Vanishing Higgs Potential in Minimal Dark Matter Models
We consider the Standard Model with a new particle which is charged under with the hypercharge being zero. Such a particle is known as one of the dark matter (DM) candidates. We examine the realization of the multiple point criticality principle (MPP) in this class of models. Namely, we investigate whether the one-loop effective Higgs potential and its derivative can become simultaneously zero at around the string/Planck scale, based on the one/two-loop renormalization group equations. As a result, we find that only the triplet extensions can realize the MPP. More concretely, in the case of the triplet Majorana fermion, the MPP is realized at the scale if the top mass is around GeV. On the other hand, for the real triplet scalar, the MPP can be satisfied for GeV and GeV, depending on the coupling between the Higgs and DM.
The discovery of the Higgs particle [1, 2] is very meaningful for the Standard Model (SM). The experimental value of the Higgs mass suggests that the Higgs potential can be stable up to the Planck scale and also that both of the Higgs self coupling and its beta function become very small around . This fact attracts much attention, and there are many works which try to find its physical meaning [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29] and implications for cosmology [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55].
In [3, 4],
the Higgs mass was predicted
to be around GeV by the requirement that and simultaneously become zero around .
By taking the fact that the MPP is realized in the SM into consideration, a natural question is whether the MPP can be also realized in the models beyond the SM. It is meaningful to consider the MPP of these models because we can understand whether the SM is actually special among them. One of the interesting extensions is adding a new weakly interacting fermion or scalar , which is a representation of with the hypercharge . Such extensions are phenomenologically well studied because they have dark matter (DM) candidates when [66, 67, 68]. In this paper, we focus on , that is, Majorana fermions and real scalars. We examine the realization of the MPP of these models, based on the one/two-loop renormalization group equations (RGEs). We use the effective Higgs self coupling and its beta function defined from the one-loop effective Higgs potential . Their definitions and the two-loop RGEs when we add a new fermion are presented in Appendix A. In the case of the new scalar (fermion), we only have to consider () since the scalar couplings ( coupling ) rapidly blow(s) up when  ( ), and the theory does not valid up to . For the septet and nonet fermion cases, we discuss this point in Appendix B.
In the following discussion, we regard the top mass as a free parameter, and the Higgs mass is varied within 
As a result, and are uniquely predicted because there is no additional free parameter. The results are
depending on .
The upper panels of Fig.1 show the runnings of the SM parameters where GeV, and is correspondingly fixed so that the MPP is realized.
Here, we also show the SM running of by the dashed green line for comparison.
Furthermore, in the middle and lower panels, we show the corresponding (left) and (right).
In these figures, the one-loop results are also shown.
One can actually see that the potential and its derivative simultaneously become zero at a high energy scale, and that the only triplet can have the other vacuum near the string/Planck scale. We note that the two-loop effects are small.
Now let us consider a new scalar. As mentioned before, the remaining possibility is . The potential of the scalar fields is
Here, is the SM Higgs doublet. The one-loop RGEs which are different from those of the SM are as follows
Furthermore, there is an additional contribution to :
In this case, the thermal abundance of depends on the value of . Here we use
for our calculation
In order to discuss the Higgs potential around the cutoff scale ,
it is meaningful to consider how the existence of a new particle changes the behavior of the bare Higgs mass as a function of .
Here, let us focus on at one-loop level. For , is given by
where the couplings are evaluated at . On the other hand, for , becomes
The left (right) panel of Fig.3 shows as a function of when a new particle is fermion (scalar). Here, the green contour is the SM prediction when GeV, and blue bands correspond to the deviation from it :
In the right panel, we change at from 0 to 0.4, and they are represented by a red band.
Depending on the values of and , one can see that the scale at which becomes zero quite changes. In both of cases, can take zero around the string scale
In conclusion, we have studied the MPP of the SM with a weakly interacting new particle with its hypercharge being zero. When a new particle is a fermion, we have found that the top mass and can be uniquely predicted. On the other hand, when a new particle is scalar, there exists a new scalar coupling . Due to this coupling, we have found that and drastically change. In both of cases, only the triplets survive from the point of view that the other vacuum should exist around the string/Planck scale and that the theory is valid up to this scale. The analysis of this paper suggests that the SM and its triplet extensions are special in that the MPP can be realized around the string/Planck scale.
