Decaying LSP in SO(10) GUT and PAMELA's Cosmic Positron

We suppose that the lightest supersymmetric particle (LSP) in the minimal supersymmetric standard model (MSSM) is the dark matter. The bino-like LSP can decay through the SO(10) gauge interactions, if one right-handed (RH) neutrino (\nu^c_1) is lighter than the LSP and its superpartner (\tilde{\nu}^c_1) develops a vacuum expectation value (VEV), raising extremely small R-parity violation naturally. The leptonic decay modes can be dominant, if the VEV scale of {\bf 16}_H is a few orders of magnitude lower than the VEV of {\bf 45}_H (\approx 10^{16} GeV), and if a slepton (\tilde{e}^c_1) is relatively lighter than squarks. The desired decay rate of the LSP, \Gamma_{\chi} \sim 10^{-26} sec.^{-1} to explain PAMELA data can be naturally achieved, because the gaugino mediating the LSP decay is superheavy. From PAMELA data, the SU(3)_c x SU(2)_L x SU(2)_R x U(1)_{B-L} breaking scale (or the {\bf 16}_H VEV scale) can be determined. A global symmetry is necessary to suppress the Yukawa couplings between one RH (s)neutrino and the MSSM fields. Even if one RH neutrino is quite light, the seesaw mechanism providing the extremely light three physical neutrinos and their oscillations is still at work.


I. INTRODUCTION
For the last three decades, many remarkable progresses in particle physics and cosmology have been made thanks to the cooperateive and intimate relation between the two fields.
In particular, the application of particle physics theory into dark matter (DM) models in cosmology was very successful. Because of the correct order of magnitude of the cross section, thermally produced weakly interacting massive particles (WIMPs) have been long believed to be DM candidates [1]. So far the lightest supersymmetric particle (LSP), which is a wellmotivated particle originated from the promising particle physics model, i.e. the minimal supersymmetric standard model (MSSM), has attracted much attentions as an excellent example of WIMP.
Recently, PAMELA [2], ATIC [3], H.E.S.S. [4], and the Fermi-LAT collaborations [5] reported the very challenging observations of positron excesses in cosmic ray above 30 GeV upto the TeV scale. In particular, PAMELA observed a positron fraction [e + /(e + + e − )] exceeding the theoretical expectation [6] above 30 GeV upto 100 GeV. However, the antiproton/proton flux ratio was quite consistent with the theoretical calculation. The ATIC, H.E.S.S., and Fermi-LAT's observations exhibit excesses of (e + + e − ) flux in cosmic ray from 100 GeV to 1 TeV. 1 They would result from the positron flux that keeps rising upto 1 TeV.
Apparently the above observational results are very hard to be interpreted in view of the conventional MSSM cold dark matter scenario: explaining the excess positrons with annihilations of Majorana fermions such as the LSP needs a too huge boost factor. Moreover, ATIC, H.E.S.S., and Fermi-LAT's observations seem to require a TeV scale DM, if they are caused indeed by DM annihilation or decay. Introduction of a TeV scale LSP, however, would spoil the motivation of introducing supersymmetry (SUSY) to resolve the gauge hierarchy problem in particle physics. In addition, TeV scale DM seems to be disfavored by the gamma ray data [4], if the excess positron flux is due to DM annihilations [7]. On the other hand, the DM decay scenario is relatively free from the gamma ray constraint [8].
In the DM decay scenario, however, there are some serious hurdles to overcome: one is to naturally obtain the extremely small decay rate of the DM (Γ DM ∼ 10 −26 sec. −1 ), and the other is to naturally explain the relic density of the DM in the Universe. The first hurdle  could be somehow resolved by introducing an extra symmetry, an extra DM component   with a TeV scale mass, and grand unified theory (GUT) scale superheavy particles, which mediate DM decay into the SM charged leptons (and the LSP) [9]. The fact that the GUT scale particles are involved in the DM decay might be an important hint supporting GUT [10,11]. However, since the interaction between the new DM and the SM charged lepton are made extremely weak by introducing superheavy particles mediating the DM decay, non-thermal production of the DM with a carefully tuned reheating temperature should be necessarily assumed. One way to avoid it is to consider SUSY models with two DM components [9,10]. In these models, the decay of the small amount of the meta-stable heavier DM component (X), which is assumed to be non-thermally produced, accounts for the cosmic positron excess, and the thermally produced lighter DM component LSP (χ), which is absolutely stable and regarded as the dominant DM [O(10 −10 ) < n X /n χ ], explains the relic density of the Universe. 2 In this paper, we suppose that the conventional bino-like LSP is the main component of the DM. Since the "bino" is a WIMP, thermally produced binos could explain well the relic density of the Universe. The bino-like LSP with a mass of about 300 -400 GeV could also explain PAMELA data, if it decays to e ± and a neutral fermion with an extremely small decay rate of order 10 −26 sec. −1 [13]. The (e + + e − ) excess observed by Fermi-LAT could be explained by astrophysical sources such as nearby pulsars [14] (and/or with the sub-dominant extra TeV scale DM component [9]). 3 In fact, pulsars can explain both the PAMELA and Fermi-LAT's data in a suitable parameter range [15]. However, this does not imply that DM in addition to pulsars can not be the source of the galactic positrons [14].
In fact, we don't know yet a complete pulsar model, in which all the free parameters would be fixed by the fundamental physical constants.
To achieve the needed extremely small decay rate of the bino-like LSP χ, we need extremely small R-parity violation. We will assume that the R-parity is broken by a non-zero vacuum expectation value (VEV) of a right-handed (RH) sneutrino ( ν 1 = 0). Since it 2 The low energy field spectrum in the models of Ref. [9] is the same as that of the MSSM except for the neutral singlet extra DM component. Moreover, the models in [9] can be embedded in the flipped SU (5) GUT and string models [10,12]. 3 Alternatively, one could assume a bino mass of 3.5 TeV in order to account for both PAMELA and Fermi-LAT with LSP decay [13]. In this case, however, the soft SUSY breaking scale should be higher than 3.5 TeV.
doesn't carry any standard model (SM) quantum number, it does not interact with the MSSM fields at all, if its Yukawa interactions with them are forbidden by a symmetry and gravity interaction is ignored. We will explore the possibility that the extremely small DM decay rate results from the gauge interaction by exchange of the superheavy gauge bosons and gauginos present in the SO(10) SUSY GUT. We will not introduce a new DM component, and will attempt to explain the PAMELA's observation within the framework of the already existing particle physics model.

