Direct detections of Majorana dark matter in vector portal

In this paper we investigate the direct detections of Majorana dark matter (MDM) in vector portal. Considering that the tree-level scattering cross sections in these models are either dark matter velocity suppressed or spin-dependent, we calculate radiative corrections to the spin-independent cross section in effective field theory approach. Wilson coefficients of effective MDM-quark interactions are calculated at the one-loop level, and the Wilson coefficient of the effective MDM-gluon interaction is derived at the two-loop level. Numerical results show that current constraints can rule out a narrow mass range of MDM when tree-level contributions are considered, and the spin-independent cross section from radiative corrections is a few orders of magnitude smaller than the tree-level contributions.


Introduction
Various observations have confirmed the existence of dark matter in our universe, whose relic density, derived by measuring the cosmic microwave background, large scale structure and galaxy formation, is about 0.1198 ± 0.0033 [1]. The standard model (SM) of particle physics contains no cold dark matter candidate, and the nature of dark matter remains elusive. There are many dark matter candidates with masses ranging from 10 −20 eV to 10 55 GeV, of which the weakly interacting massive particle (WIMP) [2][3][4][5][6][7] is well-motivated as it can naturally explain the observed relic density via the thermal freeze-out with its mass at the electroweak scale and with weak couplings to the SM particles.
There are three (direct or indirect) ways of detecting WIMPs in laboratories: looking for the scattering between WIMPs and nucleon in underground laboratories by measuring the nuclear recoil energy in the kilo-electronvolt scale, detecting the flux of cosmic rays injected by the WIMP annihilations or decays with the help of satellites or telescopes, and producing WIMPs at the Large Hardron Collider (LHC) where the signal of WIMP is missing transverse momentum or missing energy. Of these three efforts, the first detection method is most straightforward since the astrophysical sources of cosmic rays have not been clearly determined in indirect detection experiments, and LHC is actually a mediator machine in dark matter detections.
Benefiting from technological advances, direct detection experiments such as LUX [8], PandaX-II [9] and XENON1T [10] have made tremendous strides in increasing precision and detecting efficiency. In an ideal status, one can detecting arbitrarily small direct detection cross section by continuously increasing the exposure, however it is well-known that direct detection experiments will soon reach to an irreducible background from coherent elastic neutrino-nuclei scattering, the so-called "neutrino floor" [11]. The current direct detection JHEP11(2019)013 techniques will not be able to distinguish the signal of dark matter from that of neutrinos if the signal lies below the neutrino floor. That is to say the neutrino floor is the border of new and "old" direct detection techniques [12]. As a result, the precision calculation of the direct detection cross sections will be important whenever one wants to examine as many dark matter models as possible with the help of current direct detection techniques.
In this paper, we study the direct detections (DD) of a vector portal Majorana dark matter (MDM) [13]. The vector mediator model is one of the simplest dark matter models, whose phenomenology has been widely studied in refs. [14][15][16][17][18][19][20][21][22][23][24]. In this model χ is a Majorana dark matter, V µ is a vector mediator, whose mass may arise from the spontaneous breaking of certain U(1) gauge symmetry, and V µ may couple to the SM via a vector current or axial-vector current. In some models the vector portal is in associated with the Higgs portal since the scalar that causes the spontaneous breaking of the new U(1) gauge symmetry may mix with the SM Higgs. Here we assume the mixing is negligibly small, thus the scattering of χ off the nuclei is only mediated by the V µ . The direct detection cross section σ is either spin-independent but suppressed by the dark matter velocity or spin-dependent [25,26]. As a result, the spin-independent cross section σ SI , generated at the loop level [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41], turns out to be important as it may be still possible to examine these models with the current DD technique if σ SI arising from radiative corrections lies above the neutrino floor. We calculate effective operators for the evaluation of MDM-nucleon spin independent scattering cross section following [42,43]. The WIMP-gluon effective operator [44,45] raising at the two-loop level is also derived. Numerical simulations show that the σ SI from radiative corrections is reachable by the current DD technique for light MDM. Considering the constraint of pertubativty, radiative corrections are several orders of magnitude smaller than the tree level contributions.
The remaining of the paper is organized as follows: in section 2 we give a brief introduction to the vector portal MDM model. Section 3 is focused on the calculation of Wilson coefficients of effective operators. Numerical results are presented in section 4 and the last part is concluding remarks. Expressions of loop functions are listed in the appendix A. Nuclear form factors are given in the appendix B.

