Scalar quintuplet minimal dark matter with Yukawa interactions: perturbative up to the Planck scale

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 ~1014 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 ~1010 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.


Introduction
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 [1][2][3][4]. WIMP models can be easily constructed by introducing a dark sector with electroweak SU(2) L multiplets. Introducing one nontrivial SU(2) L multiplet leads to the so-called minimal dark matter (MDM) models [5][6][7][8][9][10][11][12][13][14][15][16][17][18], which only involve the minimal content of new fields. Introducing more than one SU(2) 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  [5]. For this purpose, a fermionic or scalar SU(2) L × U(1) Y multiplet in a representation is introduced. The potential DM candidate would be the electrically neutral component that should be the lightest new state. If the dimension of the SU(2) L representation is large enough to forbid dangerous decay operators, this neutral state would be able to play the role of the DM particle without imposing an artificial symmetry. In other words, the stability of DM can be protected by an accidental symmetry due to a large dimension of the SU(2) L representation and the Lorentz invariance. As proposed in the original paper [5], the minimal choices are a quintuplet ( ) for the fermionic case [6,8,[14][15][16]] and a septuplet ( ) for the scalar case [9,11,12,16,17]. Adding new electroweak multiplets will push up the SU(2) L gauge coupling when it runs to high scales. Nevertheless, the MDM model with a quintuplet fermion or a septuplet scalar can keep perturbative up to ~10 19 GeV [5] (~10 16 -10 17 GeV [43]) based on one-loop (twoloop) functions. Scalar MDM models are quite different from fermion- ic 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 will induce DM decay at loop level [43]. Therefore, the accidental symmetry in the septuplet scalar model is not totally strict. Nonetheless, one can always introduce an artificial symmetry to make the model work again, but considering would not be special any more. In this case, discussing a triplet ( ) or quintuplet ( ) real 1) scalar multiplet with would be more economic. The real scalar triplet model has been studied thoroughly in Refs. [44][45][46], while the real scalar quintuplet model is less explored and will be the main topic of this paper.
In such scalar MDM models, scalar coupling terms may lead to another problem. Solutions to the renormalization group equations (RGEs) show that the scalar selfinteraction couplings will go to infinity, i.e., a Landau pole (LP) will show up, at an energy scale far below the Planck scale [17,46,47]. With two independent septuplet self-interaction terms, the real scalar septuplet model hits a Landau pole at a scale around 10 8 GeV if the DM particle mass is fixed to satisfy the observed relic abundance. In our previous work [17], we attempted to push up the LP scale via introducing Yukawa couplings between the scalar septuplet and extra fermionic multiplets. A bonus of these extra fermions is to explain the smallness of neutrino masses through the type-III seesaw mechanism [48]. We found that such a setup can push up the LP scale to ~10 14 GeV at best. On the other hand, a real scalar quintuplet lives in a smaller representation and has only one independent selfinteraction 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 [49][50][51][52]. In this work, we will explore these possibilities in the real scalar quintuplet MDM model. For completeness, we will also discuss the constraint from the observed DM relic abundance, the bounds from direct and indirect detection experiments, and the stability of the electroweak vacuum. 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 functions and initial values of SM couplings, while Appendix B gives the detailed calculation of the Sommerfeld enhancement effect.

Model details
In the real scalar quintuplet MDM model, the dark sector only involves a real scalar quintuplet with , which can be expressed as The self-conjugate condition implies . The real scalar is a viable DM candidate. The gauge covariant derivative of is τ a where are generators for the SU(2) L representation 5: Thus, the covariant kinetic term for can be expanded as , and is defined as .
1) The term "real" means that the multiplet is self-conjugated. A electroweak multiplet with even n must be complex, and hence allows more interaction terms.
In order to protect the stability of , we require that is odd under a symmetry, while all SM fields are even. The scalar potential is constructed by and the SM Higgs doublet . Since the operator vanishes due to the self-conjugation condition, the general form of the potential respecting the symmetry is given by only five independent terms: Therefore, this model just brings in two couplings, and , and one mass parameter as new free parameters. We assume that the vacuum expectation value (VEV) of the Higgs field is nonzero, while the VEV of remains zero. Then the minimization of the potential implies two conditions, and . As in the SM, the VEV of the Higgs doublet is with =246.22 GeV.
After the Higgs field acquires a VEV, the term contributes equally to the masses of all the components. Therefore, at the tree level all components are degenerate with a shifted mass , given by Electroweak one-loop corrections break this degeneracy, making and slightly heavier than . When , the mass difference between and is [5] where 167 MeV, with . 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 [53]:

