Freeze-in Dark Matter from a Minimal B-L Model and Possible Grand Unification

We show that a minimal local $B-L$ symmetry extension of the standard model can provide a unified description of both neutrino mass and dark matter. In our model, $B-L$ breaking is responsible for neutrino masses via the seesaw mechanism, whereas the real part of the $B-L$ breaking Higgs field (called $\sigma$ here) plays the role of a freeze-in dark matter candidate for a wide parameter range. Since the $\sigma$-particle is unstable, for it to qualify as dark matter, its lifetime must be longer than $10^{25}$ seconds implying that the $B-L$ gauge coupling must be very small. This in turn implies that the dark matter relic density must arise from the freeze-in mechanism. The dark matter lifetime bound combined with dark matter relic density gives a lower bound on the $B-L$ gauge boson mass in terms of the dark matter mass. We point out parameter domains where the dark matter mass can be both in the keV to MeV range as well as in the PeV range. We discuss ways to test some parameter ranges of this scenario in collider experiments. Finally, we show that if instead of $B-L$, we consider the extra $U(1)$ generator to be $-4I_{3R}+3(B-L)$, the basic phenomenology remains unaltered and for certain gauge coupling ranges, the model can be embedded into a five dimensional $SO(10)$ grand unified theory.


INTRODUCTION
If small neutrino masses arise via the seesaw mechanism [1][2][3][4][5], the addition of a local B − L symmetry [6,7] to the standard model (SM) provides a minimal scenario for beyond the standard model (BSM) physics to achieve this goal. There are two possible classes of B − L models: one where the B − L generator contributes to the electric charge [6][7][8] and another where it does not [9][10][11]. In the first case, the B − L gauge coupling g BL has a lower limit whereas in the second case it does not and therefore can be arbitrarily small.
There are constraints on the allowed ranges of g BL from different observations [12,13] in the second case depending on whether there is or is not a dark matter particle in the theory.
In Refs. [14,15], it was shown that if we added a B −L charge carrying vector-like fermion to the minimal B − L model and want it to play the role of dark matter, new constraints emerge. In this note, we discuss an alternative possibility with the following new results.
First is that the minimal version of the B − L model itself, without any extra particles, can provide a dark matter (DM) candidate. The DM turns out to be the real part (denoted here as σ) of the complex B − L = 2 Higgs field, that breaks B − L and gives mass to the right handed neutrinos in the seesaw formula. Even though this particle is not stable, there are certain allowed parameter ranges of the model, where its lifetime can be so long that it can play the role of a decaying dark matter. We isolate this parameter range and show that in this case, the freeze-in mechanism [16] can generate its relic density. We find this possibility to be interesting since it unifies both neutrino masses and dark matter in a single minimal framework. We show how a portion of the parameter range of the model suggested by the dark matter possibility, can be probed by the recently approved FASER experiment at the LHC [17] and other Lifetime Frontier experiments.
We then show that if we replace the B−L symmetry byĨ ≡ −4I 3R +3(B−L) (where I 3R is the right handed weak isospin), the dark matter phenomenology remains largely unchanged and the model can be embedded into the SO(10) grand unified theory in five space-time dimensions. Such a symmetry breaking of SO (10) to SU (5) × U (1)Ĩ has already been shown to arise from a symmetry breaking by a particular alignment for a vacuum expectation value (VEV) of a 45-dimensional Higgs field [18]. This paper is organized as follows: in Sec. 2 after briefly introducing the model, we discuss the lifetime of the σ dark matter and its implications. In Sec. 3, we discuss the small gauge coupling g BL range where the dark matter lifetime is long enough for it to play the role of dark matter. In Sec. 4, we show how freeze in mechanism determines the relic density of dark matter and its implications for the allowed parameter range of the model. We also discuss how to test this model at the FASER and other Lifetime Frontier experiments. In Sec. 5, we show that this model can also accommodate a PeV dark matter. In Sec. 6, we discuss the SO(10) embedding of the closely allied model and in Sec. 7, we conclude with some comments and other implications of the model.
We need three right handed neutrinos (RHNs) with B − L = −1 to cancel the B − L anomaly. The RHNs being SM singlets do not contribute to SM anomalies. The electric charge formula in this case is same as in the SM i.e. Q = I 3L + Y 2 . We break B − L symmetry by giving a VEV to a B − L = 2 SM neutral complex Higgs field ∆ i.e. ∆ = v BL / √ 2. This gives Majorana masses to the right handed neutrinos (N ) via the coupling f N N ∆. The real part of ∆ (denoted by σ) is a physical field. Our goal in this paper is to show that σ has the right properties to play the role of a dark matter of the universe. There are three challenges to achieving this goal: (i) The σ field has couplings to the RHNs which in turn couples to SM particles providing a way for σ to decay. Also, the σ field has couplings to two B − L gauge bosons (Z BL ) which in turn couple to SM fields providing another channel for σ to decay. In the next section, we show that there are parameter regions of the model where these decay modes give a long enough lifetimes for σ, so that it can be a viable unstable dark matter in the universe.
(ii) The second challenge is that for σ to be a sole dark matter, it must account for the total observed relic density of the universe Ω DM h 2 0.12 [19]. We show in Sec. 4 that in the same parameter range, that gives rise to the long lifetime of σ, can also explain the observed relic density of dark matter via the freeze-in mechanism.
(iii) The σ field could mix with the standard model Higgs field h via the potential term λ H † H∆ † ∆ after symmetry breaking. However, it turns out that if we set λ = 0 at the tree level, it can be induced at the one-loop level by fermion contributions and at the two-loop level from the top loop as shown in Ref. [11]. These induced couplings can be so small that they still lead to very long lifetimes for σ in the parameter range of interest to us.

