Radiative processes in dark matter photino annihilation

Volume 225, number 4

PHYSICS LETTERS B

27 July 1989

RADIATIVE PROCESSES IN DARK MATTER PHOTINO ANNIHILATION

Lars BERGSTROM Department of Physws Unt~ersttyof Stockholm, Vanadtsvagen 9, S-113 46 Stockholm, Sneden

Received 25 April 1989

It is shown that radiative corrections to the photlno annihilation process ~?~-~e+e in the galactic halo can be enormous, with the radiative process being several orders of magnitude larger than the lowest order rate It is argued that this may give a source of observable high-energyphotons (and maybe also positrons) in forthcoming space-borne experiments The full box diagrams for the loop-induced process ~?-'77 are calculated confirming a recent conjecture that contributions from light leptons are substantlal Some estimates of expected rates for these processes are given

The nature of the dark matter of the Universe remains an unsolved problem Among the most appeahng possibihtles is the conjecture that it may be made from the llghtest neutral supersymmetric particle (photlno, or more generally, n e u t r a h n o ) that is stable or very long hved and has thus survived since the early Umverse Effects of supersymmetrtc dark matter p a m c l e s and possible experiments to detect them have been widely dtscussed in the hterature (for recent reviews, see refs [ 1-3 ] ) In p a m c u l a r , it has been noted that sharp photon lines or other structures originating from a n n i h i l a t i o n s in our galactic halo could be detectable m forthcoming space-borne experiments [ 4 - 7 ] In most previous analyses, it has been assumed for simplicity that all supersymmetrlc partners of the ordinary quarks and leptons have the same mass In this letter, we investigate some lnterestmg consequences of the ass u m p t i o n that the s u p e r s y m m e m c partners of the leptons (sleptons, ~), are lighter than those of the quarks (squarks, Cl), since this mass pattern seems to emerge from more realistic supersymmetrlc models [8 ] We also present some new results on radiative a n m h l l a U o n processes as well as a full box diagram calculation of 77 production We r e m i n d the reader that the present lower b o u n d on selectron mass, m~, from e+e colliders is a r o u n d 25 GeV [ 9 ], whereas recent Tevatron data [ 10 ] r e & c i t e that the squark masses are greater than 74 GeV We focus on the case when the neutralino, ~?, is d o m i n a n t l y a photlno, although some features of hlggsino and zino propertles in this context will also be touched upon Smce neutralinos are Majorana particles moving with typical galactic (non-relativistic) velocities in the halo, v / c ~ 10 -~, the amplitude for ~'~--,ff (~ neutralino, f f e r m l o n ) being S-wave and spin slnglet (from Fermi statistics) contains a hehctty factor mt For photino annihilation, ~ - - , f f t a k e s place through t-channel exchange of a sfermton ]' (see fig 1 ), and is given to lowest order by

(av~,

[ +fft~ )2 2 4 2 ( 2m2 - - 2 m r2 + m-~ ).Ts,.cr=2~ol Qf Ncflfmr ( m~ _ m 2 + fft2 )2( m~ _ m ~ + fft2 )2 ,

(1)

where fir = x/1 - m 2f / m , ,2 Vre~lS the relative velocity of the a n m h l l a t i n g pair, Qf ~s the electromagnetic charge of the fermlon, Nc is a colour factor ( = 3 for quarks, 1 for leptons), and where the left- and right-handed sfermions 372

0370-2693/89/$ 03 50 © Elsevier Science Publishers B V ( N o r t h - H o l l a n d Physics P u b h s h m g D i v i s i o n )

Volume 225, number 4

PHYSICS LETTERS B

~

J

27 July 1989

f

(al

{hi

(e)

Fig 2 (a)-(c) Dmgrams for the radiative process ~,9~e+e T

Fig 1 Lowest order dmgram for the annlhllatmn ofphotmos ~, into a fermmn-antffermlon palr through the exchange of a spin0 supersymmetrlc partner, ]~,of the fermmns

