Right handed Sneutrinos as Nonthermal Dark Matter

Transcription

1 ESI The Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria Right handed Sneutrinos as Nonthermal Dark Matter Shrihari Gopalakrishna André de Gouvêa Werner Porod Vienna, Preprint ESI 1788 (006) March 0, 006 Supported by the Austrian Federal Ministry of Education, Science and Culture Available via

2 IFIC/06-01 NUHEP-TH/06-01 Right-handed Sneutrinos as Nonthermal Dark Matter Shrihari Gopalakrishna and André de Gouvêa Northwestern University Department of Physics & Astronomy 145 Sheridan Road, Evanston, IL 6008, USA. Werner Porod IFIC - Instituto de Fisica Corpuscular, CSIC E Valencia, Spain. March 4, 006 Abstract When the minimal supersymmetric standard model is augmented by three righthanded neutrino superfields, one generically predicts that the neutrinos acquire Majorana masses. We postulate that all supersymmetry (SUSY) breaking masses as well as the Majorana masses of the right-handed neutrinos are around the electroweak scale and, motivated by the smallness of neutrino masses, assume that the lightest supersymmetric particle (LSP) is an almost-pure right-handed sneutrino. We discuss the conditions under which this LSP is a successful dark matter candidate. In general, such an LSP has to be nonthermal in order not to overclose the universe, and we find the conditions under which this is indeed the case by comparing the Hubble expansion rate with the rates of the relevant thermalizing processes, including self-annihilation and co-annihilation with other SUSY and standard model particles. 1 Introduction Neutrino oscillation experiments have revealed that neutrino masses are nonzero [1]. A renormalizable extension of the standard model (SM) that incorporates neutrino masses can be obtained by introducing right-handed neutrinos N R (at least two) which, in conjunction with the left-handed neutrinos ν L of the SM allow neutrino mass terms. If the masses are assumed to be of the same order of magnitude as the mass differences, this mass scale is of the order of about 0.1 ev. Various proposals have been made for explaining such a tiny mass scale, each implying a certain magnitude for the Yukawa coupling Y N []. Supersymmetry (SUSY) [3] stabilizes the electroweak scale in which the quadratic divergences in the Higgs sector due to SM quantum loops are canceled by contributions from new particles of opposite statistics. In particular, in the neutrino sector, a supersymmetric extension of the SM implies the addition (for each generation) of two new complex scalar fields - 1

3 the left-handed sneutrino ν L, and the right-handed sneutrino Ñ R - as partners of the left- and right-handed neutrinos respectively. N R and Ñ R are gauge singlets under the SM gauge group, and interact with other particles only through the Yukawa coupling. It has become evident in recent years through cosmological and astrophysical observations that there is a significant amount of non-baryonic dark matter. If R-parity (R p ) is conserved, SUSY provides a stable dark matter candidate - the lightest supersymmetric particle (LSP) 1. If SUSY is realized in nature, it must be broken, and since we do not know the exact mechanism responsible for SUSY breaking, we will parametrize this through soft SUSY breaking terms in an effective Lagrangian at the electroweak scale. SUSY breaking and electroweak symmetry breaking lead to mixing between the ν L and ÑR, and we denote the lightest such state as ν 0. We will, henceforth, explore scenarios where ν 0 is the LSP. In this work we explore such a dark matter candidate, when all SUSY breaking masses are of the order of the electroweak scale. If the ν 0 has a significant component of ν L which interacts through the electroweak coupling constants g and g, the interaction strength is big enough for it to be kept in thermal equilibrium in the early universe. The final relic abundance of such a thermal species depends on when it freezes out, varying inversely as the interaction cross-section. This scenario has been studied in Refs. [5, 6, 7, 8]. Generally, a predominantly left-handed LSP with mass around 100 GeV annihilates too efficiently in the early universe resulting in too low a relic-density today, and reducing the annihilation cross-section by increasing the LSP mass runs into conflict with direct detection experiments. This leaves the possibility of a ν L -ÑR mixed sneutrino as a viable dark matter candidate [7]. Another possibility exists, as we will argue here: (i) if the ν 0 is almost purely Ñ R, and (ii) if Y N is sufficiently small, the interaction cross-section is too small for the ν 0 to ever be thermalized, and the above analyses do not apply. The question then arises if such a ν 0 can be nonthermal dark matter, and we investigate in this work under what conditions this can be the case. The outline of the paper is as follows: In Sec. we write down the most general R p conserving renormalizable superpotential (focusing on the lepton sector), and the most general renormalizable SUSY breaking terms, including terms that violate lepton-number. We write down the sneutrino mass matrix and derive the ν L -ÑR mixing relations. We impose the constraint from neutrino masses, and specify our choice of parameters. In Sec. 3 we write down the Boltzmann equation for the ν 0 number-density, and obtain conditions that have to be met in order for the ν 0 to be thermal. For this we include self-annihilation processes of the ν 0, annihilation with other SUSY particles, and annihilation with SM particles. In Sec. 4 we point out in what cases the above conditions are not met, and thereby identify when the ν 0 can be nonthermal dark matter. We offer our conclusions in Sec. 5. In App. A, we review in general the standard computation of the relic-density of a species that was in thermal equilibrium in the early universe, and apply it to the relic-density calculation of the well-studied mixed sneutrino case in App. B. Even if the ν 0 is never in thermal equilibrium, a ν 0 relic-density could result from the decays of heavier thermal SUSY particles; we compute this relic-density in App. C. 1 In the case of R-parity violation the dark matter constraint can also be satisfied, e.g. by a very light and, thus, very long-lived gravitino LSP [4].

