Hierarchy problem, gauge coupling unification at the Planck scale, and vacuum stability

From the point of view of the gauge hierarchy problem, introducing an intermediate scale in addition to TeV scale and the Planck scale ($M_{\rm Pl} = 2.4 \times 10^{18}\,{\rm GeV}$) is unfavorable. In that way, a gauge coupling unification (GCU) is expected to be realized at $M_{\rm Pl}$. We explore possibilities of GCU at $M_{\rm Pl}$ by adding a few extra particles with TeV scale mass into the standard model (SM). When extra particles are fermions and scalars (only fermions) with the same mass, the GCU at $M_{\rm Pl}$ can (not) be realized. On the other hand, when extra fermions have different masses, the GCU can be realized around $\sqrt{8 \pi} M_{\rm Pl}$ without extra scalars. This simple SM extension has two advantages that a vacuum becomes stable up to $M_{\rm Pl}$ ($\sqrt{8 \pi} M_{\rm Pl}$) and a proton lifetime becomes much longer than an experimental bound.


Introduction
The collider experiments have discovered all particles in the standard model (SM), and properties of the SM particles are gradually revealed. Especially, masses of the Higgs boson and top quark are important to investigate a behavior of the quartic coupling of the Higgs boson at a high energy scale. The measurement of Higgs mass showed 125.6 ± 0.35 GeV [1], and a recent combined analysis of the collider experiments reported the top mass as 173.34 ± 0.76 GeV [2]. A running of the quartic coupling of the Higgs becomes negative around 10 10 GeV by use of the experimental values of the Higgs and top masses. This behavior seems to indicate that our vacuum is metastable.
There are several ways to make the vacuum stable. A simple way is to add an extra scalar to the SM. When we assign odd parity to it under an extra Z 2 symmetry, it can be a dark matter [3][4][5][6][7][8][9][10][11][12][13][14][15][16]. Another way to stabilize the vacuum is modifying runnings of the gauge coupling constants. It decreases (increases) the values of the top Yukawa (Higgs self-)coupling at a high energy, where the vacuum becomes stabilized. In this paper, we try to realize the gauge coupling unification (GCU) at the Planck scale by introducing additional particles in the TeV scale. This extension really induces the above modification of runnings of the gauge coupling constants.
The so-called hierarchy problem is related to the Higgs sector in the SM. A quadratic divergence of the Higgs mass seems to be a dangerous problem. However, the Bardeenüfs argument [17] says that it is unphysical because it can be removed by a subtractive renormalization. 1 Once it is subtracted and the Higgs mass term is vanishing at the UV scale, it continues to be zero toward the lower energy scale, since the renormalization group equation (RGE) of the Higgs mass term is proportional to itself. We assume a classical conformal symmetry to justify the vanishing Higgs mass term at the high energy scale. This symmetry can be radiatively broken by Coleman-Weinberg mechanism [20]. We can see this situation, for example, in a model with an additional U(1) gauge symmetry and three right-handed neutrinos [21]. Note that the right-handed neutrinos do not change the running of the SM gauge couplings up to the one-loop level, so they are not useful to realize the GCU at the Planck scale.
On the other hand, a logarithmic divergence remains a physical quantity after the renormalization. When there is a heavy particle with the mass, M, which couples the Higgs doublet, a quantum correction of M 2 log( /μ) causes the hierarchy problem. Thus, naively, we should not introduce any intermediate scales between TeV and UV scales. We assume here that the UV scale is the Planck scale, where all quantum corrections to the Higgs mass are completely vanishing. This assumption requires that corrections from breaking effects of the grand unification at the Planck scale are canceled by a boundary condition of the UV complete theory. Although this assumption seems to be artificial, some UV complete theories, e.g., the string theory, really provides such a boundary condition.
In addition to the above discussion about the hierarchy problem (for example, Ref. [22]), we mention gravity, which involves a specific scale, i.e., the Planck scale. In the point of view of the classical conformal symmetry, there should be no specific scales and no higher-dimensional operators at the classical level. Thus, a certain scale including the Planck scale should be generated by some dynamics. For this purpose, it is known that the Planck scale arises from the vacuum expectation value of a SM gauge singlet scalar, which has a non-minimal coupling with the curvature [23,24]. Since a mechanism of generating the VEV depends on the hidden sector, the situation is the same as the above discussion in the decoupling limit between the singlet scalar and the Higgs. Then, the hierarchy problem can be solved by a boundary condition at the Planck scale, in which the Higgs mass term is completely vanishing.
For contributions of gravity to the gauge couplings, they could not be ignored around the Planck scale. Then they might upset discussion of the GCU at the Planck scale. To solve this problem, it is known that the GCU could be realized due to the asymptotic safety of gravity, in which all gauge couplings rapidly become zero and approach the same value around the Planck scale. In this scenario, the gravitational contributions have been calculated at lowest nontrivial order in perturbation theory [25]. However, it is pointed out that this calculation depends on a regularization scheme and/or a choice of gauge fixing [26]. In addition, if one applies the dimensional regularization for the calculation, there are no gravitational corrections for the gauge couplings. Thus, we do not consider the gravitational corrections in this paper.
In this paper, we will consider that the Planck scale is the bound of the UV complete theory, in which we assume corrections of the Higgs mass term are completely vanishing at the scale. We also assume that the Higgs mass term is generated by Coleman-Weinberg mechanism and it does not cause the hierarchy problem. In this background, we will consider the GCU at the Planck scale to avoid the introduction of any intermediate scales except for the TeV scale. We introduce extra particles with masses around the TeV scale. In order to avoid the gauge anomaly, the additional fermionic particles are introduced as vector-like. A naive analysis will show that, when all extra particles are fermions and their masses are the same, the GCU at the Planck scale cannot be realized. On the other hand, when extra particles include some scalars, the GCU at the Planck scale can be realized. Then, we find that there are a number of models which can realize the GCU at the Planck scale. Next, we will consider another situation, in which extra fermions have different masses. In this case, models with only extra fermions (no scalars) can realize the GCU around √ 8πM Pl . These extensions make the gauge couplings strong enough to realize the GCU, and the top Yukawa (Higgs self-)coupling becomes smaller (larger) than that of the SM at a high energy scale. Then, the vacuum becomes stable.
This paper is composed as follows. At first, we will give a brief review of the vacuum stability and related researches in the SM in Section 2. Next, we will investigate possibilities for the realization of GCU at some high energy scales in Section 3, and show conditions of the GCU at the Planck scale in Section 4. Then, examples of extra particles, which satisfy the conditions, are given in Section 5. In addition, we will consider other possibilities, in which the GCU can be realized only by extra fermions, in Section 6. Finally, summary and discussion are given in Section 7.

