$SU(2)_X$ Vector DM and Galactic Center Gamma-Ray Excess

An unbroken $Z_3$ symmetry remains when local $SU(2)_X$ is broken spontaneously by one quadruplet. The gauge boson $\chi_\mu (\bar \chi_\mu )$ carries the dark charge and is the candidate of dark matter (DM). By the mixture of scalar boson $\phi_r$ of quadruplet and standard model (SM) Higgs, the DM can annihilate to SM particles through Higgs portal. For investigating the implications of vector DM, we study the relic density of DM, the direct detection of DM-nucleon scattering and the excess of gamma-ray spectrum, which is supported by the data from {\it Fermi} Gamma-Ray Space Telescope. We find that with the DM mass of around $70$ GeV in our model, the excess of gamma-ray could be fitted well with the data.

symmetry in the theory is necessary. However, a discrete symmetry usually is put in by hand. In order to get a stable DM naturally, we study the model in which the unbroken symmetry originates from a spontaneously broken gauge symmetry. To realize the concept, particularly we are interested in the extension of the SM with a new SU(2) X gauge symmetry where the subscript X is regarded as a dark charge. The interesting properties of a local SU(2) X group are: (1) comparing with the local U(1) case in which the U(1) charge has to satisfy some artificial tuning [57], an unbroken discrete symmetry can be naturally preserved after the spontaneous breaking of the SU(2) X gauge symmetry; (2) the massive gauge bosons from SU(2) X could be the DM candidates. The various applications of the hidden SU (2) gauge symmetry have been studied in the literature, such as a remaining Z 2 symmetry with a quintet in Ref. [58], a custodial symmetry in Refs. [32,59] and an unbroken U(1) of SU (2) in Refs. [60,61].
Since the model with the custodial symmetry discussed in Ref. [59] is similar to our proposal, it is worthy to show the difference between them. It has been noticed that without introducing any new fermions or higher multiple states in the hidden SU(2) X gauge sector, a new fundamental representation of SU(2) X could lead to three degenerate DM candidates by utilizing the SO(3) custodial symmetry [59]. Due to the custodial symmetry, the three DM candidates are stable particles. However, the symmetry could be broken easily when SU(2) X fermions and/or higher representation scalar fields are included. Although the inclusion of the new fermionic and/or higher multiple staff is not necessary, if one connects the origin of neutrino masses with the dark sector, the inclusion of the new staff becomes a relevant issue. In order to get over the possible unstable effects when more phenomenological problems in particle physics are involved, we propose to use a discrete symmetry to stabilize DM, where the discrete symmetry is not broken by higher multiplet fields or fermions under SU(2) X . Additionally, the processes for explaining the gamma-ray excess in our model are different from those dictated by the custodial symmetry [32,59]. We will see the differences in the analysis below. Moreover, we find that an Z 3 discrete symmetry indeed remains when SU(2) X is broken by a scalar quadruplet. Based on the introduced quadruplet, we summarize the characteristics of our model as follows: (a) the unbroken Z 3 symmetry is the remnant of SU(2) X , (b) two gauge bosons χ µ andχ µ carry the Z 3 charge and are the candidates of DM, (c) besides the SM Higgs (φ), only one new scalar boson (φ r ) is introduced, and (d) due to the mixture of φ r and φ, the DM annihilation is through the Higgs portal.
In the following, we briefly introduce the model and discuss the relevant interactions with the candidates of DM. To study the minimal extension of the SM that includes the staff of DM, besides the SM particles and their dictated gauge symmetry, we consider a new local SU(2) X gauge symmetry and add one quadruplet of SU(2) X to the model. The introduced quadruplet is not only responsible for the breaking of the new gauge symmetry, but also plays an important role on the communication between dark and visible sectors. Thus, the Lagrangian in SU(2) X × SU(2) L × U(1) Y is written as where L SM is the Lagrangian of the SM, µ with the representations of T a in the quadraplet, given by and T 3 = diag(3/2, 1/2, −1/2, −3/2), and the field strength tensor of SU(2) X is read by To break SU(2) X but preserve a discrete symmetry, the non-vanishing vacuum expectation value (VEV) and the associated fields fluctuated around the VEV are set to be When we regard the quadruplet as the fluctuations from the vacuum Φ 0 = (v 4 , 0, 0, v 4 )/2, Φ 4 can be parametrized by using the form In terms of scalar fields α a (x), the components of Φ 4 could be expressed as φ 1/2 = √ 3(−α 2 (x) + iα 1 (x))/2 √ 2, φ −1/2 = φ * 1/2 and ξ = 3/2α 3 (x), where we have taken the leading terms in the field expansions. Eq. (6) indeed is nothing but a local gauge transformation. Therefore, φ ±1/2 and ξ could be rotated away from the kinetic term of Φ 4 and the scalar potential; and they are the unphysical Nambu-Goldstone (NG) bosons of the local SU(2) X symmetry breaking. Consequently, we can just employΦ 4 for exploring the mass spectra of new particles.
