Vector Gauge Boson Dark Matter for the SU(N) Gauge Group Model

The existence of dark matter is explained by a new neutral vector boson, C-boson, of mass (900 GeV), predicted by the Wu mechanisms for mass generation of gauge field. According to the Standard Model (SM) W, Z-bosons normally get their masses through coupling with the SM Higgs particle of mass 125 GeV. We compute the self-annihilation cross section of the vector gauge boson C-dark matter and calculate its relic abundance. We also study the constraints suggested by dark-matter direct-search experiments. The problem on the stability of C-particle is left as an open question for future research.

a model posed by dark-matter direct-search experiments have been studied [41] and possible signals at the Large Hadron Collider (LHC) have been considered [44,45].
The goal of the present paper is to investigate the possibility that a neutral vector C-boson of mass 900 GeV (as predicted by the Wu mechanisms for mass generation of gauge fields: [46][47][48][49][50][51][52]), proposed recently by the author [53], explains the existence of dark matter. According to the SM W , Z-bosons normally acquire their masses through their coupling with the SM Higgs boson of mass 125 GeV [54][55][56][57][58]. Here we compute the self-annihilation cross section of the vector gauge C-boson-dark matter, calculate C-boson's relic abundance and study the modeling constraints generated by dark-matter direct-search experiments.

The Lagrangian of the Model
Let us suppose that the gauge symmetry of the theory is SU(N ) × U(2) group, which is written specifically as follows [53]: where SU(N ) is the special unitary group of N -dimensions, ψ(x) is a N -component vector in the fundamental representative space of SU(N ) group, and T i (i = 1, 2, . . . , N 2 − 1) denotes the representative matrices of the generators of SU(N ) group. The latter are Hermit and traceless. They satisfy the condition: where f ij k are structure constants of the SU(N ) group, and K is a constant independent of the indices i and j but dependent on the representation of the group. The representative matrix of a general element of the SU(N ) group is expressed as: with α i being the real group parameters. In global gauge transformations, all α i are independent of space-time coordinates, while in local gauge transformations α i are functions of space-time coordinates. U is a unitary N × N matrix. In order to introduce the mass term of gauge fields without violating local gauge symmetry at energy scale close to 2 TeV, two kinds of gauge fields are required: a μ and b μ [46,53].
In this version of the Wu gauge model, the gauge fields a i μ and b i μ are introduced. Gauge field a i μ is introduced to ensure the local gauge invariance of the theory. The generation of gauge field b i μ is a purely quantum phenomenon: b i μ is generated through non-smoothness of the scalar phase of the fundamental spinor fields [53,59].
From the viewpoint of the gauge field b i μ generation described here, the gauge principle is an "automatic" consequence of the non-smoothness of the field trajectory in the Feynman path integral [53,59]. a μ (x) and b μ (x) are vectors in the canonical representative space of SU(N ) group. They can be expressed as linear combinations of generators, as follows: where a i μ (x) and b i μ (x) are component fields of the gauge fields a μ (x) and b μ (x), respectively.
