Soft Leptogenesis in Higgs Triplet Model

We consider the minimal supersymmetric triplet seesaw model as the origin of neutrino masses and mixing as well as of the baryon asymmetry of the Universe, which is generated through soft leptogenesis employing a CP violating phase and a resonant behavior in the supersymmetry breaking sector. We calculate the full gauge--annihilation cross section for the Higgs triplets, including all relevant supersymmetric intermediate and final states, as well as coannihilations with the fermionic superpartners of the triplets. We find that these gauge annihilation processes strongly suppress the resulting lepton asymmetry. As a consequence of this, successful leptogenesis can occur only for a triplet mass at the TeV scale, where the contribution of soft supersymmetry breaking terms enhances the CP and lepton asymmetry. This opens up an interesting opportunity for testing the model in future colliders.

Leptogenesis is an elegant way to generate the baryon asymmetry of the Universe in connection with the origin of the observed neutrino masses and mixing through the seesaw mechanism [1]. One way of understanding a tiny neutrino mass is to relate it with the small vacuum expectation value of a Higgs triplet [2] whose decay can also induce the cosmological baryon asymmetry in the presence of at least two Higgs triplets [3] or a right-handed neutrino [4] as required by the generation of non-trivial CP and lepton asymmetry. In the minimal supersymmetric version with one pair of triplets, there is a new way of leptogenesis (called "soft leptogenesis") in which CP phases in the soft terms can contribute to generate the lepton asymmetry [5,6]. Soft leptogenesis in the minimal supersymmetric Higgs triplet model has been considered first in Ref. [7].
In this paper, we revisit this last scenario to provide a careful analysis on the quantities for the lepton and CP asymmetries and their cosmological evolution by considering the full set of Boltzmann equations including thermal masses and the temperature supersymmetry breaking effects consistently. We will also derive a set of simple Boltzmann equations from the Maxwell-Boltzmann approximation taking into account the difference between the Bose-Einstein and Fermi-Dirac statistics, and show that they provide a fairly good approximation to the full Boltzmann equations.
The most important effect included in our analysis is the contribution of the gauge annihilation processes, which lead to a significant reduction of the resulting lepton asymmetry for the low Higgs triplet mass. The dynamics of such a system is analyzed in Ref. [8] for the case of the conventional baryogenesis with heavy Higgs bosons in the SU(5) unification scheme.
Our analysis is extended to the lowest possible values of the Higgs triplet mass where, as will be shown in the following, the annihilation effect dominates over decays and inverse decays.
Another crucial ingredient of soft leptogenesis is the suppression of the asymmetry due to a small difference between boson and fermion statistics at finite temperature. We find that this effect for low values of the triplet mass becomes subleading compared to that due to soft supersymmetry-breaking terms. Let us also note that the annihilation effect becomes irrelevant for a triplet mass higher than about 10 10 GeV [9], for which, however, the lepton asymmetry in the soft leptogenesis scenario is also suppressed, as it is inversely proportional to the triplet mass [5]. As a result, we will conclude that the required baryon asymmetry can be generated only at the multi-TeV range of the Higgs triplet mass, and thus the model can lead to distinct collider signatures through, in particular, the production and decay of a doubly charged Higgs boson [10,11]. This opens up another interesting possibility for generating the neutrino masses and mixing as well as the cosmological baryon asymmetry at the TeV scale, which can be tested in future colliders [12,13].