II. SO(10) GUT
One of the appealing GUTs is the SO(10) GUT [16]. It unifies all the three SM gauge forces within the SO(10) gauge interaction. One of the nice features of SO(10) is that it predicts the existence of the RH neutrinos [or the SU(2) L singlet neutrinos], since a RH neutrino is contained in a single spinorial representation 16 of SO(10), together with one family of the SM fermions. The RH neutrinos provide a very nice explanation of the observed neutrino oscillations through the seesaw mechanism [17] and also of the baryon asymmetry in the Universe through leptogenesis [18].  [19]. 4 In this paper, we will thus identify the triplet Higgs mass scale with the VEV of 45 H .
How many and what kind of Higgs fields are needed to get the SM gauge group are quite model-dependent. Their masses would be close to the GUT scale, but they are not exactly the same as each other. Even in one Higgs multiplet, its component fields might have various mass spectra after symmetry breaking. Except for 10 h 45 H 10 h , they interact with the MSSM fields only through non-renormalizable Yukawa couplings due to their GUT scale VEVs. Such couplings can be utilized to get the realistic SM fermion masses. One might think that SO(10)-breaking superheavy fields also contribute to the mediation of DM decay through such non-renormalizable couplings with the MSSM fields. However, the extra suppression factor (1/M P ) n (n = 1, 2, 3, · · · ) makes their contributions negligible compared to those of the superheavy gauge fields and gauginos via the renormalizable gauge interactions, which will be discussed later.
The SO(10)-breaking sector could include heavy fields, which do not develop GUT scale VEVs. They are introduced in order to decouple unwanted fields in the SO(10)-breaking Higgs sector, which are absent in the MSSM, from low energy physics in non-minimal SO (10) models. Since their couplings to the MSSM fields are not essential and their masses would be heavier than the mediators leading to DM decay, we can assume that all the interactions between such SO(10)-breaking sector fields and the MSSM fields are weak enough, if they are present.
Thus, as far as the DM decay is concerned, the gauge interactions through the superheavy gauge fields and gauginos can be dominant over Yukawa interactions. They would give more predictable results, regardless of what specific SO(10) models are adopted. We will focus on the DM decay predominantly through the superheavy gauge fields or gauginos.

B. SU(5) vs. SU(2) R scale
In terms of the SM's quantum numbers, the SO(10) generator (= 45 G ) is split into the SM gauge group's generators plus We will simply write them as  If (1) R-parity is absolutely preserved and (2) χ is really the LSP, χ can never decay.
We mildly relax these two conditions: by assuming a non-zero VEV of the superpartner of the (first family of) RH neutrino,ν c 1 (i.e. R-parity violation), or its mass lighter than the χ's mass, m χ (i.e.ν c 1 LSP), χ can decay. By introducing a global symmetry, one can forbid its renormalizable Yukawa couplings to the MSSM fields. Then,ν c 1 can interact with the MSSM fields only through the superheavy gauge fields and gauginos of SO(10), since the (s)RH neutrino ν c 1 (ν c 1 ) is a neutral singlet under the SM gauge symmetry. Consequently, the decay of χ would be possible but quite suppressed. For instance, refer to the diagram of FIG.1-(a). We will discuss how this diagram can be dominant for the χ decay.  • At least one RH neutrino, i.e. the SU(2) L singlet neutrino ν c 1 (and its superpartner ν c 1 ) must be lighter than χ so that χ decays to charged leptons. It is because ν c i is always accompanied byν c i in the effective operators leading to the leptonic decay of χ, composed ofẽ c * 1 ν c 1Ẽ c ,ν c * 1 e c 1Ẽ , andν c * 1 ν c 1Ñ ,ẽ c * 1 e c 1Ñ . If all the sneutrino masses are heavier than χ,ν c 1 must develop a VEV for decay of χ. Once ν c 1 is light enough,ν c 1 can achieve a VEV much easily.