With the breaking pattern in Eq. (5), one can find that an Z 3 symmetry U 3 ≡ e iT 3 4π/3 = diag(1, e i2π/3 , e −2iπ/3 , 1) is preserved by the ground state Φ 0 . Under the Z 3 transformation, the scalar fields of the quadruplet are transformed as That is, φ ±3/2 are Z 3 blind while φ ±1 carry the charges of Z 3 . To understand the transformations of gauge fields, one can use In terms of physical states of gauge fields, one can write with T ± = T 1 ± iT 2 and χ µ (χ µ ) = (X 1 µ ∓ iX 2 µ )/ √ 2 whereχ µ is regarded as the antiparticle of χ µ . Using the identity U 3 T ± U † 3 = exp(±i4π/3)T ± , the transformations of χ µ (χ µ ) and X 3 µ under Z 3 are given by We see that χ µ (χ µ ) carries the Z 3 charge and X 3 µ is the Z 3 blind. Due to the unbroken Z 3 , the particles with the charges of Z 3 are the candidates of DM. Since φ ±1/2 are the unphysical NG bosons, the DM candidates in our model are the vector gauge bosons χ µ andχ µ .
To study the spectra of SU(2) X , we have to determine the nonvanishing VEVs of H and Φ 4 . Using Eqs. (3) and (6), we get With minimal conditions ∂V (v, v 4 )/∂v = ∂V (v, v 4 )/∂v 4 = 0, we have respectively. In terms of the parameters in the scalar potential, the VEVs could be written As known that the masses of gauge bosons arise from the kinetic term of Φ 4 , accordingly the masses of χ µ (χ µ ) and X 3 µ can be directly found by where t(t + 1) and t 3 are the eigenvalues of T 2 = T a T a and T 3 , respectively. With t = t 3 = 3/2, the masses of gauge bosons are obtained as Although there are four scalar fields in the quadruplet, three of them become the longitudinal polarizations of gauge bosons (χ µ ,χ µ , X 3 µ ). Therefore, combining with the Higgs doublet in the SM, the remaining physical scalar bosons in the model are φ and φ r . In terms of the scalar potential in Eq. (3), the mass matrix for φ and φ r is expressed by Due to the λ ′ effect, the SM Higgs φ and φ r will mix and are not physical eigenstates. The mixing angle connected with the mass eigenstates could be parametrized by where h denotes the SM-like Higgs, H 0 is the second scalar boson and tan 2θ = 2λ ′ vv 4 /(m 2 φr − m 2 φ ). According to Eq. (16), the mass squares of physical scalars are found by We note that the mass of h could be m 1 or m 2 and the mass assignment depends on the chosen scheme of the parameters. To solve the problem of the gamma-ray excess, we will focus on the case of m h > m H 0 .
Next, we derive the couplings of φ and φ r and the interactions with new gauge bosons. We first discuss the gauge interactions of φ r . From Eq. (2), we see that the gauge interactions of the quadruplet only occur in the kinetic term of Φ 4 . UsingΦ 4 defined in Eq. (6) and the covariant derivative of Φ 4 , the gauge interactions are expressed as By adopting the expression of Eq. (9), one can easily find that the gauge interactions of Eq. (19) vanish. By using the result Eq. (20) can be straightforwardly written as where the masses of gauge bosons defined in Eq. (15) have been applied. We second discuss the couplings of φ r to the SM Higgs φ where the vertices could be obtained from the scalar potential of Eq. (3). Since the derivations are straightforward, we summarize the vertices of φ r and φ in Table I. We note that although the interactions in Eq. (22) and Table I are shown in terms of φ r and φ, the expressions with h and H 0 mass eigenstates could be easily obtained when Eq. (17) is applied.
The relevant free parameters in the model are µ 2 (Φ) , λ (Φ) , λ ′ and the gauge coupling g X . Using the masses of φ and φ r and the VEVs of H and Φ 4 , the six parameters could be replaced by (g X , v, v 4 , m φ , m φr , λ ′ ). When these values of parameters are fixed, the masses of h and H 0 and the mixing angle θ are determined. According to the results measured by ATLAS [62] and CMS [63], the Higgs mass now is known to be m h = 125 GeV. Therefore, it is better to use the physical masses m h,H 0 and mixing angle θ instead of m φ,φr and λ ′ .
