Radiative processes in LSP annihilation

Volume 232, number 3 PHYSICS LETTERS B 7 December 1989

RADIATIVE P R O C E S S E S IN L S P A N N I H I L A T I O N

Ricardo FLORES Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455, USA

Keith A. OLIVE and Serge R U D A Z School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA

Received 11 September 1989

We calculate the O(a) corrections to the annihilation cross section for dark matter candidates in the minimal supersymmetric standard model. In particular, the radiative process )~z~qqg is shown to be at most 15% of the total cross sections (except for very massive binos). We apply our results to the relic abundance of neutralinos left over in the Big Bang. We also comment on the importance of these channels for dark matter annihilation in the galactic halo.

The supersymmetr ic s tandard model contains an excellent candida te for the dark mat te r in the Uni- verse [ 1-3 ]. In the min imal extension to the standard model, there is always one stable massive part i - cle, generally taken to be one of the l inear combina t ions of four neutral gauge and Higgs fermions. Despi te the uncerta int ies in the supersymmetr ic mass parameters , there is a large set of parameters in which the relic Big Bang abundance o f neutral inos is cosmological ly significant [ 1-5 ].

Let us call the lightest supersymmetr ic part icle (LSP))~, with

X = a , W + f l , B + y l I ~ l +61Iq2, (1)

where XV and B are the fermion par tners o f the SU(2 )L and U ( 1 ) r gauge bosons and H~, H2 are the par tners of the Higgs bosons giving masses to up and down quarks, respectively. Assuming a thermal state ini t ial ly in the Big Bang, the number densi ty o f z ' s is reduced when T < m x due to annihi la t ions, as in the case of neutr inos [6] . Wi thou t a pa r t i c l e -an t ipa r t i - cle asymmetry , the relic abundance depends pr imar- ily on the annihi la t ion cross section at the t ime annihi la t ions freeze-out, i.e., when the annih i la t ion rate falls below the expansion rate o f the Universe. Roughly one can write

£2xho 2-~2.7X lO- l lN~/2(Tx /Tv) 3 G e V - 2 a x + ½bx 2' (2 )

where g2x-px/pc is the ratio o f the g mass densi ty to the crit ical densi ty pc = 1 .88× 10 -29 h 2 g c m -3 and

ho = Ho/100 km M p c - 1 s - 1 is the present value o f the Hubble parameter . ArE is the number of relativist ic degrees of freedom, and (TffTv) is the rat io of the tempera ture of Z's relative to photons [7] both at freeze-out. The annihi la t ion cross section to order v 2 is writ ten as

( a v ) A = a + b x , (3)

where x = T v / m x ~ ~ at the t ime o f freeze-out. (See ref. [ 5 ] for a complete descr ipt ion regarding the cal- culat ion of £2xh 2o. )

It is well known that because Z is a Majorana particle, the s-wave part of the cross section, a, for the process Zg ~ ff is suppressed by m f2 / m ~ relative to the cont r ibut ion to (aV)A due to the p-wave part, bx, which typical ly dominates at freeze-out. In this pa- per, we consider the effect of radia t ive processes such as ZX--+ q(:lg (g, a gluon) on the annihi la t ion cross section, especially as regards the s-wave part which now need not vanish as mf-~0. The s imilar process ZX--*f/'Y was considered in ref. [8] with the hope of an observable signature in the y-ray spectrum from annihi la t ions in the galactic halo. We find that the dominan t contr ibut ion to the total s-wave cross sec- t ion can in fact come from the qqg O ( a s ) correction.

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However, the total annihilation cross-section (at freeze-out) is altered by at most ~ 15% (except for very massive LSP's) , thus changing only slightly the predicted relic abundance of LSP's. We will also comment on the likelihood of observing final state photons in halo annihilations.

Most commonly, the LSP has been assumed to be a photino [1,2], i.e., oq=sin0w, / ~ I=COS0w and 71=c~=0. In general, in the minimal model, the identity of the LSP depends on three parameters [ 3 ]: M2, the SU(2 ) supersymmetry breaking gaugino mass; e, the Higgsino mixing mass and; Vl/V2 the ratio of the VEVS of Hi and H2. When M2 is small, the LSP is indeed a photino. When e is small the LSP is the Higgsino go defined by a t = fll = 0, 71 = v2/v and fil = vl / v, where v 2 = V 2 + U22 . However, when m x > 20 GeV, it is never a ~ or g [9 ]. For larger rex, it turns out that the LSP is still typically a well-defined pure state [ 9 ]: either the bino B, fli ~ 1 and a i = 71 = 31 = 0 or a symmetric (antisymmetric) Higgsino I~(12)

(~I[121) defined by a t = f l ~ = 0 and 7 ~ = ( + ) 6 t = 1/x/~. We will for simplicity concentrate and compute the radiative annihilation cross-section for 7-'s and discuss the effects of the radiative processes for the other LSP pure states. In what follows, rn x will always refer to the mass of the LSP under consideration.

