PHYSICAL REVIEW D 101, 115022 (2020)
Freeze-in dark matter from a minimal B − L model and possible grand unification
Rabindra N. Mohapatra1 and Nobuchika Okada 2 1Maryland Center for Fundamental Physics and Department of Physics, University of Maryland, College Park, Maryland 20742, USA 2Department of Physics and Astronomy, University of Alabama, Tuscaloosa, Alabama 35487, USA
(Received 5 May 2020; accepted 5 June 2020; published 17 June 2020)
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 σ here) plays the role of a freeze-in dark matter candidate for a wide parameter range. Since the σ particle is unstable, for it to qualify as dark matter, its lifetime must be longer than 1025 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 Uð1Þ generator to be −4I3R þ 3ðB − LÞ, the basic phenomenology remains unaltered and for certain gauge coupling ranges, the model can be embedded into a five-dimensional SOð10Þ grand unified theory.
DOI: 10.1103/PhysRevD.101.115022
I. INTRODUCTION possibility with the following new results. First is that the minimal version of the B − L model itself, without any If small neutrino masses arise via the seesaw mechanism extra particles, can provide a dark matter (DM) candidate. [1–5], the addition of a local B − L symmetry [6,7] to the The DM turns out to be the real part (denoted here as σ)of standard model (SM) provides a minimal scenario for the complex B − L ¼ 2 Higgs field, that breaks B − L and beyond the standard model physics to achieve this goal. gives mass to the right-handed neutrinos in the seesaw There are two possible classes of B − L models: one where formula. Even though this particle is not stable, there are the B − L generator contributes to the electric charge [6–8] certain allowed parameter ranges of the model, where its and another where it does not [9–11]. In the first case, the lifetime can be so long that it can play the role of a decaying B − L gauge coupling g has a lower limit, whereas in the BL dark matter. We isolate this parameter range and show that second case it does not and therefore can be arbitrarily in this case, the freeze-in mechanism [17] can generate its small. There are constraints on the allowed ranges of g BL relic density. We find this possibility to be interesting since from different observations [12,13] in the second case it unifies both neutrino masses and dark matter in a single depending on whether there is or is not a dark matter minimal framework [18]. We show how a portion of the particle in the theory. There are also possible ways to look parameter range of the model suggested by the dark matter B − L for gravitational wave signals of breaking in the possibility can be probed by the recently approved FASER early Universe [14]. experiment at the LHC [19] and other Lifetime Frontier In Refs. [15,16], it was shown that if we added a B − L − experiments. charge carrying vectorlike fermion to the minimal B L We then show that if we replace the B − L symmetry by model and want it to play the role of dark matter, new ˜ I ≡ −4I3 þ 3ðB − LÞ (where I3 is the right-handed weak constraints emerge. In this note, we discuss an alternative R R isospin), the dark matter phenomenology remains largely unchanged and the model can be embedded into the SOð10Þ grand unified theory in five space-time dimensions. Published by the American Physical Society under the terms of Such a symmetry breaking of SOð10Þ to SUð5Þ × Uð1Þ˜ the Creative Commons Attribution 4.0 International license. I Further distribution of this work must maintain attribution to has already been shown to arise from a symmetry breaking the author(s) and the published article’s title, journal citation, by a particular alignment for a vacuum expectation value and DOI. Funded by SCOAP3. (VEV) of a 45-dimensional Higgs field [20].
