Gauge coupling unification in a classically scale invariant model

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 $SU(3)_C$ with masses lower than $1\,{\rm TeV}$, and the SM singlet Majorana dark matter with mass lower than $2.6\,{\rm TeV}$.


Introduction
The Higgs mass parameter m 2 h is only a dimensionful parameter in the standard model (SM), and its value is estimated by the observed Higgs mass as −2m 2 h = M h = 125.09 ± 0.21 (stat.) ± 0.11 (syst.) GeV [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 U(1) X gauge symmetry, e.g., U(1) B−L gauge symmetry, it is possible to naturally realize experimentally observed values of the Higgs mass. When the U(1) X symmetry is broken by the CW mechanism, the EW symmetry could be also broken through the scalar mixing term. If the U(1) X 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 3 × 10 16 GeV in our model, and the scale is near the typical string scale (∼ 10 17 GeV). 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 10 16 GeV and 10 17 GeV, and there are a lot of possibilities to realize the GCU at the scale. 1 For example, vector-like pairs of quark doublet Q L,R and down-type quark singlet D L,R 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 β 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 U(1) X 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 Z 2 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. 2 In the next section, we will define our model, and explain the U(1) X gauge symmetry breaking as well as the EW symmetry breaking via the CW mechanism. We also obtain the upper bound on the U(1) X 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 vectorlike fermions charged under SU(3) C 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.

Symmetry breaking mechanism
We consider the U(1) X gauge extension of the SM with three generations of the righthanded neutrinos ν R i (i = 1, 2, 3), six vector-like fermions (Q L , Q R , D L , D R , N L , and N R ), and two SM singlet scalars (Φ and S). Charge assignments of the particles are shown in Table 1. The U(1) X charge are given by B − L + 2x H Y , where x H , B, L, and Y denote a real number, the baryon and lepton numbers, and the U(1) Y hypercharge, respectively. In particular, x H = 0, −1 and −2/5 correspond to U(1) B−L , U(1) R and U(1) χ , respectively.
The vector-like fermions Q L,R , D L,R , and N L,R 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 Z 2 symmetry. Four of the vector-like fermions (Q L,R and D L,R ) play a role for achieving the GCU, and the others (N L,R ) are the DM candidates, whose stability is guaranteed by the Z 2 symmetry. These particles are not necessary for the realization of GCU and DM. We choose them for the simplest (1, 1, 0) 0 + where L SM is the SM Lagrangian except for the Higgs sector, L kinetic includes kinetic terms of the Higgs and new particles, and V (H, Φ, S) is a scalar potential of the model. Without the Z 2 symmetry, there are also additional Yukawa interactions between the SM particles and the new particles, e.g., y 1 Q L H c u R , y 2 Q L Hd R , and y 3 q L HD R . 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 Z 2 symmetry.
Since there are two U(1) gauge symmetry, U(1) kinetic mixing generally arises in the model. We can take covariant derivative as where g's are gauge couplings, T α and T a are generators of SU(3) C and SU(2) L , respectively, and V µ (V = G α , W a , B, Z ′ ) are gauge bosons. The coupling constant g mix denotes the kinetic mixing between the U(1) Y and the U(1) X gauge symmetries, and we will take g mix = 0 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 V (H, Φ, S) is given by where there is no dimensionful parameter. In the model, a complex scalar singlet Φ spontaneously breaks the U(1) X 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 Φ, the U(1) X gauge boson Z ′ , the right-handed neutrinos and the vector-like fermion N L,R become massive. After the U(1) X symmetry breaking, negative mass terms of a real scalar singlet S and the SM Higgs doublet H are generated, which induces the EW symmetry breaking. Then, S, the vector-like fermions and the SM particles become massive, and typically their masses are lighter than those obtained by the U(1) X symmetry breaking. Let us explain the symmetry breaking mechanism more explicitly. We consider the CW potential for a classical field of the singlet scalar φ as where we have taken Φ = φ/ √ 2 without loss of generality, (β functions of the model parameters are given in Appendix.) The effective potential (4) satisfies the following renormalization conditions and the minimization condition of V Φ induces where we have assumed that the scalar quartic couplings are negligibly small in the righthand side. When this relation is satisfied, the U(1) X symmetry is broken, and Φ and Z ′ become massive as respectively. Since the right-hand side of Eq. (6) should be positive, is required, and hence, M φ < M Z ′ is generally expected. In addition, the quartic terms of Majorana Yukawa couplings (Y M and Y N L,R ) are smaller than the quartic terms of g X because of λ Φ (v Φ ) > 0. The masses of right-handed neutrinos and N L,R will be discussed in Sec. 3.3.
After the U(1) X symmetry breaking, the effective potentials for s and h are approximately given by where S = s/ √ 2 and H = (0, h/ √ 2) T . Here, we have assumed that λ HS are negligibly small compared to λ ΦS and λ HΦ for simplicity. For κ 1,2 ≃ 0, λ HS is always negligibly small during renormalization group evolution [see Eq. (41)]. When λ ΦS and λ HΦ are negative, the nonzero VEVs s = v S and h = v H are obtained as Note that v S and v H is typically lower than v Φ , because the ratios of quartic couplings (λ ΦS /(2λ S ) and λ HΦ /(2λ H )) should be lower than unity to avoid the vacuum instability. The vector-like fermions and the SM particles become massive, while the masses of vectorlike fermions (Q L,R and D L,R ) 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 U(1) X breaking scale, which is described by v Φ . Since M Z ′ /g X > 6.9 TeV is required from the LEP-II experiments [45], we obtain the lower bound v Φ 3.5 TeV. On the other hand, the naturalness of the Higgs mass suggests a relatively small v Φ . A major correction to the Higgs mass is given by Z ′ intermediating diagrams, and one-loop and two-loop corrections are approximately written as respectively. When one defines requirement of the naturalness as ∆m 2 h < M 2 h , Eqs. (10) and (11) where we have taken y t ≈ 1. For |x H | < 0.1, the two-loop correction gives stronger bound than one-loop correction. In the following, we will use the stronger bound for fixed x H . Note that the mass correction from Φ is always negligible because of a small mixing coupling λ HΦ .

