Electroweak vacuum stability in classically conformal B-L extension of the Standard Model

We consider the minimal U(1)$_{B-L}$ extension of the Standard Model (SM) with the classically conformal invariance, where an anomaly free U(1)$_{B-L}$ gauge symmetry is introduced along with three generations of right-handed neutrinos and a U(1)$_{B-L}$ Higgs field. Because of the classically conformal symmetry, all dimensional parameters are forbidden. The $B-L$ gauge symmetry is radiatively broken through the Coleman-Weinberg mechanism, generating the mass for the $U(1)_{B-L}$ gauge boson ($Z^\prime$ boson) and the right-handed neutrinos. Through a small negative coupling between the SM Higgs doublet and the $B-L$ Higgs field, the negative mass term for the SM Higgs doublet is generated and the electroweak symmetry is broken. In this model context, we investigate the electroweak vacuum instability problem in the SM. It is known that in the classically conformal U(1)$_{B-L}$ extension of the SM, the electroweak vacuum remains unstable in the renormalization group analysis at the one-loop level. In this paper, we extend the analysis to the two-loop level, and perform parameter scans. We identify a parameter region which not only solve the vacuum instability problem, but also satisfy the recent ATLAS and CMS bounds from search for $Z^\prime$ boson resonance at the LHC Run-2. Considering self-energy corrections to the SM Higgs doublet through the right-handed neutrinos and the $Z^\prime$ boson, we derive the naturalness bound on the model parameters to realize the electroweak scale without fine-tunings.


