Gamma-ray line from electroweakly interacting non-abelian spin-1 dark matter

We study gamma-ray line signatures from electroweakly interacting non-abelian spin-1 dark matter (DM). In this model, $Z_2$-odd spin-1 particles including a DM candidate have the SU(2)$_L$ triplet-like features, and the Sommerfeld enhancement is relevant in the annihilation processes. We derive the annihilation cross sections contributing to the photon emission and compare with the SU(2)$_L$ triplet fermions, such as Wino DM in the supersymmetric Standard Model. The Sommerfeld enhancement factor is approximately the same in both systems, while our spin-1 DM predicts the larger annihilation cross sections into $\gamma \gamma/ Z \gamma$ modes than those of the Wino by $\frac{38}{9}$. This is because a spin-1 DM pair forms not only $J=0$ but also $J=2$ partial wave states where $J$ denotes the total spin angular momentum. Our spin-1 DM also has a new annihilation mode into $Z_2$-even extra heavy vector and photon, $Z' \gamma$. For this mode, the photon energy depends on the masses of DM and the heavy vector, and thus we have a chance to probe the mass spectrum. The latest gamma-ray line search in the Galactic Center region gives a strong constraint on our spin-$1$ DM. We can probe the DM mass for $\lesssim 25.3~$TeV by the Cherenkov Telescope Array experiment even if we assume a conservative DM density profile.


Introduction
We have overwhelming evidence that suggests dark matter (DM) in our universe. Although the nature of DM remains unrevealed, we have one quantitative piece of information about DM, the DM energy density in the current universe. This energy density is determined to be Ωh 2 = 0.120±0.001 by the Planck collaboration assuming the ΛCDM cosmological model [1].
One of the most promising DM candidates is the Weakly Interacting Massive Particles (WIMPs).
If we assume the WIMPs in our expanding universe, we can explain the current DM energy density as the thermal relic abundance through the Freeze-out mechanism [2]. In this scenario, the velocityweighted annihilation cross section for DM, σv rel , characterizes the prediction of DM abundance.
To reach the observed DM energy density, we need the value of the canonical cross section, σv rel 3 × 10 −26 cm 3 /s. This value is obtained once we assume DM mass and couplings to be the typical values of those in the electroweak theory in the Standard Model (SM). This fact is our motivation to consider DM candidates with electroweak interactions.
To reveal the general features of spin-1 DM, we construct the renormalizable four-dimensional model of electroweakly interacting non-abelian spin-1 DM [15]. This model is the minimal setup that realizes the fundamental features of the five-dimensional model of the spin-1 DM such as the Z 2 -parity and the degenerated mass spectrum. We impose the exchange symmetry between the gauge groups to realize the above features, which is inspired by the method of deconstructing dimensions [16,17]. Our spin-1 DM couples to the electroweak gauge bosons with the electroweak gauge couplings, and the DM mass should be O(1) TeV to explain the correct DM energy density.
In this region, the electroweak bosons form the approximately long-range force potential. Consequently, the Sommerfeld enhancement is relevant in the DM annihilation processes as discussed in the other electroweakly interacting DM models [18][19][20][21][22][23]. Due to this enhancement, monochromatic spectral gamma-ray lines from the DM annihilation are the striking signals for our spin-1 DM.
In this paper, we study gamma-ray line signatures from electroweakly interacting non-abelian spin-1 DM. We make a comparison with the pure Wino DM, which is the spin- 1 2 DM candidate of the SU(2) L triplet in supersymmetric models, and clarify the differences between these systems.
We derive the constraint from the latest gamma-ray observation in the Galactic Center region. We also reveal the region where we can probe in the future gamma-ray observation.
The rest of the paper is organized as follows. In Sec. 2, we briefly review the electroweakly interacting non-abelian spin-1 DM. In Sec. 3, we show the two-body effective action for the spin-1 DM system. We also show the annihilation cross section contributing to the monochromatic gammaray signals. In Sec. 4, we present our numerical results. We compare the predicted annihilation cross section with that of the Wino DM and figure out distinctive features of our spin-1 DM. We 1 The references for the other models are given in Ref. [6].
show the constraints from the latest search for a monochromatic spectral line and the prospect sensitivity in the future gamma-ray observation. Our conclusions are given in Sec. 5. We show the derivation of the two-body effective action in Appendix A.