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.

Gauge coupling unification
First, we discuss the possibility of the GCU at a high energy scale. Since four additional vector-like fermions (Q L,R and D L,R ) 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, β functions of gauge coupling constants are given by at 1-loop level. Figure 1 shows runnings of gauge couplings α −1 has been taken as M V = 800 GeV in Fig. 1. For µ < M V , the β 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 Λ GCU = (2-4) × 10 16 GeV, and the unified gauge coupling is α −1 GCU = (35. 4-35.8). 3 This is the same result as in Ref. [34], in which only Q L,R and D L,R are added into the SM. As the vector-like fermion masses become larger, the precision of the GCU becomes worse. Thus, the masses of Q L,R and D L,R 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 p → e + + π 0 , which is given by where m p is the proton mass. For Λ GCU = 3×10 16 GeV and α −1 GCU = 35.6, we can estimate τ p ∼ 10 37 yrs, which is much longer than the experimental lower bound τ p > 8.2 × 10 33 yrs [52]. Thus, the model are free from the constraint of the proton decay.

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: λ H > 0, λ Φ > 0 and λ S > 0.
The condition λ H > 0 depends on additional contributions to β λ H , i.e., κ 1,2 , g X and scalar mixing couplings. 4 If their contributions to β λ H are negligible, since the SM gauge couplings are larger compared to the SM case, running of λ H is raised and always positive.
For example, however, the EW vacuum becomes instable for κ 0.33 in the U(1) B−L (x H = 0) case. We show the running of λ H for x H = 0 in Fig. 2, where β λ H is independent of g X up to the one-loop level, and contributions of g X can be negligible. The red and blue lines correspond to κ = 0 and κ = 0.33, respectively. The black dashed line shows running of λ H in the SM. Thus, κ < 0.33 is required to realize the vacuum stability. The Higgs mass corrections from Q L,R and D L,R loops are given by where we have taken κ = κ 1 = κ 2 , which naturally arises from L ↔ R symmetry for the vector-like particles, and simplicity. Then, the naturalness requires κ < 0.1 for M V ∼ 1 TeV. Although κv H is a contribution to the vector-like fermion masses from the Higgs, it can be ignored because of κv H ≪ M V . Since the contribution of κ to β λ H , i.e., 24λ H κ 2 − 12κ 4 , is always positive for κ < 0.1, the naturalness condition also guarantees the vacuum stability. Note that κ ≃ 0 guarantees λ HS ≃ 0 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 S and T parameters, which are approximately given by [54,55] δS ≈ 43 30π where θ W and M W are the Weinberg angle and the W boson mass, respectively. For κ < 0.1, the parameters are estimated as δS < 3 × 10 −4 and δT < 2 × 10 −5 , which are consistent with the precision EW data S = 0.00 ± 0.08 and T = 0.05 ± 0.07 [52]. The condition λ Φ > 0 is almost always satisfied when g X is dominant in the right-hand side of Eq. (6), i.e., λ Φ (v Φ ) ∼ g 4 X (v Φ ). In this case, β λ Φ is positive up to the GCU scale, and then λ Φ is also positive up to the GCU scale. It is also possible to realize the critical condition λ Φ (Λ GCU ) = 0 as well as λ Φ > 0, where the running of λ Φ is curved upward as in the so-called flatland scenario [9,14,16,21,24]. Then, both g X and Majorana Yukawa couplings are dominant in β λ Φ , while λ Φ is much smaller than them. This means that there is a fine-tuning to satisfy Eq. (6).
When λ S is negligible in its β function, a solution of its RGE is approximately given by where µ is a renormalization scale. Once v S is fixed, f Q and f D are determined to realize the GCU, while f N remains a free parameter. To estimate the condition of λ S > 0, we assume f N = f Q = f D at µ = v S for simplicity. Then, we can find that λ S is positive up to the GCU scale for λ S (v S ) 0.01. This lower bound of λ S (v S ) is almost unchanged for different values of v S , because v S dependence is logarithmic. On the other hand, when λ S is dominant in β λ S , 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 where M s = 2λ S (v S )v S is a mass of the real singlet scalar field. Figure 3 shows v S dependence on the upper (red) and lower (blue) bonds of M s , which correspond to the Landau pole and vacuum stability conditions, respectively. Since the both bounds are almost proportional to v S , allowed values of λ S (v S ) are almost unchanged for different v S . We can find a strong constraint for λ S as 0.01 λ S (v S ) 0.05.  12) and (13)) bounds, respectively. For the Landau pole bound, we take Λ LP = Λ GCU = 3×10 16 GeV in Eq. (20). The shaded region (M Z ′ < 2.6 TeV) is excluded by the LHC experiments.
In the same way, the Landau pole also exists when g X (v Φ ) is sufficiently large. The energy scale where the Landau pole appears is approximately estimated by the one-loop RGE of g X as where M Z ′ is given in Eq (7). Figure 4 shows  12) and (13). The red, green, and blue colors correspond to v Φ = 10, 100, and 1000 TeV, respectively. The shaded region (M Z ′ < 2.6 TeV) is excluded by the LHC experiments [56,57]. When we define the triviality bound as Λ GCU < Λ LP , it prohibits the regions above the solid lines. One can see that the bound leads g X (v Φ ) 0.5 from Eq. (7), which is almost independent of v Φ . 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 v Φ = 10 TeV, and they exclude M Z ′ > 10 TeV.

