Radiative seesaw corrections and charged-lepton decays in a model with soft flavour violation

We consider the one-loop radiative corrections to the light-neutrino mass matrix and their consequences for the predicted branching ratios of the five lepton-flavour-violating decays $\ell_1^- \to \ell_2^- \ell_3^+ \ell_3^-$ in a two-Higgs-doublet model furnished with the type-I seesaw mechanism and soft left-flavour violation. We find that the radiative corrections are very significant; they may alter the predicted branching ratios by several orders of magnitude and, in particular, they may help explain why $\mbox{BR}(\mu^- \to e^- e^+ e^-)$ is strongly suppressed relative to the branching ratios of the decays of the $\tau^-$. We conclude that, in any serious numerical assessment of the predictions of this model, it is absolutely necessary to take into account the one-loop radiative corrections to the light-neutrino mass matrix.


Introduction
The existence of neutrino oscillations is now firmly established-see [1][2][3][4] and references therein. Therefore, the violation of the family lepton numbers L ( = e, µ, τ ) is firmly established as well. However, no violation of the L but for neutrino oscillations has been hitherto detected. In this context, the flavour-violating charged-lepton decays are of particular importance, because it is expected that in the near future the experimental bounds on the branching ratios (BRs) of those decays will be improved substantially [5][6][7][8][9] (see also section 2 of [10]). It is thus important to address those decays in specific models for the neutrino masses and lepton mixings-and the more so since, when one incorporates neutrino masses and lepton mixings in the Standard Model (SM), those BRs are so small that the decays are in practice invisible [11][12][13].
In this letter we discuss the model put forward in [14,15]. This is a multi-Higgs-doublet extension of the SM (for reviews see [16,17]) furnished with three right-handed neutrino singlets ν R and the seesaw mechanism [18][19][20][21][22], and with the additional condition that the Yukawa-coupling matrices are flavour-diagonal so that the source of lepton-flavour violation lies exclusively in the Majorana mass matrix M R of the right-handed neutrinos. In other words, the only lepton-flavour-violating (LFV) terms in the Lagrangian are in where (M R ) 1 2 = (M R ) 2 1 are coefficients with mass dimension and C is the chargeconjugation matrix in Dirac space. The salient feature of this model is the soft nature of the breaking of the L [14,23] by L ν R mass ; the softness of the breaking ensures that the one-loop amplitudes of LFV charged-lepton decays are finite.
In this letter we confine ourselves to a model with just two Higgs doublets and, without loss of generality, we work in the 'Higgs basis', wherein only the first doublet has a nonzero vacuum expectation value v √ 2, where v 246 GeV is real and positive, in its neutral component ϕ 0 1 . This allows us to parameterize the lepton Yukawa couplings as (2) where the m are the (real and positive) charged-lepton masses and d , γ , and δ are dimensionless and, in general, complex Yukawa coupling constants. We emphasize that in (2) the interactions of the scalar fields with the leptons are flavour-diagonal; this is accomplished through global U (1) symmetries associated with the L . Naturally, the lepton numbers of the two Higgs doublets are zero, therefore, the U (1) symmetries are only softly broken by (1). 1 Let S ± a (a = 1, 2) and S 0 b (b = 1, 2, 3, 4) denote, respectively, the charged-scalar and the (real) neutral-scalar mass eigenfields of our two-Higgs-doublet model (2HDM). By definition, S + 1 ≡ G + and S 0 1 ≡ G 0 are, respectively, the charged and the neutral Goldstone bosons. Again by definition, S 0 2 ≡ H is the physical scalar with mass m H 125 GeV that has been observed at the LHC. Let M 3 and M 4 denote the masses of S 0 3 and S 0 4 , respectively. It is an outstanding feature of our model that the amplitudes for the radiative decays ± 1 → ± 2 γ and Z 0 → + 1 − 2 ( 1 = 2 ) are suppressed by m −2 R , where m R is the seesaw scale [14]; one can estimate that for m R 10 3 TeV these decays are invisible, in the foreseeable future, in the context of our model [15] (another model with this feature is discussed in [24]). The same suppression occurs when the gauge bosons are off-mass shell, viz. in the one-loop diagrams for the LFV decays µ − → e − e + e − and τ − → − 2 + 3 − 3 ( 2 , 3 = e, µ) [14] where those decays are mediated by either a virtual γ or a virtual Z 0 . 