Corresponding to these two kinds of gauge fields, there are two kinds of gauge covariant derivatives: The strengths of gauge fields a μ (x) and b μ (x) are defined as respectively. Similarly, a μ (x) and b μ (x) can also be expressed as linear combinations of generators: Using relations (2) and (8), (9), we obtain The Lagrangian density of the model is Tr a μ + cb μ (a μ + cb μ ) [46] where c is a constant. The space-time metric is selected as η μν = diag(−1, 1, 1, 1), (μ, ν = 0, 1, 2, 3). According to relation (2), Lagrangian density L can be rewritten as: This Lagrangian has strict local gauge symmetry [46,53]. In Eq. (1), the gauge group U(2) = SU L (2) × U(1) Y is the known SM of electroweak (EW) interactions [54][55][56][57]. The generators of SU L (2) correspond to the three components of weak isospin T a (a = 1, 2, 3). The U(1) Y generator corresponds to the weak hypercharge Y . These are related to the electric charge by Q = T 3 2 + Y . The SU L (2) × U(1) Y invariant Lagrangian is given as follows: with field strength tensors: for the three non-Abelian fields of SU(2) L and the single Abelian gauge field associated with U(1) Y , respectively. The covariant derivative is: with g 1 , g 1 being the SU(2) L , and U(1) Y the coupling strength, respectively. The Lagrangian (16) is invariant under the infinitesimal local gauge transformations for SU(2) L and U(1) Y independently. Being in the adjoined representation, the SU(2) L massless gauge fields form a weak isospin triplet, with the charged fields being defined by The neutral component of A 3 μ mixes with the Abelian gauge field B μ to form the physical states: where tan θ w = g 1 g 1 is the weak mixing angle. Based on the gauge group SU(N ) × U(2), the final Lagrangian of the model is given as follows [53]: where c is a constant.

The Masses of Gauge Fields
Two obvious characteristics of the Wu Lagrangian equation (15) is that the mass term of the gauge fields is introduced into the Lagrangian, and that this term does not affect the symmetry of the Lagrangian. It has been proved that this Lagrangian has strict local gauge symmetry [46]. Since both vector fields a μ and b μ are standard gauge fields, this model is a gauge field model which describes gauge interactions between gauge fields and matter fields [46]. The mass term of gauge fields can be written as follows [46]: where M is the mass matrix Physical particles generated from gauge interactions are eigenvectors of mass matrix, and the corresponding masses of these particles are eigenvalues of mass matrix. The mass matrix M has two eigenvalues: The corresponding eigenvectors are: cos θ wu sin θ wu − sin θ wu cos θ wu (27) where We define C μ and F μ are eigenstates of mass matrix: they describe the particles generated from gauge interactions. The inverse transformations of (29) are: Taking Eqs. (29) and (30) into account, the Wu Lagrangian density L given by (15) changes into: In the above relations, we have used the following simplified notations: From Eq. (32) it is deduced that the mass of field C μ is μ and the mass of gauge field F μ is zero. That is: Transformations (29) and (30) are pure algebraic operations which do not affect the gauge symmetry of the Lagrangian [46]. They can, therefore, be regarded as redefinitions of gauge fields. The local gauge symmetry of the Lagrangian is still strictly preserved after field transformations. In other words, the symmetry of the Lagrangian before transformations is absolutely the same with the symmetry of the Lagrangian after transformations. We do not introduce any kind of symmetry breaking at energy scales close to 2 TeV [53].
Fields C μ and F μ are linear combinations of gauge fields a μ and b μ . The forms of local gauge transformations of fields C μ and F μ are, therefore, determined by the forms of local gauge transformations of gauge fields a μ and b μ . Since C μ and F μ consist of gauge fields a μ and b μ and transmit gauge interactions between matter fields, for the sake of simplicity we also call them gauge fields, just as W and Z are called gauge fields in the electroweak model [54][55][56][57]. This gauge field theory, therefore, predicts the existence of two different kinds of force transmitting vector fields: a massive, and a massless one.
The most general Lagrangian consistent with SU(2) × U(1) gauge invariance, Lorentz invariance, and renormalizability is [60]: where are the generators of the (φ + , φ 0 ), λ > 0, and For μ 2 < 0, there is a tree-approximation vacuum expectation value at the stationary point of the Lagrangian We can always perform a SU(2) × U(1) gauge transformation to a unitary gauge, in which φ + = 0 and φ 0 is Hermitian, with positive vacuum expectation value. In unitarily gauge, the vacuum expectation values of the components of φ are: The scalar Lagrangian (37) then yields a vector boson mass term: The masses are given as follows: The gauge fields and masses predicted by this model are summarized in Table 1.

