Texture Zeros and Majorana Phases of the Neutrino Mass Matrix

We present the generic formulas to calculate the ratios of neutrino masses and the Majorana phases of CP violation from the neutrino mass matrix with two independent vanishing entries in the flavor basis where the charged lepton mass matrix is diagonal. An order-of-magnitude illustration is given for seven experimentally acceptable textures of the neutrino mass matrix, and some analytical approximations are made for their phenomenological consequences at low energy scales.

1 The atmospheric and solar neutrino oscillations observed in the Super-Kamiokande experiment [1] have provided robust evidence that neutrinos are massive and lepton flavors are mixed. A full description of the mass spectrum and flavor mixing in the framework of three lepton families requires twelve real parameters: three charged lepton masses (m e , m µ , m τ ), three neutrino masses (m 1 , m 2 , m 3 ), three flavor mixing angles (θ x , θ y , θ z ), one Dirac-type CP-violating phase (δ) and two Majorana-type CP-violating phases (ρ and σ). So far only the masses of charged leptons have been accurately measured [2]. Although we have achieved some preliminary knowledge on two neutrino mass-squared differences and three flavor mixing angles from current neutrino oscillation experiments, much more effort is needed to determine these parameters precisely. The more challenging task is to pin down the absolute neutrino mass scale and the CP-violating phases. Towards reaching these goals, a number of new neutrino experiments have been proposed [3].
After sufficient information on neutrino masses and lepton flavor mixing parameters is experimentally accumulated, a determination of the textures of lepton mass matrices should become possible. On the other hand, the textures of charged lepton and neutrino mass matrices may finally be derived from a fundamental theory of lepton mass generation, which is unfortunately unknown for the time being. It is therefore important in phenomenology to investigate how the textures of lepton mass matrices can link up with the observables of lepton flavor mixing.
Recently Frampton, Glashow and Marfatia [4] have examined the possibility that a restricted class of lepton mass matrices may suffice to describe current experimental data. They find seven acceptable textures of the neutrino mass matrix with two independent vanishing entries in the flavor basis where the charged lepton mass matrix is diagonal.
In this paper we carry out a further study of two-zero textures of the neutrino mass matrix. Our work is different from Ref. [4] in several aspects: (a) we write out the generic constraint equations for the neutrino mass matrix with two independent vanishing entries, from which the analytically exact expressions of neutrino mass ratios can be derived; (b) the formulas to calculate the Majorana-type CP-violating phases are presented; (c) the relative magnitudes of neutrino masses, the Majorana phases, the ratio of two neutrino mass-squared differences, and the effective mass term of the neutrinoless double beta decay are estimated by taking typical inputs of the flavor mixing angles and the Dirac-type CP-violating phase; and (d) an order-of-magnitude illustration is given for seven two-zero textures of the neutrino mass matrix.
2 In the flavor basis where the charged lepton mass matrix is diagonal, the neutrino mass matrix can be written as where m i (for i = 1, 2, 3) denote the real and positive neutrino masses, and V is the lepton flavor mixing matrix linking the neutrino mass eigenstates (ν 1 , ν 2 , ν 3 ) to the neutrino flavor eigenstates (ν e , ν µ , ν τ ) in the chosen basis. A full description of V needs six real parameters: three mixing angles and three CP-violating phases. Note that V can always be expressed as a product of the Dirac-type flavor mixing matrix U (consisting of three mixing angles and one CP-violating phase) and a diagonal phase matrix P (consisting of two nontrivial Majorana phases): V = UP . Then we may rewrite M in Eq. (1) as where two Majorana-type CP-violating phases are included into the complex neutrino mass eigenvalues λ i , and the relation |λ i | = m i holds. Without loss of generality, we take In the following we shall show that both the neutrino mass ratios (m 1 /m 3 and m 2 /m 3 ) and the Majorana phases (ρ and σ) can be determined, if two independent entries of M vanish.