2 On the other hand, those five decays also have one-loop amplitudes mediated by neutral-scalar exchange, and these amplitudes are unsuppressed when m R → ∞ [14]. It is the purpose of this letter to present a theoretical and numerical study of these three-body decays while taking into account the radiative corrections to the seesaw mass matrix of the light neutrinos [25]. The latter point is new when compared to [15], and it is important because of two reasons: • For M 3 or M 4 larger than 4πv ∼ 3 TeV, and provided the Yukawa couplings d and δ are of similar order of magnitude, the radiative corrections to the neutrino mass matrix are dominant; 3 they are non-negligible even for values of M 3,4 much lower than that.
• The branching ratios BR − depend on the mass matrix M R [14,15]-see section 2. Information on M R is not directly available but has to be extracted from the mass matrix of the light neutrinos. The latter matrix may be assembled from the light-neutrino masses and from the lepton mixing obtained from fits to the neutrino oscillation data. Therefore, the radiative corrections to the mass matrix of the light neutrinos will influence the extraction of M R . Indeed, they may alter the branching ratios drastically, as we shall see later.
Henceforth, for the sake of brevity, the acronym 'BR' will always refer to the branching ratios of the five decays − 1 → − 2 + 3 − 3 ; the same will apply to the phrase 'decay rate'. Although in this letter we consider just a 2HDM, we nevertheless have a large number of parameters. In order to facilitate the numerical analysis it is useful to reduce that number. We adopt the strategy of [15] and assume the following: A. There is no mixing between the two scalar doublets.
B. All parameters are real. 2 The box diagrams for µ − → e − e + e − and τ − → − 2 + 3 − 3 are also suppressed by 1/m 2 R [14]. 3 It was already stressed in [26] (see also [27,28]) that the radiative corrections to the seesaw mechanism, in the presence of two or more Higgs doublets and heavy neutral scalars, may be quite large. Moreover, it has been demonstrated in [29] that, even with only one Higgs doublet, those corrections may be substantial for fine-tuned tree-level neutrino mass matrices.
Through assumption A, the mixing of the scalars is simplified to [15] where H + is a physical charged scalar which, however, plays no role in this letter. The advantage of assumption A is threefold: • There are in general three parameters in the mixing of the neutral scalars [15,30].
With assumption A they are reduced to only one-the phase α, which is, however, unphysical because one may freely rephase ϕ + 2 and ϕ 0 2 . 4 • The formulas for the BRs simplify considerably (see section 3).
• The couplings of H are identical to the ones in the SM, hence the experimental restrictions on the couplings of H are automatically fulfilled.
We thus consider that we are in the exact alignment limit of the 2HDM. This is in accordance with the measurements of the properties of the scalar discovered at LHC (see for instance [31,32]), which have found that that scalar behaves in a manner very similar to the SM Higgs boson; its couplings are experimentally constrained to be very close to their respective SM values. Assumption A ensures that this indeed happens in our 2HDM. In the ensuing discussions we will initially keep complex parameters, but we take into account assumption A right from the beginning. This letter is organized as follows. In section 2 we discuss the light-neutrino mass matrix, including the radiative corrections. The formulas for the decay rates are displayed in section 3, where we also derive a prediction of our model when assumption A holds. In section 4 we discuss the procedure of our numerical investigation and in section 5 some results thereof are presented. We draw our conclusions in section 6. An appendix makes a digression through the scalar potential of the 2HDM in order to demonstrate that the two new scalars S 0 3 and S 0 4 may have sufficiently different masses.

