SU(5)$\times$U(1)$_X$ grand unification with minimal seesaw and $Z^\prime$-portal dark matter

We propose a grand unified SU(5)$\times$U(1)$_X$ model, where the standard SU(5) grand unified theory is supplemented by minimal seesaw and a right-handed neutrino dark matter with an introduction of a global $Z_2$-parity. In the presence of three right-handed neutrinos (RHNs), the model is free from all gauge and mixed-gravitational anomalies. The SU(5) symmetry is broken into the Standard Model (SM) gauge group at $M_{\rm GUT} \simeq 4 \times 10^{16}$ GeV in the standard manner, while the U(1)$_X$ symmetry breaking occurs at the TeV scale, which generates the TeV-scale mass of the U(1)$_X$ gauge boson ($Z^\prime$ boson) and the three Majorana RHNs. A unique $Z_2$-odd RHN is stable and serves as the dark matter (DM) in the present Universe, while the remaining two RHNs work to generate the SM neutrino masses through the minimal seesaw. We investigate the $Z^\prime$-portal RHN DM scenario in this model context, and find that the constraints from the DM relic abundance and the search results for a $Z^\prime$ boson resonance at the Large Hadron Collider (LHC) are complementary to narrow down the allowed parameter region, which will be fully covered by the future LHC experiments (for the $Z^\prime$ boson mass $<$ 5 TeV). We also briefly discuss the successful implementation of Baryogenesis and cosmological inflation scenarios in the present model.

Despite its great success, the Standard Model (SM) suffers from several problems. The neutrino mass matrix and a candidate of dark matter (DM) are two major missing pieces of the SM, and they must be supplemented by a framework beyond the SM. The minimal U(1) B−L model [1] is a very simple extension of the SM, where the anomaly-free global B − L (baryon number minus lepton number) symmetry in the SM is gauged and only the three right-handed neutrinos (RHNs) and the U(1) B−L Higgs field in addition to the SM particle content. In the presence of the RHNs, the seesaw mechanism [2] is automatically implemented. Associated with the spontaneous U(1) B−L symmetry breaking by a vacuum expectation value (VEV) of the U(1) B−L Higgs field, the U(1) B−L gauge boson (Z ′ boson) and the three Majorana RHNs acquire their masses. The SM neutrino mass matrix is generated through the seesaw mechanism after the electroweak symmetry breaking.
Among various possibilities of introducing a DM candidate into the minimal U(1) B−L model, a way proposed in Ref. [3] would be the simplest, where instead of extending the particle content, a Z 2 -parity is introduced and a unique Z 2 -odd RHN plays the role of DM. The remaining two RHNs work to generate the SM neutrino mass matrix through the seesaw mechanism. Therefore, in this framework, the three RHNs are categorized into one Z 2 -odd RHN DM and two Z 2 -even RHNs for the so-called Minimal Seesaw [4], which is the minimal setup to reproduce the neutrino oscillation data with a prediction of one massless eigenstate. In this way, the two missing pieces of the SM are supplemented with no extension of the particle content of the minimal U(1) B−L model.
The RHN DM communicates with the SM particles in two ways: One is through exchanges of Higgs bosons in their mass basis (Higgs-portal RHN DM), where two physical Higgs bosons are realized as linear combinations of the U(1) B−L and the SM Higgs bosons. The other is through Z ′ boson exchange (Z ′ -portal RHN DM). Phenomenology of the RHN DM has been extensively studied [3,5,6], in particular, a complementarity between the RHN DM physics and the LHC physics for the Z ′ -portal RHN DM scenario has been pointed out in Ref. [6].
The minimal U(1) B−L model is easily generalized to the minimal U(1) X model [7], whose particle content is the same as the one of the minimal U(1) B−L model. The generalization appears in the U(1) X charge assignment for the SM fields: the U(1) X charge of an SM field (f ) is defined as Q where Y f and Q f B−L are the hypercharge and the B − L charge of the field, and x H is a new real parameter (see Table 1). With the charge assignment, the minimal U(1) X model is free from all the gauge and mixed-gravitational anomalies (see, for example, Ref. [8] for detailed calculations of the anomaly coefficients). The minimal U(1) B−L model is defined as the limit of x H → 0. The RHN DM is introduced in exactly the same way for the U(1) B−L model. In Ref. [9], the minimal U(1) X model with the Z ′ -portal RHN DM has been extensively studied. In the analysis, only four free parameters are involved: the U(1) X gauge coupling (α X ), the Z ′ boson mass (m Z ′ ), x H , and the RHN DM mass (m DM ). It has been found in Ref. [9] that m DM ≃ m Z ′ /2 is required to reproduce the observed DM relic density, and the number of free parameters is effectively reduced to three: α X , m Z ′ and x H . The cosmological constraint from the DM relic density leads to a lower bound on α X for fixed values of m Z ′ and x H . On the other hand, the LHC Run-2 results from the search for a Z ′ boson resonance provide an upper bound on α X for fixed values of m Z ′ and x H . Therefore, the DM physics and the LHC Run-2 phenomenology are complementary to narrow down the model parameter space.
