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Coannihilation Effects and Relic Abundance

arXiv:hep-ph/9208251v1 26 Aug 1992TU-409July, 1992Coannihilation Effects and Relic Abundanceof Higgsino-Dominant LSPsSatoshi MizutaDepartment of Physics, Tohoku University, Sendai 980, JapanandMasahiro YamaguchiDepartment of Physics, College of General EducationTohoku University, Sendai 980, JapanAbstractWe calculate the relic abundance of Higgsino-dominant lightest su-perparticles, taking account of coannihilations with the superparticleswhich are almost degenerate with the lightest one. We show that theirrelic abundance is reduced drastically by the coannihilation processesand hence they are cosmologically of no interest.

The lightest superparticle (LSP) in supersymmetric (SUSY) standardmodels, mostly taken to be a neutralino, is an intriguing candidate for darkmatter [1, 2, 3]. An important task is therefore to compute the relic abun-dance of the LSPs, which is conveniently represented by Ωχh2 where Ωχ isthe ratio of the mass density of the LSPs to the critical one to close theUniverse and h (0.4<∼h<∼1) stands for the Hubble constant in units of100 km sec−1Mpc−1.This issue has been discussed extensively in the literature [4, 5, 6, 7],assuming (1) the minimal particle content of the SUSY standard model, (2)a GUT relation on gaugino mass parameters (see below), (3) the neutralinoLSP and (4) R-parity conservation which guarantees the stability of the LSP.The abundance of the neutralino LSPs crucially depends not only on theirmass but also on the composition of them, since the LSP is a linear com-bination of neutral gauginos, ˜B, ˜W3, and neutral Higgsinos ˜H1, ˜H2, andeach of them has different pair annihilation processes which determine itsabundance.The results obtained previously are summarized as follows: let us firstconsider the case where the LSP is lighter than the W-boson.

If it is almosta pure gaugino or a pure Higgsino, its density can reach (or even exceed)the critical one and hence it is a good candidate for the dark matter of theUniverse. If the LSP is an admixture of the gauginos and the Higgsinos, therelic abundance is generally small.

On the contrary, in the case where theLSP is heavier than the W-boson, the annihilation mode to a W-pair opensif it is a pure Higgsino or a mixed LSP and the relic abundance is very smalluntil the mass increases to a TeV region where Ωχh2 of the LSPs becomesagain of order unity. If the LSP is a gaugino, there is no annihilation process1

to the W-pair and hence it is still abundant in the Universe. If its massexceeds ∼500 (GeV), it will overclose the Universe and such a parameterregion is excluded.The previous calculations have, however, included only annihilationsof the LSP pairs.

Griest and Seckel [8] have pointed out when next-to-the-lightest superparticles (NSPs) are slightly heavier than the LSP, this naivetreatment fails to give a correct answer. Namely, the abundance of the NSPsis comparable to that of the LSPs around the freeze-out epoch so that we haveto take account of annihilation processes involving the NSPs, which they havecalled coannihilations.

In ref. [8], they have considered a rather accidentalcase where squarks are degenerate in mass with the LSP, and showed that therelic abundance of the LSPs is greatly reduced due to the coannihilations ofthe squarks.

We should stress, however, the mass degeneracy rather naturallyoccurs for a Higgsino-dominant LSP [7, 9]. This is because if the electroweakgauge symmetry were not broken, all Higgsino states would be degenerate inmass.

In the broken phase, the mass splitting comes from the mixing withgauginos, which are small in the Higgsino-dominant LSP region. Thereforethe coannihilation effects will be important in the region.In this paper, we will evaluate the relic abundance of the neutralinoLSPs taking account of the coannihilation processes.

We consider only theprocesses where neutralinos and charginos pair-annihilate into a fermion pair.Since their effects are particularly important for the LSP weighing less thanthe W-boson, we will concentrate on this case. In a region where the Higgsinocomponent dominates the LSP, coannihilations of the LSP and the NSPs canbe mediated by Z- and W-boson exchange.

