Radiative Interactions between New Non-Abelian Gauge Sector and the Standard Model

We discuss one loop generation of the term connecting gauge fields from a local hidden $SU(2)_H$ and the standard model $U(1)_Y$ introducing an $SU(2)_H$ doublet fermion with non-zero hypercharge and a scalar field in adjoint representation. Then we obtain a kinetic mixing term between $SU(2)_H$ and $U(1)_Y$ gauge fields after the adjoint scalar field developing vacuum expectation value. We illustrate such a concrete scenario introducing a dark matter model in an ultraviolet (UV) completion with local $SU(2)_H$ symmetry where the scalar doublet is our dark matter candidate and its stability is guaranteed by remnant $Z_2$ symmetry from $SU(2)_H$. Relic density of dark matter is calculated focusing on the case in which dark matter annihilate into known particles via $SU(2)_H$ gauge interactions with radiatively induced kinetic mixing.


I. INTRODUCTION
A hidden/extra gauge symmetry is one of the interesting possibilities as physics beyond the standard model (SM) since it provides us rich phenomenology such as dark photon, new mediator, dark matter (DM) and so on [1][2][3]. For a hidden Abelian gauge symmetry we can always write kinetic mixing term with the SM U (1) Y as [4] − 1 2 sin δB µν B µν (1) where B µν and B µν are gauge field strengths associated with U (1) Y and the new Abelian gauge symmetry, and sin δ characterizes the size of mixing. The hidden gauge boson can interact with the SM particles through such a term even if all the SM fields are not charged under the hidden symmetry. In addition to an Abelian hidden/extra gauge symmetry, a non-Abelian one is interesting, because it can provide richer phenomenology giving both vector DM and/or mediators [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. However we cannot write any gauge invariant kinetic mixing terms between Non-Abelian hidden gauge fields and the SM ones at renormalizable level.
In fact, we can write a term generating a non-Abelian kinetic mixing at non-renormalizable level, introducing a scalar field ϕ in adjoint representation of the non-Abelian gauge group such that [6,22,25] where Λ is arbitrary cutoff scale, X µν is gauge field strength associated with new non-Abelian gauge fields, and trace is taken in representation space. It is thus interesting to investigate radiative generation of such an effective operator to realize an UV complete model with hidden non-Abelian gauge symmetry mixing with the SM one.
In this work, we discuss a simple scenario to generate non-Abelian kinetic mixing in  and discussion are given in Sec.IV.
Here we discuss one loop generation of interactions connecting SU (2) H and U (1) Y gauge fields. As a minimal setup, we introduce an SU (2) H doublet fermion with hypercharge Y = −1 and an SU (2) H adjoint real scalar field as summarized in Table I. New Lagrangian and potential are written by where L SM is the SM Lagrangian without Higgs potential, ϕ = ϕ a σ a /2 with σ a being the Pauli matrix acting on SU (2) H representation space, and H is the SM Higgs field.
Interactions connecting U (1) Y and SU (2) H gauge fields are generated from one loop diagrams given in Figs. 1 and 2. The diagrams for three point interactions in Fig. 1 are given by where p 1 (2) is momentum corresponding to B µ (X ν ) and subscript of M µν i corresponding to diagram-i in Fig. 1. Then we calculate the RHS assuming {p 2 1 , p 2 2 , p 1 · p 2 } M 2 , and obtain the following approximated formula: We also calculate diagrams for four point interactions given in Fig. 2. The analytic forms of the diagrams are where abc is anti-symmetric tensor, p 1 , p 2 and p 3 are momenta corresponding to B ρ , X µ and X ν , and subscript of M µνρ i corresponding to diagram-i in Fig. 2. As in the calculation of diagram-1 and -2, we can approximate sum of diagrams such that where we abbreviate O(1/M 3 ) terms since they are more suppressed. Finally summation of all diagrams in Figs. 1 and 2 gives effective Lagrangian terms of The first term in the RHS of Eq. (10) matches the form of Eq. (2) that can give kinetic mixing. Note that we also have extra 5-dimensional terms that give interactions including three gauge fields. Here we do not discuss these extra terms in details, since they do not contribute to kinetic mixing. After ϕ a developing VEV, we obtain Therefore we obtain small kinetic mixing between SU (2) H and U (1) Y , where components of X a µ mixing with B µ depend on configuration of the triplet VEV. In the next section, we illustrate this scenario introducing a specific UV complete model.