In this letter, we propose a grand unified SU(5)×U(1) X model, 1 into which the minimal U(1) X model with the RHN DM is embedded. The grand unified theory (GUT) has been attracting a lot of attention since its first proposal in Ref. [11], where all the three gauge interactions in the SM are embedded into the SU(5) gauge group, and all the fermions in the SM are unified into three generations of 5 * and 10-representations under the SU(5). The picture is not only mathematically beautiful, but also provides the charge quantization for the SM quarks and leptons. It seems natural to regard the GUT as a primary candidate of physics beyond the SM. However, if this is the case, we may require a GUT model to incorporate the neutrino mass matrix and a DM candidate. The model we propose satisfies this requirement with the RHN DM and two RHNs for the minimal type-I seesaw. As in the original proposal in Ref. [11], the SM gauge groups are embedded into the SU(5) group. However, note that the unification of the quarks and leptons into 5 * and 10-representations is possible only if x H = −4/5. Therefore, the U(1) X charge is quantized and x H is no longer a free parameter.
In the following, we show that our GUT model is phenomenologically viable. As we will discuss below, the SU(5) gauge symmetry is broken at M GUT ≃ 4 × 10 16 GeV, and the minimal U(1) X model with the RHN DM (x H = −4/5) is realized as low energy effective theory. The U(1) X symmetry is assume to be broken at the TeV scale. We first review this effective minimal U(1) X model at the TeV scale, and investigate phenomenological constraints on the model parameters. In our analysis, we follow Ref. [9] and identify an allowed parameter region, which will be found to be very narrow since x H = −4/5 is no longer a free parameter. Next, we discuss that the SM gauge couplings are successfully unified at M GUT with some extra fermions at the TeV scale, which originate one 5 + 5 * and one 10 + 10 * multiplets under the SU(5) gauge group. After the SU(5) breaking, a kinetic mixing between the U(1) Y and U(1) X gauge bosons is generated through the renormalization group (RG) evolution. We also discuss that this mixing is negligibly small and has little effect on our analysis.
We first define the minimal U(1) X model by the particle content listed in Table 1. The minimal B − L model is reproduced as the limit of x H → 0. The model is free from all the gauge and the gravitational anomalies in the presence of the three RHNs. Because of the Z 2 -parity assignment shown in Table 1, the N R is a unique (cold) DM candidate. Fixing x H = −4/5, we can see the unification of quarks and lepton into SU(5)×U(1) X multiplets: where the first term is the neutrino Dirac Yukawa coupling, and the second and third terms are the Majorana Yukawa couplings. Without loss of generality, the Majorana Yukawa couplings are already diagonalized in our basis. Note that only the two Z 2 -even RHNs are involved in the neutrino Dirac Yukawa coupling. After the U(1) X and the electroweak symmetry breakings, the Z ′ boson mass, the Majorana masses for the RHNs, and the neutrino Dirac masses are Table 1: The particle content of the minimal U(1) X extended SM with Z 2 -parity. In addition to the SM particle content (i = 1, 2, 3), the three RHNs (N j R (j = 1, 2) and N R ) and the U(1) X Higgs field (Φ) are introduced. The unification into SU (5)×U (1) X is achieved only for x H = −4/5, and x H is quantized in our model. generated: where g X is the U(1) X gauge coupling, GeV is the SM Higgs VEV, and we have used the LEP constraint [12] v Φ 2 ≫ v h 2 . Now we investigate phenomenological constraints for the minimal U(1) X model with the RHN DM. We follow Ref. [9] for our analysis, where only four free parameters, α X = g 2 X /(4π), m Z ′ , x H and m DM . However, the grand unification to SU(5)×U(1) X requires to fix x H = −4/5, and hence the resultant allowed parameter space is more restricted.
