Majorana Dark matter with B+L gauge symmetry
Abstract
We present a new model that extends the Standard Model (SM) with the local symmetry, and point out that the lightest new fermion , introduced to cancel anomalies and stabilized automatically by the symmetry, can serve as the cold dark matter candidate. We study constraints on the model from Higgs measurements, electroweak precision measurements as well as the relic density and direct detections of the dark matter. Numerical results reveal that the pseudovector coupling of with and the Yukawa coupling with the SM Higgs are highly constrained by the latest results of LUX, while there are viable parameter space that could satisfy all the constraints and give testable predictions.
I Introduction
The Standard Model (SM) of particle physics is in spectacular agreement with almost all experiments, but there are still observations that are not accessible solely by the SM. Accumulating evidences point to the existence of dark matter, which is neutral, colorless, stable and weakly interacting particle that accounts for about 25% Ade:2015xua () of the matter in the Universe. We know little about the nature of dark matter i.e. its mass, spin and stability as well as how it interacts with the SM particles. To accommodate the dark matter, SM has to be extended with new particle and new symmetry.
The baryon number () and the lepton number () are accidental global symmetries in the SM. must be violated to explain the baryon asymmetry of the Universe. will be violated if active neutrinos are Majorana particles, which may be tested in neutrinoless double beta decay experiments. It was showed that and FileviezPerez:2010gw (); FileviezPerez:2011pt (); Duerr:2013dza (); Arnold:2013qja (); Fornal:2015vsa () as well as Mohapatra:1980qe () can be local gauge symmetries. The case of as a local gauge symmetry was originally pointed out in Refs. Chao:2010mp (); Chao:2015nsm (), but its phenomenology was not studied in detail. Strong motivations for the local symmetry include

The 750 GeV diphoton excess observed by the ATLAS and CMS collaborations at the run2 LHC: It was showed that Chao:2015nsm () the diphoton resonance can be the fundamental scalar that breaks the local spontaneously. can be produced at the LHC via gluon fusion and then decays into diphoton with charged fermions, which were introduced to cancel anomalies, running in the loop.^{1}^{1}1For the explanation of the diphoton excess with new symmetry, flavored dark matter, neutrinos, leftright model, grand unified theory and baryon asymmetry of the Universe, we refer the reader to Refs. Chao:2015nsm (); Chao:2015ttq (); Chao:2015nac (); Chao:2016mtn (); Chao:2016aer (); Dev:2015vjd () and references cited in these papers for details.

Dark matter: As will be shown in this paper, the lightest extra fermion, introduced to cancel anomalies, is automatically stabilized by the symmetry and can naturally serve as the cold dark matter candidate.
In this paper we investigate the symmetry based on the dark matter motiviation. This scenario is interesting and economical since one does not need to introduce extra symmetry to stabilize the dark matter. We study constraints on the model from Higgs measurements and electroweak precision measurements, and then focus on the phenomenology of the Majorana dark matter. We search for the parameter space that may accommodate both the observed dark matter relic density and constraints of direct detections. Our results show that

The gauge boson mainly contributes to the annihilation of the dark matter , and the process dominates the annihilation of heavy . Its contribution to the direct detection cross section is suppressed by the velocity of the dark matter.

The pseudovector coupling with is suppressed by the latest LUX result on the spindependent cross section, and the upper limit on this coupling is about 0.058.

Yukawa couplings of with Higgs are suppressed by the LUX 2015 result on spinindependent direct detection cross section, while there are adequate parameter space that may satisfy all constraints.
It should be mentioned that the is a brandnew symmetry that deserves further detailed study in many aspects, such as the collider signature, neutrino masses, baryon asymmetry and sphaleron etc, which, interesting but beyond the reach of this paper, will be shown in the followup paper.
The remaining of the paper is organized as follows: In section II we briefly describe our model. We study constraints of Higgs measurements and oblique parameters in section III. Section IV is devoted to the study of the dark matter phenomenology. The last part is the concluding remarks.
Ii The model
When the SM is extended by the local symmetry, anomalies are not automatically cancelled as in the minimal SM. One simple way out is to introduce, in addition to the righthanded neutrinos, extra vectorlike fermions at the TeV scale so as to cancel various anomalies. To break the gauge symmetry via the Higgs mechanism, we introduce a singlet scalar with charge of . All the particle contents and their quantum numbers under the gauge group are listed in Table 1. It is easy to check that all potential anomalies are canceled in this simple framework, i.e. , , , , and . We refer the reader to Refs. Chao:2010mp (); Chao:2015nsm () for the details of anomaly cancellation.
SM particles  BSM particles  