DARK MATTER LIFETIME
As noted earlier in Sec. 2, the σ field has couplings which could make it unstable and thereby disqualify it from being a dark matter. However, we will show that there is a viable parameter range of the model where this decay lifetime is longer than 10 25 sec. [20] so that it can be a dark matter candidate. We discuss these two modes now: (i) Decay mode σ → N N → ff ff : the decay width for this process is estimated as where h ν is a neutrino Dirac Yukawa coupling, h SM is a Yukawa coupling of an SM fermion f , and m h = 125 GeV is the SM Higgs boson mass. For a GeV mass σ and TeV mass RHN, the lifetime of σ turns out to be τ σ [sec] ∼ 10 37 /(f 2 h 4 SM ), which is quite consistent with the requirement for it to be a dark matter. Here, we have used the seesaw formula This mode is sensitive to the values of g BL as well as M Z BL . The estimate of τ σ due to this decay mode is given by Imposing τ σ > 10 25 sec., this puts an upper bound on the g BL as a function of M Z BL and m σ : We find that the allowed regions where the σ field can be a dark matter correspond to a very small g BL coupling. For instance, for m σ ∼ 1 GeV and M Z BL ∼ 1 TeV, we find that (iii) We now comment on the σ-Higgs mixing effect on the DM lifetime. To keep the lifetime above limit τ σ > 10 25 sec., we set the tree-level H-∆ coupling in the Higgs potential to zero so that σ and the SM Higgs field h do not mix at the tree level. This will, for example be true if the model becomes supersymmetric at a high scale. The σ-Higgs mixing in this case is loop induced as shown in Ref. [11] and for the parameter range of interest to us, can be small enough to satisfy the DM lifetime constraint as we show below.
For the case when m σ ≤ m h , the dominant contribution to the loop induced mixing comes from a RHN fermion box diagram and the mixing angle can be estimated to be . Through this mixing, the DM particle can decay to a pair of SM fermions with a partial decay width of The lifetime constraint then translates to a limit on g BL as follows: With a suitable choice of M N (> m σ ), we can see that this limit is quite compatible with our results shown in the right panel of Figs. 1, Fig. 2 .
For the case when m σ > m h , on the other hand, the DM particle can decay to a pair of Higgs doublets through the mixing, and we find that the loop induced mixing is not small enough to be consistent with the results shown in the right panels of Figs. 1 and 3. In this case, we consider a cancellation of the mixing between the tree and loop levels contriburions.
We will now explore whether for such small parametric values for g BL , we can generate the observed dark matter relic density of the universe.

