SU-HET-10-2014 EPHOU-14-014

Gravitational effects on vanishing Higgs potential at the Planck scale

Naoyuki Haba, Kunio Kaneta, Ryo Takahashi, and Yuya Yamaguchi

Graduate School of Science and Engineering, Shimane University,

Matsue 690-8504, Japan

ICRR, University of Tokyo,

Kashiwa, Chiba 277-8582, Japan

Department of Physics, Faculty of Science, Hokkaido University,

Sapporo 060-0810, Japan

Abstract

[5mm]

We investigate gravitational effects on the so-called multiple point criticality principle (MPCP) at the Planck scale. The MPCP requires two degenerate vacua, whose necessary conditions are expressed by vanishing Higgs quartic coupling and vanishing its function . We discuss a case that a specific form of gravitational corrections are assumed to contribute to functions of coupling constants although it is accepted that gravitational corrections do not alter the running of the standard model (SM) couplings. To satisfy the above two boundary conditions at the Planck scale, we find that the top pole mass and the Higgs mass should be 170.8 GeV171.7 GeV and GeV, respectively, as well as include suitable magnitude of gravitational effects (a coefficient of gravitational contribution as ). In this case, however, since the Higgs quartic coupling becomes negative below the Planck scale, two vacua are not degenerate. We find that GeV with GeV is required by the realization of the MPCP. Therefore, the MPCP at the Planck scale cannot be realized in the SM and also the SM with gravity since GeV is experimentally ruled out.

## 1 Introduction

The ATLAS [1] and CMS [2] Collaborations of the Large Hadron Collider (LHC) experiment observed the Standard Model (SM) like the Higgs boson, whose pole mass is obtained by [3]

(1) |

The LHC data are almost consistent with the SM predictions,
and there are no signs of new physics beyond the SM at present.
In addition, the observed values of Higgs and top masses suggest the SM can be valid up to the Planck scale
in light of the vacuum (meta) stability.
The multiple point criticality principle (MPCP) requires two degenerate vacua between EW and Planck scales [4].
Necessary conditions of the MPCP (NC-MPCP) at the Planck scale is expressed
by vanishing the Higgs quartic coupling
and its function .
It was pointed out that these boundary conditions (BCs) suggest
GeV Higgs mass with the top pole mass as GeV [4].
Reference [5] showed that the Higgs mass of 126.5 GeV
with the top mass of 171.2 GeV implies both
and in a scenario of asymptotic safety of gravity
(see also [6]-[18] for more recent analyses).
Note that the MPCP could be realized at GeV
with the use of a lighter magnitude of the top mass as 171 GeV without gravitational effects.^{1}^{1}1
If one considers the realization of the MPCP at a lower energy scale than the Planck scale,
the desired values of the top mass are small compared to the MPCP at the Planck scale
as (146) GeV for
with () GeV [4]
(e.g., see also Ref. [12] for more recent analyses).

Since a gravitational interaction becomes important near the Planck scale, we must take into account gravitational effects for discussions around the Planck scale. However, general relativity is a nonrenormalizable theory by perturbation methods, and the theory of quantum gravity has not been established yet. Nevertheless, treating general relativity as an effective field theory provides a practical method to find the effects of quantum gravity [19, 20]. Because of the effective theory approach, gravitational contributions to the renormalization group equation (RGE) evolution of couplings are calculated by using different regularization schemes [21]-[28]. Since the MPCP at the Planck scale predicts close values of the Higgs and top masses to the experimental ones, it might be important for the SM with gravity to accurately analyze gravitational effects with the latest experimental results. We discuss a case that a specific form of gravitational corrections are assumed to contribute to functions of coupling constants (although it is accepted that gravitational corrections do not alter the running of the standard model couplings).

In this paper, we will first clarify regions of model parameters satisfying the NC-MPCP. To satisfy the NC-MPCP, we will find that the top pole mass should be in the region of 170.8 GeV171.7 GeV with the Higgs mass as GeV, and typical magnitude of the gravitational effects. Moreover, numerical analyses will show that the NC-MPCP can be typically satisfied in region of a coefficient of gravitational contribution to as with negative gravitational contributions to functions of the top Yukawa and gauge couplings. In this case, however, since the Higgs quartic coupling becomes negative below the Planck scale, two vacua are not degenerate. Then, we will find that GeV with GeV is required by the realization of the MPCP. Therefore, the MPCP cannot be realized in the SM and also the SM with gravity since GeV is experimentally ruled out.

## 2 Standard Model With Gravitational Effects

We discuss gravitational effects on the SM in this section. In particular, we show the effects on functions of the SM coupling constants and the resultant evolution of the couplings under the RGEs.

