1 Introduction

Gauge coupling unification in a classically scale invariant model


Naoyuki Haba, Hiroyuki Ishida, Ryo Takahashi, and Yuya Yamaguchi

Graduate School of Science and Engineering, Shimane University,

Matsue 690-8504, Japan

Graduate School of Science, Tohoku University,

Sendai, 980-8578 Japan

Department of Physics, Faculty of Science, Hokkaido University,

Sapporo 060-0810, Japan

Abstract

[5mm] There are a lot of works within a class of classically scale invariant model, which is motivated by solving the gauge hierarchy problem. In this context, the Higgs mass vanishes at the UV scale due to the classically scale invariance, and is generated via the Coleman-Weinberg mechanism. Since the mass generation should occur not so far from the electroweak scale, we extend the standard model only around the TeV scale. We construct a model which can achieve the gauge coupling unification at the UV scale. In the same way, the model can realize the vacuum stability, smallness of active neutrino masses, baryon asymmetry of the universe, and dark matter relic abundance. The model predicts the existence vector-like fermions charged under with masses lower than 1 TeV, and the SM singlet Majorana dark matter with mass lower than 2.6 TeV.

1 Introduction

The Higgs mass parameter is only a dimensionful parameter in the standard model (SM), and its value is estimated by the observed Higgs mass as  [1]. Then, a running of the Higgs quartic coupling becomes negative below the Planck scale within the SM. If the SM can be valid up to a high energy scale such as a breaking scale of a gauge symmetry in the grand unification theory (GUT), the electroweak (EW) scale should be stabilized against radiative corrections coming from the high energy physics. To solve the gauge hierarchy problem, there are a lot of works motivated by a classically scale invariance [2]-[29]. The scale invariance prohibits dimensionful parameters at a classical level, while it can be radiatively broken by the Coleman-Weinberg (CW) mechanism [30]. In addition to the classically scale invariance, with an additional gauge symmetry, e.g., gauge symmetry, it is possible to naturally realize experimentally observed values of the Higgs mass. When the symmetry is broken by the CW mechanism, the EW symmetry could be also broken through the scalar mixing term. If the breaking scale is not far from the EW scale, the Higgs mass corrections would be sufficiently small, and then the hierarchy problem can be solved. Note that these statements are based on the Bardeen’s argument [31], and we consider only logarithmic divergences in this paper (see Ref. [7] for more detailed discussions).

In this paper, we assume the classically scale invariance at the UV scale, where the SM gauge couplings are unified. We expect that some unknown mechanism, such as a string theory, realizes the classically scale invariance and the gauge coupling unification (GCU). Actually, the GCU can be realized at in our model, and the scale is near the typical string scale (). To realize the GCU, some additional particles with the SM gauge charges are needed. Conditions of the GCU can be systematically obtained by an analysis of renormalization group equations (RGEs) [32, 33]. When all additional particles are vector-like fermions with the TeV scale masses, the GCU scale can be realized between  GeV and  GeV, and there are a lot of possibilities to realize the GCU at the scale.111 For example, we can consider the origin of the vector-like fermions as the string theory, in which a number of vector-like fermions should appear above the compact scale, which is expected to be the GCU scale in our model. Some of them might have the TeV scale masses due to the fine-tuning of moduli (or Wilson line, extra-dimensional component of anti-symmetric tensor field, and so on). For example, vector-like pairs of quark doublet and down-type quark singlet can achieve the GCU [34, 35]. When there are additional fermions charged under the SM gauge symmetries, the gauge couplings and the top Yukawa coupling respectively become larger and smaller compared to the SM case, and then, both changes make the function of the Higgs quartic coupling become larger. Therefore, the vacuum can become stable when the GCU is realized.

To solve the gauge hierarchy problem, there should be no intermediate scale between the EW and the GCU scales except an energy scale, which is not so far from the EW scale, i.e., the TeV scale. Then, phenomenological and cosmological problems (e.g., smallness of active neutrino masses, baryon asymmetry of the universe, and dark matter (DM)) should be explained with sufficiently small Higgs mass corrections. The first two problems can be explained by the right-handed neutrinos, which are naturally introduced to cancel the anomalies accompanied with the gauge symmetry, via type-I seesaw mechanism [36] and resonant leptogenesis [37], respectively. In our model, the DM is identified with the SM singlet Majorana fermions, and its stability can be guaranteed by an additional symmetry [38]. In this paper, we will show that our model can explain the above problems as well as realizing the GCU without affecting the hierarchy problem.222 From theoretical point of view, there are some papers constructing a model which realizes classically scale invariance and gauge coupling unification at the same scale [39]-[41]. Furthermore, asymptotic safety of gravity [42] leads vanishing couplings at the UV scale, which suggests vanishing quartic couplings and gauge coupling unification around the Planck scale [see Fig. 1 in Ref. [43] for example]. In this paper, we simply expect such a situation comes from unknown UV physics.

