Volume 245, number 3, 4 PHYSICS LETTERS B 16 August 1990

A new dark matter candidate in the minimal extension of the supersymmetric standard model

Ricardo Flores 1 Theoretical Physics Institute, University of Minnesota, Minneapolis, M N 55455, USA

Keith A. Olive and David Thomas School of Physics and Astronomy, University of Minnesota, Minneapolis, M N 55455, USA

Received 7 May 1990

We study the cosmology of the neutral supersymmetric fermion sector in the simplest extension of the minimal supersym- metric standard model. We find that in a substantial portion of the allowed parameter space of the model, a new light fermion emerges. For this state we calculate annihilation cross-sections and determine its cosmological abundance. We find that in much of the cosmologically allowed region of parameter space its relic density could be significant enough to solve the dark matter problem.

The minimal supersymmetric standard model (MSSM) [1] has been investigated in great detail by many authors. Despite its many arbitrary soft supersymmetry breaking parameters, the model retains some predictive power: e.g. there must be a neutral Higgs scalar lighter than the Z-boson, as well as a charged one heavier

than the W-boson. An unsatisfactory feature of the MSSM is the lack of a dynamical mechanism to explain the small, but non-zero value of the mass parameter • in the superpotential W: W ~ - • H I H 2 where H1 and /-/2 are the Higgs superfields of the MSSM. Without this term, the model has an experimentally forbidden

axion-like pseudoscalar. The simplest extension of the MSSM that provides a dynamical source for • is the inclusion of an SU(3)x SU(2)× U(1) gauge singlet superfield N, such that W D - h N H t H 2 [2-5]. If the scalar N-field develops a vacuum expectation value, then in the presence of soft-supersymmetry breaking one expects ( N ) =- x = O( m w ) and therefore • = hx = O( m w ).

Though the MSSM is of course still experimentally viable, the existing constraints on the parameters may be pushed in the future to exclude the MSSM. For example, in the MSSM tan fl-= v~/v2=-(H~)/(H2) <<- 1.3 is

excluded since the mass of the lightest scalar is necessarily mH2< 23.5 GeV in contradiction with the limits imposed by LEP [6]. In the minimal extension considered here, there is no such constraint due to the added complication in the scalar sector. More importantly, if it is determined that either ran2 > mz or that the charged Higgs has a mass mc ~ < row, then the MSSM would be ruled out. Again, in this model neither of these relations hold and it would thus become the simplest viable supersymmetric model. Extensions to the MSSM such as those in many superstring-inspired models do contain an N-field, e.g. the SO(10) singlet in the 27 of E 6 [7], but this source for the mixing term - E H I H 2 is not unique and alternative mechanisms have been proposed [8].

The cosmology of the neutralino sector in the MSSM has been extensively studied [9-12]. In this letter we analyse the cosmology of the neutralino sector of this minimally extended MSSM. The scalar sector of the model has been analysed in ref. [5]. Our main result to be discussed here is that a new light neutral fermion emerges for h =O(1) and hx = O ( m w ) nearly independently of the gaugino masses. We calculate its relic abundance and determine the cosmologically allowed parameter space, in which we find its relic density could

i Address after September 1, 1990; Department of Physics, University of Missouri, St. Louis, MO 63121, USA.

0370-2693/90/$ 03.50 O 1990 - Elsevier Science Publishers B.V. (North-Holland) 509

Volume 245, number 3, 4 PHYSICS LETTERS B 16 August 1990

be significant enough to solve the dark matter problem. A complete survey of the cosmology of the neutral fermion sector of this model will be presented elsewhere [13].

The minimal non-MSSM that we shall consider here is defined by the following superpotent ia l (e~2 = +1):

W = -h%H~H~2N + ~ A N 3 + Wvuk . . . . (1)

where N is the singlet and Wyukawa contains the s tandard Yukawa couplings of the quark and lepton superfields [1]. (We follow the notat ion of refs. [9,11] in which the Higgs doublets H~ and H 2 have weak hypercharges of plus and minus one respectively.) As noted earlier, when N picks up a vacuum expectat ion value x, the first term above induces a piece -hx%Hi~H~, so hx here plays the same role as • in refs. [9,11]. The addi t ion of the cubic self-coupling of the N-field is required to avoid an unacceptable axion-l ike field.