We first consider the radiative process ~ t ' f ' / i n order to discuss the physics. It will be straightforward to add the color structure to compute the magnitude of a(~7--, Clqg) / a ( ~ i'f ) and to determine the effect of these processes in the computed relic abundance. To order ot 3 the relevant diagrams are given in figs. l a - l c (plus the corresponding diagrams with the photino lines exchanged) and we find (neglecting terms O(rn 2) and O( [p~[ 2) ):

_/ - - - - . . . . .

f

\ f

Y

. . . . . . . . .

x x x x x

(a) (b)

X (c)

Fig. 1. Feynman diagrams for the contributions to a(ZX--*t'fT) to order a 3. The corresponding diagrams with the x-lines inter- changed are not shown.

day rn z - E v dE./ = 8°L3Qf6 m x

I 2E~ × (~12+m~)(3712+rn~_2mzE~,)

- (~2+ m~_mzEv)2

( + ½ In k,~E+m 2 _2mxEv j

X \ ( M2 + m2x-mzE~)3

_ 1

mz( f42 + m~-rnzE~, ) l ' (4)

where we have assumed a common left- and right- handed sfermion mass, 3~t. (This differs from the result in ref. [ 8 ] by a factor of 2.) Unfortunately it is not possible to obtain in analytic form an expression for the total cross section (aV)A.

This differential annihilation rate is in fact o f leading order ~ t - 8 and vanishes as Ey--,0. Upon integration over the photon energy, it will give rise to a new piece of the (x-independent) s-wave part of the total ~ annihilation cross section that does not vanish when mf= 0. To see how this comes about, consider expanding the amplitude corresponding to the diagrams of fig. 1 in inverse powers of~t2: to lowest order in 1 /~ t 2 one finds, after some Fierz transforma- tions, and for finite mf,

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~,A(0) _ 23/2mf (eQf) 3 - ~t2

X [(P3"* p4"e*~ u(p3)75v(p4) L kP3 'P5 P4 "P5 /

+ ~ (p3!ps + 1 p4!ps)12(P3)7.~.7.P575v(p4) ]

(5)

to zeroth order in the initial-state relative velocity, and where the suffix 3(4, 5) refers to f(f, "/). This part of the amplitude would give rise to a typical bremsstrahlung spectrum a- ~da/dE~,~E~-~, but in fact vanishes as mf---~0. This is because in this order the initial purely s-wave ZZ state effectively behaves as an elementary pseudoscalar whose local coupling to a fermion pair is chirality-conserving: the infrared divergence arises as usual from the subsequent emission of a photon from the fermion lines.

Expanding further to order 5)--4 yields ,/~4¢A= Jf/A (0) × ( 1 -m2/5 ) z) +,~CA~), with ~1

• (eOf) 3 ~ .p ~/~ ~ =1 ~ ~ p p 5

X O(p~ )TU75u(p2 )t2(p3 )7~75v(P4 ) . (6)

Setting m f = 0 , this results in an s-wave cross section

16 m6 ot3Q6 (7) a ~ - 155) ~

This expression is quite a good approximation to the exact result obtained by integration of eq. (4), as shown in fig. 2, over most of the range mx ~< h~t of rel- evance here.

Next, we quickly discuss the radiative effects of relative order c~ on the p-wave part of the cross-section. In the local limit mx 2 << A) 2, the effect of the real photon emission process Y~--'ffT, together with virtual photon radiative corrections to Y~7-'ffin the same order, will lead by the Kinoshita-Lee-Nauenberg theo- rem (when m~ << rn~) to a finite correction to the order 5 ) -4 p-wave part of the total annihilation cross

~ The amplitude from fig. lc is not gauge invariant, but the expansion of the propagators in figs. 1 a and 1 c gives a contribution that exactly cancels that of fig. lc and leaves the gauge invariant piece (6).