Model
We briefly introduce a model of electroweakly interacting non-abelian vector DM [15]. We extend the electroweak gauge symmetry in the SM into SU(2) 0 ×SU(2) 1 ×SU(2) 2 ×U(1) Y . The gauge bosons for SU(2) 0 , SU(2) 1 , SU(2) 2 , and U(1) Y are denoted as W a 0µ , W a 1µ , W a 2µ , and B µ , respectively (a = 1, 2, 3). The gauge couplings for each symmetry are denoted as g 0 , g 1 , g 2 , and g , respectively. We summarize the matter fields and Higgs fields in Table I. Each fermion field corresponds to the SM fermion with the same SU(3) c and U(1) Y charge. We introduce bifundamental scalar fields, Φ j (j = 1, 2), expressed as the two-by-two matrices. We impose the real conditions for Φ j to reduce the degrees of freedom.
and each of Φ j contains four real degrees of freedom. The gauge transformations of Φ j and H are shown below.
where U n denote the two-by-two gauge transformation matrices of SU(2) n (n = 0, 1, 2) and θ Y is the phase of the U(1) Y . We also impose a discrete symmetry under the following transformation.
where all the other fields remain unchanged. This transformation is equivalent to the exchange of SU(2) 0 and SU(2) 2 , and thus it also requires g 0 = g 2 .
The Lagrangian for the extended bosonic sector is After all the Higgs fields develop nonzero VEVs, we still have the exchange symmetry whose symmetry transformations are shown below.
To find out a Z 2 parity from this residual discrete symmetry, we anti-symmetrize the exchanged fields and define the following states.
These fields are eigenstates of the U(1) em charge. 2 These states acquire (−1) factors under the transformation in Eq. (2.9) while all the other states remain unchanged. This is nothing but a Z 2 parity assignment. Note that the Z 2 -odd states are automatically mass eigenstates since the mass mixing terms with other states are forbidden by this Z 2 parity.
We refer to Z 2 -odd spin-1 particles, V 0 and V ± , as "V -particles." The V -particles are approximately regarded as the spin-1 triplet of SU(2) L and have the degenerated masses at the tree-level. (2.14) The electroweak radiative corrections break the degeneracy, which makes the charged component slightly heavier than the neutral one. We find the following value for mass splitting at the one-loop level.
We assume h D is heavier than V 0 to focus on the spin-1 DM scenario. Therefore, V 0 is the lightest Z 2 -odd particle and our stable spin-1 DM candidate. 2 The U(1)em symmetry generator is expressed as Q = T 3 0 + T 3 1 + T 3 2 + Y where T 3 n denote the third generators of SU(2)n (n = 0, 1, 2).

Physical spectrum
We summarize the physical spectra in our model. Diagonalizing the mass matrices, we obtain the following mass eigenstates in the Higgs sector.  We take the scalar masses, {m h , m h , m h D }, and the mixing angle, φ h , as input parameters. We can express the dimensionless couplings of V scalar in these input parameters as shown below. (2.20) In the following discussion, we focus on φ h 0.1 to evade the direct detection constraints [15].
For the charged gauge bosons, we obtain  The masses of charged Z 2 -even vectors are obtained as For the neutral gauge bosons, we obtain  We define φ 0 that satisfies the following relation. (2.33) The parameters in the Lagrangian in Eq. (2.4) are summarized below.
The first four parameters are already fixed by the experiments, while the last five parameters are free. As we mentioned before, we consider the small φ h regime. Therefore, the SM Higgs coupling with the V -particles are suppressed by the small mixing factor, sin φ h . Although we do not specify the values of m h and m h D in the following discussion, we assume these scalar masses are in the TeV scale to focus on the phenomenological aspects resulting from the electroweak interactions. The constraints on these parameters in the scalar sector are studied in Ref. [15]. For later convenience, we introduce g W as defined below.
This coupling corresponds to the SU(2) L gauge coupling approximately, and we obtain g W 0.65, which is the same value as the SU(2) L gauge coupling in the SM, in the limit of v Φ v.
We assume v Φ v throughout our study, and it is useful to derive the approximated forms for the physical values in this limit. For m W and m W , we obtain and thus we easily obtain the correct value of m W . The gauge couplings, g 0 and g 1 , are expressed These couplings are constrained by the perturbative unitarity bounds. We obtain the following constraints in the high energy regime [24].
From these bounds, we can narrow down the viable range of the mass ratio, More detailed explanations of our model are given in Ref. [15].

