Dark matter models with suppressed dark matter nuclei elastic cross section

We propose two generalizations of the dark photon model which predict the suppressed elastic dark matter nuclei cross section in comparison with the corresponding prediction of the dark photon model. In the first model the main difference from dark photon model is that the mixing parameter $\epsilon$ is nonlocal formfactor $\epsilon(q^2)= \frac{q^2}{\Lambda^2}V(\frac{q^2}{\Lambda^2}) $ depending on the square of the momentum transfer $q^2$. Here $V(\frac{q^2}{\Lambda^2})$ is an entire function of the growth $\rho \geq \frac{1}{2}$ and $\Lambda$ is nonlocal scale. In this model our world and dark world are described by renormalizable field theories while the communication between them is performed by nonlocal interaction. The second model is renormalizable model where besides dark photon field $A'$ additional vector boson $Z'$ interacts with $B - L$ current. The communication between our world and dark world is performed due to nonzero kinetic mixing between $Z'$ and $A'$ fields. The predictions of the models for the search for dark matter at the accelerators don't contain additional suppression factors.


Introduction
At present the most striking evidence in favour of new physics beyond the Standard Model (SM) is the observation of the Dark Matter (DM) (as a review see for example: [2, ?]).A lot of DM models exist [3](as a review see for example [3,4,5]) 1 .The standard assumption [2, ?] is that in the hot early Universe the DM particles were in the equilibrium with the observed particles is often used.During the Universe expansion the temperature decreases and at some point the thermal decoupling of the DM starts.Namely, at some temperature the annihilation cross section of DM particles DM particles → SM particles becomes too small to perform the equilibrium of DM particles with the SM particles and the DM particles decouple.The experimental data are in favour of the scenario with cold relic for which the freeze-out temperature is much lower the DM particle mass, i.e. the nonrelativistic DM particles decouple.The value of the DM annihilation cross section at the decoupling epoch determines the value of the current DM particles.The observed value of the DM density fraction ρ DM ρc ≈ 0.23 allows to estimate the DM annihilation cross section and hence to estimate the DM discovery potential.There are at least three possibilities to discover the DM particles.At first it is possible to use the LHC experiments ATLAS, CMS, LHCb or fixed target experiments like NA64 and BELLE.The second way is the use of the elastic DM nucleon(electron) scattering in underground experiments [7].The third possibility consists in the astrophysical detection of the DM annihilation into the SM particles [7].At present the underground experiments give bounds on the elactic DM nucleon cross section at the level O(10 −43 − 10 −47 ) cm2 [7] that strongly restricts a lot of existing DM models [3].Therefore it is interesting to consider the DM models which allow to escape the strong bounds from underground experiments and don't have any additional suppression factors for the accelerator experiments.
In this paper we propose two generalizations of the dark photon model [9] with suppressed DM nucleon(electron) elastic cross section.In our models the elastic tree level DM nucleon(electron) cross section has suppression factor 2 O(v 4 ) = O(10 −12 ) in comparison with the corresponding prediction for the dark photon model.At one loop level the suppression factor for the cross section is around O(10 −6 ).In the first model the main difference from dark photon model is that the mixing parameter ǫ is nonlocal formfactor depending on the square of the momentum transfer q 2 .Here V ( q 2 Λ 2 ) is an entire function of the growth ρ ≥ 1 2 and Λ is nonlocal scale.In the proposed model the SM world and the DM world are described by renormalizable field theories while the communication between them is performed by nonlocal interaction.The second model is renormalizable model where besides dark photon A ′ additional Z ′ vector boson interacts with B −L current of the SM.The interaction of the SM world and the DM world is performed only due to nonzero kinetic mixing of A ′ and Z ′ fields.The predictions of the models for the search for dark matter at the accelerators and in astrophysics don't contain additional suppression factors.
The predictions of the models for the search for DM at the accelerators and in astrophysics don't contain additional suppression factors.