E

O

$

O

0

-1

-2

-3

-4 0 .2 .4 .6 .8

m~/~,l

Fig. 2. The annihilation cross section, ~(7~--,ff'/)v, to order o?. Shown are the exact (solid line) result from eq. (4) and the leading contribution, eq. ( 7 ).

section. In fact, when mf----0, one should have simply the analog of the standard Jost-Luttinger result, b=bo[1 + (3ot/4~r)Q~] with bo the coefficient to lowest order of the p-wave part of the cross section for ~77~f~. In all, to relative order o~ and for m ] << m~, one has for each fermion species

av(ff+ff'/)~-a, bo I + ~ Q ~ x . (8)

Clearly, the electromagnetic radiative processes rep- resent very small corrections to the leading order re- suits: this need not be the case, however, for strong radiative processes involving a gluon. Indeed, one can then have al/bo~O(otsmaff44) which could amount to a 10% effect for mz_~5). Before investigating this in detail, we must consider other strong radiative processes which although of higher relative order in as may still be numerically important for smaller val- ues of mz/5).

We first discuss the contribution of the process pic- tured in fig. 3 to the amplitude for ~-- ' ffT: this in- volves the effective ,7~7~ 7"/vertex for one off-shell photon.

Remarkably, for massless virtual fermions in the loop, all dependence on the final state ff pair invariant mass disappears, and the form factor reduces to the quantity I ( 0 ) = ½ that appears in the matrix ele- ment for 37~-~ )q, [10]: for all fermions but the top- quark (which effectively decouples if mt 2 > m 2), one has m 2 << m 2 and they may be considered as effec-

379


f

z z

Fig. 3. Feynman diagram contributing to a(7,X~ ff~' ) to order a 5.

tively massless as far as the loop integral is con- cerned. The resulting order e 5 amplitude contributing to ??~ff~/ is then

3 4 ( ~ ) p/t(2) :12,3/2 e Qr (~r Qr )

Xeu,,,ap~,*a(p, +p2) u/l((P;3 )Tu/)(p4)+p4)2 (9)

where f ' denotes the effectively massless virtual fermions. The key feature is the appearance of a loga- rithmic enhancement factor of the form In ( m 2 / m 2) from the integral over the virtual photon propagator in the total rate. Introducing the appropriate color structure, we find from (6) and (9) , with m 2 << m 2 (the interference term vanishes for equal left and right sfermion masses)

aqog = ~ a ( ~ q q g ) v q v=O

mz .2A~ 4 _ 1 6 _ ~ u , , t ~ Q q

15 q

\ m x / d (10)

where R= ( EqQ2q )Z / ZqQ 4 and S= Zq[ln( m2 /m2q) - ~ ]. Here the sum runs over all quarks with m 2 < m 2, and the result is valid to leading order in

2 2 mq/m x. We now turn to the discussion of the effect of this process, as well as ~-- ,gg, on the abundance of relic neutralinos.

To determine the magnitude of these effects we will

adjust all particle physics parameters so as to obtain ~2zh2=J from lowest order expressions. As pure states, photinos exist as the LSP only when rne~<20 GeV [9]. For M2>>m2, we can write [1,3] a f t= (8nol2/~/[4)ZfmZQ~ and bff= (16noflm~/ A)4~Z ' q 4 / f~r . Eor m ~ = 1 0 - 2 0 GeV, )14=80-112 GeV [5]. The dominant contribution to the radiative channel is by far due to the process in fig. 3. A comparison of the s-wave cross sections is given in table 1. Also shown is a comparison of the modified total cross-section #' = 6+ ao~ ~ + 1 (as~ ~z) bfrx, with 6 = a o + ½bf, x, ao=afr+a~3+a23. The total cross section is only modified by < 15%. We have taken as ( 10 GeV) =0.17.

As was shown in ref. [ 9 ], a large portion of the supersymmetric parameter space yields a bino for the LSP. For the bino we must modify eq. (10) by (with Y = 2 ( Q - T 3 ) )

a___+O£,..~_a/COS2Ow, 2 I yq2L+ 2 Q q ~ ( (11 YqR) • )

Correspondingly when m~ << ~t, art = ( h a ' 2 / 8;Q4 ) X ~ f ( y 2 L + v 2 l fR] X2~2 t t t f and bfr~-(na'2mZa/2M4) ×~f (y4L+y4R) . When m a i M , one can take arr-~afr/4 and bf~-~bfr/8. Binos as pure states only exist when ma > l0 GeV. There are in addition other annihilation channels which can be significant. For example, depending on the masses of the Higgs sca- lars, BB-~H°H ° and B B ~ H ° H ° can occur: H ° is the pseudoscalar Higgs boson and H °, H ° are the scalar states, heavier and lighter than the Z boson respectively.

