Spin portal to dark matter
SSpin portal to dark matter.
H. Hern´andez-Arellano (1) , M. Napsuciale (1) , S. Rodr´ıguez (2) Departamento de F´ısica, Universidad de Guanajuato, Lomas del Bosque 103,Fraccionamiento Lomas del Campestre, 37150, Le´on, Guanajuato, M´exico, and Facultad de Ciencias F´ısico-Matem´aticas, Universidad Aut´onoma de Coahuila,Edificio A, Unidad Camporredondo, 25000, Saltillo Coahuila M´exico.
In this work we study the possibility that dark matter fields transform in the (1 , ⊕ (0 , Z fields to the higher multipoles of dark matter, yielding a spin portal todark matter. For dark matter ( D ) mass below a half of the Z mass, the decays Z → ¯ DD and H → ¯ DD are kinematically allowed and contribute to the invisible widths of the Z and H . Wecalculate these decays and use experimental results on these invisible widths to constrain the valuesof the low energy constants finding in general that effects of the spin portal can be more importantthat those of the Higgs portal. We calculate the dark matter relic density in our formalism, use theconstraints on the low energy constants from the Z and H invisible widths and compare our resultswith the measured relic density, finding that dark matter with a (1 , ⊕ (0 ,
1) space-time structuremust have a mass
M > GeV . I. INTRODUCTION.
The elucidation of the nature of dark matter is one of the most important problems in high energy physics [1].Although dark matter gravitational effects were noticed during the first half of the last century [2] and from recentprecise measurements of the cosmic background radiation we know that it accounts for around 26% [3] of the matter-energy content of the universe, an identification of dark matter properties is still lacking and a lot of experimentaleffort is presently being pursued in order to directly or indirectly detect dark matter particles, based mainly in theWIMP paradigm [4] . The latter is based on the fact that the proper description of the measured dark matter relicdensity, Ω expDM h = 0 . ± . , ⊕ (0 , ), ( , ) and (0 ,
0) rep-resentations of the Homogeneous Lorentz Group (HLG) respectively and it is natural that effective theories so farformulated for dark matter consider dark matter transforming in these representations.Recently, the quantum field theory of spin one massive particles transforming in the (1 , ⊕ (0 ,
1) representation ofthe HLG (spin-one matter fields), was studied in detail in [21], where the field is described by a six-component spinor,similar to the four-component Dirac spinor describing spin 1 / , ⊕ (0 ,
1) representation space, whichwas previously worked out in [23]. This basis naturally contains a chirallity operator, χ , and spin-one matter fields canbe decomposed into chiral components transforming in the (1 ,
0) (right) and (0 ,
1) (left) representations. However,the kinetic term in the free Lagrangian is not invariant under independent chiral transformations, therefore spin-one matter fields cannot have linearly realized chiral gauge interactions, hence they cannot have weak interactions.Nonetheless, it is possible to have vector-like interactions like U (1) Y or SU (3) c standard model interactions. Inaddition, spin-one matter fields can have naively renormalizable self-interactions classified also in [21].In this work we study the possibility of a (1 , ⊕ (0 ,
1) space-time structure for dark matter fields. Clearly, dark a r X i v : . [ h e p - ph ] J a n matter with standard model charges would give sizable contributions to precision measurements of standard modelobservables, thus we assume in this work that dark matter fields transform as singlets of the standard model gaugegroup.The paper is organized as follows. In the next section we review the elements of the quantum field theory of spin onematter fields needed for the calculation of the required cross sections. In Section III we discuss the leading terms inthe effective field theory. In section IV we study the mass region M < M Z /
2, calculate the decay width for Z → ¯ DD and H → ¯ DD and find the constraints on the low energy constants from the Z and Higgs invisible widths. Section Vcontains an analysis of the dark matter relic density in this formalism, when these constraints are taken into account.Finally, we give our conclusions and perspectives in section VI and close with an appendix with the required tracecalculations for operators in the (1 , ⊕ (0 ,
1) representation space.
II. QUANTUM FIELD THEORY FOR SPIN-ONE MATTER FIELDS: BRIEF REVIEW
In the standard model, matter is described by Dirac fermions which transform in the (1 / , ⊕ (0 , /
2) representationof the HLG. Spin-one matter fields are the generalization of Dirac construction to j = 1, i.e. fields transforming inthe (1 , ⊕ (0 , ψ ( x ) and the corresponding quantum field theory wasstudied in [21], taking advantage of the general construction of a covariant basis for ( j, ⊕ (0 , j ) representation spaceintroduced in [23]. For j = 1 the covariant basis is given by the set of 6 × { , χ, S µν , χS µν , M µν , C µναβ } where χ is the chirality operator, S µν , stands for a symmetric traceless ( S µµ = 0) matrix tensor transforming in the(1 ,
1) representation of the HLG, M µν are the HLG generators and C µναβ is a matrix tensor transforming in the(2 , ⊕ (0 ,
2) representation of the HLG.The spin-one matter field is written as ψ ( x ) = (cid:88) λ (cid:90) d p (cid:112) (2 π ) E [ a λ ( p ) U ( p, λ ) e − ip.x + b † λ ( p ) V ( p, λ ) e ip.x ] (1)where U ( p, λ ) ( V ( p, λ )) stands for the particle (antiparticle) solution with polarization λ respectively. In contrast withthe Dirac case, spin-one matter particle and antiparticle have the same parity. These solutions satisfy (cid:88) λ U ( p, λ ) ¯ U ( p, λ ) = S ( p ) + M M , (cid:88) λ V ( p, λ ) ¯ V ( p, λ ) = S ( p ) + M M . (2)where S ( p ) ≡ S µν p µ p ν .The spin-one matter fields free Lagrangian is given by L = 12 ∂ µ ¯ ψ ( x )( g µν + S µν ) ∂ ν ψ ( x ) − m ¯ ψ ( x ) ψ ( x ) . (3)where ¯ ψ ( x ) ≡ ( ψ ( x )) † S . The S µν operators satisfy the following anti-commutation relations { S µν , S αβ } = 43 (cid:18) g µα g νβ + g να g µβ − g µν g αβ (cid:19) − (cid:0) C µανβ + C µβνα (cid:1) . (4)Further algebraic relations of the operators in the covariant basis and the connection with the traces needed for thecalculations in this work are deferred to an appendix. The propagator for spin-one matter particles is given by iπ ( p ) = i S ( p ) − p + 2 M M ( p − M + iε ) . (5)An important outcome of this formalism is that the free field Lagrangian can be decomposed in terms of the chiralcomponents as L = 12 ∂ µ ψ R ∂ µ ψ L + 12 ∂ µ ψ R S µν ∂ ν ψ R − m ψ R ψ L + R ↔ L, (6)where ψ R = 12 (1 + χ ) ψ, ψ L = 12 (1 − χ ) ψ. (7)The right (left) field ψ R ( ψ L ) transforms in (1 ,
0) ((0 , U (1) Y or SU (3) C gauge interactionsbut not SU (2) L interactions, or simply be standard model singlets. This result motivate us to explore the possibilitythat dark matter be described by spin-one matter fields and we start with the simplest and most likely possibility:spin-one dark matter fields transforming as singlets under the standard model gauge group. III. DARK MATTER AS SPIN-ONE MATTER FIELDS: EFFECTIVE THEORY.