The model
In this section, we review the vector portal dark matter model. The simplified model contains a Majorana fermion χ and a new vector boson V µ in addition to the SM particles. The Lagrangian for χ can be written as 1 where m χ is the dark matter mass, g V is the new gauge coupling. The ultraviolet completed model contains a U(1) gauge symmetry and a complex scalar Φ(≡ 1 √ 2 (φ + iG + v Φ )) which is charged under the new U(1) and whose vacuum expectation value (VEV) v Φ leads to JHEP11(2019)013 the spontaneous breaking of the new gauge symmetry as well as the origin of masses of χ and V . In this case there will be a new Yukawa interaction, 1 2 y χ χ C P Φχ P + h.c., where the subindex P represents the chirality, and eq. (2.1) needs to extended with the 1 2χ φχ term. In the case where φ mixies with the SM Higgs, there will be Higgs portal interactions. Notice that the mixing is caused by the quartic term: λ HΦ Φ † ΦH † H. λ HΦ should be small due to constraints from precision observables as well as Higgs measurements at the LHC. Here we assume the mixing is negligible.
For interactions of new gauge boson with the SM particles, V µ may couple to vector bilinears or axial-vector bilinears, or both, depending on the U(1) charge settings of the SM fermions. For example, new gauge interactions are g Vf γ µ f in U(1) B−L [46], U(1) B+L [47] and U(1) B i −L j models. While in the model where two chiral components carry opposite U(1) charges, new gauge interactions will be g Vf γ µ γ 5 f . If only one certain chirality component carries non-zero charge, new gauge interactions will be g Vf γ µ P A V µ f [48], where P A = P L or P R . In this paper we carry out model independent study and investigate the DD cross section of vector current and axial-vector current scenarios separately. Interactions of V µ with quarks are then where g q = ζg V with ζ the U(1) hyper-charge of the quark q. Free parameters in these models are thus m χ , m V , g V and ζ.

Effective operators
In this section we calculate effective operators relevant for the spin-independent scattering cross section of χ with a nucleon. Following refs. [42,43], we write down the effective χ-quark interactions in terms of the higher-dimensional operators: where Γ = γ µ for scenario A and Γ = γ µ γ 5 for scenario B, κ ip are wilson coefficients, These effective operators are defined at the mass scale of V µ , which is assumed to be heavier than all the SM particles. In the following, we calculate Wilson coefficients at the leading and the next-to-leading order. The leading order contribution to the Wilson coefficient κ 0 arises from the tree level diagram by exchanging the vector boson V µ , κ q 1,2,3 arise from the box diagrams given in the figure 1. We calculate the diagrams in the zero-momentum transfer limit, and expand the amplitude in term of the quark momentum, which is non-relativistic, then we decompose results into effective operators following ref. [38]. For the scenario A, the relevant Wilson coefficients are For scenario B the Wilson coefficients are The effective MDM-gluon interactions arise at the two-loop level. Relevant Feynman diagrams are given in the figure 2. In this paper we only take into account the effect of twist-0 MDM-gluon operator and neglecting the contribution of higher twist MDMgluon operators. One loop correction to the two-point function of gauge boson in the gluon background field has been calculated in ref. [49] by taking the Fock-Schwinger gauge [53,54] for the gluon field, which, mapped into our cases, can be written as where g s is the coupling of the strong interaction, f indicates the flavor running in the fermion loop, q is the momentum of gauge boson. Notice that eq. (3.10) is universal for both the scenario A and the scenario B. With the help of eq. (3.10), one can write down the Wilson coefficient of the effective χ-gluon operator, which takes the form