At m ~ = 2 0 GeV, the ratio aq~f/arr depends on the presence of the BB ~ H ° H ° channel. At mi~ = 40 GeV, the o o H2 H3 channel plays a larger role, a23- (7c0l' 2/4 ) × sinZ2fl ~2/(~2 + rn~ )2 where tan fl= Vl/V2. The bino is the LSP when E > 400 GeV. When a23 = 0 (by either phase space suppression or if e is very large)

Table 1

State m~/GeV aqc~/ ao #'/6

10 0.40 1,12 20 1.2 1.11 10 0.34 1.09 20 0.47-1.1 1.07-1.08 40 0.05-3.5 1.06-1.07

100 0-27 1-1.10 300 0.57 1.28

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aa=7× 10 -12 (in units o f G e V - 2 ) , 3~t= 160 GeV. We 0 0 find then that aqqg -~ 2.5 × 10-11. Including the H2 H3

channel, a23 m a x ~--- 2.9 × 10 -- 10 and then afr-~ 10 - 1 ~ with M ~ 180 GeV: now aqq~- ~ 1.5× 10 -11 and is only a small correction to even the s-wave cross section. As one can see the total cross sections are only slightly modified.

H l H 3 channel also plays a At higher masses the o o role, a13-~ Oza'2/4)cos22fle2/ ( e2 +mZa - 1~2 ~2 2,,,zJ [9]. At m~= 100 GeV, with a12=a13=0, aft -~ 1.5× 10 -~2 with 34_~210 GeV and we find aqqg-~4.6× 10 - l l When al2, a13 :~0 , e > 250 is needed to insure a bino as a pure state. In this case £2~ho z can be completely determined by E. The radiative processes are not important.

It was found [9] that when binos are the LSP, g2~h 2 < 1 necessarily implies (at lowest order) that m~ (and 34) < 2 5 0 GeV, the reason being that no annihilation cross section is sufficient ro reduce the abundance to a cosmologically acceptable value. When m ~ = . M = 3 0 0 GeV, despite the fact that aqqg/ao< 1 the enhancement in the cross section is sufficient to bring the cosmological density to an acceptable level. Above m~-~ 320 GeV the total cross section is again too small to produce .Qzh 2 < ~.

The effect of the radiative processes on the annihilation of go is completely negligible, go annihilation is dominated by Z exchange with aft= ( G 2 / 4z0cos22fl m~. The radiative correction can affect only the squark exchange and is maximally given by aqqg=(G2a3s/g2)tan4 fl m4m~o/f f l 4 so that aqqg/ afr~ m~om~ / JgP < m~ / jgl2 << 1.

For more massive higgsinos, the effect is also negligible. Although, for pure H°2 states, there is no Z- exchange annihilation channel, for mtop>mw and

- - 0 - - 0 rnn?: < m w only when processes such a s H~2H12--}

H2H3° o are dominant are kI°2's cosmologically significant. But again the radiative process is proportional to m~. When mn% > mw annihilation to W + W - and ZZ is so strong there is no appreciable relic abundance o f I~02 until mn% > 1 TeV. We will not consider these states any further here.

Finally, we comment on the process Z)~-}gg [ 11,10]. For photinos, the process is negligible for determining the relic abundances. At m~= 10 GeV, agg/afr~-O. 1 (a f r /~=0 .2 ) and at m f = 2 0 GeV, agg/ afr-~0.3 (af t /O=0.06) . For binos, although BB-~gg does not ever significantly effect the total cross sec-

tion, when m ~ > 5 5 GeV, agg/afr> 1 ( a ~ < 4 × 1 0 - 1 2

at this point) and may dominate the s-wave part of the cross section.

Annihilations o f relic dark matter particles in the halo o f our galaxy have been considered as a poten- tial signal in cosmic ray experiments such as antipro- ton, positron and gamma ray searches [ 12-14] . The flux of neutral annihilation products is given by

2 - -2 dcP 1 .6×10 -6 { pz ~ ( m ~

dE - cm2s sr GeV \0 .3 GeV cm-3,] \G-~eV]

a day 26~3~ 1

(12)

where a is the core radius of the galactic halo and L(b, l) is a line-of-sight factor that depends on the galactic latitude b and longitude l. The background in gamma ray searches is not known at high energies, E~ > 1 GeV, however, it is expected to be dominated by decays o f neutral pions produced in cosmic ray proton inter- actions with the interstellar medium. For high galactic latitudes one expects [ 15 ]

dq~ ~1.5 dE,

" E y . - -2 .7 10-9(l eV ) cm+lsr-lOeV-', (13)

while the differential ,/-ray flux from eqs. (6) and (9) for ~-,e+e-v is given by (with xv=Ev/m, , O<~xv<~ 1 2 2 2 /me(m~_E~ ) ) - me /m~ , and f12 = 1 - me

dq5 0.9N 10 -15 dxv - cm2s sr GeV ( 1 - x ~ )x3fl

x( 2 + 0.1 i ~ _ ~ ) 2 -/ (14)

for m~= 10 GeV and M = 8 0 GeV chosen to give O,h 2 = ~. Clearly, even though the flux near x r= 1, d ~ / d E v (peak) ~ 1 0 - 9 c m - 2 s - l s r - l GeV -1, this would not be observable against the background with energy resolutions AE/E~_ 1%, such as for ASTRO- GAM and H R - G R A F (for m~= 20 GeV, the peak signal /noise only increases by ~ 3.5). A much higher flux was obtained in ref. [8] in the limit me~25 GeV << all other sfermion masses and m~/rne ~ 1. This