If we consider dark matter as spin-one matter fields (spin-one dark matter fields in the following) transforming assinglets under the standard model group, dark matter does not feel the standard model charges. On the other side,if we have more than one dark matter field, dark matter can have gauge interactions with its own (vector-like) darkgauge group. In the following we will assume a simple U (1) D structure for the dark gauge group, but the generalizationof our results to SU ( N ) D is straightforward. We remark that the only effect of this dark gauge structure in this workis to provide to dark matter particles with dark charges distinguishing particles from anti-particles and preventingthe direct decay of a dark matter particle into standard model ones.At high energies, the standard model and dark sectors couple in a yet unknown way but the low energy effects ofsuch theory can be classified in an expansion in derivatives of the fields. Each term in this expansion has a low energyconstant and the importance at low energies of each term depends on the dimension of the corresponding operator,in such a way that the most important effects are given by the lowest dimension operators.The Lagrangian must be a complete scalar operator and if dark matter fields are standard model singlets (andstandard model fields are singlets of the dark gauge group) the only possibility to have a scalar interacting Lagrangianis that it be composed of products of singlet operators on both sides. The construction of the lowest dimensioninteracting operators in this case, requires to classify the singlet operators in both sectors. The most general form ofthis interaction is L int = (cid:88) n n − O SM O DM (8)where Λ is an energy scale compensating the dimension n of the product of the standard model singlet operators O SM constructed with standard model fields and O DM made of spin-one dark matter fields.It is easy to convince one-self that the lowest dimension standard model singlet operators are ˜ φφ and B µν , where φ stands for the standard model Higgs doublet and B µν denotes the U (1) Y stress tensor. Indeed, ˜ φφ is simply thesinglet of the ⊗ product of SU (2) L ( and also a singlet under SU (3) and U (1) Y ), while in general under SU ( N )gauge transformations U ( x ), the stress (matrix) tensor operator transforms as F µν → U ( x ) F µν U − ( x ) , (9)being strictly invariant only in the U (1) case, thus, in the standard model, the U (1) Y stress tensor B µν is a singletunder the standard model gauge group. Singlet operators made of fermion fields or other combinations can also beconstructed but they are higher dimension.For spin-one matter fields with a dark gauge group U (1) D , the lowest dimension operators transforming as standardmodel and dark gauge group singlets are of the form ¯ ψOψ where O is one of the 36 matrix operators in the covariantbasis { , χ, S µν , χS µν , M µν , C µναβ } . These operators are dimension two and using the symmetry properties of S µν and C µναβ it is easy to show that the leading interacting terms in the effective theory are given by L int = ¯ ψ ( g s + ig p χ ) ψ ˜ φφ + g t ¯ ψM µν ψB µν , (10)with low energy constants g s , g p and g t . There is an effective Higgs portal to dark matter interactions with standardmodel particles given by the first two terms, the second one violating parity. The third term is an effective interactioncoupling dark matter to the photon and the Z boson. Notice however that this interaction does not involve the weakcharges (operators are standard model singlets), but proceeds through the coupling of the photon and Z fields tothe higher multipoles (magnetic dipole moment and electric quadrupole moment) of the dark matter, thus we nameit spin portal to dark matter. In addition to the interactions in Eq.(10) we have the dimension four self-interactionsdescribed in [21] which are not relevant for the purposes of this paper.In unitary gauge for the standard model fields, after spontaneous symmetry breaking and diagonalizing the gaugeboson sector we get the following Lagrangian L int = 12 ¯ ψ ( g s + ig p χ ) ψ ( H + v ) + g t cos θ W ¯ ψM µν ψF µν − g t sin θ W ¯ ψM µν ψZ µν , (11) = i ( g s + ig p χ ) = i ( g s + ig p χ ) vk, µγ = 2 g t cos θ W M µν k ν k, µZ = − g t sin θ W M µν k ν FIG. 1: Feynman rules from the leading terms in the effective theory. where H stands for the Higgs field, v denotes the Higgs vacuum expectation value and F µν , Z µν are the electromagneticand Z stress tensors respectively. The Feynman rules arising from the Lagrangian in Eq. (11) are given in Fig. 1. IV. DARK MATTER WITH A MASS
M < M Z / : Z → ¯ DD AND H → ¯ DD DECAYS.