The light-neutrino mass matrix
The Majorana mass matrix M ν of the light neutrinos is diagonalized as where U L is 3 × 3 unitary and the m j (j = 1, 2, 3) are real and non-negative. Since the charged-lepton mass matrix is diagonal from the start, cf.
(2), U L is just the lepton mixing matrix. The matrix M R , defined in (1), is diagonalized as 4 Due to assumption B, later on we will set e −iα = 1 in (3).
where the m 3+j are real and positive and the matrix U R is 3 × 3 unitary. For the decay rates-see (17) in the next section-we need the quantities [14,15] for 1 = 2 . 5 This requires us to know both U R and the heavy-neutrino masses m 4,5,6 . Note that, since U R is a 3 × 3 unitary matrix, X 1 2 cannot be large; one has |X 1 2 | 0.1 if 10 9 GeV ≤ m 4,5,6 ≤ 10 19 GeV.
We parameterize U L as In (7a), e iα and e iβ are diagonal matrices of phase factors while U PMNS is the Pontecorvo-Maki-Nakagawa-Sakata matrix [4]. Out of the three phases in e iβ , one may be absorbed intô α and the remaining two are the so-called Majorana phases, which are physically meaningful quantities. In (7b) and (7c), c ij = cos θ ij and s ij = sin θ ij for ij = 12, 13, 23. The matrix U R may be parameterized in the same way as U L . The matrix M ν is the sum of two parts: where the tree-level part M tree ν is given by the seesaw mechanism and the one-loop-level part δM L is generated by the radiative corrections. As is well known, where M D is the neutrino Dirac mass matrix. Referring to (2), let us define the diagonal matrices Because we use the Higgs basis, M D is given by hence it is diagonal. Thus, The radiative part of M ν is given by [25] where the four lines correspond successively to the contribution of the Z 0 gauge boson with mass m Z , of the SM scalar H, and of the new scalars S 0 3 and S 0 4 . In (13) we have already taken into account assumption A of section 1. We have also used m 4,5,6 m Z , m H , M 3,4 . It is clear that for M 3,4 4πv and provided ∆ 1 and ∆ 2 are of identical orders of magnitude, the contributions (13c) and (13d) dominate over the contribution (12). 6 Lines (13c) and (13d) coincide with the well-known scotogenic mechanism; however, in the scotogenic model proper [33] the Yukawa couplings d and γ are zero (because of an additional symmetry), while in this letter they are nonzero.
We reformulate (4) to Equation (14) is the basis for our numerical computations. The diagonal matrix e iα in the left-hand side of (14) is irrelevant; indeed, it can be absorbed into U R in the right-hand side, since ∆ 1 and ∆ 2 are diagonal matrices. In principle we use as input the Majorana phases, U PMNS ,m, ∆ 1 , ∆ 2 , α, M 3 , and M 4 (and additionally the fixed values v = 246 GeV, m Z = 91 GeV, and m H = 125 GeV) and we solve (14) to find the three m 3+j and the nine parameters of the 3 × 3 unitary matrix U R . All the matrices in (14) are 3 × 3 symmetric and complex; therefore, equation (14) is in effect a system of 12 real equations for the 12 unknowns-m 4,5,6 and the nine parameters of U R -that we need for the computation of the X 1 2 . We stress that this parameter counting only serves to demonstrate the theoretical possibility of obtaining the X 1 2 from equation (14); when one attempts to do it numerically, equation (14) may sometimes prove difficult or impossible to solve.
Using the assumptions (15), the symmetric matrix in the left-hand side of (14) is real, hence it has just six degrees of freedom. Then, the matrix U R may be written where U R ∈ SO(3) is a real matrix parameterized by three angles and the 3+j may be either 1 or i. Equation (14) is then used to determine the three angles of U R and the three 2 3+j m 3+j ; the latter are either positive, if 3+j = 1, or negative, if 3+j = i. Let us take stock of the (real) parameters in the game, after having performed the simplification stated in the previous paragraph. From the neutrino oscillation data, both the two mass-squared differences among the three light-neutrino masses and the three mixing angles in U PMNS are known and they are used as input. There are then 15 unknown parameters in (14): the lightest neutrino mass, viz. m 1 for normal ordering and m 3 for inverted ordering of the neutrino masses, M 3,4 , d , δ , 2 3+j m 3+j for j = 1, 2, 3, and the three angles in U R . As we shall see in the next section, there are in addition the three parameters γ , which do not appear in (14) but occur in the BRs.