As M is symmetric, it totally has six independent complex entries. If two of them vanish, i.e., M ab = M αβ = 0, we obtain the following constraint relations: where each of the four subscripts runs over e, µ and τ , but (α, β) = (a, b). Solving Eq. (4), we find and One can observe that the left-hand sides of Eqs. (5) and (6) are associated with the Majoranatype CP-violating phases, while the right-hand sides of Eqs. (5) and (6) are associated with the Dirac-type CP-violating phase hidden in the elements of U. Therefore two Majorana phases must depend upon the Dirac-type CP-violating phase. This dependence results simply from the texture zeros of M that we have taken.
Comparing Eq. (5) or Eq. (6) with Eq. (3), we arrive at the expressions of two neutrino mass ratios as follows: Furthermore, the expressions of two Majorana phases are found to be With the inputs of three flavor mixing angles and the Dirac-type CP-violating phase, we are able to predict the relative magnitudes of three neutrino masses and the values of two Majorana phases. This predictability allows us to examine whether the chosen texture of M with two independent vanishing entries is empirically acceptable or not.
Indeed the prediction for m 1 /m 2 and m 2 /m 3 in a given pattern of M is required to be compatible with the hierarchy of solar and atmospheric neutrino mass-squared differences: The magnitude of R ν depends upon the explicit solution to the solar neutrino problem.
For the large-angle Mikheyev-Smirnov-Wolfenstein (MSW) oscillation of solar neutrinos [5], which is most favored by the present Super-Kamiokande [1] and SNO [6] data, we have , the atmospheric neutrino oscillation is approximately decoupled from the solar neutrino oscillation.
With the help of Eqs. (7) and (8), one can calculate the effective mass term of the neutrinoless double beta decay, whose magnitude amounts to |M ee |. The explicit expression of |M ee | reads as follows: The Heidelberg-Moscow Collaboration has reported |M ee | < 0.34 eV at the 90% confidence level [8]. Useful information on the absolute mass scale of neutrinos could in principle be extracted from a more accurate measurement of |M ee | in the future.
3 As already pointed out in Ref. [4], there are totally fifteen logical possibilities for the texture of M with two independent vanishing entries, but only seven of them are in accord with current experimental data and empirical hypotheses. The seven acceptable patterns of M are listed in Table 1, where all the non-vanishing entries are symbolized by ×'s. To work out the explicit expressions of λ 1 /λ 3 and λ 2 /λ 3 in each case, we adopt the following parametrization for the Dirac-type flavor mixing matrix U: where s x ≡ sin θ x , c x ≡ cos θ x , and so on. The advantage of this phase choice is that the Dirac-type CP-violating phase δ does not appear in the effective mass term of the neutrinoless double beta decay [9]. In other words, the latter depends only upon the Majorana phases ρ and σ in our phase convention. Without loss of generality, three mixing angles (θ x , θ y , θ z ) can all be arranged to lie in the first quadrant. Three CP-violating phases (δ, ρ, σ) may take arbitrary values from −π to +π (or from 0 to 2π). Now let us calculate λ 1 /λ 3 and λ 2 /λ 3 for each pattern of M with the help of Eqs. (5), (6) and (11). The instructive results for m 1 /m 3 , m 2 /m 3 , ρ, σ, R ν and |M ee | may then be obtained.
Pattern A 2 : M ee = M eτ = 0 (i.e., a = b = e; α = e and β = τ ). We obtain In the lowest-order approximation, we explicitly obtain Using the same inputs as above, we get m 1 /m 3 ≈ 0.06, m 2 /m 3 ≈ 0.18, ρ ≈ 135 • (or −45 • ), σ ≈ 45 • , and R ν ≈ 0.03. We see that the phenomenological consequences of patterns A 1 and A 2 are nearly the same [4]. However, pattern A 2 seems to be more interesting for model building [10], in particular when the spirit of lepton-quark similarity is taken into account.