Couplings of V -particles
We show the couplings of the V -particles in the limit of v Φ v. The vector triple couplings are shown below.
We define s W ≡ sin θ W and c W ≡ cos θ W where θ W is the Weinberg angle. We also define g Z in Eq. (2.47) to characterize the couplings between Z and the V -particles. The vector quartic couplings are shown below.

Annihilation cross section for V -particles
As studied in the previous work [15], we need m V O(1) TeV to explain the correct DM relic abundance in our model. In this region, the W and Z bosons effectively form the long-range force potential for the V -particles. Consequently, we have the sizable Sommerfeld enhancement in the DM annihilation processes. For this class of models, the monochromatic gamma-ray line from the DM annihilation is an excellent probe.
In this section, we summarize the formulas to discuss the gamma-ray line signature from our spin-1 DM. We apply the same formalism as those for the spin-0 or spin-1/2 DM with electroweak interactions [4,20]. First, we define the non-relativistic (NR) field operators for the V -particles to fix our notations. Second, we show the NR two-body effective action for the V -particles. We focus on the electrically neutral two-body states to study the DM annihilation signals. We derive the effective action in Appendix A.

Non-relativistic field operators for V -particles
The asymptotic field operators for the V -particles are defined below.
where A µ (p) denote the physical polarization vectors. These polarization vectors satisfy the transverse conditions, and the orthogonal relations.
We impose the canonical commutation relations between the annihilation and creation operators.
To derive the NR two-body effective action, we perform the NR expansion for these operators and integrate out the large momentum modes. Since the zeroth component of the polarization tensor is sub-leading, O(|p|/m V ), we focus on the spacial components (i = 1, 2, 3) for the Vparticle operators.
where we define We also define the NR one particle states for the V -particles with the momentum p and the polarization A.

Two-body effective action
We show the two-body effective action for the NR V -particles. The electrically neutral two-body . We change the space-time coordinates into the center of mass coordinate, R = (R 0 , R), and the relative coordinate, r.
As shown later, the NR leading-order potential has a spherically symmetric form. Therefore, the total spin angular momentum, J, and z-component of the spin angular momentum, J z , are good quantum numbers. We obtain the following effective action in the decomposed form into each partial wave state.
where r ≡ |r| and J = 0, 1, 2 with |J z | ≤ J. TheV andΓ J denote the real part and imaginary part of the potential, respectively. We define the two-body fields for each (J, J z ) as shown below. (3.20) The upper and lower components are formed by V − V + and V 0 V 0 , respectively.
In the above definition, we introduce the basis of three-by-three matrices.
These matrices satisfy the following orthogonal relation.
Note thatŜ J,Jz is symmetric matrices for J = 0, 2 and anti-symmetric for J = 1. We have no φ 1,Jz N because φ N is composed of identical particles. As mentioned before, J is conserved as long as we focus on the NR leading order contributions. Therefore, J = 1 partial wave modes are irrelevant to discuss the DM annihilation signals. We only consider J = 0, 2 states in the following discussion.
The normalization factors for the two-body states are fixed to realize the canonical weights for the two-body propagators. The two-body propagator of φ J,Jz C is defined as the time-ordered product. The leading-order expression forV is obtained as follows.
. This potential is induced by the NR processes between V -particles. The offdiagonal elements are induced by the W boson exchange processes, which cause the mixing between , and g Z is defined in Eq. (2.47). We show the derivation ofΓ J in Appendix 2.