Let us first evaluate the DM relic density by integrating the Boltzmann equation given by [13] where x = m DM /T is the ratio of the DM mass to the temperature of the Universe (T ), H(m DM ) is the Hubble parameter at T = m DM , Y is the yield (the ratio of the DM number density to the entropy density s) of the DM, and Y EQ is the yield of the DM particle in thermal equilibrium. The thermal average of the DM annihilation cross section times relative velocity, σv rel , is calculated from the total cross section of the DM pair annihilation process NN → Z ′ → ff (f denotes the SM fermions) given by where and the total decay width of Z ′ boson is given by where β = 1 − 4m 2 DM /m 2 Z ′ , θ(z) is the unit step function, and the masses of all SM fermions are neglected. Here, we have assumed m N 1,2 R > m DM , m Z ′ /2, for simplicity. By solving the Boltzmann equation numerically, the present DM relic density is evaluated by where s 0 = 2890 cm −3 is the entropy density of the present Universe, and ρ c /h 2 = 1.05 × 10 −5 GeV/cm 3 is the critical density. We identify a parameter region to reproduce the observed DM relic density in the range of 0.1183 ≤ Ω χ h 2 ≤ 0.1213 (68% confidence level), measured by the Planck satellite experiment [14]. As shown in Ref. [9], an enhancement of the DM pair annihilation cross section via the Z ′ boson resonance in the s-channel necessary to reproduce the observed DM density. Hence, we always find m DM ≃ m Z ′ /2, and thus our results are effectively described by only two free parameters: α X and m Z ′ .
Next we consider the current LHC constraints from the search for a narrow resonance with dilepton final states at the LHC Run-2. In the analysis by the ATLAS and CMS collaborations, the so-called sequential SM Z ′ (Z ′ SSM ) has been studied as a reference model. We interpret the current LHC constraints on the Z ′ SSM boson into the Z ′ boson in our U(1) X model to identify an allowed parameter region. Although our analysis follows that in Ref. [9], we will update the LHC constraints presented in Ref. [9] by employing the latest ATLAS results in 2017 [15].
The cross section for the process pp → Z ′ + X → ℓ + ℓ − + X is given by where M ℓℓ is the invariant mass of a final state dilepton, f q is the parton distribution function for a parton (quark) "q", and √ s = 13 TeV is the center-of-mass energy of the LHC Run-2. In our numerical analysis, we employ CTEQ6L [16] for the parton distribution functions with the factorization scale Q = m Z ′ . The cross section for the colliding partons is given bŷ where the function F qℓ (x H ) is given by for q being the up-type (u) and down-type (d) quarks, respectively. By integrating the differential cross section over a range of M ℓℓ set by the ATLAS analysis, we obtain the cross section to be compared with the upper bounds obtained by the ATLAS collaboration. Only two free parameters, α X and m Z ′ , are involved in our analysis.
In Figure 1, we show our combined results from the DM relic abundance and the search results for a Z ′ boson resonance at the LHC. We can see these two constraints are complementary to narrow down the allowed parameter region. In Figure 1, we also show the perturbativity bound (see the discussion around Eq. (15)), which provides the upper bound on the U(1) X gauge coupling. Combining all three constraints, we have obtained the Z ′ boson mass bound in the range of 3.0 ≤ m Z ′ [TeV] ≤ 9.2.
Although in our grand unified model x H = −4/5 is not a free parameter, we present the combined results as a function of x H in Figure 2 for m ′ Z = 4 TeV, in order to show that x H = −4/5 has an interesting phenomenological implication. The (black) convex-downward solid line shows the cosmological lower bound on α X as a function of x H . The (red) convex-upward dashed line shows the upper bound on α X presented in Ref. [9], where the results are obtained from the LHC 2016 data with a 13/fb luminosity. We have updated the results by employing the latest ATLAS results with 36/fb [15], and our result is shown by the (red) convex-upward solid line. The (green) shaded region is the final result for the allowed parameter space after combining the cosmological and the LHC constraints when x H is a free parameter. Interestingly, the plot  ) shows the upper bound on α X obtained in Ref. [9] by employing the LHC data with a 13/fb luminosity. Our update of the results by employing the ATLAS results with 36/fb is shown by the (red) solid line.
indicates that x H = −4/5 required by the grand unification into SU(5)×U(1) X is almost the best value within the allowed region.
Let us now consider the gauge coupling unification. To realize the grand unification picture, the SM gauge coupling must be unified at a high energy scale. It has been shown in Ref. [17] that the SM gauge couplings are successfully unified around M GUT ≃ 4 × 10 16 GeV in the presence of two pairs of vector-like quarks, D L + D R and Q L + Q R , with their mass of O(1 TeV) in the representations of (3, 1, 1/3) and (3, 2, 1/6), respectively, under the SM gauge group of SU(3) C ×SU(2) L ×U(1) Y . Here, we adopt this simple case to our scenario with a common mass (M) for D L + D R and Q L + Q R . We simply take M = m Z ′ /2 not to change the Z ′ boson decay width that we have used in our DM analysis. The unification scale leads to proton lifetime to be τ p ≃ 10 38 years, which is consistent with the current experimental lower bound obtained by the Super-Kamiokande [18], τ p (p → π 0 e + ) 10 34 years.