have masses r~L and/~/R, respectively The eventual non-degeneracy o f the sfermIon masses has very little influence on the results in this paper, so for s l m p h c i t y we set ritE= rhr---- mT In the following F o r the ~ ? ~ ff process, since mr is large, the polntlyke limit (neglecting the m o m e n t u m transfer in the t-channel exchange) is often a p p r o p r i a t e The terms o f order V{el, neglected m eq ( 1 ) (for the full expression, see e g ref [ 6 ] ), are not very i m p o r t a n t for a n n i h i l a t i o n o f the slow ~'s in the halo, but have an i m p o r t a n t effect in the early universe when relic a b u n d a n c e s o f n e u t r a h n o s are d e t e r m i n e d As a first simple model, we start with the interesting scenario when ]' masses follow the fermion mass hierarchy ( m e a n i n g in practice that the selectron is lighter than the other sfermlons by a factor o f l0 or so) The lowest order channel ~ - ~ e + e - is heavily suppressed due to the exphclt factor m r in eq ( 1 ) Taking the m i n i m u m allowed ~ mass 25 GeV, we still find a value o f aVrel o f only a r o u n d 10-32 cm 3 s--l, which is far too small to give a detectable electron or positron signal from annihilations in the halo At first one would think that bremsstrahlung from the p r o d u c e d electrons would be even smaller, suppressed by an extra factor o f c~ Indeed, shrinking the ~ p r o p a g a t o r in figs 2a and 2b to a point, the four-fermion effective interaction becomes axial vector times axial vector and one m a y readily calculate the bremsstrahlung distribution o f the p r o d u c e d photons We find (d~)po,n, - (a(992+e-) _

o~ { [ ( I _ E , ) hEy ~-

2

da(Y?--'e+ dEv e-7))po,n,

+1---

m~

m~d

In

(l+fle~

\l-floJ

-2

(

1-

Ev~fl~ t m~/ j

,

(2)

where Ev is the p h o t o n energy and fl~ = x/1 - m 2/ m~ ( m, - E~ ) This shows the typical bremsstrahlung behavIour ~ 1/E~, and is not very i m p o r t a n t as a source o f high-energy photons G o i n g b e y o n d the p o i n t h k e a p p r o x i m a t i o n , a strikingly different b e h a v i o u r is obtained from the d i a g r a m in fig 2c It has an extra 8 p r o p a g a t o r tending to suppress it, but on the other h a n d there is no hehclty suppression for large p h o t o n energies (in the soft p h o t o n limit, the factor m~ reappears, of course) F o r cosmologically interesting particle masses (m~ between a few GeV and 100 GeV and m~ less than a few h u n d r e d G e V ) this d i a g r a m gives a cross section several orders o f magnitude larger than the lowest-order result in eq ( 1 ) This is a quite r e m a r k a b l e situation with the first order Q E D radiative correction being much larger than the lowest order rate It is, however, explained by these hehcity considerations which means, for instance, that the second o r d e r radiative correction is, as we will see, not very different from the first order result The calculation o f the full d i a g r a m s in figs 2 a - 2 c is rather straightforward The only technical subtlety to r e m e m b e r is that the M a j o r a n a nature of~? means that only the slnglet S initial state contributes This condition is conveniently h a n d l e d by introducing a spin projector for this state (see ref [6 ] ) The differentml distribution is found to be 1 da(?~--,e+e-3,) o'(99-,e+ e - ) dE~ dEe =

2o:m~,(m~+m2--m~) 2 nm2fl~ ~(Ev'Ee'm~'m~'m~)'

(3)

373

Volume 225, number 4

PHYSICS LETTERS B

27 July 1989

where Ee is the positron energy and ~ is a c o m p l i c a t e d rational function o f its arguments The i m p o r t a n t piece surviving for me = 0 is given by ~ ( E v , Ee, m~, me, 0 ) =

( m , - Ev) ( 2 m ~ - 4m~Ee - 2m~E 7 + 2E 2 + 2EeEv + E~ ) (3m 2 -2m, Ee-2m~Ev+m2)2(m 2 -2m~Ee-m~) 2

(4)