III. AN UV COMPLETE DM MODEL WITH NON-ABELIAN KINETIC MIXING
In this section we consider a simple DM model under SU (2) H symmetry with radiatively generated kinetic mixing discussed in previous section. In addition to the field contents in Table I, we introduce a second SU (2) H triplet real scalar ϕ with non-zero VEV and a doublet scalar χ with vanishing VEV as shown in Table II. We need two real triplet scalars to break SU (2) H into Z 2 spontaneously [22]. The doublet χ ≡ [χ 1 , χ 2 ] T is our DM candidate whose component has Z 2 odd parity; after SU  where L and V are the same as given in Eqs. (3) and (4). Here we choose the VEV alignments of two scalar triplets as where the configurations are equivalent to those of discussed in ref. [22]. Then SU (2) H is broken to Z 2 symmetry by these VEVs where the components of SU (2) H doublets have odd parity and the other fields have even parity. The scalar potential including only ϕ and ϕ is the same as in ref. [22] and we do not discuss details here. Also we assume parameters associated with χ satisfy inert condition taking µ 2 χ > 0.
In general, χ 1 and χ 2 mix by the last term of Eq. (15) but we assume µ to be much smaller than the other mass dimension parameters so that they are approximated to be mass eigenstates. We then obtain physical masses such that where we choose m χ 1 < m χ 2 assuming µ > 0. Thus χ 1 is our DM candidate.
Hidden gauge bosons: In this model, we have two kinetic mixing terms between SU (2) H and U (1) Y , since we introduce two SU (2) H triplet scalars. From discussion in previous section, we obtain where we omit extra terms appearing in Eq. (10) 1 . After ϕ and ϕ developing VEVs, the kinetic mixing terms can be obtained as where sin . For small kinetic mixing parameter, kinetic terms can be approximately diagonalized bỹ where B µ and X a µ are gauge fields under the basis with diagonalized kinetic terms. In our DM analysis below, we assume δ δ for simplicity. Ignoring small kinetic mixing effect, we obtain masses of X a µ such that Thus X 2 is always heavier than the other components. We also have Z-X 3 mixing via the kinetic mixing effect. The mixing angle can be written as where m Z is the SM Z boson mass. In Fig. 3, we show sin θ X as a function of m X 3 for several values of δ. We find that sin θ X is sufficiently small and allowed by experimental constraints [26].
DM physics: here we discuss DM in our model and estimate its relic density. In our analysis, we focus on gauge interactions and assume scalar portal interactions are suppressed by small couplings in the potential. The relevant interactions for DM are written by where f denotes the SM fermion, s X (c X ) ≡ sin θ X (cos θ X ) and Q is electric charge. We calculate relic density of DM using micrOMEGAs 5.2.4 [27] implementing the interactions to search for parameter region realizing observed value. The parameters are scanned in the following ranges: where we fix δ = 10 −4 , δ = 10 −5 and m χ 2 = 1.5m χ 1 to suppress coannihilation process.
In Fig. 4, we show parameter region, satisfying observed relic density of DM [28], where we apply approximated region of 0.11 < Ωh 2 < 0.13. We find that relic density can be explained by O(0.1)-O(1) gauge coupling g X in the region of m DM > m X 3 , since cross section of χ 1 χ 1 → X a X b is sizable. In the region of m DM < m X 3 relic density can be explained only around 2m DM ∼ m X 3 , since we need resonant enhancement of annihilation cross section because of small kinetic mixing. Note that DM-nucleon scattering cross section is suppressed by small kinetic mixing δ and it is safe from current direct detection constraints.

IV. SUMMARY AND DISCUSSION
In this paper, we have discussed one loop generation of the term connecting gauge fields from the local hidden SU (2) H and the SM U (1) Y , introducing an SU (2) H doublet fermion with non-zero hypercharge and a scalar field in adjoint representation. Then we have obtained the kinetic mixing term between SU (2) H and U (1) Y gauge fields after the adjoint scalar field developing VEV.
We have introduced a DM model in an UV completion with SU (2) H , where the scalar doublet is our DM candidate and its stability is guaranteed by remnant Z 2 symmetry from SU (2) H . Relic density of DM has been calculated focusing on the case in which DM an-nihilates into the SM fields via SU (2) H gauge interactions with radiatively induced kinetic mixing. Then we have shown parameter region satisfying observed relic density in Fig. 4.
Before closing our letter, it would worthwhile to mention another application of extra fields.
E fermion [29,30] or its extended field to SU (2) L doublet L [31] are applied to generate small mass terms such as neutrinos at loop levels. In fact, it is possible to induce tiny neutrino masses, retaining our main result of radiative kinetic mixings. This types of models also provide a lot of intriguing phenomena and we will proceed this direction as another projects.