The vacuum stability
We give a brief review of the vacuum stability and related researches in the SM. Realization of the vacuum stability depends on a value of the Higgs quartic coupling λ. A running of λ is obtained by solving the RGE dλ/d ln μ = β λ , in which μ is a renormalization scale and β λ is the β-function of λ. The β-function of λ up to two-loop level is given by [27,28]   If the minimal value of λ is zero with β λ = 0 at some high energy scales, the vacua at the EW and the high energy scale are degenerate. This requirement is known as the multiple point criticality principle (MPCP) [29]. Note that the MPCP can be realized at O(10 17 ) GeV by use of a lighter top mass as 171 GeV (see also Refs. [28,[30][31][32][33][34][35][36][37][38][39][40] for more recent analyses).
From Fig. 1, we can show a minimum of the Higgs potential. It is given by V eff (φ) = 1 4 λφ 4 , where φ is a field value of the Higgs, and its stationary condition satisfies β λ + 4λ = 0. This equation is satisfied when |λ| becomes almost zero, and its solutions are classified in three cases as follows: • λ = 0 and β λ = 0: this is just the MPCP condition, where the height of the potential becomes zero. • λ > 0 and β λ < 0: this point is a local maximum before λ becomes a minimal value. If there is another solution for λ > 0 and β λ < 0, the point is a local minimum. • λ < 0 and β λ > 0: this point is a global minimum.
For (M h , M t ) = (125.7 GeV, 173.3 GeV), the Higgs potential has a local maximum and global minimum at φ 9.5 × 10 9 GeV and φ 3.9 × 10 29 GeV, respectively. 2 When M h is larger than 125.7 GeV and/or M t is smaller than 173.3 GeV, the points of local maximum and global minimum are larger and smaller, respectively. For M t 171.2043 GeV, the potential is positive in any energy scale, and there are no global minimum in the high energy scale. Only for Table 1 c 2 (R) for irreducible representations of SU(2) (left) and SU(3) (right).

Representation of SU(2)
171.2041 GeV M t 171.2043 GeV, the potential has a local minimum at 4.7 × 10 17 GeV φ 6.1 × 10 17 GeV. When the potential has a plateau around the local minimum, the Higgs inflation can be realized. However, if the Higgs potential includes new contributions as higher order terms of φ, they can significantly affects the vacuum stability [41][42][43].