3  
3  
3  
1  3  
1  3  
1  3  
0  6 
The most general Higgs potential takes the following form:
(1) 
where is the SM Higgs with its vacuum expectation value (VEV), and with the VEV of . After imposing the minimization conditions, one has and . Due to the last term in Eq. (1), is mixed with to form mass eigenstates , and the relations between mass eigenstates and interaction eigenstates take the following form,
(2) 
where and , with the mixing angle that diagonalizes the CPeven scalar mass matrix. The parameters , and in the potential can be reconstructed by the physical parameters , , , and as:
(3) 
Trilinear scalar interactions are listed in Table. 3. After the spontaneous breaking of the , the bosons obtain its mass:
(4) 
where is the gauge coupling of .
Due to their special charge new fermions do not couple directly to the SM fermion. New Yukawa interactions can be written as
(5)  
where and we have neglected Yukawa interactions of charged fermions. One might write down the mass matrix of neutral fermions in the basis of :
(6) 
where the mass matrix is symmetric. can be diagonalized by a unitary transformation: , where . Yukawa couplings in Eq. (6) can then be reconstructed by the mass eigenvalues and mixing angles, which are collected in Table 2. In this case the relation between interaction eigenstates and mass eigenstates can be written as
(7) 
We identify the fermion as the cold dark matter candidate and define the Majorana field as . Interactions of with the mediators , , and can be written as
(8) 
where and are given in interaction eigenstates. When studying the phenomenology of the dark matter, e.g. the relic abundance and direct detection, they need to be rotated to the mass eigenstates and the corresponding couplings (for and respectively) turn to:
(9) 
Iii Constraints
Before proceeding with the dark matter phenomenology, we study first constraints on the model from Higgs measurements as well as oblique parameters. The mixing angle between the two scalars and is constrained by the data from Higgs measurements at the LHC. Performing a universal Higgs fit Giardino:2013bma () to the data of ATLAS and CMS collaborations, one has at the 95% confidence level (CL) Chao:2015nsm (); Chao:2016vfq (), which is slightly stronger than the result of global fit preformed in Ref. Profumo:2014opa ().
A heavy with SM couplings to fermions was searched at the LHC in the dilepton channel, which is excluded at the 95% CL for Aad:2014cka () and for Khachatryan:2014fba (). Considering the perturbativity and RG running constraints on the gauge coupling (see the discussions below), the lower limit on might imply a lower bound on the . Phenomenological constraints also require the mixing angle to be less than Erler:2009jh (). In our model mixes with only through loop effect. This constraint can easily be satisfied, and we refer the reader to Ref. Chao:2012pt () for the calculation of the mixing angle in detail.
The function of can be written as
(10) 
which is very different from the function of : . Thus one may distinguish from by studying the running behavior of the gauge coupling. It should be mentioned that the function can be modified by changing the representation of extra fermions, see for instance Dev:2015vjd (); Mohapatra:2014qva (). There is also constraint on from perturbativity. A naive assumption of at the results in . With a looser constraint on the value of at the Planck scale , is allowed to take a larger value at the TeV scale.
We consider further the constraint from oblique observables Peskin:1991sw (); Peskin:1990zt (), which are defined in terms of contributions to the vacuum polarizations of the SM gauge bosons. One can derive the following formulae of and using gauge boson self energies , and as given in Ref. Peskin:1991sw (),
(11) 
In our model there are two separate contributions to the oblique parameters: the scalar sector and the new fermions. The dependence of and parameters on the new scalars can be approximately written as Grimus:2008nb ()
(12)  
(13)  
where is the mixing matrix of the CPeven scalar mass matrix, with the weak mixing angle, and the expression of is given in Chao:2016cea (). The contribution of vector like fermions to the oblique parameters are a little bit complicated. If we work in the basis wherein the charged heavy fermions are in their mass eigenstates, then the lefthanded and righthanded mixing matrices diagonalizing the charged fermion mass matrix are diagonal, then the expressions can be simplified to Joglekar:2012vc ()
(14)  
(15)  
The expressions of can be found in Ref. Joglekar:2012vc ().
We show in Fig. 1 corrections to oblique observables in the plane, where we set , taken from the universal fit to the data of Higgs measurements at the LHC, and set the largest mass splitting between the heavier neutral(charged) fermions and the dark matter to be 200 GeV. The dashed and solid red curves correspond respectively to the contours at the 68% and 95% C.L., which comes from the recent electroweak fit to the oblique parameters performed by the Gfitter group Baak:2012kk (). Obviously in a large parameter space of our model, the constraint of oblique observables can be satisfied.
Iv dark matter
The fact that about of the Universe is made of dark matter has been established. The weakly interacting massive particle (WIMP) is a promising dark matter candidate, since it can naturally get the observed relic density for a WIMP with mass around and interaction strength with SM particles similar to that of the weak nuclear force. In this section we take as the WIMP and study its implications in relic abundance and direct detections^{2}^{2}2For the indirect detection signal of this kind of dark matter, we refer the reader to Ref. Duerr:2015vna () for detail.. Due to its special charge, is automatically stabilized by the , whose interactions are given in Eq.(8). The phenomenology of is a little similar to that of the dark matter with the symmetry, which was wellstudied in many references Basso:2012gz (); Guo:2015lxa (); Baek:2013fsa (); SanchezVega:2015qva (); Rodejohann:2015lca (); ElZant:2013nta (); Li:2010rb (); Okada:2010wd (); Burell:2011wh (); Okada:2016gsh (). Briefly speaking the thermal dark matter is in the thermal equilibrium at the early Universe and freezes out as the temperature drops down. The Boltzmann equation, governing the evolution of the dark matter density , can be written as Bertone:2004pz ()