In the supersymmetric form of the Higgs triplet model [14], one needs to introduce a vector-like pair of ∆ = (∆ ++ , ∆ + , ∆ 0 ) and ∆ c = (∆ c−− , ∆ c− , ∆ c0 ) with hypercharge Y = 1 and −1, allowing for the renormalizable superpotential as follows: where hLL∆ contains the neutrino mass term, hνν∆ 0 . The soft supersymmetry breaking terms relevant for us are Note that we have used the same capital letters to denote the superfields as well as their scalar components. We will consider the universal boundary condition of soft masses; A L = A 1 = A 2 = A and m ∆ = m ∆ c = m 0 . In the limit M ≫ m 0 , A, the Higgs triplet vacuum expectation value ∆ 0 = λ 2 H 0 2 2 /M gives the neutrino mass The mass matrix of the scalar triplets is diagonalized by where ∆ ± are the mass eigenstates with the mass-squared values, M 2 ± = M 2 +m 2 0 ±BM, and the mass-squared difference, ∆M 2 = 2BM. In terms of the mass eigenstates, the Lagrangian becomes The heavy particles∆ ± decay to the leptonic final states, LL,LL, as well as the Higgs final states, H 1 H 1 ,H 1H1 andH 2H2 ,H 2H 2 . Thus, the out-of-equilibrium decay∆ ± → LL,LL can lead to lepton asymmetry of the universe.
In order to discuss how to generate a lepton asymmetry in the supersymmetric triplet seesaw model let us first consider the general case of a charged particle X (X) decaying to a final state j (j) and generating tiny CP asymmetric number densities, n X − nX and n j − nj. The relevant Boltzmann equations in the approximation of Maxwell-Boltzmann distributions are where Y 's are the number densities in unit of the entropy density s as defined by Y X ≡ n X /s ≈ nX/s, Y x ≡ (n X − nX )/s and Y j ≡ (n j − nj)/s. Here, the CP asymmetry ǫ j in the decay X → j is defined by In Eq. (5), K ≡ Γ X /H 1 with the Hubble parameter H 1 = 1.66 √ g * M 2 /m P l at the temperature T = M, and B j is the branching ratio of the decay X → j. For the relativistic degrees of freedom in thermal equilibrium g * , we will use the Supersymmetric Standard Model value: The evolution of the X abundance is determined by the decay and inverse decay processes, as well as by the annihilation effect described by the diagrams of FIG. 1, and are accounted for by the functions γ D and γ A , respectively. Note that the triplets are charged under the Standard Model gauge group and thus have nontrivial gauge annihilation effect which turns out to be essential in determining the final lepton asymmetry. Moreover, as a consequence of unitarity, the relation 2Y x + j Y j ≡ 0 holds, so that one can drop out the equation for Y x , taking the replacement: in the last of Eqs. (5). In our model, the heavy particle X can be either of the six charged particles; X = ∆ ++ ± , ∆ + ± or ∆ 0 ± . Each of them follows the first Boltzmann equation in Eq. (5) where γ D and γ A are given by with where t w ≡ tan(θ W ) with θ W the Weinberg angle, and We use this latter evaluation when we solve the full Boltzmann equations for the lepton asymmetry numerically. The last figure shows that the Boltzmann approximation is well justified as expected for the region of our relevance, z > 10.
Given γ D and γ A , we can now analyze the thermal evolution of Y X [8]. In FIG. 3, we plot the quantity (Y X − Y eq X )/Y X , which quantifies the departure of the triplet density from its equilibrium value. In particular, the higher line shows the result when only the processes of decay and inverse decay to light particles are included in the calculation. As expected, since K >> 1, Y X follows closely the equilibrium density Y eq X with a slight deviation of order 10 −1 . However, annihilation is indeed important in our case and cannot be neglected. This is shown in the same figure by the lower curves, which represent the departure of the triplet density from its equilibrium value when annihilation is included. The importance of annihilation can be understood in the following way. The inverse decay freezes out at z f ≈ 9 for K = 32 as Kz

5/2
f e −z f = 1. On the other hand, the thermal averages of the annihilation and decay rate can be compared by considering the following ratio [8]: where α X = KH 1 /M. Thus, the annihilation effect becomes negligible for M > ∼ 10 9 GeV. But in our case of soft leptogenesis, higher M suppresses the lepton asymmetry asǫ l ∝ A/M, so there is a tension between these two effects, and lower values of M turn out to be favored.
