Planck scale boundary conditions in the standard model with singlet scalar dark matter

We investigate Planck scale boundary conditions on the Higgs sector of the standard model with a gauge singlet scalar dark matter. We will find that vanishing self-coupling and Veltman condition at the Planck scale are realized with the 126 GeV Higgs mass and top pole mass, 172 GeV $\lesssim M_t\lesssim$ 173.5 GeV, where a correct abundance of scalar dark matter is obtained with mass of 300 GeV $\lesssim m_S \lesssim$ 1 TeV. It means that the Higgs potential is flat at the Planck scale, and this situation can not be realized in the standard model with the top pole mass.


Introduction
The Higgs particle has just been discovered at the Large Hadron Collider (LHC) experiment [1,2]. In addition, the results from the experiment are consistent with the standard model (SM), and an evidence of new physics such as supersymmetry (SUSY) is not obtained. Currently, the experimental results strongly constrain the presence of SUSY at low energy although the minimal supersymmetric standard model (MSSM) is an attractive candidate for new physics beyond the SM. Thus, a question, "How large is new physics scale?", might become important for the SM and new physics. One can consider several scenarios such as high scale supersymmetric models or a scenario without SUSY in which the SM is valid up to the Planck scale M pl , etc.
As an example of the later scenario, it was pointed out that imposing a constraint that the SM Higgs potential has two degenerate vacua, in which one of them is at the Planck scale, leads to the top mass 173 ± 5 GeV and the Higgs mass 135 ± 9 GeV [3]. More recent work [4] showed that an asymptotic safety scenario of gravity predicts 126 GeV Higgs mass with a few GeV uncertainty. In these two scenarios, the boundary conditions (BCs) of the vanishing Higgs self-coupling (λ(M pl ) = 0) and its β-function (β λ (M pl ) = 0) are imposed at the Planck scale. In addition to these two BCs, the work [5] also discussed the Veltman condition (StrM 2 (M pl )=0) and the vanishing anomalous dimension of the Higgs mass (γ m h (M pl ) = 0) at the Planck scale. It was found that the four BCs yield a Higgs mass range of 127 − 142 GeV. Thus, combining these BCs can interestingly predict values of the Higgs and top masses in the SM close to the experimental ones but a slightly heavier Higgs mass and/or lighter top mass than experimental ones are generally predicted from these BCs as shown in [6] (see also [7,8,9,10,11,12,13] for the latest analyses). BCs of λ(M pl ) = 0 and StrM 2 (M pl )=0 mean that there exists an approximately flat direction in the Higgs potential, 1 which might be adopted to the Higgs inflation [14,15,16,17,18,19,20,21,22,23]. In addition, the quadratic (logarithmic) divergence for the Higgs mass disappear at the Planck scale under the Veltman condition (the vanishing anomalous dimension γ m h (M pl ) = 0). In this work, we investigate the three BCs in a gauge singlet extension of the SM.
One important motivation for the gauge singlet extension of the SM is that the SM does not include a dark matter (DM). In the extension, the gauge singlet scalar can be DM when the scalar has odd parity under an additional Z 2 symmetry [24] (see also [25,26,27]). Once a gauge singlet scalar is added to the SM, an additional positive contribution from new scalar coupling appears in the β-function of the Higgs self-coupling and the Veltman condition (and the anomalous dimension of the Higgs mass). 2 This means that the discussion of the three BCs at the Planck scale is modified from the SM. Since it actually seems difficult to reproduce 126 1 In order to confirm the existence of the flat direction, one should know the full ultraviolet (UV) completion. In the work, we assume an UV theory yielding the BCs for the flat direction at the Planck scale.
2 See [28,29] for discussions of the vacuum stability and triviality in the SM with a gauge singlet real scalar. See also [30] and references therein for implications of the LHC data to models with an extra singlet scalar, [31,32] for the classically conformal U (1) B−L extended SM, [33] for a model of electroweak and conformal symmetry breaking.
GeV Higgs mass and 173.07 ± 1.24 GeV top pole mass [34] (see also [35,36]) at the same time (i.e. a slightly heavier Higgs and/or a lighter top masses than the experimental center values are required) under the above three BCs at the Planck scale in the SM, it is interesting to investigate if the BCs could be realized with the center values of the Higgs and top masses in the singlet scalar DM extension of the SM. In this work, we take the following setup: (i) We consider a simple framework, in which only one gauge singlet real scalar is added to the SM. (ii) The gauge singlet scalar is DM. (iii) All scalar quartic couplings in the model can be perturbatively treated up to the Planck scale.
The paper is organized as follows: In Section 2, we investigate the three BCs at the Planck scale in the above framework. As a result, we will find that the vanishing self-coupling and Veltman condition at the Planck scale are realized with the 126 GeV Higgs mass and top pole mass, 171.8 GeV M t 173.5 GeV, where a correct abundance of scalar dark matter is obtained with mass of 300 GeV m S 1 TeV. It means that the Higgs potential is flat at the Planck scale, and this situation cannot be realized in the SM with the top pole mass. Section 3 is devoted to the summary.