We will show in this region, thecoannihilations drastically change the previous conclusions: the mass density2

of them becomes very small, typically Ωχh2 being much less than 10−2.The mass matrix for the four neutralinos is given byM10−mZ sin θW cos βmZ sin θW sin β0M2mZ cos θW cos β−mZ cos θW sin β−mZ sin θW cos βmZ cos θW cos β0−µmZ sin θW sin β−mZ cos θW sin β−µ0(1)in the ( ˜B, ˜W3, ˜H1, ˜H2)T basis. Here M1 and M2 are the mass parameters of˜B and ˜W3, respectively, µ is the supersymmetric Higgs mass parameter andtan β represents the ratio of the expectation values of the two Higgs bosons[10].

As is done in most of the literature, we will impose the GUT relationfor the gaugino mass parametersM1 = 53 tan2 θWM2 ≈0.51M2,(2)which is derived by solving one-loop renormalization-group equations withM1 = M2 at the GUT scale. As a phase convention, we take M2 > 0 andµ both positive and negative.

It is straightforward to diagonalize the massmatrix. The lightest of the neutralinos denoted by χ1, which is also assumedto be the LSP, isχ1 = Z11 ˜B + Z12 ˜W3 + Z13 ˜H1 + Z14 ˜H2,4Xi=1Z 21i = 1.

(3)Using Z1i, we define the Higgsino purity asp = Z 213 + Z 214. (4)When p is nearly one, say p>∼0.99, χ1 is almost a pure Higgsino, whichis realized if M2 ≫|µ|.

If M2 ≪|µ| the LSP is gaugino-like, whilst forM2 ≈|µ| ≈mZ it is a general mixture of the four neutralinos.3

In fig. 1, we have plotted the Higgsino purity of the LSP, p, defined inEq.

(4) with tan β = 2 fixed. In this paper, we will call the LSP (almost)a pure Higgsino if p>∼0.99, and a Higgsino-dominant LSP if p>∼0.9.The region where 0.99>∼p>∼0.9 will be called the Higgsino-dominantmixed region.

Fig. 2 shows the ratio of the mass difference between the LSPand the NSP (which is a chargino or a neutralino) to the LSP mass: ∆=(mχ2 −mχ1)/mχ1 where mχ1 and mχ2 are the masses of the LSP and NSP,respectively.

We can see the severe mass-degeneracy in the pure HiggsinoLSP region. In most of the parameter space, the NSP is the chargino.

Thisis also observed when we take a different value for tan β.In calculating the relic density including the effects of the coannihi-lations, we use the method developed by Griest and Seckel [8].Here wesummarize it briefly. Let χi (i = 1, · · ·, N)1 be superparticles with mass mχi(mχ1 < · · · < mχN) and suppose that they are nearly degenerate in masswith the LSP, χ1.

Due to the mass degeneracy the number density ni of thei-th particle (i > 1), which eventually decays to the LSP, is comparable withn1 around the freeze-out epoch. The Boltzmann equation for the total of thenumber densities n = Pi ni is writtendndt = −3Hn −Xi,j⟨σijvrel⟩(ninj −neqi neqj ),(5)where σij is the pair annihilation cross section of the particles χi and χj,vrel is their relative velocity, ⟨· · ·⟩denotes the thermal average and neqiisthe number density of χi in thermal equilibrium.

Since the reactions whichinterchange the superparticles χi’s with each other occur much more rapidlythan their annihilations, the ratio of χi density ni to the total density n is1 Note that a particle and its anti-particle should be counted separately when theyconstitute a Dirac fermion.4

well approximated by its equilibrium value: ni/n ≈neqi /neq. This greatlysimplifies the Boltzmann equation asdndt = −3Hn −⟨σeffvrel⟩(n2 −(neq)2).

(6)Therefore, we can solve eq. (6) by the standard method [11] using the effectivecross section defined byσeff =Xi,jσijrirj,(7)where ri represents the Boltzmann suppression of the density of the heavierparticle χi.

Explicitlyri=neqineq =gi(1 + ∆i)3/2e−∆ixPj gj(1 + ∆j)3/2e−∆jx ∝e−∆ix,(8)∆i=(mχi −mχ1)/mχ1,(9)where gi is the degree of freedom of the χi and x = mχ1/T with T being thephoton temperature.The relic abundance of the LSPs at the present day can be calculatedasΩχh2 =1.07 × 109GeV−1g1/2∗mP lR ∞xf ⟨σeffvrel⟩x−2dx,(10)where xf = mχ1/Tf, Tf is the freeze-out temperature, g∗is the effectivedegree of freedom at the freeze-out epoch [11] and mP l denotes the Planckmass, 1.22 × 1019 (GeV). In eq.

