Simplified Dirac Dark Matter Models and Gamma-Ray Lines
We investigate simplified dark matter models where the dark matter candidate is a Dirac fermion charged only under a new gauge symmetry. In this context one can understand dynamically the stability of the dark matter candidate and the annihilation through the new gauge boson is not velocity suppressed. We present the simplest Dirac dark matter model charged under the local gauge symmetry. We discuss in great detail the theoretical predictions for the annihilation into two photons, into the Standard Model Higgs and a photon, and into the gauge boson and a photon. Our analytical results can be used for any Dirac dark matter model charged under an Abelian gauge symmetry. The numerical results are shown in the dark matter model. We discuss the correlation between the constraints on the model from collider searches and dark matter experiments.
- I Introduction
- II Simplified Models
- III Gamma-Ray Lines
- IV Summary
- A Experimental Constraints on the Annihilation Cross Sections
- B Loop Functions
The existence of dark matter (DM) in the Universe has motivated the particle physics community to investigate extensions of the Standard Model (SM) of particle physics, with countless dark matter candidates on the market which need to be probed against experiments. There are basically three ways to search for these candidates: one looks for possible signals in dark matter direct detection experiments, for gamma-ray lines and other signals from dark matter annihilation in indirect detection experiments, and for large missing energy signatures together with a mono-jet or a mono-photon at colliders. See Refs. Jungman:1995df (); Bergstrom:2000pn (); Feng:2010gw (); Bringmann:2012ez () for a detailed discussion of these possibilities.
Most of the effort so far has been focused on studies of complete models and their dark matter candidates, as for example the neutralino in the Minimal Supersymmetric Standard Model. Also effective field theory (EFT) has been used to constrain the scale of the new physics, particularly using the data from the LHC run one. However, given the large center-of-mass energy of the LHC in its second run, it is obvious that the EFT approach is prone to fail in large fractions of parameter space. Therefore, it is sensible to consider simplified dark matter models which can capture the main features of the dark matter sector with a limited number of parameters. For a discussion of simplified models see Refs. Abdallah:2015ter (); Buchmueller:2014yoa ().
A simplified model should contain a mediator and a dark matter candidate, and it should not violate generically any low energy observables. These criteria are necessary but not sufficient, since a naive model can result in misleading statements. We argue that the simplified model must be self consistent, i.e., not violate gauge invariance and be anomaly free. Only then is it possible to perform a full study and compute for example higher order processes leading to gamma-ray lines relevant for indirect dark matter searches.
In this article we investigate in detail simplified models for Dirac dark matter, which have the following features:
The dark matter is charged under a local gauge symmetry. This symmetry can be spontaneously broken at the low scale and a remnant discrete symmetry guarantees the dark matter stability. In the simplest case one has a local symmetry broken to a discrete symmetry. In the case of an unbroken gauge symmetry one can use the Stueckelberg mechanism and the dark matter stability is ensured by the choice of quantum numbers.
One can generate mass for the new gauge boson using the Stueckelberg or the Higgs mechanism. In the Stueckelberg scenario the dark matter candidate interacts only with the new neutral gauge boson in the theory. However, if one uses the Higgs mechanism the dark matter could also have interactions with the new physical Higgs boson.
The existence of a new gauge boson is key for the testability of the mechanism for dark matter stability. Therefore, in the ideal case the new gauge boson should define all the properties of the dark matter candidate. In particular, the main annihilation channels must proceed through the interaction between the dark matter and the new gauge boson. One can show that in this case the dark matter annihilation through the new gauge boson is not velocity suppressed.
For a Dirac dark matter, we investigate in detail the relic density constraints, the predictions for direct detection and the dark matter annihilation channels producing monochromatic photons, . We compute the one-loop generated vertices , and needed for the annihilation cross sections. We discuss all the technical details for the computation of the loop graphs and we stress the need to check the Ward and Slavnov–Taylor identities to make sure the final results are correct. We point out that the effective coupling is possible only in models where the charged fermions inside the loop have an axial coupling to the . The effective couplings and are always present when the Standard Model fermions are charged under the new gauge symmetry. Our results can be used for the study of the gamma-ray lines in any theory with Dirac dark matter charged under a new gauge symmetry.
In order to illustrate the main results, we discuss the simplest possible self-consistent dark matter model which is most relevant for studying the connection between direct and indirect dark matter searches, as its annihilation cross section is not velocity suppressed. In this model, dark matter is charged under the gauge symmetry and one has only two annihilation channels into photons, . We discuss the parameter space for direct detection in agreement with the relic density and collider constraints, and we show the experimental limits on the indirect searches for the gamma-ray lines and . We show that the interplay between the relic density, collider searches and indirect dark matter detection experiments sets non-trivial bounds on these simplified models.