In fig 3a is shown the differential p h o t o n energy spectrum o b t a i n e d by integrating eq (3) over positron energy, for rn~ = 20 G e V and with the lowest allowed ~ mass, me = 25 G e V As can be seen, the " r a d i a t i v e correct i o n " IS e n o r m o u s as advertised, a r o u n d 1 6 × l 0 s times the lowest o r d e r result Since the d i a g r a m m fig 2c contains two ~ propagators in the amplitude, the rate for this radiative process decreases quite rapidly with me However, even at m e = 100 GeV, which is much larger than genertc calculations indicate [8 ], the radiative process is still several h u n d r e d times larger than the lowest o r d e r result as is shown in fig 3b In fig 3b can also be seen the small effect o f o r d i n a r y bremsstrahlung at low p h o t o n energies An appealing feature o f this process is the fact, evident from fig 3, that it tends to favour high-energy photons, something that m a y facilitate eventual e x p e r i m e n t a l detection o f this r a d i a t i o n against the cosmic g a m m a ray b a c k g r o u n d Later in this p a p e r we will estimate the signal to b a c k g r o u n d ratio for this channel Perhaps even m o r e r e m a r k a b l e is the positron (or electron) spectrum integrated over p h o t o n energies, also shown in fig 3 The d i s t r i b u t i o n o f positron energies is very flat all the way to the m a x i m u m possible energy, where there is even a rise in the n u m b e r d i s t r i b u t i o n The characteristic experimental signature o f this channel would then be a flat positron energy spectrum with a break at the p h o t i n o mass (although this sharp break would be softened by various r a d i a t i o n processes when the positrons propagate through the galactic m e d i u m ) It is an amusing fact that present ( a d m i t t e d l y scarce) positron d a t a i n d e e d do show some surplus o f positrons in the region between a r o u n d 5 G e V and 30 G e V [ 11 ] Although we have now found a m e c h a n i s m that could give such an enhancement, we prefer to await better data before taking this too seriously ( F o r other attempts o f explaining this surplus as being due to s u p e r s y m m e t r i c dark matter, see refs [ 12,13 ] ) It m u s t be realised that the magnitude o f the r a d i a t i v e effect is caused by the extraordinary suppression o f the lowest o r d e r processes ??--, e + e - If the s u p e r s y m m e t r i c partners o f the m u o n and the tau lepton are as light as the selectron, then the lowest o r d e r cross section will be d o m i n a t e d by x + z - final states, and the e + e - y channel just discussed will d r o p to the level o f a few percent In the z + x - y process, the diagram in fig 2c adds little extra (since the hehclty suppression o f the d i a g r a m s in figs 2a and 2b is not very severe), and eq ( 2 ) gives a good

104

aposltrons

~ " Ib~s~irons

103

,o1

\ photons

102

t

~

"

,,- /

\

/I I

m, m~:20 =25Ge GevV

//

Z"O 10 I

10-1 ,,,,I,,,,I, ,,,I,L,,I,,,,J,,,,I,,,,]J,,,I,

o

z5

5

75

1o ~2s

Energy (GeV)

~s ivs

20

0

2s

s

7s

10

izs

15 17s

2o

Energy (6eV)

Ftg 3 (a) Energy distribution ofposatrons (dashed curve) and photons (sohd curve) produced m the anmhllatlon process ~--,e+e y The photmo mass rn~ has been set to 20 GeV and the selectron mass me =25 GeV (b) Same as m (a), but with the selectron mass me-- 100 GeV

374

Volume 225, number 4

PHYSICS LETTERS B

27 July 1989

a p p r o x i m a t i o n to the radiative process Since this o r d i n a r y bremsstrahlung spectrum is p e a k e d at low p h o t o n energies where cosmic b a c k g r o u n d g a m m a r a d i a t i o n is very large, it does not p r o v i d e a useful experimental signature Paradoxically the best o p p o r t u n i t y to get an observable p h o t o n signal from annihilation in this situation is p r o v i d e d by a yet higher o r d e r process, namely ~ - ' T ) ' , to which we now turn The ~dea that ~Y~T)' might be a p r o m i s i n g channel was first put forward in ref [6], where the crucial observation was that this process IS not hehclty suppressed in contrast to ~ - ~ ff Moreover, in contrast to the radiative process discussed above, it gives a very sharp line In the g a m m a spectrum at the highest possible energy Ev=rn~, (the intrinsic width o f the line is o f the o r d e r ofv/c~ 10 - 3 ) In ref [6], the p o m t h k e a p p r o x i m a t i o n to the ~ ? ~ f f vertex was used, and in a d d i t i o n it was assumed that there be no c o n t r i b u t i o n from the axial a n o m a l y (this is certainly true if the neutralino is a pure hlggsino which only couples through a Z ° to fermlons) Recently, it has been argued by R u d a z [ 14] that m the case o f p h o t i n o anmhxlatlon there is effectively an i n d u c e d a n o m a l y that could make the rate for YY~TT even larger than estim a t e d in ref [ 6 ] However, this suggestion was based on the p o i n t h k e a p p r o x i m a t i o n for the ~,?~ffvertex, and in view o f the inadequacy o f that a p p r o x i m a t i o n in the radiative process discussed above, we have been p r o m p t e d to p e r f o r m the full one-loop box d m g r a m calculation given by the diagrams In figs 4 a - 4 c The d i a g r a m in fig 4b actually vanishes up to small CP violating terms so that we are left with calculating d m g r a m s 4a a n d 4c Writing the a m p h t u d e J (~--'YY) = ~'~'~P~ T~~k~pk2,,,

