Vector Dark Matter through a Radiative Higgs Portal

We study a model of spin-1 dark matter which interacts with the Standard Model predominantly via exchange of Higgs bosons. We propose an alternative UV completion to the usual Vector Dark Matter Higgs Portal, in which vector-like fermions charged under SU(2)$_W \times$ U(1)$_Y$ and under the dark gauge group, U(1)$^\prime$, generate an effective interaction between the Higgs and the dark matter at one loop. We explore the resulting phenomenology and show that this dark matter candidate is a viable thermal relic and satisfies Higgs invisible width constraints as well as direct detection bounds.

2 coupling. But this form, while invariant under the SM gauge symmetries, is misleading.
Just like the SM W and Z bosons, a well-behaved UV description of V requires that it be associated with a gauge symmetry (the most simple construction of which would be an Abelian U(1) , though one could also consider non-Abelian theories as well), spontaneously broken to give V a mass. The term in Eq. (1) violates the U(1) , and must be engineered via its spontaneous breaking.
One tempting avenue would be to charge the Higgs itself under U (1) . In that case the Higgs kinetic term (D µ H) † (D µ H) contains Eq. (1), and the mass of V will arise as part of the vacuum expectation value (VEV) of H, naturally connecting the scale of the V mass to the electroweak scale. However, this construction contains other terms which mix V with the SM Z boson, with the result that V will inevitably end up unstable and contribute unacceptably to precision electroweak measurements unless it is very light (implying that it is very weakly coupled). This regime, though worth pursuing, is not very interesting for particle physics at the weak scale, and not very amenable to exploration through Higgs measurements at the LHC.
The situation is very different when the V mass is the result of a VEV living in a different scalar particle Φ which is a SM gauge singlet. In that case, there is no dangerous mixing with the SM Z boson, and the gauge coupling can be relatively large, where D µ Φ ≡ ∂ µ Φ − gQ Φ V µ Φ is the usual covariant derivative for a particle of charge Q Φ and V (Φ) is a U(1) -invariant potential designed to induce a VEV Φ = v φ , producing a mass for V , We have also included a scalar Higgs portal coupling λ P , which leads to tree-level mixing between the SM Higgs boson and the Higgs mode of Φ, effectively implementing the Higgs portal. As a construction implementing the Higgs portal, it is well motivated and has been extensively explored in the literature 1 [6][7][8][9][10][11][12][13][14][15]. A radiative model often has multiple paths to the same low energy physics, since the mediating particles are not themselves involved in the initial and final states. Starting with the basic module of Eq. (2), we aim for a construction which adds fermions mediating an interaction of the form (1) such that: • the vector particle V remains stable at the radiative level, which in particular requires that it does not kinetically mix with the SM electroweak interaction; • the full gauge structure SU(3) C × SU(2) W × U(1) Y × U(1) remains free from gauge anomalies; • there are no large contributions to the SM Higgs coupling to gluons or photons in contradiction with LHC measurements [16].
The first of these is the most subtle. Generically, communication between the SM Higgs and V requires that the mediator fermions be charged under both U(1) and the Standard Model, which typically will induce processes involving an odd number of V 's, resulting in their decay.
The simplest example of such a process is the kinetic mixing between V and hypercharge.
Such dangerous processes can be forbidden by a charge-conjugation symmetry, under which V is odd. In analogy with Furry's theorem of QED [17], this symmetry forbids processes involving an odd number of V 's at energies below the masses of the mediator fermions.
Cancelling gauge anomalies further suggests that the additional fermions appear in vectorlike pairs under both the SM and U(1) gauge symmetries, whereas renormalizable coupling to the Higgs requires fields in SU(2) W representations of size n and n+1 (and have hypercharges differing by 1/2). A minimal set of particles satisfying these conditions is shown in Table I the analysis of their mass eigenstates. We will restrict ourselves to other values for Q Φ , which avoids these features, and serves simply to adjust the mass of V . It's worth pointing out that this implies that the lightest of the fermionic states is also stable, and will be present in the Universe to some degree as a second component of dark matter. However, provided its mass is much larger than m V , fermion anti-fermion pairs will annihilate efficiently into weak bosons and V 's, leaving it as a negligible fraction of the dark matter.
In 2-component Weyl notation, the Lagrangian contains mass terms and Yukawa interactions for the new fermions, where a and b are SU(2) W indices, the SM Higgs H is defined to transform as a (2, −1 /2, 0), and spin indices have been suppressed. The U(1) charge conjugation symmetry, f 1 ↔ f 2 is manifest. After electroweak symmetry-breaking, the mass terms can be written as, where assemble collections of the electrically neutral (N and N ) and charged (E and E ) components of the fermions, and the mass matrices are given by, In the mass basis, there are three electrically neutral and two charged Dirac fermions, all of which interact with the dark matter V diagonally, since the states that mix all carry the same U (1) charge. Their coupling to the SM Higgs will involve the mixing matrices which transform from the gauge to the mass basis.
Note that by construction the electrically charged fermions receive no contributions from H , implying that they do not interact with the Higgs boson and lead to no one-loop correction to its effective coupling to photons. Our choice to arrange N such that they also receive no contributions from Φ implies that the fermions do not renormalize the usual Higgs portal coupling λ P of Eq.
(2) at one-loop (starting at two loops, there are contributions mediated by a mixture of the fermions and V itself). In order to better extract the features of the radiative model, we self-consistently assume that λ P is small enough to be subdominant in the majority of the remainder of this work.
Representative triangle diagram contributing to the Higgs-dark matter interaction.