RELIC DENSITY
4.1 Allowed range of g BL from pre-conditions to freeze-in First point to notice is that for GeV scale DM (σ), for values of g BL that satisfy the lifetime constraint, the σ field is out of equilibrium from the SM particles. Therefore, the standard thermal freeze-out mechanism for creation of DM relic density does not apply and one has to explore the freeze-in mechanism. For this to work, we need the Z BL field, whose annihilation will produce the DM, to be in equilibrium with the SM fields. This question was explored in Ref. [14] and it was pointed out that the most efficient process for Z BL to be in equilibrium with SM particles is via the process ff → Z BL + γ. The condition on g BL for this to happen is g BL > 2.7 × 10 −8 M Z BL 1 GeV An upper bound on g BL comes from the fact that the DM particle σ is out of equilibrium in the early universe. The first process to consider is Z BL Z BL ↔ σσ for which the out-ofequilibrium condition is given by n σ σv < H,. Here n σ ∼ T 3 is the number density of the DM σ, σv ∼ g 4 BL /(4πT 2 ), and the Hubble parameter H = π 2 90 g * T 2 /M P with the reduced Planck mass M P = 2.43 × 10 18 GeV and the effective total number of relativistic degrees of freedom g * (we set g * = 106.75 for the SM particle plasma in our analysis throughout this paper). Requiring that this inequality is satisfied until T ∼ M Z BL , we find that g BL < . Combining with the equilibrium condition for Z BL , we find that we have to work in the range of g BL values to generate the relic density.
There is another upper bound on g BL that arises from the fact that the process N N → σσ should also out of equilibrium. The reason is that in the early universe, the right handed neutrinos are always in equilibrium with SM particles via processes such as N + t ↔ ν + t etc. and N ↔ H for M N > m h . If N N ↔ σσ is also in equilibrium, the freeze-in mechanism for relic density generation of σ will not work. To get this upper bound on g BL using this condition, we use n σ σ N N →σσ v < H at T ∼ M N and find 1 4π Using M Z BL = 2g BL v BL , this leads to Note that for M N ∼ M Z BL , this upper limit is about the same level as in Eq. (6) so that indeed the freeze-in mechanism is called for in creating the relic density build-up. In the following, we consider M N < M Z BL , for which the upper bound is determined by the B − L gauge interaction. Incidentally, we note that If M N < m h , the interactions of the RHNs with the SM particles are too week for them to be in thermal equilibrium, and the above discussion is not applicable. 1 1 Note also that, as a general possibility, if M N is greater than the reheating temperature after inflation (T RH ), the RHN is irrelevant to our DM physics discussion.

Relic density build-up
In order to calculate the relic density build-up via the freeze-in mechanism, we solve the where Y is the yield of the DM σ, Y eq is Y if the DM σ is in thermal equilibrium, and s(m σ ) and H(m σ ) are the entropy density and the Hubble parameter, respectively, evaluated at T = m σ . For the DM particle creation process Z BL Z B → σσ, we approximate σv for this is that for T ≤ M Z BL , the number density of Z BL is Boltzmann suppressed and σ particle creation stops. Using S(mσ) 14 m σ M P and Y eq 2.2 × 10 −3 and integrating the Then taking Y (∞) Y (x BL = m σ /M Z BL ), we estimate the DM relic density, where s 0 = 2890/cm 3 is the entropy density of the present universe, and ρ c /h 2 = 1.05 × 10 −5 GeVcm 3 is the critical density. This leads to the following expression for g BL : to reproduce the observed DM relic density Ω DM h 2 = 0.12.
Considering all the constraints from Eqs. (6), (12) and (13), we show the allowed parameter region in Fig. 1 (Right Panel). The region between two diagonal black lines satisfies the In the right panel of Fig. 1, we can see that there is an allowed parameter region for g BL = O(10 −5 ) and M Z BL = 1 MeV−1 GeV. For the parameter region, Z BL boson can be long-lived and such a long-lived neutral particle can be explored in the near future by the Lifetime Frontier experiments, such as FASER [17], SHiP [21], LDMX [22], Belle II [23], and LHCb [24,25]. The Z BL boson search of the FASER experiment at the LHC is summarized in Ref. [17] along with the search reaches of other experiments as well as the current excluded region [26]. In Fig. 2, we show our results of the right panel of Fig. 2 along with the summary plot in Ref. [17]. other words, through the seesaw formula, N is sufficiently light. The discussion for the DM production process of H → N σ is applicable even if the RHN is not in thermal equilibrium.

