Criticality and Inflation of the Gauged B-L Model

We consider the multiple point principle (MPP) and the inflation of the gauged B-L extension of the Standard Model (SM) with a classical conformality. We examine whether the scalar couplings and their beta functions can become simultaneously zero at $\Lambda_{\text{MPP}}:=10^{17}$ GeV by using the two-loop renormalization group equations (RGEs). We find that we can actually realize such a situation and that the parameters of the model are uniquely determined by the MPP. However, as discussed in \cite{Iso:2012jn}, if we want to realize the electroweak symmetry breaking by the radiative B-L symmetry breaking, the self coupling $\lambda_{\Psi}$ of a newly introduced SM singlet complex scalar $\Psi$ must have a non-zero value at $\Lambda_{\text{MPP}}$, which means the breaking of the MPP. We find that the ${\cal{O}}(100)$GeV electroweak symmetry breaking can be achieved even if this breaking is very small; $\lambda_{\Psi}(\Lambda_{\text{MPP}})\leq10^{-10}$. Within this situation, the mass of the B-L gauge boson is predicted to be \begin{equation} M_{B-L}=2\sqrt{2}\times\sqrt{\frac{\lambda(v_{h})}{0.10}}\times v_{h}\simeq 696\hspace{1mm}\text{GeV},\nonumber\end{equation} where $\lambda$ is the Higgs self coupling and $v_{h}$ is the Higgs expectation value. This is a remarkable prediction of the (slightly broken) MPP. Furthermore, such a small $\lambda_{\Psi}$ opens a new possibility: $\Psi$ plays a roll of the inflaton \cite{Okada:2011en}. Another purpose of this paper is to investigate the $\lambda_{\Psi}\Psi^{4}$ inflation scenario with the non-minimal gravitational coupling $\xi\Psi^{2} {\cal{R}}$ based on the two-loop RGEs.

Well before the discovery of the Higgs, it was argued that the Higgs mass could be predicted to be around 130 GeV by the requirement that the minimum of the Higgs potential becomes zero at M pl [3,4]. Such a requirement (not always at M pl ) is generally called the multiple point principle (MPP). One of the good points of the MPP is its predictability: the low-energy effective couplings are fixed so that the minimum of the potential vanishes; see, e.g., Refs. [23,24].
By taking the fact that the MPP can be realized in the SM into consideration, a natural question is whether such a criticality can also be realized in the models beyond the SM. One interesting extension is the gauged B − L (baryon number minus lepton number) model with a classical conformality [25][26][27][28]. Here, "classical conformality" means there is no mass term at the classical level without gravity. This model can be obtained by gauging the global U(1) B−L symmetry of the SM with three right-handed neutrinos and an SM singlet complex scalar . As discussed in the following, if we PTEP 2015, 073B04 K. Kawana neglect the Yukawa couplings between the Higgs and neutrinos, there are six unknown parameters in this model. In particular, two of them are new scalar couplings: κ and λ . Therefore, in principle, these six parameters can be uniquely fixed by the MPP conditions: where MPP is the scale at which we impose the MPP. The analyses in this paper are based on the following assumptions: 1. We consider the MPP at MPP = 10 17 GeV. 2. As well as the analyses in Refs. [25,26], we do not include mass terms in the Lagrangian. As a result, all the low-energy scales are radiatively generated.