381


choice of s fe rmion mass spec t rum, however , gives ~2~<0.05, which is marg ina l ly suff ic ient to account for dark ma t t e r in halos. For mass ive b inos the sig- n a l / n o i s e becomes larger (O (1) for ms > 250 G e V ) bu t the pho ton de tec t ion rate is exceedingly small ( ~ 10 -4 pho tons y r - ~ for an acceptance o f ~ 1 m E sr

such as for A S T R O G A M an d H R - G R A F ) . Halo a n n i h i l a t i o n s in to q u a r k - a n t i q u a r k pairs

could yield a de tec table flux of cosmic ray an t ip ro- tons as well, a l though the cur ren t level o f sens i t iv i ty is far f rom the level o f expected flux [ 14]. G l u o n product ion , we have seen, could significantly alter the pred ic ted fluxes, e.g., by as m u c h as a factor o f 2 for r r~= 20 GeV. The pred ic ted flux, however , is still a factor ~ 2 × 10 -3 be low the cur ren t level o f sensi t iv- ity (for ~2eh g = ¼ ), a n d for larger masses the correc-

t ions are m u c h larger, b u t the fluxes are yet smaller . As always, larger fluxes are possible wi th lower relic

abundances .

The work of Ke i th A. Ol ive a n d Serge R u d a z was suppor ted in par t by D O E grant AC02-83ER-40105 at the Un ive r s i t y of M i n n e s o t a a n d by Pres iden t ia l

Young Inves t iga tor Awards.

References

[ 1 ] H. Goldberg, Phys. Rev. Lett. 50 ( 1983 ) 1419. [2] L.M. Krauss, Nucl. Phys. B 227 (1983) 556. [3] J. Ellis, J.S. Hagelin, D.V. Nanopoulos, K.A. Olive and M.

Srednicki, Nucl. Phys. B 238 (1984) 453. [4] K. Griest, Phys. Rev. D 38 (1988) 2357. [ 5 ] M. Srednicki, R. Watkins and K.A. Olive, Nucl. Phys. B 310

(1988) 693. [6] P. Hut, Phys. Lett. B 96 (1977) 85;

B. Lee and S. Weinberg, Phys. Rev. Lett. 39 (1977) 165; M.I. Vysotskii, A.D. Dolgov and Ya.B. Zeldovich, JETP Lett. 26 (1977) 188.

[7 ] K.A. Olive, D.N. Schramm and G. Steigman, Nucl. Phys. B 180 (1987) 497.

[ 8 ] L. Bergstrom, University of Stockholm preprint USITP-89- 04 (1989).

[9] K.A. 0live and M. Srednicki, University of Minnesota preprint UMN-TH-801 / 89 (1989).

[ 10] S. Rudaz, Phys. Rev. D 39 (1989) 3549. [ 11 ] L. Bergstrom, University of Stockholm preprint USITP-88-

12 (1988). [12]J.E. Gunn, B.W. Lee, I. Lerche, D.N. Schramm and G.

Steigman, Astrophys. J. 223 (1978) 1015; F.W. Stecker, Astrophys. J. 223 (1978) 1032.

[ 13 ] J. Silk and M. Srednicki, Phys. Rev. Lett. 53 (1984) 624; F.W. Stecker, S. Rudaz and T.F. Walsh, Phys. Rev. Lett. 55 ( 1985 ) 2622; S. Rudaz and F.W. Stecker, Astrophys. J. 325 (1988) 16.

[ 14] J. Ellis, R. Flores, K. Freese, S. Ritz, D. Seckel and J. Silk, Phys. Lett. B 214 (1988) 403; F.W. Stecker and A.J. Tylka, Astrophys. J. 336 (1989) 151.

[ 15 ] F.W. Stecker, The large scale characteristics of the galaxy, ed. W.B. Burton (Reidel, Dordrecht, 1979) p. 475; C.D. Dermer, Astron. Astrophys. 157 (1986) 223.

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Radiative processes in LSP annihilation