One–loop corrections for Higgs–portal dark matter
Models endowed with Higgs portals can probe into the hidden sectors of particle physics while providing stable dark matter candidates. Previous tree–level computations in such scenarios have shown that experimental bounds constrain dark matter to a very narrow region in parameter space. Aiming at improving the study of the implications of those constraints, we inspect one–loop corrections to the annihilation cross section for scalar dark matter into observable fermions. We find that these loop contributions might be enough to drastically change those results by deforming in about 10% the allowed parameter space for dark matter particles with masses even below 1 TeV. These findings encourage further investigation.
Since the early days of cosmology , the quest for an understanding of dark matter (DM) and its origin has led to several proposals. These include massive astrophysical compact halo objects (MACHOs) , weakly interacting massive particles (WIMPs)  and axion-like particles (ALPs) . WIMPs have been considered as optimal DM candidates mainly because of their natural appearance in many models that could explain other longstanding issues of particle physics. Their theoretical appeal has triggered an exhaustive experimental search by, among others, the XENON100 and LUX collaborations, that has resulted in severe constraints on the WIMP parameter space [5, 6]. Verifying that existing models comply with these strict bounds requires now that all their predicted observables, such as the production and decay rates, be computed with maximal precision.
On the other hand, the recent confirmation of the existence of the Higgs particle with a relatively large mass opens up a new set of possibilities. Especially, WIMPs might well couple directly to the Higgs field, establishing a connection known as a Higgs portal  between the dark sector and the Standard Model (SM). If there were such couplings, WIMPs would e.g. decay into visible matter via the exchange of a Higgs boson, affecting direct and indirect detection signals.
Beyond DM, Higgs portals may be relevant to address other intriguing questions, such as the nature of cosmological inflation , the instability in the electroweak vacuum , and more recently, the 750 GeV diphoton excess reported at the LHC . Despite their great potential, in this work we shall only focus on the features of DM in these models.
The additional particles required in Higgs portals are considered to be hidden in the sense that they do not carry SM charges. They are dynamical (real) scalars , spinors or vectors , confined to interact with the SM via Lagrangian couplings with the Higgs field , such as
and affected by their own (in principle, arbitrary) potential energy density. The scalar case clearly requires the inclusion of a symmetry (or another similar mechanism) to avoid tadpole contributions and to ensure DM stability. These different particles are good DM candidates as long as they comply with experimental bounds. In this respect, fermions are disfavored because they do not only yield a non–renormalizable model, but they are already excluded by XENON data when considering only thermal DM .
The phenomenological implications of Higgs portals with scalars and vectors as DM have been studied in [11, 12] (and a recent global analysis for the scalar case has been done in ), where only tree–level contributions to the decay and production rates have been considered. It was shown in those works that experimental bounds of WMAP  and XENON100  allow for a very reduced set of values in the parameter space of the models, so that radiative corrections may become important to falsify them. Although one–loop corrections to DM–nucleon scattering have been already analyzed , the corresponding study for DM annihilation processes is still missing. To motivate and to exhibit the importance of completing this task is one of the aims of the present work.
Based on the results obtained in , we discuss here the main aspects of Higgs portals and explore the importance of one–loop corrections to the cross sections of the processes associated with DM detection. The structure of this contribution is as follows. In section 2, we give a brief review of the scalar and vector Higgs portals; in section 3 we present the one–loop analysis for the cross section in the context of the scalar portal done with the aid of FeynArts  and FormCalc  and finally, in section 4 we discuss the obtained results and ongoing work.
2 Scalar and vector Higgs portals
The simplest example of these models is the scalar portal. It is achieved by introducing a (real) massive scalar field subject to a symmetry, under which is odd and all SM fields are even. The full renormalizable Lagrangian then reads
where , and are the portal and the two self–interaction real parameters. After electroweak symmetry breaking (EWSB), the Higgs–portal potential becomes
where is the physical mass of , denotes the electroweak vacuum expectation value (VEV) of the Higgs field , such that (and has vanishing VEV) in the vacuum, with the small Higgs field perturbation .
Similarly, for the vector Higgs portal we introduce a vector field , probably stemming from a U(1) gauge symmetry, whose Higgs–portal potential is given by
Since the mass term is not gauge invariant if arises from a gauge symmetry,111 may have a different origin. E.g. it could be a composite field, whose effective potential is eq. (4). some gauge–restoring mechanism, such as the Stueckelberg mechanism,222For a review on the Stueckelberg mechanism, see  must be invoked. After EWSB, the physical mass of the vector is given by .