Neutrino masses and baryon asymmetry of the universe
From the Lagrangian (1), the neutrino mass terms are given by There is no mixing term between ν L,R and N L,R due to the Z 2 symmetry. The active neutrino masses can be obtained by the usual type-I seesaw mechanism [36], i.e., m ν ≈ The heavier mass eigenvalue is nearly equal to M M , whose upper bound is given by the naturalness of the Higgs mass. Neutrino one-loop diagram contributes the Higgs mass as where we have used the seesaw relation. For m ν ∼ 0.1 eV, the naturalness requires M M 10 7 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 M M 10 9 GeV [59].
However, the resonant leptogenesis can work even at the TeV scale, where two righthanded neutrino masses are well-degenerated [37]. In our model, additional U(1) X 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 Y N and Y M 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 (N L,R ), we consider M N = M N L = M N R , which naturally arises from L ↔ R symmetry for the vector-like fermions. Then, the mass eigenvalues The lighter mass eigenstate N 1 is a DM candidate, because its stability is guaranteed by the Z 2 symmetry. In the limit of m N → 0 (M N 1 = M N 2 ), N 1 and N 2 are degenerate, and N 2 is also effective for a calculation of the DM relic abundance.
In the next subsection, we will investigate the degenerate N 1,2 case. In our model, the U(1) X gauge symmetry is successfully achieved via the CW mechanism. It requires λ Φ (v Φ ) > 0 in Eq. (6), that is, where n ν is a relevant number of right-handed neutrinos, which is defined as trY 4 Thus, the Majorana masses must be lighter than the Z ′ boson mass. We have made sure that this constraint is always satisfied when N 1,2 explain the DM relic abundance.

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 N L,R (or N 1,2 ), and the SM fermions do not have U(1) X charges. The annihilation processes are tchannel NN → φφ, t-channel NN → Z ′ φ, and Z ′ mediated s-channel NN → Z ′ φ. 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 N. Thus, t-channel NN → ss process and s mediated s-channel NN → ν R ν R process does not occur at tree level. From Eq. (23), (2M N ) 2 < M 2 Z ′ is always required. Then, the annihilation cross section σ(NN → Z ′ * → ff ), where f is some U(1) X charged fermion, is suppressed by 1/M 2 Z ′ . 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 h and φ, which has been considered in Ref. [19]. However, our model has an additional contribution due to Z ′ exchange diagrams, which is given by [61] where m n is the nucleon mass, and µ n = m n M N /(m n + M N ) is the reduced nucleon mass. For the DM with the masses of 100 GeV and 1 TeV, the small v Φ regions such as v Φ < 11 TeV and v Φ < 6 TeV are excluded by the LUX experiment, respectively [63]. These bound are stronger than the LEP bound, where v Φ < 3.5 TeV is excluded. On the other hand, there are two conditions Y N L = Y N R and Eq. (9), and we require that N explains the DM relic abundance Ω DM h 2 = 0.1187 [62]. Thus, we have three free parameters for the DM analysis.

Conclusion
To solve the gauge hierarchy problem, we have constructed a classically scale invariant model with a U(1) X 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 Q L,R and D L,R , which respectively have the same quantum number as the SM quark doublet and down-type quark singlet but distinguished by the additional Z 2 symmetry, and their masses lie in 800 GeV M V 1 TeV. The GCU scale is Λ GCU = 3×10 16 GeV with α −1 GCU = 35.6, and the proton life time is estimated as τ p ∼ 10 37 yrs, which is much longer than the experimental lower bound τ p > 8.2 × 10 33 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 β 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 N 1,2 , and stability of the DM is guaranteed by the additional Z 2 symmetry. We have analyzed the DM relic abundance in the degenerate case (M N 1 = M N 2 ), and found the upper bound on the DM mass as M N 2.6 TeV.