PEV DARK MATTER FROM B − L BREAKING
So far we have explored the lower mass range of the dark matter. In this section, we explore the possibility that the σ mass is in the PeV range so that one could attempt to explain the 100 TeV to PeV neutrinos observed in IceCube Neutrino Observatory [27] by using σ decay. We do not attempt to explain the IceCube signal here but simply to raise the possibility that a PeV mass σ can also qualify as the dark matter in our model in a different parameter range. For this purpose, let us go through all the constraints on the model discussed above for this case.

Lifetime constraint
This constraint is same as in the case of light σ in Eq. (4) except that in the right-hand side, the masses of σ and Z BL are now higher and the new constraint can be written as If we restrict the B − L breaking VEV v BL ≤ 10 16 GeV, then the lifetime constraint can be translated to M Z BL ∼ 10 10 GeV for g BL as large as 10 −5 .
We note that the one-loop σ − h mixing contribution in this case leads to a very strong upper limit on the g BL value and much too small to generate enough relic density for the dark matter. In this case tyherefore, we fine tune the tree-level and one-loop σ-Higgs coupling to zero.

Relic density constraints
We next explore the constraints of relic density on the heavy DM case. For such low g BL values, a heavy PeV scale DM and the 10 10 GeV or higher mass Z BL would never have been in equilibrium. The relic density must arise as in the first case via the freeze-in mechanism. Since Z BL is not in thermal equilibrium, the production takes place via the process ff → Z BL σ through the SM fermion pair annihilations in the thermal plasma. In this case, the Boltzmann equation is given by where Y BL eq is the yield of Z BL in thermal equilibrium and the cross section for the process ff → Z BL σ is estimated as Recall that the DM production stops at T M Z BL due to kinematics. Using Y BL eq 2Y eq where we have used Y (x RH ) = 0 and x RH x BL . We now use, as before, and estimate the DM relic density, In order to reproduce Ω DM h 2 = 0.12, we find We require that the Z BL is not in equilibrium which gives the consistency condition In Fig. 3  The condition of v BL ≤ M P is depicted by the right diagonal black line.
of v BL ≤ M P , which is depicted by the right diagonal black line. We thus see that there is enough parameter range in the model for the dark matter to be in the PeV range so that it can be relevant to the PeV neutrinos observed in IceCube experiment. This is possible for M Z BL > ∼ 10 10 GeV and v BL > ∼ 10 14 GeV.

