SUHET072014
Vanishing Higgs potential at the Planck scale in a singlet extension of the standard model
Naoyuki Haba, Hiroyuki Ishida, Kunio Kaneta, and Ryo Takahashi
Graduate School of Science and Engineering, Shimane University,
Matsue, Shimane 6908504, Japan
ICRR, University of Tokyo, Kashiwa, Chiba 2778582, Japan
Abstract
[5mm] We discuss the realization of a vanishing effective Higgs potential at the Planck scale, which is required by the multiplepoint criticality principle (MPCP), in the standard model with singlet scalar dark matter and a righthanded neutrino. We find the scalar dark matter and the righthanded neutrino play crucial roles for realization of the MPCP, where a neutrino Yukawa becomes effective above the Majorana mass of the righthanded neutrino. Once the top mass is fixed, the MPCP at the (reduced) Planck scale and the suitable dark matter relic abundance determine the dark matter mass, , and the Majorana mass of the righthanded neutrino, , as within current experimental values of the Higgs and top masses. This scenario is consistent with current dark matter direct search experiments, and will be checked by future experiments such as LUX with further exposure and/or the XENON1T. and
1 Introduction
The Higgs particle was discovered at the LHC experiment [1, 2], but one finds no evidence to support the existence of physics beyond the standard model (SM) so far. Thus, the question “How large is new physics scale?” is important for the SM and new physics. One simple answer is that the SM is valid up to the Planck scale; i.e., there is no new physics between the electroweak (EW) and the Planck scales. In that case, the current experimental values of the Higgs and top masses might imply a vanishing effective Higgs potential at the Planck scale. In fact, there are intriguing researches about this possibility. For instance, Ref. [3] proposed the multiplepoint criticality principle (MPCP). This principle means that there are two degenerate vacua in the SM Higgs potential, with , where is the effective Higgs potential, is the vacuum expectation value (VEV) of the Higgs, and is the Planck scale. One is at the EW scale where we live, and another is at the Planck scale, which can be realized by the Planckscale boundary conditions (BCs) of the vanishing Higgs selfcoupling [] and its function []. As a result, Ref. [3] pointed out that the principle predicts a Higgs mass and a top mass, which are close to the experimental values but not the current center values. Furthermore, an asymptotic safety scenario of gravity [4] predicted Higgs mass with a few GeV uncertainty, and this scenario also pointed out and (see also Refs. [5][14] for more recent analyses). In this paper, we discuss the realization of a vanishing effective Higgs potential at the Planck scale, which is required by the MPCP, in the SM with singlet scalar dark matter (DM) and a righthanded neutrino.
An important motivation of the gauge singlet extension of the SM is to explain DM and the tiny active neutrino mass. In this extension, the scalar particle can be DM when it has odd parity under an additional symmetry [15] (see also Refs. [16][26]). The righthanded Majorana neutrino can generate the tiny active neutrino mass via the typeI seesaw mechanism. Once the scalar (righthanded neutrino) is added to the SM, an additional positive (negative) contribution appears in .^{1}^{1}1See also Refs. [18, 23, 24] for researches of the vacuum stability and the coupling perturbativity in the SM with scalar DM, and Refs. [27, 28, 29, 30] for explaining the recent BICEP2 result [31] in the framework of the Higgs inflation [32] with gauge singlet fields. In addition, since it is difficult to reproduce the 126 GeV Higgs mass and the GeV top pole mass [33] at the same time under the MPCP at the Planck scale in the SM, it is intriguing to study whether the principle can be realized with the center values of the Higgs and top masses in the singlet extension of the SM, or not.
In this paper, we discuss the realization of the vanishing effective Higgs potential at the Planck scale, which is required by the MPCP, in the SM with singlet scalar DM and the righthanded neutrino. Intriguingly, both the scalar DM and the righthanded neutrino are necessary to realize the MPCP which predicts the DM mass and the Majorana mass of the righthanded neutrino : within current experimental values of the Higgs and top masses. and
2 Singlets extension of the SM
The relevant Lagrangians of the singlet extension of the SM are given by
(1)  
(2)  
(3) 
where the SM Lagrangian including the effective Higgs potential is given by , and is the Higgs doublet (), is a gauge singlet real scalar field, is the lefthanded lepton doublet of the SM, is the righthanded neutrino, is the neutrino Yukawa coupling, and is the Majorana mass of the righthanded neutrino. In the model, since only the singlet real scalar is assumed to have odd parity under an additional symmetry, it can be DM with suitable mass and couplings. The DM mass is given by . The righthanded neutrino generates the small active neutrino mass through the typeI seesaw mechanism.
We utilize the renormalization group equations (RGEs) at twoloop level in this model, which were first given in Ref. [30]. Here, we mention the features of RGE runnings of the scalar quartic couplings at the twoloop level:

Since the function of is proportional to itself, an evolution of is tiny when is close to zero. Note that is the SM limit.

becomes negative within when the experimental center values of the Higgs and top masses are taken; this is known as the vacuum instability or metastability in the SM. This is induced from the dominant negative contribution of the top Yukawa coupling, . NNLO computations [7] indicate that can be positive within for the Higgs mass as with a top mass of or , with (see also Ref. [14]).