The Higgs mass is fixed at
and we regard the top mass M t as one of the free parameters. 4. We assume that small neutrino masses are produced by the seesaw mechanism via radiative breaking of the B − L symmetry. As a result, we can neglect the Yukawa couplings y ν between the Higgs and neutrinos because the typical breaking scale is very small 10 13 GeV .
In Sect. 2.2, we will see that Eq. (1) can be actually realized at MPP = 10 17 GeV. One of the good features of this model is that electroweak symmetry breaking can be triggered by U(1) B−L symmetry breaking via the Coleman-Weinberg (CW) mechanism. In Ref. [26], it was argued that we can naturally obtain v h = O(100) GeV by imposing λ M pl = 0 and κ M pl = 0. Here, the important point is that λ MPP = 0 is needed to realize such B − L breaking. 1 Therefore, if we try to combine this fact and the MPP, a natural question arises: • Is O(100) GeV electroweak symmetry breaking possible even if λ ( MPP ) is small?
In Sect. 2.3, we will see that this is actually possible even if λ MPP ≤ 10 −10 . The reason for this is very simple: By tuning the parameters of the model, we can obtain the favorable scale at which U(1) B−L breaks so that v h becomes O(100) GeV. Therefore, the B − L model is a phenomenologically very interesting model in that it can explain the natural-scale electroweak symmetry breaking while satisfying the (slightly broken) MPP. Furthermore, within this situation, we find that the mass of the B − L gauge boson is predicted to be where v B−L is the expectation value of and we have used the typical value λ(v h ) 0.1. This is a remarkable prediction of the (slightly broken) MPP, and it is surprising that the predicted value of M B−L depends only on the SM parameters. 2 On the other hand, there are many observational results from the cosmological side. One of the reliable possibilities to explain them is cosmic inflation. As is well known, Higgs inflation is possible in the SM where the criticality of the Higgs potential plays an important role in realizing the inflation naturally [17]. Of course, such a Higgs inflation is possible in the B − L model, but we can also consider the inflation scenario where plays the role of the inflaton [28]. In this paper, we PTEP 2015, 073B04 K. Kawana study λ 4 inflation with non-minimal gravitational coupling ξ 2 R. Our analysis is based on the following condition: • We consider inflation in the situation where the minimum of the Higgs potential vanishes at MPP = 10 17 GeV and electroweak symmetry breaking occurs at O(100) GeV.
In the following discussion, we will see that this condition strongly constrains the parameters, and, as a result, we can obtain unique cosmological predictions 3 that are consistent with the recent values observed by Planck [32] and BICEP2 [33]. This paper is organized as follows. In Sect. 2, we study the MPP and the B − L symmetry breaking from the point of view of the slightly broken MPP. In Sect. 3, we investigate the inflation scenario where the SM singlet complex scalar plays the role of the inflaton. In Sect. 4, we give a summary.

MPP of the B − L model and symmetry breaking
The flow of this section is as follows. In Sect. 2.1, we briefly review the gauged B − L model. In Sect. 2.2, we consider the MPP of this model. In Sect. 2.3, we study whether O(100) GeV electroweak symmetry breaking can be realized even if λ ( MPP ) is very small.

Short review of the B − L model
In this subsection, we briefly review the B − L extension of the SM. Here, our discussion is mainly based on Ref. [30]. As mentioned in the introduction, this model can be obtained by gauging the global U(1) B−L symmetry. The kinetic terms of the two U(1) gauge fields are given as follows: where ω(∈ R) represents the kinetic mixing. The U(1) part of the covariant derivative of a matter field φ k is given by where A 1 μ and A 2 μ are the gauge fields of U(1) Y and U(1) B−L , respectively, Y i k are the U(1) charges, and g i j represent the U(1) gauge couplings. We can remove the mixing term by changing A 1 μ and A 2 μ to the new fields A Y μ and A B−L μ : We simply express Eq. (6) as A i μ = α R i α A α μ . By this transformation, the new gauge couplings are We denote g iα as g YY , g YE , g EY , and g EE without a prime in the following discussion. Only three of them are meaningful because we can further rotate the gauge fields without producing PTEP 2015, 073B04 K. Kawana the mixing term: Thus, we can choose the angle θ so that one of g αβ vanishes. For convenience, we take the following bases: In these bases, the second term of Eq. where As a result, B μ plays the role of the ordinary U(1) Y gauge field, and E μ is a new gauge field that can have a mass if the B − L symmetry is broken. We use Eq. (9) for the calculations of the RGEs in Appendix A. The particle contents (except for the gauge bosons) and their charges are presented in Table 1. In addition to the SM particles, there are three right-handed neutrinos and a SM singlet complex scalar whose U(1) B−L charge is +2. The relevant terms of the renormalizable Lagrangian are In the following discussion, we use the bases such that y i j ν and Y i j R are real and diagonalized, and assume that they are equal, respectively, for the three generations. As a result, by including the top mass M t , there are seven unknown parameters in this model: If we assume that small neutrino masses ( 1 eV) are generated by the ordinary seesaw mechanism triggered by U(1) B−L symmetry breaking at a low-energy scale 10 13 GeV , y ν should be very small, and its effects on the RGEs are negligible. In this paper, we assume such a situation.