( 5)

where e ~ and e2~ are the polarization vectors o f the photons and kl and k2 are their m o m e n t u m vectors, the cross section is given by O.Vrel(~__~,)t~) _ m~ I~¢Zl2 327r

(6)

As before, we e m p l o y the v ~ 0 h m l t a n d use a spin singlet projector for the initial state We calculate the resulting a m p l i t u d e for a given fermlon species f mrculating the loop to be

j / = 8x/2 N~a2Q~ mf

F,

(7)

with 1

F= 2(l+r~-rr)

1

o N l ( x ) d~v+ ~r~

~2(x,y) dxdy,

(8)

w h e r e t o = rn~/rnr, 2 2 r r _- m f2/ m r, 2 and the functions N~ 2 are given by

r__~f [L~(x)+4L2(x)]+l - x1' cffI (X) = L 2 ( x ) -L~ (x) + 4r~cx2

(9)

2L3 (x, Y) L4(X,Y) ~2(x,y)- 2x-y-2 + 2x-y-1 '

(Q)

(b)

(10)

to)

Fig 4 (a)-(c) Diagramsfor the one-loop process 9~'/~, 375

Volume 225 number 4

PHYSICS LETTERS B

27 July 1989

with

4r~x(x~l)+rt , ( r~x(l_-x)+_rf(l_-x)+x.~ L2(x)=log _ r ~ x ( l _ x ) + r f ( l _ x ) + x j ' (r~(4x2--4xy--4x+y2+3y) + rr( 1 --y) + y ) L3(x,y) = l o g \ r ~ +--~f(1 - y ) + y Ll(X)=log

(11)

(12)

( 13)

'

and

L4(x, y) = l o g

(r>(2xy-y2-y) +rr( 1 - y ) + y -r~(2xy-y2-y) +rf( 1 - y ) + y J

(14)

The term containing L4 c o m e s from dmgram 4c, all the others are from dmgram 4a The singularities m the arguments of the logarithms are handled by adding the usual l¢ terms to the masses of the internal particles in the loop It is interesting to note, however, that the diagram in fig 4c is never singular in the physical region for the external m o m e n t a On the other hand, its real (dispersive) part plays an important role We find m fact that this, together with non-vanishing parts of diagram 4a m the hmlt when m r , 0 , simulates well the anomalous contrlbunon of ~'?-'77 obtained in the pomthke hmlt This calculation then gives an example of how a lowenergy effective anomaly may arise m a theory that is anomaly-free at the fundamental level In fact, this result could perhaps have been anticipated from the treatment in ref [ 15 ], where it was shown that the anomalous a m p h t u d e is actually a low-energy phenomenon, independent of the eventual divergence or non-divergence of the triangle diagram Another mstrucnve observation to make is that the diagram in fig 4c, although it looks related to dmgram 2c, does not have an extra suppressmn due to the double ]'propagator This is caused by the loop lntegranon and Is crucial for the successful reproduction of the anomaly term The numerical results of our calculation show that the pomt-hke approximation m this case is excellent (provided the anomaly term is kept) even for ? masses qmte close to the ]~mass, where a priori this approx~manon need not make much sense As a consequence, for most purposes the complicated expressmn m eq (7) may be replaced by

~f~po,nt-

8x/2Nc°~2Q4 Fp (r~ ) m~ ~f , ....

(15)

w~th Fpo,m(x) = 1 arcsln2 (vFx) _ 1

X< 1,

x

= ~r2-41nZ(x/x+x/x----1) - l + l - ~ l n ( x ~ + x ~ - l ) 4x x

(16) x>l

Here the term - 1 in the real part is the anomaly contribution For ? masses larger than mT the exact amphtude drops rapidly, for other mass combinations the amplitude is well described by eq (16) The anomalous feature of the constant term an eq (16) is the fact that ~t renders the amplatude non-zero as mf-~0 In the pomthke approximation, the process ~'~'-~77 is obtained from ~,~--,ff through a rescattermg of the produced fermlons Since the latter process is forbidden for real fermlons in the hmlt m ~ 0 , the ~magmary part of the amplitude has to be zero, as is indeed the case m eq (16) In the loop integration over virtual fermlons there are, however, regions where the mtegrand is singular as 1/m~, and it is this dispersive part that creates the anomaly term 376

Volume 225 number 4

PHYSICS LET FERS B

a

27 July 1989

!b

I

poinf

/

10-)

omf

10-2

eXClCf ~ ~ \ % \ \

N

U_

exacf ~

I0-3

(m'~/m~-) : 16/25

(mr/m~) 2 : 1/16

i0-I i0-~. 10-s ,,,,,if

........