Renormalization of the Model
In this paper we use two mechanisms that can make gauge field to gain nonzero mass. One is the Wu mechanism [46][47][48][49][50][51][52][53], by which the mass term of the gauge field is introduced by using another set of gauge fields. In this mechanism, the mass term of the gauge field does not affect the symmetry of the Lagrangian. We can imagine the new interaction picture: when matter fields take part in gauge interactions, they emit or absorb one kind of gauge field which is not eigenstate of mass matrix. Perhaps this is the gauge field consisted with the existence of dark matter in our Universe. This gauge field would appear in two states, a massless and a massive one, which correspond to two kinds of vector fields. The second mechanism of mass generation is the Higgs mechanism [54][55][56][57][58], which can make gauge field-W , Z to gain nonzero mass and to guarantee renormalizability by means of the interactions of the Higgs boson (h) with gauge bosons W, Z, C [53]: where are the dimensionless and dimension of mass coupling constants. For instance, the C-boson readily derives its mass through the Wu mechanism; yet renormalizability is ensured via the Higgs mechanism [53].
However, as we have stated above, the Wu gauge field theory has maximum local SU(N ) gauge symmetry [46,53]. When we quantise the Wu gauge field theory in the path integral formulation, we have to select gauge conditions first [46,53]. To fix the degree of freedom of the gauge transformation, we must select two gauge conditions simultaneously: one for the massive gauge field C μ , and another for the massless gauge field F μ . For instance, if we select temporal gauge condition for massless gauge field F μ , there still exists a remainder gauge transformation degree of freedom, because the temporal gauge condition is unchanged under the following local gauge transformation: where To render this remainder gauge transformation degree of freedom completely fixed, we have to select another gauge condition for gauge field C μ , for instance: If we select two gauge conditions simultaneously, when we quantise the theory in path integral formulation, there will be two gauge fixing terms in the effective Lagrangian. The effective Lagrangian can then be written as: where If we select then the propagator for massive gauge field C μ is: If we let k approach infinity, then In this case, according to the power-counting law, the Wu gauge field theory suggested in this paper is a renormalizable theory [46,53].
Neutralino annihilations to fermions are chirality-suppressed by a factor of m 2 f /m 2 χ , and thus do not produce electrons-positrons e + e − pairs directly [37]. By contrast, C-dark matter, being a boson, is not similarly suppressed and can annihilate directly to leptons e + e − , μ + μ − and τ + τ − pairs. Each of these particles yield a large number of high energy electrons and positrons.

Relic Density of Vector Gauge Boson C-Dark Matter
Let us now review the standard calculation of the relic abundance of a particle species, denoted C-particle, which was at thermal equilibrium in the early universe and decoupled when it was non relativistic [61,62]. The evolution of its number density n in an expanding universe is governed by the Boltzmann equation: where σ u rel is the total annihilation cross section multiplied by velocity. Brackets denote thermal average, H = (8πρ/3M pl ) 1/2 is Hubble constant, M pl = 10 19 GeV is the Planck mass, and n eq is the number density at thermal equilibrium. For C-particle, in the nonrelativistic limit, and in the Maxwell Boltzmann approximation, it is where m c is the C-particle mass and T is temperature. Next we introduce the variables where s is the entropy density s = 2π 2 g * T 3 /45 and g * counts the number of relativistic degrees of freedom. Using the conservation of entropy per co-moving volume (sa 3 = constant), it follows thatṅ + 3H n = sẎ . Equation (56), therefore, reads: If we further introduce the variable x = m/T , Eq. (59) can be expressed as For heavy states, we can approximate σ u rel with the non-relativistic expansion in powers of u 2 rel : which leads to our final version of Eq. (60) in terms of the variable Δ = Y − Y eq : where prime denotes d/dx and Following [61], we introduce the quantity x F ≡ m/T F , where T F is the freeze-out temperature of the relic particle. We notice that Eq. (62) can be solved analytically in the two extreme regions x x F and x x F : These regions correspond to conditions long before freeze-out and long after freeze-out, respectively. Integrating the last equation between x F and ∞ and using Δ x F Δ ∞ , we can derive the value of Δ ∞ and arrive at The present density of a generic relic, C, is simply given by ρ c = m c n c = m c s 0 Y ∞ , where s 0 = 2889.2 cm −3 is the present entropy density. The relic density can finally be expressed in terms of the critical density (Ω c = ρ c /ρ crit ): where a and b are expressed in GeV −2 , and g * is evaluated at the freeze-out temperature. It is conventional to express the relic density in terms of the Hubble parameter: To estimate the relic density, one is thus left with the calculation of the annihilation cross sections (in all of the possible channels) and the extraction of the parameters a and b, which depend on the particle mass. The freeze-out temperature x F can be estimated through the iterative solution of the equation where c is a constant of order one, determined by matching the late-time and early-time solutions.