In FIG. 3, one can see that, due to annihilation which freezes out at z ≈ 20, Y X follows more closely its equilibrium density Y eq X compared with the previous case, with a deviation which is now of order 10 −3 . In particular, this implies that the approximation is a good one, since decoupling occurs indeed at high z. Nevertheless, in our numerical analysis, we solve the full Boltzmann equations where the Bose-Einstein and Fermi-Dirac distributions as well as thermal masses, are included properly.
To find out the cosmological lepton asymmetry by the decay of X = ∆ ± , one needs to calculate Y j with the states j = LL andLL and thus the corresponding CP asymmetry: Recall that one cannot rely on the the above Boltzmann equation (5) where dΠ's are the phase space integration factors and A j (Āj) is the amplitude of the decay for the bosonic or fermionic state j. Using the effective field-theory approach of resummed propagators for unstable particles [15], the effective vertices of ∆ + (∆ + ) and the states j (j) are where ∆M 2 = 2BM. For ∆ − , one takes the interchange of + ↔ − and ∆M 2 → −∆M 2 .
Here, Π's are the absorptive part of two point functions; Calculating |A j | 2 − |Āj| 2 ∝ |S j X | 2 − |Sj X | 2 , we get for Eq. (13), Here, R k include the thermal propagator effect in the cutting rule [16] and C j are the thermal phase space factor of the final states. For the bosonic and fermionic states, we have where x B,F = m 2 B,F (T ) 2 /T 2 are the thermal masses of the bosons or fermions. Let us note in Eq. (16) that the relation j,k holds for any final states of j 1,2 and intermediate states in the loop k 3,4 . The same is true for the second part of Eq. (13). In fact, this is nothing but the unitarity relation j Γ(X → j) = j Γ(X →j) from Eq. (13). Therefore, the lepton asymmetry in the integrand of the Boltzmann equation (13) is found to be Note that the terms proportional to |h| 4 , which do not break lepton number, disappear because of the previous relation of C L RL − CLR L = 0. Thus, the asymmetry in Eq. (18) obviously contains only the mixed terms with hλ 1,2 , signaling a lepton number violation.
With the universality condition for the soft terms (A = A L = A 1 = A 2 ), we get a simple equation for the lepton asymmetry as follows: where and putting R = C = 1 in the denominator. In the limit M ≫ m 0 , |A|, we have ignoring the small thermal effect and thus putting R k = 1. One thus finds that, the quantity inside the integrand of Eq. (13) is proportional to Here, one has the approximation of δ sof t = (m which gives a further suppression compared to the conventional contribution with the Bessel Function K 1 (z). The above expression, which is monotonically decreasing in z, is valid for z ≫ 1, and is compared in FIG. 4 to a numerical calculation including the effect of thermal masses, which cause δ BF to vanish at small z. The latter calculation of δ BF is obtained by numerically evaluating the thermal average of the absorption part of the two-point function Π ±∓ . Concluding the above discussions, we find that the total lepton asymmetry density Y l = Y L + YL follows the approximate Boltzmann equation: where g ∆ = 6 counts the total number of triplet components generating the lepton asymmetry and δ(z) ≡ δ BF (z) + δ sof t . In the above equation, the number K = Γ ± /H 1 takes the minimal value of K = 32 for |h| = |λ 2 | ≫ |λ 1 | as we have the relation [7]; As one goes away from the minimum value of K with |h| = |λ 2 |, the quantityǫ l in Eq. (21) gets suppressed. Furthermore, one realizes that the resulting lepton asymmetry is maximized in case of B L,L = B H 2 ,H 2 ≫ B H 1 ,H 1 with |h| = |λ 2 | ≫ |λ 1 |, in which case the Boltzmann equation for the lepton asymmetry takes the simplest form of Let us now note that, taking the resonance condition B = Γ ± , one finds the maximal value ofǫ l = Im(A) M , which becomes order one for A ∼ M ∼ TeV. It is now easy to find the approximate solution for Y l from Eq. (25) with the insertion of Y X given in Eq. (11). Both are found to be a fairly good approximation to the numerical solution of the full Boltzmann equations, as expected from our previous discussions. The results of our numerical calculation