In the next section, we will define our model, and explain the gauge symmetry breaking as well as the EW symmetry breaking via the CW mechanism. We also obtain the upper bound on the breaking scale from the naturalness. In Sec. 3, we will discuss the GCU, vacuum stability, smallness of active neutrino masses, baryon asymmetry of the universe, and the DM relic abundance. Our model predicts the existence vector-like fermions charged under with masses lower than 1 TeV, and the SM singlet Majorana dark matter with mass lower than 2.6 TeV. We summarize our results in Sec. 4.

2 Symmetry breaking mechanism

We consider the gauge extension of the SM with three generations of the right-handed neutrinos (), six vector-like fermions (, , , , , and ), and two SM singlet scalars ( and ). Charge assignments of the particles are shown in Table 1. The charge are given by , where , , , and denote a real number, the baryon and lepton numbers, and the hypercharge, respectively. In particular, , and correspond to , and , respectively. The vector-like fermions , , and respectively have the same charges as the SM quark doublet, the SM down-quark singlet, and the right-handed neutrino, while only the vector-like fermions are odd under an additional symmetry. Four of the vector-like fermions ( and ) play a role for achieving the GCU, and the others () are the DM candidates, whose stability is guaranteed by the symmetry. These particles are not necessary for the realization of GCU and DM. We choose them for the simplest extension.

(3, 2, 1/6)
(3, 1, )
(3, 1, )
(1, 2, )
(1, 1, )
(1, 1, 0)
(1, 2, 1/2)
(3, 2, )
(3, 1, )
(1, 1, 0)
(1, 1, 0)
(1, 1, 0) 0
Table 1: Charge assignment of particles, where is a real number.

The relevant Lagrangian is given by

(1)

where is the SM Lagrangian except for the Higgs sector, includes kinetic terms of the Higgs and new particles, and is a scalar potential of the model. Without the symmetry, there are also additional Yukawa interactions between the SM particles and the new particles, e.g., , , and . However, these coupling constants have to be very small due to constraints from the precision electroweak data [44]. To forbid these terms, we have imposed odd parity to only the vector-like fermions under the symmetry.

Since there are two gauge symmetry, kinetic mixing generally arises in the model. We can take covariant derivative as

(2)

where ’s are gauge couplings, and are generators of and , respectively, and () are gauge bosons. The coupling constant denotes the kinetic mixing between the and the gauge symmetries, and we will take at the GCU scale. This boundary condition naturally arises from breaking a simple unified gauge group into .

We impose the classically scale invariance at the GCU scale, and hence, the scalar potential is given by

(3)

where there is no dimensionful parameter. In the model, a complex scalar singlet spontaneously breaks the gauge symmetry due to radiative corrections, i.e. the CW mechanism. Since the complex scalar field obtains the nonzero vacuum expectation value (VEV), the SM singlet scalar , the gauge boson , the right-handed neutrinos and the vector-like fermion become massive. After the symmetry breaking, negative mass terms of a real scalar singlet and the SM Higgs doublet are generated, which induces the EW symmetry breaking. Then, , the vector-like fermions and the SM particles become massive, and typically their masses are lighter than those obtained by the symmetry breaking.

Let us explain the symmetry breaking mechanism more explicitly. We consider the CW potential for a classical field of the singlet scalar as

(4)

where we have taken without loss of generality, and is the VEV of . functions of , , almost depends on quartic terms of , and for . ( functions of the model parameters are given in Appendix.) The effective potential (4) satisfies the following renormalization conditions

(5)

and the minimization condition of induces

(6)

where we have assumed that the scalar quartic couplings are negligibly small in the right-hand side. When this relation is satisfied, the symmetry is broken, and and become massive as

(7)

respectively. Since the right-hand side of Eq. (6) should be positive, is required, and hence, is generally expected. In addition, the quartic terms of Majorana Yukawa couplings ( and ) are smaller than the quartic terms of because of . The masses of right-handed neutrinos and will be discussed in Sec. 3.3.

After the symmetry breaking, the effective potentials for and are approximately given by

(8)

where and . Here, we have assumed that are negligibly small compared to and for simplicity. For , is always negligibly small during renormalization group evolution [see Eq. (41)]. When and are negative, the nonzero VEVs and are obtained as

(9)

Note that and is typically lower than , because the ratios of quartic couplings ( and ) should be lower than unity to avoid the vacuum instability. The vector-like fermions and the SM particles become massive, while the masses of vector-like fermions ( and ) have to be lower than 1 TeV to realize the GCU as we will show in Sec. 3.1.

