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Derivative Portal Dark Matter

We propose a new kind of Dark Matter: Derivative Portal Dark Matter. This kind of Dark Matter connects to the Standard Model through mediators which link to Standard Model in the derivative form. Derivative of mediator in momentum space corresponds to the mediated momentum, which vanishes in the zero momentum transfer limit. As a result this kind of Dark Matter can evade stringent constraint from the Dark Matter direct detection while fitting the Dark Matter relic density observation naturally.


INTRODUCTION
The ever-improving sensitivities of Dark Matter (DM) direct detection experiments have put the famous Weakly Interacting Massive Particle (WIMP) DM models under pressure. In recent years there are works studying models where direct detection interaction is cancelled by two scalar mediators in zero momentum transfer limit [1][2][3][4][5]. In our previous work [6] we have generalised the cancellation mechanism from scalar mediators to vector mediators. Inspired by the proof of cancellation mechanism in [6], we propose a new kind of Dark Matter which possesses the cancellation mechanism and can thus avoid stringent direct detection constraint.
The cancellation happens when the scattering between DM and Standard Model (SM) fermion mediated by two mediators cancel out each other, leading to a momentum transfer proportional amplitude. Therefore the amplitude vanishes in DM direct detection since we usually adopt zero momentum transfer limit in direct detection, while the DM relic density is not diminished since the momentum transfer in the annihilation process is larger than two times of the DM mass and thus can not be neglected. The usual way of proving the cancellation mechanism is calculating the amplitude directly. Noticing the momentum transfer is actually the momentum of mediators which equals to the derivative of mediators in momentum space, we can denote interactions in models with cancellation mechanism in the form of derivative of mediators (i.e., in the form of kinetic mixing between mediators) and thus see the cancellation property immediately.
In our previous work [6], we have constructed a cancellation model by adding one U(1) gauge symmetry to SM. Where the extra gauge boson will mix with the Z boson from the mass matrix. And the direct detection mediated by the Z boson will be cancelled by the extra gauge boson. However, the kinetic mixing between Photon and the extra gauge boson will ruin the cancellation. While in this work we study a kind of DM model where the DM and SM fermion is linked by the kinetic mixing between the Z boson and a dark vector boson 1 , and we will show that this kind of DM model possesses the cancellation property. There are lots of works have studied models where interaction between DM and SM fermion comes from kinetic mixing term [7][8][9][10][11][12][13][14][15][16]. While the distinctive point in our construction is that the kinetic mixing between Photon and the dark vector boson should be negligible naturally (e.g., the kinetic mixing between Photon and the dark vector boson comes from 2-loops corrections). The reason why Photon should be out of the picture is that the propagator of Photon contains a momentum transfer t in its denominator, which will cancel the momentum transfer in the numerator and thus ruin the cancellation.

DERIVATIVE PORTAL DARK MATTER
The key Lagrangian of the Derivative Portal Dark Matter (DPDM) model is where Z µ and Z µ are massive vector mediators, while J µ f and J µ DM are the current of SM fermion and DM respectively. 2 Z µν Z µν is the derivative portal which connects the SM and dark sector. Then the dark matter SM fermion scattering is depicted by Fig. 1, where we use χ and f to denote DM and SM fermion respectively. Since the derivative portal contains derivative of mediators, the scattering amplitude will be proportional to the mediators' momentum and thus the momentum transfer t: .
Where m Z and m Z represent the mass of Z and Z boson. Therefore the amplitude goes to zero in the zero momentum transfer limit 2 . From Eq. (2) we see that when the mass of one mediator goes to zero, there will be a t in the denominator which will cancel the t in the numerator and thus ruin the t-proportional property. Hence the massless Photon is not suitable for building the derivative portal. Now let us look back at the derivative portal at Eq. (1), it is actually a kinetic mixing term between Z boson and Z boson. For Abelian gauge bosons, one can write down the kinetic mixing term directly. For non- 2 The usual way of proving the cancellation mechanism can be seen in Appendix B.
Abelian gauge bosons the kinetic mixing term can originate from loop corrections as shown in Fig. 2 and Ψ represent the scalar and fermion which contribute to the kinetic mixing respectively. The kinetic mixing can thus be estimated as [6,7,16] Where the first term and the second term represent contribution from scalars and fermions respectively, and g i , g i and m i are the couplings and mass of the i th particle which contribute to the kinetic mixing. One thing should be kept in mind is that the kinetic mixing between the Photon and the massive gauge boson which couples to DM should be naturally small, therefore the leading loop corrections to this kinetic mixing should be at least two-loops.

