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Absolute neutrino mass scale from flavor symmetries
Available online at www.sciencedirect.com
Nuclear Physics B (Proc. Suppl.) 237–238 (2013) 40–42 www.elsevier.com/locate/npbps
Absolute neutrino mass scale from flavor symmetries S.Morisi AHEP Group, Institut de F´ısica Corpuscular – C.S.I.C./Universitat de Val` encia Edificio Institutos de Paterna, Apt 22085, E–46071 Valencia, Spain and Institut f¨ ur Theoretische Physik und Astrophysik, Universit¨ at W¨ urzburg, 97074 W¨ urzburg, Germany
Abstract The 2012 has been an important year for neutrino physics, current data are reviewed and compared with 2011 global fit. Models based on discrete flavor symmetries predict neutrino mass mass sum-rule (MSR), which is useful to reduce the number of independent model parameters. We found that only four classes of MSR are possible. Such neutrino MSRs constrain the absolute neutrino mass scale. We study the implications of these mass relations for the lightest neutrino mass and for the lower bound of the effective mass mee of the neutrinoless double beta decay. The 2012 represents a fantastic year for neutrino physics since the reactor angle has been measured to be different from zero for about 10 σ. We compare in Fig. 1 the 2011 and 2012 global analysis provided but three different groups. We can see how data have impressively changed in only one year. The measure of the reactor angle is not the only important 2012 discovery in neutrino physics. Indeed data suggest a large atmospheric angle, but not maximal as it was in 2011. As we can see in Fig. 1 the atmospheric mixing angle can be either in the first and second octant for two of the global fits, while only in the first octant for one the global fit. But in all the cases a deviation from the maximality is suggested. Latest 10 years of neutrino data biased us to tri-bimaximal mixing (TBM) where the solar angle is trimaximal, the atmospheric angle is bimaximal and the reactor angle is zero. Nowaday TBM is ruled out. Deviation of the atmospheric and reactor angles yield to new possible ansatz like the Bi-Large [1] or the Bi-trimaximal [2]. However in 0920-5632/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nuclphysbps.2013.04.053
principle we can accommodate data by taking deviation from TBM. Therefore TBM as fist order approximation is not ruled out yet. Moreover at 3 σ the atmospheric angle can be maximal. But new data have changed completely our theoretical approach to neutrino phenomenology. Paradoxically a better measure of neutrino mixing parameters does not correspond to an improvement in our theoretical understanding. Indeed much more options like for instance Bi-Large, Bi-trimaximal, bi-maximal (BM) [3] or even anarchy [4] are possible. However, when it is assumed that the neutrino mass matrix has some particular structure, like the case of TBM, the three mixing angles are predicted. Then it can occur that the three neutrino mass eigenvalues are given in terms of only two free parameters giving a prediction for the absolute neutrino mass scale. Here we give a classification of the neutrino mass relations [5, 6] assuming that neutrino mixing is TBM. We assume that deviations arising from the charged sector can yield to the observed value of the reactor angle. The class of mass
S. Morisi / Nuclear Physics B (Proc. Suppl.) 237–238 (2013) 40–42
41
Figure 1: (Upper plots): In each plot the three bands refer to three different global 2011 neutrino data analysis, namely [7] (top), [8] (medium) and [9] (bottom) for normal mass hierarchy. The light gray bands are for 3σ values, the dark gray bands are for 1σ values and the black dots are the best fit values. The vertical dashed lines are for the tri-bimaximal values. (Bottom plots): In each plot the three bands refer to three different global 2012 neutrino data analysis, namely [10] (top), [11] (medium) and [12] (bottom) for normal mass hierarchy.