Annihilation cross section
We derive the s-wave spin-averaged velocity-weighted cross section for V 0 V 0 → XX annihilation process through the optical theorem.
where C = 2 for the initial states composed of the identical particles. We introduce E as the NR kinetic energy of the V -particles, and d αβ (E) (α, β = 1, 2) as the Sommerfeld enhancement factor. We numerically obtain d αβ (E) by solving the Schrödinger equation [20].
Before we show the numerical results, we remark the distinctive features of our spin-1 DM compared with the Wino DM, which is SU(2) L triplet spin-1 2 DM. We focus on the leading-order results in both DM systems to show the comparison. 5 The Sommerfeld enhancement factor is approximately the same in both systems since we have the sameV and the mass splitting at the leading-order. This is because the leading-order interactions with W and Z bosons in the NR limit are independent of the DM spin. We have contributions other than W and Z interactions but they are sub-leading as mentioned above. The SM Higgs contributions forV , which is zero for pure Wino DM at the leading-order, are suppressed by the small mixing angle φ h . The contributions from W , Z , and h are suppressed by their masses. 6 Consequently, we have the same resonance structure as that in the Wino system. On the other hand, the annihilation cross section into γγ and Zγ are larger than those for the Wino by 38 9 4.22 · · · . This is because spin-1 DM forms both J = 0 and J = 2 partial wave states while spin-1 2 DM forms only J = 0 state. Our spin-1 DM also has a new annihilation mode into Z γ. If this new channel is kinematically opened, a photon is emitted with the energy depending on both m V and m Z . This channel also contributes to the monochromatic gamma-ray line signals, which we study further in Sec. 4.

Gamma-ray line signatures
We study the gamma-ray line signatures from our spin-1 DM. We derive the constraints from the search for monochromatic gamma-ray lines in the Galactic Center region. The derived constraints highly depend on the DM density profiles, and thus we show our result by taking some types of benchmark profiles to show the uncertainty.

Line cross section
The latest analysis for the gamma-ray line signals is performed by the H.E.S.S. collaboration using the ten years data of the gamma-ray observation in the Galactic Center region [32]. We constrain the gamma-ray line signatures from the DM annihilation by using this result. In our model, we have three annihilation modes involving the gamma-ray line signals, {γγ, Zγ, Z γ}. For the γγ/Zγ modes, the photon energy is approximately equal to the DM mass, E γ m V , where we take the NR limit for the initial DM pair and neglect m Z . For the Z γ mode, we can not neglect m Z because m Z must be heavier than m V , see Eq. (2.42). The photon energy in the Z γ annihilation mode depends on both m V and m Z .
If the Z γ mode is kinematically allowed, namely for m Z 2m V , we have a chance to observe the double-peak gamma-ray spectrum at E γ m V − ∆E γ and E γ m V . To distinguish between these two peaks, ∆Eγ m V should be larger than the instrumental energy resolution. In the H.E.S.S. experiment, the energy resolution is about 10 % for m DM 300 GeV. Our interesting region is to discuss the signal discrimination. The lower and upper values come from the g 0 perturbative unitarity and the kinematical suppression of the Z γ annihilation mode, respectively. If we focus on this region, the condition ∆Eγ m V 0.1 is always satisfied. Therefore, we can discriminate the gamma-ray peak originated from the γγ/Zγ modes and the peak from the Z γ mode. This doublepeak gamma-ray spectrum is an outstanding feature of our DM model, and we can read out the values of m V and m Z from this double-peak spectrum. 7 We define the "line cross section", which contributes to the gamma-ray line signal. We introduce two types of line cross sections that predict the different final photon energy. We derive the excluded region by using the experimental bound shown in Fig. 6 of Ref. [32]. In their analysis, all the final state particles are assumed to be massless, and thus we can directly compare these constraints with σv rel line γγ,Zγ . We can also derive the constraint on σv rel line Z γ by noting that the horizontal axis of Fig. 6 in Ref. [32] corresponds to E γ . These constraints are derived by assuming the three cuspy DM density profiles as defined below.
Another choice is the cored DM density profile. The core radius, r c , depends on the model of baryonic physics, and cores extending to r c ∼ 5 kpc can potentially be obtained [36]. The cored Einasto profile is defined as shown below.
where ρ Einasto (r) is defined in Eq. (4.5). For the cored profile, ρ s is chosen to realize the value of the local DM density. We show the current excluded region with the cusped DM density profiles assumed in the analysis of the H.E.S.S. collaboration [32]. The sensitivity of the H.E.S.S. experiment is studied for the cored profiles in Ref. [37], which is focusing on the pure Wino DM search. 8 From this study, the upper bound on the line cross section will be weakened by a factor of O(10 − 100) if we use the cored DM density profile.