The Higgs potent ial comes from F-terms, D-terms and soft supersymmetry breaking terms

<of, = m2,1H, I 2 + m2~l H212 + m2l NI 2 + (hAh %Hi~H{N - ] A A A N 3 + h.c.). (2)

The minimizat ion of the scalar potential is discussed in detail in refs. [5,13]. One can perform a gauge t ransformat ion so that

( H i ) = 0, ( H °) = v, ~ a +. (3)

In addi t ion the global phases of H1 and N can be chosen so that

hAh, A A A E R +. (4)

One can also take the vacuum expectat ion values

( H ° ) = v 2 ¢ R +, ( N ) = x E R ÷ (5)

and ( H ~ ) = 0 must be enforced. As discussed in ref. [5], a sufficient condi t ion to avoid spontaneous violation of CP in the scalar sector is

that - h A ~ R +. We find, however, that with a mild restriction on a combinat ion of soft supersymmetry breaking parameters and vacuum expectat ion values of the scalars, the vacuum considered in ref. [5] remains a global minimum even if hA ~ ~+, as we shall consider here. A detai led discussion of CP violation and the minimizat ion of the potent ial is given in ref. [13]. The condi t ion reduces to

AAAh > 3hv'v2 mh + AxAa. (6) X

If we now demand that the charged Higgs have nonnegative squared-mass we get another condit ion

A , > A x + )(2h 2 - g2) vl v2/hx. (7)

There are other condit ions imposed by the Higgs sector before we can start choosing values for parameters. The requirement that the physical scalars have nonnegative squared-mass leads to five condit ions on the parameters of the model. From our numerical work it appears that the most critical of these is that the determinant of the scalar mass matrix be positive. For a given set of values for h, A, Ah, AA and tan fl we can solve d e t ( M 2) = 0 and use eqs. (6) and (7) to find the range of x in which these three condit ions are met. In calculating cross-sections we checked at each point that the Higgs masses were all positive, thus ensuring that the other condi t ions were met. The Higgs mass matrices are given in ref. [5].

We now turn to the neutral ino sector of the model. As in the MSSM there are the two neutral gauginos I~ 3 and /~, and the two neutral higgsinos /4, and /42. There is in addi t ion the singlet fermion N. In this basis the

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mass matrix is given by

[ M2 0 -(1/,,,~)g2v, (1/4~)g~v~ ~1 M , ( 1 / v / 2 ) g l / ) , - (1 /x /2)g ,v 2

[-(1/,/2)g2v, (1/x/2)g,v, 0 hx hv2

~ (1/x/2)g2v2 -(1/x/2)g, v2 hx 0 hv, I \ o 0 hi) 2 hv~ 2Ax]

(8)

where M1 and ME are Majorana masses for t he /} and ~ 3 fields. As usual, upon grand unification one expects M1 = ~ tan20w ME. We shall assume this relation here, al though our main results do not crucially depend on it. We take the lightest mass eigenstate X to be the lightest supersymmetric particle (LSP), which is thus stable, since a variety of experiments exclude other candidate LSPs in the mass range of interest here.

To calculate relic densities we fix h, A, Ah, AA and tan/3 (but note that the neutralino masses and composi t ions are independent of A h and AA) and then cycle through points in the ME, hx plane. At each point we diagonalise the neutral ino mass matrix to find the mass and composi t ion of the LSP. It is convenient to write

~?0 = . a,3 +/3# + ~#o + 8#0 + ~ , , (9)

where e should not be confused with the superpotent ial term that we d ropped from eq. (1), nor /3 with tan/3 = vl/v2. We have taken h = g = 2A here. The motivation for these values is in the renormalizat ion group equations (RGEs) , (which were first given for this model in full in ref. [4]) for h and A, which have infra-red stable point solutions. As pointed out in ref. [5], if the value of h or A is of order unity or larger at some grand unification scale, the effective value at the electroweak scale is very close to the infra-red stable point value, h 20 .87 (A-= 0) and A---0.63 (h---0). However, if both h u - A u = O(1) or larger at the unification scale, we find that at the electroweak scale h -~ 2A = g. Note that because the sign of hA is an RGE invariant, it is enough to choose huAu ~ R + to enforce hA ~ R +.