Interactions of the MSSM fields and heavy gauginos
To be consistent with PAMELA's observations on high energy galactic positron excess [2], the DM mass should be around 300 -400 GeV [13]. Thus, one can simply take the following values; Consequently, SO(10) is broken first to LR, which would be the effective gauge symmetry valid below the GUT scale. As seen from TABLE I, the gauge interactions by the LR gauginos (and also gauge fields) preserve the baryon numbers. Even if the masses of the LR gauginos and gauge fields are relatively light, their gauge interactions don't give rise to proton decay. We will show later that the decay channels of χ through the mediation of the superheavy gauge fields are relatively suppressed.

B. Seesaw mechanism
Although one RH neutrino is light enough, the seesaw mechanism for obtaining the three extremely light physical neutrinos still may work. Let us consider the following superpotential; where the Majorana mass term of ν c i could be generated from the non-renormalizable superpotential 16 H 16 H 16 i 16 j /M P (i, j = 1). Thus, M ij (≫ h u ) could be determined, if the LR breaking scale by 16 H is known. In this superpotential, we note that one RH neutrino ν c 1 does not couple to the MSSM lepton doublets and Higgs. For instance, by assigning an exotic U(1) R-charge to ν c 1 , one can forbid its Yukawa couplings to the MSSM superfields. Thus, ν c 1 would be decoupled from the other MSSM fields, were it not for the heavy gauge fields and gauginos of the SO(10) SUSY GUT.
Taking into account only Eq. (2), one neutrino remains massless. The two heavy Majorana mass terms of ν c 2 and ν c 3 are sufficient for the other two neutrinos to achieve extremely small physical masses through the constrained seesaw mechanism [22]: where v ij ≡ y ij could make leptogenesis possible [22].

C. Heavy gauginos' masses
The gauge interactions between the gauginos and an SU(2) R lepton doublet (2 1 ) in the LR model is described by where {Ñ R ,Ẽ,Ẽ c } andÑ BL are the superpartners of the SU(2) R and U(1) B−L gauge fields, respectively. (−gÑ R + g ′Ñ BL )/ g 2 + g ′ 2 is identified with "Ñ" discussed above. Hence, its orthogonal component (g ′Ñ R + gÑ BL )/ g 2 + g ′ 2 corresponds to the bino of the MSSM. The hypercharge of the MSSM is defined by where + (−) for 2 (2). It is straightforward to write down the interaction between the LR gauginos and 2 −1 . When the LR model embedded in the SO(10) GUT, the LR and B − L gauge couplings, g and g ′ can be expressed in terms of the SO(10) gauge coupling, By introducing a pair of SU(2) R doublet Higgs [or 16 H and 16 H in SO(10)], and, for instance, the superpotential where M E ≡ vg 10 /2 and M N ≡ v g 2 + g ′ 2 /2 = vg 10 5/8 = M E 5/2. We note here that M N is heavier than M E . The other combination (ν c H + ν H )/ √ 2 (≡ ν c + ) and S get a mass from the superpotential Eq. (8) at the SUSY minimum. The second line of Eq. (9) contains the soft mass terms. Since S can develop a VEV of order the gravitino mass m 3/2 due to the "A-term" corresponding to W of Eq. (8), the last two mass terms of Eq. (9) [⊂ S (e c H e H − ν c H ν H )] are induced. We rewrite Eq. (9) in terms of the four component spinors as follows; where λ −(0) and ψ −(0) are the Dirac (Majorana) spinors constructed with the two components' Weyl spinors for the gauginos and higgsinos: where the "bar" denotes the complex conjugates of the fermionic fields. λ + and ψ + are respectively given by (λ − ) C and (ψ − ) C , and λ 0 and ψ 0 satisfy (λ 0 ) C = λ 0 and (ψ 0 ) C = ψ 0 .
The mass eigenstates and their eigenvalues turn out to be 1,2 = ∓M N + where ǫ ≡ [m
Eq. (4) includes also the neutral interactions of the SU(2) L lepton singlets withÑ and the bino. One can extract the part interacting only withÑ: They are actually reminiscent of the Z boson interactions in the SM. By contracting λ 0 and λ 0 , the decayẽ c * 1 → e − 1 + ν 1 +ν c 1 is possible. See FIG.2-(a). However, since M 2 N is 5 2 times heavier than M 2 E as shown from Eq. (9), and the effective coupling is √ 5 2 × 1 √ 20 = 1 4 times smaller than that of the charged interaction case, the amplitude mediated by λ 0 is just 1 10 of that by λ − .
As seen in TABLE I, the MSSM Higgs and higgsinos also couple toẼ c ,Ẽ orÑ . Since the MSSM charged Higgs and higgsinos are assumed to be much heavier thanẽ c 1 and χ, the decay channels through them are quite suppressed or kinematically forbidden.
So far we did not discuss the case in which χ decays through the mediation of the superheavy gauge bosons. The potentially dominant diagram is displayed in FIG.2-(b). e c 1 is coupled to χ and e c 1 . The scalar-scalar-gauge boson vertex is basically a derivative coupling. Accordingly, this diagram is suppressed compared to FIG.1-(b), only if the bino is much lighter than the soft mass of {Ẽ,Ẽ c }. As presented above, in this paper, we assume that m χ ∼ 300 -400 GeV and the soft mass of {Ẽ,Ẽ c } is of O(1) TeV.