The paper is organized as follows.In the next section we describe nonlocal generalization of the dark photon model.In the third section we consider renormalizable model with additional U(1) Z ′ vector boson interacting with dark photon field A ′ .Section 4 contains concluding remarks.Appendix contains the main formulae used for the DM density calculations.

Nonlocal generalization of dark photon model
In dark photon model [9] the additional light vector boson A ′ interacts with the gauge SU c (3) ⊗ SU L (2) ⊗ U(1) fields of the SM due to nonzero mixing with the U(1) SM gauge field.The Lagrangian of the model is represented in the form where L SM is the SM Lagrangian and Here ǫ is the mixing parameter, and L dark is the DM Lagrangian3 .At present scalar, Dirac, pseudo-Dirac and Majorana DM models are often considered.
In dark photon model with Dirac or scalar DM the electron DM elastic cross section has the form [10] σ(DM where µ χe = mχme mχ+me and . The analogous formula is valid for nucleon.For m χ ≈ 10 3 GeV experimental bounds on σ(DM + nucleon → DM + nucleon) are at the level 10 −9 pb [7] that restricts rather strongly dark photon mass, namely m A ′ ≥ 3.5 T eV at α D = 0.1 and ǫ = 0.1.
In nonlocal generalization of the dark photon model we assume that both our world and dark world are described by local renormalizable field theories while the communication between our world and dark sector is performed by nonlocal interaction [11,12,13,14].We propose to use nonlocal generalization for the mixing term (2), namely In nonlocal field theory the formfactor ǫ( q 2 Λ 2 ) is an entire function on q 2 of the growth ρ ≥ 1 2 [11,12,13,14].Moreover we require that nonlocal interaction ǫ( q 2 Λ 2 ) has to dissapear in the limit of the infinite nonlocal scale Λ, i.e. ǫ( q 2 Λ 2 ) → 0 at Λ → ∞.In other worlds it means that the communication between our world and dark world switches off for infinite nonlocal scale.As a consequence we find that ǫ( As an example we shall use the formfactor The existence of exponential multiplier exp(− (q 2 ) 2 Λ 4 ) in the formfactor (5) leads to ultraviolet convergence of the corresponding Feynman diagrams.The Feynman rules for the nonlocal dark photon model are the same as in original dark photon model except the use of the formfactor ǫ( q 2 Λ 2 ) instead of the constant ǫ.It should be stressed that nonlocal formfactor ǫ( q 2 Λ 2 ) → 0 for q 2 → 0 that is crusial for the suppression of the elastic DM nucleon(electron) cross section.In nonlocal model with the formfactor (5) vanishing at q 2 → 0 we have the suppression factor k nl = ∼ O(10 −12 ) for elastic cross section (3).In nonlocal dark photon model the production cross section of For nonlocal dark photon model the formula for the annihilation cross section σ(DM + DM → SM particles) coincides with the corresponding formula for the standard dark photon model except the replacement A ′ = 5m 2 χ and α D = 0.1.For dark photon mass m A ′ ≤ O(1) GeV the NA64 [16] and BABAR [17] experiments give the most strongest bounds on ǫ( . As a consequence of the assumed equilibrium of the LDM with the SM particles at the early Universe one can find that For pseudo-Dirac LDM with m A ′ = 3m χ and α D − 0.1 we find that The obtained values (6, 7) for ǫ( Λ 2 ) don't contradict to experimental bounds [8], [16,17] at m A ′ ≤ 1 GeV .The predicted value of nonlocal scale Λ depends on dark photon mass m A ′ and it is rather small, for instance Λ ∼ 10 GeV at m A ′ = 100 MeV .