Additionally, the VEV of v ≈ 246 GeV is determined from the Fermi constant G F and v 4 can be replaced by m χ . Hence, the involving unknown parameters in the model are g X , m χ , m H 0 and θ.
To constrain the free parameters, two observables have to be taken into account: one is the relic density [1] and another one is the DM-nucleon scattering cross section [2,3]. The number density of DM is dictated by the well-known Boltzmann equation, expressed by where H is the Hubble parameter, n = n χ + nχ, and n eq is the equilibrium density, defined by n χ,eq = nχ ,eq = g χ m 2 with g χ the internal degrees of freedom of DM, T the temperature and K i the modified Bessel function of the second kind [64]. For the vector DM, we take g χ = 3. The thermally averaged annihilation cross section is given by In the model, the DM annihilating into the SM particles is through the Higgs portal, where the associated Feynman diagrams are presented in Fig. 1. We note that in contrast to Ref. [32], the DM semi-annihilation processes such as χχ → χ(H 0 , h) are absent in our model. To study the DM abundance after the freeze-out, usually it is more convenient to where H 2 = 8π 3 Gg * T 4 /90 and M 2 P = 1/(8πG) have been used. If we set Y ∞ to be the present value after the freeze-out, the current relic density of DM is given by where H 0 and s 0 are the present Hubble constant and entropy density, respectively. For numerical calculations, we employ micrOMEGAs 4.1.5 [65] to solve the Boltzmann equation and get the present relic density of DM defined in Eq. (27).
Although the direct detection of DM via the DM-nucleon scattering has not been observed yet, the sensitivity of the current experiment could give a strict constraint on the free parameters. In the model, the sketch of a vector DM scattering off a nucleon is shown in Fig. 2. By neglecting the small momentum transfer, the scattering amplitude of the χ µ (χ µ )-nucleon is written as By assuming that the effective couplings of DM to the proton and neutron are the same, we parametrize the nucleon transition matrix element to be N|m qq q|N = f N /( where the range of f N is [1.1, 3.2] × 10 −3 [66,67]. As a result, the scattering cross section of the DM-nucleon is formulated by Before discussing the numerical analysis, we set up the possible schemes for the values of m χ and m H 0 . Since χχ → W + W − , ZZ are the dominant channels in the case of m H 0 > m χ > m W and in disfavor with the gamma-ray spectrum [28], we assume χ is lighter than W and Z. To explain the excess of the gamma-ray spectrum, it has been pointed out that the preferred channels via the Higgs portal are χχ → SS → bbbb with S being the possible scalar and χχ → bb [28,32,37], where the former produces the on-shell S and subsequently S decays into SM particles while the latter utilizes the resonant enhancement of m S ∼ 2m χ . As a result, we focus on the following two schemes: In scheme (a), as the main DM annihilating processes are from Figs. 1I and 1II and the produced H 0 pairs are on-shell, the results are insensitive to the mixing angle θ. To understand the constraint of the observed Ω DM h 2 , we present Ω χ h 2 as a function of g X in Fig. 3(a). From the results, we see that for matching the observed relic density of DM, the value of the gauge coupling g X should be around 0.23(0.21) for m χ = 70(60) GeV and m H 0 = 69(59) GeV. We note that for explaining the excess of the gamma-ray via the DM annihilation, we adopt m χ ≈ m H 0 in scheme (a). We will clarify this point later. In scheme Fig. 1III becomes dominant. Since h and H 0 both contribute to the DM annihilation, besides the gauge coupling g X and m H 0 , the results are also sensitive to the mixing angle θ. Since there are three free parameters involved in this scheme, in Fig. 3(b) we show the correlation between sin θ and m H 0 when g X = 1 is taken and the observed Ω DM h 2 is simultaneously satisfied. show the 90%-CL upper limits by XENON100 [2] and LUX [3] Collaborations on the plots.