It is sometimes useful to perform an order-of-magnitude estimate using an approximate version of Eq. (67) [63]. For vector gauge boson C mass of 900 GeV, predicted by the Wu mechanisms for mass generation of gauge field, the relic density should be where σ u rel is calculated by Eq. (55). Analysis of the three-year Wilkinson Microwave Anisotropy Probe (WMAP) data tells us that the density of dark matter is Ω dm h 2 = 0.102 ± 0.009, where Ω dm is ρ dm /ρ crit , ρ crit is the density corresponding to a flat universe [64] and h is the Hubble constant in units of 100 km s −1 Mpc −1 [65]. A cold dark matter candidate produced at the Large Hadron Collider (LHC) should, therefore, have this annihilation cross section.
This quantity leads us to the second method of measuring the coupling of dark matter from Standard Model particles: through the search for the annihilation or decay products of dark matter coming from high-density regions of the Universe, such as the centre of galaxies [66]. Since WMAP results provide good information about σ u rel , uncertainties in this approach stem from our sketchy knowledge of the exact density of dark matter in the centre of galaxies and the difficulty in separating the signal from dark-matter annihilation from possible background signals.
In the SM there already exists an example of a particle which is accidentally stable: the proton [67]. There is, therefore, no reason why the C-particle could not also be stable [67][68][69].
It thus follows that no dimension five operator between the dark matter vector field candidate and SM fields is allowed by the SM (and hidden sector) gauge symmetries [67][68][69][70]. If, on the contrary, they allow dimension 6 operators, it turns out that the lifetime of the vector dark matter candidate is of the order 10 26 sec if 10 14 GeV [68], which is close to the GUT scale. In other words, an accidentally stable C-dark matter candidate which can be destabilized by a dimension six GUT scale induced interaction results in a flux of cosmic rays of the order of the observed order; therefore, potentially to a rich phenomenology [67].
If the C-particle is accidentally stable, and since it also interacts weakly with baryonic matter, it can be a good Weakly Interacting Massive Particles (WIMP) candidate. We do not discuss the stability of C-particle further in this article; we leave this intriguing topic as an open question for future research.
In any case, however, the C-particle of mass around 900 GeV predicted by the C-dark matter model, also provides the correct abundance of dark matter in the universe. This encouraging theoretical suggestion is testable through LHC studies.