The RGE evolution of can be raised by the additional positive term in the function of . There is also a negative contribution to the function of from the neutrino Yukawa coupling, which pushes down the RGE evolution of . We will investigate whether the MPCP can be realized by considering these two contributions in the model, or not.
3 Multiple point criticality principle in singlets extension of the SM
The MPCP requires that there exist two degenerate vacua in the effective Higgs potential. One is at the EW scale where we live and another is at the Planck scale. This principle is described as . In terms of and , this principle is written as
(4) 
which is obtained from the stationary condition, . The conditions cannot be realized in the SM within the current experimental ranges of top and Higgs masses; i.e., the MPCP in the SM requires a lighter top mass and/or a heavier Higgs mass [3]. Thus, we need to consider an extension of the SM anyhow.
We investigate a realization of Eq. (4) in the singlet extension of the SM by solving the twolooplevel RGEs. The scalar DM (neutrino Yukawa) coupling lifts up (pushes down) the running of . Thanks to these two contributions in this extension, the positive contribution from the scalar to can avoid the metastable vacuum of the SM with the current experimental values of the top and the Higgs masses. The scalar contribution becomes dominant in at the Planck scale, which can realize and . And a negative contribution from the neutrino Yukawa coupling to above the Majorana mass scale can successfully achieve and . This is the essence of the realization of the MPCP in this singlet extension of the SM, and the realization is nontrivial. For the RGEs, decoupling effects of the scalar and the righthanded neutrino should be taken into account below their mass scales by taking away their relevant couplings from the corresponding functions. In particular, it is very important that the neutrino Yukawa becomes effective above the Majorana mass of the righthanded neutrino. For the neutrino sector, the active neutrino mass is induced from the seesaw mechanism [] and is taken as eV. With these relations, the value of is given by . This is an example in which one active neutrino mass is obtained. Two other neutrino masses can also be effective in the RGE analyses, but here we assume that other neutrino Yukawa couplings are small enough to be neglected in the analyses.
Our results are summarized in Fig. 1, where the conditions of and are depicted by blue and orange curves, respectively.
(a)  (b) 
(c)  (d) 
We analyze the realization of the MPCP at both the Planck and the reduced Planck scales , which are shown by the dashed and solid curves in all figures, respectively. In the regions above (below) the blue and orange solid curves, and [ and ], respectively. These correspondences are the same for the case of the reduced Planck scale. Figures 1 (a)1 (d) are the cases of , , , and , respectively, with [34], where is the density parameter of , is for DM, and is the Hubble constant. Since is DM in the model, the value of is determined by and . We utilize micrOMEGAs [35] to estimate the relic abundance of , and we take the Higgs mass as 126.1 GeV in the calculation.^{2}^{2}2We also take the strong coupling as . For the matching terms of and at the top pole mass scale, we take twoloop results, shown in e.g. Refs. [5, 7]. In the region above the pink dashed and solid lines, selfcoupling exceeds at the Planck and reduced Planck scales, respectively, while perturbative calculation is valid in the parameter space below the pink lines. At an intersection point of the blue and orange solid (dashed) curves below the horizontal pink solid (dashed) line, the MPCP can be satisfied within the experimentally allowed region of the Higgs, top, and DM masses with suitable scalar quartic couplings up to the (reduced) Planck scale. One can really see that there are some intersection points in Fig. 1. We mention parameter dependences for the realization of the MPCP as follows:

When is relatively light, as GeV, the contribution from the neutrino Yukawa coupling to is negligible. Thus, once is fixed, is realized by taking suitable value for only . This is shown by flat regions of blue curves in the figures. In this region, is always positive. A similar case, i.e. the decoupling limit of the righthanded neutrino , was discussed in Ref. [24], and our analysis is consistent with the results of Ref. [24].

When becomes large, we can successfully achieve with . The correlation between and is seen in the slanting regions of the blue curves. One can see that a larger value of is required to balance with the large scalar contribution.

Regarding the coupling perturbativity of , it strongly depends on values of and but not on , because the neutrino Yukawa does not contribute to at the oneloop level. Thus, when one takes a larger , the bound of the coupling perturbativity of becomes severe for .