4 The Model To the field content of the MSSM, we add (for each generation) the right-handed neutrino superfield N = (ÑR, N, F N ). Written as left-chiral fields, the superfields are: Q, Ûc, D c, L, Êc, N c. As usual, the MSSM Higgs doublet superfields are Ĥu and Ĥd. The most general R p conserving renormalizable superpotential is W = Ûc Y U Q Ĥu D c Y D Q Ĥd + N c Y N L Ĥu Êc Y E L Ĥd + N cm N N c + µĥu Ĥd, (1) where A B denotes the antisymmetric product of the fields A and B, Y are the Yukawa couplings that are 3 3 matrices in generation space, and, M N breaks lepton number. In general the Y s are arbitrary complex matrices, and M N is complex symmetric. However, without loss of generality, we can rotate N c by a unitary matrix to make M N real and diagonal with this rotation matrix absorbed into a redefinition of Y N. The SUSY breaking terms are L SUSY Br = q Lm q q L ũ Rm uũ R d Rm d d R + ( ũ RA u q L h u + d RA d q L h d + h.c.) l Lm l l L Ñ Rm NÑR ẽ Rm eẽ R + ( Ñ RA N ll h u + ẽ RA e ll h d + h.c.) [ + ( l h u ) T c ] l ( l h u ) + ÑR T b N M N Ñ R + h.c. + (bµh u h d + h.c.), () where c l and b N M N break lepton number, and h u and h d are the scalar components of Ĥu and Ĥd respectively. Note that A can be an arbitrary complex matrix, and without loss of generality, m is Hermitian while b N M N and c l are complex symmetric..1 Sneutrino masses The sneutrino mass terms resulting from Eqs. (1) and () are (the generation structure is suppressed) L ν mass = ( ν L M ν = 1 Ñ R ν T L ν L ÑR T ) M ν ÑR ν L Ñ R m LL m RL vu c l v u Y N M N m RL m RR v u MNY T N (b N M N ) vuc l v u YN T MN m LL m T RL v u M NY N b N M N m RL m RR, (3) where m LL = (m l+vuy NY N + ν), m RR = (M N MN+m N+vuY N Y N), m RL = ( µ v d Y N +v u A N ), and ν = (m Z /)cos β is the D-term contribution. Based on the ν mass matrix, we make the following observations: m RL = ( µ v d Y N + v u A N ) results in ν L Ñ R mixing. c l breaks lepton number and results in ν L ν L mixing [10, 6]. 3

5 M N breaks lepton number and results in ν L Ñ R mixing. b N M N breaks lepton number and results in ÑR Ñ R mixing. The phases present in Eq. (3) lead to CP violating effects which we do not explore in this work. We can write ν L and ÑR in terms of real fields ν 1, ν, Ñ 1 and Ñ as ν L = ( ν 1 + i ν )/, Ñ R = (Ñ 1 + iñ )/. (4) In this basis, if m LL, m RR, m RL, c l, Y N M N, and b N M N are all real, Eq. (3) reduces to block diagonal form [10, 11] as L ν mass = 1 ( νt 1 M r ν = Ñ T 1 ν T ν 1 Ñ T ) M r ν Ñ1 ν Ñ m LL c l m RL T + v uyn TM N 0 0 m RL + v u M NY N m RR b N M N m LL + c l m RL T v u YN T MN.(5) 0 0 m RL v u M NY N m RR + b N M N We can diagonalize Eq. (5) by performing the unitary rotation ( ) νi Ñ i = ( cosθ ν i sinθ ν ) ( sinθ ν i ν ) i i cos θ ν i Ñ i ( ) ( C ν ν ) i i Ñ i, (6) where the mixing angle is given by tan θ ν i = ( m RL ± v u M NY N ) (m LL c l ) (m RR b N M N ), (7) with the top (bottom) sign for i = 1 (i = ). We will use the notation: s i sin(θ ν i ), c i cos(θ ν i ) and t i tan(θ ν i ). We see from Eq. (5) that the lepton-number-violating parameters c l, b N and M N split the ν 1 ν degeneracy, and also the Ñ1 Ñ degeneracy. We denote the lightest of the sneutrino mass eigenstates as ν 0, and its mass M LSP. We will also assume that ν 0 is the lightest supersymmetric particle (LSP), with R p conserved, so that it is a stable particle. For definiteness, we will assume that the LSP is in the upper-left block. We denote the heavier sneutrino mass eigenstates collectively as ν H. The hidden generation structure in Eq. (5) in general implies new flavor changing neutral current (FCNC) contributions, which are experimentally constrained to be small [1]. These contributions are small for instance if the fermion and scalar mass matrices are suitably aligned. Moreover, an alignment at the high (GUT) scale is sufficient, since we expect that renormalization-group loop-induced violations of this alignment to be proportional to Y N, which, as will be discussed shortly, we assume to be very small. 4