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where n f is the number of quarks that carry nonzero U(1) charge. Wilson coefficients given above are matched to the simplified model at the scale µ ≈ m V . The energy scale for the DM direct detections is about the nuclear energy scale. It has been shown in ref. [29] that effects from the running of renormalization group equations might be sizable in certain vector-portal scenario. We use the public code RUNDM [29] to evolve the running of κ 0 and evaluate running effects of other Wilson coefficients following refs. [45,[55][56][57][58].

Results
In this section we present results for the MDM-nucleon scattering cross section. We start by determining the g V using the observed relic density. As was mentioned in the previous section, there are four parameters in the vector portal, which are all relevant for both the relic abundance and the direct detection cross section. The thermal relic abundance is determined by the following processes:χχ →f f andχχ → V µ V µ , where f indicates the SM fermion. The first channel depends on the parameter ζ, while the second channel does not. For m χ < m V , the channelχχ → V µ V µ is kinematically forbidden, and thus the combination ζg 2 V can be determined in this mass range. We show in the figure 3, the new gauge coupling g V as the function of the dark matter mass m χ determined by the observed relic abundance Ωh 2 = 0.1198, by setting m V = 3 TeV and ζ = 1. The solid and dashed lines are the cases of scenario A and scenario B, respectively. The first dip of the plot appears at m χ ∼ m V /2, where the annihilation χχ →f f is resonantly enhanced. The second dip of the plot at m χ ∼ m V is due to the opening of the annihilation channelχχ → V µ V µ . Notice that g V becomes almost scenario independent as m χ ∼ m V . We show in the right panel of the   correspond to cases of scenario A and B, respectively. As can be seen, the Wilson coefficient of the scalar type interaction is largest and is comparable to that in pseudo-scalar portal model [38]. Wilson coefficients in scenario B is similar to these in scenario A except κ B2q at m χ ∼ m V , which is due to the cancellation of various contributions. The Wilson coefficient κ g is of the order O(10 −11 ) by setting g V = ζ = 1. Now we calculate the MDM-nucleon scattering cross section. For scenario A, there is velocity suppressed spin-independent scattering cross section at the leading order, where µ is the reduced mass of χ and nucleon system, v is the velocity of the dark matter. Effective interactions given in the eq. (3.1) contribute to the scattering cross section at the next-to-leading order, where N stands for (p, n) with m N its mass, f N Tq is the quark matrix element defined by are second moments for quark distribution functions of N . Numerical values of f N Tq , f N Tg as well as q N (2) andq N (2) are listed in the appendix B. Notice that there is no interference between the leading oder and the next to leading oder contributions. As a result, the total cross section for scenario A can be written as σ tot SI = σ LO SI + σ NLO SI . For scenario B, the χ-nucleon scattering cross section is spin-dependent at the leading order, where ∆ N q is the spin fraction of quark q, defined by 2∆ N q s µ = N |qγ µ γ 5 |N with s µ the nucleon spin four-vector, J N is the angular momentum of the nucleon. ∆ N q are measured in DIS and one has ∆ p u = 0.77, ∆ p d = −0.47 and ∆ p s = −0.15 [50]. The χ-nucleon scattering cross section at the next-to-leading order can be spin independent, whose expression is the same as eq. (4.2), up to following replacements, κ Aiq → κ Biq (i = 1, 2, 3).
As an illustration, we show in the figure 5 the direct detection cross section as the function of the dark matter mass m χ for scenario A, by setting m V = 3 TeV and ζ = 1. The red solid and blue dashed lines correspond to σ LO SI and σ NLO SI , respectively. The black dashed and green dashed lines are separately constraints given by the PandaX-II [9] and XENON1T [8] experiments. One can conclude from the plot that even though the σ LO SI is suppressed by the dark matter velocity it is still sizable and the current constraints have excluded the low dark matter mass region (6 GeV < m χ < 120 GeV). The next-to-leading contribution is reachable by the current DD technique for very light MDM, and it is about JHEP11(2019)013 5 orders of magnitude smaller than the LO contribution for heavy MDM. The reason that σ NLO being sensitive to light MDM is that σ NLO is proportional to As a result, a slight increase of g V may enhance the σ NLO obviously. The vertical dashed line in the plot is the constraint of perturbativity [64,65], to the left of which one has g V > √ 4π, that is excluded by the perturbative unitarity. We show in the left-panel of the figure 6 the spin-dependent χ-neutron scattering cross section as the function of m χ for scenario B, by setting ζ = 1 and m V = 3 TeV. The blue and green dotted lines are constraints of PandaX-II [59] and XENON1T [60] experiments, respectively. One can see that the mass region (8,90) GeV is already excluded. The vertical dashed line is the constraint of perturbative unitarity. We show in the right-panel of the figure 6 the spin-independent cross section as the function of m χ for scenario B. The solid and dashed lines correspond to m V = 3 TeV and 0.5 TeV, respectively. The black solid line is the neutrino floor. It is clear that the spin-independent cross section is too small to be detected in the near future for heavy MDM. Vertical lines are constraints of perturbativity, where the blue and red vertical lines correspond to m V = 3 TeV and 0.5 TeV, respectively.
Notice that we did not consider constraints of LHC [23,63,64] since we are only focusing on direct detections in underground laboratories, however constraints from LHC are definitely important when considering a specific vector portal MDM model. Direct search for the spin-1 mediator, especially the search for dijet resonances at the LHC, may put strong constraint on various vector portal models. After producing dijet events in a Monte Carlo generator, applying the cuts, and comparing the predicted dijet invariant mass distributions with the experimental results, one may get limits on a Z model. The resulting largest value of g q (at the 95% confidence level) depends on m V and Γ V , and takes the form [23] where a(3000 GeV) = 0.30428, b(3000 GeV) = 1.315 and c(3000 GeV) = 7.375 × 10 −4 [23].