I0 -2

J

........

I0 -i

(mf/m]) 2

L

I

t

) ,'llHl

I0

10-2

10-I

1

io

( m~ i rnr ] 2

Fig 5 (a) The values of IF[ 2, with F defined in eq ( 7 ), for the exact calculation eq ( 8 ) (solid line) and for the pomtlike four-fermlon approximanon eq (16) (dashed line) as a function of (mr~my)2, for (mv/m-f)2=16/25 (b) Same as m (a) but as a function of ( m~/ mr) 2, for ( m,/ my) - = l / 16 In fig 5a we show the values of ]FI 2 from eq (8) together with the approximating function J fpo,n I ]2 from eq ( 16 ) as a function of (mr/m~)2 Even for the high value of (rn~/rnr)2= ( 2 0 / 2 5 ) 2 chosen it can be seen from fig 5a that there is very little difference between the two expressions ~ For smaller values of this mass ratio the two curves rapidly become Indistinguishable In fig 5b are shown the corresponding values as a function of (m,/mT) 2, with a fixed value of (mJm~)2= 1/16 As can be seen, the constant value lmphed by eq (16) is a good a p p r o x i m a t i o n to the full results all the way up to (mJrnv) 2~ 2, where the exact expression starts to fall off rapidly To summarize this part, we have verified that the full box diagram calculation is well approximated by the simple analytic result in eq ( 16)for all values of mr and m, less than mr In fact, one may make the even simpler step function a p p r o x i m a t i o n ]F[2=l =0

m~andm~
(17)

otherwise

(This near constancy of IF] 2 is, however, not reflected in the real and imaginary parts of F separately As can be gathered from eq (16), these fluctuate rapidly near the threshold for ff production ) Finally, we make some estimates of rates from these processes An appropriate estimate of the photon flux from dark matter annihilation m the galactic halo is given by [ 16]

cl)v(Ev) -

1 6 × 1 0 -2 m2s sr GeV

P~ ~ ( avrd ~(1GeV-']( a '](aN'] 0 3 Ge-V/cm~J \ 1 0 2--Uc-cm~/sJ\~J\7k~pcpcJ\~J

I(b, l)

(18)

Here a halo densaty of the form p,(r)=p~(a-+r~)/(a2+r h 2) has been assumed, and l(b, l) is a line of sight integral that depends on galacttc latitude b a n d longitude l In the direction of the galacttc pole, I = 1 24 The background flux for photon energies above 100 MeV is unfortunately poorly known experimentally [ 17 ] Before better m e a s u r e m e n t s at high energies are at hand we are thus confined to making theoretical extrapolations and esnmates The most recent analysis has been performed by Stecker and Tylka [ 18 ] who argue that the #~ On the other hand, for such large my~mrthe polnthke approximation to the lowest order process "~ffglves an overestimation of the cross section by approximately a factor of 2 7 according to eq ( l ) 377

Volume 225, number 4

PHYSICS LETTERS B

27 July 1989

extragalacttc ( l s o t r o p l c ) b a c k g r o u n d m a y be small at high energies a n d that the galactic b a c k g r o u n d at high latitudes can be p a r a m e t r l s e d as

--rtbga~...... ~ 1 5 × 1 0 - s

--27

( 10-~ )

1

sr- GeV-',

(19)

for p h o t o n energies above 3 GeV Keeping in m i n d the uncertainty o f the size of this background, we will take the dtstrlbutlon m eq ( 1 9 ) as our estimate Before presenting results o f full numerical calculations, we note that a rough estimate o f the order o f magnitude o f the various p h o t l n o a n n i h i l a t i o n processes can be o b t a i n e d through the simple hellclty arguments above 2

f~ f~7 7 7 ~ 1 a

m~ 7~ m f2 ?n]~4

~/]