Requirement for the GCU
In this section, we investigate possibilities for the realization of GCU at some high energy scales. In order to see the behavior of the gauge couplings in an arbitrary high energy scale we have to solve the corresponding RGEs. The one-loop level RGEs of the gauge couplings α i = g 2 i /4π are given by where i = Y , 2, and 3, and the coefficients of U(1) Y , SU(2) L , and SU(3) C gauge couplings are is obtained by multiplying a GUT normalization factor 3/5 to b SM Y as b SM 1 = 41/10. 3 Once particle contents in the model are fixed, values of b i are calculated by [44] where j = Y , 2, and 3. The meanings of the notation are as follows: Some values of c 2 (R) are given in Table 1 in a convention [45,46]. The factor κ is 1 or 1/2 for Dirac or Weyl fermions, respectively. In addition, the factor η is 1 or 1/2 for complex or real scalars, respectively. Using the values, we can obtain contributions to b i from fermions and scalars. Table 2 Contributions to b i from anomaly free fermions. U(1) Y hypercharge "a" can take different values for different representations, and an electric charge is given by Since the GCU is not realized in the SM, one has to extend the SM for the realization of GCU. We will consider adding extra particles with the TeV scale mass to the SM without any additional gauge symmetry. The extra particles with the TeV scale mass are motivated by avoiding the gauge hierarchy problem. Once we fix extra particles, we can easily calculate the values of b i by using Table 1. However, we have to take care of gauge anomalies induced from extra fermions. The simplest way to avoid the anomalies is to add extra fermions as a vector-like form. Thus, in this paper, we will introduce the extra Weyl fermions as a vector-like form except for real representations such as (1, 1, 0), (1, 3, 0), (8, 1, 0), and (8, 3, 0), which do not yield any gauge anomaly. Although the anomalies can be accidentally canceled as in the SM, we do not consider such cases. Contributions of anomaly free fermions to b i are given in Table 2, which shows only small representations up to an adjoint representation, (8, 3, a). In the same way, contributions from complex scalar particles to b i are given in Table 3. For real scalar particles, contribution to b i is half of the value in Table 3 because of η (see Eq. (3)).
Next, we investigate conditions for the GCU. The solution of Eq.
(2) are given by where M * is the mass scale of extra particles and M GUT is the GUT scale, in which the GCU can be realized. The GCU conditions are given by α −1 i (M GUT ) = α −1 j (M GUT ) ≡ α −1 GUT for i, j = 1, 2, and 3. Then, it can be written by Table 3 Contributions to b i by complex scalar particles. U(1) Y hypercharge "a" can take different values for different representations, and an electric charge is given by 16 3 , 3) where where the lower and upper bounds correspond to M GUT = M Pl and √ 8πM Pl , respectively. The RGEs and their boundary conditions in this analysis are given in Appendix A.
In addition to these constraint, we impose the conditions of α −1 i (M GUT ) > 0 to avoid the Landau pole (divergence of gauge couplings). Then, these conditions lead where the values correspond to the M GUT = M Pl ( √ 8πM Pl ) case. Since all b i are positive, gauge couplings become strong compared to those in the SM. In particular, extra fermions of large rep-resentations such as (6, 3, a) ⊕ (6, 3, −a) in Table 2 cannot be added to the SM because both b 2 and b 3 are larger than the upper bound. Similarly, extra particles with some large representations cannot also be added. Thus, since we need not to consider higher representations than the adjoint representation, extra fermions in Table 2 are sufficient to investigate the realization of GCU.

The GCU at the Planck scale by extra fermions
When all extra particles are fermions, one can see that the smallest value of b 2 and b 3 are 2/3 from Table 2, and then b 3 − b 2 ∝ 2/3. Thus, the cases of only extra fermions cannot satisfy Eq. (7), and unfortunately the GCU occurs at M GUT 9.0 × 10 16 GeV or 7.8 × 10 19 GeV, for b 3 − b 2 = 0 or 2/3, respectively. This is the same result in Ref. [47]. Note that, however, if we use two-loop RGEs and one-loop threshold corrections, the above results could be changed. In fact, there exists O(1) uncertainty in values of gauge couplings at a high energy scale. Thus, the GCU could be realized at the Planck scale even for b 3 − b 2 = 2/3. In addition, we can consider other possibility, in which extra fermions have different masses. In Section 6, we will show that the GCU at the Planck scale can be realized in this situation.