For the mass region m χ ∼ O(1) T eV consider as an example fermion dark matter with α D = 0.1 and m 2 A ′ = 5m 2 χ .The analog of the formula (6) reads The predicted value of nonlocal scale Λ depends on the dark photon mass m A ′ , for instance Λ = 7 T eV at m A ′ = 2 T eV .Note that the obtained value (8) for the mixing parameter does not contradict to the LEP1 data since the mixing parameter ǫ( q 2 Λ 2 ) strongly depends on the mass scale.In the energy region of the Z-boson we have to use ǫ( The CMS and ATLAS bounds for nonlocal dark photon model with a mass m A ′ ≥ O(200) GeV coincide with the corresponding bounds for dark photon model [7,15] and they are not very strong.The reason is that dark photon decays mainly into invisible modes A ′ → χ χ that makes its detection not very easy in contrast to most Z ′ models(for instance the Z ′ model with B−L current) which have visible decays into e + e − or µ + µ − .Moreover in dark photon model the cross section of the dark photon production is suppressed by factor ǫ 2 ( Within nonlocal approach it is possible to use nonlocal interaction between the SM fields and the DM fields without dark photon.For instance, nonlocal interaction of the (vector × vector) type can play the role of messenger between the SM world and dark sector.Here J µ SM is the current made up of the SM fields, J µ,dark is the current made up of the dark particles and V (− ∂ µ ∂µ Λ 2 ) is nonlocal formfactor.For the model (9) with the formfactor (5) the elastic tree level DM nucleon(electron) cross section is suppressed by the same factor O(v 4 ).

Renormalizable extension of the dark photon model with additional vector Z ′ boson
In dark photon model dark photon field A ′ interacts with DM particles due to nonzero kinetic mixing between the dark photon field A ′ and the U(1) gauge field B of the SM model.Here we consider the extension of the dark photon model with additional U(1) gauge field Z ′ interacting with the SM fields.As the simplest possibility we consider the interaction of the Z ′ with (B − L) current [18,19,20,21], namely We assume that dark photon A ′ interacts with the DM matter in standard way.For instance, for the Dirac fermion DM the interaction is Note that in DM model with (B − L) Z ′ vector boson the DM nucleon elastic cross section is [3] σ(DM Here g χ is the coupling constant of DM with Z ′ .From the experimental bound σ(DM + nucleon → DM + nucleon) ≤ 10 −9 pb [7] and the formula (3) for the elastic cross section we find rather strong bound In our model we assume that Z ′ does not interact directly with the DM.The interaction of Z ′ with DM is performed due to nonzero kinetic mixing of Z ′ with dark photon A ′ , namely where As a consequence of the interaction (13) and nonzero Z ′ mass we find that the tree level amplitudes with the interaction of Z ′ and A ′ bosons contain the multiplier , where q is the momentum transfer.Note that in dark photon model the role of the Z ′ -boson plays massless photon field A and the multiplier is q 2 q 2 = 1.As a consequence of nonzero Z ′ boson mass for |q 2 | ≪ m 2 Z ′ we have the suppression factor q 2 m 2 Z ′ for tree level amplitudes.As it was explained in the previous section the existence of the factor q 2 m 2 Z ′ leads to the suppression factor O(v 2 ) = O(10 −12 ) for the tree level elastic DM nucleon(electron) cross section.For the model with additional Z ′ boson consider two mass regions for DM.For the case of the LDM with O(1) MeV ≤ m χ ≤ O(1) GeV there are rather strong bounds on coupling constant g B−L for the Z ′ boson, see ( [22]) and [24,25,23].Consider as an example the scalar LDM.The annihilation cross section into electron positron pair in nonrelativistic approximation has the form In comparison with the B − L LDM model we have additional factor ) 2 for the cross section (14).For the particular case m 2 A ′ = 3m 2 χ and ǫ Z ′ A ′ = 0.25 the additional factor k ad = 1 and the predictions for g B−L for both models coincide.Also the predictions of the considered model with For the mass region with m χ = O(1) T eV the model also does not contradict to existing accelerator bounds for some parameters.Consider the model with Dirac DM.For m χ = O(1) T eV the equation for the determination of the DM density leads to where k = 6.5 for m χ ≫ m top .As a numerical example we use As a consequence of the equation ( 15) we find that [26,27] 4 we find that the mixing parameter ǫ Z ′ A ′ ≥ 0.14.As a second numerical example consider For this set of parameters we find that ǫ m Z ′ ≈ 0.54 • 10 −5 GeV −1 and as a consequence of the LEP bound [26,27] ǫ Z ′ A ′ ≥ 0.038.As in previous example at one-loop level we have the suppression factor ( gχg B−L ǫ Z ′ A ′ 8π 2 ) 2 ∼ O(10 −6 ) in comparison to tree level cross section and the bound on Z ′ -boson mass is weaker by factor ∼ 30 the corresponding bound in (B − L) DM model.The Z ′ -boson phenomenology in considered model is similar to the phenomenology of the Z ′ -model without dark photon.All LHC and fixed target bounds are valid for the model with additional dark photon A ′ .In contrast to the Z ′ -boson dark photon A ′ decays mainly into invisible modes.As a consequence the bounds on dark photon are much weaker the bounds on Z ′ -boson.