Constraint from gamma-ray line signatures
We compare the predicted line cross section between our spin-1 DM and the Wino system. In  [32] for the Einasto profile [33]. We also show green dashed and green dotted curves as the upper limits on the cross section for the cusped NFW profile [35] and the Einasto2 profile [34], respectively. Since our spin-1 DM and the Wino DM have the SU(2) L triplet features,  [32] for the Einasto profile [33]. The green dotted curve and dashed curve show the upper limit for the NFW profile [35] and Einasto2 profile [34], respectively. The upper bound is expected to be weakened by a factor of O(10 − 100) for the cored DM density profile.
the Sommerfeld resonance structures are almost the same. The line cross section for the spin-1 DM is larger than that for the Wino DM by 38 9 , and thus we obtain more severe constraints on the spin-1 DM. We find the following excluded regions for our spin-1 DM depending on the DM density profiles. 9 If we take m h mV , the parameter region in Fig. 2 may be constrained by the perturbative unitarity bounds on the scalar couplings, which is studied in our previous collaboration [15]. We can evade these unitarity bounds by taking sufficiently small values of φ h and m h .  Fig. 3, we show the constraints for the NFW [35], and Einasto2 DM density profiles [34]. The derived constraints depend on the DM density profiles. The blue excluded regions from σv rel line Z γ give stronger constraints for m Z m V . This is because we have the enhancement factor in the coupling of Z , g Z , defined in Eq. (2.47). Note that g 0 gets larger in the same region, and thus we expect relatively large higher-order correction for our perturbative calculations.
We have a further chance to explore the parameter region in the upcoming Cherenkov Telescope Array (CTA) [38,39]. The prospect sensitivity for the line gamma-ray signals is studied by Ref. [40] for the Wino and Higgsino DM. In Fig. 4, we show the current bound and the future sensitivity expected in the CTA. The green region is excluded by the H.E.S.S. observation for the cusped Einasto2 profile shown in the lower panel of Fig. 3. We use the prospect bound derived in Ref. [40] to show the CTA sensitivity. The orange region with the dashed boundary shows the most conservative sensitivity assuming the core radius r c = 5 kpc, and we will obtain m V 25.3 TeV as the prospect bound. The whole mass range in Fig. 4 will be covered if we take r c 2 kpc.

Conclusions
In this paper, we study the gamma-ray line signatures from the electroweakly interacting nonabelian vector DM model. We derive the two-body effective action for the spin-1 DM in the NR limit. Since our DM has the SU(2) L triplet-like features, the Sommerfeld enhancement factor is almost the same as that for the pure Wino DM. The predicted annihilation cross sections into γγ/Zγ are larger by 38 9 than those of the Wino. This is because our spin-1 DM pair has the additional partial wave contributions with the total spin angular momentum J = 2. Therefore, the gamma-ray line signals provide stronger constraints on our spin-1 DM compared with the Wino DM.
We show the constraints on the line cross section for the γγ, Zγ, and Z γ modes by the H.E.S.S. observation in the Galactic Center region. The photon from the γγ/Zγ modes and the Z γ mode are separable through the final photon energy. Therefore, we may observe a double-peak in the gamma-ray energy spectrum, which provides a unique signature of our spin-1 DM. The constraints strongly depend on the DM density profiles, and we obtain m V 14.4 TeV (11.7 TeV) for the Einasto (Einasto2) profile. The annihilation into Z γ also gives a strong constraint for m V m Z where couplings between DM and Z are enhanced. We also show the future sensitivity in the CTA experiment. We can probe m V 25.3 TeV even if we take the conservative cored DM density profile with the core radius of 5 kpc.
We have possible extensions of our study. We can derive a more robust constraint on our model by studying the continuous gamma-ray search in observations of dwarf spheroidal galaxies by the Fermi-LAT experiment [41,42]. We also expect a viable constraint from the antiproton observations as studied for the pure Wino DM [43]. We need to follow the decay chain including the decay of the W /Z bosons in our model. Although we expect the strongest bound comes from the gamma-ray line signatures studied in this paper, we suffer a large uncertainty in the DM density profiles. Therefore, it is worth evaluating the more robust constraints from the other channels. We also expect the Sommerfeld enhancement affects the calculations of the thermal relic abundance, which we need to derive the effective action for the electrically charged two-body states [31,44].
These studies are completed in our future collaborations.  [34] shown in the lower panel of Fig. 3. The orange region with the dashed boundary is the prospect sensitivity in the CTA experiment for the cored Einasto profile with r c = 5 kpc, which is the most conservative one [40]. If we take r c 2 kpc, the prospect sensitivity will cover the whole region of this figure.