### 2.1 Gravitational effects on functions of coupling constants

We consider the SM and quantized general relativity, in which the graviton couples with the SM particles. The gravitational coupling constant is given by , in which is the Newton constant, and the Planck mass is GeV. This coupling is small enough to neglect far below the Planck scale, while it is effective around the Planck scale. To see energy dependences of couplings, we have to solve the corresponding RGEs. The RGE for a coupling constant is given by

(2) |

where and are the Callan-Symanzik
function in the SM and a gravitational contribution to the functions, respectively.^{2}^{2}2
functions in the SM at the two-loop level are given in the Appendix.
In our numerical analysis, we utilize the SM functions at the two-loop level
and include the leading contributions from gravity given in Eq. (3).
The gravitational contribution at the one-loop level is written by

(3) |

where is a constant that denotes the gravitational contribution to the coupling , and it can be obtained by calculating the corresponding graviton one-loop diagrams.

In this paper, we assume for the SM gauge couplings as taken in [5]. This assumption seems to be valid due to the universality of the gravitational interactions. If is negative, all gauge couplings approach zero near the Planck scale. Thus, the gauge interactions are asymptotically free. In addition, the Landau pole problem for the gauge coupling in the SM can be solved due to the presence of the fixed point. References [21]-[24] obtained negative values of with . For the top Yukawa coupling , the Yukawa interaction also becomes asymptotically free when . Regarding the Higgs quartic coupling , the gravitational contribution can be dominant around the Planck scale. Thus, the vacuum stability ( up to the Planck scale) can be more easily realized compared to the SM by taking the gravitational contribution with into account. In fact, was utilized in Refs. [24, 25]. We note that Refs. [26, 27] and [28] showed that the values of and depend on the gauge fixing, respectively. In addition, Ref. [29] showed that gravitational effects of the gravitational constant depend on physical processes. Therefore, we treat , and as free parameters in this work.

### 2.2 Gravitational effects on RGE evolution

Next, we show typical gravitational effects on the RGE evolution of coupling constants. To solve the RGEs, the following BCs [12, 13] are taken,

(4) | |||

(5) | |||

(6) | |||

(7) | |||

(8) | |||

(9) |

where and are the pole masses of the top quark and the Higgs boson, respectively. The BCs of the gauge and top Yukawa couplings are determined by Eqs. (4)-(7) for given and . After solving the RGEs, Eq. (8) gives as

(10) |

In Fig. 1, we show an example of the RGE evolution of the SM couplings that include gravitational effects, in which the above BCs are taken.

Figure 1 (a) shows typical RGE evolution of the SM couplings. The solid (dashed) lines show the RGE evolution with (without) the gravitational contributions. In the figure, we take [5, 23], [24, 25], GeV, GeV, and , which realize . The RGE evolution of gauge and top Yukawa couplings indicates asymptotic freedom of the interactions due to the gravitational contributions as mentioned above. If we take and , the gauge and top Yukawa couplings become large around the Planck scale. One can see that the gravitational contributions become effective above GeV. In this work, we concentrate on the energy region up to as a physical scale, which is depicted by the vertical solid line. On the other hand, Ref. [24] mentioned that the analyses can be valid in the region of , which reads . In this case, an effective energy region for the model is larger than the Planck scale when . As a result, the gravitational effects with may assist the gauge coupling unification since all gauge couplings unify at an energy scale lower than .

Figure 1 (b) shows the and dependence of the evolution of . We take , GeV, and in the figure. The solid, dashed, and dotted lines correspond to , , and , respectively. The red, black, and blue lines are the cases of GeV, 171.2 GeV, and 173 GeV, respectively. Note that the sign of in the Planck scale strongly depends on the top pole mass, e.g. for GeV when we take GeV and . One can understand the behaviors of around the Planck scale by considering as follows. Now can be approximately written as

(11) |

around the Planck scale. When both and are negative, the gauge and top Yukawa couplings approach zero as becomes large. Then, the last term (the gravitational contribution term) of Eq. (11) dominates when is sufficiently large. Thus, is approximately given by

(12) |

where is an energy scale at which the gravitational contribution becomes dominant, and is a value of at . As a result, is larger (lower) for () while keeping the sign of .

One can also see that the gravitational effects and the value of slightly change the RGE evolutions of the SM couplings around the Planck scale, while they significantly change the functions. It should be remarked that can be induced from when the gravitational contributions are taken into account in the SM as seen in Eq. (11). Thus, two independent BCs of and predict almost the same value of the Higgs mass for a given . In the next section, we quantitatively clarify regions of model parameters for the realization of the MPCP in this scenario.

## 3 Numerical Analyses

In this section, first we will clarify regions of the model parameters satisfying the NC-MPCP and . Next, we will discuss whether those parameter regions realize the MPCP, which requires two degenerate vacua.

### 3.1 Realization of the NC-MPCP

We show the results of numerical analyses satisfying the NC-MPCP in Figs. 2 and 3.
Figure 2 shows predictions for the Higgs and the top pole masses from the realizations of and .
In the figure, we take and .
To realize GeV,
the top pole mass should be in the region of .^{3}^{3}3
On the other hand, the experimentally favored value of the top quark mass is GeV [30].
In Ref. [13], the authors suggest this discrepancy might not be problematic,
and just the discrepancy between the pole mass and the mass obtained at the collider experiments [30].
Our predicted value is a pole of a colored quark,
while the experimentally obtained value is an invariant mass of the color singlet final states.
Since the observed pair is dominantly color octet at the hadron collider,
the discrepancy of a few GeV from the singlet final states may be caused [31, 32].
The result is consistent with the previous result including gravitational effects [5].