Pattern B 1 : M µµ = M eτ = 0 (i.e., a = b = µ; α = e and β = τ ). We obtain The smallness of s 2 z allows us to make a similar analytical approximation as before. To lowest order, we find where t 2y ≡ tan 2θ y and c δ ≡ cos δ. Note that in the next-to-leading order approximation, where s δ ≡ sin δ and s 2x ≡ sin 2θ x . Typically taking θ x = 30 • , θ y = 40 • , θ z = 5 • and δ = 89 • , we arrive at m 1 /m 3 ≈ m 2 /m 3 ≈ 0.7 with a difference of about 0.007, σ ≈ ρ ≈ 179 • (or −1 • ) with a difference of about 3 • , R ν ≈ 0.02, and |M ee |/m 3 ≈ 0.7. One can see that |δ| ≈ 90 • is required in this texture of M for plausible inputs of θ y and θ z , such that R ν gets suppressed sufficiently. If pattern B 1 is realistically correct, large CP-violating effects may be observable in long-baseline neutrino oscillations. It is also worth mentioning that a typical upper bound on three nearly degenerate neutrino masses can be extracted from the Heidelberg-Moscow experiment [8]: m 1 ≈ m 2 ≈ 0.7m 3 ≈ |M ee | < 0.34 eV. This bound is certainly compatible with the present direct-mass-search experiments [2], in particular for the electron neutrino.
Pattern B 3 : M µµ = M eµ = 0 (i.e., a = b = µ; α = e and β = µ). We obtain The approximate expressions for the neutrino mass ratios, the Majorana phases and the observables R ν and |M ee | turn out to be In addition, in the next-to-leading order approximation. Taking the same inputs as in pattern B 1 , we find m 2 /m 3 ≈ m 1 /m 3 ≈ 0.7 with a difference of about 0.005, ρ ≈ σ ≈ 179 • (or −1 • ) with a difference of about 2 • , R ν ≈ 0.014, and |M ee |/m 3 ≈ 0.7. One can see that the phenomenological consequences of pattern B 3 are essentially the same as those of pattern B 1 . This point has been observed in Ref. [4].
Pattern B 4 : M τ τ = M eτ = 0 (i.e., a = b = τ ; α = e and β = τ ). We obtain To lowest order, we get the following approximate results: together with Using the same inputs as in pattern B 1 , we obtain m 1 /m 3 ≈ m 2 /m 3 ≈ 1.4 with a difference of about 0.01, ρ ≈ σ ≈ 179 • (or −1 • ) with a difference of about 2 • , R ν ≈ 0.03, and |M ee |/m 3 ≈ 1.4. One can see that the phenomenological consequences of patterns B 1 , B 2 , B 3 and B 4 are nearly the same. Therefore it is very difficult, even impossible, to distinguish one of them from the others in practical experiments. Nevertheless, one of the four textures might be more favored than the others in model building, when underlying flavor symmetries responsible for those texture zeros are taken into account.
Pattern C: M µµ = M τ τ = 0 (i.e., a = b = µ and α = β = τ ). We obtain Assuming s 2 z ≪ 1 and t x ∼ t y ∼ O(1), we may make an analytical approximation for the exact result in Eq. (28). To lowest order, we get To give an order-of-magnitude illustration of the neutrino mass matrix, we calculate the elements of M for each pattern by using the formula and the typical inputs taken before. The rough results are listed in Table 1. We see that there is no clear hierarchy among the non-vanishing elements of M, unlike the familiar case of quark mass matrices [11].
Of course, the specific textures of lepton mass matrices cannot be preserved to all orders or at any energy scales in the unspecified interactions which generate lepton masses [4]. Nevertheless, those phenomenologically favored textures at low energy scales may shed light on the underlying flavor symmetries responsible for the generation of lepton masses at high energy scales. It is expected that more precise data of neutrino oscillations in the future could help select the most favorable pattern of lepton mass matrices.
The author would like to thank P.H. Frampton for useful discussions during WIN2002 in Christchurch. He is also grateful to D. Marfatia for helpful communications via e-mail. This work was supported in part by National Natural Science Foundation of China. Table 1: Seven patterns of the neutrino mass matrix M with two independent vanishing entries, which are in accord with current experimental data and empirical hypotheses. An order-of-magnitude illustration of M is given by using typical inputs of θ x , θ y , θ z and δ, as explained in the text.

Pattern
Texture of M Order of Magnitude