6 . Neutrino masses The superpotential in Eq. (1) leads to the neutrino mass terms (N c and ν are -component spinors) L ν mass = N c v u Y N ν N cm N Nc + h.c.. (8) When v u Y N M N, the lowest neutrino mass eigenvalue is given by the standard seesaw relation.3 Choice of parameters m ν = v uy N M N. (9) The masses of the light neutrinos, m ν, are required to be less than about 0.1 ev to correspond to the mass scale inferred from neutrino oscillation experiments. The usual Type I seesaw results if M N GeV and Y N O(1). However, this is not the only possibility; M N could be at the electroweak scale [7, 8, 9], or even at the ev scale [13]. If lepton number is a good symmetry (i.e., if M N = b N = c l = 0), a Dirac neutrino results, and Y N 10 1 in order to have m ν 0.1 ev. We will parameterize A N such that A N = a N Y N m l, (10) where a N is a dimensionless constant. We further assume a N to be order one, i.e., if the neutrino Yukawa couplings turn out to be very small (which is the case here), the neutrino A N also turn out to be significantly smaller than the weak scale. This assumption is automatically satisfied in the constrained MSSM [1] (sometimes referred to as minimal supergravity MSSM). Other SUSY breaking mechanism that yield zero A-terms at a low-enough SUSY-breaking scale would also qualify. In this case, Eq. (7) then implies that s 1 Y N v u m l α m ; α m ( a N + M N µ cot β m l ). (11) In this work we consider the case when all other relevant mass-scales are at the electroweak scale, including the neutrino Majorana mass (M N ). In order to obtain m ν 0.1 ev, we see from Eq. (9) that we require Y N We take the mass of Ñ R to also be around the electroweak scale, i.e., (m RR b N M N ) v (cf. Eq. (5)). For Y N 10 6 and v u, m l 10 GeV, from Eqs. (10) and (11), we have: Y N 10 6 ; A N a N 100 kev ; s α m. (1) Note that, as long as a N is indeed order one (as we assume above), α N is also expected to be of order one (see Eq. 11), and hence s 1 is guaranteed to be very small. Hence, we will be interested in exploring such a predominantly right-handed LSP ( ν 0 ÑR) as a dark matter candidate, with s 1 1 and c

7 3 When is the ν 0 thermal? Depending on the ν 0 interaction strength with itself and other particles, it can either thermalize if the interaction rate is bigger than the Hubble expansion rate, or remain nonthermal if the interactions are too weak. In this section we identify what conditions have to be consistently satisfied for ν 0 to be in thermal equilibrium with the SM thermal bath at some moment in the thermal history of the universe. 3.1 Boltzmann equation The ν 0 number density n ν0 is governed by the Boltzmann equation d ( ) dt n ν 0 = 3Hn ν0 σv SA n ν0 n ν 0 eq σv CA (n ν0 n φ n ν0 eqn φ eq ) + C Γ, (13) where the subscripts SA and CA on the thermally averaged cross-section σv denote selfannihilation and co-annihilation (annihilation with another species φ) respectively, n eq denotes the equilibrium number density, and C Γ is the contribution due to decay of heavier particles into ν 0. C Γ includes the contribution from all heavier SUSY particles that can decay into ν 0. In App. C, we identify such heavier SUSY particles, and, assuming they are in thermal equilibrium, compute the present relic density of the ν 0 by integrating the Boltzmann equation. It turns out that these contributions could contribute too big of a ν 0 relic-density, and have to be dealt with appropriately. In the remainder of this section we will consider the effects of the other terms in the Boltzmann equation. The ν 0 number density will be driven to its equilibrium value provided that the interaction rate is bigger than the Hubble expansion rate, i.e., if σv SA n ν0 > 3H ; σv CA n φ > 3H, (14) for the self- and co-annihilation channels respectively. For annihilation of non-relativistic scalar particles ( ν 0 here), we can make the Taylor expansion [16] σv = a +bv + a + 6bT/M, where the first (second) term is the s-wave (p-wave) contribution. Using Eq. (17) for H, we get ( ) ( ) σv SA n ν0 > g 0.5 T M Pl ; σv CA n φ > g 0.5 T M Pl, (15) as the condition for thermal equilibrium for the self- and co-annihilation channels respectively. Next, we check for various processes whether the inequality Eq. (15) is satisfied for T M LSP ; if we find that equilibrium is not established for T M LSP, it will not be for T M LSP. This is because the cross-section at most goes like σ 1/T, while the number density goes like n T 3, which implies that the left-hand-side of Eq. (15) goes like T while the right-hand-side goes like T. Therefore, if the inequality is not satisfied at T M LSP it will never be satisfied for T M LSP. For this reason we only need to check if the condition is satisfied for T < M LSP. 3. Self-annihilation The dominant ν 0 self-annihilation processes are : The process, ν 0 ν 0 ψψ (where ψ is a SM fermion) due to s-channel Z-boson exchange, is not present since the coupling of a pair of identical scalars to the Z-boson is zero. 6

