Neutrino parameters in the Planck-scale lepton number breaking scenario with extended scalar sectors

Two-loop effects on the right-handed neutrino masses can have an impact on the low-energy phenomenology, especially when the right-handed neutrino mass spectrum is very hierarchical at the cut-off scale. In this case, the physical masses of the lighter right-handed neutrinos can be dominated by quantum effects induced by the heavier ones. Further, if the heaviest right-handed neutrino mass is at around the Planck scale, two-loop effects on the right-handed neutrino masses generate, through the seesaw mechanism, an active neutrino mass which is in the ballpark of the experimental values. In this paper we investigate extensions of the Planck-scale lepton number breaking scenario by additional Higgs doublets (inert or not). We find that under reasonable assumptions these models lead simultaneously to an overall neutrino mass scale and to a neutrino mass hierarchy in qualitative agreement with observations.

1 Introduction the Z 2 -odd scalar (the "inert" doublet), or in some regions of the parameter space the singlet fermion. The predicted mass hierarchy among the active neutrinos is again too large in general. However, a mild mass hierarchy can be generated in the presence of more than one Z 2 -odd inert doublet [14,15].
In this paper we will analyze the Planck-scale lepton number breaking scenario, which is successful in predicting the correct overall neutrino mass, with an extended scalar sector, which is successful in explaining the observed mild neutrino mass hierarchy. We will show that, under plausible assumptions, it is possible to reproduce simultaneously the correct neutrino mass scale and mass hierarchy. In Section 2, we calculate the quantum effects on the right-handed neutrino masses in the two-Higgs doublet model. In Section 3, we explore the implications for the active neutrino masses in the general two-Higgs doublet model, and in Section 4 for the scotogenic model. Finally, in Section 5 we present our conclusions.
2 Quantum effects on the right-handed neutrino mass matrix in the two-Higgs doublet model We consider in this section the two-Higgs doublet model (2HDM) extended by three RHNs, N i , i = 1, 2, 3. The scalar potential reads where Φ a , a = 1, 2, are scalar SU (2)-doublets with hypercharge Y = 1. The hermiticity of the potential requires µ 2 ab = (µ 2 ba ) * and λ abcd = λ * badc (for a comprehensive review of the 2HDM, see [16]). The part of the Lagrangian involving the RHNs reads: where L α (α = e, µ, τ ) are the lepton doublets. Further, Φ a = iσ 2 Φ * a denotes the charge conjugated scalar fields, and N c i = −iγ 2 N * i the charge conjugated RHN fields. The parameters of the Lagrangian in Eq. (2) are subject to quantum corrections, which can have significant impact on the phenomenology. The leading quantum effects can be calculated using the renormalization group equations (RGEs). Including up to two-loop effects, the RGE of the RHN mass matrix reads: where for convenience, we have defined G = 1 16π 2 Here, g 1 and g 2 are the U (1) Y and SU (2) L gauge couplings and Y (a) u are the up-quark Yukawa coupling matrices to both Higgs doublets (we assume that the Yukawa couplings of the other SM fermions to both Higgs doublets are negligible). For the purposes of this paper, it is sufficient to consider the one-loop RGE of the neutrino Yukawa coupling. Using SARAH [17], we obtain: We will work in the basis where the RHN mass matrix is real and diagonal at the cut-off energy scale Λ: Integrating Eq. (3), one can calculate the RHN mass matrix at the scale µ < Λ. Keeping terms up to the order O P (ab)2 we obtain: where we have denoted t = log(µ/Λ). We are interested in the scenario where the mass matrix at the cut-off scale is approximately rank-1, M 1 , M 2 M 3 and where M 3 ∼ M P , being M P = 1.2 × 10 19 GeV the Planck mass. To emphasize the main features of the RGE, let us consider the limiting scenario where M 1 = M 2 = 0, namely when the mass matrix is exactly rank-1 (our conclusions, however, apply to a wider class of scenarios, as we will discuss below). One can readily check that at O P (ab) , i.e. keeping just the first line of Eq. (9), the mass matrix at the scale µ is also rank-1. However, at O P (ab)2 , the mass matrix in general becomes rank-3: the RGE evolution generates radiatively non-zero values for M 1,2 (µ) proportional to M 3 , through the diagram shown in Fig. 1. This effect was explored in [9] for the seesaw scenario with one Higgs doublet (see also [18]). In that case, however, a rank-1 mass matrix at the scale Λ remained rank-1 at order O(P ), became rank-2 at order O(P 2 ), and became rank-3 only at order O(P 4 ). However, the existence of an additional RHN Yukawa coupling in the 2HDM (and thereby the existence of additional flavor symmetry breaking parameters), allows to increase the rank of the mass matrix at lower order in perturbation theory. Here, we have considered the limiting case where the mass matrix is exactly rank-1 at the cut-off scale. For an approximately rank-1 mass matrix, one finds that the physical masses of the two lightest RHNs can be dominated by the quantum contribution induced by the heaviest RHN. Correspondingly, their tree-level masses would not play any role in the phenomenology, thus rendering a more predictive scenario.