The Lagrangian in Eq.(11) induces transitions between the standard model and dark sectors. Annihilation ofdark matter into standard model particles such as ¯ DD → ¯ f f, γγ, W + W − , Z Z , HH, Z γ, Hγ, Z H which couldbe important in the description of dark matter relic density are induced by these interactions under appropriatekinematical conditions. Also, for dark matter mass below half the Z mass ( M < M Z / Z → ¯ DD and H → ¯ DD are kinematically permitted and contribute to the invisible Z and H widths respectively. In this work weconsider this mass region and work out the predictions of the formalism for the dark matter relic density.A straightforward calculation yields the following invariant amplitude for the Z ( k, (cid:15) ) → D ( p ) ¯ D ( p ) decay − i M = 2 g t S W ¯ U ( p , λ ) M µν k ν V ( p , λ )) (cid:15) µ ( k ) , (12)where S W = sin θ W . The calculation of the average squared amplitude can be reduced to a trace of products ofoperators in the covariant basis of (1 , ⊕ (0 ,
1) representation space, in a procedure similar to conventional calculationswith Dirac fermions. We obtain | ¯ M| = 43 g t S W T r (cid:20) S ( p ) + M M M µν S ( p ) + M M M αβ (cid:21) k ν k β ( − g µα + k µ k α M Z ) . (13)The trace-ology of matrices in (1 , ⊕ (0 ,
1) space is deferred to an appendix. Using results in the appendix we obtainthe corresponding decay width asΓ( Z → ¯ DD ) = g t S W πM ( M Z − M ) / ( M Z + 2 M ) . (14)The invisible width Γ invexp ( Z ) = 499 . ± . M eV reported by the Particle Data Group [5], includes the decay to ν ¯ ν .We use the SM prediction for the latterΓ SM ( Z → ¯ νν ) ≡ (cid:88) i Γ SM ( Z → ¯ ν i ν i ) = (cid:88) i,α U iα M Z πv (cid:113) M Z − m ν i = M Z πv = √ G F M Z π . (15)where in the last step we neglected the neutrino masses and used the unitarity of the PMNS matrix elements. TheParticle Data Group report the value M Z = 91 . ± . GeV while the µ − Lan collaboration reported the mostprecise measurement of the Fermi constant as G F = 1 . × − GeV − [24] . Using these values we getΓ SM ( Z → ¯ νν ) = 497 . ± . M eV. (16)Subtracting this quantity from the PDG reported value for the invisible width we get the constraint Γ( Z → ¯ DD ) < Γ invZ ≡ Γ invexp ( Z ) − Γ SM ( Z → ¯ νν ) = 1 . ± . M eV . This width depends on the coupling g t and the dark matter mass M , hence the invisible Z width constrain these parameters to the region shown in Fig. 2.Similar calculations for the H → ¯ DD decay yield the following decay widthΓ( H → ¯ DD ) = v πM H M (cid:113) M H − M (cid:2) g s (cid:0) M H (cid:0) M H − M (cid:1) + 6 M (cid:1) + g p M H (cid:0) M H − M (cid:1)(cid:3) , (17)The H → ¯ DD width depends on the unknown g s , g p couplings and on the dark matter mass. This channel contributesto the invisible Higgs width which has been recently reported in [5, 25] as Γ invH = 1 . ± . M eV . In this case, thecontribution of the ν ¯ ν channel is negligible. The constraints on g s , g p arising from the Γ( H → ¯ DD ) < Γ invH conditionare also shown in Fig. 2. The solid lines correspond to the central values and the shadow regions to the one sigmaregions. We conclude from this plot that the coupling of the spin portal g t in general can be larger than those of theHiggs portal g s or g p , by at least one order of magnitude. FIG. 2: Parameter space for g t , g s and g p consistent Γ( Z → ¯ DD ) < Γ invZ = 1 . ± . MeV and Γ( H → ¯ DD ) < Γ invH =1 . ± . MeV for
M < M Z /
2. Solid lines correspond to the central values of the invisible decay widths.
V. DARK MATTER RELIC DENSITY.A. Boltzman equation.
The evolution of the dark matter comoving number density n D ( T ) is described by the Boltzmann equation [26] dYdx = − λ ( x ) x ( Y − Y eq ) , (18)where x = M/T , Y ( x ) = n D ( x ) /T and λ ( x ) ≡ M (cid:104) σv r (cid:105) H ( M ) . (19) H Z, γ
FIG. 3: Feynman diagrams for ¯ DD → ¯ ff . Here, H ( M ) = M (cid:113) π G N g ∗ ( M )90 stands for the Hubble parameter at the dark mass scale, M , with G N = 6 . × − GeV − denoting the Newton gravitational constant [5], g ∗ ( M ) standing for the relativistic effective degrees offreedom at T = M in the thermal bath and Y eq ( x ) = n eqD T = g D T (cid:90) d p (2 π ) e ET − π (cid:90) ∞ x u √ u − x due u − ≈ π (cid:90) ∞ x e − u u (cid:112) u − x du. (20)The thermal average (cid:104) σv r (cid:105) includes all channels for the annihilation D ( p ) ¯ D ( p ) → X ( p ) Y ( p ) of dark matter intostandard model particles X, Y in the thermal bath and it is given by (cid:104) σv r (cid:105) = 1 n eqD n eq ¯ D (cid:90) g D d p (2 π ) e − E /T (cid:90) g ¯ D d p (2 π ) e − E /T σv r , (21)where g D ( g ¯ D )denotes the number of internal d.o.f of the dark matter particle (antiparticle), v r stands for the darkmatter particle-antiparticle relative velocity and σ is the conventional cross section for the D ( p ) ¯ D ( p ) → X ( p ) Y ( p )process.A qualitative analysis of the solution of Eq. (18) assuming the freezing of dark matter at some temperature whichwould explain dark matter relic density, shows that dark matter must be non-relativistic at the time of its decouplingfrom the cosmic plasma [26]. This is consistent with data on dark matter relic density extracted from precisionmeasurement of the cosmic background radiation [3, 5]. In this case, it is a good approximation to perform a non-relativistic expansion of (cid:104) σv r (cid:105) keeping only the leading terms in the expansion in powers of v r <<
1. This expansionrequires the calculation of the flux for dark matter particles in the thermal bath, which can be written as [27, 28] F = 4 (cid:112) ( p · p ) − M = 2( s − M ) v r (22)where v r is related to s as s = 2 M (cid:32) (cid:112) − v r (cid:33) = 4 M + M v r + .... (23)In the last step we performed the non-relativistic expansion for v r <<
1. The cross section σ is a function of s thususing Eq.(22) the leading terms in the expansion are σv r = a + bv r , (24)and performing the thermal average we obtain (cid:104) σv r (cid:105) = a + 6 bx . (25)For non-relativistic dark matter with M < M Z /
2, the kinematically allowed channels are ¯ DD → ¯ f f for fermionswith m f < M and ¯ DD → γγ . In the following we calculate the corresponding cross sections in our formalism, performthe non-relativistic expansion and work out the predictions for the a, b coefficients. B. Annihilation of dark matter into a fermion-antifermion pair.