Boundary conditions at the Planck scale
We consider the SM with a gauge singlet real scalar S, and investigate the values of scalar quartic couplings at the Planck scale by solving renormalization group equations (RGEs) in the model. The relevant Lagrangian of the model and the RGEs for the scalar quartic couplings are given by and with respectively, where we assume that the Higgs mass m H is smaller than DM mass m S . 3 H is the SM Higgs doublet, v is the vacuum expectation value (VEV) of the Higgs as 246 GeV, y is the top Yukawa coupling, t is defined as t ≡ ln µ, and µ is a renormalization scale within the range of M Z ≤ µ ≤ M pl . We also impose an additional Z 2 symmetry on the model. Only the gauge singlet scalar has odd parity while all the SM fields have even parity under the symmetry. We give some comments on properties of the three scalar quartic couplings obeying Eqs. (4)∼(7): • The right-hand side of Eq. (6) is proportional to k itself. Thus, if we take a small value of k(M Z ), where M Z is the Z boson mass, a change of value in the running of k(µ) is also small and remained in a small value. As a result, the running of λ closes to that of the SM.
• It is known as the vacuum instability that the value of λ becomes negative before the Planck scale in the SM with the experimental center values of the Higgs and top masses. This is due to the negative contribution from the top Yukawa coupling to the β-function of λ as in Eq. • Once the gauge singlet scalar is added to the SM, the additional contribution of k 2 /2 with the plus sign appears in the β-function of λ. This contribution can lift the running of λ, and thus, λ can be around zero at the Planck scale.
• The position of the minimum in the running of λ comes to lower energy scale than O(10 17 ) GeV by adding the gauge singlet scalar because the contribution of k 2 /2 in Eq. (5) becomes large at a high energy scale compared to the electroweak (EW) scale.
• The realization of the vanishing λ around the Planck scale by adding the gauge singlet scalar means that λ becomes negative before the Planck scale due to the above third and fourth properties of λ. Then, λ returns to zero.
• The running of λ S is an increasing function of t (or µ). There is not a direct contribution from λ S to the β-function of λ but the running of λ S affects on that of λ through the running of k.
We investigate the case that the gauge singlet scalar is DM with the three BCs (λ(M pl ) = 0, β λ (M pl ) = 0, and StrM 2 (M pl ) = 0) in this model. Since we impose the odd-parity on the singlet scalar under the additional Z 2 symmetry, the singlet can be a candidate for DM. Thus, Ω S h 2 = 0.119 must be reproduced in the case, where Ω S is the density parameter of the singlet and h is the Hubble parameter.

Vanishing Higgs self-coupling: λ(M pl ) = 0
First, we consider the BC that λ is zero at the Planck scale M pl = 10 18 GeV, λ(M pl ) = 0. The BCs of the Higgs self-coupling and top Yukawa coupling at low energy are taken as It can be seen from Fig. 1 (a) that |λ(M pl )| < 10 −2 can be satisfied in a region of 85 GeV m S 1.1 × 10 3 GeV with the corresponding top pole mass, 171.8 GeV M t 173.8 GeV. Such a DM mass region will be checked by the future XENON100 experiment with 20 times sensitivity of the current data [37]. One can also see that a larger top mass requires a larger DM mass in the region of m S 10 2 GeV. This is due to the following reason: A larger top mass needs a larger value of k in order to realize the tiny value of λ at the Planck scale. And a larger k requires a larger DM mass to give the correct abundance in the range of m S 10 2 GeV (e.g., see [29,37]).
In order to realize the correct abundance of DM in m S 10 3 GeV, k(M Z ) 0.3 is needed. Thus, k(M pl ) is not also large (k(M pl ) < 1) for the realization of DM. Since we have also imposed the condition of 0 < λ S (M pl ) < 1 in the analyses, the model can be described by a perturbative theory up to the Planck scale. On the other hand, the value of λ S does not actually affect on the abundance of DM. Thus, the region described by the gray dots in Fig. 1 (a) is not changed even with the condition of λ S (M pl ) > 1.
One might suggest the Higgs inflation by the use of the region satisfying λ(M pl ) = 0, which is included in gray dots of Fig. 1 (a), but it is not possible. 5 Since λ(µ) < 0 (10 8 GeV µ < M pl ) and λ(M pl ) = 0, there is a global minimum of the potential between the EW and Planck scales. If one identifies the Higgs with the inflaton, the inflaton rolls downslope to the global minimum not to the EW one. Thus, one must consider the other inflation models.
The smallness of λ(M pl ) predicting close values of the Higgs and top masses to experimental ones at low energy motivates one to investigate the BC of λ(M pl ) = 0 and/or the possibility of the Higgs inflation. On the other hand, the value of λ S does not strongly affect the SM (Higgs and top masses), DM sectors, and other cosmology compared to that of λ (and k which determines the abundance of DM). Thus, we focus only on the BC of λ(M pl ) = 0 in this work. If there could be phenomenological and/or cosmological motivations to impose λ S (M pl ) = 0, the discussions of the realization of the BC might also be interesting.