E. LSP decay rate and the seesaw scale
Now let us estimate the decay rate of FIG.1-(a), which is the dominant decay channel, and determine the LR breaking scale such that it is consistent with PAMELA data. Indeed, if mνc 1 < m χ , a non-zero VEV ofν c 1 is not essential: χ can decay to the four light particles, e ± , ν c 1 , andν c 1 . However, just for simplicity, we will assume that a non-zero VEV ofν c 1 is developed. For instance, let us consider the following terms in the superpotential; where M P = 2.4 × 10 18 GeV and κ is a dimensionless coupling constant. Σ is an SO (10) singlet. We assign e.g. the U(1) R-charge of 2/3 to 16 1 and Σ, and 0 to 16 H . The scale of can be determined such that it is consistent with PAMELA data. The soft mass term of Σ and the A-term corresponding to κΣ 3 in the scalar potential permit a VEV Σ ∼ m 3/2 /κ. Then, the scalar potential generates a linear term ofν c 1 coming from the A-term corresponding to the first term of Eq. (18), V ⊃ m 3 3/2 ( 16 H /κ 2 M P )ν c 1 . The linear term and the soft mass term ofν c 1 in the scalar potential can induce a non-zero VEV ofν c 1 : Thus, the decay rate of χ in FIG.1-(a) can be estimated: where α 10 (≡ g 2 10 /4π) and α Y [≡ g 2 Y /4π = (3/5) × g 2 1 /4π, where g 1 is the SO(10) normalized gauge coupling of g Y ] are approximately 1/24 and 1/100, respectively. Here, we ignore the RG correction to α 10 . 300 -400 GeV fermionic DM decaying to e ± and a light neutral particle can fit the PAMELA data [13]. For m χ ≈ 300 -400 GeV, (m 3/2 /κmẽc 1 ) ∼ 10, M E or 16 H is estimated to be of order 10 14 GeV. This is consistent with the assumption 16 H ≪ 45 H ∼ 10 16 GeV. Therefore, the masses of the other two RH neutrinos, which do not contribute to the process of FIG.1-(a), are around 10 10 GeV or smaller in this case: W ⊃ y ij ( 16 H 16 H /M P )16 i 16 j (i, j = 1) ⊃ y ij (10 10 GeV) × ν c i ν c j (i, j = 1). So the Yukawa couplings of the Dirac neutrinos should be a bit small (∼ 10 −2 ).
If m χ ≈ 3.5 TeV and the model is slightly modified such that χ decays dominantly to µ ± , ν c 2 rather than to e ± , ν c 1 , which is straightforward, the Fermi-LAT's data as well as the PAMELA's can be also explained [13]. In this case, M E or 16 H should become somewhat heavier (∼ 10 15 GeV), and the seesaw scale should be replaced by 10 12 GeV. However, the motivation of introducing SUSY to resolve the gauge hierarchy problem in the SM would become more or less spoiled.

IV. CONCLUSIONS
In this paper, we have shown that the bino-like LSP in the MSSM can decay through the SO(10) gauge interactions, if a RH neutrino is light enough (m ν c 1 m χ ) and its superpartner develops a VEV ( ν c 1 = 0). The Yukawa couplings between the RH (s)neutrino and the MSSM fields can be suppressed by a global symmetry such as the U(1) R-symmetry. It gives rise to an extremely small R-parity violation very naturally. If the LR breaking scale or the seesaw scale is low enough compared to the GUT scale (i.e.