A minimal model for ${\rm SU}(N)$ vector dark matter

We study an extension of the Standard Model featuring a hidden sector that consists of a new scalar charged under a new SU$(N)_D$ gauge group, singlet under all Standard Model gauge interactions, and coupled with the Standard Model only via a Higgs portal. We assume that the theory is classically conformal, with electroweak symmetry breaking dynamically induced via the Coleman-Weinberg mechanism operating in the hidden sector. Due to the symmetry breaking pattern, the SU$(N)_D$ gauge group is completely Higgsed and the resulting massive vectors of the hidden sector constitute a stable dark matter candidate. We perform a thorough scan over the parameter space of the model at different values of $N=2$, $3$, and $4$, and investigate the phenomenological constraints. We find that $N=2,3$ provide the most appealing model setting in light of present data from colliders and dark matter direct search experiments. We expect a heavy Higgs to be discovered at LHC by the end of Run II or the $N=3$ model to be ruled out.


INTRODUCTION
In addition to the Higgs boson [1,2], LHC has so far not discovered any signals for new physics at the terascale. This result has recently led to explore novel possible solutions to the naturalness problem [3][4][5][6]. The essential assumptions of these approaches are the absence of physical mass scales above the electroweak (EW) scale and that the boundary conditions at the Planck scale lead to the vanishing of the quadratic divergence to the Higgs boson mass.
Within a classically conformal theory, one sets all explicit mass terms to zero in the tree level Lagrangian. One must then address the question of how the weak scale arises.
One possibility is that the weak scale is generated radiatively [7], but this does not work quantitatively for the Standard Model (SM). On the other hand, motivated by the lack of the SM to explain the observed dark matter abundance or matter-antimatter asymmetry, one may introduce additional sectors very weakly coupled with the SM. Maintaining the classical conformality also in the hidden sector, one can then generate a nontrivial scale radiatively and this is transmitted to the SM sector via interactions between the two sectors [8]. This is the mechanism which we consider in this paper.
More concretely, we extend the SM by a hidden sector consisting of a scalar transforming nontrivially under a new non-abelian gauge symmetry. All SM fields are singlet under this new gauge symmetry, and the radiatively generated vacuum expectation value of the hidden sector scalar leads to a complete breaking of the hidden gauge symmetry. The resulting massive gauge bosons are mass degenerate and due to a residual global symmetry, they constitute a dark matter candidate [9]. We set up the theory for general hidden gauge group SU(N ) D , extending earlier work [10,11] where the N = 2 case was considered. We then investigate the phenomenological viability of the model numerically for N = 2, 3, and 4 by imposing the stability of the potential up to the Planck scale, requiring perturbativity of all couplings, and imposing the constraints from the LHC data. Furthermore we compute the dark matter relic density and impose constraints from the presently known abundance [12], as well as from the direct searches for dark matter [13][14][15].
The paper is organised as follows: the model and computation of the EW symmetry breaking as well as the dark matter relic density are presented in Sec. 2. Various phenomenological constraints are considered in Sec. 3, and in Sec. 5 we present the conclusions and outlook.

Preliminary
Before the EW gauging, the global symmetry of the SM scalar sector is SU(2) L ×SU(2) R , which can be made explicit by assembling the Higgs fields into a matrix H transforming as a bifundamental of this symmetry. We now generalise this as follows: Consider extending the matter content of the SM by a scalar fields assembled into a matrix Φ, singlet under the SM gauge group and transforming as a bi-adjoint under the global SU(N) L ×SU(N) R symmetry. Then we gauge the SU(N) L symmetry and denote this new gauge group by SU(N ) D . Explicitly, we then have with real fields σ and φ a , I the identity matrix in N 2 − 1 dimensions, T a a generator of the adjoint representation 1 of SU(N ) D , and their numerical factors chosen to preserve canonical normalization for any N . On the other hand we assume the SM matter fields to be singlets under SU(N ) D , so that the Lagrangian includes all the SM kinetic, gauge and Yukawa terms, While mass terms for both scalar fields are allowed, we set them to zero at tree level to make the model classically conformal: The last term of the potential, generally referred to as the portal coupling [8,[16][17][18]

