Supported by the National Natural Science Foundation of China (NSFC) (11375277, 11410301005, 11647606, 11005163, 11775086, 11875327, 11805288), the Fundamental Research Funds for the Central Universities, the Natural Science Foundation of Guangdong Province (2016A030313313), and the Sun Yat-Sen University Science Foundation.

^{3}and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd

We confront the perturbativity problem in the real scalar quintuplet minimal dark matter model. In the original model, the quintuplet quartic self-coupling inevitably hits a Landau pole at a scale ~10^{14} GeV, far below the Planck scale. In order to push up this Landau pole scale, we extend the model with a fermionic quintuplet and three fermionic singlets which couple to the scalar quintuplet via Yukawa interactions. Involving such Yukawa interactions at a scale ~10^{10} GeV can not only keep all couplings perturbative up to the Planck scale, but can also explain the smallness of neutrino masses via the type-I seesaw mechanism. Furthermore, we identify the parameter regions favored by the condition that perturbativity and vacuum stability are both maintained up to the Planck scale.

Article funded by SCOAP^{3}

One of the biggest mysteries of nature, dark matter (DM) has drawn much attention from astrophysicists, cosmologists, and particle physicists. Among various guesses at the identity of the DM particle, the most extensively studied class of DM candidates is weakly interacting massive particles (WIMPs), because they can naturally explain the observed DM relic abundance via the thermal production mechanism in the early Universe [_{L} multiplets. Introducing one nontrivial _{L} multiplet leads to the so-called minimal dark matter (MDM) models [_{L} multiplet results in a richer phenomenology, but the models are much more complicated [

The philosophy of the MDM models is to extend the standard model (SM) in a minimal way to involve dark matter [_{L} × _{Y} multiplet in a representation
_{L} representation
_{L} representation and the Lorentz invariance. As proposed in the original paper [_{L} gauge coupling
^{19} GeV [^{16}-10^{17} GeV [

Scalar MDM models are quite different from fermionic ones, since scalars will bring in more coupling terms. Such complexity has caused the neglect of a dangerous decay operator for the septuplet scalar model in the original consideration: the dimension-5 operator
^{①)}

The term “real” means that the multiplet itself is self-conjugated. A electroweak multiplet with even

In such scalar MDM models, scalar coupling terms may lead to another problem. Solutions to the renormalization group equations (RGEs) show that the scalar self-interaction couplings will go to infinity, i.e., a Landau pole (LP) will show up, at an energy scale far below the Planck scale [^{8} GeV if the DM particle mass is fixed to satisfy the observed relic abundance. In our previous work [^{14} GeV at best.

On the other hand, a real scalar quintuplet lives in a smaller representation and has only one independent self-interaction term. Consequently, the quintuplet couplings should evolve slower and reach a Landau pole at a higher scale. If extra fermionic multiplets are introduced, we may even push the LP scale above the Planck scale. Besides, such fermions could be used to explain the tiny neutrino masses via the type-I seesaw mechanism [

The paper is organized as follows. In Section 2, we introduce the quintuplet MDM model, and discuss its phenomenological constraints and the LP scale. In Section 3, we study an extension with extra fermions for pushing up the LP scale, and discuss the constraints from perturbativity and vacuum stability. Conclusions and discussions are given in Section 4. Appendix A gives the

In the real scalar quintuplet MDM model, the dark sector only involves a real scalar quintuplet

The self-conjugate condition implies

where
_{L} representation 5:

Thus, the covariant kinetic term for

where

In order to protect the stability of

Therefore, this model just brings in two couplings,

After the Higgs field acquires a VEV, the

Electroweak one-loop corrections break this degeneracy, making

where

Vacuum stability (VS) sets a stringent constraint on the model. The philosophy is that the potential should remain bounded from below as the couplings evolve to high energies. The VS conditions can be obtained by means of the copositive criteria [

The observation of the DM relic abundance sets a constraint on the

where

For

We take

A more proper treatment is to consider the Sommerfeld enhancement (SE) effect for DM freeze-out, following the strategy in Refs. [

Assuming thermally produced
^{②)}

This value may be slightly modified if the bound state effect is also considered [

Other constraints come from DM direct and indirect detection experiments. Direct detection uses the response to DM-nucleon scattering. The only tree-level diagram of the spin-independent
^{−46} cm^{2} for a fermionic quintuplet. As the gauge interactions of the scalar quintuplet are similar to those of the fermionic one, we may expect that the cross section for the scalar case would also be around this value^{③)}

In order to give an accurate DM-nucleon cross section, a detailed calculation for loop diagrams is needed. But such a calculation would be beyond the scope of this paper. We will leave it to a further study.

. Current direct detection experiments, such as PandaX-II [^{−45}cm

^{2}on the DM-nucleon cross section for a DM particle mass of 7 TeV. Thus, the scalar quintuplet model can evade current direct searches, but should be well tested in near future experiments.