We also show the parameter dependence of the realizations of and for a typical top mass as 171.2 GeV in Fig. 3. As mentioned above, GeV cannot realize the MPCP although the NC-MPCP is satisfied. The red circles and blue crosses in Fig. 3 indicate the necessary conditions of the MPCP, and , respectively. Hence, the NC-MPCP can be satisfied at the overlapped points of red circles and blue crosses in the figure.

Let us mention the results of parameter dependence for the NC-MPCP. First, the NC-MPCP can be satisfied in almost all regions of and [see Figs. 3 (a) and 3 (b)]. Since gauge and top Yukawa couplings can be asymptotically free in this regime, is dominated by the gravitational contribution. As a result, is induced from as mentioned above.

Second, regarding the value of for the NC-MPCP, sufficiently large value of such as is typically required [see Figs. 3 (a) and 3 (b)]. When is sufficiently large, the gravitational term is dominant in . Thus, the conditions and are satisfied at the same time. Note that cannot be realized for .

Third, when is positive, gauge couplings become larger as becomes large. Since the terms positively contribute to , the condition leads . Similarly, top Yukawa coupling becomes larger for , and the terms negatively contribute to . Then, the condition leads . As a result, for () the NC-MPCP cannot be satisfied in most cases. However, the NC-MPCP can be satisfied when both and are positive, because the contributions of gauge and top Yukawa couplings cancel each other [see upper right region in the first quadrant of Fig. 3 (c)].

Finally, there are also regions satisfying the NC-MPCP, in which and [e.g., see the second and fourth quadrants of Fig. 3 (c)], when both and are relatively small with sufficiently large . Since the last term of Eq. (11) can still dominate the other terms also in the regions, can be induced from .

### 3.2 Realization of the MPCP

In the previous subsection, we investigate the regions of the model parameters satisfying the NC-MPCP. Figure 4 shows examples of the RGE evolution of satisfying . We take , GeV, and in the figure. The red, green and blue lines correspond to GeV, 174 GeV and 177 GeV, respectively. The behavior around the Planck scale is understood by Eq. (12), that is, is larger for as keeping the sign of . This figure implies that the Higgs potential has a local maximum at the Planck scale for GeV, and thus the MPCP cannot be realized for GeV. We find that GeV with GeV is required by the realization of the MPCP. Since the gravitational contribution is effective for GeV, these values of the Higgs and the top masses for the MPCP in our scenario are not drastically changed from the ones predicted by the past research without taking into account gravity [4] (see also [3, 6]). Therefore, we conclude that the MPCP cannot be realized in the SM and also the SM with gravity since GeV is experimentally ruled out.

## 4 Summary and Discussion

We have investigated the realization of the MPCP at the Planck scale, which requires the degenerate SM vacua at the Planck scale, in the SM with gravitational effects. The NC-MPCP is expressed by the vanishing Higgs quartic coupling and its function . The vanishing function can be induced from the vanishing Higgs quartic coupling with a suitable magnitude of and the asymptotic free of Yukawa and gauge interactions due to the gravitational effects around the Planck scale. To satisfy the NC-MPCP, the top and the Higgs pole masses should be in the regions of 170.8 GeV171.7 GeV and GeV, respectively, with , . Moreover, numerical analyses have shown that the NC-MPCP can be typically satisfied in the region of with and . In this case, however, since the Higgs quartic coupling becomes negative below the Planck scale two vacua are not degenerate. We have found that GeV with GeV is required by the realization of the MPCP. Therefore, the MPCP cannot be realized in the SM and also the SM with gravity since GeV is experimentally ruled out.

Finally, we comment on a nonminimal coupling of the Higgs to the Ricci scalar which was not considered in this paper. This nonminimal interaction is useful in the Higgs inflation [13], [33]-[38], and the RGE evolution of the SM couplings might be modified around the Planck scale. Then, it is expected that the Higgs and top masses and allowed regions of might be also modified. The analyses will be given in a separate publication.

### Acknowledgment

This work is partially supported by the Scientific Grant by Ministry of Education and Science, No. 24540272. The works of R.T. and Y.Y. are supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists [Grants No. 24801 (R.T.) and No. 262428 (Y.Y.)].

## Appendix

## Functions in the Sm

The RGE of coupling is given by , in which is a renormalization scale. The functions for coupling constants of the SM are given by

(13) | |||

(14) | |||

(15) | |||

(16) | |||

(17) |

up to the two-loop level [12]. We have only included the top quark Yukawa coupling, and omitted the other Yukawa couplings, since they do not contribute significantly to the Higgs quartic coupling and gauge couplings.

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