In order to construct the low-energy effective theory of the Planck-scale lepton number breaking scenario, we first integrate out the heaviest RHN at the scale µ = M 3 . The effective Lagrangian reads: with Yukawa and mass matrices given by with i, j = 1, 2. The first term in Eq. (10) is a Weinberg operator giving rise to a contribution to the active neutrino masses suppressed by M 3 . In the Planck-scale lepton number breaking scenario M 3 is close to the Planck scale. Hence, this term gives a negligible contribution to the neutrino masses. We will drop this term henceforth, and describe the effective theory as a two-right handed neutrino model with the Lagrangian: with Y and M given in Eq. (11). Using Eq. (9), and using that M i3 /M 33 | µ=M 3 is a small parameter, one can further approximate where we have assumed generic Yukawa couplings at the scale Λ, and that the running does not significantly modify the Yukawa couplings. However, two-loop quantum effects lift the zeroes in the RHN mass matrix and generate radiatively two mass eigenvalues. The eigenvalues can be calculated from the tensor invariants: 1 1 In this paper we assume all parameters to be real for simplicity. In the complex case, the corresponding invariants are I1 = Tr M † M = M 2 1 + M 2 2 and I2 = det Assuming a hierarchy between the eigenvalues, one obtains which are complicated expressions that depend on the Yukawa couplings.
In order to gain some analytical understanding of the results, let us consider for simplicity rank-1 Yukawa matrices. The Yukawas Y (a) can be expressed in terms of the non-vanishing eigenvalues, y a , and the tensor products of two vectors in flavor space u In a similar notation, the RHN neutrino mass matrix can be cast as: Clearly, the physical RHN masses can only depend on invariant quantities. In our simplified scenario, we have three vectors in the RHN flavor space, ω and u L ), as well as the three eigenvalues y 1 , y 2 and M 3 . In terms of these invariants, the radiatively generated RHN masses M 1 and M 2 at the scale µ = M 3 read: where the square of the triple product explicitly reads: From these equations one concludes that in order to generate a non-vanishing M 2 either u R must be non-orthogonal to ω. Further, in order to generate a non-vanishing M 1 , the three following conditions must be simultaneously fulfilled: i) both u      L must also point in different directions. In more generality, generating M 2 requires at least two independent directions in the RHN flavor space, and generating M 1 requires three independent directions in the RHN flavor space, as well as two independent directions in the LHN flavor space. This can also be understood from the breaking of the global flavor group, U (3) L × U (3) R → nothing, by the Yukawa couplings [8]. In the Standard Model extended with RHNs, a rank-1 Yukawa matrix and a rank-1 mass matrix provide two directions in the RHN flavor space, and therefore this model generates only M 2 , but not M 1 (due to a residual global U (1) symmetry). A rank-2 Yukawa matrix can generate via quantum effects a non-vanishing M 1 , although suppressed by the next-to-largest Yukawa eigenvalue and only beyond two-loop order. In the 2HDM extended with RHNs there are many more directions in flavor space, and therefore it is possible to generate radiatively both M 1 and M 2 with rank-1 Yukawa couplings. From these expressions, one can also construct the limit where one of the RHNs, say N 1 , has no coupling to the left-handed doublets; this would correspond to a model with only two RHNs. In this case, u R and ω are coplanar and therefore M 1 | µ=M 3 = 0.