8 Table 1: Self-annihilation cross-sections and limits. We convert the limit in the third column to an implied limit shown in the last column using Eqs. (10) and (11). Process Cross-section Limit Implied limit ( ) s (a s ) 4 1 g + g s 16π M W M 1 > 10 3 α m Y N > 10 3 ( B ) s (b s ) 4 1 g +g + g +g 16π M ν M M LSP s 1 > 10 α m Y N > 10 H ẽ L (c s ) (d s ) s 1 c 1 Y ψ 16π Y 4 N c4 1 16π A N µ Y N M 4 h 1 M H f PS Y ψ s 1 A N µ Y N f PS > 10 kev Y ψ α N α m Y N f PS > 10 T M LSP Y N > Y N > (a s ) ν 0 ν 0 ν L ν L via t-channel exchange of (Majorana) W 3 and B. (b s ) ν 0 ν 0 ZZ ; W + W via t-channel exchange of ν H ; l. (c s ) ν 0 ν 0 ψψ (where ψ = (c, t, b)) via s-channel exchange of h u ; h d. 3 (d s ) ν 0 ν 0 ν L ν L ; e L e L via t-channel exchange of H + u ; H0 u. In Table 1 we show the cross-sections for the above processes for T < M LSP, under the simplifying assumption that the mass of the exchanged particle is much bigger than T. Y ψ is the Yukawa coupling of ψ. The limit shown in the table is got from the inequality Eq. (15), and if the limit is satisfied, that particular process can thermalize ν 0. The inequality is evaluated at T M LSP, with n ν0 T 3 MLSP 3, with all SUSY masses taken to be around 100 GeV, and for c 1 1. In process (d s ), we include the factor (T/M LSP ) due to the p-wave annihilation, and ignore the s-wave contribution that is helicity-suppressed by the factor (m e /M LSP ) 10 4 (for the τ lepton). In deriving the limit for process (c s ) we assume M hu M hd M h, and make use of the definition α N (a N µ /m l ). For large tan β, the bottom final-state can be important since Y ψ = Y b can be O(1). f PS denotes the phase-space factor which can be a severe suppression for the top final-state. For processes that limit parameters other than Y N, we show in the last column of Table 1 implied limits on Y N, assuming Eqs. (10) and (11). Among all the self-annihilation processes, (d s ) leads to the strongest limit Y N > Therefore, for the case of interest, Eq. (1), we infer that self-annihilation processes are not effective in thermalizing ν Co-annihilation with SUSY particles The ν 0 can thermalize by co-annihilation with other SUSY particles through processes such as (ψ denotes a SM fermion, and s denotes SUSY particles other than ν): (a c ) ν 0 ν H ψψ via s-channel exchange of Z-boson. (b c ) ν 0 s ẽ L s via t-channel exchange of W ±. (c c ) ν 0 t R e Rb L via t-channel exchange of H ±. 3 Associated with channels involving h u, h d exchange, there are analogous channels with a longitudinal gaugeboson (Goldstone boson) exchange. We do not explicitly show these since their contributions are of the same order of magnitude as the h u, h d contributions discussed. 7

9 Table : Co-annihilation with SUSY - cross-sections and limits. We convert the limit in the third column to an implied limit shown in the last column using Eqs. (10) and (11). Process Cross-section Limit Implied limit (a c ) (b c ) (c c ) (d c ) (e c ) (g +g ) s 1 c 1 16π g 4 s 1 MLSP 16π MZ 4 Y e Y t s 1 16π Y N c 1 Y t 16π (c 1 s 1 ) Y ψ 16π 1 M H 1 M Z T M LSP ζ νh s 1 > ζ νh α m Y N > f PS ζ s s 1 f PS > ζ s α m Y N f PS > µ M 4 H ζ t R s 1 > 10 5 ζ t R α m Y N > 10 5 T M LSP ζ t R Y N > ζ t R Y N > A N µ Y N f Mh 4 PS ζ νh Y ψ A N µ Y N f PS > 10 kev Y ψ ζ νh α N Y N f PS > 10 7 (d c ) ν 0 t R e Lb L ; ν L t L via t-channel exchange of H + u ; H 0 u. (e c ) ν 0 ν H ψψ (where ψ = (c, t, b)) via s-channel exchange of h u ; h d. We recall here that if a heavier species φ is in thermal equilibrium, its number density at a temperature T M LSP is Boltzmann suppressed (cf. Eq. (1)) compared to the thermal number density of ν 0 by the factor ζ φ e ( M φ/t), where M φ (M φ M LSP ). In Table we show cross-sections and limits that arise from the above co-annihilation processes with heavier SUSY particles. We denote the Boltzmann suppression of the heavier species φ as ζ φ (see above), the phase-space suppression factor due to heavy particles in the final state as f PS, and the ψ Yukawa coupling as Y ψ. Processes (a c ) and (d c ), like (d s ) in the previous subsection, has the p-wave annihilation factor (T/M LSP ), and we ignore the helicity-suppressed s-wave contribution. The implied limit if Eqs. (10) and (11) hold is shown in the last column. We have defined, as before, α N (a N µ /m l ). As we see from Table, though the processes impose strong constraints, if phase-space and Boltzmann suppression factors are strong enough, for the case of interest here (Eq. (1)), we expect co-annihilation processes with heavier SUSY particles not to be effective in thermalizing ν 0. We note that if tan(β) is large, co-annihilation with b R can also be important, and for Y b 1 will lead to limits similar to that from processes (c c ) and (d c ). Also, in this case, in process (e c ) the bottom final state is important. 3.4 Co-annihilation with SM particles The ν 0 can thermalize through interactions with thermal populations of SM particles. The important processes are (ψ denotes a SM particle): (a M ) ν 0 ψ R,L ν H ψ L,R (where ψ = (c, t, b)) via t-channel h u ; h d exchange. (b M ) ν 0 ψ L ν L ψr (where ψ = (c, t, b)) via t-channel H exchange. (c M ) ν 0 ψ ẽ L ψ via t-channel W ± exchange. (d M ) ν 0 ψ ν H ψ via t-channel Z exchange. (e M ) ν 0 ψ L ψ via t-channel W exchange. 8