Decay rates
Repeating the result of [15], the decay rates are given by where we have used the approximation that the final-state charged leptons are massless, and We stress that assumption A is responsible for the relatively simple form of the decay rates.

A prediction
Taking ratios of decay rates of the τ , we obtain ratios of BRs. Defining we obtain The maximum of the function in the right-hand side of (21) is 1 at x = 1; its minimum is 3/4 at x = 0 and x = ∞. Therefore, we have the following prediction: should lie between   9 16 and 1.
This is a non-trivial result of our model, provided assumption A holds.

Suppressing
With the mean lives τ µ and τ τ of muon and tau, respectively, it follows from (17) that where The extant experimental upper bounds on the BRs are given in table 1. In the future, it is expected that the experimental sensitivity on BR (µ − → e − e + e − ) will reach ∼ 10 −16 [5], while the sensitivity on the BRs of the four LFV τ decays may be increased by one order of magnitude to ∼ 10 −9 either at a Super B factory [6] or at the High Luminosity LHC [7,8], and even reach ∼ 10 −10 at Belle II [9].
We will be interested in obtaining parameter-space points for which all the BRs are below the extant experimental bounds but above the expected future sensitivities. Using the present experimental upper bound 10 −12 on BR (µ − → e − e + e − ) and taking, for definiteness, the future sensitivity on the BRs of the τ − decays to be 10 −9 , we obtain from (22) that for such points. This may happen either because R X is very small, or R A is very small, or both. Focussing specifically on R A , by using m 2 m 1 v together with the assumption that all the Yukawa couplings are real, we read off from (18) the dominant terms Therefore, in order to obtain a small R A both d e δ µ − d µ δ e and γ 2 e − γ 2 µ should be small.