PROSPECTS FOR SO(10) EMBEDDING
In this section, we like to point out that a slight variation of the model leads to its possible embedding into SO(10) grand unified theory (GUT), which we believe should add to its theoretical appeal as a minimal GUT model that unifies neutrino masses and dark matter.
The starting point of this discussion is the observation that the hypercharge generator Y is a linear combination of the I 3R and the normalized B − L generators I BL of SO(10) as follows: where The B − L generator in the main body of the paper is not orthogonal to the Y generator defined above. Therefore, it cannot emerge from SO(10) breaking since I BL is not orthogonal to Y defined above. Instead if we consider the generatorĨ ≡ −4I 3R + 3(B − L), we get Tr(ĨY ) = 0 (i.e. they are orthogonal) for any irreducible representation of SO(10) and can therefore emerge from SO(10) breaking. This generator was also identified in Ref [28] as the generator U (1) X for x H = −4/5. Indeed, it has been shown in Ref. [18] that such a generator emerges out of SO(10) breaking by a 45 Higgs field. To see this note that (1, 1, 3, 0) which can take VEVs ω Y and ω BL , respectively. If we fine-tune the parameters of the Higgs potential, we can get ω Y = ω BL in which case the unbroken generators are As it turns out, the dark matter phenomenology discussed above remains unchanged if we use the Higgs field σ to break the U (1)Ĩ symmetry. The σ field then emerges from the 126-dimensional representation of SO(10) and our dark matter field σ hasĨ = √ 10 2 and therefore has all the properties required above for our dark matter.
where bĨ = −49/10 at a scale µ below the SU (5) unification while bĨ = −5 in SU (5) × U (1)Ĩ theory by considering that the SM Higgs doublet is embedded into a 5-representation in SU (5). For simplicity, we have assumed that in each step of the gauge symmetry breaking, only the minimal sets of Higgs fields are light.
To see our coupling unification strategy in this model, we first discuss the SU (5) unification without supersymmetry. As is clear, in this case, we will need extra fields beyond the SM fields below the SU (5) unification scale. For this purpose, we introduce n 3 real scalar SU (2) L triplets with Y = 0 and n 8 real scalar color octets with Y = 0. The coupling evolution equations in this case are the following: where M 3,8 stand for the masses of the triplet (1, 3, 1) and octet (8, 1, 0) fields, respectively.
Solving these equations with n 3 = 5 with mass M 3 = 5 TeV and n 8 = 3 with mass M 8 = 200 TeV, we find that the SU (5) gauge coupling unification is achieved at M U = 6.8 × 10 15 GeV.
Let us now proceed to SO(10) unification i.e. the running of the gĨ coupling from its breaking scale (which does not affect very much) to where it unifies with the SU (5) coupling evolving after the SU (5) unification scale. We see that due to the small value of gĨ required to get the relic density from the freeze-in mechanism, the SO(10) gauge coupling unification in 4-dimensions is hard to obtain. We therefore assume that above the SU (5) GUT scale, the model becomes five dimensional [29] with the fifth dimension compactified on S 1 /Z 2 orbifold with a radius R = M U −1 . In that case if we assume that the gauge fields are in the bulk while all the matter and Higgs fields are on a brane at an orbifold fixed point, their Here, in the parenthesis of the right-hand side, 43/3 is the contribution from the zero-mode SU (5) gauge boson and the SM fermions, −1/6 from the 5-representation Higgs field, and − 5 6 (1 + n 3 + n 8 ) from one adjoint Higgs to break the SU (5) symmetry and n 3 + n 8 adjoint Higgs field into which the triplet and octet scalars are embedded, and the last term is the contribution from the SU (5) gauge boson KK modes. For the KK mode mass spectrum, we have simply added the contribution from the SU (5) symmetry breaking. Once the extra dimension opens, the contribution from the KK modes changes the scale dependence of the running gauge coupling from a log to a power [29]. Thus it is possible to unify the SU (5) and U (1)Ĩ couplings into SO(10) coupling as desired. This is shown in Fig. 4. In the figure, the SO(10) gauge coupling unification is achieved at M P with a unified coupling g SO(10) 0.1.
This result corresponds to an allowed parameter set, m σ 100 keV and M Z BL = 10 14 GeV, in the right panel of Fig. 3.
As far as proton decay is concerned, the primary mode is p → e + + π 0 mediated by the SU (5) gauge boson. The proton decay amplitude gets contribution from all the KK excitations of the SU(5) gauge fields, and we estimate the modification of a coefficient of the 4-Fermi operator to be Then, (ignoring threshold effects) the proton lifetime is estimated as where m p = 0.938 GeV. Using α 5 (M U ) 0.026 and M U 6.8×10 15 GeV from Fig. 4, we find that τ p 2.1×10 34 years, which is consistent with the lower bound τ p ≥ 1.6×10 34 years from the Super-Kamionkande results [30]. More importantly, we would expect that p → e + π 0 should be observable in the next round of proton decay searches at Hyper-Kamiopkande [31] or the model will be ruled out. interesting LHC phenomenology [32,33]. Discussion of this phenomenology is beyond the scope of this paper. There are also ranges for the RHN masses in the model where resonant leptogenesis can generate the baryon asymmetry of the universe. This will be the subject of a forthcoming publication.