10 Table 3: Co-annihilation with SM - cross-sections and limits. We convert the limit in the third column to an implied limit shown in the last column using Eqs. (10) and (11). Process Cross-section Limit Implied limit (a M ) (b M ) (c M ) (d M ) (e M ) Yψ (c 1 s 1 ) A N µ Y N f 16π Mh 4 PS YN Y ψ c π M H g 4 s 1 MLSP 16π MW 4 g 4 s 1 c MLSP 16π MZ 4 g 4 s 1 16π Y ψ ζ ψ A N µ Y N f PS > 10 kev Y ψ ζ ψ α N Y N f PS > 10 7 f PS Y ψ ζ ψ Y N f PS > 10 7 Y ψ ζ ψ Y N f PS > 10 7 f PS s 1 f PS > α m Y N f PS > f PS s 1 f PS > α m Y N f PS > MLSP f M W PS s 1 f PS > α m Y N f PS > Table 3 shows the cross-sections and the limits due to co-annihilation of ν 0 with SM particles. As before, in process (a M ) we define α N (a N µ /m l ), and we compute the limit assuming M hu M hd M h. In (e M ), the B channel can also contribute but is less important given the smaller g. In (a M ) and (b M ), the charm and bottom channels are suppressed by the smaller Yukawa couplings compared to the top, and for T < 100 GeV the top contribution is Boltzmann suppressed. In general, processes (a M ) and (c M )-(e M ) are suppressed further due to f PS 1 if the left-handed SUSY particles in the final-state are all much heavier. If tan β is large, coannihilation with bottom, especially (b M ), will lead to the strongest constraint since it is not Boltzmann suppressed (unless T < m b ). However, (b M ) may be suppressed if M H 100 GeV. Thus a combination of suppressions from f PS, ζ ψ and M W, H can render these processes incapable of thermalizing ν 0. 4 Nonthermal ν 0 If the ν 0 is never in thermal equilibrium, the standard relic density calculation presented in App. A cannot be applied. We consider in this section such a situation when an almost purely right-handed LSP ( ν 0 ÑR) is not in thermal equilibrium, and the possibility of it being nonthermal dark matter. In order for such a situation to be realized, it should be true that all the ν 0 thermalization conditions that we discussed in Sec. 3 are not satisfied. For the case specified in Eq. (1), this happens for instance in the inflationary paradigm for a low reheat temperature (T RH 100 GeV), whereby the reheating produces only ÑR (and SM particles), and the number-densities of heavier SUSY particles (and top) are severely Boltzmann suppressed. We will elaborate on this in the following. First, as we found in Sec. 3., none of the self-annihilation processes can keep the ν 0 in thermal equilibrium for the parameter values in Eq. (1). Second, from Sec. 3.3, the co-annihilation processes can be important in thermalizing the ν 0 for the parameter values in Eq. (1). However, (b c ) is suppressed further due to f PS 1 which we expect given the heavy particles in the final state, (d c ) is p-wave suppressed for T RH M LSP, and, (a c ) and (e c ) are Boltzmann suppressed due to the heavy ν H ; hence we expect these processes also to be ineffective in thermalizing ν 0. Third, the processes of Sec. 3.4 can all be important. However, phase-space suppression f PS 1 due to the heavy particles in the final state, the Boltzmann suppression due to the 9

11 heavy top (for T RH < M t ), and suppression due to M W, H > 100 GeV can render these processes ineffective in thermalizing the ν 0. Fourth, for the parameter values in Eq. (1), the ν 0 relic-density from decays of heavier SUSY thermal particles are problematic and tend to overclose the universe if thermal populations of heavier SUSY particles are present (cf. App. C). However, for T RH 100 GeV, with no significant number-densities of heavier SUSY particles, this problem is avoided. If such a situation of a nonthermal Ñ R is realized, the relic abundance is determined by how the ÑR is produced in the early universe. For example, if the universe went through an inflationary epoch driven by a scalar field Φ (the inflaton), the relic abundance today becomes directly related to the coupling of Φ to the ÑR (and the other particles), and the details of reheating. 5 Conclusions and future directions We analyzed the sneutrino sector after writing down the most general R p conserving renormalizable effective theory, including lepton-number violating terms. We considered the case when the neutrino Majorana mass term M N, and all SUSY breaking masses are at the electroweak scale, which implied that the neutrino Yukawa coupling Y N Assuming that the A-term is proportional to Y N we were led to a tiny sneutrino mixing angle s Y N. We analyzed the cosmological implications of having such a predominantly right-handed sneutrino LSP ( ν 0 Ñ R ), for values given in Eq. (1). Since such a ν 0 interacts mostly through the tiny Y N, its self- and co-annihilation crosssections are tiny. Therefore, the ν 0 should not be in thermal equilibrium at any time in the early universe, for if it is, the standard thermal relic-density calculation (reviewed in Apps. A and B) indicates that it would severely overclose the universe since it freezes-out too early. Moreover, we argue in App. C that decays into ν 0 from heavier thermal SUSY particles, for these parameters, also overcloses the universe (see also [14]). The main focus of the paper is in investigating in which case the ν 0 remains nonthermal, so that the problems mentioned in the previous paragraph for a thermal relic do not apply. In order to answer this, we start with the Boltzmann equation for the ν 0 number-density, and compare its self- and co-annihilations with the Hubble expansion rate; as shown in Eq. (15). For a given process, if the interaction cross-section is smaller than the Hubble rate, this process cannot thermalize the ν 0. We check to see whether dominant interaction processes of the ν 0 with itself, other SUSY particles, and SM particles, satisfy this condition. In Sec. 3. we find from Table 1 that the self-annihilation processes are ineffective in thermalizing such a ν 0 for the values in Eq. (1). Next, in Sec. 3.3, Table we find that ν 0 co-annihilation processes with other SUSY particles (if they are present in thermal number-densities) are quite effective in thermalizing the ν 0. However including phase-space and Boltzmann suppression factors shown in the table can render these processes ineffective. Lastly, in Sec. 3.4, Table 3 we find that co-annihilation processes with SM particles are significant in thermalizing the ν 0 ; however, the phase-space, Boltzmann, and heavy intermediate particle mass suppression factors can render them ineffective. In Sec. 4 we argued that in the inflationary paradigm, if the reheat temperature T RH 100 GeV, all of these constraints are rendered harmless and the ÑR is nonthermal. The relic number density will then be connected to details of reheating and the ν 0 coupling to the inflaton. Exploring these details and of how baryogenesis works in this scenario is, however, beyond the 10