2470-0010=2020=101(11)=115022(9) 115022-1 Published by the American Physical Society RABINDRA N. MOHAPATRA and NOBUCHIKA OKADA PHYS. REV. D 101, 115022 (2020)
We note that there are examples of other models in the symmetry breaking. However, it turns out that if 0 literature connecting neutrino mass generation mechanisms we set λ ¼ 0 at the tree level, it can be induced at the to dark matter; see, for example, Refs. [21,22]. one-loop level by fermion contributions and at the This paper is organized as follows: in Sec. II, after briefly two-loop level from the top loop as shown in introducing the model, we discuss the lifetime of the σ dark Ref. [11]. These induced couplings can be so small matter and its implications. In Sec. III, we discuss the small that they still lead to very long lifetimes for σ in the parameter range of interest to us. gauge coupling gBL range where the dark matter lifetime is long enough for it to play the role of dark matter. In Sec. IV, we show how freeze-in mechanism determines the relic III. DARK MATTER LIFETIME density of dark matter and its implications for the allowed σ parameter range of the model. We also discuss how to test As noted earlier in Sec. II, the field has couplings this model at the FASER and other Lifetime Frontier which could make it unstable and thereby disqualify it from experiments. In Sec. V, we show that this model can also being a dark matter. However, we will show that there is a accommodate a PeV dark matter. In Sec. VI, we discuss the viable parameter range of the model where this decay 25 SOð10Þ embedding of the closely allied model and in lifetime is longer than 10 seconds [24] so that it can be a Sec. VII, we conclude with some comments and other dark matter candidate. We discuss the following two implications of the model. modes now: (i) Decay mode σ → NN → lff¯lff¯: The decay width for this process is estimated as II. BRIEF OVERVIEW OF THE MODEL ð 2 2 Þ2 13 ð1Þ fhνhSM mσ Our model is based on the U B−L extension of the SM Γ ≃ ; ð1Þ ð1Þ NN ð4πÞ8 M4 m8 with gauge quantum numbers under U B−L determined N h by the baryon or lepton number of the particles. The gauge where hν is a neutrino Dirac Yukawa coupling, hSM group of the model is SUð3Þ ×SUð2Þ ×Uð1Þ ×Uð1Þ − , c L Y B L is a Yukawa coupling of an SM fermion f, and mh ¼ where Y is the SM hypercharge. We need three right- 125 GeV is the SM Higgs boson mass. For a GeV − ¼ −1 handed neutrinos (RHNs) with B L to cancel the mass σ and TeV mass RHN, the lifetime of σ B − L anomaly. The RHNs being SM singlets do not τ ½ ∼ 1037 ð 2 4 Þ turns out to be σ sec = f hSM , which is contribute to SM anomalies. The electric charge formula in quite consistent with the requirement for it to be a ¼ þ Y this case is same as in the SM, i.e., Q I3L 2. dark matter. Here, we have used the seesaw formula − 2 2 We break B L symmetry by giving a VEV to a hνv =M ≃ mν with v ¼ 246 GeV and a − ¼ 2 Δ hΔi¼ EW N EW B Lpffiffiffi SM neutral complex Higgs field , i.e., typical neutrino mass scale mν ≃ 0.1 eV. 2 ¯ ¯ vBL= . This gives Majorana masses to the right-handed (ii) Decay mode σ → ZBLZBL → ffff: The decay neutrinos (N) via the coupling fNNΔ. The real part of Δ width for this process is σ (denoted by ) is a physical field. Our goal in this paper is 4 2 4 7 6 7 σ ð2g Þ v g mσ g mσ to show that has the right properties to play the role of a Γ ≃ BL BL BL ¼ BL : ð2Þ ZBLZBL ð4πÞ5M8 256π5 M6 dark matter of the universe. There are three challenges to ZBL ZBL achieving this goal which are as follows: (i) The σ field has couplings to the RHNs which in turn This mode is sensitive to the values of gBL as well as M . The estimate of τσ due to this decay mode is couple to SM particles providing a way for σ to ZBL σ B − L given by decay. Also, the field has couplings to two gauge bosons (Z ) which in turn couple to SM 6 7 BL −20 1 1 GeV σ τσ ≃ 5.2 × 10 fields providing another channel for to decay. In g m the next section, we show that there are parameter BL σ M 6 regions of the model where these decay modes give a × ZBL seconds: ð3Þ long enough lifetimes for σ, so that it can be a viable 1 GeV unstable dark matter in the universe. τ 1025 (ii) The second challenge is that for σ to be a sole dark Imposing σ > seconds, this puts an upper bound on the gBL as a function of MZ and mσ, matter, it must account for the total observed relic BL Ω 2 ≃ 0 12 7 6 density of the universe DMh . [23]. We show M 1 GeV = ≤ 4 2 10−8 ZBL ð Þ in Sec. IV that in the same parameter range, that gBL . × : 4 1 GeV mσ gives rise to the long lifetime of σ, can also explain the observed relic density of dark matter via the We find that the allowed regions where the σ field freeze-in mechanism. can be a dark matter correspond to a very small gBL σ ∼ 1 ∼ (iii) The field could mix with the standard model Higgs coupling. For instance, for mσ GeV and MZBL 0 † † −5 field h via the potential term λ H HΔ Δ after 1 TeV, we find that gBL ≲ 4 × 10 .