Not only does EWSB generate the physical masses and of the DM particles, but, as eq. (3) evidences, it also “produces” the cubic interactions and . These interaction terms allow for the possibility of the invisible decay of the Higgs particle into DM. LHC bounds  on the associated decay width do not constrain DM with masses over few hundred GeV ; hence, this decay shall not be investigated in this work.
Additionally, as depicted in Fig. 1, the vertex allows for DM annihilation into SM fermions as well as DM–fermion scattering. The thermally–averaged cross section of the first process, (with referring either to or ), relates to Planck’s measurement of DM relic abundance , , roughly by
where denotes the DM relative velocity or Møller velocity. On the other hand, DM–fermion scattering is important for direct DM detection, because the spin–independent DM–nucleon scattering cross section can be computed from it under certain assumptions, allowing for a comparison with LUX and XENON100 data.
For scalar and vector Higgs portals, the newest version of micrOMEGAs  performs the numerical computation of up to one–loop level and compares it with the LUX and XENON100 bounds. Therefore, only the computation of radiative corrections to may be relevant if those contributions are sizable, as we shall show that they turn out to be.
At tree–level, the thermally–averaged annihilation cross sections are given by
for the scalar and vector case, respectively. In these eqs., is the mass of the fermions in the final state. Note that the cross section eq. (6) (eq. (7)) depends solely on and . It then follows that Planck’s data bound on leads to constraints on these parameters.
3 One–loop corrections in the scalar portal
Sizable one–loop contributions to DM cross sections in Higgs portals may become important as they may close further the narrow region of allowed parameters, thereby finding the model already falsified by data.
In a previous analysis , we studied the WIMP mass evolution of the ratio between the simplest one–loop amplitude correction and the tree–level amplitude for DM annihilation into quarks,
where, as before, stands for either or WIMP candidates. This ratio lets us estimate how subdominant one–loop corrections are. As it is evident from Fig. 2 (left), in the scalar case this ratio shows that, for DM masses around 1 TeV and larger, the simplest one–loop contribution analyzed can be as large as of the tree–level amplitude. This preliminary result encourages further study, since other one–loop contributions may enlarge this already sizable correction. In the vectorial case, this approach reveals (see Fig. 2 (right)) that one–loop contributions may not be so relevant. Thus, we can safely focus on the corrections to the scalar Higgs portal.
We have extended that previous analysis in two ways. First, we completed the computation of the one–loop cross section for annihilation into ’s, including all possible contributions. Secondly, we further studied the complete cross section at one–loop level for the annihilation of scalar DM into , and quarks and and leptons. The Feynman rules were generated with LanHEP ; the computation of amplitudes, one loop integrals and the cross sections have been performed by using the public numerical and analytical tools FeynArts , LoopTools and FormCalc .
Our results are displayed in Figs. 3 and 4. Fig. 3 shows the comparison of the tree–level and full up–to–one–loop cross sections for scalar DM annihilation into quarks and . We note that for TeV, this contribution is negative and significantly larger than , becoming as large as for scalar Higgs–portal WIMPs with TeV. A similar comparison has been done for DM annihilation into light quarks and heavy leptons (Fig. 4), to confirm our expectation that top contributions are dominant. For different values of , we find similar results.
4 Punchline and outlook
We have computed the one–loop corrections to the annihilation cross section of Higgs–portal DM into SM fermions. Our preliminary results stress the need to carefully determine the influence of radiative corrections in the computation of the cross sections in models with Higgs portals. We notice that, particularly for the scalar Higgs portal and certain parameter values, the one–loop contributions can be negative and as large as () of the tree level cross section for admissible DM with masses of about 1 TeV (2 TeV).
The resulting thermally averaged cross sections are functions of the two parameters of the Higgs portal, and , such that for each value of an interval of is still allowed by Planck’s data, according to eq. (5). Since the computed cross section is reduced by a sizable amount for heavy WIMPs, so is . Thus, the predicted relic abundance increases, stressing the tension between Higgs portals and measurements by WMAP, LUX and XENON100, when the one–loop corrections to the DM–nucleon scattering amplitude are additionally included. The detailed report of this analysis shall be presented elsewhere .
We are grateful for useful discussions with T. Hahn and A. Ibarra. This work was partly supported by CONACyT grant 151234. S. R-S. would like to thank the ICTP for the kind hospitality and support received through its Junior Associateship Scheme during the realization of this work.
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