Electroweak symmetry breaking
While the tree level potential in Eq. (3) has its minimum at the origin of field space, the scalar vevs acquire non-zero values via dimensional transmutation because of quantum corrections [7]. The one loop contribution to the effective potential in the MS scheme [19] can be written as where the sum over p includes scalars (ϕ), fermions (ψ), and vectors (A). The factor s p denotes the spin of the particle in question, and m p its field dependent tree-level mass. The resulting one loop effective potential, reaches a minimum at provided that the values of the vevs, assumed to be real, satisfy the minimization conditions for the tree level potential, The scalar mass matrix at the minimum of the potential is then defined as (with no sum over indices) where the last term represents the shift generated by the one loop correction on the otherwise zero tree level mass terms [20].
The vevs in Eq. (6) ensure the breaking of the SM gauge group following the usual pattern, and of SU(N ) D entirely. Consequently, all the dark gauge bosons A a acquire the same mass This degeneracy is a consequence of the residual SO(N ) global symmetry of the Lagrangian, which guarantees the stability of the SU(N ) D gauge boson multiplet. These massive gauge bosons are therefore suitable dark matter candidates [9].
The pseudoscalars φ a provide the longitudinal degree of freedom to A a , while π 0 and π ± are absorbed by EW gauge bosons Z and W ± , respectively. In Appendix A we provide the analytical result for the one loop scalar mass matrix, Eq. (8), in the (h, σ) basis. From Eqs. (A1) the one loop masses and the corresponding mass eigenstates, h 1 and h 2 , can be easily derived analytically.
From the results above we see that viable EW symmetry breaking is possible without the intervention of any mass term at the tree level. Under renormalisation then, the higher order corrections to the scalar masses depend on the renormalization scale only logarithmically, and therefore are in principle natural [6,8]. In this sense classical conformality trades the SM fine tuning problem with finding justification for taking the mass terms equal to zero to begin with.

Dark matter abundance
As we pointed out in the previous subsection, the residual SO(N ) makes the massive A a vector bosons stable and therefore suitable dark matter candidates. Their annihilation and semi-annihilation cross sections can be easily calculated in the limit of zero portal coupling λ p . This approximation for the dark matter analysis is consistent, as in the next section it turns out that λ p g D in the viable region of parameter space.
For the thermally averaged annihilation (AA → σσ) and semi-annihilation (AA → σA) cross section times relative velocity we find which for N = 2 reproduce the results of [10,11]. This approximation is sufficient when working away from the resonance thresholds, where the full thermal average [21] should be used; see e.g. [22].
The dark matter abundance is determined by where Y = n/s, Y eq the corresponding equilibrium density, x = m A /T , and with g * denoting the effective number of degrees of freedom and M P l the Planck mass.
Using the standard approximations, the dark matter abundance is determined by The value of where δ determines the deviation of the distribution from the equilibrium one, δ = Y /Y eq −1, before the freeze out. The value of δ is expected to be O(1), and in the numerical analysis we choose δ = 1.

The LHC data fit
To perform the quantitative analysis of the viability of the model, we start by scanning the parameter space for data points producing a viable mass spectrum. Between the two possible hierarchy choices, we focus on the case when the Higgs scalar is heavier than the dark scalar; we comment on the viability of the alternative possiblity in the next section.
Given the tight experimental constraints on the masses of the SM particles, we set all the SM couplings (except λ h ) as well as the Higgs vev to their standard values (v h = 246 GeV).
We fix λ h and λ φ via Eqs. (7) and v φ by setting m h 1 = 125 GeV. This leaves us with only two free parameters: g D and λ p . We then collect 10 5 data points for each value of N = 2, 3, 4 in the region We will see in the next subsection that the scanned range of values of g D and λ p is sufficient to cover the phenomenologically viable region.
The off-diagonal terms in the scalar mass matrix, Eq. (A1), are proportional to the portal coupling, λ p , which therefore controls the amount of mixing between the SM Higgs field h and the dark scalar σ in the mass eigenstate h 1 , parametrized by the angle α according to Given that σ does not couple to SM particles, the physical Higgs h 1 couplings turn out to be suppressed as compared to their SM values by a factor of cos α. We constrain this factor, and consequently λ p , by determining for each data point the goodness of fit of the Higgs coupling strengths to their corresponding measured values for the γγ, ZZ, W W , bb, τ τ inclusive processes [1, 2, 23]. To calculate χ 2 , we follow the procedure detailed in [24], and here we present directly the results of the statistical analysis. Among the 10 5 scanned data points about 39%, 40%, 39%, for N = 2, 3, 4, respectively, satisfy the 95%CL constraint As expected, given that the measured Higgs couplings are SM like, the portal coupling is constrained by collider data to acquire small values. The quartic coupling λ φ turns out to be even smaller than λ p , from Eqs. (7), given that the dark vev v φ is much larger than v h .
In the next section we further constrain the viable data points by requiring perturbativity of all the couplings and stability of the potential up to the Planck scale.