115022-2 FREEZE-IN DARK MATTER FROM A MINIMAL B − L MODEL … PHYS. REV. D 101, 115022 (2020)
FIG. 1. Left panel: the dark matter σ lifetime as a function of MZ . The diagonal solid lines correspond to mσ ¼ 1 MeV, 10 MeV, BL 2 100 MeV, and 1 GeV from left to right, along which the observed DM relic density of ΩDMh ¼ 0.12 is reproduced. Right panel: the gBL values as a function of MZBL from the requirement of relic density buildup. Different red lines correspond to different DM masses (mσ starting with 10 keVat the top and as we go below, we go in steps of a factor of 10 to 100 keV, 1 MeV, etc., till 100 GeV) that satisfy the Ω 2 ¼ 0 12 relic density constraint, i.e., DMh . . Two diagonal black lines denote the condition of Eq. (6), and the horizontal black line corresponds to Eq. (14).
(iii) We now comment on the σ-Higgs mixing effect on For the case when mσ >mh, on the other hand, the DM the DM lifetime. To keep the lifetime above limit particle can decay to a pair of Higgs doublets through the 25 τσ > 10 seconds, we set the tree-level H − Δ mixing, and we find that the loop induced mixing is not 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 mσ ≤ mh, the dominant contribution to the loop induced mixing comes from an RHN fermion box diagram. This contribution is logarithmically diver- gent. Using the Planck mass as the cutoff, we can estimate 2 2 3 1 mνM 2 θ ∼ f hν vEWvBL ∼ N gBL the mixing angle to be 16π2 2 16π2 2 M . mh vEWmh ZBL Through this mixing, the DM particle can decay to a pair of SM fermions with a partial decay width of 2 θ mf 2 Γσ→ ¯ ∼ ð Þ mσ. The lifetime constraint then translates ff 4π vEW to a limit on gBL as follows: v 1 GeV 1=2 1 GeV 3 M 2 8 10−6 EW ZBL FIG. 2. FASER reachable region of the parameter space of our gBL < . × 1 : mf mσ MN GeV model. The black lines at the top and bottom denote the upper and ð5Þ lower limits on the gBL [Eq. (6)]. The red lines correspond to mσ ¼ 10 keV, 100 keV, 1 MeV, and 10 MeV from top to bottom, respectively, along which ΩDM ¼ 0.12 is satisfied. The parameter With a suitable choice of M ð>mσÞ, we can see that this 10 ≲ ≲ 1 10 ≲ ≲ N region of keV mσ MeV and MeV MZBL afew limit is quite compatible with our results shown in the right GeV can be tested by various Lifetime Frontier experiments in the panel of Figs. 1 and 2. near future.
115022-3 RABINDRA N. MOHAPATRA and NOBUCHIKA OKADA PHYS. REV. D 101, 115022 (2020)
2 FIG. 3. Left panel: the red line corresponds to DM mass mσ ¼ 1 PeV with Ω h ¼ 0.12. This corresponds to the case where the DM ¯ DM is produced by ff → ZBLσ. The left of dashed line corresponds to the DM being in thermal equilibrium and therefore is not the area for 25 freeze-in case. The left of black solid line corresponds to τσ < 10 seconds and is excluded. Right panel: the red lines represent the DM Ω 2 ¼ 0 12 masses from top 100 keV, 10 MeV, 1 GeV (jump of 100 times) till 100 PeV being the lowest red line. Along the red line DMh . is satisfied. The lower black line comes from Eq. (21). The upper black line corresponds to ZBL not being in equilibrium. The condition of vBL ≤ MP is depicted by the right diagonal black line.