In the grand unified SU(5)×U(1) X picture, D L +D C R and Q L +Q C R are unified into F 5 +F 5 = (5, 3/5) + (5 * , −3/5) and F 10 + F 10 = (10, 1/5) + (10 * , −1/5), respectively. The way to realize the mass splittings among the components in the SU(5) multiplets and leave only D L + D R and Q L + Q R light is analogous to the triplet-doublet mass splitting for the 5-plet Higgs field in the usual SU(5) GUT model. We introduce the Yukawa couplings and the mass terms such as where Y 5,10 are Yukawa coupling constants, M 5,10 are masses for the vector-like fermions, Σ is a U(1) X charge-neutral SU(5) adjoint Higgs field, whose VEV of Σ = v GUT diag(1, 1, 1, −3/2, −3/2) breaks the SU (5) gauge group into the SM ones, and we have used antisymmetric 5 ×5 matrices to express F 10 and F 10 . We tune the Yukawa coupling to realize Y 5 v GUT − M 5 = O(1 TeV) for F 5 + F 5 , so that the vector-like SU(3) C color triplets (D L + D R ) become light, while the vector-like SU(2) L doublets in the multiplets are heavy with mass of (5/2)M 5 . By tuning Y 10 to realize Y 10 v GUT + 4M 10 = O(1 TeV) for F 10 and F 10 , we obtain Q L + Q R light, while the rest in the 10-plets are heavy with mass of 5M 10 . Taking suitable values for Y 5,10 = O(1), we can obtain the common TeV scale mass for D L + D R and Q L + Q R , while the other components have GUT-scale masses. Once the SU (5) symmetry is broken to the SM gauge groups at M GUT , a kinetic mixing between the U(1) Y and the U(1) X gauge bosons is generated at low energies through the RG evolutions. Following standard techniques in Ref. [19], we generally set a basis where the gauge boson kinetic terms are diagonalized and a covariant derivative of a field is defined as Here, Y and Q X are U(1) Y and U(1) X charges of the field, respectively, B µ is the SM U(1) Y gauge field, and g Y is the U(1) Y gauge coupling. Originating from the gauge kinetic mixing, a new parameter, namely, "mixed gauge coupling" g mix is introduced. In this basis, the RG evolution of the SM U(1) Y gauge coupling remains the same as the SM one at the one-loop level, while g X and g mix evolve according to their coupled RG equations. At the one-loop level, the coupled RG equations for µ > O(TeV) are given by where β g X = 1 6 g X 80 + 64x H + 45x 2 H g 2 X + 2 (32 + 45x H ) g X g mix + 45g 2 mix , β g mix = 5 3 g 2 Y 32 5 + 9x H g X + 9g mix + 1 6 g mix (80 + 64x H + 45x 2 H )g 2 X + 2(32 + 45x H )g X g mix + 45g 2 mix .
Here we have taken into account all particle contributions to the beta functions at the TeV scale. Numerically solving the RG equations with g mix = 0 and various values of g X at µ = M GUT , we have found that g mix /g X ≃ 0.034 at the TeV scale for any input values of g X at M GUT . Therefore, we can safely neglect effects of g mix in our analysis and set g mix = 0 as a good approximation. Neglecting g mix in the RG equations, we find the following analytic solution for the U(1) X gauge coupling: where b X = (80 + 64x H + 45x 2 H ) /6 = 48/5 for x H = −4/5. Since the running U(1) X gauge coupling α X (µ) is asymptotically non-free, we now impose the "perturbativity bound" that α X (M GUT ) must be in the perturbative regime. Adopting a condition of α X (M GUT ) ≤ 4π, we find α X (m Z ′ ) ≤ 0.022 for m Z ′ ≤ 10 TeV. In Figure 1, we see that this perturbativity bound is more severe than the LHC bound for m Z ′ 4.5 TeV.
Finally, our grand unified SU(5)×U(1) X model can also account for the origin of the Baryon asymmetry in the Universe through leptogenesis [20] with two Z 2 -even RHNs if they are almost degenerate (resonant leptogenesis [21]). Introducing non-minimal gravitational couplings, the U(1) X Higgs field plays the role of inflaton. We can achieve the successful cosmological inflation scenario with a suitable choice of the non-minimal gravitational coupling constant. See, for example, Ref. [22].