Experimental constraints
The observation of the DM relic abundance sets a constraint on the mass . Assuming DM is thermallyproduced in the early Universe, its relic abundance can be expressed as [54] where is the Planck mass, is the freeze-out parameter, is the total number of effectively relativistic degrees of freedom, and the effective thermally averaged annihilation cross section accounting for the coannihilation effect. (11) We take and for (TeV), and calculate the prediction of the relic abundance in the quintuplet MDM model.
A more proper treatment is to consider the Sommerfeld enhancement (SE) effect for DM freeze-out, following the strategy in Refs. [6,15]. Annihilation and coannihilation channels in the dark sector are categorized by the electric charges of the two-body states. Enhancement factors can be computed in various categories via numerically solving the Schrödinger equations for the two-body states. The inclusion of the SE effect would increase the effective annihilation cross section, and hence reduce the relic abundance for fixed model parameters. Details of the calculation are summarized in Appendix B.
Assuming thermally produced particles in the early Universe fulfill the observed relic abundance, [55], the mass can be constrained within a narrow range, depending on the coupling . As shown in Fig. 1 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 -nucleon scattering process is mediated by the Higgs boson, but its cross section is suppressed by and thus is insignificant unless is very large. On the other hand, according to the discussions in Ref.
[58], the gauge loops lead to a DM-nucleon cross section of ~2×10 −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 2) .
Current direct detection experiments, such as PandaX-II [59] and XENON1T [60], have set an upper bound of 5×10 −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 . The SE effect on such a process is even more significant, since the velocities of Galactic DM particles are much lower than those at the freeze-out epoch. Details of the related calculation are also given in Appendix B. Using the 95% CL exclusion limit on annihilation cross section obtained by the combined MAGIC and Fermi-LAT -ray observations of dwarf satellite galaxies, assuming DM totally annihilating into [56], we find that indirect detection experiments have given very stringent constraints, as shown by the red regions in Fig. 1. Nonetheless, three windows around 7 TeV, 8-10 TeV, and 13-14 TeV survive. As a result, the parameter regions suggested by the observation of relic abundance are still available. Renormalization group equation (RGE) evolution of couplings are determined by functions. In Appendix A, we list the functions in the SM, as well as initial values of SM couplings. When the renormalization scale goes above a scale , the effect of the real scalar quintuplet should be involved. Hereafter we will adopt =10 TeV, as suggested by the observed relic abundance. At one-loop level, the real scalar quintuplet MDM model introduces new terms in the functions of the SU(2) L gauge coupling and the Higgs quartic coupling , while the functions of the other SM couplings do not change. Here we list the functions that are relevant in the following analysis: [47] .
Note that is related to the U(1) Y gauge coupling by . The solution to for is just and are coefficients corresponding to in Eq. (A1) and in Eq. (12), respectively. Below we analyze the Landau pole scale. The large coefficients of the and terms in the beta function of drive to divergence at high energy scales. At oneloop level, and do not enter the beta function of , while and do not contribute to the beta functions of and . Therefore, and remain small at high energy scales. For an approximate analysis, we can safely neglect and in Eqs. (13) and (14). Thus, the RGEs for , , and become dλ dt ′ = 24λ 2 + 9 8 . For convenience, we define three functions by , , and , and obtain the equations for them: 1) This value may be slightly modified if the bound state formation effect is also considered [57].
2) 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.
In order to eliminate the linear terms, we further define functions and by the shifts and . Then we have 104 Noting that runs much faster than and , we can simply neglect in the second equation and find its solution asf where and . The Landau pole is reached when . Thus, the corresponding scale is

Setting
, which corresponds to 0 at , we find that the maximal LP scale for is =5.6×10 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.