18 small enough to be consistent with the results shown in the MP ¼ 2.43 × 10 GeV and the effective total number of right panels of Figs. 1 and 3. In this case, we consider a relativistic degrees of freedom g (we set g ¼ 106.75 cancellation of the mixing between the tree- and loop-levels for the SM particle plasma in our analysis throughout contributions. this paper). Requiring that this inequality is satisfied until M 1=4 We will now explore whether for such small parametric T ∼ M , we find that g < 6.4 × 10−5ð ZBL Þ . ZBL BL 1 GeV values for gBL, we can generate the observed dark matter Combining with the equilibrium condition for Z ,we relic density of the universe. BL find that we have to work in the range of gBL values IV. RELIC DENSITY 1=2 1=4 −8 MZBL −5 MZBL 2.7 × 10 115022-4 FREEZE-IN DARK MATTER FROM A MINIMAL B − L MODEL … PHYS. REV. D 101, 115022 (2020) 1=4 3=4 1=4 1 1=4 −5 MZBL MZBL −6 MZBL GeV gBL < 3.2 × 10 : ð8Þ gBL ≃ 2.4 × 10 ð12Þ 1 GeV MN 1 GeV mσ ∼ to reproduce the observed DM relic density Ω h2 ¼ 0.12. Note that for MN MZBL, this upper limit is about the same DM level as in Eq. (6) so that indeed the freeze-in mechanism is Using Eq. (12) in Eq. (3), we show the lifetime for called for in creating the relic density buildup. In the various values of mσ in Fig. 1 (left panel). The diagonal lines from left to right correspond to mσ ¼ 1 MeV, following, we consider MN Considering all the constraints from Eqs. (6), (12), and where Y is the yield of the DM σ, Yeq is Y if the DM σ is in (14), we show the allowed parameter region in Fig. 1 (right thermal equilibrium, and sðmσÞ and HðmσÞ are the entropy density and the Hubble parameter, respectively, evaluated panel). The region between two diagonal black lines ¼ satisfies the condition of Eq. (6) and the horizontal black at T mσ. For the DM particle creation process Ω 2 ¼ 0 12 4 4 2 line corresponds to Eq. (14). The observed DMh . gBL gBL x Z Z → σσ, we approximate hσvi ≃ 2 ¼ 2. Note BL B 4πT 4π mσ is reproduced along the red lines each of which corresponds ≥ ≫ that this formula is applicable for T MZBL mσ. The to a fixed mσ value. In the right panel, the region for ≤ ≲ 10 ∼ 10−5 reason for this is that for T MZBL , the number density of MZBL MeV and gBL is excluded by the long- ZBL is Boltzmann suppressed and σ particle creation stops. lived ZBL boson search results. See Fig. 2 for details. Sðmσ Þ −3 Using ≃ 14mσM and Y ≃ 2.2 × 10 and integrat- In the right panel of Fig. 1, we can see that there is an Hðmσ Þ P eq ¼ Oð10−5Þ ¼ allowed parameter region for gBL and MZBL ing the above equation from xRH to x (where xRH ¼ 1 MeV–1 GeV. For the parameter region, ZBL boson can mσ=T with the reheating temperature after inflation RH be long-lived and such a long-lived neutral particle can be T ≫ M ), we obtain RH ZBL explored in the near future by the Lifetime Frontier experiments, such as FASER [19],SHiP[25], LDMX −6 4 MP ð Þ − ð Þ ≃ 5 1 10 ð − Þ ð Þ [26], Belle II [27], and LHCb [28,29]. The ZBL boson Y x Y xRH . × gBL x xRH : 10 mσ search of the FASER experiment at the LHC is summarized in Ref. [19] along with the search reaches of other ð∞Þ ≃ ð ¼ Þ Then taking Y Y xBL mσ=MZBL , we estimate the experiments as well as the current excluded region [30]. DM relic density, In Fig. 2, we show our results of the right panel of Fig. 2 along with the summary plot in Ref. [19]. The red lines ¼ 10 mσs0Yð∞Þ mσ 1 GeV correspond to mσ keV, 100 keV, 1 MeV, and 10 MeV Ω h2 ≃ ≃ 3 4 1021g4 ; DM ρ 2 . × BL 1 from top to bottom, respectively. The parameter region of 0=h GeV MZBL 10 keV ≲ mσ ≲ 1 MeV and 10 MeV ≲ M ≲ afewGeV ð Þ ZBL 11 can be tested by various Lifetime Frontier experiments in the near future. 