The GCU at the Planck scale by extra fermions and scalars
When extra particles include some scalars such as (1, 2, a), we can see that the smallest value of b 2 and b 3 are 1/6 from Table 3, and then b 3 − b 2 ∝ 1/6. Then, there are two cases to satisfy Eq. (7) in which the GCU is realized at the Planck scale as follows: • One is b 3 − b 2 = 1/3, which corresponds to M GUT M Pl . 4 In this case, b 1 is determined by the lower bound of Eq. (6). As a result, the GCU at M Pl can be realized by extra particles satisfying b 3 = 17 6 + n 6 (n = 0, 1, 2, · · · , and 35), where the minimum value of b 3 is determined to satisfy b 1 ≥ 0, and the largest value of n is determined by Eq. (9). • Another is b 3 − b 2 = 1/2, which corresponds to M GUT √ 8πM Pl because b 3 − b 2 = 1/2 corresponds to upper bound of Eq. (7). In this case, b 1 is determined by the upper bound of Eq. (6). Thus, the GCU at √ 8πM Pl can be realized by extra particles satisfying b 3 = 10 3 + n 6 (n = 0, 1, 2, · · · , and 33), (11) where the minimum value of b 3 is determined to satisfy b 1 ≥ 0, and the largest value of n is determined by the values in parentheses in Eq. (9).  Table 4 Contributions to b i by the SM fermions (with vector-like partners) and adjoint fermions.

Realization of the GCU at the Planck scale
According to the above discussions, we systematically investigate possibilities of the realization of GCU at the Planck scale, and find that a number of combinations of extra particles satisfy Eq. (10) or (11). For simplicity, we consider representation of extra fermions are the same as the SM fermions (with vector-like partners) and an SU(2) L adjoint fermion as in Table 4. Then, when we consider extra scalars are two SU(2) L doublets (1, 2, 0), the GCU can be realized at Table 5 The leftmost column shows representations of extra fermions as (SU(3) C , SU(2) L , U(1) Y ). With two SU(2) L doublets as (1, 2, 0), these extra fermions satisfy Eq. (10). In all cases, we take M * = 1 TeV, and the GCU is realized at M Pl . In the rightmost column, n is given in Eq. (10). M Pl by extra fermions shown in Table 5. 5 In all cases, masses of extra particles are 1 TeV. The values of the gauge couplings at M GUT are calculated by Eq. (4). They are characterized by n given in Eq. (10), which is shown in the rightmost column. The larger n (equivalently b i ) becomes, the smaller α −1 GUT becomes. We denote the pair of singlets (1, 1, a) ⊕ (1, 1, −a) could be used for tuning the running of g 1 because it only affects b 1 . In addition, we did not list a complete gauge singlet fermion (1, 1, 0), which is usually considered as a right-handed neutrino, because this fermion does not affect the GCU.
For a typical example, we consider the first one of Table 5. In Fig. 3 we show the runnings of gauge and top Yukawa couplings in the extended SM model. Here, we assume that coupling constants of extra particles to the SM particles are negligibly small, and thus introductions of the particles do not significantly change the runnings of top Yukawa and Higgs quartic couplings. The solid and dashed lines correspond to the cases of the extended SM and the SM, respectively. 5 Stable TeV-scale particles with fractional electric charge such as SU(2) L doublet scalar (1, 2, 0) might cause cosmological problems. In order to avoid the problems, the reheating temperature after the inflation should be about 40 times lower than the particle masses [48]. In the case, the corresponding particles cannot be thermally produced in the universe. Thus, since the reheating temperature should be larger than the QCD scale, we consider that it is O(10) GeV in the case. Table 6 Examples of combinations of extra fermions which realize the GCU around √ 8πM Pl . In the leftmost column, the characters show extra fermions as in Table 4, and the values in bracket show the fermion masses with a unit of TeV.
We can see that the GCU is realized at M Pl as mentioned above. In addition, the value of gauge couplings at M GUT is α −1 GUT 28.0 as in Table 5. From Fig. 3, we can expect that the Higgs quartic coupling λ is positive up to the Planck scale. This reason is understood as follows. In the extended SM, all gauge couplings are large compared to those in the SM because of b i ≥ b SM i . Then, y t becomes smaller due to the large gauge couplings (see Eq. (19)). Moreover, since β λ almost depends on quartic terms of y t and g i , the smaller y t and the larger g i make β λ become larger (see Eq. (20)