Ii Simplified Models
We discuss models where the dark matter is a Dirac fermion charged only under a new gauge force. In this context we can understand why the dark matter is stable. For simplicity, we consider the case where one has an Abelian force, i.e., a . The part of the Lagrangian relevant for our discussion is
Here we neglect the kinetic mixing between the new Abelian symmetry and . For the moment we do not discuss the anomaly cancellation and how the masses are generated but will address these issues later in a well-motivated model. Notice that in general the Standard Model fermions can be charged under the new symmetry such that new fermions are needed for anomaly cancellation.
In these models the relevant interaction of the dark matter candidate to the gauge boson is given by
where we use the standard projection operators, and . The interactions of all other fermions in the theory, the SM fermions or new fermions needed for anomaly cancellation, to the can be parametrized as
As usual, all charged fermions couple to the photon according to their electric charge ,
and the coupling of the fermions to the Standard Model can be parametrized as
where is the gauge coupling and is the Weinberg angle.
ii.1 Dirac Dark Matter
The local symmetry is anomaly free once we add three copies of right-handed neutrinos to the Standard Model particle content. It is well known that this symmetry could play a major role in neutrino physics. Here we focus on a very simple model with Dirac dark matter charged under . The relevant part of the Lagrangian is given by
where , , and . One can generate the gauge boson mass through the Higgs mechanism or the Stueckelberg mechanism. Let us discuss both cases here:
Stueckelberg Mechanism: The mass of the gauge boson can be generated through the Stueckelberg mechanism as discussed in Ref. Nath ():
where the gauge transformations are given by
In this case the neutrinos are Dirac fermions because the symmetry is never broken, and the dark matter stability is a result of the choice of the quantum number for the dark matter candidate.
Higgs Mechanism: One can generate the mass for the gauge boson through the Higgs mechanism and at the same time we can generate masses for the SM neutrinos through the see-saw mechanism TypeI-1 (); TypeI-2 (); TypeI-3 (); TypeI-4 (); TypeI-5 () via the following interactions:
Here is a Standard Model singlet and has charge two. Notice that if the charge of the new Higgs is different from two, the neutrinos will be Dirac fermions. After the is broken, there is a remnant symmetry which is the reason for the dark matter stability.
These simple models have only four relevant parameters for the dark matter study: the gauge coupling , the dark matter mass , the gauge boson mass , and the charge of the dark matter candidate. The relevant interactions needed to compute the dark matter annihilation channels are
where is the charge of the Standard Model fermion .
ii.1.1 Relic Density
The dark matter candidate can annihilate into all the Standard Model particles and the gauge boson . Therefore, one can have the annihilation channels
in both the Stueckelberg and the Higgs scenario discussed above. In the Higgs scenario, one has the additional annihilation to right-handed neutrinos. There are two main regimes for our study:
: when the dark matter candidate is lighter than the gauge boson, we have the following channels,
: when the dark matter candidate is heavier than the gauge boson, one has a new open channel which is not velocity suppressed,
In order to test this model at the collider, the invisible decay is crucial to establish the connection between the existence of the new gauge boson and the dark matter candidate. Therefore, we focus on the first regime.
The annihilation cross section for is given by
Here is the color factor of the fermion with mass , is the square of the center-of-mass energy, and is the total decay width of the gauge boson. In order to compute the relic density we use the analytic approximation Gondolo:1990dk ()
where is the Planck scale, is the total number of effective relativistic degrees of freedom at the time of freeze-out, and the function reads as
The thermally averaged annihilation cross section times velocity is a function of , and is given by
where and are the modified Bessel functions. The freeze-out parameter can be computed using
where is the number of degrees of freedom of the dark matter particle. In Fig. 1 we show the numerical predictions for the relic density vs. the dark matter mass for two values of . In the left (right) panel we show the results for when , which is in agreement with the collider bounds Carena (). As expected, for small values of the gauge coupling one needs to rely on the resonance to achieve the right relic density. However, generically one can be far from the resonance and in agreement with relic density constraints.
ii.1.2 Direct Detection
The direct detection constraints must be considered in order to understand which are the allowed values of the input parameters in this theory. The elastic spin-independent nucleon–dark matter cross section is given by
where is the nucleon mass. Notice that is independent of the matrix elements. The cross section can be rewritten as
where is the reduced mass and .