Introduction
The stability of the electroweak scale is one of the biggest mysteries in the Standard Model (SM), since the self-energy of the SM Higgs doublet field receives quantum corrections which are quadratically sensitive to the ultraviolet cutoff of the SM. A fine-tuning of the Higgs mass parameter is required to reproduce the correct electroweak scale if the ultraviolet cutoff scale is far above the electroweak scale (the gauge hierarchy problem). This problem can be solved if new physics beyond the SM makes the self-energy of the SM Higgs doublet insensitive (or logarithmically sensitive) to the ultraviolet cutoff. It is well-known that supersymmetric extension of the SM can achieve this insensitivity. Despite lots of efforts of searching for supersymmetry at the Large Hadron Collider (LHC) experiments, current LHC data include less indications for productions of supersymmetric particles. Hence we may seek other possibilities to solve the gauge hierarchy problem without supersymmetry.
According to the argument by Bardeen [1] once the classical conformal invariance and its minimal violation by quantum anomalies are imposed on the SM (or the general Higgs model), the model can be logarithmically sensitive to the ultraviolet cutoff. If this is the case, we introduce the classically conformal symmetry to the SM to make the model free from the quadratic corrections. 4 In this system, there is no mass parameter in the original Lagrangian, and the mass scale must be generated by quantum corrections. The massless U(1) Higgs model discussed by Coleman and Weinberg [3] nicely fits this picture, where the model is defined as a massless, conformal invariant theory, and the U(1) gauge symmetry is radiatively broken by the Coleman-Weinberg (CW) mechanism, generating a mass scale through the dimensional transmutation.
Recently, the extension of the SM with the classically conformal invariance has received a fair amount of attention, and many models in this direction have been proposed [4]- [6]. Among them, the classically conformal U(1) B−L extension of the SM [7,8] is a very simple and well-motivated model, since the B − L (baryon number minus lepton number) is a unique anomaly-free global symmetry and it can be easily gauged. Once the U(1) B−L is gauged, we need new chiral fermions to cancel the U(1) B−L gauge and the mixed gravitational anomalies. The simplest possibility is to introduce three right-handed neutrinos, which are nothing but the particles that we need to incorporate the neutrino mass in the SM. In this conformal symmetric model, the B − L gauge symmetry is broken by the vacuum expectation value (VEV) of the B − L Higgs field developed by the CW mechanism, and the masses for Z ′ boson and three right-handed neutrinos are generated. This radiative B − L gauge symmetry breaking is the sole origin of mass scale in this model, and the negative mass squared for the SM Higgs doublet is generated by this symmetry breaking [7].
The SM Higgs boson is finally discovered at the LHC, and the experimental confirmations of the Higgs properties in the SM has just begun. According to the SM, we can read off the value of the quartic Higgs coupling at the electroweak scale from the measured Higgs boson 1 mass, and we can investigate the behavior of the Higgs potential toward high energies by extrapolating the quartic coupling through its renormalization group evolution. It turns out that the running quartic coupling becomes negative around 10 10 GeV [9], and this fact means that the electroweak vacuum is not stable. Practically, this instability may not be a problem, since the lifetime of our electroweak vacuum is estimated to be much longer than the age of the universe [10]. However, in our context of the classically conformal extension of the SM, this electroweak vacuum instability seems to cause a theoretical inconsistency. The instability indicates that the electroweak symmetry is radiatively broken at a very high energy, which in turn generates a large mass term for the B − L Higgs field. Therefore, with such a large mass, the B − L symmetry breaking is no longer trigged by the CW mechanism.
In this paper, we investigate the electroweak vacuum stability in the context of the classically conformal U(1) B−L extension of the SM. It is known that the electroweak vacuum is still unstable in this context in the renormalization group analysis at the one-loop level [5,6]. We extend the analysis to the two-loop level and find that there exist parameter regions which can keep the electroweak vacuum stable. In our analysis, we use the result from the combined analysis by the ATLAS and the CMS experiments for the Higgs boson mass measurement in the range of m h =125.09± 0.21 (stat.)± 0.11 (syst.) GeV [11] and the recent result of top quark mass measurement m t = 172.38± 0.10± 0.65 [12] by the CMS experiments. We also consider the current collider bounds, namely, a lower bound on the B −L gauge symmetry breaking scale from the LEP electroweak precision measurements, and a lower bound on the Z ′ boson mass from the recent ATLAS [13] and CMS [14] results at the LHC Run-2. In addition, we evaluate self-energy corrections to the SM Higgs doublet from the heavy states, the Z ′ boson and the right-handed neutrinos associated with the B − L symmetry breaking, and find naturalness bounds to reproduce the electroweak scale without any fine-tunings of model parameters.
This paper is organized as follows. Our model is defined in the next section. In Sec. 3, we discuss the radiative B −L symmetry breaking through the CW mechanism and the electroweak symmetry breaking triggered by it. In Sec. 4, we analyze the renormalization group evolutions of the couplings at the two-loop level, and find a parameter regions which can keep the quartic SM Higgs coupling to be positive anywhere between the electroweak scale and the Planck scale. We also consider the current collider bounds of the model parameters, in particular, the recent ATLAS and CMS results of search for Z ′ boson resonance at the LHC Run-2 are interpreted to our B − L model. In Sec. 5, we evaluate self-energy corrections to the SM Higgs doublet, and derive the naturalness bounds to reproduce the electroweak scale without fine-tunings for the model parameters. We summarize our results in Sec. 6. Formulas we used in our analysis are listed in Appendices.  Table 1. In addition to the SM particle contents, we introduce the B − L Higgs field with the B − L charge 2 (Φ) and three right-handed neutrinos (N i R ) for cancelation of all the gauge and gravitational anomalies. The covariant derivative In addition to the SM particle contents, the right-handed neutrino N i R (i = 1, 2, 3 denotes the generation index) and a complex scalar Φ are introduced.
where Q Y and Q BL are U(1) Y and U(1) B−L charges of a particle, respectively, and gs are the gauge couplings. Because of the kinetic mixing between the two U(1) gauge bosons, the off-diagonal elements (g Y B and g BY ) are introduced. In the following analysis, we take the boundary condition, g Y B = g BY = 0, at the B − L symmetry breaking scale, where the two U(1) gauge bosons are diagonal with each other, for simplicity. The Yukawa sector of the SM is extended to have where the first term is the neutrino Dirac Yukawa coupling, while the second term is the Majorana Yukawa coupling. Without loss of generality, we have already diagonalized the Majorana Yukawa coupling. The B − L gauge symmetry breaking generates the Majorana neutrino mass term in the second term. The seesaw mechanism [15] is automatically implemented in the model after the electroweak symmetry breaking. We apply the classically conformal invariance to the model, and the scalar potential is given by Note that the mass terms are all forbidden by the conformal invariance. If λ 3 is negligibly small, we can analyze the Higgs potential separately for Φ and H. This will be justified in the 3 next section. When the Majorana Yukawa coupling Y i N is negligible compared to the U(1) B−L gauge coupling, the Φ sector is identical with the original Coleman-Weinberg model [3], so that the U(1) B−L gauge symmetry is radiatively broken. The mass term for the SM Higgs doublet is generated through λ 3 with the non-zero VEV of Φ, and the electroweak symmetry is broken when we choose λ 3 < 0 [7]. Therefore, the electroweak symmetry breaking is driven by the radiative B − L symmetry breaking.