are shown in FIG. 5 where we plot the final lepton asymmetry as a function of the triplet mass M for Im(A) from 1 to 5 TeV. When |A| ∼ M, one needs to recover the contributions of order |A|/M which were neglected, e.g., in Eqs. (15) and (21) annihilation rate, increasing substantially the value of the parameter K compared to the amount given by Eq. (24), which is obtained in the limit |A|/M ≪ 1. As a consequence of this, the consequent additional wash-out effect could in principle suppress the ensuing lepton asymmetry. However, this is not the case due to the fact that, as we have already discussed, the annihilation process freezes out later than inverse decays, so the latter play almost no role in the determination of the epoch when lepton asymmetry production can start. Actually, this epoch starts when decays eventually overcome annihilations, so a higher value of K can slightly anticipate it, leading so to a higher asymmetry instead than a suppression, although this effect is quite mild. This is what we observe in the numerical calculation shown in FIG. 5, where we have assumed as before |h| = |λ 2 | ≫ |λ 1 | (in order to maximize the amount of CP violation given by Eq. (21)) and maximized the CP -violating phase (i.e., we have assumed Re(A) = 0).
We also remark that sphaleron interactions are kept in thermal equilibrium even after the electroweak phase transition and freeze out around T sp = M/z sp ≃ 90 GeV, so that only the lepton asymmetry produced for z < z sp can be efficiently converted into a baryon asymmetry [13]. As a result, it is evident that the required baryon asymmetry can be reached whenever A and M are in the multi-TeV region.
Before concluding our work, let us remark some experimental consequences of the model at future colliders. As shown above, successful baryogenesis requires a TeV-scale triplet mass and Yukawa couplings of the same order, h ∼ λ 2 ∼ m ν M/v 2 2 ∼ 10 −6 . Thus, all the low-energy lepton flavor violating processes like µ → eγ or µ → 3e are highly suppressed [11]. On the other hand, future accelerators have a potential to produce such Higgs triplets, in particular, the peculiar doubly charged component through the Drell-Yan processes [10].
Then, various features of the model can be checked by observing the branching ratios of the triplet decay to lepton and Higgsino pairs, in particular, 2 , allowing also to study neutrino mass patterns [11].
In conclusion, we have investigated baryogenesis assuming the minimal supersymmetric Higgs triplet model as the origin of neutrino masses and mixings. This model, with only one pair of triplets, can provide a mechanism for soft leptogenesis employing a CP violating phase and a resonant behavior in the supersymmetry breaking sector. Our analysis shows that the original soft leptogenesis, relying on the supersymmetry breaking effect proportional to the small difference between boson and fermion statistics at finite temperature cannot produce the right amount of baryon asymmetry due to the gauge annihilation effect. In particular, we have calculated the full gauge-annihilation cross section including all the relevant supersymmetric intermediate and final states, as well as coannihilations with the fermionic superpartners of the triplets, finding that this effect strongly suppresses the resulting lepton asymmetry. On the other hand, the contribution of soft supersymmetry breaking terms, particularly a sizable value for the Im(A) parameter, can enhance the lepton asymmetry to provide successful leptogenesis if the triplet mass is in the TeV range. In this case, the model predictions can be tested in future colliders by searching for a very clean signal, e.g., from the production and decay of doubly charged Higgs bosons.
In the last equation we have kept within squared parentheses quantities that vanish for β → 0, and the subscripts refer to the contributing Feynman diagrams listed in Fig. 1. In Eq. (26), T a and τ a /2 are the SU(2)× U(1) group generators for the triplet and doublet representations and f abc are the structure constants. Assuming the minimal supersymmetric Standard Model particle content, the traces are given by: 1 3 a tr(T a ) 2 tr( τ a 2 ) 2 = g 4 2 (14 + 11t 4 w ),