Our results are presented as "pur i ty" contours in the (M2, hx) plane in fig. 1 for h = 2A = g and v2/v2 ~ 2. A more complete survey for other values of these parameters is presented in ref. [13]. We define a vector g = (a,/3, 3,,8, E) with )~-)~=1 and ot=a(M2,hx) , f l=f l (M2, hx), etc. We then define the project ion of )~ onto a s ta te /3 ( /3. /3 = 1) of constant composi t ion coefficients, p =- )~./3, and refer to )~ as a / 3 of purity p. Fig. 1

~ I I I I I I I I I I I [ I I I 10000 y//.F//6.-d. ~ | 7 I ! I ! 16~,~/20 50 ! 100i 50£I 250(;I

3000 ~ ' ' '

~ 300 X

e -

30 : :

10 ' 10 30 100 300 1000 3000 10000

M 2 (GeV)

Fig. 1. Mass contours (given by dashed lines, at 10, 20, 50, 100, 500, 2500 GeV) and 99% purity contours for the lightest neutralino, for the case h = 2A = g.

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Volume 245, n u m b e r 3, 4 PHYSICS LETTERS B 16 August 1990

~/= (sin 0w, cos 0w, 0, 0, 0),

,4 = (0, 0, sin/3, - cos/3, 0),

as well as the new states

shows X mass contours and purity contours inside which X is at least a 99% pure state. The pure states of

impor tance are

/~ = (0, 1, 0, 0, 0), l&" = (1, 0, 0, 0, 0),

*o = (O, O, l , l , O)/v/~, HE,2]_ H(12 ) ^o _ (0, O, 1 , - 1, 0)/,,/2,

= (0, O, cos 3, sin/3, 0),

~r = (0, O, O, O, 1) (10)

D = (0, 0, sin fl, cos f l , - sign(x)x/sin/3 cos/3 h / A ) , (11)

= (0, 0, I, 1,_ 1/4~). (12)

(Note that D is not a unit vector. The normal ised expression is somewhat more messy.) The region where the LSP is a near ly pure ,~o is ruled out by the presence of a light (<45 GeV) chargino. The bino and phot ino are in roughly the same posi t ion as in the minimal model (remember, our hx is equivalent to E there) and with roughly the same masses; however, what was a symmetric or ant isymmetric higgsino/412 in the minimal model has now become two states. One of t h e m , / 9 is a light state with m x < 65 GeV. We concentrate on this state in looking for dark matter candidates .

Before calculating relic densities we have to choose values for Ah and Aa. For a given pair of values we can calculate the range in x for which our results will be valid from the condit ions that the physical Higgs must have nonnegative squared-masses. For this case we chose A h = 430 GeV and A a = 240 GeV. These values are not the most natural choice (the renormalisat ion group equations imply that A h ~ 2Aa); however, they give a range in hx which most nearly covers t h e / ) region. Varying these figures affects the exact slice of the plane in which our results are valid; however, values o f / 2 h 2 appear to vary slowly. We will study the effect of adjust ing these parameters in more detail in a later paper [13].