5-5-1 model
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.

Yukawa interactions
There are three minimal ways to construct Yukawa interactions with the quintuplet scalar: introducing fermions in , , and The first and second options have potential for explaining the tiny neutrino masses via the type-I and type-III seesaw mechanisms, respectively. In order to keep stable, one of the two fermions participating the Yukawa interaction should be odd under the symmetry. Particularly, in the second option, the two (3, 0) fermions should be different: one is -odd, and the other one is -even. Consequently, in order to give correct neutrino oscilla- tion properties, we have to introduce at least one more triplet which is -even. On the other hand, in the third option, the representation should correspond to a -odd Dirac fermion for avoiding anomalies [61][62][63][64]. The fermions can just be the SM lepton doublets which are -even. However, this case is less interesting for us, as it cannot explain the neutrino masses.
For these reasons, in this work we only concentrate on the case. The resulting model is dubbed the "5-5-1" model. Minimally, we introduce a left-handed self-conjugated fermionic quintuplet and several righthanded fermionic singlets . It is convenient to use the tensor notation for writing down the interaction terms. The tensor notation can be translated to the familiar vector notation using the following dictionaries of and : At the renormalizable level, the Yukawa interactions can be expressed as where denotes the SM lepton doublets. are totally symmetric SU(2) L indices. and are family indices and at least two singlets are required for generating the realistic neutrino mixing. Equation (26) respects the symmetry, with being -odd and beingeven. Consequently, the new terms do not endanger stability of the scalar quintuplet, which has mass far below the fermionic one.
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 , and the third generation has the largest Yukawa coupling, which is close to the Yukawa coupling, i.e., . Other elements in the matrix are much smaller but should be consistent with neutrino oscillation data. Next, we assume all the new fermions, and have the same mass , which is a characteristic scale of the 5-5-1 model. Thus, the neutrino masses given by the seesaw mechanism are . According to the current cosmological constraint [65], 0.2 eV, which implies that ~10 10 GeV. Hereafter, =10 10 GeV will be set as a benchmark scale of the 5-5-1 model. In the con-Chinese Physics C Vol. 43 crete 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 , we only consider the coupling to , neglecting the other two couplings. Thus, we just need to deal with one 5-5-1 Yukawa coupling, . This 5-5-1 Yukawa term can be expanded as where and . The gauge couplings of the fermionic quintuplet are given by The contributions of the quintuplet and singlet fermions at further modify the one-loop functions of the SU(2) L gauge coupling and scalar couplings: In these expressions, we have neglected the effect of the neutrino Yukawa couplings . This will be justified later. Note that the function of becomes positive for . Thus, one may worry about up to which scale can remain perturbative. By requiring , we find the non-perturbative scale of in the 5-5-1 model as where . Setting =10 TeV, we find that almost any will give . Thus, would still be perturbative at the Planck scale.
By solving the RGEs, we obtain the exact values of and at as where leads to a critical value of , , i.e., , the 5-5-1 Yukawa coupling will reach a Landau pole at Then the condition for is Note that the perturbative condition will give a smaller upper bound on .
As grows at high scales, the effect of may become important. The function of with all Yukawa coupling included is ab Therefore, the growing will boost the running of , and hence the LP scale of becomes lower. Nonetheless, is important only when is very large (not far from its LP), so the LP scale would not change too much. Thus, it is still reasonable to neglect the effect of .
On the other hand, if , i.e., , as goes up will increase at the beginning, and then turn its direction at some scale, and then exponentially drop down to zero. If the decreasing behavior happens at a scale lower than the Planck scale, the effect of the 5-5-1 Yukawa coupling is not significant. In this case, the model is quite similar to the original quintuplet MDM model, where blows up before the Planck scale. If the decrease happens at some scale higher than the Planck Chinese Physics C Vol. 43 scale, then might be large enough to keep the finite.

Numerical calculation
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 on the running of couplings. Figure 2 shows the evolution of the couplings with different values of for fixing =10 TeV, =10 10 GeV, , and . We can see that the evolution behavior dramatically depends on the delicate input value of .

