Extended Empirical Fermion Mass Relation

It is known that the charged lepton masses obey to high precision an interesting empirical relation (Koide relation). In turn, the light neutrino masses cannot obey such a relation. We note that if neutrinos acquire their mass via the seesaw mechanism, the empirical mass relation could hold for the masses in the Dirac and/or heavy Majorana mass matrix. Examples for the phenomenological consequences are provided. We furthermore modify the mass relation for light neutrino masses including their Majorana phases, and show that it can be fulfilled in this case as well, with interesting predictions for neutrinoless double beta decay. Finally, we remark that while the relation does not hold for the up- and down-quarks, it may be valid for the u, d, s quarks, and for the c, b, t quarks.

Motivated by the idea of grand unification, one may wonder whether a similar mass relation exists for other fermions.Numerical analysis has shown that neither the up-quark masses, down-quark masses nor the light neutrino masses could satisfy such an empirical relation, even if the renormalization group (RG) running effects are taken into account [15,16].Consider for illustration the charged leptons and ignore the small ratios m e /m τ , m e /m τ and m µ /m τ in Eq. (1).One approximately obtains By using the pole masses of charged leptons given in Particle Data Group [17], one can estimate that m µ /m τ ≃ 0.25, which roughly yields K ℓ = 2/3.Obviously, if the mass of the lightest particle can be neglected, the empirical relation relies on the mass ratio between the two heavier masses.In the extreme case of a highly hierarchical mass spectrum, in which the ratio of the second heaviest and the heaviest mass is very small, one has K → 1.This is the case for the up-and down-quarks, for which the values K up ≃ 0.89 and K down ≃ 0.75 apply (both evaluated at M Z = 91.2GeV [18]).The other extreme limit applies when the mass spectrum is nearly degenerate, in which case K → 1/3.In this sense, the empirical mass relation Eq. ( 1) appears interestingly exactly in the middle of all its possible values 1 .One can understand also why light neutrinos cannot obey the empirical relation.If they are quasi-degenerate, K ν ≃ 1  3 , and if they obey a normal hierarchical spectrum, where ∆m 2 ⊙ and ∆m 2 A are the solar and atmospheric mass-squared differences.
We now extend the expression in Eq. ( 1) for arbitrary three masses, m x , m y and m z , i.e., where ǫ 1 = m x /m z and ǫ 2 = m y /m z .The allowed range of K is shown in Fig. 1.We have in particular indicated the values of ǫ 1 and ǫ 2 , which give K = 2 3 .One can read from the plot that the minimum value K = 1/3 appears at the position ǫ 1 = ǫ 2 = 1, while the maximum value is vanishing, a lower bound K ≥ 1/2 can be achieved for ǫ 2 = 1 (ǫ 1 = 1).As mentioned above, in the quark sector both up-type and down-type quark mass spectra are quite hierarchical, and the K-value is generally larger than 2/3.However, if the quarks are divided in light and heavy quarks, instead of up-and down-like quarks (i.e., according to mass instead of their electric charge or isospin), the empirical relation could be well satisfied.For example, considering the pole masses of the heavy quarks at 1σ C.L., the ratio is found to be 0.66 < K heavy < 0.68.This is in amazing agreement with Eq. ( 1).Due to the non-perturbative nature of quantum chromodynamics at low energies, the pole masses of light quarks are not well defined.Therefore, we make use of the running light quark masses at the scale M Z = 91.2GeV, and find that a similar relation for light quarks is located in the interval 0.49 < K light < 0.65, indicating a small deviation from the exact empirical relation at about 1σ C.L. We will focus on the application of the mass relation to the neutrinos in the rest of the paper.As mentioned above, their mass spectrum is not hierarchical enough to reproduce the empirical relation, and 1/3 < ∼ K ν < ∼ 3/5 typically holds, regardless of the neutrino mass ordering [16].However, we would like to note here that if neutrinos are Majorana particles their mass presumably originates from the seesaw mechanism [19][20][21][22], i.e., the light neutrino masses appear as a combination of Dirac and Majorana mass terms: In this case, it is not surprising that the empirical mass relation does not directly apply to neutrino masses.Instead, it is much more natural to assume that the relation is fulfilled in M D and/or M R .Therefore, in this paper we will extend the empirical mass relation to the seesaw framework, and study some of the phenomenological consequences resulting from this hypothesis.Another crucial aspect of Majorana neutrinos is the presence of Majorana phases, and we will also modify the empirical relation taking this into account.It is shown that the relation can work in this case as well.
The rest of the paper is organized as follows: In Sec.II, we introduce the seesaw extended mass relation, and describe four typical scenarios, in which very distinctive predictions on the light and heavy neutrino mass spectrum can be gained.The phenomenological consequences of these four scenarios are figured out, and the parameter spaces are illustrated.In Sec.III, we generalize the empirical mass relation by including contributions from CP-violating phases, and present the constraints on the Majorana phases as well as the light neutrino masses.In particular, we present the predictions on the effective mass relevant for neutrinoless double beta decay.Finally, in Sec.IV, we summarize our work and state our conclusions.