matrices with two free parameters and diagonalized by TBM matrix implies the mass relations A) B) C) D)
χ mν2 + ξ mν3 = mν1 , ξ 1 χ + ν = ν, mν2 m3 m1 χ mν2 + ξ mν3 = mν1 , ξ 1 χ ν + ν = ν. m2 m3 m1
where χ and ξ are free parameters that are fixed once a specific model has been selected. The relation of type (A) is valid when neutrino mass matrix M ν arises from dimension five operator or type-II seesaw, type (B) for type-I seesaw, type (C) for type-I seesaw and type (D) for inverse seesaw (for details see [5]). As explained above, MSR like (A),(B),(C) and (D) can be obtained in flavour models where the neutrino mass matrix only depends on two independent parameters and the resulting mixing angles are fixed, like for TBM or BM mixings. Here we consider the case where the rotation in the neutrino sector is of TBM form even though, after the T2K,
MINOS and DayaBay results a vanishing reactor angle is no longer a good approximation. Since corrections from higher dimensional operators as well as from the charged lepton sector can bring θ13 to be different from zero, we retain the TBM approximation as a useful starting point to get our MSR relations but, when evaluating a lower bound on the effective neutrino mass mee appearing in neutrinoless double-β decay, we include the effects of nonvanishing θ13 . Such a lower bound can be obtained from the following procedure. We first consider that the neutrino masses are complex with Majorana phases encoded in mν2 and mν3 , that is mν1 = m01 , mν2 = m02 eiα and mν3 = m03 eiβ . The MSR can then be splited into two independent equations for the real and imaginary part of each term. This allows to extract the two Majorana phases α and β as a function of the absolute values of the three masses and the fixed parameters χ and ξ. Two masses can be further expressed in terms of the measured square mass differences Δm2sol and Δm2atm and the lightest neutrino mass, m01 in case of normal hierarchy (NH) or m03 in case of inverted hierarchy (IH). The relations obtained can then be inserted into
42
S. Morisi / Nuclear Physics B (Proc. Suppl.) 237–238 (2013) 40–42
the general expression of mee . The lower bound for the lightest neutrino mass can be obtained from our MSR, using the triangle inequality in the complex plane as suggested by Rodejohann and Barry in [6]. We first have to select the biggest side of the triangle; calling them x1 , x2 and x3 , then the inequality |xi | ≤ |xj | + |xk | must be fulfilled, where |xi | ≡ Max(|x1 |, |x2 |, |x3 |) and i = j = k 1 . In the case (A) and assuming NH for the neutrino mass spectrum, we always have (χm02 , ξm03 ) > m01 and the largest side can be either χm02 or ξm03 ; so we have to study separately the two different cases. After rewriting two masses in terms of the two square mass differences, we can obtain a lower limit for the lightest neutrino mass from the triangle inequality |xi | ≤ |xj | + |xk |. For the other cases we follow the same procedure. The lower bound on the lightest neutrino mass obtained in this way is then used to estimate the lower bound for mee . The output of the procedure discussed above is shown in Table 1 and 2 where we give the lower limits of mee corresponding to different integer choices of (χ, ξ) between 1 and 3 and for any of the four MSR, for both normal and inverted hierarchies. For the cases already studied in literature the corresponding reference are given in Ref. [5]. The entries denoted with the symbol represent situations that do not satisfy the inequality for any value of mlight . χ, ξ 1,1 1,2 1,3 2,1 2,2 2,3 3,1 3,2 3,3
A–NH 0.010 0.006 0.019 0.004 0.011 0.023
A–IH 0.044 0.046 0.011 0.026 0.046 0.061
B–NH 0.008 0.008 0.030 0.006 0.023 0.007 0.004 0.004 0.029
B – IH 0.036 0.027 0.005 0.007 0.008 0.008 0.008 0.021 0.031
Table 1: Minimal values for mee in eV for the cases (A) and (B) for normal and inverse hierarchies. The means that the triangular inequality can not be valid.
We observe that most of the values can be tested in future neutrinoless double beta decay experiments. 1 Notice that there are three inequality of the type |x | ≤ i |xj | + |xk | obtained permuting the three index i, j and k, but only one gives constraints on the lightest neutrino mass.
χ, ξ 1,1 1,2 1,3 2,1 2,2 2,3 3,1 3,2 3,3
C– NH 0.006 0.000 0.017 0.000 0.011
C –IH 0.029 0.014 0.014 0.031 0.019
D –NH .005 .004 .018 .003 .005 .018
D– IH .008 .026 .025 .007 .015 .026 .007 .016
Table 2: Minimal values for mee in eV for the cases (C) and (D).