Numerical procedure
Solving (14) means finding m 4 , m 5 , m 6 and the matrix U R . The latter is parameterized just as U L in (7), i.e. its elements are given by where S ij = sin θ R ij and C ij = cos θ R ij . However, following the reality assumption (15), the matrix in the left-hand side of (14) is real, hence U R is real as well, apart from possible imaginary factors 3+j in (16). In order to avoid finding the same solutions of (14) several times in different conventions, we fix the phases in For the mass-squared differences and the mixing angles in U PMNS we take the best-fit values of [34]: ∆m 2 21 = 7.39 × 10 −5 eV 2 , ∆m 2 31 = 2.525 × 10 −3 eV 2 , sin 2 θ 12 = 0.310, sin 2 θ 13 = 0.02241, sin 2 θ 23 = 0.580.
Our fitting program consists of two parts. In the first part, the matrix equation (14) is solved by using a minimization procedure wherein the function χ 2 eq , given in (29) below, is adjusted to zero with high precision. In this part of the program all the parameters that occur in the branching ratios, except the Yukawa couplings γ , are determined. In the second part of the program, we use the parameters obtained in the first part and we search for γ such that either several or all five branching ratios are within the future experimental reach; this is done with the help of the function χ 2 br given in (31) below. The function χ 2 eq is constructed in the following way. Let (M exp ν ) ij and M theor ν ij be the matrix elements of the matrices in the left-hand and right-hand sides, respectively, of (14). Then, the function that we minimize is 9 In the first part of our fitting program we proceed in the following way. The masssquared differences ∆m 2 21 and ∆m 2 31 and the lepton mixing angles θ 12 , θ 13 , and θ 23 are fixed to their best-fit values [34]. In section 2 we have already stated the values of v, m Z , and m H used in our code. Nine parameters-the masses M 3 and M 4 of the new scalars, the mass m 1 of the lightest neutrino, and the real Yukawa couplings d and δ for = e, µ, τ -are inputted into (14); that matrix equation is solved by minimizing χ 2 eq with respect to the six parameters θ R ij and 2 3+j m 3+j , which form the output of (14). We consider (14) to be solved when χ 2 eq < 10 −16 ; the resulting set of 15 parameters is then saved for usage in the second part of the fitting program.
Note that, since we use a minimization procedure, we may as well explore the full parameter space and minimize χ 2 eq with respect to all 15 parameters simultaneously. It is also possible to choose any subspace in the 15-dimensional parameter space and to perform the minimization of χ 2 eq in that subspace; indeed, in the following we shall do precisely this, by either fixing or imposing restrictions on the ranges of some of the input parameters prior to minimization of χ 2 eq . The function χ 2 br is constructed in the following way: where the index i runs over the five BRs, Θ is the Heaviside step function, BR bound i denotes the experimental upper bound on each BR (these are the bounds given in table 1), BR theor i is the calculated value of the BR, and k is a small number that is meant to give a kick to the minimization algorithm whenever the calculated value is larger than the experimental upper bound. Note that the minimum possible value of χ 2 br is five, which materializes in the limit where all five calculated BRs are just a little smaller than the experimental bound on the corresponding BR. 10 The minimization function (31) can handle even situations when the calculated BRs and the upper experimental bounds differ by many orders of magnitude. The function χ 2 br is minimized only with respect to the Yukawa couplings γ , because the other parameters have been fixed already in the first part of the fitting program. We stress that, in contrast to χ 2 eq , it is not necessary to minimize χ 2 br with high precision, since our objective is to obtain BRs that are below but not necessarily close to their respective experimental bounds. We use 10 −16 and 10 −9 as the future experimental sensitivities on BR (µ − → e − e + e − ) and the BRs of the τ − decays, respectively.
Often, we want to compare the results of equation (14) with the ones of its tree-level counterpart Whenever we perform such a comparison, we use the superscript "(loop)" on quantities that arise from the solution of (14) and the superscript "(tree)" on quantities that arise from the solution of (32). It is one objective of this letter to show that the quantities with superscript "(loop)" may be substantially different from the corresponding quantities with superscript "(tree)".

Evolution of the X 1 2
In this subsection we give two examples of the way the quantities X 1 2 may change when the input parameters are varied. In our first example we fix eight inputs as follows: m 1 = 30 meV, M 3 = 1.5 TeV, M 4 = 1.6 TeV, 11 d e = 0.01, d µ = 0.1, d τ = 0.001, δ e = 1, and δ µ = 0.001. We vary δ τ from 0.005 to 0.5 and compute X µe , X τ e , and X τ µ for each value of δ τ . In this case ∆ 1 is kept fixed, hence the solution of (32) is always the same and produces X

Scatter plots of BRs
In figure 4 we present scatter plots of the branching ratios of 12 µ − → e − e + e − , τ − → e − e + e − , and τ − → µ − µ + µ − as functions of m 1 . Here as elsewhere in this section we always assume, for the sake of simplicity, the neutrino mass ordering to be normal. To produce the scatter plots, we chose m 1 at random in betweeen 10 −2 meV to 30 meV, prior to the minimization of χ 2 eq ; larger values of m 1 would violate the Planck 2018 cosmological bound on the sum of the light-neutrino masses [38]. Then the BRs are computed, as described in section 4, by consecutive minimization of χ 2 eq and χ 2 br with respect to the remaining parameters. We restrict the parameter space by adopting the boundary conditions 750 GeV < M 3,4 < 2 TeV, (33a) 0.05 ≤ |d | , |δ | , |γ | ≤ 0.5 ( = e, µ, τ ), (33c) and 10 11 GeV ≤ m 4,5,6 ≤ 10 16 GeV, with the 2 3+j m 3+j being either positive or negative. It is worth making a number of comments concerning (33) and (34)  3. Sometimes the solution of (14) requires one of the m 3+j to be very large, even divergent. This is not surprising because, when e.g. m 6 → ∞, the contribution of m 6 to (14) simply vanishes. Unfortunately, though, when m 6 → ∞ the X 1 2 diverge. We avoid this problem by discarding, through the upper bound in (34), those points where the solution of (14) requires very large m 3+j .