There are three contributions to the process D ( p ) ¯ D ( p ) → f ( p ) ¯ f ( p ) shown in Fig. 3. The correspondingamplitudes are given by − i M H = i m f s − M H ¯ u ( p ) v ( p ) ¯ V ( p ) ( g s I + ig p χ ) U ( p ) , − i M γ = − Q f g t M W S W C W vs ¯ u ( p ) γ µ v ( p ) ¯ V ( p ) M µβ ( p + p ) β U ( p ) , (26) − i M Z = g t M Z S W v ( s − M Z ) ¯ u ( p ) γ µ ( A f + B f γ ) v ( p ) ¯ V ( p ) M µβ ( p + p ) β U ( p ) . Here, C W = Cosθ W , Q f stands for the fermion charge in units of the proton charge e , while the A f , B f factors arerelated to the corresponding fermion weak isospin T f as A f = 2 T f − Q f S W , B f = − T f . (27)A straightforward calculation yields the following average squared amplitude in terms of the Mandelstam variables (cid:12)(cid:12) M ¯ ff (cid:12)(cid:12) = − g t M Z S W M v ( s − M Z ) (cid:2) M (cid:0) A f + B f (cid:1) m f (cid:0) M − s (cid:1) + 4 m f (cid:0) M − s (cid:1) (cid:0) A f M (cid:0) M + s − t − u (cid:1) + B f (cid:0) M − M ( s + t + u ) − s (cid:1)(cid:1) + (cid:0) A f + B f (cid:1) (cid:0) M − M ( s + 4( t + u )) + 4 M ( t + u )( s + t + u ) + M (cid:0) s − s (cid:0) t + u (cid:1)(cid:1) + s (cid:0) ( t − u ) − s (cid:1)(cid:1)(cid:3) + 8 A f C W Q f g t M W M Z S W M sv ( s − M Z ) (cid:2) M m f (cid:0) M − s (cid:1) (cid:0) M + s − t − u (cid:1) + 4 m f (cid:0) M − M s (cid:1) + 16 M − M ( s + 4( t + u )) + 4 M ( t + u )( s + t + u ) + M (cid:0) s − s (cid:0) t + u (cid:1)(cid:1) + s (cid:0) ( t − u ) − s (cid:1)(cid:3) + A f m f g s g t M Z S W M v ( s − M H ) ( s − M Z ) s (cid:0) M − s (cid:1) ( t − u ) − C W m f Q f g s g t M W S W M v ( s − m H ) (cid:0) M − s (cid:1) ( t − u ) − C W Q f g t M W S W M s v (cid:2) M m f (cid:0) M − s (cid:1) (cid:0) M + s − t − u (cid:1) + 4 m f (cid:0) M − M s (cid:1) +16 M − M ( s + 4( t + u )) + 4 M ( t + u )( s + t + u ) + M (cid:0) s − s (cid:0) t + u (cid:1)(cid:1) + s (cid:0) ( t − u ) − s (cid:1)(cid:3) + m f M ( s − M H ) (cid:0) s − m f (cid:1) (cid:2) g p s (cid:0) s − M (cid:1) + g s (cid:0) M − M s + s (cid:1)(cid:3) . (28)Integrating the final state phase space finally we obtain the following cross section for ¯ DD → ¯ f f where we can easilyidentify the individual contributions from H, Z and γ exchange as well as the Z − γ interference: σ ¯ ff ( s ) = 172 πM √ s (cid:113) s − m f F m f (cid:16) s − m f (cid:17) (cid:0) g p s (cid:0) s − M (cid:1) + g s (cid:0) M − M s + s (cid:1)(cid:1) ( s − M H ) + 2 g t M Z S W s (cid:0) s − M (cid:1) (cid:0) M + s (cid:1) (cid:16) (cid:16) A f − B f (cid:17) m f + s (cid:16) A f + B f (cid:17)(cid:17) v ( s − M Z ) + 32 C W Q f g t M W S W (cid:0) s − M (cid:1) (cid:0) M + s (cid:1) (cid:16) m f + s (cid:17) v s − A f C W Q f g t M W M Z S W (cid:0) s − M (cid:1) (cid:0) M + s (cid:1) (cid:16) m f + s (cid:17) v ( s − M Z ) . (29)Notice that the H − Z and H − γ interferences vanish after integration of phase space. D ( p )¯ D ( p ) γ ( p , (cid:15) ) γ (cid:48) ( p , η ) D ( p )¯ D ( p ) γ (cid:48) ( p , η ) γ ( p , (cid:15) )FIG. 4: Feynman diagrams for ¯ DD → γγ . C. Dark matter annihilation into two photons
This process is induced by the t and u channel dark matter exchange shown in Fig. 4. The corresponding amplitudesare given by − i M t = i g t C W M ¯ V ( p , λ ) M αβ S ( p − p ) − t + 2 M t − M M µν U ( p , λ ) p α η β ( p ) p µ (cid:15) ν ( p ) , (30) − i M u = i g t C W M ¯ V ( p , λ ) M µν S ( p − p ) − u + 2 M u − M M αβ U ( p , λ ) p α η β ( p ) p µ (cid:15) ν ( p ) . (31)The average squared amplitude is given by |M γγ | = (cid:18) g t C W M (cid:19) T r (cid:20) S ( p ) + M M T αβµν S ( p ) + M M ¯ T β νσ ρ (cid:21) p µ p ρ p α p σ , (32)where T αβµν = M αβ S ( p − p ) − t + 2 M t − M M µν + M µν S ( p − p ) − u + 2 M u − M M αβ , (33)¯ T αβµν = M µν S ( p − p ) − t + 2 M t − M M αβ + M αβ S ( p − p ) − u + 2 M u − M M µν . (34)A straightforward calculation using the algebraic relations in the appendix yields |M γγ | = 2 C W g t M ( t − M ) ( u − M ) (cid:104) tu ) + 2 ( tu ) (cid:0) − M + 11 M s + 2 s (cid:1) + ( tu ) (cid:0) M − M s + 33 M s + 4 M s + 2 s (cid:1) + 2 M tu (cid:0) − M + 43 M s − M s + 17 M s − M s + 2 s (cid:1) + M (cid:0) M − M s + 51 M s − M s + 25 M s − M s + 2 s (cid:1)(cid:3) (35)Integrating the final state phase space we get the following cross section σ γγ ( s ) = 1 F (cid:113) − M s C W g t πM (cid:34) M (cid:0) M − M s − s (cid:1) tanh − (cid:114) − M s + s (cid:114) − M s (cid:0) − M + 228 M s − M s + 43 s (cid:1)(cid:35) . (36) D. Dark matter relic density