3 where s0 ¼ 2890=cm is the entropy density of the present Before moving on to the next section, we comment on 2 −5 3 universe, and ρc=h ¼ 1.05 × 10 GeV cm is the critical the dark matter production processes involving the RHN. If density. This leads to the following expression for gBL: the RHN is in thermal equilibrium, the DM particles can also be created through NN → σσ. The estimate of ð∞Þ 1Note also that, as a general possibility, if M is greater than Y from this process is analogous to the process N → σσ the reheating temperature after inflation (T ), then RHN is ZBLZBL , and resultant density is roughly given by RH → → irrelevant to our DM physics discussion. Eq. (11) with replacing gBL f and MZBL MN. Thus, 115022-5 RABINDRA N. MOHAPATRA and NOBUCHIKA OKADA PHYS. REV. D 101, 115022 (2020) we take M B. Relic density constraints In Fig. 3 (left panel), we show our result for We next explore the constraints of relic density on the mσ ¼ 1 PeV. The dashed line denotes the upper bound heavy DM case. For such low gBL values, a heavy PeV on gBL from the out-of-equilibrium condition of Eq. (22). 10 scale DM and the 10 GeV or higher mass ZBL would The diagonal black line shows the lifetime constraint of never have been in equilibrium. The relic density must arise Eq. (4), or equivalently Eq. (15). Along the red line, the as in the first case via the freeze-in mechanism. Since ZBL is observed DM relic density is reproduced [see Eq. (20)]. In ¼ not in thermal equilibrium, the production takes place via the figure, we find the lower bound on MZBL ¯ 4 5 109 the process ff → ZBLσ through the SM fermion pair . × GeV. In the right panel of Fig. 3, we show the annihilations in the thermal plasma. In this case, the results for various values of mσ. The red lines from top to Boltzmann equation is given by bottom correspond to the results for mσ ¼ 100 keV, 115022-6 FREEZE-IN DARK MATTER FROM A MINIMAL B − L MODEL … PHYS. REV. D 101, 115022 (2020) 10 MeV, 1 GeV, 100 GeV, 10 TeV, 1 PeV, and 100 PeV, Our scenario for SOð10Þ breaking is as follows: we use respectively. The left diagonal black line denotes the out- 45-dimensional Higgs field to break SOð10Þ down to ð5Þ ð1Þ ω ¼ ω of-equilibrium condition of Eq. (22), while the horizontal SU × U I˜ by choosing the vacuum with Y BL, black line corresponds to Eq. (21). We also impose a as noted above. The I˜ quantum numbers of fermions are condition of v ≤ M , which is depicted by the right ˜ð10Þ¼ p1ffiffiffiffi ˜ð5¯Þ¼ p−3ffiffiffiffi ˜ð1Þ¼ p5ffiffiffiffi BL P then given by I 2 10, I 2 10 and I 2 10, diagonal black line. We thus see that there is enough where 10, 5¯, and 1 are the SUð5Þ representations in SOð10Þ parameter range in the model for the dark matter to be in the spinor 16. For a 10-representation Higgs field in SOð10Þ, PeV range so that it can be relevant to the PeV neutrinos which is decomposed into 5 þ 5¯ under SUð5Þ and includes observed in IceCube experiment. This is possible for ˜ 10 14 the SM Higgs doublet, the I quantum numbers are given by M ≳ 10 GeV and v ≳ 10 GeV. ZBL BL ˜ð5Þ¼ p−2ffiffiffiffi ˜ð5Þ¼ p2ffiffiffiffi I 2 10 and I 2 10. Let us now discuss the evolution of the I˜ gauge coupling. VI. PROSPECTS FOR SO(10) EMBEDDING ð1Þ The evolution of U I˜ gauge coupling (gI˜) is given by In this section, we like to point out that a slight variation α−1 of the model leads to its possible embedding into SOð10Þ d ˜ b˜ μ I ¼ I ; ð24Þ grand unified theory (GUT), which we believe should add dμ 2π to its theoretical appeal as a minimal GUT model that ¼ −49 10 μ ð5Þ unifies neutrino masses and dark matter. The starting point where bI˜ = at a scale below the SU uni- ¼ −5 ð5Þ ð1Þ of this discussion is the observation that the hypercharge