Neutrino mass matrices leaving no trace

We point attention to the fact that in SO(10) models with non-canonical (type II) see-saw mechanism and exact b-tau unification the trace of the neutrino mass matrix is very small, in fact practically zero. This has the advantage of being a basis independent feature. Taking a vanishing trace as input, immediate phenomenological consequences for the values of the neutrino masses, the CP phases or the predictions for neutrinoless double beta decay arise. We analyze the impact of the zero trace condition for the normal and inverted mass ordering and in case of CP conservation and violation. Simple candidate mass matrices with (close to) vanishing trace and non-zero U_{e3} are proposed. We also compare the results with the other most simple basis independent property, namely a vanishing determinant.


Introduction
The neutrino mass matrix m ν contains more parameters than can be measured in realistic experiments. This concerns in particular the lightest of the three mass eigenstates and one if not both of the Majorana phases [1]. In addition, if the mixing matrix element |U e3 | is too small, also the Dirac phase will be unobservable. Thus, the presence of certain conditions or simplifications of the neutrino mass matrix is more than welcome. What comes first to one's mind is of course the presence of zeros in the mass matrix [2]. However, zeros in a certain basis must not be zeros in another one, so that basis independent conditions are advantageous to consider. Any matrix possesses two basis independent quantities, namely its trace and its determinant. The most simple situation is present if these quantities are zero. The condition det m ν = 0 [3] leads to one neutrino with vanishing mass and courtesy of this fact one gets also rid of one of the notorious Majorana phases. A vanishing determinant can be motivated on various grounds [4,5]. The second, most simple basis independent requirement is a vanishing trace, i.e., Tr m ν = 0. Its consequences have first been investigated in [6] applying a three neutrino framework that simultaneously explains the anomalies of solar and atmospheric neutrino oscillation experiments as well as the LSND experiment. In [7], the CP conserving traceless m ν has been investigated for the more realistic case of explaining only the atmospheric and solar neutrino deficits. Motivation of traceless mass matrices can be provided by models in which m ν is constructed through a commutator of two matrices, as it happens in models of radiative mass generation [8]. More interestingly, and stressed here, a (close to) traceless m ν can be the consequence of exact b−τ unification at high scale within type II see-saw models [9], which in this framework is also the reason for maximal atmospheric neutrino mixing [10,11]. The type II see-saw mechanism was the original motivation of the traceless m ν condition as investigated in [6].
In this letter we shall investigate the outcomes of the requirement Tr m ν = 0 for the values of the neutrino masses and in case of CP violation also of the CP phases. We investigate the predictions for observables such as the effective mass measured in neutrinoless double beta decay and compare them with the ones stemming from the zero determinant case. Simple forms of m ν that accomplish the traceless condition and allow for simple correlations between the mixing parameters, mass squared differences and the effective mass as measurable in neutrinoless double beta decay are presented.

Data
The light neutrino Majorana mass matrix m ν is observable in terms of Here m diag ν is a diagonal matrix containing the light neutrino mass eigenstates m i . For the normal mass ordering (NH) one has |m 3 | > |m 2 | > |m 1 |, whereas the inverted mass ordering (IH) implies |m 2 | > |m 1 | > |m 3 |. Mixing is described by U, the unitary Pontecorvo-Maki-Nagakawa-Sakata [12] lepton mixing matrix, which can be parametrized as where c ij = cos θ ij and s ij = sin θ ij . The phases are usually distinguished as the "Dirac phase" δ and the "Majorana phases" [13] α and β. The former can be measured in oscillation experiments, whereas the latter show up only in lepton number violating processes. Their influence on the values of the mass matrix elements is known [14,15], however, only the ee element of m ν can realistically be expected to be measured [16,14]. In case of CP conservation, different relative signs of the masses m i are possible, corresponding to the intrinsic CP parities of the neutrinos [17,18]. Four situations are possible, with m i = η i |m i | one can write these cases as (+ + +), (+ − −), (− + −) and (− − +), where the (± ± ±) correspond to the relative signs of the mass states. Special values of the phases correspond to these sign signatures [18]: Observation implies the following values of the oscillation parameters [19]: (14) . . . 10 (19)) × 10 −5 eV 2 , ∆m 2 A ≃ (1.5 . . . 3.9) × 10 −3 eV 2 , where the 90 % C.L. ranges for the respective quantities are given. For ∆m 2 ⊙ two upper and lower limits are given, corresponding to the so-called LMA-I and LMA-II solutions [20]. The best-fit points are located in the LMA-I parameter space and are [19] tan 2 θ 12 = 0.45, ∆m 2 ⊙ = 7.1×10 −5 eV 2 . For the atmospheric sector one finds the best-fit points tan 2 θ 23 = 1 and ∆m 2 A = 2.6×10 −3 eV 2 [19]. At the moment no information at all about the CP phases or relative CP parities exists.