Direct Search for C-Dark Matter
Following, J. Hisano et al. [71], the elastic cross section is calculated in terms of effective coupling constants, which are given by coefficients of effective interactions of vector dark matter with light quarks (q = u, d, s) and gluons. Since the scattering process is non relativistic, all terms that depend on the velocities of C-dark matter particles and nuclei are subdominant in the velocity expansion. In this study, therefore, we neglect the operators suppressed by the velocities of the C-dark matter particles or nuclei. Since ∂ μ C μ = 0, physical degrees of freedom of the C-dark matter particle (assumed to be a real vector field) are then restricted to their spatial components. As a consequence, in the expansion of a strong coupling constant and in the non relativistic limit of the scattering process, the leading interaction of the vector field with quarks and gluons is given as with eff q = f m q m q C μ C μq q + f i/ D q C μ C μq i/ Dq + d q m c ε μνρσ C μ i∂ ν C ρq γ σ γ 5 q eff where m q is the mass of a quark, m c the mass of a C-particle and ε μνρσ is the totally antisymmetric tensor defined as ε 0123 = +1. f q , d q and g q are the scalars-, axial vector-, and twist 2-type couplings of C-dark matter particle with quarks, respectively. Here, the covariant derivative is defined as D μ = ∂ μ + igA a μ T a , with g s , T a , and A a μ being the SU(3) C coupling constant, generator, and gluon field, respectively. O q μν and O g μν are the twist-2 operators (traceless parts of the energy-momentum tensor) for quarks and gluons, respectively. These are given by: G a μν is the field strength tensor of a gluon.
To derive the effective Lagrangian, we use the equations of motion for vector field, i.e., (∂ 2 − m 2 c )C μ = 0. The third term in the right-hand side of Eq. (71) contributes to the spindependent interaction in the scattering of dark matter with nuclei, whereas the other terms in the effective Lagrangian contribute to the spin-independent interaction. The scattering amplitude is given by the matrix element of the effective interaction between initial and final states. In this evaluation, we use the equation of motion for light quarks.
To prove the validity of the application we evaluate the operator using on-shell hadron states [72]. In this case, denoting |N (N = p, n) as the on-shell nucleon state, we can evaluate the second term in Eq. (71) as Now, let us first examine the spin-independent scattering. As in the scattering process momentum transfer is negligible; the matrix elements between initial and final nucleon states with the mass m N (N = p, n) are obtained by: In the matrix elements of twist-2 operators, q(2),q (2), and G(2) are the second moments of the Parton Distribution Functions (PDFs) of a quark q(x), antiquarkq(x) and gluon g(x), respectively, Then, after the trivial calculation of matrix elements for vector dark matter states, we obtain a spin-independent effective coupling of vector dark matter with nucleons, f N as where f q = f m q + f i/ D q and a s = g 2 s /4π . Note that the effective scalar coupling of dark matter with gluons, f G , gives the leading contribution to the cross section even if it is suppressed by a one-loop factor compared with those of light quarks.
Taking into account twist-2 operators and gluonic contributions calculated recently [71], we find that: To obtain f n /m n , the numerical coefficients (i.e. 0.052, 0.222, 0.925) in the above equation are replaced with (0.061, 0.330, 0.922) [42]. The spin-independent elastic cross section for C-dark matter scattering off a nucleus of Z protons and A − Z neutrons normalized to one nucleon is then given by: where m N and m c are the masses of the nucleon and C-dark matter, respectively.
For the scalar (η) scattering off a nucleus by the Higgs exchange (h) [43,75], the terms f p and f n in cross section (70) Assuming an effective m η = 125 GeV and using Z = 54 and A − Z = 77 for 131 Xe, we find that σ 0 ≈ 3.2 × 10 −10 pb. This value is far below the current upper bound, calculated by the 2011 XENON100 experiment [76].

Conclusion
We suggest that a new, neutral C-boson of mass 900 GeV, predicted by the Wu mechanisms for mass generation of gauge field, can explain the dark matter in our Universe.
In the suggested model, the Standard Model W and Z-bosons acquire their masses through coupling with the Standard Model Higgs boson of mass 125 GeV.
C-dark matter readily derives its mass by the Wu mechanism; yet renormalizability occurs only via the Higgs mechanism.
We also compute the thermally averaged cross section for C-dark matter self-annihilation into SM particles, calculate the relic abundance of C-dark matter and study the constraints suggested by dark-matter direct-search experiments. In view of these constrains, scalar (η) is not a suitable dark-matter candidate.
The problem on the stability of C-particle is left as an open question for future research.