For our analysis we will find convenient to use as invariants the following four angles: as well as the three eigenvalues y 1 , y 2 and M 3 . With this parametrization, the eigenvalues in Eq. (18) can be written as with s i = sin θ i and c i = cos θ i (i = 1, 2, L, R). Below the scale M 3 , both RHN masses are subject to additional quantum effects, although in this case they amount to small corrections. Therefore, one can approximate the physical masses for N 1 and N 2 by the running masses at the scale µ = M 3 in Eq. (21).
The overall mass scale of both M 1 and M 2 is determined by the parameter Numerically, in the Planck-scale lepton number breaking scenario Let us note that the same conclusion holds whenever the physical masses of N 1 and N 2 are dominated by quantum contributions proportional to M 3 , even if they do not vanish at the cut-off scale. In this case, quantum effects can milden the hierarchy between the two lighter eigenvalues, leading to M 2 | µ=M 3 ∼ M 1 | µ=M 3 . The consequences for the light neutrino mass spectrum are expected to be dramatic. If two-loop effects had been neglected and M 1 M 2 at the decoupling scale, the generation of a mild mass hierarchy m 3 ∼ m 2 would look rather accidental. However, the quantum effects induced at two loops by the two Higgs doublets generically lead to a mild hierarchy between M 1 and M 2 , and therefore it will be easier to generate m 3 ∼ m 2 .

Active neutrino masses
At energy scales below the mass of the lightest RHN, the phenomenology of the model can be properly described by the following effective Lagrangian: where the Wilson coefficients at the scale µ = M 1 can be calculated in the usual manner by integrating-out the heavy RHNs: with Y (a) and M given in Eq. (13) (for calculating κ (ab) , we neglect the contribution from integrating-out N 3 , which as mentioned above is subdominant). For rank-1 Yukawa couplings one finds with P (ab) ≡ Y (a) T Y (b) . Using the notation of Eq. (16) we explicitly find: Furthermore, to determine the low-energy neutrino parameters, we include quantum contributions to κ (ab) . As in the rest of this paper, and due to the large separation between M 1 and the energy scale of neutrino oscillation experiments, the dominant quantum contributions to κ (ab) can be encoded in the RGE [19][20][21][22]: with dominant terms of the β function at one loop given by: where we have neglected all gauge and Yukawa couplings except for the top-quark coupling for the analytical treatment. The Wilson coefficient of the Weinberg operator at the scale m H is: Finally, after the electroweak symmetry breaking through the expectation value of the neutral components of the Higgs fields, Φ 0 a = v a / √ 2, with v 2 1 + v 2 2 = v 2 , a 3 × 3 neutrino mass matrix is generated:  by O(1) factors. Second, and more importantly, the different Wilson coefficients κ (ab) can mix through the running, due to "Higgs changing interactions" in the Weinberg operators, induced by Higgs quartic couplings. This effect is characteristic of the model with an extended Higgs sector, and can significantly affect the low-energy phenomenology [11,12,23,24].
In order to better differentiate the impact on the phenomenology of the quantum effects above and below the RHN decoupling scale, two scenarios are analyzed. We first discuss a scenario where the operator mixing between the scales M 1 and m H is negligible, and then a scenario where the operator mixing is sizable.