II. MASS RELATION IN THE SEESAW MODEL
Without loss of generality, one can always work in a basis in which the right-handed neutrino Majorana mass matrix is diagonal, i.e., M R = diag(M 1 , M 2 , M 3 ) with M i (for i = 1, 2, 3) being the masses of righthanded neutrinos.If lepton mixing stems entirely from the charged lepton sector, as possible in flavor symmetry models, the Dirac mass term M D is diagonal, i.e., M D = diag(D 1 , D 2 , D 3 ).This is a strong assumption, but very helpful for illustrating the point we wish to make in this note.We will comment later on the general case.The light neutrino masses are simply given by We stress again that though the empirical relation is shown to be not compatible with the light neutrino masses, the exact relation may exist in the mass matrices M D and/or M R .We will discuss 4 different cases.
• Case I: The empirical relation exist in M R whereas M D is an identity matrix, i.e., D i = D 0 .In this limit, the right-handed neutrino masses are proportional to the inverse of the light neutrino masses.
A hierarchical mass spectrum of right-handed neutrinos can be achieved if one of the light neutrino masses is extremely small.For example, in the normal hierarchy case m 1 ≪ m 2 ≪ m 3 , one obtains which gives K R ≃ 2/3 if m 1 /m 2 ≃ 0.03.In the inverted hierarchy case one has m 3 ≪ m 1 < m 2 , and can be expected.The empirical relation requires We show in the upper plot of Fig. 2 the dependence of K R on the lightest neutrino mass in case I.In the numerical computations, we make use of the values of neutrino mass-squared differences from a globalfit of current neutrino oscillation experiment [23], and allow them to vary in their 1σ interval.In ad- dition, the dotted line indicates the exact empirical relation, i.e., K R = 2/3.One directly finds that the empirical relation in right-handed neutrino masses can be achieved for m 1 ≃ 2.5 × 10 −4 eV in the normal hierarchy case and m 3 ≃ 8 × 10 −4 eV in the inverted hierarchy case.These numerical results are in good agreement with the analytical results.
• Case II: The empirical relation exists in M D whereas M R is an identity matrix, i.e., M i = M 0 .According to Eq. ( 8), one has D i ∼ √ m i indicating that the hierarchy of D i is milder than the light neutrino masses.Therefore, the empirical relation cannot hold, and this case is then ruled out.
In the lower plot of Fig. 2, we show the value of K D with respect to the lightest neutrino mass in case II.The maximum value of K D cannot exceed 0.5, indicating that case II is incompatible with the experimental data.
• Case III: Both M D and M R fulfill the empirical relation.In this case, we have in total six free parameters in M D and M R , out of which five can be fixed by using Eqs.( 1) and (8).One can freely choose one of the parameters, e.g., M 1 , since the absolute mass scale of right-handed neutrinos cannot be determined from the empirical relation.The light neutrino masses are constrained by the relation, and we will show in what follows that a nearly degenerate spectrum is unfavorable.
In our numerical analysis of case III, we fix the mass of one right-handed neutrino, namely, we take M 1 = 10 9 GeV in the normal hierarchy case and M 3 = 10 9 GeV in the inverted hierarchy case.Then, the other two right-handed neutrino masses can be determined from the empirical relation and Eq. ( 8) for a given set of light neutrino masses.The allowed ratios between right-handed neutrino masses are shown in Fig. 3 for both the normal and inverted hierarchies.One reads from the plots that there exist stringent upper bounds on the light neutrino masses, i.e., the mass of the lightest neutrino cannot exceed 4 × 10 −3 eV.Furthermore, M 2 lies below M 1 and M 3 in the inverted hierarchy case, whereas, in the normal hierarchy case, there exists a flip between M 2 and M 3 for m 1 ∼ 1.5 × 10 −4 eV.A simple estimate shows that, at the flip point, M 2 /M 1 = M 3 /M 1 ≃ 0.015.We stress that the results for this case do not depend on the chosen value of M 1 .
• Case IV: The eigenvalues of M D are the same as the charged-lepton masses m e , m µ and m τ , respectively, whereas M R remains unconstrained.Such a special structure of M D could arise in some grand unification theories.In this case, the right-handed neutrino masses can be directly obtained according to Eq. ( 8) once the light neutrino masses are chosen.
The predicted right-handed neutrino masses in case IV are illustrated in Fig. 4. In both the normal and inverted hierarchy cases, the right-handed neutrinos possess a hierarchical spectrum with M 1 ≪ M 2 ≪ M 3 .The mass of lightest right-handed neutrino M 1 could be located around the TeV scale well within the scope of current colliders, while the heaviest right-handed neutrino mass M 3 is greater than 10 9 GeV.