In our case , and using the collider lower bound Carena () one finds an upper bound on the elastic spin-independent nucleon–dark matter cross section given by
for a given value of . There is also a simple way to find a lower bound on the spin-independent cross section. The minimal value of the gauge coupling in agreement with relic density constraints corresponds to the case when one sits on the resonance, i.e., . Therefore, the lower bound on the cross section for a given value of the dark matter mass is given by
In Fig. 2 we show the numerical predictions for the direct detection cross section vs. the dark matter mass compatible with the relic density constraints. The colored dashed lines show the values of for different choices of compatible with current collider limits. The black dash-dotted line shows the minimal direct detection cross section. We show the bounds from the LUX Akerib:2013tjd () and XENON100 experiments Aprile:2012nq (), as well as the prospects for XENON1T Aprile:2012zx (). One can see that for , the scenario for is allowed by the LUX experiment for a large part of the parameter space. However, for the case , the ratio needs to be larger than 20 in order to satisfy the experimental bounds. Therefore, only the scenario when could be tested at the LHC. Unfortunately, the minimal value of the cross section is below the neutrino background Billard:2013qya () and it is very difficult to test this part of the parameter space in the current direct detection experiments.
ii.2 Upper Bound on the Dark Matter Mass
In this section we show that in these simple models it is possible to derive an upper bound on the dark matter mass. We focus on the case when the is heavier than the dark matter because only then one can test the main properties of these models. The argument is based on the observational requirement that the relic density of dark matter produced in the freeze-out must not overclose the Universe. Today, we know that and since the relic density scales as , one has a lower bound on the annihilation cross section
Here is the minimal value of the cross section compatible with observations. In order to guarantee the validity of a given theory we have to make sure that the maximal value of the cross section in the theory obeys the condition . This is a necessary condition, since if it is not fulfilled there is no parameter choice in the model which can make it compatible with observations.
The corresponding cross section has the following structure,
It is obvious that the maximum of this expression given a fixed value of is realized when . Examining the functional dependence of the obtained expression one finds that
In the model with local and , this leads to an upper bound on the mass of
Note that this bound is conservative because at the resonance one would need to perform the full average as discussed in Sec. II.1.1. This bound is useful to understand the possibility to test this type of model.
Iii Gamma-Ray Lines
In this section, we discuss the predictions for gamma-ray lines in detail. First, we give the general results for any simplified model with an Abelian gauge symmetry and a corresponding , then we move on to study numerically the predictions for the minimal model discussed before.
iii.1 Loop-Induced Couplings
coupling: The coupling is generated by a loop of electrically charged fermions charged also under , see Fig. 3, and is given by
Since we are interested in processes with two external photons, one has and the terms in the second line of Eq. (III.1) proportional to do not contribute to the amplitude. The relevant coefficient functions are given by
Notice that this coupling can be generated only if the axial coupling of the fermions in the loop to the is different from zero. See Appendix B for the explicit form of the loop functions and . In the above equations , is the fermion mass, and is the color factor and the electric charge of the fermion. We checked that the Ward identities are satisfied.
coupling: In Fig. 4 we show the coupling generated by a top loop. The other Standard Model fermions have smaller Yukawa couplings and their contributions are therefore negligible. For the top quark , and the coupling is given by
Here is the vacuum expectation value of the Standard Model Higgs. Notice that for processes with an external photon, the term does not contribute to the amplitude. See Appendix B for the explicit form of the loop functions. For this vertex, one can show that the Ward identity is satisfied, .
coupling: The coupling between the , the photon, and the can be generated at one-loop level as shown in Fig. 5. This coupling can be written as
For processes with external and only, the terms in the second line of Eq. (III.1) proportional to and do not contribute and the relevant functions are given by
(34) (35) (36) (37) (38)
Here the coefficients and are given by
Using the Ward identity one can write as a function of :
which is the generalization of the Ward identity. Here, is the Goldstone boson. From this identity, one finds the relation
We have checked that these identities are satisfied in our calculations. Since for our case we find , one can write
It can easily be checked that the following additional relation between the coefficients holds,
These relations are very useful to cross-check the results and simplify the final expressions for the cross sections. We have used Package-X Hiren () to perform all one-loop calculations and have cross-checked the results.
Using the above calculations for the loop-induced couplings, we compute the dark matter annihilation cross sections for the different channels:
: The amplitude for the dark matter annihilation into two photons is given by
The cross section times velocity for this channel in the non-relativistic limit is given by
: For the dark matter annihilation into the SM Higgs and a photon one finds the amplitude
where is the angle between and in the center-of-mass system. The corresponding annihilation cross section is given by
: In general, the explicit form of the amplitude for the dark matter annihilation into a and a photon is very involved and cannot be given here. Here we list the result for where the integrated amplitude is given by
while the cross section is given by
Notice that these results are very general and can be used for any dark matter model with the features discussed above. For simplicity, we show the results for the only for a vector coupling of the to dark matter. In Refs. Jackson:2009kg (); Jackson:2013pjq (); Jackson:2013tca (), some of these effective couplings have been computed and the implications for dark matter models have been investigated in detail.
iii.2 Gamma-Ray Lines and Symmetry
In the simple dark matter model one has only vector couplings to the gauge boson and there are only two relevant channels for the annihilation into gamma-ray lines:
The energy of the line signal for the process is given by
where is the mass of the particle . The cross section for the dark matter annihilation into the Standard Model Higgs and a photon is given by