3) If
, all couplings will remain perturbative up to the Planck scale, and the VS conditions will be satisfied at the same time. The above fine-tuning of is expected. In the function of one has to arrange a delicate cancellation between and at so that is well under control in a wide energy region. Otherwise, the effect of Yukawa damping on is insufficient for a slightly smaller and too much for a slightly larger that renders a negative too early.
Secondly, we study the parameter regions where the perturbativity and VS conditions are satisfied. We choose as three typical inputs, and perform scans in the plane with =10 TeV and =10 10 GeV. The results are shown in Fig. 3. In the blue regions, all parameters can remain perturbative up to the Planck scale, while in the orange region, the vacuum remains stable up to the Planck scale. The overlap regions simultaneously satisfy the perturbativity and VS conditions.
As we vary the initial value of the Yukawa coupling , the overlap region varies. We find that the perturbativity and VS conditions constrain within a range of , and within a range of This favored range of is also indicated as a green strip in Fig. 1, for comparing with other phenomenological constraints.
The VS condition seems to allow a small negative . However, the results in Fig. 3 exclude the whole regions with negative . The reason is that the vacuum stability also requires the Higgs quartic coupling remaining positive when running to higher scales. In the SM, will turn negative at a scale ~10 9 GeV. The existence of the quintuplet scalar could change this behavior, because the coupling has a positive contribution to the function of , as shown in Eq. (13). If is not large enough to turn around the trend of , however, the vacuum will be unstable at some scale. We set , , and , and demonstrate the evolution of in Fig. 4. We can see that goes to negative values at scales ~10 9 -10 14 GeV. In this case, the vacuum stability cannot be ensured along the whole way to the Planck scale.
To end up this section, we would like to make a comment on the possible impacts from a different scale set for GeV is also disfavored, because would be too large to be stopped by a perturbative . One can employ a parallel analysis for other choices of , and the configuration of the viable parameter space will not change significantly as long as is not very far from the benchmark value.

Discussion and conclusions
Perturbativity puts a strong constraint on scalar MDM models, especially when the multiplet lives in a large SU(2) 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 functions of the self-couplings. Once the self-couplings obtains a modest value, such terms will drive the self-couplings to grow exponentially and soon violate perturbativity. Another reason is that there are also significant terms contributed by other couplings in the functions of the self-couplings, e.g., a term with a large coefficient. These terms ensure that even when their initial values are very tiny, the self-couplings will quickly obtain modest values after a short journey of running.
In a previous work [17], we found that the perturbativity problem in the real septuplet scalar MDM model is quite stubborn: even after some fermions with Yukawa λ 2 interactions are introduced to slow down the running of scalar self-couplings, the model is still unable to remain perturbative up to the Planck scale. Nonetheless, this may be achievable if the MDM scalar lives in a smaller representation. Therefore, we have studied the real scalar quintuplet MDM model in this work. The observed relic abundance implies that the scale for introducing such a quintuplet should be near 10 TeV. Our calculation suggests that the quintuplet self-coupling will hit a Landau pole at a scale of 5.6×10 14 GeV, which is consistent with Ref. [47]. 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 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 function of and hence slow down the growing of at high scales.
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 the quintuplet scale within a range of 0.4, and the quintuplet self-coupling within a range of . 17 20 . The related functions are: 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. [6,15]. The twobody states can be categorized by their charges, denoted as . For the scalar quintuplet, there are three types of initial states ( ), and each of them is endowed with an annihilation rate matrix and a potential matrix. The potential matrices account for the non-relativistic effect of exchanging electroweak gauge bosons between the two incoming particles, with the following forms: .

(B1)
Here and corresponds to neutral and charged gauge bosons, respectively.

(B3)
After solving these equations, annihilation and coannihilation cross sections with the SE effect are given by where , and is the tree-level annihilation rate matrix. 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 (2.11). Note that when computing annihilation cross sections, the states with contribute twice since there are contributions from the (++, −) pair as well as from the (−−, +) pair.
In order to estimate the constraint from the MAGIC and Fermi-LAT experiments, whose result was obtained by assuming DM totally annihilating into [56], we also utilize the annihilation rate matrix for ,