III. VARIATION OF THE EMPIRICAL RELATION
As shown above, the pure light neutrino masses cannot fulfill the empirical mass relation.However, due to the Majorana nature of light neutrinos, the deviation of light neutrino masses from the empirical relations might be viewed as the effects of non-vanishing Majorana phases.In this sense, it is worth investigating a variation of the empirical relation, in which Majorana CP-violating phases are included, i.e., Allowing φ 1 and φ 2 to vary between 0 and 2π, Kν = 2/3 can be easily achieved, since there are two more free parameters entering the relation.In particular, Kν could be larger than 1 or close to zero if there exists strong cancellation in the denominator or nominator, which is quite relevant in the case of a nearly degenerate light neutrino mass spectrum.
The allowed parameter spaces of φ 1 and φ 2 required for Kν = 2/3 are shown in Fig. 5.We see that the two Majorana phases are constrained by the empirical relation, e.g., φ 1 = φ 2 = 0 is unfavored in both hierarchies.In the nearly degenerate limit, the correlation between φ 1 and φ 2 is basically the same for normal and inverted ordering, since in the limit holds for both cases.
In case of a normal and hierarchical mass spectrum, Kν turns out to be only sensitive to the phase difference φ = φ 2 − φ 1 , and according to the plot φ should be very close to −π.This can be understood from Eq. (11) which, in the limit m 1 ∼ 0, can be reduced to where r = m 2 /m 3 ≃ 0.4, and φ = φ 1 − φ 2 .Taking φ = −π, one can estimate that Kν ≃ 0.7, roughly in agreement with the empirical relation.An interesting result is found for the inverted hierarchy (m 2 ≃ m 1 ≫ m 3 ).One has no dependence on φ 2 , and finds This expression gives Kν = 2 3 for a phase which corresponds to sin 2 φ 1 /2 = 21 25 .
The allowed parameter spaces of φ1 and φ2 for normal hierarchy (upper plot) and inverted hierarchy (lower plot).The red, green, blue and black lines in the plot correspond to the lightest mass m1(m3) = 0, 0.02 eV, and 0.2 eV, respectively.
Since the Majorana nature of neutrinos is aimed to be revealed in future neutrinoless double decay experiments, there is a need to address some comments on the connection between the empirical mass relation and the neutrinoless double beta decay process, in which the decay amplitude is proportional to Here the leptonic flavor mixing matrix V is given by V e1 = cos θ 12 cos θ 13 , V e2 = sin θ 12 cos θ 13 , and V e3 = sin θ 13 , with θ ij being the lepton mixing angles.We illustrate in Fig. 6 the constraints on m ee when Kν = 2/3 is satisfied.In the normal hierarchy case, the predicted m ee almost saturates all of the experimentally allowed range for m 1 1 eV.However, in the inverted hierarchy case, the allowed range of m ee shrinks a lot if m 3 0.01 eV.This is a consequence of Eq. ( 14), and the value sin 2 φ 1 /2 = 21 25 following from the requirement Kν = 2  3 .Inserting this phase in the expression for the effective mass, whose general value is which gives the remarkable result This fixes the effective mass to about 0.025 eV, with little dependence on the oscillation parameters.One should compare this value with the general lower and upper limits of 0.017 and 0.05 eV, respectively.