Expanding the ¯ DD → ¯ f f and ¯ DD → γγ cross sections we get σv r ≡ σ γγ v r + (cid:88) f σ ¯ ff v r = a + bv r (37)where the sum runs over all the kinematically allowed fermion states ( m f < M ) and a = 29 C W g t πM + (cid:88) f N f g s m f (cid:16) M − m f (cid:17) πM ( M H − M ) b = 365 C W g t πM + (cid:88) f N f (cid:113) M − m f πM M g t M Z S W (cid:16)(cid:16) A f − B f (cid:17) m f + 2 M (cid:16) A f + B f (cid:17)(cid:17) v ( M Z − M ) (38)+ 192 A f M C W Q f g t M W M Z S W (cid:16) m f + 2 M (cid:17) v ( M Z − M ) + 96 C W Q f g t M W S W (cid:16) m f + 2 M (cid:17) v − M m f (cid:16) g p (cid:0) M − M H (cid:1) (cid:16) M − m f (cid:17) + g s (cid:16) − m f (cid:0) M − M H (cid:1) − M M H + 20 M (cid:17)(cid:17) ( M H − M ) − M m f g s (cid:16) M − m f (cid:17) ( M H − M ) , with N f = 3 for quarks and N f = 1 for leptons. We can see in these equations that for the mass region M < M Z / m f /M H .In Fig. (5) we analize the Higgs and spin portal contributions to (cid:104) σv r (cid:105) as a function of the couplings for differentvalues of the dark matter mass. In general, we find that Higgs portal contributions are negligible compared to thecontributions of the spin portal. Therefore, we will neglect the contribution of the Higgs portal for the calculation ofthe relic density in the following. FIG. 5: Individual contributions of the spin portal ( g t = g, g s = g p = 0) and the Higgs portal ( g t = 0 , g s = g p = g ) to (cid:104) σv r (cid:105) .Similar results are obtained in the second case when varying independently g s or g p . Using Eqs. (25,38), we numerically solve Boltzman equation (18) for different values of g t and M , matching thesolution Y ( x ) with the equilibrium solution Y eq ( x ) in Eq.(20) at high temperatures, i.e., in the relativistic regime x <<
1. In Fig.(6) we show the solutions for some specific values of g t and M . Clearly, at some temprature T f thesolution Y ( x ) departs from the equilibrium solution Y eq ( x ) and dark matter decouples from the cosmic plasma in thenon-relativistic regime, x >> Y for the present temperature T . This can bedone from the numeric solution to Boltzman equation for specific values of g t and M scanning the parameter space0 FIG. 6: Solution of the Boltzman equation for different values of M and g t showing that dark matter decouples in the non-relativistic regime. The solid line corresponds to Y eq ( x ). consistent with the measured relic density. It is however more illustrative to follow the semi-analytic procedure thattakes advance of the freezing mechanism. For x > x f we have Y ( x ) >> Y eq ( x ) an we can find an approximate solutionneglecting Y eq ( x ) in the r.h.s of Eq.(18) and integrating from T f to a given temperature T , which for our purposeswe take as the present temperature T , to obtain1 Y ( x ) = 1 Y ( x f ) + (cid:114) π G N M (cid:90) x x f (cid:104) σv (cid:105) (cid:112) g ∗ ( x ) x dx. (39)The relic dark matter density is given byΩ DM = ρ DM ( x ) ρ c = ( n D ( x ) + n ¯ D ( x )) Mρ c = 2 n D ( x ) Mρ c = 2 M Y ( x ) T ρ c , (40)where we used n ¯ D = n D and ρ c = H πG N = 1 . × − h GeV /cm = 8 . × − h GeV is the criticaldensity [5]. Neglecting the term Y ( x f ) − in Eq. (39) which turns out to be small compared with the second term weget Ω DM h = 2 T h ρ c (cid:114) π G N (cid:32)(cid:90) x x f (cid:104) σv (cid:105) (cid:112) g ∗ ( x ) x dx (cid:33) − = 4 . × − GeV − (cid:32)(cid:90) x x f (cid:104) σv (cid:105) (cid:112) g ∗ ( x ) x dx (cid:33) − (41)where we used T = 2 . K = 2 . × − GeV [5]. Notice that the r.h.s. of this equation depends on g t and M . For a given M we can find the values of g t consistent with the measured value of the relic density. In ourcalculations we use the complete function g ∗ ( x ) but our results are quite similar if we use the average over the rangeof energies considered, ¯ g ∗ = 33.The freezing value x f can be found from the condition that the annihilation rate equals the expansion rate of theuniverse n eq ( x f ) (cid:104) σv (cid:105) ( x f ) = H ( x f ) , (42)which using the non-relativistic form for n eq ( x ) and Eq. (25) leads to (cid:18) a + 6 bx f (cid:19) √ x f e − x f = (2 π ) M (cid:114) G N g ∗ ( x f )90 . (43)1The value of x f depends also on g t and M , so we have two conditions, Eqs. (41,43), for the three variables x f , g t , M which are solved numerically to obtain the set of values g t ( M ) consistent with the measured dark matter relicdensity. The set of values g t ( M ) is shown in Figure 7. We checked also that these solutions are consistent with theapproximations used, i.e. that decoupling occurs when dark matter is non-relativistic. The values of x f correspondingto g t ( M ) lie in the range 23 . < x f < .
