of p× that appears in eq. ( 1 ) is the mean value ofpx within the dark halo weighted by the relative probabilities that O's produced at different locations in the halo will diffuse through the galactic magnetic field to reach the Earth. We shall take px = p~ in eq. ( 1 ), and put the effects o f the inhomogeneity o f Px into To, thereby following previous work [7,8]. The propagation of charged particles in the galaxy is not well understood (see e.g. ref. [25 ] ), and their c o n f n e m e n t time is uncertain and may well be energy-dependent [ 26 ]. We will discuss two essentially different models for confinement: disk and halo confinement. For disk confinement, r~--- 107 yr, as suggested by the abundances o f both stable [26] and unstable (e.g., '°Be) isotopes [27 ]. We contemplate two types of halo confinement. For type I halo confinement, the local density of cosmic rays is dominated by an old halo population, in which case the age of l°Be is given by the geometric mean of the '°Be half-life and the halo confinement time, z h [28]. This suggests zh-~ 5 × 10 7 yr and a halo thickness ~ 1-2 kpc, roughly five times the disk scale height. For type II halo confinement, the local density of cosmic rays is dominated by a young disk population, as in a disk model. However, a longer z h is not ruled out so long as the older population does not contaminate the isotope abundances successfully given by the disk model. We approximate the average residence time for an old cosmic ray in the disk b the probability of finding it there, rres= ThvO/V h. Requiring zres< 107 yr to avoid contamination, we conc l u d e r h < 3 × 10 s yr, where we estimate v d / v h ~ 1/ 30 as the ratio of the disk scale height to the core size of the dark matter distribution. Another model is motivated by evidence [ 15,29 ] that a substantial fraction ( 1 0 - 5 0 % ) o f the cosmic rays must have passed through much higher grammage ( ~ 100 g c m - 2 ) , if the observations [30] of 0's 407

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at several G e V ~5 are interpreted as due to secondary production by nuclear intractions o f cosmic ray primaries with interstellar gas. This enhancement in path length m a y be ascribed to halo confinement in a closed box model for the galactic cosmic rays that leak out o f the spiral arms where they are p r o d u c e d and spend ~ 107 yr. The resulting old c o m p o n e n t has a lifetime ~ 7 × 109 (0.01 c m - 3/nH ) yr, where nH is the neutral hydrogen density in a t o m s cm -3, which is close to the m a x i m u m confinement time ~ 1.5 × 101° yr given by the age o f the galaxy ~6. We shall not restrict our attention to a particular confinement model, but instead consider a range for z that embraces the three m o d e l s considered above. These vary through 0.02 ~< ~p ~<200 for ~ p - Zp/2 × 10 ~5 s. P r o b a b l y 0.15 < fo < 5, but the confinement time zo remains as the most uncertain o f the ingredients in eq. ( 1 ) .

We present in fig. 2 numerical estimates o f the I~/P ratio derived ~7 from the ~ flux in a sampling o f sup e r s y m m e t r i c models, based on the default options p h = 0 . 3 G e V cm -3 and z p = 2 × 1 0 ~5 s. We see that because o f the large reduction factors m e n t i o n e d earlier, due mainly to the assumed values o f (axzV~)A a n d p~, the i m p r o v e d u p p e r limit f 6 4 0 MeV

205M~V dEp~p(Ep) < 4 . 6 × 10 -5 ( 8 5 % C L ) , 2o5 M~v dEpCp ( E o)

f 6 4 0 MeV

recently reported [ 12,13 ] does not i m p o s e any significant constraint on s u p e r s y m m e t r i c particle models. Indeed, they might even give fluxes considerably below the curves in fig. 2, since some o f the halo d a r k m a t t e r could be baryonic, and Zp could be shorter than 2 × 10 t 5 s, as discussed above. F o r this reason, we prefer to use the d a t a o f refs. [ 12,13 ] to ~h quote u p p e r limits on the quantity pxv/~p, where fih = p h / 0 . 3 G e V c m - 3 . Fig. 3 shows u p p e r limits on / ~ h x ~0 for ~(=~ a n d h, as functions o f m z. These resuits were derived assuming an energy-independent

~5 It is desirable to repeat these observations with better particle identification. ~6 A possible test of this model could be the observation of a halo component in the 7 data, with an energy spectrum consistent with that for the decay of~°'s from nuclear interactions. 10-4 5×I0 -5

~7 We use the p flux from ref. [23 ] with AE=495 MeV [ 12,13].

l .... i

24 November 1988

,

, ....

i

I

10-5 5xi0-6

~ ......... 6 o~v .... ..