B. σ SI and Higgs Invisible Width
Both the direct detection cross-section and the Higgs invisible decay width result from triangle diagrams (see Fig. 1). Integrating out the fermion ψ running in the loop, the h − V − V interaction can be encoded by two form factors: with coefficients A and B which are (in the on-shell DM limit, k 2 1 = k 2 2 = m 2 V ) functions of the fermion masses and mixings, m V , and the momentum through the Higgs line, p 2 .
Reasonably compact analytic expressions for A and B are derived in Appendix A. We observe that B(p 2 ) → 0 in the limit m V → 0 (i.e. when the U (1) symmetry is restored), as is required by gauge invariance, see Appendix A.
In terms of A and B, the cross section for non-relativistic scattering of V with a nucleon n is given by, where the momentum transfer through the Higgs is approximated as p 2 ≈ 0, and we use the hadronic matrix elements f T q , from DarkSUSY [18]. Because of the tiny up and down Yukawa couplings, scattering mediated by a Higgs is to good approximation iso-symmetric.
Representative box diagrams which contribute to DM annihilation into pairs of Higgs or electroweak bosons.
The same three point vertex function also describes the invisible decay width of the Higgs boson, where the Higgs is on-shell, p 2 = m 2 h . Note that because for small m V the coefficient B(p 2 ) ∝ m 4 V , this expression is finite in the limit m V → 0, as it should be.

C. Annihilation Cross Section and Relic Abundance
Pairs of dark matter can annihilate through the three point coupling of Fig. 1 through an (off-or on-shell) SM Higgs, leading to final states containing heavy quarks and/or weak bosons. These contributions exhibit a strong resonant behavior when m V m h /2. The gauge and Higgs boson final states also receive contributions at the same order from box diagrams (see Fig. 2), which contribute to processes including V V → hh, ZZ, W W, γγ, hZ, Zγ.
These box diagrams are sensitive to more of the details of the UV theory, receiving contributions from the charged fermions as well as the neutral ones. As a result, simple analytic forms are not particularly illuminating, and we evaluate them using FeynArts [19], Form-Calc, and LoopTools [20]. In the following section, we compute the full annihilation cross section including all of the accessible SM final states.

III. EXPERIMENTAL CONSTRAINTS AND PARAMETER SPACE
In this section, we examine the interesting parameter space, finding the regions consistent with the LUX limits on the spin independent DM-nucleon scattering cross-section [21]; and the invisible decay width of the Higgs produced via vector boson fusion (VBF) as constrained by CMS with 19.7 fb −1 at 8 TeV [22]. In the latter, we include the off-shell Higgs contribution following the technique presented in [23], simulating VBF Higgs production with HAWKv2.0 [24]. We also identify the regions leading to the correct thermal relic abundance for a standard cosmology, computing the loop diagrams with FeynArts [19], FormCalc, and LoopTools [20], which is then linked into micrOMEGAsV4.0 [25].
Because of the relatively large number of parameters, we build up insight into the phenomenology gradually by considering three different limits of the full theory. Initially in Sec. III A, we consider the limit in which one of the neutral fermions is much lighter than both the other two neutral states and both of the charged ones, and the coupling λ P is small enough to be neglected. We follow this in Sec. III B by allowing λ P to be large enough that there is relevant mixing between h and the Higgs mode of Φ. Finally, in Sec. III C we switch off λ P once more, but consider the case where all mediator fermions have comparable masses.