Acknowledgments This work was supported by the Spanish MICINN under grants FPA2008-00319/FPA, FPA2011-22975 by Prometeo/2009/091 (Generalitat Valenciana), DFG grant WI 2639/4-1 and by a Juan de la Cierva contract. References [1] S. M. Boucenna, S. Morisi, M. Tortola and J. W. F. Valle, Phys. Rev. D 86, 051301 (2012). [2] S. F. King, C. Luhn and A. J. Stuart, Nucl. Phys. B 867, 203 (2013). [3] G. Altarelli, F. Feruglio and L. Merlo, JHEP 0905, 020 (2009). [4] L. J. Hall, H. Murayama and N. Weiner, Phys. Rev. Lett. 84, 2572 (2000). [5] L. Dorame, D. Meloni, S. Morisi, E. Peinado and J. W. F. Valle, Nucl. Phys. B 861, 259 (2012). [6] J. Barry and W. Rodejohann, Nucl. Phys. B 842, 33 (2011). [7] M. C. Gonzalez-Garcia, M. Maltoni and J. Salvado, JHEP 1004, 056 (2010). [8] G. L. Fogli, E. Lisi, A. Marrone, A. Palazzo and A. M. Rotunno, Phys. Rev. D 84, 053007 (2011). [9] T. Schwetz, M. Tortola and J. W. F. Valle, New J. Phys. 13, 109401 (2011). [10] M. C. Gonzalez-Garcia, M. Maltoni, J. Salvado and T. Schwetz, JHEP 1212, 123 (2012). [11] G. L. Fogli, E. Lisi, A. Marrone, D. Montanino, A. Palazzo and A. M. Rotunno, Phys. Rev. D 86, 013012 (2012). [12] D. V. Forero, M. Tortola and J. W. F. Valle, Phys. Rev. D 86, 073012 (2012).
Nuclear Physics B (Proc. Suppl.) 237–238 (2013) 40–42 www.elsevier.com/locate/npbps
Absolute neutrino mass scale from flavor symmetries S.Morisi AHEP Group, Institut de F´ısica Corpuscular – C.S.I.C./Universitat de Val` encia Edificio Institutos de Paterna, Apt 22085, E–46071 Valencia, Spain and Institut f¨ ur Theoretische Physik und Astrophysik, Universit¨ at W¨ urzburg, 97074 W¨ urzburg, Germany
Abstract The 2012 has been an important year for neutrino physics, current data are reviewed and compared with 2011 global fit. Models based on discrete flavor symmetries predict neutrino mass mass sum-rule (MSR), which is useful to reduce the number of independent model parameters. We found that only four classes of MSR are possible. Such neutrino MSRs constrain the absolute neutrino mass scale. We study the implications of these mass relations for the lightest neutrino mass and for the lower bound of the effective mass mee of the neutrinoless double beta decay. The 2012 represents a fantastic year for neutrino physics since the reactor angle has been measured to be different from zero for about 10 σ. We compare in Fig. 1 the 2011 and 2012 global analysis provided but three different groups. We can see how data have impressively changed in only one year. The measure of the reactor angle is not the only important 2012 discovery in neutrino physics. Indeed data suggest a large atmospheric angle, but not maximal as it was in 2011. As we can see in Fig. 1 the atmospheric mixing angle can be either in the first and second octant for two of the global fits, while only in the first octant for one the global fit. But in all the cases a deviation from the maximality is suggested. Latest 10 years of neutrino data biased us to tri-bimaximal mixing (TBM) where the solar angle is trimaximal, the atmospheric angle is bimaximal and the reactor angle is zero. Nowaday TBM is ruled out. Deviation of the atmospheric and reactor angles yield to new possible ansatz like the Bi-Large [1] or the Bi-trimaximal [2]. However in 0920-5632/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nuclphysbps.2013.04.053
principle we can accommodate data by taking deviation from TBM. Therefore TBM as fist order approximation is not ruled out yet. Moreover at 3 σ the atmospheric angle can be maximal. But new data have changed completely our theoretical approach to neutrino phenomenology. Paradoxically a better measure of neutrino mixing parameters does not correspond to an improvement in our theoretical understanding. Indeed much more options like for instance Bi-Large, Bi-trimaximal, bi-maximal (BM) [3] or even anarchy [4] are possible. However, when it is assumed that the neutrino mass matrix has some particular structure, like the case of TBM, the three mixing angles are predicted. Then it can occur that the three neutrino mass eigenvalues are given in terms of only two free parameters giving a prediction for the absolute neutrino mass scale. Here we give a classification of the neutrino mass relations [5, 6] assuming that neutrino mixing is TBM. We assume that deviations arising from the charged sector can yield to the observed value of the reactor angle. The class of mass
S. Morisi / Nuclear Physics B (Proc. Suppl.) 237–238 (2013) 40–42
41
Figure 1: (Upper plots): In each plot the three bands refer to three different global 2011 neutrino data analysis, namely [7] (top), [8] (medium) and [9] (bottom) for normal mass hierarchy. The light gray bands are for 3σ values, the dark gray bands are for 1σ values and the black dots are the best fit values. The vertical dashed lines are for the tri-bimaximal values. (Bottom plots): In each plot the three bands refer to three different global 2012 neutrino data analysis, namely [10] (top), [11] (medium) and [12] (bottom) for normal mass hierarchy.