The GCU only with extra fermions
Next, we consider other situations, in which extra fermions have different masses. In the same way as before, we consider extra fermions within Table 4. Moreover, their masses are taken as 0.5 TeV ≤ M ≤ 10 TeV. Actually, we take only lepton masses 0.5 TeV, since lower bounds of vector-like lepton and quark masses are around 200 GeV and 800 GeV, respectively [49][50][51]. Unfortunately, we find that the GCU at M Pl cannot be realized only by extra fermions. In Table 6, we show extra fermions which can realize the GCU around √ 8πM Pl . Here, we relax the GCU condition as √ 8πM Pl M GUT 2 √ 8πM Pl because one-loop analyses always have O(1) ambiguity. In the table, for example, "W × 1 (0.5)" shows one (1, 3, 0) fermions with a mass of 0.5 TeV. The reason why the GCU can be realized around √ 8πM Pl is understood by runnings of couplings as a following discussion.
In Fig. 4 we show the runnings of gauge, top Yukawa, and Higgs quartic couplings in the extended SM model which correspond to the first one of Table 6. Here, we assume couplings between the Higgs doublet and extra fermions are negligibly small, and extra fermions do not significantly change running of top Yukawa and Higgs quartic couplings. The solid and dashed lines correspond to the extended SM and the SM, respectively. We can see that the GCU is realized around √ 8πM Pl . When extra fermions have different masses, β-functions of gauge couplings change several times. Then, our previous naive analyses are modified, and values of M GUT shown in Fig. 2 have O(1) uncertainty. Thus, the GCU can be realized around √ 8πM Pl by extra fermions with b 3 − b 2 = 2/3. Note that, to realize the vacuum stability, couplings between the Higgs boson and extra fermions should be small as mentioned above. Finally, we mention the GCU at the string scale (M GUT = s ≈ 5.27 × 10 17 GeV). Fig. 2 shows that the GCU at the string scale could be realized by b 3 − b 2 = 0, 1/6, and 1/3. The O(1) difference could come from two-loop RGEs and one-loop threshold corrections. On the other hand, another possibility is discussed in Ref. [52]. In this paper, the authors consider several possible string-GUT models. Then, the GCU condition is given by where G N and α are the gravitational constant and the Regge slope, respectively. The factor k i (i = Y , 2, and 3) is the so-called Kac-Moody levels, and the values are different for the considering GUT models [53]. Particularly, k 2 and k 3 should be positive integer, and we take Kac-Moody levels as (k Y , k 2 , k 3 ) = (5/3, 1, 1), which are given in GUT models such as SU (5) and SO (10). However, for k 2 = 1 and/or k 3 = 1, the GCU conditions of our analyses are changed. When the new physic scale is M * = 1 TeV, the GCU at s can be realized by for (k Y , k 2 , k 3 ) = (13/3, 1, 2), which is given in the GUT model as SU(5) × SU (5) and SO(10) × SO (10). In the same way, the GCU at s can be realized by for (k Y , k 2 , k 3 ) = (2/3, 2, 1), which is given in the GUT model as E 7 . Both conditions can be satisfied only by extra fermions due to b 3 /2 − b 2 ∝ 1/3 and b 3 − b 2 /2 ∝ 1/3. Thus, in some string-GUT models, the GCU at s can be realized only by extra fermions.

Summary and discussion
We  Table 5.
Moreover, we have considered other situations, in which extra fermions have different masses. In this case, extra fermions can realize the GCU around √ 8πM Pl as in Table 6. Since β-functions of gauge couplings change several times by extra fermions with different masses, our previous naive analyses are modified, and the GCU can be realized around √ 8πM Pl . Note that, if we use the two-loop RGEs and one-loop threshold corrections, these results could change, and other possibilities could exist.
If there are no intermediate scales between the TeV scale and the GCU scale, and quantum corrections to the Higgs mass term are completely vanishing at the GCU scale due to a UVcomplete theory, the Higgs mass receives quantum corrections only from TeV scale particles. In this paper, we have assumed that the GCU scale is the Planck scale, and the Higgs mass term are vanishing at the scale. More detailed discussion has been done in the introduction and Ref. [22]. When the GCU at the Planck scale is realized, gauge couplings become larger compared to the SM case. Then, top Yukawa and Higgs quartic couplings become smaller and larger, respectively. As a result, the vacuum can be stable up to the Planck scale.
Finally, we mention the proton lifetime in a GUT model. Although we do not discuss any specific GUT model, the proton lifetime should be long enough to avoid the experimental lower bound. The proton lifetime is usually given by This is derived from a four-fermion approximation for the decay channel p → e + + π 0 . For M GUT M Pl , we obtain τ proton ∼ α −1 i (M GUT ) 2 × 10 42 yrs. Since α −1 i (M GUT ) is larger than 1 (see Table 5), the proton lifetime is much longer than the experimental lower bound.