Radiative gauge symmetry breakings
Assuming a negligibly small λ 3 , we first analyze the U(1) B−L Higgs sector. Without mass terms, the CW potential [3] at the one-loop level (in the Landau gauge) is found to be , and we have chosen the renormalization scale to be the VEV of Φ ( φ = M). Here, the coefficient of the one-loop quantum corrections is given by where in the last expression, we have used λ 2 2 ≪ g 4 BL as usual in the CW mechanism. The stationary condition dV /dφ| φ=M = 0 leads to and this λ 2 is nothing but the renormalized quartic coupling at M defined as For more detailed discussion, see [5]. Associated with this radiative U(1) B−L symmetry breaking, the Z ′ boson and the righthanded Majorana neutrinos acquire their masses as In this paper, we assume degenerate masses for the three Majorana neutrinos, Y i N = y N (equivalently, M i N = M N ) for all i = 1, 2, 3, for simplicity. The U(1) B−L Higgs boson mass is given by When the Majorana Yukawa coupling is negligibly small, this reduces to the well-known relation derived in the radiative symmetry breaking by the CW mechanism [3]. For a sizable Majorana mass, this formula indicates that the potential minimum disappears for M N > M Z ′ /2 1/4 , leading to the upper bound on the right-handed neutrino mass in order for the U(1) B−L symmetry to be broken radiatively.
Once the U(1) B−L gauge symmetry is radiatively broken by the CW mechanism, the electroweak symmetry is subsequently triggered through the coupling λ 3 . With φ = M, the SM Higgs potential is given by where H = 1/ √ 2 (0 h) T in the unitary gauge. Choosing λ 3 < 0, the electroweak symmetry is broken in the same way as in the SM [7]. However, the crucial difference from the SM is that in our model the electroweak symmetry breaking originates from the radiative breaking of the U(1) B−L gauge symmetry. At the tree level, the stationary condition V ′ | h=v = 0 leads to the relation |λ 3 | = 2λ(v/M) 2 , and the Higgs boson mass m h is given by In the following renormalization group analysis, this relation, λ 3 = −m 2 h /M 2 , is used as the boundary condition for λ 3 at the normalization scale µ = M. Since M 3 TeV by the LEP constraint [16,17,18], |λ 3 | 10 −3 . With such a small λ 3 , the back reaction to the B − L Higgs sector through λ 3 v 2 is negligibly small, and this fact allows us to treat the two Higgs sectors separately. 5