A Derivation of effective action
We give the derivation of the effective action shown in Sec. 3.
1 Real part of potential 2 Potential Real Part In the NR limit, we obtain the following amplitude at the leading-order.
where M Z,γ and M W corresponds to the neutral and charged boson exchange processes, respectively. The labels of the polarization are implicit. We obtain the effective action composed of the NR V -particle operators, which are defined in Eqs. (3.12)-(3.14), to realize the above amplitudes.
To express the action in the decomposed form into each partial wave mode, we reform the vector indices (i, j) by using the Fierz identity for Grassmann-even operators. 2 Imaginary part of potential a Matching procedure To derive the imaginary part of the potential,Γ J , we perform the operator matching between the two-body field operators and the calculations of the one-loop amplitudes of the V -particles.
Since we focus on the annihilation modes into the neutral vectors, the (1, 1)-component ofΓ J only has the nonzero value. Therefore, we can use the optical theorem to calculate the imaginary part of the one-loop amplitude.
where the left-hand side denotes the imaginary part of the forward scattering amplitude for 11 . In the right-hand side, σv rel J V − V + →XX denotes the partial wave annihilation cross section of V − V + → XX for the initial state with J. In the evaluation of the annihilation cross section, we only leave the leadingorder terms in the NR limit and take the massless limit for all the SM particles in the final states while we leave the masses of Z . We deriveΓ J through the above procedure as summarized in In this section we calculate the cross section for the V − and V + annihilation to two photons. The tree-level diagrams in the unitarity gauge are shown in Fig.??
amplitude for each diagram is given as follows.
To calculate the Sommerfeld effects, we need to know the cross section for each total spin state.
In this case, the initial state (two spin-1 particles) can be total spin, J = 0, 1, 2. We can obtain the cross section for each total spin by replacing the initial state polarization vectors with the initial spin state matrix for the total spin, J and the z-component of the spin, J z , as follows.
We consider the non-relativistic limit for the initial state. The polarization vectors of massive gauge bosons in the initial state are given by The time-like components of these polarization vectors are 0, so S 00 = 0 and S σ0 = 0 = S 0ρ . Then We show the derivation ofΓ J γγ as a concrete example. We can focus on the nonzero component, (Γ J γγ ) 11 . First, we calculate the velocity-weighted annihilation cross section for V − V + → γγ. The tree-level diagrams in the unitarity gauge are shown in Fig. 6. The amplitudes that correspond to each diagram are obtained as follows.
In the NR limit for the initial V -particles, the zeroth components of the polarization vectors are sub-leading, and we focus on the spatial component, i (p − ) j (p + ). We decompose the amplitude into each partial wave contribution by replacing the initial state polarization vectors with S J,Jz ij defined in Eqs. where |J z | ≤ J and k i denotes the spatial components of k 1µ . We use S 2,Jz ii = 0 to obtain the above result. The polarization vectors for the photons are given by where θ is the scattering angle for the photon with its momentum k 1 .
After taking the sum over the final state spins, the squared amplitudes can be expressed as follow. The spin-averaged total cross section is obtained by adding up all the partial wave cross sections. This cross section is larger than that of the Wino DM by a factor of 38 9 . For the Zγ mode, we take the massless limit of the Z boson, and the calculation procedure is the same. For the Z γ mode, we do not neglect m Z , and the longitudinal mode also contributes to the final result.
Next, we calculate the imaginary part of the forward scattering amplitude using the two-body effective action shown in Eq. (3.19). Namely, we use the following coupling to calculate the γγ contribution to ImM. This result is summarized in Eqs. (3.30)-(3.31). We can also determineΓ J Zγ andΓ J Z γ in the same procedure.