Multiple point principle
To understand how these couplings behave at a high-energy scale, we need to know the RGEs. The two-loop RGEs of this model are presented in Appendix A. Furthermore, the one-loop effective PTEP 2015, 073B04 K. Kawana Table 1. The particle contents of the B − L model and their charges except for the gauge bosons. Here, i represents the generation.
potentials in the Landau gauge are as follows: 4 where Here, μ is the renormalization scale and , are the wave function renormalizations. To minimize the one-loop contributions, we take μ = φ ( ) in the following discussion. 5 From these results, we PTEP 2015, 073B04 K. Kawana can define the effective self-couplings and their effective beta functions as follows: (17) Figure 1 shows the typical behaviors of λ eff (φ) and its parameter dependences. Here, for later convenience, the initial values of λ , κ, g B−L , g mix , and Y R are given at MPP = 10 17 GeV, and their typical values are chosen to be 0.1, respectively. One can see that λ eff (φ) depends weakly on g B−L and Y R because they appear in β λ at the two-loop level. Now, let us consider the MPP. By including the top mass M t and neglecting y ν , there are six parameters in this model: Therefore, in principle, they are uniquely determined by the MPP conditions: Among these, λ eff MPP = κ MPP = 0 are just the initial conditions of λ and κ, and other conditions give us constraints between the remaining parameters. We can understand such constraints qualitatively from the one-loop RGEs: • β eff λ MPP = 0 mainly relates M t and g mix because they appear in β λ at the one-loop level (see Eq. (A12) in Appendix A). As a result, we can fix M t and g mix by λ eff according to 0 ≤ g B−L MPP ≤ 0.4. 6 • We can obtain a relation between g B−L MPP and Y R MPP by β λ MPP = 0 because the one-loop part of β λ at MPP is • Finally, g B−L MPP (or Y R MPP ) can be fixed at 0 by β κ MPP = 0 because the one-loop part of β κ at MPP is In Fig. 2, we show the effective potentials (upper) and the runnings (lower) of λ eff and κ that satisfy the above MPP conditions. Here, in the lower panels, we leave g B−L MPP as a free parameter. One can see that the flat potentials can be actually realized at MPP .

Electroweak symmetry breaking by breaking the MPP
We first explain how electroweak symmetry breaking is triggered by B − L symmetry breaking. If has an expectation value produces the mass term of H : This is a relation between v h and v B−L . We must consider a few questions to realize electroweak symmetry breaking at O(100) GeV: Question 1: Does B − L symmetry breaking actually occur? In particular, is it possible to realize it in the situation where the MPP is exactly satisfied?
See the lower-left panel of Fig. 2 once again. This shows the running of λ eff when the MPP conditions are satisfied. One can see that λ eff is a monotonically decreasing function in the μ ≤ MPP region. Thus, we cannot obtain B − L symmetry breaking if the MPP is realized exactly. However, as discussed in Ref. [26], the situation changes when λ eff MPP > 0 and β eff λ MPP > 0, which mean the breaking of the MPP. See the upper-and middle-left panels of Fig. 3. They show the runnings of λ eff when λ eff MPP = 10 −10 and 10 −12 , respectively. 7 One can see that λ eff can cross zero, and its scale strongly depends on g B−L MPP . For convenience, we also show the corresponding effective potentials of in the upper-and middle-right panels. Here, we have normalized the vertical axes so that the minimums of the potentials can be easily understood. In the following discussion,  besides λ eff MPP > 0 and β λ eff MPP > 0, we consider the situation such that only λ eff , β eff λ , and κ satisfy the MPP conditions: where we have used g mix 0.2, which was obtained from λ eff MPP = β eff λ MPP = 0. Thus, κ at a low-energy scale μ is approximately given by where c is a constant and we have used the fact that g B−L does not change significantly. This is the qualitative expression of κ(μ). In the lower-left (-right) panel of Fig. 3, we show κ vs g B−L at μ = M t = 171.8 GeV in the case of λ eff MPP = 10 −10 (10 −12 ). One can see that Eq. (28) nicely explains the numerical results when c is 1.0. As a result, v h is given by where we have used Eq. (29) and c = 1.0. By using the experimental value v h = 246 GeV and the typical value λ(v h ) 0.1, this leads to Although this is a remarkable prediction of the MPP, this value is already excluded by the ATLAS experiment [29] because g mix is too large.