9, thus x f >>
1. Finally, we directly calculate Y ( x ) from the numericgeneral solution of the Boltzman equation for the set of values g t ( M ), matching the solution with Y eq ( x ) for x << x f finding indeed that 1 /Y ( x f ) is small compared to 1 /Y ( x ) in Eq.(39). FIG. 7: Values of g t and M consistent with the measured dark matter relic density, Ω expDM h = 0 . ± . Z → ¯ DD ) < Γ invZ = 1 . ± . MeV . These constraintsexclude masses below 43
GeV for dark matter with a (1 , ⊕ (0 ,
1) space-time structure.
Our results are summarized in Figure 7, where it is clear that taking into account constraints from the data onthe Z invisible width and from the measured dark matter relic density, dark matter with a (1 , ⊕ (0 ,
1) space-timestructure must have a mass
M > GeV . VI. CONCLUSIONS AND PERSPECTIVES
Effective theories for the interaction of dark matter with standard model fields has been done mainly assumingspace-time structures for dark matter similar to those of the standard model fields, i.e., dark matter fields transformingin the (0 , , ⊕ (0 , ) or ( , ) representations of the HLG.In this work we study the possibility of a (1 , ⊕ (0 ,
1) space-time structure for dark matter fields. Assumingthat dark matter fields are standard model singlets, we find three lowest order terms which are dimension-four in thecorresponding effective theory. Two of them couple the Higgs to dark matter and the third one couples the photonand Z fields to higher multipoles of the spin-one dark matter fields, yielding a spin portal to dark matter.We start the study of the phenomenology derived from our proposal considering dark matter mass M < M Z /
2, inwhose case the H → ¯ DD and Z → ¯ DD are kinematically permitted and contribute to the Higgs and Z invisibledecay widths. We use experimental results on these widths to put upper limits to the corresponding low energyconstants. In general we find stringent constraints for the couplings of the Higgs portal: g s , g p ≤ − and lessstringent constraints on the spin portal coupling g t .For dark matter mass in this region, non-relativistic dark matter can annihilate into a photon pair or into afermion-anti-fermion pair if M > m f . We calculate these processes in our formalism and use them to calculate thecorresponding dark matter relic density. We find that the contribution of the Higgs portal to the dark matter relicdensity is negligible and the main contribution comes from the spin portal. Taking into account the constraints from2the Z invisible width, we find that a proper description of the measured dark matter relic density imposes the lowerbound M > GeV for dark matter with a (1 , ⊕ (0 ,
1) space-time structure.The spin portal yields a new avenue for the possible transitions between the dark matter and standard modelsectors whose phenomenological consequences are worthy to explore further. Here, we study the low mass regime,
M < M Z /
2, where low energy constants can be constrained from the H and Z invisible widths. For M > M Z / Z → ¯ DD decay is kinematically forbidden and we loose the corresponding constraints on g t . Furthermore,in this regime, depending on the kinematics, new channels for the annihilation of dark matter such as ¯ DD → Z γ, Hγ, W + W − , Z Z , Z H, HH, ¯ tt open and must be considered in the analysis of the dark matter relic density.On the other hand, some experiments of direct detection of dark matter attempt to detect nuclear recoil due to thescattering of nuclei with dark matter, ultimately related to the quark-dark matter scattering, which takes place inour formalism. It is important to calculate these effects in order to further constrain the possible values of the massand couplings of spin-one dark matter. Finally, it would be important to study all processes involving dark matter sofar analyzed at the LHC on the light of spin-one dark matter fields. Acknowledgments
Work supported by CONACyT M´exico under project CB-259228. H.H.A. acknowledges CONACyT for a scholarshipand DAIP-UG for a grant under the Call for Support to Graduate Studies 2017.