A. Single Fermion Limit
We begin with the case where the charged fermions and the two heavier neutral states are much heavier than the lightest neutral state, effectively decoupling from the phenomenology, and λ P can be ignored. As before we assume the physical scalar contained in Φ is heavy enough to be ignored. In this limit, the relevant parameters are the U (1) gauge coupling g, Yukawa coupling to the light fermion y, light fermion mass m ψ , and the vector dark matter mass m V . As we will see below, the correct thermal relic density can only be achieved for annihilation in the Higgs funnel region, for which one can neglect the box diagram contributions. In that case, the gauge and Yukawa couplings always appear in the combination yg 2 , leaving only three relevant parameter combinations. relevant fermion has a mass of 400 GeV. Despite the fact that the limits on the couplings are relatively weak, the conclusion is nonetheless that aside from a narrow region in the Higgs funnel region, additional interactions would be required to deplete the dark matter relic density enough to saturate the observed relic density.

B. Single Fermion with Scalar Mixing
Building on the single fermion limit, we now allow for substantial λ P such that the radial modes of H and Φ experience significant mixing, resulting in two CP even scalars we denote by h and h 2 . Describing this limit requires three additional free parameters, which we take to be the mass of the second scalar m h 2 , Φ = v φ , and the Higgs-scalar mixing angle α.
For small α, the form factors of Eqn. (8) are shifted: where the additional contribution is the tree level contribution to B(p 2 ) from the induced

C. Full Matter Content
As our final limit, we return to λ P 1 but allow for all of the fermions to have comparable masses. We consider three benchmark sets of masses and Yukawa interactions summarized in Table II, which contains the model parameters associated with the fermion sector, m, m n , y ψ , and y χ , as well as the resulting spectrum of neutral state masses M N and the coefficient of the h-N i -N j coupling in the mass basis, Y ij , with the mass eigenstates ordered With these quantities fixed, we explore the plane of the U(1) gauge coupling g and the mass of the dark matter m V .
In Yukawa couplings are stronger. In terms of the dominant contribution to the effective h-V -V coupling, in the first and third models, the lightest neutral state is the dominant contribution, whereas in the second benchmark model the lightest state has a small Yukawa coupling and is less important than the second lightest state, which has a much larger coupling.
In Fig. 6, we plot the relic abundance for the benchmark parameters with a large, fixed gauge coupling of g = 3.5, to make comparisons between the benchmarks more apparent.
Note that for our second and third benchmark models, this value is mildly excluded by limits on the invisible width of the Higgs for m V ≤ 60 GeV. All benchmarks can be thermal relics when the vector can resonantly annihilate through a Higgs, causing the sharp dip at m V ∼ m h /2. We also find that the second benchmark can attain a thermal relic for vector masses above 100 GeV, and third may be a thermal relic above 80 GeV. The success at larger DM masses is due to annihilation channels with two bosons in the final state. Of the three benchmarks, the second has the lightest charged states. This allows efficient annihilation through loops involving the charged fermions, such as those which result in the W W and ZZ final states. The third benchmark, also benefits from this with slightly heavier charged states. However, this case also has large Yukawas causing a marked drop in the relic abundance when DM is heavy enough to annihilate to two Higgs bosons.

IV. CONCLUSION
We have explored a simplified model in which the dark matter is a spin one vector particle which interacts with the Standard Model predominantly through Higgs exchange. Unlike the more usually considered Higgs portal based on the quartic interaction λ P , we mediate the interaction radiatively, via a loop of heavy fermions charged under both the dark U(1) as well as the SM electroweak interaction. By construction, the theory is anomaly free, has a heavy vector particle which is effectively stable, and leads to no large deviations in the properties of the SM Higgs. This last feature, together with the possibility to completely decouple the U(1) -breaking Higgs Φ from the SM are the primary features which distinguish the radiative model from the quartic-induced Higgs portal as far as dark matter phenomenology is concerned.
Of course, the UV structure of the radiative model is also far richer, with a family of electroweakly charged particles whose decays produce gauge bosons and missing momentum, a signature already under study in the context of the neutralinos and charginos of a supersymmetric theory. These states are the true avatars of the radiative Higgs portal. The thermal relic density suggests that their masses are at most around TeV, raising the hope that they could be found at the LHC run II or a future high energy collider. The contribution to the matrix element from a single fermion of mass m and Yukawa coupling y is given by: where, Evaluating the trace in the numerator and making use of the fact that k 1 · (k 1 ) = k 2 · (k 2 ) = 0 for on-shell vectors results in, Tr[...] = 4m g µν (m 2 − k 1 ·k 2 − k 2 ) + 4k µ k ν + k ν 1 k µ 2 . (A3) After Passarino-Veltman decomposition [26] we find, where the arguments of the C functions are (uniformly) C 0 (k 1 , k 2 ; m, m, m), etc.