- flux ratio P 10-6

. . . . . . . . . . . \" '~ -" - " ~

5xi0-7 16 GeV

10-7 0.1

~..\. \~

\

. . . . ..._~ . . . . . . . . . . . . . . . . . . "~. . . . . . . . .

I

I

0.5

1.0

" .

m~

'~ ~.....

\

'\

I"N"

5.0

T

15

GeV

\

,l I0.0

(GeV)

Fig. 2. Ratios ofl~ and p fluxes from different relic candidates, after modulation by the solar wind. Also shown is the recent upper limit on the p flux from refs. [ 12,13]. Previous measurements 128] at higher energies arc off-scale beyond the top of the figure. The curves are calculated with the default options pxh =0.3 G e V c m -3 [ 19 ] and r~ = 2 × I0' 5 s. The solid/dotted/dashed/dash-dotted curves arc for

m~= 3 OeV/m~ = 6 GeV/m~ = 10 GeV/m~ = 15 OeV. 408

Volume 214, number 3

PHYSICS LETTERS B I00.0

I

.

.

.

24 November 1988 I

.

50.0

....../:-<> h : ~+

...... , ~ :

:.<::55S / ~

@

/

IO.O

~¢f

s. o

1.0

0.5

,

I 5.0

,

I

,

,

I0.0

m (GeV) X Fig. 3. Upper limits on l~h~p from thc uppcr limit of rcfs. [ 12,13] on the l~ flux, for Z = ~ and h as functions of mx. Also shown are "upper limits" on/~ ~ . + from mcasurcmcnts of the e + cosmic ray flux above 2 G e V [20 ]. Lower-energy e + data are not competitive for constraining rclicsin thc mass ranges considcrcd here.

mean free path-length which is likely to be the case for Ep ~<5 GeV [ 25 ]. Although these limits are not as stringent as the upper limits on ph from the non-observation of high-energy solar neutrinos [ 31 ], they are much stronger than the available upper limits on the local halo density of axions [ 32 ]. It should be noted that even a positive signal for low-energy f~'s would not provide a "smoking gun" for relic annihilations. This is because of a possible background from supernovae buried deep in dense interstellar clouds [33]. They could generate highenergy primary cosmic rays which produce ~'s that are initially above the kinematic threshold of several GeV but subsequently decelerated adiabatically. One could produce in this manner a low-energy 1~flux close to the new upper limit even if only (3 or 4%) of cosmic ray nucleons were produced in this manner. This translates into requiring ~ 10% of all supernova explosions to occur in such dense clouds, which is well within the experimental uncertainties [ 34 ]. Also shown in fig. 3 are upper limits on/~v/~¢+ from a comparison of the observed flux of high-energy cosmic ray e +'s with the flux expected from ~X

annihilation. The analysis ~8 of the e + spectrum is similar to that of the ~'s discussed above, but the following two points merit comment. (a) There is a significant contribution to the e + spectrum from particles (e.g., n +, IX+, fi) which appear stable in e+e experiments, but must be allowed to decay in our case ~9. (b) Energy loss during propagation through the interstellar medium is significant for e+'s, unlike the ~ case. The loss rate can be written as [ 36 ] dE/dt=

-

[2.7 nH q - 7 . 3 nHEd- ( Urad 71- U m a g ) E 2 ]

X 10 -16 GeV s -1,

(9)

where Urad is the ambient photon energy density in eV cm -3, and E is the e + energy in GeV. The four terms in (9) represent the energy losses due to ionization, bremsstrahlung, inverse Compton scattering ,s Similar Monte Carlo calculations of the e ÷ spectra have been made by Tylka and Eichler [33], so our discussion here will be brief. ,9 For ease of computation, we have neglected polarization effects on these decay spectra, which could change them by a factor < 2.