12 scope of this work [18]. In colliders, any heavy SUSY particle that is produced undergoes cascade decays into the ν 0 LSP, which exits the detector as missing energy. An important signature associated with a predominantly ÑR LSP is the occurrence of a displaced vertex in the detector in the case of Majorana neutrinos, or even a very long-lived charged particle leaving the detector in the case of Dirac neutrinos as the life-time scales like Y N. Further details will be discussed in an up-coming paper [19]. In summary, if all new physics scales, including the SUSY breaking and the lepton number breaking scales are small, there is a good chance that the LSP is the lightest sneutrino ν 0. Furthermore, depending on the mechanism of SUSY breaking, the ν 0 could naturally be an almost-pure right-handed sneutrino. Here, we argue that, if the ν 0 is stable, such a scenario is not only cosmologically allowed, but can also properly fit our understanding of dark matter, as long as very stringent requirements on initial conditions are met. Such a scenario may also address some of the problems with the standard cold dark matter (CDM) paradigm [0]. Moreover, collider signatures and signatures associated with such a dark matter candidate seem unusual enough to render such a scenario phenomenologically intriguing. Acknowledgments We thank the organizers and participants of the LC workshop Snowmass 005 where this work was initiated. We thank Csaba Balazs, Marcela Carena, Martin Hirsch, Dan Hooper and James Wells for valuable discussions. W.P. thanks the organizers of the nd Vienna Central European Seminar for their hospitality as well as for the corresponding support by the Austrian Federal Ministry for Education, Science and Culture, the High Energy Physics Institute of the Austrian Academy of Sciences and the Erwin Schrödinger International Institute of Mathematical Physics. The work of AdG and SG is sponsored in part by the US Department of Energy Contract DE-FG0-91ER40684, while WP is supported by a MEC Ramon y Cajal contract and by the Spanish grant BFM A Thermal relic density calculation Observations indicate at a high confidence level that the universe is flat, and we therefore assume that such is the case in presenting all the formulas here. In order to derive the relic abundance of the dark matter candidate χ, we start with the Friedmann equation for a flat universe H = 8πG 3 ρ, (16) which can be written during the radiation-dominated era in terms of the temperature as H(T) = 1.66g 0.5 T M Pl, (17) where g is the effective number of relativistic degrees of freedom at T. The species freezes out at the temperature T f when the self-annihilation rate of χ becomes roughly equal to the Hubble expansion rate H, i.e., σv n χ H(T f ), (18) 11

13 where σv is the thermally averaged cross-section and n is the number density of χ. The number density is given as [15] (ζ(3)/π )gt 3 (Boson), T M, n(t) = (3/4)(ζ(3)/π )gt 3 (Fermion), T M, g ( ) 3/ MT π e (M µ)/t T M, where g is the number of degrees of freedom of χ (for example, g = 1 for a real scalar, g = for a Dirac fermion, etc.). If χ can co-annihilate with another species φ then we can generalize Eq. (18) to (19) σv SA n χ + σv CA n φ H(T f ), (0) where the subscripts SA and CA on the cross-sections denote self- and co-annihilation respectively. When χ and φ are in thermal equilibrium, and furthermore if M φ > M χ, Eq. (19) allows us to write ( ) (3/) Mφ n φ = e ( Mφ/T) n χ T M χ, (1) M χ with M φ (M φ M χ ). Then Eq. (0) becomes when χ freezes-out. The entropy density is given as [15] ( ) (3/) Mφ σv SA + σv CA e φ/t M n χ H(T f ), () M χ s(t) = π 45 g T 3 = 0.44g T 3, (3) where g is the number of relativistic degrees of freedom at T. Defining the ratio of n to s Y n s, (4) we find, by writing the Boltzmann equation Eq. (13) in terms of Y, that it is conserved (i.e. dy/dt = 0) provided: (a) the σv term is negligible, and (b) C Γ is negligible 4. If the above two conditions are satisfied, the conserved value after freeze-out is given by Y f = n f = 3.77 x f 1 s f g f MM Pl σv, (5) where M is the mass of χ and x f M/T f. The entropy density today s 0 is given by s 0 = π 45 g 0T 3 0 ( ev) 3 = 3678 cm 3, (6) where we have used g 0 = 6.3 and T 0 =.7 K = 10 4 ev is the photon temperature today, from which we get the number density of n today 4 If C Γ is not negligible see Eq. (35). n 0 = Y f s 0. (7) 1