Operator mixing between κ (ab) negligible
Following our analysis of Section 2, we assume that the RHN Yukawa coupling and mass matrices are rank-1 at the cut-off scale Λ. Then, from Eq. (27), and neglecting the effects of the running below the scale µ = M 1 , one obtains: where The active neutrino mass eigenvalues can be calculated using the tensor invariants: From Eqs. (32) and (33) one obtains: Therefore, m 1 = 0 in the approximation that only two RHNs contribute to the mass matrix. 2 The other two eigenvalues read, under the assumption m 3 m 2 : where we have used v 1 = v cos β and v 2 = v sin β, and we have defined the overall mass scale with M 0 defined in Eq. (22). Numerically, which is in the right ballpark if M 3 is around the Planck scale, and the largest Yukawa eigenvalues are O(1). It is evident in Eq. (36) that a necessary condition to generate a non-vanishing m 2 is to have a misalignment between u  which is ∼ 1 − 10 under the assumptions listed above. This expectation is confirmed by our numerical analysis. We consider different realizations of our scenario at the cut-off Λ = M P , assuming a rank-1 RHN mass matrix with M 3 = M P / √ 8π and rank-1 Yukawa matrices with eigenvalues y 2 = 1, and y 1 = 1 or 0.01. The Yukawa eigenvectors u (a) R,L are chosen randomly. We then solve numerically the two-loop RGEs for the RHN parameters above the scale M 1 , and the one-loop RGEs for the Wilson coefficients below the scale M 1 , neglecting the terms in Eq. (29) that mix the different κ (ab) . Finally, we calculate the neutrino mass matrix assuming tan β = 1 or 0.01. The resulting values for m 3 and |m 3 /m 2 | are shown in the scan plot in Fig. 3, for the cases i) y 1 = 1 and tan β = 1 (green points), ii) y 1 = 1 and tan β = 0.01 (orange points), and iii) y 1 = 0.01 and tan β = 1 (blue points). When the Yukawa eigenvalues are y 2 ∼ y 1 ∼ 1 and tan β ∼ 1 the predicted neutrino parameters are in the ballpark of the experimental values. However, when y 1 /y 2 and/or tan β are very different from 1, the predicted neutrino mass hierarchy is generically too large. It is remarkable that this simple scenario can already reproduce the observations for reasonable parameters. Further, and as we will see in the next subsection, the allowed parameter space widens when including the operator mixing induced by quantum effects below the scale M 1 .

Operator mixing between κ (ab) non-negligible
We consider in what follows the phenomenologically interesting case of the 2HDM in the decoupling limit, where the lighter CP-even scalar resembles the Standard Model Higgs, while the other scalars are very heavy. In this case, tan β = v 2 /v 1 0, so that M ν = κ (11) v 2 1 /2, with v 1 = v. If quantum effects between M 1 and m H were negligible, the decoupling limit would lead to a very large neutrino mass hierarchy, cf. Eq. (39). However, operator mixing below the scale M 1 can significantly modify this conclusion.
The relevant Wilson coefficient κ (11) at the scale m H is calculated from κ 11 | m H = κ 11 | M 1 +δκ 11 where Using the explicit form of the β function, this correction can be schematically written as [11]: which makes clear the operator mixing through the RGE running. Here B 1a denote 3 × 3 matrices whereas b is a number. Explicitly, which depends linearly on the coefficient of the potential term 1 where we have taken m H v to implement the decoupling limit. Expressing M ν in terms of mass parameters m ab as in Eq. (32), we find for the case tan β = 0: Using the invariants from Eq. (35), we obtain for the largest active neutrino mass: and the mass hierarchy Therefore, for y 1 ∼ y 2 , generic angles θ 1 , θ 2 , θ L and λ 5 ∼ O(1) (so that b = O(1)), a mild neutrino mass hierarchy is generically expected. The effect of λ 5 in the running is illustrated in Fig. 4, which considers the same scenarios as in Fig. 3, but including the running between M 1 and m H setting λ 5 = 1. Clearly, for plausible values of λ 5 the operator mixing has a significant impact on the phenomenology and widens the allowed parameter space of the model.

Planck-scale lepton number breaking scotogenic scenario
We consider now a variant of the previous scenario where the SM symmetry group is extended with a discrete Z 2 symmetry, assumed to be exact in the electroweak vacuum. All SM particles are even under the Z 2 symmetry. Further, the SM particle content is extended with fermion singlets, N i , and scalar doublets with identical gauge quantum numbers as the SM Higgs boson, η a , all odd under the Z 2 symmetry.