Regarding the total neutrino mass scale, only upper limits exist. Three different observables are at one's disposal, the effective Majorana mass m as measurable in neutrinoless double beta decay, the sum of neutrino masses Σ as testable through cosmology and the mass parameter m νe as testable in direct kinematical experiments. Their definitions and current limits read m ≡ | U 2 ei m i | < ∼ 1 eV [21] , Σ ≡ |m i | < 1.01 eV [22] , Regarding m , a factor of ∼ 3 for the uncertainty in the nuclear matrix element calculations was included. The Heidelberg-Moscow collaboration gives -using the results of one specific calculation for the nuclear matrix elements -a limit of m < 0.35 eV at 90% C.L. [21].

Theory
This letter is supposed to analyze the impact of a traceless m ν . There exists a very simple and phenomenologically highly interesting explanation for this possibility [6]. The neutrino mass matrix is given by the see-saw mechanism [24] in general as where m D is a Dirac mass matrix and M R (M L ) a right-handed (left-handed) Majorana mass matrix. In SO(10) models, choosing Higgs fields in 10 and 126 and the B − L breaking being performed by a 126 Higgs, one can write (see e.g. [25]): where m down ( . Suppose now that the first term in the see-saw formula (6) dominates. The mass matrix reads in this case: Suppose now that m down and m lep are hierarchical, i.e., they contain small off-diagonal entries and the diagonal entries correspond roughly to the down quark and charged lepton masses, respectively. Then, the 23 sector of m ν is diagonalized by This mixing becomes maximal when b − τ unification takes place, i.e., m b = m τ . This simple and appealing argument was first given in [10]. In [11] the idea was generalized to the full 3 flavor case and shown to be fully consistent with existing neutrino data.
Here we wish to emphasize that due to the same fact, b − τ unification, the trace of m ν is proportional to m b − m τ and therefore, to a good precision, the trace vanishes [6]. We can quantify the smallness of the trace as Here, m s ≃ 0.2 GeV and m µ ≃ 0.1 GeV are the masses of the strange quark and muon, respectively. For the typical values of [26] v L < ∼ 0.1 eV and v down 10 > ∼ 1 GeV we can expect the trace to be less than 10 −3 eV. We shall take the fact as the starting point of our purely phenomenological analysis. Note that most of our results should be a specific case of a more detailed, but model-dependent analysis as performed in [11]. They can serve as a simple insight of the physics involved and results obtained.

The CP conserving case
We shall investigate now the consequences of the requirement Tr m ν = 0 on the mass states in the CP conserving case.
The numbers of course coincide with the ones presented in [7]. The characteristic relation |m 3 | ≃ 2 |m 2 | ≃ 2 |m 1 | holds as long as Tr m ν < ∼ 10 −3 eV. The mass spectrum corresponds to a "partially degenerate" scheme.
The different relative signs of the mass states correspond to the (− − +) configuration, for which α = 2β = π. The effective mass m reads for these values and for which is independent on tan 2 θ 12 . Varying θ 13 leads to values of 0.025 eV < ∼ m < ∼ 0.030 eV, thus predicting a very narrow range within the reach of running and future experiments [27].
Direct kinematical measurements will have to measure which is one order of magnitude below the limit of the future KATRIN experiment [28]. The sum of the absolute values of the neutrino masses is Σ ≃ 0.12 eV. As shown in [29], this is the lowest value (at 95 % C.L.) measurable by combining data from the PLANCK satellite experiment and the Sloan Digital Sky Survey. Galaxy surveys one order of magnitude larger could reduce this limit by a factor of two [29] and thus test the prediction.