In the end of this section, we mention the breaking scale, which is described by . Since  TeV is required from the LEP-II experiments [45], we obtain the lower bound  TeV. On the other hand, the naturalness of the Higgs mass suggests a relatively small . A major correction to the Higgs mass is given by intermediating diagrams, and one-loop and two-loop corrections are approximately written as

(10)
(11)

respectively. When one defines requirement of the naturalness as , Eqs. (10) and (11) lead the upper bound on as

(12)
(13)

where we have taken . For , the two-loop correction gives stronger bound than one-loop correction. In the following, we will use the stronger bound for fixed . Note that the mass correction from is always negligible because of a small mixing coupling .

3 Phenomenological and cosmological aspects

In this section, we will discuss phenomenological and cosmological aspects of the model: the GCU, vacuum stability and triviality, smallness of active neutrino masses, baryon asymmetry of the universe, and dark matter. We will also restrict the model parameters from the naturalness of the Higgs mass.

3.1 Gauge coupling unification

First, we discuss the possibility of the GCU at a high energy scale. Since four additional vector-like fermions ( and ) have gauge charges under the SM gauge groups as shown in Tab. 1, runnings of the SM gauge couplings are modified from the SM. Then, functions of gauge coupling constants are given by

(14)

at 1-loop level. Figure 1 shows runnings of gauge couplings , where gauge coupling is normalized as .

Figure 1: Runnings of gauge couplings . The dashed and solid lines correspond to the SM and the extended model cases, respectively. The vertical lines express  GeV and  GeV.

The calculation has been done for with using 2-loop RGEs. We note that the running of gauge couplings are almost independent of . In the figure, the horizontal axis is the renormalization scale and the vertical axis indicates value of . The red, green, and blue lines show , , and , respectively. The dashed and solid lines correspond to the SM and our model, respectively. The left vertical line stands for a typical scale of vector-like fermions, which has been taken as  GeV in Fig. 1. For , the functions are the SM ones, and we take boundary conditions for the gauge couplings such that experimental values of the Weinberg angle, the fine structure constant, and the strong coupling can be reproduced [46]. The GCU can be achieved at  GeV, and the unified gauge coupling is .333 The GCU can be achieved by adjoint fermions as in Ref. [47, 48]. This is the same result as in Ref. [34], in which only and are added into the SM. As the vector-like fermion masses become larger, the precision of the GCU becomes worse. Thus, the masses of and should be lighter than 1 TeV, while vector-like fermion masses are constrained by the LHC experiments [49, 50, 51]. Since the lower bound of vector-like quark lies around 700 GeV, the possibility of the GCU can be testable in the near future.

We note that the proton lifetime in a GUT model. The proton lifetime is roughly derived from a four-fermion approximation for the decay channel , which is given by

(15)

where is the proton mass. For  GeV and , we can estimate yrs, which is much longer than the experimental lower bound yrs [52]. Thus, the model are free from the constraint of the proton decay.

3.2 Vacuum stability and triviality

Next, we discuss the vacuum stability. However, it is difficult to investigate exact vacuum stability conditions, since there are three scalar fields and each of them has nonzero VEVs. Therefore, we simply investigate three necessary conditions: , and .

Figure 2: Running of in the () case. The red and blue lines correspond to and , respectively. The black dashed line shows running of in the SM. The vertical lines express  GeV and  GeV.

The condition depends on additional contributions to , i.e., , and scalar mixing couplings.444 Running of also depends on mass (or Yukawa coupling) of the top quark. We will use the central value of world average, i.e.,  [53]. If we change this value of top quark mass, the following numerical results can slightly change. If their contributions to are negligible, since the SM gauge couplings are larger compared to the SM case, running of is raised and always positive. For example, however, the EW vacuum becomes instable for in the () case. We show the running of for in Fig. 2, where is independent of up to the one-loop level, and contributions of can be negligible. The red and blue lines correspond to and , respectively. The black dashed line shows running of in the SM. Thus, is required to realize the vacuum stability.

The Higgs mass corrections from and loops are given by

(16)

where we have taken , which naturally arises from symmetry for the vector-like particles, and ( and ) for simplicity. Then, the naturalness requires for  TeV. Although is a contribution to the vector-like fermion masses from the Higgs, it can be ignored because of . Since the contribution of to , i.e., , is always positive for , the naturalness condition also guarantees the vacuum stability. Note that guarantees at any energy scale, which is required to justify our potential analysis for Eq. (8).