Electroweak vacuum stability
In the context of the classically conformal U(1) B−L extended model discussed in the previous sections, we now investigate a possibility to solve the electroweak vacuum instability problem. The electroweak vacuum stability has been investigated in the minimal B − L model [19] (see also [20]), and the parameter regions for which the electroweak vacuum is stable have been identified. A crucial difference in our analysis from the previous one is that our model is classically conformal and the gauge symmetry breaking originates from the CW mechanism. Hence, we have constraints on the initial values of λ 2 and λ 3 at the scale M, and it is nontrivial to solve the electroweak vacuum instability problem. In the classically conformal extension of the SM the electroweak vacuum stability has been investigated though the renormalization group analysis at the one loop level in [5,6], it turns out that there is no parameter region to keep the electroweak vacuum stable. In the following, we extend the renormalization group analysis to the two-loop level, and examine if the vacuum instability can be resolved by the higher order corrections.
In our analysis, we employ the SM renormalization group (RG) equations at the two-loop level [9] from the top quark pole mass to the U(1) B−L Higgs VEV (M), and connect the RG Here we have fixed the other parameters as g BL = 0.314, g Y B = g BY = 0 and y N = 0 at µ = M = 4 TeV. The solid lines denote the RG evolutions of the Higgs quartic coupling in our model, while the dashed lines denote those in the SM. We can see that in our model, the Higgs quartic coupling remains positive up to the Planck scale, M P l = 1.2 × 10 19 GeV, and therefore the electroweak vacuum becomes stable. As the same as in the SM [9], the situation becomes better with an increasing (decreasing) value of m h (m t ) for a fixed value of the m t (m h ).
In order to identify parameter regions to keep the electroweak vacuum stable, we perform parameter scans for the free parameters M Z ′ and M N with fixed values of M = 3.5, 4.0 and 6 To generate the RG equations at the two-loop level for the minimal U(1) B−L model, we have used SARAH [21]. For a complete RG analysis at the two-loop level, we need to take into account the threshold corrections at the 1-loop level to match the 2-loop RG evolutions at M . The most important corrections is to top Yukawa coupling at M since the electroweak vacuum instability problem is very sensitive to the input of top Yukawa coupling. We have estimated the threshold corrections to be of the order of y t × (1/3) 2 α BL /(4π) through the Z ′ boson loop diagrams, which changes the top Yukawa input at M by O(0.01%) for α BL = 0.012 (see Fig. 4), or equivalently O(0.01GeV) in terms of top quark mass. Since we have neglected the threshold corrections in our analysis, our results in this paper have a theoretical uncertainty of O(0.01 GeV) in the top quark mass. As can be seen from Fig. 3, the uncertainty at this size is negligibly small.  Fig. 1. In this analysis, we impose the following conditions for the running couplings at M ≤ µ ≤ M P l : the stability of the Higgs potential (λ, λ 2 > 0 and |λ 3 | 2 < 4λλ 2 ), and the conditions that all the running couplings remain in the perturbative regime, namely, g 2 i (i = 1, 2, 3), g 2 BL , g 2 Y B , g 2 BY < 4π and λ, λ 2,3 < 4π. The results are shown in Fig. 2. In this Figure, we also show the B − L Higgs boson mass by using Eq. (3.6). As we expect, the allowed region becomes larger as m h is increased.
We TeV is more restricted than the one for M = 2 TeV. When we increase the M value further, the allowed region disappears (see Fig. 4).
Finally, we show in Fig. 4 the results of our parameter scans for various values of g BL and M, with m h = 124.77 GeV (left panel) and 125.09 GeV (right panel) for m t = 171.63 GeV. In this Figure, we present the results with α BL = g 2 BL /(4π) and M Z ′ by using the mass formula M Z ′ = 2g BL M. Here we have considered not only the conditions of the electroweak vacuum stability and the perturbativity, but also the current collider bounds. The search for effective 4-Fermi interactions mediated by the Z ′ BL boson at the LEP leads to a bound [16] (see also [17,18]) state. For the so-called sequential SM Z ′ model [22], where the Z ′ boson has exactly the same couplings with the SM fermions as those of the SM Z boson, the cross section bounds lead to lower bounds on the Z ′ boson mass as M Z ′ ≥ 2.90 TeV from the ATALS analysis [23] and M Z ′ ≥ 2.96 TeV from the CMS analysis [24], respectively. Very recently, these bounds have been updated by the ATLAS [13] and CMS [14] analysis with the LHC Run-2 at √ s = 13 TeV as M Z ′ ≥ 3.4 TeV (ATLAS) and M Z ′ ≥ 3.15 TeV (CMS), respectively. We interpret theses ATLAS and CMS results to the B −L Z ′ boson case. In our model, the U(1) B−L gauge coupling is a free parameter, and for a fixed gauge coupling we can read off the lower limit on the Z ′ boson mass from the ATLAS and CMS cross section bounds. In this way, we can find an upper (lower) bound on the the U(1) B−L gauge coupling α BL = g 2 BL /(4π) (Z ′ boson mass M Z ′ ) as a function of M Z ′ (α BL ). In interpreting the ATLAS and the CMS results to the B − L model, we follow a strategy presented in detail in [25] (see also [26]). In Fig. 4, the vertical solid lines correspond to the bounds from the LEP result, the ATLAS with the LHC Run-1, the CSM with the LHC Run 1, the CMS with the LHC Run-2 and the ATLAS with the LHC Run-2, from left to right. The parameters inside the shaded triangles satisfy all the constraints. Naturalness bound, which will be obtained in the next section, is also shown as the dashed lines. In, for example, Ref. [27], the search reach of the Z ′ boson at the LHC Run 2 with a 14 TeV collider energy and a 100/fb luminosity is obtained as M Z ′ ≃ 5 TeV for α BL ≃ 0.01. A large potion of the allowed regions presented in Fig. 4 can be tested in the near future. The (indirect) search reach of the future e + e − linear collider with a 1 TeV collider energy can be as large as 10 TeV (see, for example, [8]), and almost of all allowed regions presented in Fig. 4 can be covered.