We calculate next the X annihi la t ion cross-section into fermion-ant i fe rmion pairs for )t" = / 9 in order to determine its relic density, and hence the cosmological ly al lowed region in fig. 1. (The expression below is actually valid for any gaugino-less neutralino.) Using the thermal averaging technique of ref. [14] to first order in x we compute a and b, where (~vro~)= a + bx, with x = T/mx , T is the photon temperature and m x the LSP mass. We present below (in the limit a =/3 = 0 ) the expressions for a and for terms in b due to Z ° and pseudosca la r Higgs exchange (including interference terms), since these are the dominant contr ibutions; in our numerical work we included all other terms in b, due to scalar and sfermion exchange:

1 2 m2 / m 2 1 f ( y / s i n / 3 ) 2) , 1

/ ~ g 2 - ~ ~(6/cos/3)2~ -~g2 mw a = lx/~S_~f ~ g 2 f I i f T3(,~2 _ t~2)

m w \ m w

+,/~mx ~C°t /3 } ~ ri[ (hy6+ Ae2)si + h( ~c° s /3 + y sin /3 )er,]) 2, t t an /3 ~=~ 4 m ~ - m ~ ,

1 4 mE 1 (y2_62)2[(l+½1~f)(T3_2eysin20)2+(l_sCf)T2] ' bzz = 4 1 - sol 1 ~ g2 m 4 ( 1 - 4 ~ / ) 2

2 m2xm} [ cot2 /3 } ( 2 r,[ ( hy6 + Ae2)s, + h( 6 cos /3 + y sin /3 )er,] 2 bpp = x/1 - ¢y 4zr g2 m 2 [ t an 2/3 ,~, 4rn2x - m~,

- 4 m 2 ~ r ' [ ( h y ~ + a ' 2 ) s i + h ( t S c ° s / 3 + y s i n / 3 ) ' r l ] ~ r j [ ( h Y 6 + Z ' 2 ) s j + h ( 6 c ° s / 3 + y s i n / 3 ) ' r J ] ) (13) 2 2 (4m2x _ m2j)2 , i=~ 4m x -- m p~ j=l

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Volume 245, number 3, 4 PHYSICS LETI~ERS B 16 August 1990

3 3m}mx(y2_82)T31co t f l } ~ ri[(h78+Ae2)si+h(Scosfl+ysin~)er,] bzp = lx / ] -~ - ~y 8---~-~ g2 ~ ( t an /3 ,=, (4m2x - m2) 2

where the upper (lower) factors are for decay into up (down) type fermions, A = m } - rex2 _ rn I~ and other parameters are defined through

H° = v ' + ~ 2 ( ,= , ~ x'S°+i cos fl ,=,Y" r,P °),

N = X + ~ i ° + i i = 1 S i e ~ '

1 3 ) H°=v2+=( Y~ y ,S°+is in f l ~ riP~i 42 \~=l i=, '

m2p, (13 cont 'd)

= my/mx , r I = rex~ mz, ~ f 2 2 2 2

(14)

where the S O and the po are the three scalar and two pseudoscalar physical Higgs particles respectively. Note that according to the prescript ion of ref. [14] there is a further term to add to the p-wave part:

b=bzz+bzp+bpp+(otherterms)+(-3+ 3- #Y )a. (15) 4 1 - s c f

To get the final values for a and b we then sum over all fermions with mass less than that of the LSP:

atota,='~, a, btota,=~ b. (16) f f

From the thermally averaged cross-section we then calculate /2h 2 (where 1"2 is the mass density in LSPs divided by the critical density and h is the Hubble parameter in units of 100 km s - ' Mpc -x) by numerical ly integrating the Boltzmann equation.

Fig. 2 shows contours o f / 2 h 2 - x ( - ~ , r6, ], 1) for the region in which the LSP is a nearly pure /~ . The hatched areas at the top and bot tom of the wedge (hx> 168 GeV and h x < 7 8 GeV respectively) are regions in which our chosen vacuum is not a minimum of the Higgs potential (i.e. one of the scalars has negative squared-mass). In the region where the /2h 2 = ~ line is dashed one of the pseudoscalars has a mass about twice that of the LSP and the thermal expansion here is inval idated by divergence of the pseudoscalar propagator . The central part of the wedge is e l iminated since it gives excessively large relic densities. The lower part has Oh 2 < ~ which is too small to account for dark matter. This leaves a region of the (M2, hx) plane w h e r e / ~ would be the dark matter. In that region we find that rnt~ = 13 to 64 GeV. We are in the process of investigating further regions of the parameter space of this model.