Here, we check contributions of vector-like fermions to the and parameters, which are approximately given by [54, 55]

(17)

where and are the Weinberg angle and the boson mass, respectively. For , the parameters are estimated as and , which are consistent with the precision EW data and [52].

The condition is almost always satisfied when is dominant in the right-hand side of Eq. (6), i.e., . In this case, is positive up to the GCU scale, and then is also positive up to the GCU scale. It is also possible to realize the critical condition as well as , where the running of is curved upward as in the so-called flatland scenario [9, 14, 16, 21, 24]. Then, both and Majorana Yukawa couplings are dominant in , while is much smaller than them. This means that there is a fine-tuning to satisfy Eq. (6).

When is negligible in its function, a solution of its RGE is approximately given by

(18)

where is a renormalization scale. Once is fixed, and are determined to realize the GCU, while remains a free parameter. To estimate the condition of , we assume at for simplicity. Then, we can find that is positive up to the GCU scale for . This lower bound of is almost unchanged for different values of , because dependence is logarithmic.

Figure 3: dependence on the upper (red) and lower (blue) bounds of , which correspond to the Landau pole and vacuum stability conditions, respectively. For the Landau pole bound, we take  GeV in Eq. (19).

On the other hand, when is dominant in , the Landau pole might exist, at which the theory is not valid from the point of view of perturbativity (triviality). The energy scale where the Landau pole appears is approximately estimated as

(19)

where is a mass of the real singlet scalar field. Figure 3 shows dependence on the upper (red) and lower (blue) bonds of , which correspond to the Landau pole and vacuum stability conditions, respectively. Since the both bounds are almost proportional to , allowed values of are almost unchanged for different . We can find a strong constraint for as .

Figure 4: The upper bound of for fixed , which depends on . The solid and dashed lines show the Landau pole (20) and the naturalness (Eqs. (12) and (13)) bounds, respectively. For the Landau pole bound, we take  GeV in Eq. (20). The shaded region ( TeV) is excluded by the LHC experiments.

In the same way, the Landau pole also exists when is sufficiently large. The energy scale where the Landau pole appears is approximately estimated by the one-loop RGE of as

(20)

where is given in Eq (7). Figure 4 shows the upper bound of for fixed , which depends on . The solid lines show the maximal value of allowed in the model, which are calculated by Eq. (20) for  GeV. Note that the peak of solid lines at corresponds to the orthogonal basis of two gauges. The dashed lines show the naturalness bound estimated by Eqs. (12) and (13). The red, green, and blue colors correspond to , 100, and 1000 TeV, respectively. The shaded region ( TeV) is excluded by the LHC experiments [56, 57]. When we define the triviality bound as , it prohibits the regions above the solid lines. One can see that the bound leads from Eq. (7), which is almost independent of . Since the naturalness requires the stronger constraints than the triviality bound in almost all parameter space, we can say that the naturalness guarantees no Landau pole below the GCU scale. Note that the both bounds are almost the same for  TeV, and they exclude  TeV.

3.3 Neutrino masses and baryon asymmetry of the universe

From the Lagrangian (1), the neutrino mass terms are given by

(21)

where , , , and . There is no mixing term between and due to the symmetry. The active neutrino masses can be obtained by the usual type-I seesaw mechanism [36], i.e., . The heavier mass eigenvalue is nearly equal to , whose upper bound is given by the naturalness of the Higgs mass. Neutrino one-loop diagram contributes the Higgs mass as

(22)

where we have used the seesaw relation. For  eV, the naturalness requires  GeV.

We mention the baryon asymmetry of the universe. In the normal thermal leptogenesis [58], there is a lower bound on the right-handed neutrino mass as  GeV [59]. However, the resonant leptogenesis can work even at the TeV scale, where two right-handed neutrino masses are well-degenerated [37]. In our model, additional gauge interactions make the right-handed neutrinos be in thermal equilibrium with the SM particles [60]. A large efficiency factor can be easily obtained, and the sufficient baryon asymmetry of the universe can be generated by the right-handed neutrinos with a few TeV masses. Since the neutrino Yukawa coupling and almost do not depend on the other phenomenological problems, we can do the same analysis as in Ref. [60], and hence, the result is also the same as in Ref. [60].

For the vector-like neutrinos (), we consider , which naturally arises from symmetry for the vector-like fermions. Then, the mass eigenvalues are respectively and for and . The lighter mass eigenstate is a DM candidate, because its stability is guaranteed by the symmetry. In the limit of (), and are degenerate, and is also effective for a calculation of the DM relic abundance. In the next subsection, we will investigate the degenerate case.