For larger , the MPCP is satisfied in lighter [compare Fig. 1(a) and 1(c), or Figs. 1(b) and 1(d)]. Since coupling gives a positive contribution to the function of , grows more rapidly for larger . As a result, smaller (or ) is favored for canceling the negative contribution from the top Yukawa coupling in a larger case.
Next, Fig. 2 shows the positions of the intersection points in the [] plane for the case.^{3}^{3}3One might also find another intersection point around GeV in each figure [(, , , and in Figs. 1(a), 1(b), 1(c), and 1(d), respectively]. Since these points are close to the lines of the coupling perturbativity bound on , we do not consider these solutions around GeV in this paper anymore. But this could also be the solution for the MPCP. The solid and dashed curves indicate the MPCP solutions at and , respectively. We can show that and have onetoone correspondence ( and also have one to one correspondence). When one takes larger , larger and are required to achieve and at the same time. To summarize, there are seven independent parameters; i.e., five coupling constants (, , , , and ) and two mass scales of the singlets ( and ), in the scalar and Yukawa sectors of the model, in which is determined by . The suitable DM relic abundance relates with and the seesaw mechanism relates with . Thus, there are four independent parameters (, , , and ). When the top mass and are fixed, the two conditions of the MPCP ( and ) uniquely determine and .^{4}^{4}4By extending the model, and could be induced dynamically from a dimensional transmutation, which could have a conformal or shift symmetry in the framework of conformal gravity as a UV theory.
As our result, we find that the MPCP at the (reduced) Planck scale predicts the following mass regions:
(5)  
(6) 
within GeV and . They are obtained by maximal and minimal values of and on the intersection points of the two contours of the and lines which are located at (, , and for the case of the MPCP at the (reduced) Planck scale shown in Figs. 1(a), 1(b), 1(c), and 1(d), respectively.^{5}^{5}5There are also intersection points around GeV in the reduced Planck case as (, , , and in Figs. 1(a), 1(b), 1(c), and 1(d), respectively. ,
Finally, we draw Fig. 3, which shows the current experimental bounds on and [XENON100 225, livedays (blue solid line) and LUX, 85.3 livedays (orange solid line)] and the future detectability by the LUX (orange dashed line) and the XENON1T (blue dashed line) experiments [36, 37]. The black solid curve indicates the contour of . One can find that the DM mass region in Eq. (5) for the realization of the MPCP can be consistent with the current DM direct detection experiments, and it will be checked by future DM direct searches, e.g., the future XENON1T experiment.
4 Summary and discussions
We have discussed the realization of the vanishing effective Higgs potential at the Planck scale, which is required by the MPCP, in the SM with the singlet scalar DM and the righthanded neutrino. We have found that the scalar DM and the righthanded neutrino play crucial roles for realization of the MPCP, where the neutrino Yukawa becomes effective above the Majorana mass of the righthanded neutrino. Once the top mass is fixed, the MPCP at the (reduced) Planck scale and the suitable DM relic abundance determine the DM mass and Majorana mass of the righthanded neutrino as region is allowed by the current experimental results of the DM direct searches. Moreover, it is of importance that this scenario is testable by the future direct search experiments such as the LUX with further exposure and/or the XENON1T. within the current experimental values of the Higgs and top masses. The and
Finally, we also show other solutions of the MPCP as examples of different shares of for ; i.e., , shown in Figs. 4(a)4(d), and 0.2 in Figs. 4(e)4(h). The meanings of the lines and colors are the same as in Fig. 1. One can see that the MPCP can be realized in a lighter region compared to the case. At the same time, the bound from the coupling perturbativity of on becomes more severe when the value of becomes smaller than unity (see Figs. 1 and 4). This is because a smaller needs a larger value of for the same DM mass [e.g., see the green () and the yellow () dashed curves in Fig. 3]. Thus, a lighter gives a solution for the MPCP, and a heavier region is constrained by the coupling perturbativity of in a smaller case. We also show excluded (shaded) regions of GeV by the LUX 85.3 liveday WIMP search [36] in Figs. 4(e)4(h). Regarding an experimental bound on DM, although there are intersection points around GeV in the case of with GeV [see Figs. 4(e) and 4(g)], the LUX experiment has ruled out GeV. As a result, the MPCP is satisfied in the regions and for within GeV, and and for with GeV. One can find that these regions for the realization of the MPCP can also be consistent with the current DM direct detection experiments, and they will be checked by future DM direct searches.
(a)  (b) 
(c)  (d) 
(e)  (f) 
(g)  (h) 
Acknowledgement
This work is partially supported by Scientific Grant by the Ministry of Education and Science, No. 24540272. The work of R.T. is supported by research fellowships of the Japan Society for the Promotion of Science for Young Scientists.
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