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To constrain the dark matter coupling using the dijet results, one may first reconstruct the dark matter coupling using the observed relic abundance, which is the function of g p , m V and m χ . Then we get its lower bound from the combination of the observed relic density and LHC dijet results. Since the decay width Γ V is model dependent (there might be invisible decays of heavy vector boson) and eq. (4.4) is valid for Γ V /m V ≤ 0.3, we will not apply these constraints to our plots.

Conclusion
In this paper we have studied the direct detection cross section of Majorana dark matter χ in the vector portal. At the leading order, the cross section is either velocity suppressed or spin-dependent, and current constraints given by XENON1T and PandaX-II experiments can only rule out a narrow mass range depending on the inputs. Future direct detection experiments may improve the detection sensitivity to higher level. Next-to-leading order corrections may turn out to be important. We have derived the effective χ-quark interactions at the one-loop level and the effective χ-gluon interaction at the two-loop level. We got the full formula of SI direct detection cross section at the next-to-leading order. Our numerical results show that the next-to-leading order corrections to the SI cross section is several orders of magnitude smaller than the leading order contribution for the heavy mediator case.

JHEP11(2019)013 B Nuclear form factor
To calculate the WIMP-nucleon scattering cross section, one needs following nuclear form factors: N |m qq q|N = m N f N Tq , (q = u, d, s), N |− 9αs 8π G a µν G aµν |N = m N f N Tg , N |O q µν |N = 1 m N (p N µ p N ν − 1 4 m 2 N g µν )(q N (2) +q N (2)), where m N is nucleon mass, f N Tq and f N Tg are form factors taken from micrOmegas [61], q N (2) andq N (2) are the second momentum for quark distribution functions evaluated at µ = m Z by using CTEQ PDF [62]. Specific inputs are [61]