In our model, the gauge symmetry is successfully achieved via the CW mechanism. It requires in Eq. (6), that is,

(23)

where is a relevant number of right-handed neutrinos, which is defined as . Thus, the Majorana masses must be lighter than the boson mass. We have made sure that this constraint is always satisfied when explain the DM relic abundance.

3.4 Dark matter

Figure 5: Annihilation processes of the dark matter ().

To calculate the DM relic abundance, we use the same formula for the DM annihilation cross sections as in Ref. [19], where a new vector-like fermion is only (or ), and the SM fermions do not have charges. The annihilation processes are -channel , -channel , and mediated -channel . The corresponding diagrams are shown in Fig. 5. Although our model has other contributions to the annihilation cross sections, they are all negligible in the following setup. We consider the degenerate case for simplicity, in which there is no vector-like mass term of . Thus, -channel process and mediated -channel process does not occur at tree level. From Eq. (23), is always required. Then, the annihilation cross section , where is some charged fermion, is suppressed by . As a result, we can use the same formula for the DM annihilation cross sections as in Ref. [19].

The spin independent cross section for the direct detection is almost dominated by t-channel exchange of scalars and , which has been considered in Ref. [19]. However, our model has an additional contribution due to exchange diagrams, which is given by [61]

(24)

where is the nucleon mass, and is the reduced nucleon mass. For the DM with the masses of 100 GeV and 1 TeV, the small regions such as and are excluded by the LUX experiment, respectively [63]. These bound are stronger than the LEP bound, where is excluded.

In the following, we consider () case. There are six new parameters in the model: the gauge coupling , the two Majorana Yukawa coupling , , the two quartic couplings , , and the VEV of the complex scalar field . On the other hand, there are two conditions and Eq. (9), and we require that explains the DM relic abundance  [62]. Thus, we have three free parameters for the DM analysis.

Figure 6: Scatter plots in (, ) plane (left) and (, ) plane (right), which realize the DM relic abundance , and satisfy all constraints as discussed in this paper as well as the LUX bound. The horizontal line shows the lower bound on by the LHC experiments. The red, pink green, cyan, and blue dots correspond to , , , , , respectively.

Figure 6 shows scatter plots in (, ) plane (left) and (, ) plane (right), which realize the DM relic abundance , and satisfy all constraints as discussed above as well as the LUX bound. The parameter space starts from the initial values , , and . Although the two figures in Fig. 6 are very similar, is always satisfied. The region of is excluded by the current LHC bound [56, 57]. Since is required to avoid the Landau pole, the upper bound on is given by , while the upper bound in the region is given by the naturalness (13). In the region, the lower bound on is given by the LUX bound. To realize the DM relic abundance, sufficiently large annihilation cross sections are required, which induce the lower bound on in the region. From Fig. 6, we can see the upper bound on the DM mass as , and the bound of is almost the same as .

4 Conclusion

To solve the gauge hierarchy problem, we have constructed a classically scale invariant model with a gauge extension. We have assumed the classical scale invariance at the GCU scale, where the Higgs mass completely vanishes even with some quantum corrections. The scale invariance is violated around the TeV scale by the CW mechanism, and the Higgs mass can be naturally generated through the scalar mixing term. The GCU is realized by vector-like fermions and , which respectively have the same quantum number as the SM quark doublet and down-type quark singlet but distinguished by the additional symmetry, and their masses lie in . The GCU scale is  GeV with , and the proton life time is estimated as yrs, which is much longer than the experimental lower bound yrs.

In addition, we have shown that the model can explain the vacuum stability, smallness of active neutrino masses, baryon asymmetry of the universe, and dark matter relic abundance without inducing large Higgs mass corrections. Since there are additional fermions with the SM gauge charges, the SM gauge couplings become larger than the SM case, which leads smaller top Yukawa couplings. Then, the function of the Higgs quartic coupling becomes larger, and hence the EW vacuum becomes stable. The smallness of active neutrino masses and the baryon asymmetry of the universe can be explained by the right-handed neutrinos via the type-I seesaw mechanism and resonant leptogenesis, respectively. The DM candidate is the SM singlet Majorana fermions , and stability of the DM is guaranteed by the additional symmetry. We have analyzed the DM relic abundance in the degenerate case (), and found the upper bound on the DM mass as .

Acknowledgment

The authors thank Y. Kawamura for helpful discussion on the origin of the vector-like fermions. This work is partially supported by Scientific Grants by the Ministry of Education, Culture, Sports, Science and Technology (Nos. 24540272, 26247038, and 15H01037). The work of Y.Y. is supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists (Grants No. 262428).

Appendix

functions in the extended SM

We give one-loop -functions in our model:

(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)