Stabilization of the SM potential
The SM potential turns out to be metastable for the measured Higgs mass, since the The average values of the only two free parameters, λ p and g D , with their respective standard deviations for the viable data points featuring perturbativity and stability are In Fig. 2 we plot the values of the dark scalar and dark vector masses for all the data points satisfying the LHC constraint, for N = 2 (left panel) and N = 3 (right panel). The color coding is the same as in Fig. 1. As a final comment we point out that no data point featuring a dark Higgs lighter than 125 GeV satisfies all the collider, stability, and perturbativity constraints simultaneously.
We therefore do not investigate further the possibility that h 2 might be the Higgs boson discovered at LHC.
In the next section we implement in our analysis the dark matter abundance, as determined for the present model in Subsection (2.3), and direct detection constraints to further test the model's phenomenological viability.

Dark matter abundance and direct detection
We compare the numerical result produced by Eq. (13) at each viable data point with the 95% experimental interval [12] Ωh 2 = 0.1193 ± 0.0028 , and find 23 data points (or 1% of the total) for N = 2 (Figs. 1,2, left panels) and 39 (or 2% of the total) for N = 3 (Figs. 1,2, right panels) that satisfy the dark matter constraint above as well as those in Eqs. (17,19). Interestingly, as can be seen from Fig. 1, for increasing Finally, we also impose the direct detection constraints. The spin independent cross section for the elastic scattering off a nucleon N of the vector dark matter candidate, A a , mediated by either h 1 or h 2 is, in the limit m A m N , where f N = 0.303 is the effective Higgs nucleon coupling [22] 2 , m N = 0.939 GeV is the average nucleon mass. In the mass range of interest to us here, m D ∼ O(100) GeV, the most stringent bounds are provided by the LUX experiment [15]. We evaluate Eq. (23) at each data point and compare with the experimental constraint in [15], as a function of the mass of the dark matter candidate, m A : all the data points satisfying the LHC constraint on the Higgs couplings, Eq. (17), and consequentially also the points satisfying the stability, perturbativity, and dark matter abundance constraints, are viable. In more detail we obtain that the experimental constraint on the spin independent cross section is on average an order of magnitude larger than the predicted value To summarize the results of this section, the classically conformal SU(3) bi-adjoint scalar extended SM turns out to be an even more appealing model than its SU(2) version, given that the former features a larger region of parameter space that satisfies collider, stability, perturbativity, and dark matter abundance constraints than the latter.
Vector DM in SU(3) gauge theory has been considered also in [25]. There SU(3) is broken completely by a pair of scalar triplets, which introduce four new physical scalars, compared to just one in the present minimal model.