4.
We have obtained points with values of the heavy-neutrino masses as low as 10 9 GeV. However, those points have very low BRs for the decays of the τ − , of order 10 −12 .
In (34) we have discarded those points by enforcing a lower bound on the heavyneutrino masses.
In figure 5 we display the ratios BR (loop) BR (tree) for the same points as in figure 4 and with the same colour notation. One sees that for the τ − decays the BRs derived from (14) may easily be one or two orders of magnitude either above or below the corresponding BRs derived from (32). For the decay µ − → e − e + e − things may be much more dramatic, with differences of several orders of magnitude; this happens because either X (loop) µe or X (tree) µe frequently become zero. It is worth mentioning that by allowing for wider ranges of the Yukawa couplings (for example, allowing |d |, |δ |, and |γ | to be between 0.001 and 1) would lead to the ratios BR (loop) BR (tree) being sometimes much larger; those ratios could be two  (33) and (34). In all the displayed points, all five BRs satisfy the present experimental bounds given in table 1. Blue points have all five BRs larger than the expected future sensitivities, while red points allow one or more (but not all) BRs to be below the future sensitivities. The shadowed bands show the ranges between the present experimental bounds and the future experimental sensitivities, viz. 10 −16 for BR (µ − → e − e + e − ) and 10 −9 for the BRs of the τ − decays. orders of magnitude larger or smaller than is shown in figure 5.

The suppression of
In figure 6 we reuse the blue points of the previous figures 4 and 5, viz. points for which all five BRs are in between the respective present upper bounds and future expected sensitivities. For those points, we display R BR defined in (22), and R X and R A defined in (23). In the right panel one sees that the inequality (24) holds and that R X R A is proportional to R BR as stated in (22). In the left panel one sees that the smallness of R X R A most of the time occurs because both R A and R X are small, but there is a non-negligible fraction of points where one of them is extremely small and the other one is not small.
The discussion at the end of section 3.2 suggests that the smallness of R A is correlated with the smallness of the asymmetries Using the same points as in figure 6, these asymmetries are displayed in figure 7. One sees that A 1 and A 2 are indeed very small when R A 10 −7 , but they may be largish for R A above that value; we remind the reader that, like we saw in figure 6, the smallness of R X R A is often due to the smallness of R X and not to the smallness of R A , or vice-versa.

Benchmark points
In table 2 we produce three benchmark points. The first nine lines of that table contain the input to (14), viz. the matrices ∆ 1 and ∆ 2 , the lightest-neutrino mass m 1 , and the new-scalar masses M 3 and M 4 . In the next six lines of table 2 one sees the output of (14), viz. the heavy-neutrino masses 2 3+j m 3+j and the angles θ R ij that parameterize the matrix U R . In the next three lines of table 2 one finds the parameters γ that we have fitted in order to obtain the desirable branching ratios which are in the ensuing five lines of the table. Finally, in the last three lines of table 2 we compare the values of the quantities X (loop) 1 2 that were obtained  from the solution of the one-loop equation (14) to the quantities X (tree) 1 2 that result from the solution to the tree-level equation (32).
All the points in table 2 have small A 2 asymmetries. The asymmetry A 1 is also small for point 1, but not for points 2 and 3; the latter points rely on very small X (loop) µe to suppress BR(µ − → e − e + e − ).