409

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and synchrotron radiation respectively. None of the terms is negligible for e+'s o f energy E ~ 1 GeV in the galactic disc ( n , ~ 0.3 atoms c m - 3, Gad ~ 2 eV c m - 3, U m a g ~ 0 . 5 e V c m - 3 ) . However, the e+'s spend most of their time, and lose most o f their energy, in the halo outside the disc, where nil~0.01 atoms cm -3, Ur,d ~ 0.25 eV cm - 3 and U m ag ~ 0.05 eV c m - 3, so that inverse C o m p t o n scattering is dominant for E > 1 GeV. The e ÷ spectra produced in)~x annihilation have been corrected for these energy losses assuming halo confinement ~,o, and for the modulation by the solar wind [ 23 ], and then compared with observation [ 20 ] to produce the e + constraints on/~Zhx~e+ shown in fig. 3. We only use the data for E~+ > 2 GeV, since the model curves at lower energies rise much more slowly than the observed e + flux, so that the lower energy data would give a less stringent bound on/~zhX/~e+ ~'' Our " b o u n d s " were obtained by demanding that the models give a smaller flux than a simple power-law fit to the data above 2 GeV. We see that they are somewhat less stringent than the 0 limits. Moreover, for the reasons discussed above, more care should be exercised in interperting the e + limits. For one thing, as mentioned above the diffusion of relativistic cosmic ray nucleons may be energy-dependent [26], suggesting that relativistic electrons could have an energy-dependent path length ocE-°-5-+° ,. Another point is that there must be a secondary e ÷ yield from the nuclear interactions of primary matter cosmic rays [37]. Therefore an anomalous e + / ( e - + e ÷) ratio also could not be interpreted as a "smoking gun" for relic annihilations. Finally, we discuss the y spectra from Z)~ annihilation. As in the cases of the ~'s and e+'s, we include in our y spectra f(Ev) the decays o f all unstable particles. Clearly there are no energy losses on the way to the Earth. The confinement time r in eq. ( 1 ) is replaced by an integral over the line-of-sight which depends on the galactic latitude b and longitude I [38 ]. The y-ray flux may be approximated by

~'CThe effect of neglecting energy losses in the disk is to overestimate somewhat the positron flux. nh~This would not be the case for smaller values o f m x, but closure cannot be reached for rn~ < 5 GeV, and values of the slepton and squark masses which conflict with experimental limits would be required to obtain closure for m, < 3 GeV [ 16 ].

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G(E) =

,

~

<~zzvz>f~(E~)

1.6×10-6 cm2 s sr GeV

{

(

aI(b, l)

ph

)2

Z

0.3 GeV cm -3

A , { mx ~-2[, a "U"f(Er)

t,

s-,) t v) t

(lO)

×I(b,l) , where we assume a halo density of the form

(a2+r2"~ px=p~\a2 +r: ] ,

(11)

so that /~ ( 1 "{'-OL2 X~2F7'g

I(b, l) = ~ k - - ~ - Y L2 + ~

+tan-'(~ cos b cos l ) ]

Odj~

cos b cos l

,

(12)

where a = ro/a and f12= 1 + a 2 - o~2 cosZb cos2/. The factor I(b, l) (12) varies between 1.24 ( b = 90°: the galactic pole), 9.01 (b = 0 °, l = 0 °: the galactic centre) and 0.61 ( b = 0 °, l = 180°: the anticentre). We show in fig. 4 the y fluxes for b = 90 ° and ~ = ~ and h. These predicted diffuse y-ray fluxes are to be compared with that observed [39] for Ev> (35 to 200) MeV, which consists o f two components: an isotropic component fitted by ~v(E) = 2 . 8 X 10-8E -34 cm -2 s -1 sr - l GeV -1, and a disc component fitted by ~v(E) = 2 . 8 × 10-6E -1"6 × ( s i n l b l ) - 1 c m - 2 s - ' sr -z G e V - L The galactic disc component dominates the observed background for E > 80(70) MeV at I bl = 9 0 ° (60 ° ) and is plotted in fig. 4 for I b l = 90 °. Since the slopes of the theoretical spectra become equal to that of the background spectrum at energies E > 0 . 1 GeV, we compare theory with experiment in this range o f energies. Demanding that the predicted flux spectra be bounded above by this disc component implies / ~ x / a / 7 k p c ~ < 5 . 3 ( 1 4 , 60) for m , = 3 ( 6 , 16) GeV, which is less stringent than the corresponding p limit for fp = 1. This is a conservative limit: e.g., taking ac-

Volume 214, number 3

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24 November 1988

10-5 5×10 -6

disc

i0-6 5xlO -7

"'"................m~ = 6 GeV 10 -7 y

flux

"""'4,,.,,,.,,,,,,,

5xlO -8

(cm-2s-lsr-IGev-I)

"'--L

10-8 SxlO -9

.............. .........