VII. APPENDIX: TRACE-OLOGY FOR (1 , ⊕ (0 , . In this appendix we collect the trace relations necessary for the calculations in this work. The covariant basisfor the (1 , ⊕ (0 ,
1) representation space is given by the set of 6 × { , χ, S µν , χS µν , M µν , C µναβ } where is the identity matrix. The first principles construction of these matrices can be found in [23] and their explicitform depends on the basis chosen for the states in the (1 , ⊕ (0 ,
1) representation. All the calculations in this workare representation independent and rely only on their algebraic properties. The starting point are first principlesconstruction of the rest-frame parity operator (Π), the Lorentz generators J i = (cid:15) ijk M jk and K i = M i and thechirality operator χ entering the projectors on the chiral subspaces (1 ,
0) and (0 ,
1) which satisfy { χ, Π } = 0 , [ χ, M µν ] = 0 , χ = . (44)The S µν tensor is the covariant version of the rest-frame parity operator (Π) such that S = Π and other componentscan be written as S µν = Π (cid:0) g µν − i ( g µ M ν + g ν M µ ) − { M µ , M ν } (cid:1) . (45)This is a symmetric traceless ( S µµ = 0) tensor with nine independent components. As a consequence of Eqs.(44) weget { χ, S µν } = 0 . (46)The C tensor is given by C µναβ = 4 { M µν , M αβ } + 2 { M µα , M νβ } − { M µβ , M να } − g µα g νβ − g µβ g να ) . (47)with the symmetry properties C µναβ = − C νµαβ = − C µνβα ; C µναβ = C αβµν . It satisfies the Bianchi identity C µαβν + C µβνα + C µναβ = 0 and the contraction of any pair of indices vanishes C ν ναβ = 0. These constraints leaveonly 10 independent components. Clearly it satisfies [ χ, C µναβ ] = 0.The covariant basis is orthogonal with respect to the scalar product defined as (cid:104) A | B (cid:105) = T r ( AB ), thus these matricessatisfy the following relations T r ( χ ) = T r ( S ) = T r ( M ) = T r ( χS ) = T r ( C ) = 0 ,T r ( χM ) = T r ( χC ) = T r ( M S ) =
T r ( M χS ) =
T r ( M C ) =
T r ( SχS ) =
T r ( SC ) = T r ( χSC ) = 0 . (48)where we suppressed the Lorentz indices.Calculations in this work requires traces of products of the S µν tensor and other elements in the covariant basis.Let us consider first T r ( SM M ) =
T r (cid:0) χ SM M (cid:1) = − T r ( χSχM M ) = − T r ( χSM M χ ) = − T r ( SM M ) ⇒ T r ( SM M ) = 0 , (49)3where we used Eqs. (44,46) and the cyclic property of a trace. Since χ commutes also with C , this procedure can beused to show that in general if we have a term with an odd numbers of S tensors the trace of this term will vanish T r (term with an odd S ’s) = 0 . (50)The trace of terms with an even number of S factors can always be reduced to a linear combination of terms withthe trace of the product of two S or two M factors using the following (anti)commutation relations[ M µν , M αβ ] = − i (cid:0) g µα M νβ − g να M µβ − g µβ M να + g νβ M µα (cid:1) (51) { M µν , M αβ } = 43 ( g µα g νβ − g µβ g να ) − iε µναβ χ + 16 C µναβ , (52)[ M µν , S αβ ] = − i (cid:0) g µα S νβ − g να S µβ + g µβ S να − g νβ S µα (cid:1) , (53) (cid:8) M µν , S αβ (cid:9) = ε µνσβ χS α σ + ε µνσα χS β σ , (54)[ S µν , S αβ ] = − i (cid:0) g µα M νβ + g να M µβ + g νβ M µα + g µβ M να (cid:1) , (55) (cid:8) S µν , S αβ (cid:9) = 43 (cid:18) g µα g νβ + g να g µβ − g µν g αβ (cid:19) − (cid:0) C µανβ + C µβνα (cid:1) . (56)The simplest case appears in the calculation of H → ¯ DDT r (cid:0) S µν S αβ (cid:1) = T r (cid:18)
12 [ S µν , S αβ ] + 12 { S µν , S αβ } (cid:19) = 4 (cid:18) g µα g νβ + g µβ g να − g µν g αβ (cid:19) ≡ T µναβ . (57)Similarly, the calculation of Z → ¯ DD requieres T r (cid:0) M µν M αβ (cid:1) = T r (cid:18)