We turn now to a simple form of the mass matrix that accomplishes the requirement of being traceless. We concentrate on mass matrices with three parameters, sizable U e3 and no zero entries. For hierarchical neutrinos one might expect a quasi-degenerated and dominant 23 block of the mass matrix. Thus, one is lead to propose where ǫ 1,2 ≪ a. Note that with ǫ i = 0 the mixing angles are θ 23 = π/4, θ 13 = 0 and tan θ 12 = 1/ √ 2, which is a widely discussed scheme [30]. We find with the mass matrix (14) that the mass states are and the observables are given by together with maximal atmospheric mixing. For ǫ 1 = ǫ 2 = 0 the solar mixing angle vanishes. Comparing the last two equations with the data from Eq. (4), one finds that a 2 ≃ 10 −3 eV 2 , ǫ 1 − ǫ 2 ≃ 10 −3 eV and ǫ 1 + ǫ 2 ≃ 10 −2 eV in order to reproduce the observations. It is seen that |U e3 | should be sizable; we can express this element in terms of the other observables as which becomes smaller, the larger the solar neutrino mixing angle θ 12 becomes. Inserting the data from Eq. (4) in the right-hand side of the equation, the range of U 2 e3 is found for the LMA-I (LMA-II) case to lie between 0.0007 (0.002) and 0.0022 (0.04) in accordance with its current limit. The best-fit point predicts U 2 e3 ≃ 0.0056. The effective mass is given by where we inserted the best-fit value of ∆m 2 A . The allowed range of m lies between 0.022 eV and 0.036 eV, with a best-fit prediction of 0.029 eV. Both observables should thus be measurable with the next round of experiments. An alternative formulation of the correlation of observables reads which could be used as a further check if both m and U 2 e3 were measured.
The question arises if the results are stable against radiative corrections. As known, the 12 sector is unstable for quasi-degenerate neutrinos with equal relative CP parity [31], which is what happens here. The effect of radiative corrections can be estimated by multiplying the αβ element of m ν with a term (1 + δ α ) (1 + δ β ), where Here m α is the mass of the corresponding charged lepton, M X ≃ 10 16 GeV and c = −(1 + tan 2 β) (3/2) in case of the MSSM (SM). We checked numerically that for the SM there is no significant change of θ 12 and ∆m 2 ⊙ but for the MSSM and tan β > ∼ 20 the results become unstable. Also, the relation between |U e3 | and the other observables remains its validity for the SM and for the MSSM with tan β < ∼ 20.
One can relax the traceless condition a bit by adding a term proportional to ½ × ξ/3 to m ν , where ξ = Tr m ν . The mixing angles are of course unaffected by this term but the masses are changed by m i → m i + ξ/3. Thus, the new mass squared differences read The correlation of the observables U e3 and m also changes, it is now given by For ξ = 0 the previous two equations reproduce (16,17,18). The formula for the correlations of the observables, Eq. (19), holds also in the case of ξ = 0. As long as ξ does not exceed 10 −3 eV, the corrections due to Tr m ν = ξ = 0 increase (reduce) the predictions for U 2 e3 ( m ) by ∼ ξ/ ∆m 2 A ≃ 3 %. 7
The signs of the mass states correspond to the (+ − −) configuration for which α = β = π/2. The effective mass then reads for |m 2 | ≃ |m 1 | ≫ |m 3 | m ≃ |m 2 | cos 2θ 12 cos 2 θ 13 ≃ |m 2 | 1 − tan 2 θ 12 1 + tan 2 θ 12 , which lies between 0.005 eV and 0.028 eV, thus predicting a range with the upper (lower) limit within (outside) the reach of running and future experiments [27]. The lower limit is however reachable by the 10t version of the GENIUS [32] project. The best-fit prediction is m ≃ 0.020 eV. In contrast to the normal mass ordering, m has a crucial dependence on tan 2 θ 12 and thus a rather large allowed range. The mass measured in direct kinematical experiments is m νe ≃ |m 2 | ≃ 0.05 eV, which is larger than the corresponding quantity in the normal hierarchy but still almost one order of magnitude below the limit of the future KATRIN experiment. The sum of the absolute values of the neutrino masses is Σ ≃ 0.10 eV, lower than the corresponding quantity in the normal hierarchy and thus still requiring larger galaxy surveys, as shown in [29].