We have shown that in the simplest extension to the minimal supersymmetric s tandard model, there exists a light (m x < 30 GeV) dark matter candidate /~, composed of ff/~, /42 and ~f. Note that this candidate is light

200

150 > 0 (.9 -if- 100 r-- 9 0

8o 70 60 50

I i I I I I I I I I I I I I I I ~ : / . - / / / / / / / / . / / / / / / / / / / / / / . / / / / / ~

" " " " \ \ \ "- \ " \ \ ",:

~ . ¢ , - ¢ , . ~ L Z / / / / / / / / / / / / / / / / / / z # . . . . . . " . . . . .

, , , I , , I [ I I , , , L I , I 300 1000 3000 10000

M2(GeV) Fig. 2. Contours of/2h 2 (=~, ~6, 4 t, 1) for the region of the hx, M 2 plane in which the lightest neutralino is a 99% pure /9, for h = 2A = g, A h = 430 GeV and A A = 240 GeV. The /~ mass in this region varies from 0 to 73 GeV.

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Volume 245, number 3, 4 PHYSICS LETTERS B 16 August 1990

wi thou t requi r ing small mass paramete rs w h e n h - A ~ g. The ent ire pa rame te r space a d m i t t i n g / 9 as the LSP

has a cosmolog ica l ly re levant rel ic a b u n d a n c e which varies f rom g2h 2 = & to 3. The m o d e l also has a s l ightly

e n h a n c e d possibi l i ty for the pho t ino as the LSP over the min ima l mode l , and as in the m i n i m a l mode l the b ino

remains a v iable (massive) dark mat te r candida te . F ina l ly we stress again that such an ex tens ion to the M S S M

w o u l d be necessary i f e i ther no scalar Higgs boson is f ound with m , 2 < m z or if there exists a charged Higgs

boson with m , + < row.

We wou ld like to thank M. Srednicki for impor t an t communica t ions . This work was suppor t ed in par t by

D O E grant DE-83ER-AC02-40105 , and by K.A.O.s Pres ident ia l Young Inves t iga tor Award .

References

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[3] H.P. Nilles, M. Srednicki and D. Wyler, Phys. Lett. B 120 (1983) 346; J.M. Frere, D.R.T. Jones and S. Raby, Nucl. Phys. B 222 (1983) 11.

[4] J.P. Derendinger and C.A. Savoy, Nucl. Phys. B 237 (1984) 307. I-5] J. Ellis, J.F. Gunion, H.E. Haber, L. Roszkowski and F. Zwirner, Phys. Rev. D 39 (1989) 844. 1-6] ALEPH Collab., D. Decamp et al., Phys. Lett. B 241 (1990) 141. [7] M. Dine, V. Kaplunovsky, M. Mangano, C. Nappi and N. Seiberg, Nucl. Phys. B 259 (1985) 549. [8] L. Hall, J. Lykken and S. Weinberg, Phys. Rev. D 27 (1983) 2359;

J. E. Kim and H.P. Nilles, Phys. Lett. B 138 (1984) 150; G.F. Giudice and A. Masiero, Phys. Lett. B 206 (1988) 480.

[9] J. Ellis, J.S. Hagelin, D.V. Nanopoulos, K. Olive and M. Srednicki, Nucl. Phys. B 238 (1984) 453. [10] K. Griest, Phys. Rev. D 38 (1988) 2357; D 39 (1989) 3802 (E). [11] K.A. Olive and M. Srednicki, Phys. Lett. B 230 (1989) 78; University of Minnesota preprint UMN-TH-805/90 (1990). [12] K. Griest, M. Kamionkowski and M.S. Turner, Fermilab preprint 89/239 (1989). [13] R. Flores, K.A. Olive and D. Thomas, in preparation (1990);

D. Thomas, in preparation (1990). [14] M. Srednicki, R. Watkins and K.A. Olive, Nucl. Phys. B 310 (1988) 693.

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