fication while bI˜ in SU × U I˜ theory by con- generator Y is a linear combination of the I3R and the sidering that the SM Higgs doublet is embedded into a 5 ð5Þ normalized B − L generators IBL of SOð10Þ as follows: -representation in SU . For simplicity, we have assumed that in each step of the gauge symmetry breaking, rffiffiffi ð10Þ → ð5Þ ð1Þ → ð3Þ ð2Þ ð1Þ 2 SO SU × U I˜ SU c × SU L × U Y× ¼ þ ð Þ ð1Þ → ð3Þ ð2Þ ð1Þ Y I3R 3IBL; 23 U I˜ SU c × SU L × U Y, only the minimal sets of Higgs fields are light. qffiffi To see our coupling unification strategy in this model, we 3 B−L where IBL ¼ 2 2 . The B − L generator in the main first discuss the SUð5Þ unification without supersymmetry. body of the paper is not orthogonal to the Y generator As is clear, in this case, we will need extra fields beyond the ð5Þ defined above. Therefore, it cannot emerge from SOð10Þ SM fields below the SU unification scale. For this purpose, we introduce n3 real scalar SUð2Þ triplets with breaking since IBL is not orthogonal to Y defined above. L ˜ Y ¼ 0 and n8 real scalar color octets with Y ¼ 0. The Instead, if we consider the generator I ≡ −4I3 þ R coupling evolution equations in this case are the following: 3ðB − LÞ, we get TrðIY˜ Þ¼0 (i.e., they are orthogonal) for any irreducible representation of SOð10Þ and can −1 dα 1 41 therefore emerge from SOð10Þ breaking. This generator μ 1 ¼ − ; ð1Þ dμ 2π 10 was also identified in Ref. [32] as the generator U X for −1 xH ¼ −4=5. Indeed, it has been shown in Ref. [20] that dα2 1 19 n3 μ ¼ − θðμ − M3Þ ; such a generator emerges out of SOð10Þ breaking by a 45 dμ 2π 6 3 Higgs field. To see this note that 45 Higgs under SUð3Þ × −1 dα3 1 n8 SUð2Þ × SUð2Þ × Uð1Þ − group has multiplets μ ¼ 7 − θðμ − M8Þ ; ð25Þ L R B L dμ 2π 2 (1,1,1,0) and (1,1,3,0) which can take VEVs ωY and ω BL, respectively. If we fine-tune the parameters of the 1 3 1 Higgs potential, we can get ω ¼ ω in which case the where M3;8 stand for the masses of the triplet ( ; ; ) and Y BL octet (8; 1; 0) fields, respectively. Solving these equations Uð1Þ Uð1Þ˜ unbroken generators are Y × I. The normalized ¼ 5 ¼ 5 ¼ 3 ˜ ¼ p1ffiffiffiffi ð−4 þ 3ð − ÞÞ with n3 with mass M3 TeV and n8 with mass I 2 10 I3R B L . M8 ¼ 200 TeV, we find that the SUð5Þ gauge coupling As it turns out, the dark matter phenomenology dis- 15 unification is achieved at MU ¼ 6.8 × 10 GeV. cussed above remains unchanged if we use the Higgs field Let us now proceed to SOð10Þ unification, i.e., the σ Uð1Þ˜ σ to break the I symmetry. The field then emerges running of the g˜ coupling from its breaking scale (which 126 ð10Þ I from the -dimensional representationpffiffiffiffi of SO and does not affect very much) to where it unifies with the ˜ 10 our dark matter field σ has I ¼ 2 and therefore has all the SUð5Þ coupling evolving after the SUð5Þ unification scale. properties required above for our dark matter. Again, as We see that due to the small value of gI˜ required to get the before, we fine-tune parameters to get the desired dark relic density from the freeze-in mechanism, the SOð10Þ matter (σ). Our goal here is not to construct a natural model gauge coupling unification in four dimensions is hard to but rather to see whether it is phenomenologically possible obtain. We therefore assume that above the SUð5Þ GUT to have σ play the role of dark matter. scale, the model becomes five dimensional [33] with the 115022-7 RABINDRA N. MOHAPATRA and NOBUCHIKA OKADA PHYS. REV. D 101, 115022 (2020) 1 fifth dimension compactified on S =Z2 orbifold with a −1 radius R ¼ MU . 