Conclusions
In this paper we proposed two generalizations of the dark photon model which predict the suppression for the elastic DM nucleon(electron) cross section in comparison with the corresponding prediction of the dark photon model.In the first model the main difference from dark photon model is that the mixing parameter ǫ is nonlocal formfactor ǫ(q 2 ) = q 2 Λ 2 V ( q 2 Λ 2 ) depending on the square of the momentum transfer q 2 .Here V ( q 2 Λ 2 ) is an entire function of the growth ρ ≥ 1 2 and Λ is nonlocal scale.In this model our world and dark world are described by renormalizable field theories while the communication between them is performed by nonlocal interaction.In the second model besides dark photon field A ′ we introduce additional Z ′ vector boson interacting with B−L current.The interaction between our world and dark world is performed only due to nonzero kinetic mixing of the A ′ and Z ′ fields.Both models allow the existence of the LDM with a mass m χ = O(1) GeV and the DM with the mass in TeV region.The predictions for the search for dark matter at the accelerators and in astrophysics don't contain additional suppression factors.
I am indebted to my colleagues from INR RAS for discussions.
5 Appendix.The main formulae for the DM density calculations In the nonrelativistic approximation with < σv rel >= σ 0 x −n f one can find that [2,28] takes place for the x f calculation: Here x f = mχ T d , T d is the decoupling temperature, Ω DM is the DM density of the Universe, h is the value of Hubble parameters in special units and < σv rel > is the product of the DM annihilation cross section and the relative velocity of the DM particle .The value of n = 0 corresponds to the s-wave annihilation and the n = 1 corresponds to the p-wave annihilation.The following approximate formula [2,29,30] takes place for the x f calculation: 1/2 * )dT , g * s (g * ) is the number of relativistic degree of freedom for entropy(energy) and g is the number of internal degree of fredom for DM particle χ.If DM particle differs from DM antiparticle σ o = σan 2 where σ an is the annihilation cross section of the reaction χ + χ → SM particles.In our numerical estimates we take Ω DM h 2 = 0.12.For s-wave annihilation cross section with n = 0 < σv rel >= 7.3 • 10 −10 GeV −2 1 g For dark photon model with Dirac DM particles the annihilatio cross section into electron positron pair has the form Here α = e 2 4π = 1/137, m A' is the dark photon mass, α D = e 2 D 4π and ǫ is the mixing parameter.For the p-wave annihilation in nonrelativistic approximation < σv rel >=< Bv 2 rel >= 6B • T d mχ .In dark photon model with scalar DM particles the annihilation cross section is

4m 2 χΛ 2 ) 2 χΛ 2 )
. Using the formulae of the Appendix we can estimate the product α D ǫ 2 ( 4m as a function of m χ and m A ′ .Consider at first the case of scalar LDM with m 2