From the results, we clearly see that to satisfy the DM direct detection experiments, we need sin θ < 0.1. For scheme (b), we present σ χN as a function of m H 0 in Fig. 5 with g X = 1 and m χ = 50 (40) GeV for the left (right) panel. In order to fit the data of Ω DM h 2 together, in the figure we have applied the results shown in Fig. 3(b). By the plots, we find that current DM direct detection experiments further limit the mass relation to be m H 0 ∼ 2m χ . After analyzing the constraints of the DM relic density and direct detection, we now study the gamma-ray which is originated from the DM annihilation. It is known that the flux of the gamma-ray from the DM annihilation is expressed by where dN γ /dE γ is the gamma-ray spectrum produced per annihilation, ψ is the observation angle between the line-of-sight and the galactic center, ρ(r) is density of DM, and the integration of the density squared is carried out over the line-of-sight. The general DM halo profile could be parametrized by where r s = 20 kpc is the scale radius, ρ ⊙ = 0.3 GeV/cm 3 is the local dark matter density at r ⊙ = 8.5 kpc and r is the distance from the center of the galaxy. Note that (α, β, γ) = (1, 3, 1) corresponds to the Navarro-Frenk-White (NFW) profile. In our numerical estimations, we set α = 1 and β = 3, but γ to be a free parameter. Since ρ(r) is proportional to r −γ , we see that the change of the parameter γ can only shift the entire gamma-ray spectrum but not the shape of gamma-ray flux. For executing the numerical calculations of Eq. (30), we implement our model to micrOMEGAs 4.1.5 [65] and use the program code to estimate the gamma-ray spectrum.
In the model, the processes to produce the gamma-ray by the DM annihilation are similar to those for the relic density, except that the gamma-ray is emitted in the final states. In scheme (a), we present the flux of the gamma-ray as a function of the photon energy E γ in we see that when the mass difference m χ − m H 0 becomes larger, due to the boosted H 0 , the flux after the peak of the excess tends to be enhanced and disfavors with the data.
Hence, we only focus on m χ ≈ m H 0 . In scheme (b), χχ → bb is dominant. The result of the gamma-ray flux as a function of E γ is given in Fig. 6(b), where g X = 1 is taken, the solid and dashed lines stand for (m χ , m H 0 ) = (50, 101) and (40, 101) GeV, respectively, and the value of sin θ ≃ 0.02 is read from Fig. 3(b) for both cases when the observed Ω DM h 2 is satisfied. In addition, the value of m H 0 has been chosen to follow the constraint of the direct detection, i.e. m H 0 ∼ 2m χ . For the case of m H 0 2m χ , due to the produced H 0 being an on-shell particle, the annihilation cross section becomes too large to explain the gamma-ray excess. Hence, we adopt m H 0 2m χ . Finally, we make some comparisons with the study in Ref. [32] where the stable DM candidates are dictated by the custodial symmetry [59]. Since the trilinear couplings of gauge bosons exist in the model given by Ref. [59], besides the annihilation processes which we only have in our model, there are also semi-annihilation processes in Refs. [32,59]. With the taken values of parameters and the best-fit approach, the authors of Ref. [32] have found that the gamma-ray excess is dominated by the semi-annihilation. As a result, DM with its mass around 39 − 76 GeV could fit the measured gamma-ray spectrum of the Galactic Center. However, the resulted σv rel is a factor of 2-3 larger than that of the observed Ω DM h 2 . In our approach, with the selected values of m χ , e.g. m χ = (70, 60) GeV in scheme (a) and m χ = (50, 40) GeV in scheme (b), we first constrain the free parameters by using the observed Ω DM h and the upper limit of the DM direct detection. With the allowed values of parameters, we subsequently estimate the gamma-ray spectrum from the DM annihilation.
Although the best-fit approach is not adopted in the analysis, our results from the on-shell H 0 production in scheme (a) and m H 0 ∼ 2m χ in scheme (b) are morphologically consistent with the gamma-ray spectrum of the Galactic Center.
In summary, to interpret the excess of the gamma-ray through the DM annihilation, we have studied the DM model in the framework of SU(2) X gauge symmetry. To break the gauge symmetry, we have used one quadruplet of SU(2) X . As a result, the remnant Z 3 symmetry of SU(2) X leads to the stable DMs, which are the gauge bosons of SU(2) X . Due to the mixture of the quadruplet and SM Higgs doublet in the scalar potential, the DM annihilation to SM particles is through the Higgs portal. When the observed relic density of DM and the limit of the DM direct detection are both satisfied, we find that m χ < m W could give a correct pattern for the gamma-ray spectrum. For more specific numerical studies, we classify the values of parameters to be scheme (a) with (m χ , m H 0 ) = (70, 69) and (60,59) GeV and scheme (b) with (m χ , m H 0 ) = (50, 101) and (40, 81) GeV. We show that for matching the gamma-ray excess, in scheme (a) it is better to take m χ ≈ m H 0 . If m χ −m H 0 is increasing, due to the boosted H 0 , the gamma-ray flux at the photon energy over the peak of the gamma-ray spectrum is enhanced and the resulted flux tends to be away from the data.
In scheme (b), for avoiding the constraint from the DM direct detection and the production of the on-shell H 0 which causes too large cross section, the condition of m H 0 2m χ is adopted. Based on our current analysis, we see that the results of scheme (a) fit the data well.