2 ~2f2

Beginning wtth the simple m o d e l where only e + e - i n t e r m e d i a t e states are i m p o r t a n t , we get from eq ( 1 ) aVre~(??-~e+e - ) ~ 0 4 × l0 -32 c m 3 s - ~, for m~ = 20 G e V a n d m e = 25 G e V The d o m i n a n t process is ~ - , e + e - 7 , which has a cross section times velocity value o f 0 6 × 10 -27 cm 3 s - 1 This is perhaps visible against the cosmic background, eq (19), but as can be seen in fig 6 q m t e high statistics would be necessary In an e x p e r i m e n t with an acceptance o f 1 m2sr, as has been discussed for the Space Station [ 19-21 ], several h u n d r e d events from this channel would be detected in a one-year exposure which gives more than a 4tr effect p r o v i d e d that the background estimate eq (19) is correct It has also been p o i n t e d out that the lsotroplc b a c k g r o u n d could have been o v e r e s t i m a t e d and in patches o f the sky the b a c k g r o u n d could then be much lower In addition, there could be local density fluctuations o f the d a r k m a t t e r that could increase the signal considerably [ 18 ] Unfortunately, the rate for the radiative process decreases very rapidly with increasing me, so that it ts hardly observable above m e = 50 G e V On the other hand, this m e a n s that the possibility o f this m e c h a m s m for producing observable high-energy g a m m a s and positrons could p r o b a b l y be established or ruled out with forthcoming d a t a from LEP The case of??--,),), seems much better In the simple e+e - m o d e l the value o f a v ~ for ??~),), is 4 3 X 10 -29 cm3s - ~ The corresponding signal is also shown ]n fig 6, where this peak is seen to clearly stand out against b a c k g r o u n d I f me ~ m~ ~ m~, then aVr~l is m u l t i p l i e d by a factor o f 9 (all contributions a d d coherently in the a m p l i t u d e ) m a k i n g the signal even m o r e striking p r o v i d e d that the detectors have the excellent energy resolutlon envisaged in refs [ 19-21 ] It should also be n o t e d that due to the remarkable rise with m~2 o f the process ??-,),% as seen in eq ( 6 ) , for fixed m~ the inverse squared b e h a v l o u r o f the annihilation cross section as a function o f p h o t i n o mass, eq ( 18 ), is cancelled Since the width o f the ), line goes as m~, the signal to b a c k g r o u n d

m~ = 20 QeV m~~=25 GeV 10-5 ,-r'~ T

2T

o N

i0-6 10

12

14

16

18

20

22

Photon Energy (GeV) 378

24

Fig 6 Photon flux coming from the anmhllatmn processes ~?~ e+e 7 and ? ~ y 7 in the galactic halo, for rnv=20 GeV and m~=25 GeV Also shown ]s the background expected from the d~rectxon of the galactic pole, as estimated in eq (19)