"-

10-9

i

. \

5xlO -IO

0.

0.5

1.0 E

5.0

10.0

(GeV) Y

Fig. 4. Fluxes ofy's produced in relic X,Zannihilations calculated with the default options p~ =0.3 GeV cm -3 and a = 7 kpc. The solid/ dotted/dashed/dash-dotted curves are for m, = 3 G e V / m , = 6 GeV/m~ = 16 GeV/m~ = 15 GeV. Also shown is the disc component of the cosmic y-ray background [39].

count of the angular dependence could improve our constraint on #x/a, but this would require knowledge of the halo density close to the core. Future experiments such as the G a m m a Ray Observatory ( G R O ) will enable one to improve further this limit. In addition to the broad band spectrum, it has been suggested [40] that there might be significant DM annihilation into a vector meson V and a single photon, thus producing a narrow-band photon spectrum that might observable. The branching ratio for this process has been estimated [41] to be l'vv/ Fqq~2X10 -4 (3 G e V / m z ) 2 for V = J / ~ , and 1.5X 10 -5 ( 10 GeV/mx) 2 for V=Y, where/'q~ is the total branching ratio for % annihilation into the corresponding qCl pair. Unfortunately, we estimate that the signal/background for this process, relative to that for the broad band, n°-produced y's, is (S/ B)v/(S/B),~,,~36 ( G e V / m z ) 2, thus only for relatively light % does the narrow-band signal look at all promising. The new low-energy cosmic ray ~ limit [12,13] does not set, by itself, any significant constraint on relic halo particles. Current models could give a p sig-

nal just below the new limit, but astrophysical uncertainties, especially in containment time, render our predictions uncertain by some four orders of magnitude. Storage times in the halo could even be as long as ~ 101° yr, in which case relic signatures should be present, though buried in the various backgrounds we have discussed. We do not believe it will ever be possible to derive significant limits on supersymmetric relic particles from cosmic ray f) data lone. A possible strategy for future experiments is to combine searches with those for e+'s and "/s. By ensuring that comparable sensitivities to a given relic candidate could be attained for both p and e ÷, one might circumvent some of the background problems. The merit of the "/search is to remove much of the uncertainty in containment time, but disentangling a signal will require persistence.

References [ 1 ] V. Trimble, Ann. Rev. Astron. Astrophys. 25 (1987) 425. [2] D. Hegyi and K. Olive, Astrophys. J. 303 (1986) 56.