DISCOVERABILITY AT LHC RUN II
The bounds on additional Higgs-like resonances in the mass region defined by Eq. (21) are rather stringent [26], given that such a heavy Higgs can decay into a pair of EW bosons almost at rest. The amplitude for production and decay of the heavy Higgs h 2 is equal to that of the SM Higgs suppressed by a factor s 2 α = sin 2 α, and there are no hidden decays to new particles or to a pair of light Higgs bosons. In Fig. 4 we plot the CMS constraint on s 2 α (Fig. 8 in [26]), together with the viable data points (in green those stable and with viable light Higgs couplings, and in black those that satisfy also DM constraints): of the universally viable data points, about half for N = 3 (right panel) and all for N = 2 (left panel) satisfy the CMS constraint. Notice that even assuming that not all the DM relic density is generated by the dark vectors A a , only in the region above the strip of black data points the corresponding relic density satisfies the 95% CL upper bound in Eq. (22), while the region below is ruled out. To estimate the improvement of this upper bound at LHC Run II, we assume the corresponding constraint on the cross section to be dominated by data statistical uncertainty, and therefore to depend on the square root of the total number N of events observed: where σ h 2 is the production rate of a SM-like Higgs boson of mass m h 2 , ef f is the efficiency of the trigger, and L tot the total integrated luminosity. Assuming the efficiency at Run II to be unchanged, and taking the total integrated luminosity at the end of Run II in 2019 to be 150 f b −1 , the upper bound on the production rate of h 2 should be reduced by a factor of 2.497 × 150 25 where the first coefficient under square root is equal to the ratio of production rates for a 175 GeV SM-like Higgs at 13 and 8 TeV, respectively. This result corresponds to a reduction of 1/2 of the upper limit on s 2 α , which changes like the h 2 production amplitude. Assuming the limits in Fig. 4 to be simply shifted down by a factor of 1/2, we expect a large portion of viable parameter space of the SU(2) model to be tested at LHC Run II, and a heavy Higgs to be discovered by the end of Run II or the SU(3) model to be ruled out.

CONCLUSIONS
In this paper we presented a novel extension of the SM, featuring a new scalar Φ in the bi-adjoint representation of SU(N ) L ×SU(N ) R , with only SU(N ) L gauged. The vev of such N 2 −1 dimensional matrix field is proportional to the identity matrix, and breaks completely the new gauge group, providing the corresponding vector bosons A a with the same mass.
Because of the residual SO(N ) global symmetry, A a is stable and a viable dark matter candidate. The dark sector couples to the SM only via a Higgs portal term. We set the mass parameters to zero, and let the EW symmetry be broken via dimensional transmutation due to quantum corrections. This choice has two motivations: 1) Reducing the number of free parameters to just two (the gauge coupling g D and the portal coupling λ p ), and 2) Allowing only for logarithmic quantum corrections to the scalar mass, and as a consequence solving in principle the SM hierarchy problem. In this extension of the SM indeed the fine tuning problem is traded with that of finding an ultraviolet boundary condition that motivates the choice of zero mass parameters.
The resulting model provides a general setup for SU(N ) vector dark matter with a minimal number of free parameters and matter fields. We studied quantitatively the phenomenology of the model for N = 2, 3, 4 by scanning the two-dimensional parameter space for data points producing a viable mass spectrum. We then selected, by performing a goodness of fit analysis, the data points that match at 95% CL the LHC measured coupling coefficients of the physical Higgs to SM particles. We then calculated the beta functions for all the couplings and showed that for about 5% of the LHC viable data points the Higgs field quartic coupling (as well as the dark scalar one) stay positive up to the Planck scale, therefore solving the metastability problem of the SM. The same data points feature also successful EW symmetry breaking and perturbativity up to the Planck scale. Finally, we calculated the dark matter relic abundance and selected the data points that satisfy the experimental 95% CL bound: for N = 2, 3 we found that about 1% and 2%, respectively, of the LHC viable, stable data points produce also a viable relic abundance, while for N = 4 no such data point exists. The constraint from the dark matter direct detection experiments is comfortably satisfied by the same viable data points: they produce a cross section for the scattering of the dark matter candidate on nuclei which is an order of magnitude smaller than the experimental lower limit. To assess the discoverability of the predicted heavy Higgs, we imposed the CMS constraint on additional Higgs-like resonances and found that about half of the N = 3 universally viable data points are actually already ruled out, while none is for the N = 2 model. To summarize, we have shown that the minimal vector dark matter extension of the SM presented in this paper leads to EW symmetry breaking through radiative corrections, stabilizes the scalar potential while providing an experimentally viable dark matter candidate and satisfying direct search and LHC Higgs coupling constraints. All this is achieved within a two dimensional parameter space. As such, these models represent a valid and attractive scenario beyond the SM. We find that N = 2, 3 provide the most appealing model setting in light of present data from colliders and dark matter direct search experiments. We expect a heavy Higgs to be discovered at LHC by the end of Run II or the N = 3 model to be ruled out.