Veltman condition: StrM
The Veltman condition, which indicates a disappearance of the quadratic divergence on the 1-loop radiative correction to the bare Higgs mass, is modified to where the term k/2 in Eq. (9) is new contribution from k|H| 2 S 2 /2 interaction in the SM with gauge singlet scalar. In the SM, the value of the left-hand side of Eq. (9) without k/2 term at the Planck scale is −0.291 when one takes m h = 126 GeV and M t = 173.07 GeV.
We also show the region satisfying |StrM 2 (M pl )/v 2 | < 10 −2 and Ω S h 2 = 0.119 for the SM with the singlet DM in Fig. 1 (a) We will return to this point later. This DM mass region will also be checked by the future XENON100 experiment with 20 times sensitivity of the current data [37]. One can also see that a larger top mass requires a larger DM mass. The reason is similar to the case of the vanishing λ condition, i.e. a larger top mass needs a larger value of k in order to cancel the negative contribution from −6y 2 term at the Planck scale, and thus a larger k requires a larger DM mass to give the correct abundance in the range of m S 10 2 GeV. We also comment on the anomalous dimension for the Higgs mass defined by which indicates the logarithmic divergence. It is also modified to In fact, the deep and light blue dots in Fig. 1 (a) show a region satisfying |γ m h /m 2 h (M pl )| < 10 −2 and Ω S h 2 = 0.119 at the same time. The deep and light blue dots indicate 10 −5 < λ S (M Z ) < 0.1 and 0.1 < λ S (M Z ) < 1, respectively. One can see that |γ m h /m 2 h (M pl )| < 10 −2 can be satisfied in a region of 200 GeV m S 300 GeV with the corresponding top pole mass, 171.8 GeV M t 174.3 GeV. A larger top mass leads a smaller value of anomalous dimension due to 12λ term in Eq. (12). Therefore, a larger top mass requires a larger value of k (equivalently to m S ). However, the magnitude of the decrease of the anomalous dimension by a larger top mass is smaller than those of λ and StrM 2 /v 2 because the sign of contribution from the top Yukawa coupling only in the anomalous dimension is positive unlike the λ and StrM 2 /v 2 cases (see Eqs. (5), (9), and (12)). As a result, a top mass dependence of the anomalous dimension is weaker compared to those of the vanishing λ and the Veltman condition. It should be mentioned that there is also a region in which two conditions of the vanishing λ and γ m h can be realized at the same time.

Vanishing beta-function of the self-coupling: β λ (M pl ) = 0
In the SM, the β-function of λ becomes tiny at the Planck scale. The value is about β SM λ (M pl ) ≃ 8.42 × 10 −4 when one takes the Higgs and top pole masses as m h = 126 GeV and M t = 173.07 GeV at low energy. Thus, this condition may also be meaningful for theories beyond the SM around the Planck scale.
For instance, when one takes m S = 800 GeV, k(M Z ) = 0.24, and λ S (M Z ) = 0.34 in addition to m h = 126 GeV and M t = 173.07 GeV as an example in the context of the SM with the gauge singlet field, the corresponding values of the β-function at the Planck scale become β λ (M pl ) ≃ 6.32 × 10 −2 . Therefore, the vanishing β-function of λ at the Planck scale in the SM with the gauge singlet cannot be satisfied because the runnings of λ is increasing from a negative value due to the effect of the singlet field as shown in Fig. 1 (b). In this extension of the SM, β λ (µ) becomes zero at µ ∼ O(10 15−17 GeV) not the Planck scale.
According the above analyses, the BC of β λ (M pl ) = 0 cannot be realized but two BCs of λ(M pl ) = StrM 2 (M pl ) = 0 can be satisfied in the model. Since the result might indicate that all the Higgs potential is induced from a quantum correction under the current circumstances, one has no warrant for β λ (M pl ) = 0. Thus, the non-vanishing β-function can be compatibly understood. Furthermore, there are also two additional β-functions (β k and β λ S ) in this model. Since values of β k (M pl ) and β λ S (M pl ) cannot be zero when we impose λ(M pl ) = 0 and the correct abundance of DM, the vanishing condition for only β λ (M pl ) might be meaningless. Thus, in this work we take a stance of giving up the vanishing β-function at the Planck scale to predict the Higgs and top masses, although β λ (M pl ) = λ(M pl ) = 0 condition adopted in [3] predicted the values of the Higgs and top masses roughly close to experimental magnitudes.