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 Kaluza-Klein (KK) modes contribute to the running of the SUð5Þ ð1Þ coupling whereas U I˜ being Abelian its coupling running does not get any extra contribution from the opening of fifth dimension. The evolution of the SUð5Þ gauge cou- pling (α5) obeys −1 dα5 1 43 1 5 μ ¼ − − ð1 þ n3 þ n8Þ dμ 2π 3 6 6 55 X pffiffiffiffiffiffiffiffiffiffiffiffiffi þ θðμ − 1 þ 2 Þ ð Þ 3 n MU : 26 n¼1 Here, in the parenthesis of the right-hand side, 43=3 is the contribution from the zero-mode SUð5Þ gauge boson and the SM fermions, −1=6 from the 5-representation Higgs − 5 ð1 þ þ Þ FIG. 4. Unification of gauge couplings in the presence of one field, and 6 n3 n8 from one adjoint Higgs to α−1 extra dimension. The horizontal blue line denotes I˜ , while solid SUð5Þ n3 þ n8 −1 −1 −1 break the symmetry and adjoint Higgs field black lines from top to bottom denote α1 , α2 , and α3 , into which the triplet and octet scalars are embedded, and ð1Þ respectively. Here, we have set the U I˜ gauge boson mass the last term is the contribution from the SUð5Þ gauge 1014 (corresponding to MZBL in the previous sections) to be GeV −1 boson KK modes. For the KK mode mass spectrum, we as an example. The red curve represents the running of α5 in the have simply added the contribution from the SUð5Þ presence of the gauge boson KK modes. For a comparison with symmetry breaking. Once the extra dimension opens, the four-dimensional theory, we show the dashed line for the SUð5Þ contribution from the KK modes changes the scale without the KK mode contributions. dependence of the running gauge coupling from a log to apower[33]. Thus, it is possible to unify the SUð5Þ and bound τ ≥ 1.6 × 1034 years from the Super-Kamionkande Uð1Þ˜ couplings into SOð10Þ coupling as desired. This is p I → shown in Fig. 4. In the figure, the SOð10Þ gauge coupling results [34]. More importantly, we would expect that p eþπ0 should be observable in the next round of proton unification is achieved at MP with a unified coupling decay searches at Hyper-Kamiopkande [35] or the model gSOð10Þ ≃ 0.1. This result corresponds to an allowed param- ≃ 100 ¼ 1014 will be ruled out. eter set, mσ keV and MZBL GeV, in the right panel of Fig. 3. VII. CONCLUDING REMARKS As far as proton decay is concerned, the primary mode is þ 0 ð1Þ p → e þ π mediated by the SUð5Þ gauge boson. The We have presented a minimal model based on a U B−L proton decay amplitude gets contribution from all the KK extension of the standard model where the B − L breaking excitations of the SU(5) gauge fields, and we estimate Higgs field plays the role of a decaying dark matter. We the modification of a coefficient of the four-Fermi operator discuss two regions of the DM masses: one light mass to be region in the keV to MeV range and another where the DM mass is in the PeV range. In both cases, due to the stability ∞ 1 1 X 1 2.08 1 requirement of the dark matter, the freeze-in mechanism is → 1 þ ≃ ≡ : ð27Þ M2 M2 1 þ n2 M2 Λ2 required to understand the observed relic density of DM. U U n¼1 U We then discuss how the model can be tested in the FASER and other Lifetime Frontier experiments. Finally, we show Then, (ignoring threshold effects) the proton lifetime is how the model can emerge from an SOð10Þ GUT model. estimated as Coupling unification in this case requires that the model be Λ4 part of a five-dimensional space-time with the compacti- τ ≃ ; ð28Þ fication radius being of the order of the inverse of the p α2 m5 U p SUð5Þ unification scale MU. This embedding reflects itself in an enhanced decay rate for the proton due to extra gauge ¼ 0 938 α ð Þ ≃ 0 026 where mp . GeV. Using 5 MU . and KK mode contributions, which we have estimated. The 15 MU ≃ 6.8 × 10 GeV from Fig. 4, we find that model may have TeV scale hypercharge neutral weak 34 τp ≃ 2.1 × 10 years, which is