(10), we can see the relic density is roughlyproportional to the inverse of the effective cross section.The LSPs freeze out at xf ∼20, so when ∆i (i > 1) is less than about0.1 the coannihilation effects are in general important. When the magnitudesof the annihilation cross sections involving the heavier particles are similarto that of the LSP pair, the coannihilation effects change the relic densityat most by several factors.

On the other hand, if some cross sections with5

the heavier particles are much larger than that of the LSP pair, the relicabundance can be reduced by several orders of magnitude. It will turn outthat this is the case when the LSP is Higgsino-dominant.Before giving our numerical result on the relic density, it is illustrativeto give a crude estimate for the effects of the coannihilations in the degeneratelimit M2 ≫|µ| (see eq.

(1)). In this limit, the lightest neutralino is nearlyeither of the following states:˜HS,A = 1√2( ˜H1 ± ˜H2).

(11)The coupling of (neutralino)-(neutralino)-(Z-boson) is given14(g/ cos θW)Zµ( ¯˜H1γµγ5 ˜H1 −¯˜H2γµγ5 ˜H2)= 12(g/ cos θW)Zµ ¯˜HAγµγ5 ˜HS,(12)while that of (neutralino)-(chargino)-(W-boson) isg2W −µ ( ¯˜HAγµ ˜H+ + ¯˜HSγµγ5 ˜H+) + h.c.,(13)where g is the SU(2)L gauge coupling constant. Eq.

(12) implies that ina region of the Higgsino-dominant LSP, the coupling of the LSPs to the Z-boson is very suppressed: it is proportional to Z 213 −Z 214 ∼m2Z/(M2µ), whichvanishes at M2 →∞limit.The annihilation of the LSP pair to fermions is dominated by this sup-pressed Z-boson exchange process, since the couplings of the other processes,i.e. the Higgs boson exchange and the sfermion exchange, are even smaller.Moreover the s-wave annihilation to a fermion pair is suppressed as the initialstate is a pair of the identical Majorana fermions.

Therefore the authors ofthe previous papers have concluded that the Higgsino-dominant LSPs whichare lighter than the W-boson are (too) rich in the Universe.6

The reality is, however, they disappear through the coannihilations. Tosee the coannihilation effects, consider the coannihilation process ˜HS,A ˜H± →(fermions) mediated by the W-boson exchange in the s-channel.

The crosssection for this process is much larger than that of the annihilation of theLSP pair discussed above, because there is no suppression factor such asZ 213 −Z 214 in the couplings to the weak boson and the annihilation occurs inthe s-wave. Indeed it is estimated as⟨σvrel⟩∼9g416πm2Wf(µ) ∼5f(µ) × 10−6GeV−2(14)with f(µ) = (µ/mW)2/(4(µ/mW)2 −1)2.

The contribution to the effectivecross section is therefore given by⟨σvrel⟩rC ∼5f(µ)rC × 10−6GeV−2,(15)where rC ∼e−∆Cxf ∼e−20∆C. ∆C and rC represent the mass degeneracy andthe Boltzmann suppression factor for the chargino, respectively (see eqs.

(9)and (8)). A quite similar estimate can be obtained for the cross section withthe second lightest neutralino, which is in the same order of the magnitude.Since, as we mentioned above, the chargino is the NSP in a large part of theparameter space, the effective cross section is dominated by eq.

(15), whichyields the estimate of the relic abundanceΩχh2 ∼5 × 10−10GeV−2⟨σeffvrel⟩∼10−4rCf(µ). (16)Therefore, we can expect that for ∆C<∼0.2 (i.e.

rC>∼10−2), the relicabundance of the LSPs will become very small due to the coannihilations.It is tedious but straightforward to calculate the effective cross sectionaccurately for a general set of the parameters (µ, M2). In fig.