(16) 
where is the Hubble constant, is the thermal average of reduced annihilation cross sections.
One can approximate with the nonrelativistic expansion: and the contributions from various channels are
(17)  
(20)  
(21) 
where , , ; (for ) and (for ); the trilinear couplings are given in Table. 3; , with , , being the electric charge, weak mixing angle and the third component of the isospin respectively. If the mediator is close to its mass shell, one needs to do the replacement . Notice that Eq. (21) is simplified by neglecting terms proportional to , which is lengthy, but we keep them in numerical calculations.
Given these results, the final relic density can be written as
(22) 
where is the planck mass, , with the freezeout temperature, is the degree of the freedom at . We show in Fig. 2 the scattering plot of as the function of the dark matter mass by fixing and TeV. For the sake of clarity, we set the widths = 1 GeV and GeV. We work in the basis where the mass matrix of heavy charged fermions is diagonal, while the masses of heavy neutral fermions (including the dark matter candidate ) and the mixing angles among them are random parameters. The VEV also varies from 1.5 TeV to 3 TeV, which renders that the gauge coupling goes from 0.17 to 0.33 via equation (4). Note that can not be too small, or some of the Yukawa couplings are pushed to be unacceptably large by , c.f. Table 2. In this plot the constraints of oblique parameters are also taken into consideration, at the 95% C.L.. The horizontal line is the observed relic density value of . The first and second valleys around GeV and TeV come respectively from the resonance enhancement of and to the annihilation cross section.
The particle dark matter can be tested directly via scattering on target nuclei. In our model the dark matter has both spindependent (SI) and spinindependent (SD) scattering with nuclei mediated by scalars () and respectively. The effective Lagrangian for the scalar interactions can be written as
(23) 
It leads to the following expression for the cross section of a Majorana dark matter particle at the zeromomentum transfer,
(24) 
where is the reduced mass of WIMPnucleus system, . One has , , , , and Ellis:2000ds ().
We show in the left panel of Fig. 3 the scattering plot of the SI cross section as the function of for the general case, where inputs are given as , , and , with the mixing angles in . For each point in the plot one has , while the oblique parameters and lie in the contour as shown in Figure 1. The magenta, green and blue points correspond to cases where , and final states dominate the annihilation of respectively. The red solid line is the exclusion limit of the LUX 2015 Akerib:2015rjg (). We show in the right panel of Fig. 3 the as the function of for a simplified case, where are decoupled from the singlets . We set and when making the plot, while and are free parameters. All the points in the plot give a relic density within the deviation from the observed central value of 0.1197. One can conclude from the plot that this scenario is available only for , where the annihilation cross section is resonantly enhanced, and for where new annihilation channel is open.
The effective Lagrangian for the axialvector interaction, which is relevant to the SD scattering of DM from nuclei, can be written as
(25) 
There are also effective interaction of the form: . It turns out that the corresponding matrix elements are suppressed by the tiny dark matter velocity, whose contribution to the direct detection is thus negligible considering . The expression for the SD cross section for the Majorana particle takes the form
(26) 
The value of depends on the nucleus. It reduces to , for scattering off free proton(neutron). is the total angular momentum quantum number of the nucleus, which equals to for free nucleons.
In Fig. 4 we show constraint on from the latest spindependent WIMPnucleon cross section limits given by the LUX experiment Akerib:2016lao (). The solid line is the constraint of neutron, while the dotted line is the constraint of proton. Since the majority of nuclear spins are carried by the unpaired neutron, the neutron sensitivity is much higher than the proton case. The smallest WIMPneutron cross section is at Akerib:2016lao (). It corresponds to an upper limit of 0.058 on the coupling , which, when translated to constraint on mixing matrix elements, gives at the 90% confidence level. It is a quite loose constraint, while future measurement of from LUXZEPLIN may improve the current limit by a factor Akerib:2015cja ().
V Conclusion
In this paper, we extended the SM with a local symmetry and showed that the lightest extra fermion, which was introduced to cancel anomalies, can serve as a cold dark matter candidate. Constraints on the model from Higgs measurements and electroweak precision measurements were studied. Further applying these constraints to the dark matter, we searched for available parameter space that can give the correct relic density and satisfy the constraints of spinindependent and spindependent direct detections in the meanwhile. The model possesses adequate parameter space that satisfies all constraints. This model is complementary to the extension of the SM, and deserves further study on either the model itself or the collider phenomenology. It will be also interesting to investigate the baryon asymmetry of the Universe in this model, which, although interesting but beyond the reach of this paper, will be shown in the future study.
Acknowledgements.
This work of W.C. and H.G. were supported in part by DOE Grant DESC0011095. H.G. was also supported by the China Scholarship Council. Y.Z. would like to thank the IISN and Belgian Science Policy (IAP VII/37) for support.References
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