IV. CONCLUSION
In this work we have added several new points regarding the properties of an empirical mass relation (Koide relation), which frequently is discussed in the literature.
We first extended the empirical mass relation of charged leptons to the other fermion sectors.In particular, we have noted that there could exist an universal empirical relation in the quark sectors once the quarks are classified by their mass scales instead of their electric charge or isospin.We then noted that if light neutrinos acquire their mass via the seesaw mechanism, it is not surprising that they fail to obey the empirical relation.Instead, we applied the relation to the Dirac and/or heavy Majorana mass matrix.We illustrated the consequences for neutrino masses in a simplified seesaw framework by assuming all lepton mixing to stem from the charged lepton sector, and analyzed four typical scenarios realizing the empirical relation in different ways.Furthermore, we generalized the empirical mass relation with Majorana phases being included and found that in case of an inverted hierarchy mass spectrum, the effective mass is strongly constrained.The relevant Majorana phase is fixed and a value of about m ee ≃ 0.025 eV exists, being a factor of 1.5 larger than the general lower bound.
Let us remark here that in a more general seesaw framework, there will be much more free parameters entering the expression of M D , and there are in principle enough degrees of freedom to reproduce the empirical relation in both M D and M R for any choice of light neutrino masses.In addition, thermal leptogenesis cannot work in our simplified framework even if flavor effects are taken into account.Note that the empirical mass relation discussed in this work could be re-scaled up to a higher and more fundamental scale, i.e., the grand unification scale, by using RG equations.However, as shown in Ref. [16], the fermion mass ratios tend to be very stable against radiative corrections even in the supersymmetric model with a large tan β.If there are intermediate scales in the RG evolution, the running fermion masses could be modified by the threshold effects, in particular for neutrinos.Such a study could be done in a model-dependent manner and lies beyond the scope of current work.

2 KFIG. 1 :
FIG.1:The allowed range of K with respect to the ratios ǫ1 and ǫ2.The red and blue curves on the ǫ1-ǫ2 plane indicate K = 1/2 and K = 2/3, respectively.

m 1 m 1 FIG. 2 :
FIG. 2:The dependence of KR (upper plot) and KD (lower plot) on the lightest neutrino mass in case I (case II).The red lines correspond to the normal mass hierarchy, while the blue lines to the inverted hierarchy.The dotted line corresponds to the exact empirical mass relation KR(KD) = 2/3.

1 FIG. 3 :
FIG.3: Ratios of heavy right-handed neutrino masses in case III for normal hierarchy (upper plot) and inverted hierarchy (lower plot).The red and green lines correspond to the allowed ranges of M2/M1 and M3/M1, respectively.

FIG. 4 :
FIG.4: Ratios of heavy right-handed neutrino masses in case IV for normal hierarchy (upper plot) and inverted hierarchy (lower plot).The red and green lines correspond to the allow ranges of M2/M1 and M3/M1, respectively.

FIG. 6 :
FIG. 6: The allowed range of the effective mass mee as a function of the lightest neutrino mass in the normal hierarchy (red line) and inverted hierarchy (green line) at 1σ C.L. with Kν = 2/3 being satisfied.The dashed lines correspond to the parameter range without assuming Kν = 2/3.