consistent with the lower isotriplet and color octet scalars, which have interesting 115022-8 FREEZE-IN DARK MATTER FROM A MINIMAL B − L MODEL … PHYS. REV. D 101, 115022 (2020) LHC phenomenology [36,37]. Discussion of this phenom- ACKNOWLEDGMENTS enology is beyond the scope of this paper. There are also The work of R. N. M. is supported by the National ranges for the RHN masses in the model where resonant Science Foundation Grants No. PHY-1620074 and leptogenesis can generate the baryon asymmetry of the No. PHY-1914631, and the work of N. O. is supported universe. This will be the subject of a forthcoming by the U.S. Department of Energy Grant No. DE- publication. SC0012447. [1] P. Minkowski, Phys. Lett. 67B, 421 (1977). [21] E. Ma, Phys. Rev. Lett. 115, 011801 (2015). [2] R. N. Mohapatra and G. Senjanović, Phys. Rev. Lett. 44, [22] L. Canetti, M. Drewes, and M. Shaposhnikov, Phys. Rev. 912 (1980). Lett. 110, 061801 (2013). [3] T. Yanagida, Conf. Proc. C7902131, 95 (1979). [23] N. Aghanim et al. (Planck Collaboration), arXiv:1807 [4] M. Gell-Mann, P. Ramond, and R. Slansky, Conf. Proc. .06209. C790927, 315 (1979). [24] M. G. Baring, T. Ghosh, F. S. Queiroz, and K. Sinha, Phys. [5] S. L. Glashow, NATO Sci. Ser. B 61, 687 (1980). Rev. D 93, 103009 (2016). [6] R. E. Marshak and R. N. Mohapatra, Phys. Lett. 91B, 222 [25] S. Alekhin et al., Rep. Prog. Phys. 79, 124201 (2016). (1980). [26] T. Akesson et al. (LDMX Collaboration), arXiv:1808 [7] R. N. Mohapatra and R. E. Marshak, Phys. Rev. Lett. 44, .05219. 1316 (1980). [27] M. J. Dolan, T. Ferber, C. Hearty, F. Kahlhoefer, [8] A. Davidson, Phys. Rev. D 20, 776 (1979). and K. Schmidt-Hoberg, J. High Energy Phys. 12 (2017) [9] L. Basso, A. Belyaev, S. Moretti, and C. H. Shepherd- 094. Themistocleous. Phys. Rev. D 80, 055030 (2009). [28] P. Ilten, J. Thaler, M. Williams, and W. Xue, Phys. Rev. D [10] A. A. Abdelalim, A. Hammad, and S. Khalil, Phys. Rev. D 92, 115017 (2015). 90, 115015 (2014). [29] P. Ilten, Y. Soreq, J. Thaler, M. Williams, and W. Xue, Phys. [11] S. Iso, N. Okada, and Y. Orikasa, Phys. Lett. B 676,81 Rev. Lett. 116, 251803 (2016). (2009); Phys. Rev. D 80, 115007 (2009). [30] M. Bauer, P. Foldenauer, and J. Jaeckel, J. High Energy [12] J. Heeck, Phys. Lett. B 739, 256 (2014). Phys. 07 (2018) 094. [13] M. Bauer, P. Foldenauer, and J. Jaeckel, J. High Energy [31] M. Aartsen et al. (IceCube Collaboration), Phys. Rev. Lett. Phys. 07 (2018) 094. 111, 021103 (2013). [14] S. Blasi, V. Brdar, and K. Schmitz, arXiv:2004.02889. [32] N. Okada, S. Okada, and D. Raut, Phys. Lett. B 780, 422 [15] R. N. Mohapatra and N. Okada, arXiv:1908.11325. (2018). [16] S. Heeba and F. Kahlhoefer, Phys. Rev. D 101, 035043 (2020). [33] K. R. Dienes, E. Dudas, and T. Gherghetta, Phys. Lett. B [17] L. J. Hall, K. Jedamzik, J. March-Russell, and S. M. West, 436, 55 (1998); Nucl. Phys. B537, 47 (1999). J. High Energy Phys. 03 (2010) 080. [34] K. Abe et al. (Super-Kamiokande Collaboration), Phys. [18] For different B-L models with right-handed neutrino as dark Rev. D 95, 012004 (2017). matter, see A. Biswas and A. Gupta, J. Cosmol. Astropart. [35] K. Abe et al., arXiv:1109.3262. Phys. 09 (2016) 044; 05 (2017) A01; K. Kaneta, Z. Kang, [36] See for example, J. Gunion, R. Vega, and J. Wudka, Phys. and H.-S. Lee, J. High Energy Phys. 02 (2017) 031. Rev. D 42, 1673 (1990); H. E. Logan and M. Roy, Phys. [19] J. L. Feng, I. Galon, F. Kling, and S. Trojanowski, Phys. Rev. D 82, 115011 (2010); C. Englert, E. Re, and M. Rev. D 97, 035001 (2018). Spannowsky, Phys. Rev. D 88, 035024 (2013). [20] S. Bertolini, L. Di Luzio, and M. Malinsky, Phys. Rev. D 81, [37] T. Han, I. Lewis, and Z. Liu, J. High Energy Phys. 12 (2010) 035015 (2010). 085. 115022-9