Volume 225, number 4

PHYSICS LETTERS B

27 July 1989

ratio in fact improves as m~ 4 The lines should then be measurable in an experiment of the a b o v e - m e n t i o n e d type for ~ masses not larger than about mw ~2 A word of caution has, however, to be given concerning the rehablhty of the background estimate eq (19) With the soon to be launched G a m m a Ray Observatory, we will be in a better position to judge this crucial point (The even more spectacular rise with m~6 of the ~ ? ~ e + e - y process IS unfortunately h m l t e d by the low value of me needed to make the rate observable ) It may be r e m i n d e d that sample calculation in supersymmetric models [8 ] give ~ masses in the range 2 0 - 5 0 GeV, whereas squarks are much heavier ( 100-200 GeV) The llghtest supersymmetrlc particle in these models is in general a mixture of photlno, htggslno a n d zano However, we believe that the estimates given here should give the right order of magnitude even in this general case In particular, the mixed hlggsino-zino part of the n e u t r a h n o has a significant coupling to the neutral pseudoscalar Hlggs bosons, which tend to be quite light in these models, and could thus give an Important, perhaps even resonant c o n t r i b u t i o n ( R e m e m b e r that the initial ~ state, being S wave a n d spin sanglet, is effectively pseudoscalar ) Although we have argued that these radiative processes should be observable for certain mass parameters, it should be noted that the situation could in reality be less favourable For instance, If all sfermions are very heavy except E which is still heavier than presently believed, say 100 GeV, then a photlno with a mass of 30 GeV would provide closure density of the Universe a n d would be an attractive candidate for halo dark matter Such a particle would not be detectable at LEP experiments, it would be essentially impossible to detect in direct scatterlngs in cryogenic detectors (since interactions with nuclei are then negligible) There would be no high-energy neutrinos produced in the a n n i h i l a t i o n s m e a n i n g that it will pass unnoticed also in u n d e r g r o u n d detectors The d o m i n a n t a n n i h i l a t i o n channel would in fact be YY-~YT, but the line rate is only 2 × 10 9 s - i sr-~ on a background at least ten times a large, m e a n i n g that it is beyond observablhty with presently proposed detectors The only hope would be for a local e n h a n c e m e n t of the halo density, e g close to the galactic centre, that could increase a n n i h i l a t i o n rates considerably To conclude, we have shown that radiative corrections to ~ - ~ f f show quite remarkable features, making the radiative processes a promising means to detect a n n i h i l a t i o n of supersymmetric dark matter The process ~ 77 which we have calculated beyond the pointhke approximation, seems to offer the best hope in this context, with observable lines resulting from a favourable but realistic range of mass parameters We have also shown the reaction ~ - ~ e + e - ' f to be free from hehclty suppression, but it needs a value of the E mass close to the present experimental lower b o u n d to give a detectable flux of high-energy photons or positrons In any case, the latter process is theoretically remarkable as an example of a radiative process being orders of magnitude larger than the lowest order rate The author wishes to thank P Carlson, H Rubxnsteln, S Rudaz and H Snellman for interesting discussions This work was supported by the Swedish Natural Science Research Council ( N F R )

~2 In this paper, we do not reqmre that the observed density of dark matter m the local halo corresponds to (2~~ 1 m the whole Umverse However, we have checked that in the parameter intervals mentioned there is no overclosure caused by photmos Typical values of .Qv range from a couple of percent to several tens of percent (Taking the Hubble parameter to be H0= 50 km s-l Mpc-' ) Closure can indeed be obtained for certain parameter values, e g my= 7 GeV and m~= 50 GeV

References [ 1] v Tnmble, Annu Rev Astron Astrophys 25 (1987) 425 [2] D Hegyland K A Ohve, Astrophys J 303 (1986) 56 [3] J R Prlmack, D Seckel and B Sadoulet, Annu Rev Nucl Part Scl 38 (1988) 751 [4 ] M Sredmckl, S Thelsen and J Silk, Phys Rev Len 56 (1986) 263, 1883 (E) [5] S Rudaz, Phys Rev Len 56 (1986) 2128 [6] L Bergstrom and H Snellman, Phys Rev D 37 ( 1988)3737 379

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PHYSICS LETTERS B

27 July 1989

[ 7 ] L Bergstrom, preprlnt USITP-88-12, to be pubhshed [8] J Ellis, J F Gunlon, H E Haber, L Roszkowskl and F Zwlrner, Phys Rev D 39 (1989) 844 [ 9 ] TOPAZ Collab, I Adachi et al, Phys Lett B 218 (1989) 105 [ 10] F Abe et al, Plays Rev Left 62 (1989) 1825, m Conf on High energy physics, eds R Kotthaus and J Kuhn (Springer, Berhn, 1989) [ 11 ] D Mueller and K -K Wang, Astrophys J 312 ( 1987 ) 183 [ 12] S Rudaz and F W Stecker, Astrophys J 325 (1988) 16 [ 13 ] M S Turner and F Wflczek, preprlnt FNAL PUB 89/44A ( 1989 ) [ 14 ] S Rudaz, in Proc Workshop on High resolution gamma ray cosmology, Nucl Phys B (Proc Suppl ) 11A (1989), in press [ 15 ] A D Dolgov and V l Zakharov, Nucl Phys B 27 ( 1971 ) 525 [16] J Ellis et al, Phys Lett B 214 (1988)403 [ 17 ] D J Thompson and C F Flchtel, Astron Astrophys 109 (1982) 352 [ 18 ] F W Stecker and A J Tylka, preprlnt LHEA/Th-88-46 (1988) [ 19] J H Adams et al, ASTROGAM proposal (1988), unpublished [20 ] P Carlson et al, SUSY-SKY proposal ( 1988 ), unpublished [21 ] T Doke et al, in Proc Workshop on High resolution gamma ray cosmology, Nucl Phys B (proc Suppl ) 1 IA (1989), in press, O Clme, private commumcatlon

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