12 [ M µν , M αβ ] + 12 { M µν , M αβ } (cid:19) = 4( g µα g νβ − g µβ g να ) ≡ G µναβ . (58)The first example of the reduction mentioned above is faced in the calculation of Z → ¯ DD which also requires tocalculate T r (cid:0) S µν S αβ M ρσ (cid:1) = T r (cid:18) (cid:8) S µν , S αβ (cid:9) M ρσ + 12 (cid:2) S µν , S αβ (cid:3) M ρσ (cid:19) = − i T r (cid:0)(cid:0) g µα M νβ + g να M µβ + g νβ M µα + g µβ M να (cid:1) M ρσ (cid:1) = − i (cid:0) g µα G νβρσ + g να G µβρσ + g νβ G µαρσ + g µβ G ναρσ (cid:1) . (59)and T r (cid:0) S αβ M µν S ρσ M γδ (cid:1) = T r (cid:18) ( 12 [ S αβ , M µν ] + 12 { S αβ , M µν } )( 12 [ S ρσ , M γδ ] + 12 { S ρσ , M γδ } ) (cid:19) = T r (cid:18)(cid:18) i g µα S νβ − g να S µβ + g µβ S να − g νβ S µα ) − ε µντβ χS α τ − ε µντα χS β τ (cid:19)(cid:18) i g γρ S δσ − g δρ S γσ + g γσ S δρ − g δσ S γρ ) − ε γδλσ χS ρ λ − ε γδλρ χS σ λ (cid:19)(cid:19) = − g µα g γρ T νβδσ + g µα g δρ T νβγσ − g µα g γσ T νβδρ + g µα g δσ T νβγρ + g να g γρ T µβδσ − g να g δρ T µβγσ + g να g γσ T µβδρ − g να g δσ T µβγρ − g µβ g γρ T ναδσ + g µβ g δρ T ναγσ − g µβ g γσ T ναδρ + g µβ g δσ T ναγρ + g νβ g γρ T µαδσ − g νβ g δρ T µαγσ + g νβ g γσ T µαδρ − g νβ g δσ T µαγρ − (cid:0) ε µντβ ε γδλσ T α ρτ λ + ε µντβ ε γδλρ T α στ λ + ε µντα ε γδλσ T β ρτ λ + ε µντα ε γδλρ T β στ λ (cid:17) (60)Similarly it can be shown that T r (cid:0) M µν M αβ M ρσ (cid:1) = − i (cid:0) g µα G νβρσ − g να G µβρσ − g µβ G ναρσ + g νβ G µαρσ (cid:1) (61) T r (cid:0) χS γδ S αβ M µν (cid:1) = − (cid:0) ε µνσβ T γδα σ + ε µνσα T γδβ σ (cid:1) , (62) T r (cid:0) χM µν M αβ (cid:1) = − iε µναβ . (63)4The calculation of the trace of terms involving six or eight S or M factors (with an even number of S factors)needed in this paper are reduced in a similar way.There is a simpler way to obtain these results however, which is specially useful for terms with six or more factors.Since the result rests only on the algebraic properties in Eqs. (51, 52,53,54,55,56) we can use any representation ofthese operators for the calculation of the trace. In this concern the use of the representation where the internal matrixindices transform as Lorentz indices is convenient, since in this case the calculation of the trace reduces to contractionsof Lorentz indices which can be easily done using conventional algebraic manipulation codes like FeynCalc. In thisrepresentation, each internal matrix index a is replaced by a pair of antisymmetric Lorentz indices αβ [29]. Theexplicit form of the operators in the covariant basis is given by( ) αβγδ = 12 ( g αγ g βδ − g αδ g βγ ) , (64)( χ ) αβγδ = i ε αβγδ , (65) (cid:0) M µν (cid:1) αβγδ = − i ( g µγ αβνδ + g µδ αβγν − g γν αβµδ − g δν αβγµ ) , (66) (cid:0) S µν (cid:1) αβγδ = g µν αβγδ − g µγ αβνδ − g µδ αβγν − g γν αβµδ − g δν αβγµ . (67)The explicit form of C µναβ can be constructed from Eq.(47) and the above relations. [1] G. Arcadi et al. , arXiv:1703.07364 (2017).[2] F. Zwicky, Helv. Phys. Ac. , 110 (1933), [Gen. Rel. Grav.41,207(2009)].[3] P. A. R. Ade et al. , Astron. Astrophys. , A13 (2016).[4] G. Steigman and M. S. Turner, Nucl. Phys. B253 , 375 (1985).[5] C. Patrignani et al. , Chin. Phys.
C40 , 100001 (2016 and 2017 update).[6] E. W. Varnes, Acta Phys. Polon.
B47 , 1595 (2016).[7] D. G. Charlton, PoS
ICHEP2016 , 004 (2017).[8] T. Camporesi, PoS
ICHEP2016 , 005 (2017).[9] V. Silveira and A. Zee, Phys. Lett. , 136 (1985).[10] J. McDonald, Phys. Rev.
D50 , 3637 (1994).[11] C. P. Burgess, M. Pospelov, and T. ter Veldhuis, Nucl. Phys.
B619 , 709 (2001).[12] S. Kanemura, S. Matsumoto, T. Nabeshima, and N. Okada, Phys. Rev.
D82 , 055026 (2010).[13] S. Andreas et al. , Phys. Rev.
D82 , 043522 (2010).[14] A. Djouadi, O. Lebedev, Y. Mambrini, and J. Quevillon, Phys. Lett.
B709 , 65 (2012).[15] Y. Mambrini, Phys. Rev.
D84 , 115017 (2011).[16] A. Djouadi, A. Falkowski, Y. Mambrini, and J. Quevillon, Eur. Phys. J.
C73 , 2455 (2013).[17] L. Lopez-Honorez, T. Schwetz, and J. Zupan, Phys. Lett.
B716 , 179 (2012).[18] J. Kearney, N. Orlofsky, and A. Pierce, Phys. Rev.
D95 , 035020 (2017).[19] G. Bambhaniya et al. , Phys. Lett.
B766 , 177 (2017).[20] R. C. Cotta, J. L. Hewett, M. P. Le, and T. G. Rizzo, Phys. Rev.
D88 , 116009 (2013).[21] M. Napsuciale, S. Rodr´ıguez, R. Ferro-Hern´andez, and S. G´omez- ´Avila, Phys. Rev.
D93 , 076003 (2016).[22] P. Dirac,
Lectures on Quantum Mechanics (Belfer Graduate School of Science, Yeshiva University, New York, 1964).[23] S. G´omez- ´Avila and M. Napsuciale, Phys. Rev. D , 096012 (2013).[24] D. M. Webber et al. , Phys. Rev. Lett. , 041803 (2011), [Phys. Rev. Lett.106,079901(2011)].[25] V. Khachatryan et al. , JHEP , 135 (2017).[26] S. Dodelson, Modern Cosmology (Academic Press, Amsterdam, 2003).[27] P. Gondolo and G. Gelmini, Nucl. Phys.
B360 , 145 (1991).[28] M. Cannoni, Int. J. Mod. Phys.
A32 , 1730002 (2017).[29] E. Delgado-Acosta, M. Kirchbach, M. Napsuciale, and S. Rodriguez, Phys.Rev.