14 We can then compute the present relic energy density of the species χ from Ω 0 n 0M = x f ev 3 ρ c g f 0.5 M Pl ρ c σv, (8) where ρ c = ( h ev) 4 is the critical energy density. With g f 100, we get (cf. Eq. (3.4) of Ref. [16]) ( ) ( ev cm Ω 0 h = 10 9 x f = ) /s x f. (9) σv σv For a species that is non-relativistic at freeze-out, i.e., cold dark matter (CDM), the number density is given by (cf. Eq. (19)) ( ) M 3/ Tf n f = g e M/T f. (30) π From Eq. (18), x f is implicitly given by (cf. Eq. () of Ref. [17]) ( ) 0.038gMMPl σv x f = ln g f 0.5 x 0.5. (31) f x f depends logarithmically on M, as can be seen from Eq. (31). In order to obtain the observed Ω (with h 0.5), for M 100 GeV, g = 1, we therefore need x f 1, or equivalently, σv GeV 0.45 pbarn. B Mixed sneutrino thermal dark matter We found in Sec. 3 that if s 1 > 10 3 then the ν 0 will be kept in thermal equilibrium by selfand co-annihilations. A thermal relic results if this condition is satisfied, and in this section, we compute the present relic density resulting from such a thermal population of ν 0. This has been explored earlier in [7]. With the lepton-number-violating parameters nonzero the ν 0 ν 0 Z coupling is zero [6]. This prevents the too efficient ν 0 self-annihilation via s-channel Z-boson exchange and therefore leads to an adequate relic-density as we will show here. Also, due to the s 1 suppression in the ν 0 -nucleon interaction cross-section, direct-detection constraints are easily evaded [7]. Given that there are three generations, the ν 0 are really three states, and the presence of these co- LSP states may lead to interesting effects depending on mass-splittings between them. For instance, the ν 0 ν 0 Z coupling may now be non-zero (where the ν 0 is a heavier co-lsp state), and have implications in direct-detection experiments. We will not explore this further in this work. In App. A we gave the details of estimating the present relic abundance, and Eq. (9) is of particular relevance here. In principle, σv is the sum of the cross-sections of all the processes that we identified in Sec. 3. and 3.3. However, for simplicity we consider the case when H, ν H, l are all much heavier in which case the only processes that are relevant are (a s ) and (c s ). We thus have σv σ (as) SA + σ(cs) SA, and for x f 1, the present relic abundance of ν 0 is Ω 0 h = 10 4 s 4 1 [ g ( 100 GeV M W ) ( )] 100 GeV + g + Y ψ M B t 1 13 ( AN µ ) (100 ) Y N GeV M h M h 1. (3)

15 If ν 0 is predominantly right-handed (s 1 1, c 1 1, ν 0 Ñ R ), if A N µ Y N > M h, and if M νh (M νh M LSP ) is not too large compared to T, co-annihilation of the Ñ R with ν H by process (e c ) can be effective, with the cross-section given in Table. In this case, the present relic-density is Ω 0 h = 1 ( e ( M ν c 4 H /T) 1 M h 100 GeV ) ( M h A N µ Y N ) (33) We can see from Eq. (7) that in order to have s 1 1 when A N µ Y N > M h requires m l v. This implies that M νh is necessarily large, leading to too big a relic-density, and excluding this possibility. In general we find that if a thermal population of Ñ R is generated due to the processes of Secs. 3. and 3.3, then the resulting relic density is too big. The reason for this is that the co-annihilation cross-section is big enough to thermalize the ÑR, but is too small to keep it in thermal equilibrium long enough to deplete the number-density to acceptable levels before freeze-out. Therefore, if the ν 0 is in thermal equilibrium we must have s 1 not too small, as dictated by Eq. (3). C Relic from decay of thermal SUSY particles At some point in the history of the universe, if there exists a thermal population of heavier SUSY particles, they could decay into the ν 0 giving rise to a relic abundance [14]. Here we focus on the case s 1 1 i.e., ν 0 Ñ R. The important decay channels are: (a D ) H u ν 0 L (where L is the lepton doublet). (b D ) ν H ν 0 ψψ via h u, h d (where ψ is a SM fermion). (c D ) l ν 0 ψψ via W ±. We find the cumulative relic abundance of Ñ R today by integrating the Boltzmann equation, Eq. (13). We take as an estimate for C Γ, C Γ n χ Γ(χ ν 0 X), (34) where Γ is the decay rate of the SUSY particle χ in processes (a D )-(c D ), and X denotes SM particles in these processes. In this estimate, we ignore the angular dependence in the decays. The Boltzmann equation written in terms of Y ν0 is d dt Y = C Γ s, (35) in the limit where the σv term can be neglected in Eq. (13). In the radiation dominated era we can change the independent variable from t to T using after which Eq. (35) can be written as dt = dt HT, (36) d dt Y = C Γ HTs. (37) 14