We present again a simple 3 parameter mass matrix with the traceless feature, no zero entries and non-vanishing U e3 . One is naturally lead to propose where b > a > η. We find with the mass matrix (14) that the mass states for b 2 ≫ η 2 are The observables are found to be 8 Again, the observed values of the quantities are easy to reproduce with, in this case, e.g., b > a > η ∼ 0.01 eV. There is again a simple correlation of the observables, namely Putting again the data from Eq. (4) in the right-hand side of this equation leads to U 2 e3 > ∼ 0.013 with a best-fit prediction of U 2 e3 ≃ 0.036. For large values of ∆m 2 ⊙ , i.e., in the less favored LMA-II solution, which corresponds to ∆m 2 ⊙ > ∼ 10 −4 eV 2 , the value of U e3 is above its current experimental limit. Comparing the expressions for U e3 in the normal (Eq. (17)) and inverted (Eq. (28)) ordering leads to the observation that for the first case the value is smaller by a factor of ≃ 1/2 (1 − tan 2 θ 12 ) 2 /((1 + tan 2 θ 12 ) tan θ 12 ) < ∼ 0.35. The effective mass is given by Comparing this result with our prediction for m in the normal mass ordering, Eq. (16), one finds that the inverted mass hierarchy predicts an effective mass smaller than a factor √ 3 tan 2θ 12 > ∼ 4. This is larger than the typical uncertainty of the nuclear matrix elements that usually tends to spoil extraction of information from neutrinoless double beta decay. If both, m and U e3 are measured, one can further check the mass matrix by the relation We checked numerically that the results are stable under radiative corrections in the SM and in the MSSM for tan β < ∼ 50.
For ξ = 0 the results for exact zero trace given above are re-obtained.
Interestingly, the same mass matrix, Eq. (25), has been found in [5]. In this work a local horizontal SU(2) symmetry has been applied to the charged leptons. A consequence was a vanishing determinant of m ν and an inverted hierarchy for the neutrino masses (i.e. m 3 = 0) with opposite signs for m 2 and m 1 . In this case, both the trace and the determinant of m ν are vanishing, which explains that our results are identical to the ones in [5].
To put this Section in a nutshell, the requirement of a vanishing trace of m ν leads in the CP conserving case to values of m , larger in the NH than in the IH. Due to the dependence on tan θ 12 , m in case of IH has a broad range. Simple mass matrices were proposed which reproduce the values found by the traceless condition and in addition predict larger U e3 in the IH case. Relaxing the traceless condition does not significantly change the predicted values as long as the trace stays below the expected 10 −3 eV.

The CP violating case
Now we shall investigate the more realistic case of CP violation and the consequences of the traceless m ν condition. Within the parametrization (2) one finds -using Eq. (1)for the trace of m ν that Tr m ν ≃ m 1 + m 2 e 2iα + m 3 e 2i(β+δ) .
The minimal values of m 1 or m 3 that fulfill the condition (34) are the ones that correspond to the CP conserving case discussed in the previous Section. As a check, one can convince oneself that for δ = 0 and m 1 = m 2 = m 3 /2 the solution of the two equations in (34) is given by α = 2β = π while for δ = 0 and m 1 = m 2 ≫ m 3 ≃ 0 one finds that α = π/2, which is in accordance with the previous Section. This means that in case of the normal hierarchy and the LMA I (LMA II) solution a lower limit on the neutrino mass of 0.019 (0.021) eV can be set, which is obtained by inserting the lowest allowed ∆m 2 A and the largest ∆m 2 ⊙ . In case of inverted hierarchy, one finds that |m 3 | ≥ 0.0013 (0.0024) eV for the LMA I (LMA II) solution. Due to the zero trace condition one can write Interestingly, this implies that in the expressions for Σ and m νe the phases appear. In particular, m 2 νe = 1 1 + tan 2 θ 12 m 2 3 + m 2 2 (1 + tan 2 θ 12 ) + 2 m 2 m 3 cos φ + m 2 3 sin 2 θ 13 .