Multi coincidence
It is remarkable that there is a region, given in Eq. (10), satisfying two independent BCs at the Planck scale (λ(M pl ) ≃ 0 and StrM 2 (M pl ) ≃ 0 (or γ m h (M pl ) ≃ 0)) at the same time with the correct abundance of the gauge singlet DM, 126 GeV Higgs mass, experimentally allowed top pole mass, and the coupling perturbativity. This double coincidence in the above BCs, λ(M pl ) ≃ 0 and StrM 2 (M pl ) ≃ 0, with the correct DM abundance is just a non-trivial result. The double coincidence means that the Higgs potential becomes flat at the Planck scale. The gauge singlet scalar plays a crucial role for the realization, and it becomes DM with the correct abundance in the universe at present. The double coincidence with DM might be an alternative principle to "multiple point criticality principle" discussed in Ref. [3], where a condition that the SM Higgs potential has two degenerate vacua was imposed. 6 In the above analyses, we have limited the values of k(M pl ) and λ S (M pl ) to be less than 1. But, when one allows the values up to 4π, the two regions for the Veltman condition and the vanishing anomalous dimension are changed. We also weaken the conditions of (|StrM 2 (M pl )/v 2 |, This means that the triple coincidence for the three BCs occurs. The triple coincidence requires that the logarithmic divergence of the Higgs mass also disappear at the Planck scale instead of allowing a fine-tuning between the bare Higgs mass and a quadratic correction.

Summary and discussions
We Next, we discuss some points, which are related with this work and an extension. The BC of λ(M pl ) = 0 implies that our EW vacuum is false and the true vacuum is at a high energy scale slightly smaller than the Planck scale like the SM with the center values of the Higgs and top masses. We have checked that the quantum tunnelling probability p through out the history of the universe, which is estimated by p ≃ V U H 4 exp(−8π 2 /(3|λ(H)|)) (e.g., see [38]), can be much smaller than 1, where V U = τ 4 U , τ U is the age of the universe as τ U ≃ 13.7 Gyrs, and we took |λ(H)| ≃ 9.59 × 10 −5 with H ≃ 8 × We comment on the realization of the BCs of β λ (M pl ) = 0 with λ(M pl ) = 0, which were firstly considered in [3], in this single extension of the SM. β λ (µ) cannot be zero at the Planck scale with λ(M pl ) = 0 in the extension because there is an additional positive contribution from kS 2 |H| 2 interaction to β λ (µ). Finally, we also comment on other issues such as the existence of the tiny neutrino mass and the baryon asymmetry of the universe (BAU), which cannot be explained in the SM. One popular explanation is given by adding heavy right-handed Majorana neutrinos into the SM. These are known as the seesaw mechanism and the leptogenesis for generating the tiny neutrino mass and BAU, respectively. In this example of the extension, there exist additional contributions from the neutrino Yukawa couplings to β λ , StrM 2 , and γ m h . If the magnitude of the neutrino Yukawa couplings is smaller than O(10 −2 ), which corresponds to the right-handed neutrino Majorana neutrino mass smaller than O(10 10 ) GeV, the contributions are negligible in the BCs like the Yukawa couplings of the bottom quark and tau. On the other hand, if the neutrino Yukawa couplings are larger than O(0.1), the contributions should be taken account in the BCs. For the BC of λ(M pl ) = 0, a larger k(M Z ) (equivalently a heavier DM mass) is required because of a negative contribution from the neutrino Yukawa coupling to β λ . Such a negative contribution may well cancel other positive contributions in β λ such that β λ (M pl ) = 0 can be realized at the same time. A larger k(M Z ) is needed also for the BC of StrM 2 (M pl ) = 0 because the contribution from the neutrino Yukawa to StrM 2 is negative. Finally, an effect of the neutrino Yukawa coupling for γ m h is relatively non-trivial compared to the other BCs because the positive contributions from (top and neutrino) Yukawa couplings to γ m h compete with the negative one from 12λ term, i.e. larger (positive) Yukawa couplings lead smaller (negative) value of λ at the Planck scale. Thus, an accurate numerical analysis is required. Effects in the BCs from additional particles and their mass scales strongly depend on a model for generating the tiny neutrino mass and BAU, e.g. adding right-handed neutrinos, but such a model dependent analysis of the BCs with explanations of the neutrino mass and BAU in addition to DM might also be interesting.