Conclusions
The predictions for the lepton-flavour-violating charged-lepton decays may be used to discriminate among theoretical models. For instance, it has been found [41] that, in a model with a heavy charged gauge boson, the present bounds on µ − → e − γ and µ − → e − e + e − restrict the parameters of the model in such a way that the decays τ − → −    is proportional to |X 1 2 | 2 , where the quantities X 1 2 defined in (6) depend on m 4,5,6 and U R . We have shown that the one-loop radiative corrections to the light-neutrino mass matrix may modify that matrix so much that the model's predictions for µ − → e − e + e − and τ − → − 2 + 3 − 3 change drastically. This is especially true for BR (µ − → e − e + e − ), which may shift by several orders of magnitude when one (dis)considers the effect of the radiative corrections on the determination of the heavy-neutrino masses and mixings. This happens, in particular, because X µe may be zero for different values of the model's parameters at the tree level and at the one-loop level. The predictions for the four decays τ − → − 2 + 3 − 3 usually change by no more than two orders of magnitude when one takes into account the radiative corrections, but for values of the Yukawa couplings larger than the ones displayed in figure 4 the effects on BR τ − → − 2 + 3 − 3 may be dramatic too. Our work highlights the necessity of taking into account the one-loop radiative corrections to the light-neutrino mass matrix when making any numerical assessment or prediction of an effect that involves the masses m 4,5,6 and the mixing matrix U R . Usage of the standard seesaw formula (9) is not adequate when one looks for detailed numerical predictions because the 'scotogenic-type' contributions to δM L in (13c) and (13d) may be non-negligible or even dominant. This happens even when one takes into account the restrictions posed by unitarity of the scalar potential on the squared-mass differences among the neutral scalars; though those differences are rather small, the effects of the radiative corrections are nevertheless large in general. In this appendix we study in detail the scalar potential of the 2HDM with alignment. Our purpose is to demonstrate that the difference between the squared masses of the two new neutral scalars of that model may reach v 2 (8π/3) ≈ 5.07 × 10 5 GeV 2 . We do not claim this to be an absolute upper bound; simply, we were able to demonstrate analytically that it may be reached. On the other hand, numerical scans that two of us have performed [40] suggest that 8π/3 is indeed the maximum possible value of the parameter λ 5 of the scalar potential, even in the general case without alignment.
The coefficients λ 1,...,7 are subject to two types of conditions: the unitarity conditions and the boundedness-from-below (BFB) conditions.

A.5 Additional conditions
One must guarantee that our assumed vacuum state is indeed the state with the lowest value of V , viz. that we are not in the situation where there are two local minima of the potential and we are sitting on the local minimum with the highest value of V instead of being at the true vacuum; this undesirable situation has been called 'panic vacuum'. This produces the following condition [47][48][49]54]. Let ζ ≡ 2m 2 C /v 2 and let us order the eigenvalues of Λ E as Λ 0 > Λ 1 > Λ 2 > Λ 3 . Then, either ζ > Λ 1 or Λ 2 > ζ > Λ 3 .
There is also a phenomenological condition arising from the oblique parameter T . With alignment [30], where s 2 w = 0.22 is the squared sine of the weak mixing angle, m W = 80.4 GeV is the mass of the W ± gauge bosons, and The phenomenological constraint is T = 0.03 ± 0.12 [4].
With (A15), We are interested in the situation where is rather small, so that |λ 5 | is not very far from 8π/3. When is small, m 2 C lies in between the M 2 3 and M 2 4 given in (A17), but it is very close to one of them because λ 1 is so small. Then, T is negative but very small, automatically satisfying the phenomenological constraint on that parameter.
With (A15) one has Therefore, if we choose we avoid the undesirable situation of panic vacuum. Thus, we must have m C 707 GeV. This lower bound on m C coincides with an analogous bound obtained in a recent phenomenological analysis [39]. Our solution (A15) explicitly demonstrates that |M 2 3 − M 2 4 | may reach v 2 (8π/3) without violating the unitarity and BFB conditions and with a very small oblique parameter T . Moreover, the inequality (A19) provides a way to choose the mass of the physical charged scalar such as to evade panic vacuum.