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Once the U(1) B−L gauge symmetry is radiatively broken by the CW mechanism, the masses for the Z ′ boson and the Majorana neutrinos are generated, which in general create self-energy corrections to the SM Higgs doublet. If the B − L gauge symmetry breaking scale is very large, the self-energy corrections may exceed the electroweak scale and require us to fine-tune the model parameters in reproducing the correct electroweak scale. Two major corrections have been discussed in [7,8]: one is one-loop corrections with the Majorana neutrinos, and the other is two-loop corrections involving the Z ′ boson and the top quark. In the calculations of the self-energy corrections in [8], the cutoff procedure with the Planck scale cutoff is applied to derive the naturalness bounds. Although this treatment is good for rough estimates, in order to derive more accurate naturalness bounds we will renormalize the loop corrections properly in this section.
Since the original theory is classically conformal and defined as a massless theory, the selfenergy corrections to the SM Higgs doublet originates from corrections to the quartic coupling λ 3 . Thus, what we calculate to derive the naturalness bounds is quantum corrections to the term λ 3 h 2 φ 2 in the effective Higgs potential. For the one-loop diagram involving the Majorana neutrinos (for the Feynman diagram, see Fig. 3 in [8]), we calculate the effective potential as where the logarithmic divergence and the terms independent of φ are all encoded in C. By adding a counter term, we renormalize the coupling λ 3 with the renormalization condition, where V eff is the sum of the tree-level potential and ∆V 1−loop , and λ 3 is the renormalized coupling. As a result, we obtain Substituting φ = M, we obtain the SM Higgs self-energy correction as where we have used the seesaw formula, m ν ∼ Y 2 D v 2 /M N [15]. If ∆m 2 h is much larger than the electroweak scale, we need a fine-tuning of the tree-level Higgs mass (|λ 3 |M 2 /2) to reproduce the correct Higgs VEV, v = 246 GeV. Here, we introduce the naturalness condition as For example, when the light neutrino mass scale is around m ν ≃ 0.1 eV after the seesaw mechanism, we have an upper bound for the Majorana mass as M N 4 × 10 6 GeV.
For the two-loop diagrams involving Z ′ boson and top quark (for the Feynman diagrams, see Fig. 4 in [8]), we have where the logarithmic divergence and the terms independent of φ are all encoded in C. Following the same strategy as the above, we obtain The dashed lines shown in Fig. 4 are plotted by using the condition δ = 1 in Eq. (5.5).

Conclusions
We have considered the minimal B − L extension of the Standard Model, where the anomalyfree global B − L symmetry in the Standard Model is gauged and three right-hand neutrinos and a B − L Higgs field are introduced. This model is very simple and well-motivated, since the right-handed neutrinos acquire their Majorana masses associated with the B − L gauge symmetry breaking, and the seesaw mechanism for the neutrino mass generation is automatically implemented. Motivated by the argument that the Higgs model can be free from the the gauge hierarchy problem once the classically conformal symmetry is imposed in the model, we have introduced the classically conformal symmetry to the minimal B − L model. In this context, the B − L symmetry is radiatively broken by the Coleman-Weinberg mechanism and this breaking is the sole origin of all mass parameters in the model. The electroweak symmetry breaking is realized by the negative mass term for the Higgs doublet, which is subsequently generated through the B − L gauge symmetry breaking. Therefore, the electroweak symmetry breaking originates from the radiative B − L gauge symmetry breaking.
In the context of the classically conformal B −L model, we have investigated the electroweak vacuum instability problem. With the measured Higgs boson mass around 125 GeV, it turns out that the electroweak vacuum is not the true minimum in the the effective Higgs potential of the Standard Model. In other words, the electroweak symmetry is radiatively broken at some energy much higher than the electroweak scale. This ruins the theoretical consistency of our model that the radiative B − L symmetry breaking is the sole origin of the mass. We have analyzed the renormalization group evolutions of the model couplings at the two-loop level with the recent results of the Higgs boson mass and top quark mass measurements at the LHC. We have identified parameter regions which satisfy the conditions of the stability of the electroweak vacuum and the perturbativity of the running couplings, as well as the current collider bounds from the search for the B − L gauge boson, in particular, at the LHC Run-2.
In addition, we have considered the naturalness of the electroweak scale against self-energy corrections for the Higgs doublet. We have refined the previously obtained results in a theoretically consistent way for the Coleman-Weinberg effective potential, and derived the naturalness bounds on the B − L gauge boson and the right-handed neutrino masses. The allowed regions satisfying the naturalness bounds can be tested in the future collider experiments.

Acknowledgements
This work is supported in part by the United States Department of Energy Grant, No. de-sc0013680.

A The beta functions for the SM couplings
A. 1 The one-loop beta functions for the SM gauge couplings A. 2 The one-loop beta function for the top Yukawa coupling A. 3 The one-loop beta function for the quartic Higgs coupling A. 4 The two-loop beta functions for the gauge couplings A. 5 The two-loop beta function for the top Yukawa coupling A. 6 The two-loop beta function for the Higgs quartic coupling In our analysis, we numerically solve the SM RG equations with the following boundary conditions at µ = m t [9]  We have used the inputs, α 3 (m Z ) = 0.1184 and m W = 80.384 GeV. 7 We have employed the boundary conditions in arXiv:1307.3536v4.

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B The beta functions for the couplings in the U(1) B−L extended SM B.1 The one-loop beta functions for the gauge couplings (B.1)

B.2 The one-loop beta function for the top Yukawa coupling
B. 3 The one-loop beta function for the Majorana Yukawa coupling .