16 Integrating this from T down to T, we have T Y (T) = dt C Γ T HTs. (38) Using Eqs. (34), (17), (19) and (3) in the above equation we get Y (T) ΓM Pl M χ dx e 1/x. (39) T/M χ x9/ Owing to the exponential suppression in the integrand for T < M χ, Y freezes in at Y f Y (M χ ). The decay rate from process (a D ) (in the H rest-frame) is given by ( 1 M x /M y ) Γ (ad ) c 1Y N 16π M Hf PS (M LSP, M H), f PS (M x, M y ) = ( 1 + M x /M y ), (40) where we have taken the lepton in the final state as massless, c 1 = cos θ ν 1, and f PS is the phase-space function. Due to this decay, from Eq. (39), we get Y (ad )(T) 10 c 1YNf PS M Pl M H T/M H Using Eqs. (6), (7) and (8) we find the present relic density to be Ω 0 (ad )h 10 6 c 1 Y N The decay rate of process (b D ) is given by dx e 1/x. (41) x9/ M LSP fps M. (4) H Γ (bd ) (c 1 s 1 ) Y ψ 56π 3 A N µ Y N M 3 ν H M 4 h f 3PS, (43) where f 3PS is the 3-body phase-space, and following the same procedure as in the previous case, we find Ω 0 (bd )h = 10 4 (c 1 s 1) Yψ A N µ Y N M νh M LSP f Mh 4 3PS. (44) The decay rate of process (c D ) is given by and we find Γ (cd ) s 1g 4 M 5 l f 56π 3 MW 4 3PS, (45) M Ω 0 (cd )h = 10 4 s 1 g4m3 l LSP f MW 4 3PS. (46) From these estimates, we infer that for Ω 0 h < O(1), with all masses about 100 GeV, we need: Y N < 10 13, A N µ Y N < 10 ev, and s 1 < 10 1 (where we ignore possible phase-space suppression factors). This is realized for instance for a purely Dirac neutrino, and has recently been considered in Ref. [14]. In this case the heavier SUSY particles are very long lived, and one has to worry if it decays during big-bang nucleosynthesis (BBN) causing photo-dissociation of 15

17 the light elements, and spoiling its successful predictions. We find from Eq. (36) that in order for heavy particles to have decayed away before BBN, we need the life-time of any such particle to be τ < 1 s, which implies that the width Γ > 10 5 GeV. Considering for example the decay of H, we find from Eq. (40) that we need Y N > in order for the H to have decayed away before BBN. It is worth mentioning that a scenario similar to the one explored in [14] might be realizable even if the neutrinos are Majorana fermions. It seems, for example, that an ev-seesaw [13], which imposes M N 1 ev and Y N might satisfy the conditions discussed above. A more detailed analysis of this possibility is beyond the ambitions of this paper. References [1] For a pedagogical overview and many references see, for example, A. de Gouvêa, hepph/ [] For a review of different ideas related to the origins of neutrino masses and new physics see, for example, R. N. Mohapatra et al., hep-ph/ and many references therein. [3] For reviews see for example: S. P. Martin, hep-ph/ ; M. Drees, hep-ph/ [4] S. Borgani, A. Masiero and M. Yamaguchi, Phys. Lett. B 386 (1996) 189 [hep-ph/9605]; F. Takayama and M. Yamaguchi, Phys. Lett. B 485 (000) 388 [hep-ph/000514]; M. Hirsch, W. Porod and D. Restrepo, JHEP 0503, 06 (005) [hep-ph/ ]. [5] J. S. Hagelin, G. L. Kane and S. Raby, Nucl. Phys. B 41, 638 (1984); L. E. Ibanez, Phys. Lett. B 137, 160 (1984); T. Falk, K. A. Olive and M. Srednicki, Phys. Lett. B 339, 48 (1994) [hep-ph/940970]. [6] L. J. Hall, T. Moroi and H. Murayama, Phys. Lett. B 44, 305 (1998) [hep-ph/971515]. [7] N. Arkani-Hamed, L. J. Hall, H. Murayama, D. R. Smith and N. Weiner, Phys. Rev. D 64, (001) [hep-ph/000631]. [8] D. Hooper, J. March-Russell and S. M. West, Phys. Lett. B 605, 8 (005) [hepph/ ]. [9] F. Borzumati and Y. Nomura, Phys. Rev. D 64, (001) [hep-ph/ ]. [10] M. Hirsch, H. V. Klapdor-Kleingrothaus and S. G. Kovalenko, Phys. Lett. B 398, 311 (1997) [hep-ph/970153]. [11] Y. Grossman and H. E. Haber, Phys. Rev. Lett. 78, 3438 (1997) [hep-ph/97041]. [1] S. Eidelman et al. [Particle Data Group], Phys. Lett. B 59, 1 (004). [13] A. de Gouvêa, Phys. Rev. D 7, (005) [hep-ph/ ]. [14] T. Asaka, K. Ishiwata and T. Moroi, hep-ph/ [15] See for example: E. Kolb and M. Turner, The Early Universe, Addison-Wesley, 1994; S. Dodelson, Modern Cosmology, Academic Press,

18 [16] G. Jungman, M. Kamionkowski and K. Griest, Phys. Rept. 67, 195 (1996) [hepph/ ]. [17] K. Griest and D. Seckel, Phys. Rev. D 43, 3191 (1991). [18] For related discussions see: B. Garbrecht and A. Pilaftsis, [hep-ph/ ]; A. Pilaftsis and T. E. J. Underwood, Phys. Rev. D 7, (005) [hep-ph/ ]; C. Pallis, Astropart. Phys. 1, 689 (004) [hep-ph/040033]. [19] Work in progress. [0] See for example: W. B. Lin, D. H. Huang, X. Zhang and R. H. Brandenberger, Phys. Rev. Lett. 86, 954 (001) [astro-ph/ ]; J. A. R. Cembranos, J. L. Feng, A. Rajaraman and F. Takayama, Phys. Rev. Lett. 95, (005) [hep-ph/ ], and references therein. 17