For indirect detection of DM annihilation in space, the dominant process is

Renormalization group equation (RGE) evolution of couplings are determined by
_{L} gauge coupling

Note that
_{Y} gauge coupling

where

Below we analyze the Landau pole scale. The large coefficients of the

where

where

In order to eliminate the linear terms, we further define functions

Noting that

where

Setting
^{14} GeV, which is far below the Planck scale. Such a Landau pole implies that other new physics may exist between the quintuplet mass scale and the Planck scale, rendering all the couplings finite.

In this section, we will attempt to push up the LP scale obtained above. A lesson we can learn from the standard model is that the top Yukawa coupling gives a negative contribution to the self-coupling of the Higgs boson. As the Landau pole is induced by the self-coupling of the quintuplet, it is straightforward to introduce extra fermions with a Yukawa coupling to the quintuplet to shift the Landau pole. Such a motivation leads to the 5-5-1 model studied below.

There are three minimal ways to construct Yukawa interactions with the quintuplet scalar: introducing fermions in

For these reasons, in this work we only concentrate on the

At the renormalizable level, the Yukawa interactions can be expressed as

where
_{L} indices.

The above Yukawa interactions involve many new parameters. To illustrate the point, we adopt some working simplifications. In the first, we assume that there are three generations of
^{10} GeV. Hereafter,
^{10} GeV will be set as a benchmark scale of the 5-5-1 model. In the concrete numerical analysis, we will comment on the situation of deviation from this scale setup. Finally, in the 5-5-1 Yukawa interaction term, i.e., the second term of

where

The contributions of the quintuplet and singlet fermions at
_{L} gauge coupling and scalar couplings:

In these expressions, we have neglected the effect of the neutrino Yukawa couplings

Note that the

where

By solving the RGEs, we obtain the exact values of

where

If

Then the condition for

Note that the perturbative condition

As

where

As

Therefore, the growing

On the other hand, if

The above analysis is based on analytic calculations. Below we present the results obtained by solving the RGEs numerically .

Firstly, we investigate the impact of
^{10} GeV,

(color online) Evolution of the couplings
^{10} GeV,

1) If

2) If

3) If

4) If

The above fine-tuning of

Secondly, we study the parameter regions where the perturbativity and VS conditions are satisfied. We choose
^{10} GeV. The results are shown in

(color online) Regions favored by the perturbativity and VS conditions in the
^{10} GeV. The blue and orange regions correspond to the parameter regions satisfying the perturbativity and VS conditions, respectively. The overlap regions are favored by both conditions.

As we vary the initial value of the Yukawa coupling

(color online) Constraints on the real scalar quintuplet MDM model in the

The VS condition
^{9} GeV. The existence of the quintuplet scalar could change this behavior, because the
^{9}-10^{14} GeV. In this case, the vacuum stability cannot be ensured along the whole way to the Planck scale.

(color online) Evolution of
^{10} GeV,

To end up this section, we would like to make a comment on the possible impacts from a different scale set for
^{10} GeV. If

Perturbativity puts a strong constraint on scalar MDM models, especially when the multiplet lives in a large _{L} representation. The scalar self-couplings usually reach a Landau pole at an energy scale far below the Planck scale, in spite of the initial values. There are two reasons leading to such a disaster. One is that the quadratic self-interaction Lagrangians result in terms with large coefficients in the

In a previous work [^{14} GeV, which is consistent with Ref. [

In order to push up this LP scale, we have extended the model with Yukawa couplings of the scalar quintuplet to a fermionic quintuplet and three fermionic singlets, resulting in the so-called 5-5-1 model. The new singlets can also play the role of right-handed neutrinos, explaining the smallness of neutrino masses by the type-I seesaw mechanism. We have found that if such Yukawa couplings are involved after a scale of
^{10} GeV, all couplings can remain perturbative up to the Planck scale. The reason is that the Yukawa couplings contribute a large negative term to the

We have also investigated the parameter regions favored by the perturbativity and vacuum stability conditions up to the Planck scale. It has been found that these conditions constrain the Higgs-quintuplet coupling

At one-loop level, the

As most of the Yukawa couplings are negligible, only the top Yukawa coupling

In the RGE calculation, we use the following

The measured values of

setting

Here the constants

In order to account for the SE effect on the DM relic abundance and the annihilation cross section related in indirect detection, we consider the two-body Schrödinger equations for (co-)annihilation pairs of dark sector particles, following Refs. [

Here

The Schrödinger equations for the two-body wave functions

where

After solving these equations, annihilation and coannihilation cross sections with the SE effect are given by

where

The annihilation rate matrices account for annihilation and coannihilation of dark sector particles into SM particles at tree level. Dominant final states are pairs of gauge bosons and of Higgs bosons. Using the optical theorem, the annihilation rate matrices are given by

These matrices are utilized in the calculation of relic abundance. One may check that the sum of all the diagonal elements can reproduce the tree-level effective annihilation cross section (11). Note that when computing annihilation cross sections, the states with

In order to estimate the constraint from the MAGIC and Fermi-LAT experiments, whose result was obtained by assuming DM totally annihilating into