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[3] H. Goldberg, Phys. Rev. Lett. 50 (1983) 1419; J. Ellis, J.S. Hagelin, D.V. Nanopoulos, K. Olive and M. Srednicki, Nucl. Phys. B 238 (1984) 493. [4] L. Ibafiez, Phys. Lett. B 137 (1984) 160; J.S. Hagelin, G.L. Kane and S. Raby, Nucl. Phys. B 241 (1984) 638. [5 ] M. Goodman and E. Witten, Phys. Rev. D 31 ( 1985 ) 3059. [6 ] J. Silk and M. Srednicki, Phys. Rev. Lett. 55 ( 1985 ) 257; M. Srednicki, K. Olive and J. Silk, Nucl. Phys. B 279 (1987) 804. [ 7] J. Silk and M. Srednicki, Phys. Rev. Lett. 53 (1984) 624. [8] F.W. Stecker, S. Rudaz and T. Walsh, Phys. Rev. Lett. 55 (1985) 2622; S. Rudaz and F.W. Stecker, Astrophys. J. 325 (1988) 16. [9] S.P. Ahlen et al., Phys. Lett. B 195 (1987) 603; D.O. Galdwell et al., Santa Barbara preprint UCSB-HEP88-6 (1988). [10]Fr6jus Collab., B. Kuznik, Orsay preprint LAL 87-21 (1987); IBM Collab., J.M. Losecco et al., Phys. Lett. B 188 (1987) 388. [ I I ] J . S . Hagelin, K.-W. Ng and K. Olive, Phys. Lett. B 180 (1987) 375; K.-W. Ng, K. Olive and M. Srednicki, Phys. Lett. B 188 (1987) 138. [ 12] S.P. Ahlen et al., Phys. Rev. Lett. 61 (1988) 145. [ 13 ] T. Bowen et al., University of Arizona preprint ( 1988 ). [ 14] A. Buffington, S.M. Schindler and C.R. Pennypacker, Astrophys. J. 248 (1981) 1179. [ 15] R.J. Protheroe, Astrophys. J. 251 ( 1981 ) 387. [ 16] R. Brandenberger, Rev. Mod. Phys. 57 (1985) 1. [ 17] T. Sj~strand, University of Lund preprint LUTP 85-10 (1985). [18]T. Sj/Sstrand, Intern. J. Mod. Phys. A 3 (1988) 751, and references therein. [ 19] R. Flores, CERN preprint TH.4736/87 (1987). [20] A. Buffington, C.D. Orth and G.F. Smoot, Astrophys. J. 199 (1975) 669; R.L. Golden et al., submitted to Astron. Astrophys. ( 1986 ).

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[ 21 ] C. Peterson, D. Schlatter, I. Schmitt and P.M. Zerwas, Phys. Rev. D27 (1983) 105. [ 22 ] TASSO Collab., paper submitted to the 24th Intern. Conf. on High energy physics (Munich, 1988). [23] J.S. Perko, Astron. Astrophys. 184 (1987) 119. [24] J. Binney, A. May and J.P. Ostriker, Mon. Not. R. Astron. Soc. 226 (1987) 149. [ 25 ] V.L. Ginzburg and V.S. Ptuskin, Rev. Mod. Phys. 48 (1976) 161; C.J. Cesarsky, Ann. Rev. Astron. Astrophys. 18 (1980) 289. [26] J.F. Ormes and R.J. Protheroe, Astrophys. J. 272 (1983) 756. [27] M.E. Weidenbeck and D.E. Greiner, Astrophys. J. 239 (1980) L139. [28] V.L. Prishchep and V.T. Ptuskin, Astrophys. Space Sci. 32 (1975) 265. [29] S.A. Stephens, Nature 289 ( 1981 ) 267. [30] R.L. Golden et al., Phys. Rev. Lett. 43 (1979) 1196. [ 31 ] J. Ellis, R. Flores and S. Ritz, Phys. Lett. B 198 ( 1987 ) 393. [32] S. de Panfilis et al., Phys. Rev. Lett. 59 (1987) 839. [ 33 ] V.L. Ginzburg and V.S. Ptuskin, Sov. Astron. Lett. 7 ( 1981 ) 325; L.C. Tan and L K . Ng, Astrophys. J. 269 (1983) 751; S.A. Stephens and B.C. Manger, High energy astrophysics, eds. J. Audouze and J. Tran Thanh Van (Editions Fronti~res, Gif-sur-Yvette, 1984) p. 217. [34] J.M. Shull, Astrophys. J. 237 (1980) 269. [ 35 ] A.J. Tylka and D. Eichler, Cosmic ray positrons from photino annihilation in the galactic halo, preprint ( 1988 ). [36] M.S. Longair, in: High energy astrophysics (Cambridge U.P., Cambridge, 1981 ) p. 278. [37] R.J. Protheroe, Astrophys. J. 254 (1982) 391. [ 38 ] J.E. Gunn et al., Astrophys. J. 223 ( 1978 ) 1015; M.S. Turner, Phys. Rev. D 34 (1986) 1921. [39] C. Fichtel et al., Astrophys. J. 217 (1977) L9. [40] M. Srednicki, S. Theisen and J. Silk, Phys. Rev. Lett. 56 (1986) 263. [41 ] S. Rudaz, Phys. Rev. Lett. 56 (1986) 2128.