We thank Hikaru Kawai and Koji Tsumura for valuable discussions and useful comments. This work is supported by the Grant-in-Aid for Japan Society for the Promotion of Science (JSPS) Fellows No.251107 (YH) and No.271771 (KK).
Appendix Appendix A Two-loop renormalization group equations and one-loop effective Higgs potential
The two-loop RGEs of the SM with a new fermion which is a representation of with the hypercharge are as follows
Here, with being the renormalization scale, is the wave function renormalization of the Higgs, and are the Casimir and Dynkin index, and for Dirac and Weyl fermion. The two-loop RGEs of and are agreement with  by putting .
The one-loop effective Higgs potential is
In Eq.(21), we have neglected the contribution from the Higgs quartic term because it is small when we consider the MPP. In principle, should be determined as a function of so that is minimized. However, in this paper, is taken to be for simplicity. It is known that this is a good approximation . From , we define and as follows:
Appendix Appendix B Landau pole in septet and nonet fermion
As mentioned in the introduction, in cases of and , there exists a scale at which becomes infinity below , which is well known as the Landau Pole. Therefore, these theories are not favored from the point of view of perturbativity (triviality) up to the string/Planck scale. For completeness, we give numerical results of the Landau pole in Fig.4. Here, the two-loop results are shown by dashed lines. As is known, the one-loop Landau pole can be analytically solved:
where is the Dynkin index. From Fig.4, one can see that the two-loop effect is relatively important.
Appendix Appendix C New Fermion with
Here, we consider a new fermion with . As well as the real and 9 cases, the Landau pole of exists below when . So, let us here focus on
Appendix Appendix D Electroweak symmetry breaking by Coleman-Weinberg mechanism
Here, we discuss a possibility to realize the electroweak symmetry breaking by the Coleman-Weinberg mechanism in the case of the SU(2) triplet scalar. The one-loop effective Higgs potential is
where and are given by Eq.(10) and Eq.(21) respectively, and we have assumed that the quadratic term vanishes at the tree-level. In the following, we choose . Then, develops the vacuum expectation value because the negative quadratic term appears from the second term in Eq.(25). The resultant vacuum expectation value is
where we have neglected the 1-loop correction to the quartic term. It is interesting that the successful electroweak symmetry breaking is realized for which is also favored by the MPP around the Planck scale.
- It is interesting that the quadratic divergent bare Higgs mass also vanishes around this scale .
- These values of are consistent with the recent analyses: GeV  and GeV  at level. However, the relation between these masses and the pole mass is not clear. In the following calculation of the bare Higgs mass, we use more conservative value of determined by the total cross section .
- As one can see from the results of the fermion cases, the two-loop effects are small when we consider the MPP. This is why we consider the one-loop beta functions here.
- The mass of a new scalar suffers from fine-tuning problem. However, because our motivation in this paper is to distinguish the minimal dark matter models in the context of the MPP, we take Eq.(11) as the dark matter mass.
- Within field theory, the quadratic divergence does not appear after the renormalization. However, it can have the physical meaning if we consider the scale around the Planck/string one, because the SM couples with the gravity. In this paper, we assume that the physics around the Planck scale is described by string theory, which is the cutoff theory whose universal cutoff scale is given by the string scale. This is why we take the universal cutoff in the calculation of . See  for the detail.
- The vanishing bare mass is so-called Veltman condition . From the point of view of low energy field theory, is accidental and seems to require the fine-tuning at the Planck scale. We hope that comes from some mechanism related to the physics at the Planck scale.
- In order to obtain the correct electroweak symmetry breaking, we need to add small negative mass term to the Higgs potential, which is much small than . However, in the case of the triplet scalar, it may be possible to realize the electroweak symmetry breaking by the Coleman-Weinberg mechanism. See Appendix D. We thank the referee for pointing this out.
- Our calculations are based on [75, 76, 77, 78].
- For and , the LP of the gauge coupling also appears below respectively when and . This is why we only show when in Fig.5.
- Furthermore, when =1, 2 and 3, there are additional Yukawa couplings among the SM leptons , the Higgs and . However, we can neglect these effects because the lepton masses are small.
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