Non-minimal inflation: The SM singlet scalar as the inflaton
As is well known, Higgs inflation is possible in the SM [14][15][16][17][18]. There, the criticality of the Higgs potential plays a crucial role in realizing inflation naturally; we can obtain sufficient e-foldings and cosmic microwave background (CMB) fluctuations even if ξ is O(1) by making the running Higgs self-coupling arbitrarily small (see Ref. [17] for more details). In other words, the smallness of the self-coupling is needed to realize the inflation naturally. Such a Higgs inflation is, of course, possible in our B − L model; however, the conclusion of the previous section indicates a new possibility: The newly introduced SM singlet complex scalar plays the role of the inflaton [28]. We study this scenario in this section. The action with the non-minimal gravitational coupling ξ 2 R in the Jordan frame is given by where is the physical (real) field, and we have written the relevant terms for later discussion. To study the inflation, it is convenient to move to the Einstein frame. Namely, by the conformal PTEP 2015, 073B04 K. Kawana transformation g E μν := 2 g μν , and the field redefinition the action becomes This is the canonically normalized form, and the potential in this frame is given by For large values of so we have In this limit, the potential in the Einstein frame, Eq. (37), becomes This is an exponentially flat potential (see, e.g., Fig. 6), so we can use the slow-roll approximations. The slow-roll parameters are  where a prime represents a derivative with respect to . By using these quantities, the number of e-foldings N , the spectral index n s , its running dn s /d ln k, and the tensor-to-scalar ratio r are given by where ini ( end ) represents the initial (end) value of . In the following discussion, we denote ini simply as .
Here, we give the current cosmological constraints by Planck TT + lowP [32]. The overall normalization of the CMB fluctuations at the scale k 0 = 0.05 Mpc −1 is There has been discussion suggesting that this result may be consistent with r = 0 due to the foreground effect [34,35]. Our calculations are based on the following conditions: (1) Although there are six parameters, we consider the situation where Eq. (26) is satisfied. Namely, M t , g mix MPP , and κ MPP are fixed, respectively, at 171.8 GeV, 0.2, and 0.   (3) The remaining two parameters g B−L MPP and Y R MPP are chosen so that v h becomes O(100) GeV. As discussed at the end of Sect. 2, the allowed region is quite limited in this case. We have checked that the cosmological predictions do not change very much even if we change these parameters within such a region (see Fig. 8). Figure 7 shows our numerical results when we fix g B−L MPP and Y R MPP . Our results are, of course, consistent with previous results such as Refs. [28,36]. The left (right) panels show r (dn s / ln k) vs n s . Here, the solid blue (red) lines represent ξ ( ) = constant, and the contours that correspond to N = 50 and 60 are represented by orange and black, respectively, from ξ = 0 to ξ = 100. In the left panels, we also show the contours of A s = 2.2 × 10 −9 in green. These results are consistent with the observed results (49) and (50) of Planck and BICEP2. In particular, as one can see from the behaviors of the green lines, the values of λ eff MPP that can simultaneously explain A s = 2.19 × 10 −9 , sufficient e-foldings (N ≥ 50), and the BICEP2 result r = 0.2 are quite limited: Among the three quantities n s , r , and dn s / ln k, one might think that the predicted values of dn s / ln k are small compared with the observed values O(−0.01). It might be possible to improve this situation by including a higher-dimensional operator; see, e.g., Ref. [17]. In Fig. 8, we also show how n s , r , and dn s / ln k depend on g B−L MPP when λ eff MPP = 10 −10 . Here, we change g B−L MPP within the region such that electroweak symmetry breaking occurs at O(100) GeV. Furthermore, and ξ are chosen so that they explain both the observed value of A s and N = 50 when g B−L MPP = 0.0020. One can see that n s and r hardly depend on g B−L MPP and that the change in dn s / ln k is at most O(0.0001). As a result, in the situation where the minimum of the Higgs potential vanishes at MPP and electroweak symmetry breaking occurs at O(100) GeV, the gauged B − L model uniquely predicts the cosmological observables. This is also one of the benefits of the (slightly broken) MPP.

Summary
In this paper, we have considered the MPP and the inflation of the gauged B − L extension of the SM. We have found that the scalar couplings and their beta functions can simultaneously become zero at MPP = 10 17 GeV and that the parameters of the model can be uniquely fixed by these conditions. However, from the point of view that electroweak symmetry breaking should be realized by radiatively broken B − L symmetry, it is necessary to break the MPP: we need λ eff MPP > 0 and β λ eff MPP > 0. In Sect. 2.3, we found that small values of λ eff MPP are compatible with electroweak symmetry breaking at O(100) GeV. In particular, we have found that the mass of the B − L gauge boson can be predicted to be from the MPP of the Higgs potential and κ. This is one of the remarkable predictions of the MPP. In Sect. 3, we have studied inflation, where the SM singlet scalar plays the role of the inflaton. We have calculated the cosmological observables based on the assumptions that the minimum of the Higgs potential vanishes at MPP = 10 17 GeV and electroweak symmetry breaking occurs at O(100) GeV. The results in this paper are consistent with the observations by Planck and BICEP2. Among these, the predicted values of the running of the spectral index dn s / ln k are small compared with the observed values O(−0.01). It might be interesting to consider whether we can improve this situation. One such possibility is to include a higher-dimensional operator [17]. In conclusion, the gauged B − L extension of the SM is a phenomenologically very interesting model in that it can explain both the cosmological observations and electroweak symmetry breaking at O(100) GeV by breaking the MPP.