REALISTIC MODELS AND DM RELIC DENSITY
A simple and direct construction of DPDM model is extending an U(1) B−L model with an extra U(1) X gauge symmetry. The relevant Lagrangian can be given by: Where C and X are gauge bosons of U(1) B−L and U(1) X symmetry 3 . And the kinetic mixing term can be written directly or generated from loop corrections. In this section we will mainly focus on adding the SM with an extra gauge boson. We adopt the SM Z boson and a dark boson Z as mediators, then the relevant Lagrangian can be written as: Where the first, the second, and the third lines represent the kinetic terms, the coupling terms, and the mass terms respectively. Since the kinetic mixing term comes from loop corrections, this Lagrangian is not UV complete. A possible UV complete theory can be seen in Appendix A.
To normalize the kinetic terms, we can apply the following transformation: After this operation there will be a mixing term in the mass matrix ofZ boson andZ boson. After diagonalizing these bosons to their mass eigenstates, one can prove the cancellation in the usual way. See Appendix B for more details. We implemented this model in FeynRules 2 [17], and utilized the MadGraph [18] plugin MadDM [19] to calculate the DM relic density. The results are shown in Fig. 3. Where the lines are contours that saturate the Planck experiment [20] observation of DM relic density. There are four free parameters in the DPDM model, which are m χ , , g χ and m Z . And we adopt the DM mass m χ vs the kinetic mixing coupling and the gauge coupling g χ in the left and right panel of Fig. 3 respectively. In the left panel of Fig. 3 we fix g χ = 0.1 and use green line and blue line to denote cases where m Z = 1000 GeV and m Z = 2000 GeV respectively. In the right panel of Fig. 3 we fix m Z = 1000 GeV and use green line and blue line to denote cases where = 0.01 and = 0.1 respectively. Area below the lines is parameter space where DM relic density is larger than the Planck experiment observation, and thus these area is excluded by the Planck experiment. And there are dips which correspond to the resonant annihilation that happens when DM is around half of m Z . There are also dips around m Z which are caused by DM coannihilating with the dark mediator Z .
From Fig. 3 we see that the kinetic mixing coupling should be in the order similar or larger than O(0.01) to not being constrained severely by DM relic density. Also the kinetic mixing will change the mass of Z boson from m Z to

CONCLUSION AND DISCUSSION
In this work we have proposed a new kind of DM model where DM interacts with SM fermion through the derivative portal. We have proved in this kind of model the scattering amplitude between DM and SM fermion is proportional to the momentum transfer, therefore the DM direct detection goes to zero in the zero momentum transfer limit. We have also studied the DM relic density predicted by this kind of DM model and discussed the possible UV completion of this variety of model. In this work we focused on the framework of the DPDM model, and the derivative portal can link to a vast kind of DM models. Also there are lots of works can be done in the future: the possible UV completion model in Appendix A might be able to give mass to neutrino; the derivative portal might come from scalars in UV complete theory. And phenomenology studies like electroweak oblique parameters constraints, and collider search can be explored in the future.

Appendix A: Possible UV complete model
One possible UV complete model can be written as Where L could be the SM lepton or an extra fermion doublet, and N R is a right-handed "neutrino" that will give mass to either ψ L or ν L , which is the neutral component of L. And the covariant derivatives are given by: Where g χ and n χ are the gauge coupling and the quantum number of χ. After H and Φ get their vacuum expectation value v H and v Φ , we can write the mass matrix of ν L , N R and ψ L as: In these three particles the Z boson couples to ν L and the Z boson couples to ψ L , therefore after diagonalizing these three particles to mass eigenstates, they all couple to Z and Z boson simultaneously. Thus these particles can generate the kinetic mixing between Z and Z boson through one loop corrections. While the kinetic mixing between Photon and Z can be generated through two loop corrections. If we assume the kinetic mixing and its leading loop corrections are in the same order of magnitude, then the kinetic mixing between Photon and the extra vector boson will be naturally negligible. Alternatively, we can set Z to be a member of a multiplet, then there will be no tree level kinetic mixing between Photon and Z boson.
Appendix B: Proof of cancellation mechanism in mass eigenstates In the proof of cancellation mechanism in the DPDM model, the mass term of the DM is irrelevant. Therefore the relevant Lagrangian reads: After the following transformation the Lagrangian becomes Define k 1 = 1/ √ 2 − 2 and k 2 = 1/ √ 2 + 2 , then the Lagrangian will be simplified to: And the mass matrix of the vector mediators can be written as: Where we have defined M 1 = (m 2 Z + m 2 Z ), M 2 = (m 2 Z − m 2 Z ), and O is an orthogonal matrix that diagonalizes the mass matrix which can be defined as: and tan 2θ can be calculated as: After diagonalization, the Lagrangian becomes +g χ ((k 1 cos θ + k 2 sin θ)Ẑ µ + (k 2 cos θ − k 1 sin θ)Ẑ µ )χγ µ χ Knowing that the scattering amplitude is proportional to the coupling factors and the propagators, we can write the scattering amplitude between SM fermions and DM as: iM ∝ (k 2 sin θ − k 1 cos θ)(k 1 cos θ + k 2 sin θ) t − m 2 Z + (k 1 sin θ + k 2 cos θ)(k 2 cos θ − k 1 sin θ) t − m 2 .
If the amplitude is proportional to the momentum transfer t, then it goes to zero in the zero momentum transfer limit. Thus the cancellation is valid when the last line of the above equation equals to zero. Which means: k 2 2 sin 2 θ − k 2 1 cos 2 θ k 2 2 cos 2 θ − k 2 and we will prove this is right. From Eq. (14) we can write: −m 2 Z m 2 Z = k 1 k 2 M 2 cos 2 θ − k 2 2 M 1 sin θ cos θ k 1 k 2 M 2 sin 2 θ + k 2 2 M 1 sin θ cos θ .