3, we show theabundance of the neutralino LSPs, including the coannihilation processes.7

We have chosen tan β = 2, all squark and slepton masses equal to 1 TeVand the pseudoscalar mass mA = 1 TeV. Since the coannihilations occurmainly through the Z- or W-exchange, their effects do not depend on thechoice of the masses of the sfermions and the pseudoscalar.

We can obtainqualitatively a similar plot for a different value of tan β. For comparison,we have given in fig.

4, the same plot but without taking account of thecoannihilation processes. By comparing the two figures, the importance ofthe coannihilations is manifest.

In the pure Higgsino region with the purityp>∼0.99, the mass degeneracy is severe, i.e. ∆<∼0.1, and the coannihi-lations greatly reduce the neutralino density.

In most of the region, we cansee Ωχh2 ≪10−2. Thus the LSPs cannot constitute any form of the darkmatter, i.e.

not only the dark matter of the universe but also the dark matterof the galactic halos. This contrasts with the conclusion obtained previouslywhere the density of the light Higgsinos can reach the critical one.

Evenfor the Higgsino dominant-mixed region (0.9<∼p<∼0.99) where the massdegeneracy is not so severe, the coannihilation effects are significant. Theyreduce the density with an order of magnitude or more.In this paper, we have considered the coannihilation effects to calculatethe present abundance of the neutralino LSPs whose mass is less than thatof the W-boson.

We have shown that in the Higgsino-dominant LSP regionthe relic density gets very small. On the other hand, as mentioned above fora Higgsino-dominant LSP heavier than the W-boson, the annihilation to theW-pair is kinematically allowed and Ωχh2 is much smaller than order unityuntil the LSP mass reaches the TeV region.

Combining these two results,we can conclude that the Higgsino-dominant LSP is no longer an interestingcandidate for the dark matter.8

We would like to make two comments: firstly our calculation does notcontain annihilation processes whose final state is other than a fermion pair.When Higgs bosons are light, the LSPs and NSPs will annihilate to them sig-nificantly. If this is the case, the relic density becomes even smaller.

Moreoverwe may consider a process such as˜HS,A ˜H± →W ± →γW ±,(17)which would enhance the effective cross section by several factors. Althoughit will not change our result drastically, a more detailed and accurate analysisis significant [12].

Secondly there are other cases where the coannihilationsare important. For example, when the GUT relation for the gaugino masses(2) is relaxed, the neutral Wino can become the LSP [9].

Since it is highlydegenerate in mass with its charged counterparts, the coannihilations involv-ing them dominate the effective annihilation cross section and hence theirrelic abundance is very small.This issue will be discussed in a separatepublication [13].We would like to thank T. Yanagida for careful reading of the manuscriptand useful comments. We are also grateful to H. Murayama and T. Yanagidafor discussions.9

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[13] S. Mizuta, D. Ng and M. Yamaguchi, Tohoku University preprint TU-410 (August 1992).10

Figure captionsFig. 1.

The Higgsino purity of the LSP, p ,defined by p = Z213+Z214; (a)µ > 0and (b)µ < 0. We have chosen tan β = 2.

“LEP” means the excludedregion by LEP constraints and “Mχ > MW” means the region wherethe lightest neutralino is heavier than W-boson with which we are notconcerned.Fig. 2.

The ratio of the mass difference between the LSP and the NSP tothe LSP mass, ∆,defined by ∆= (mχ2 −mχ1)/mχ1; (a) µ > 0 and(b) µ < 0. We have chosen tan β = 2.

The meanings of “LEP” and“Mχ > MW” are the same as in fig. 1.Fig.

3. The relic abundance of the neutralino LSPs, including the coanni-hilation processes; (a) µ > 0 and (b) µ < 0.

We have chosen tan β = 2,all squark and slepton masses equal to 1TeV and the pseudoscalar massmA = 1TeV. Note that the coannihilation effects are independent of thesquark and slepton masses and the pseudoscalar mass.

The meaningsof “LEP” and “Mχ > MW” are the same as in fig. 1.

By comparingthis with fig. 4, we can find that the coannihilations greatly reducethe relic abundance of the LSPs in the Higgsino-dominant LSP region(M2 ≫|µ|).Fig.

4. Same as fig.

3 but without taking account of the coannihilationprocesses.11

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