With this set-up, all lepton number violating interactions involving only SM particles vanish at tree level. In particular, the Weinberg operator arises at the one-loop level. Further, the lightest particle of the Z 2 -odd sector constitutes a dark matter candidate. This is the so-called "scotogenic model" [13]. This model, however, presents the same drawbacks as the type-I seesaw model in regards of predicting the correct neutrino mass and neutrino mass hierarchy. It was argued in [14,15] that the extension of the scotogenic model by an additional Z 2 -odd scalar doublet leads in general to a mild neutrino mass hierarchy. In this section, we will investigate whether the breaking of the lepton number at the Planck scale in a variant of the scotogenic model, extended by an additional Z 2 -odd scalar doublet, can simultaneously lead to the correct neutrino mass and mass hierarchy.
The part of the Lagrangian containing the Z 2 -odd fermions and scalars reads: with a = 1, 2. Here we have chosen without loss of generality to work in the basis for the singlet fermions where the mass matrix is diagonal with eigenvalues M k . The scalar potential can be split into three separate parts with a = 1, 2. Here, V Φ (Φ) = µ 2 Φ † Φ + λ 2 (Φ † Φ) 2 is the potential for the Z 2 -even scalar doublet (the SM Higgs doublet), V η has the form of Eq. (1), replacing Φ a by η a , and is the interaction potential between the Standard Model Higgs doublet and the inert doublets. The masses of the neutral components of the inert doublets will be denoted by m ηa . As in the rest of this paper, we assume the RHN mass matrix to be approximately rank-1 at the cut-off scale of the theory, for which we take Λ = M P . We set for simplicity M 3 ∼ M P and M 1 , M 2 = 0. Two-loop quantum effects induced by the inert doublets generate non-zero values for M 1 and M 2 , given by Eq. (18), with the appropriate substitutions. Integrating-out the heavy particles, a single Weinberg operator arises, shown in Fig. 5, corresponding to the effective Lagrangian Using that in our scenario M k m η 1 , m η 2 , one can simplify: where in the logarithm we have approximated both scalar masses by m η , and both fermion masses by M 0 (see Eq. (22)).
It is now straightforward to calculate approximate expressions for the largest active neutrino mass, using the rank-1 assumption for the Yukawa and RHN mass matrices as previously  10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 Figure 6: Scan plot showing the mass hierarchy between the two heavier active neutrinos |m 3 /m 2 | vs. |m 3 | in the scotogenic Planck-scale lepton number breaking scenario with two inert doublets, assuming Λ = M P , M 3 = M P / √ 8π, and inert doublet masses m η 1 = 100 TeV and m η 2 = 2m η 1 . The Yukawa couplings have eigenvalues y 1 = y 2 = 1 and random misalignment angles, and the quartic couplings are λ lepton number breaking seesaw scenario, has a larger predictive power compared to the general seesaw framework, since the whole neutrino mass spectrum is determined by only two mass scales, the Planck scale and the electroweak symmetry breaking scale, which are known.
We have shown that under fairly general conditions, this scenario leads to active neutrino masses in the ballpark of the experimental values. At very high energies, two-loop quantum effects induced by the two Higgs doublets, generate comparable masses for the lighter right-handed neutrinos. Integrating-out the right-handed neutrinos leads to small neutrino masses through the seesaw mechanism. One of the active neutrinos is predicted to be m 3 ∼ (16π 2 ) 2 v 2 /M P ∼ 0.1 eV. Further, the mild hierarchy between the two lighter right-handed neutrino masses generically leads to mild hierarchies between the two largest eigenvalues of the Wilson coefficients of the Weinberg operators. This already suggests the generation of solar and atmospheric mass scales with a mild hierarchy. Further quantum effects due to the operator mixing among the Weinberg operators assist in the generation of a mild neutrino mass hierarchy, which then becomes a fairly generic expectation of the model. This scenario does not require a light exotic Higgs sector and the same conclusions apply in the decoupling limit, where lepton flavor and CP-violating processes have suppressed rates.
We have finally considered a "scotogenic" variant of this scenario with three fermion singlets and two scalar doublets carrying a Z 2 charge. The same conclusions apply for the neutrino phenomenology. Further, the "inert" doublets in this case make for a viable dark matter candidate.