Applying the condition Tr m ν = 0 to Eq. (35) and inserting it in the expression for m one finds where we neglected sin 2 θ 13 . Courtesy of the zero trace condition, m depends effectively only on one phase. The CP conserving cases in the previous Section should come as special cases of the last formula. Indeed, for δ = 0, m 2 = m 3 /2 and α = 2β = π one recovers Eq. (12) and for δ = 0 and m 3 = 0 one re-obtains Eq. (24). For quasi-degenerate neutrinos m 0 ≡ m 2 ≃ m 3 the above formula simplifies. Then, since cos φ ≃ −1/2: which can be used to set an upper limit on m 0 . For m < ∼ 1 eV we have m 0 < ∼ 1.96 eV with a best-fit limit of 1.67 eV. Therefore, the zero trace condition implies a limit stronger than the one stemming from direct kinematical experiments. Using the less conservative limit given by the Heidelberg-Moscow collaboration, the above limits are reduced by a factor of roughly 2.9 and the limits come nearer to the ones from cosmological observations. To be precise, for m < 0.35 eV one finds m 0 < ∼ 0.69 eV and, for the best-fit value, m 0 < ∼ 0.58 eV. The values are testable by the KATRIN experiment. Thus, together with the lower limit (about 0.02 eV for NH and 0.002 eV for IH) from the beginning of this Section, a neutrino mass window is defined.
One can compare the predictions for m in case of zero trace with the ones in case of zero determinant [3]. This corresponds for NH (IH) to zero m 1 (m 3 ), which results in particular simple forms of m , see [3] for details. Regarding the inverted hierarchy, we already commented that in case of an opposite relative sign of the two quasi-degenerate neutrinos and a very small m 3 both the trace and the determinant vanish and the situation is identical. We use the data from Eq. (4) for our predictions. For the normal mass ordering strong cancellations are possible [18,33], and m is in general predicted to be below 0.01 eV. In case of the inverted mass ordering, m lies between 0.004 and 0.034 eV, independent on sin 2 θ 13 . Unlike the zero trace case, the zero determinant conditions allows no statements about the phases, at least not before the limit on m is significantly improved.
We also performed a numerical analysis of the zero trace condition. For this exercise the mass squared differences and solar neutrino mixing angle were fixed to their best-fit points and the smallest neutrino mass and the phases α as well as β − δ were varied within their allowed range. The results in the form of scatter plots for the normal hierarchy is shown in Fig. 1 and for the inverted scheme in Fig. 2. One recognizes for example in Figs. (1,2)c the correlation of Σ with α − β − δ as implied by Eq. (35). For the inverted hierarchy, the spread of the phases is rather different from the case of normal hierarchy. This can be understood from the fact that for small m 3 the dependence on β − δ practically vanishes.

Summary and Conclusions
The condition of a zero trace of the neutrino mass matrix m ν was reanalyzed in case of CP conservation and violation for both possible mass orderings. The motivation for this purely phenomenological analysis was given by exact b − τ unification in connection to the non-canonical type II see-saw mechanism in SO(10) models. This situation has recently gathered renewed attention because of its ability to produce large atmospheric neutrino mixing in a simple way. In case of CP conservation, the values of the neutrino masses and their relative CP parities are fixed and allow to give simple expressions for the effective mass as measurable in neutrinoless double beta decay. The masses are given by m 1 ≃ m 2 ≃ −m 3 /2 ≃ 0.03 eV for the normal mass ordering and ∆m 2 A ≃ m 2 ≃ −m 1 ≫ −m 3 for the inverted mass ordering. In case of the normal hierarchy, m does not depend on the solar neutrino mixing angle and is predicted to be around 0.03 eV. In case of inverted hierarchy, m depends rather strongly on the solar neutrino mixing angle and its range is between 0.005 eV and 0.03 eV; the best-fit prediction is 0.02 eV. The presence of CP violation and therefore non-trivial values of the Majorana phases allows for larger values of the masses. In case of quasi-degenerate neutrinos a peculiar relation between the phases exists: α − β − δ = ±π/3 ± nπ. The minimal values of the masses correspond to the CP conserving case and are in case of the normal (inverted) hierarchy roughly 0.02 (0.002) eV. The upper limit comes from non-observation of neutrinoless double beta decay and is for m < 0.35 eV roughly 0.7 eV. Correlations of various parameters are possible, some of which are shown in Figs. 1 and 2.