matrices with two free parameters and diagonalized by TBM matrix implies the mass relations A) B) C) D)
χ mν2 + ξ mν3 = mν1 , ξ 1 χ + ν = ν, mν2 m3 m1 χ mν2 + ξ mν3 = mν1 , ξ 1 χ ν + ν = ν. m2 m3 m1
where χ and ξ are free parameters that are fixed once a specific model has been selected. The relation of type (A) is valid when neutrino mass matrix M ν arises from dimension five operator or type-II seesaw, type (B) for type-I seesaw, type (C) for type-I seesaw and type (D) for inverse seesaw (for details see [5]). As explained above, MSR like (A),(B),(C) and (D) can be obtained in flavour models where the neutrino mass matrix only depends on two independent parameters and the resulting mixing angles are fixed, like for TBM or BM mixings. Here we consider the case where the rotation in the neutrino sector is of TBM form even though, after the T2K,
MINOS and DayaBay results a vanishing reactor angle is no longer a good approximation. Since corrections from higher dimensional operators as well as from the charged lepton sector can bring θ13 to be different from zero, we retain the TBM approximation as a useful starting point to get our MSR relations but, when evaluating a lower bound on the effective neutrino mass mee appearing in neutrinoless double-β decay, we include the effects of nonvanishing θ13 . Such a lower bound can be obtained from the following procedure. We first consider that the neutrino masses are complex with Majorana phases encoded in mν2 and mν3 , that is mν1 = m01 , mν2 = m02 eiα and mν3 = m03 eiβ . The MSR can then be splited into two independent equations for the real and imaginary part of each term. This allows to extract the two Majorana phases α and β as a function of the absolute values of the three masses and the fixed parameters χ and ξ. Two masses can be further expressed in terms of the measured square mass differences Δm2sol and Δm2atm and the lightest neutrino mass, m01 in case of normal hierarchy (NH) or m03 in case of inverted hierarchy (IH). The relations obtained can then be inserted into
42
S. Morisi / Nuclear Physics B (Proc. Suppl.) 237–238 (2013) 40–42
the general expression of mee . The lower bound for the lightest neutrino mass can be obtained from our MSR, using the triangle inequality in the complex plane as suggested by Rodejohann and Barry in [6]. We first have to select the biggest side of the triangle; calling them x1 , x2 and x3 , then the inequality |xi | ≤ |xj | + |xk | must be fulfilled, where |xi | ≡ Max(|x1 |, |x2 |, |x3 |) and i = j = k 1 . In the case (A) and assuming NH for the neutrino mass spectrum, we always have (χm02 , ξm03 ) > m01 and the largest side can be either χm02 or ξm03 ; so we have to study separately the two different cases. After rewriting two masses in terms of the two square mass differences, we can obtain a lower limit for the lightest neutrino mass from the triangle inequality |xi | ≤ |xj | + |xk |. For the other cases we follow the same procedure. The lower bound on the lightest neutrino mass obtained in this way is then used to estimate the lower bound for mee . The output of the procedure discussed above is shown in Table 1 and 2 where we give the lower limits of mee corresponding to different integer choices of (χ, ξ) between 1 and 3 and for any of the four MSR, for both normal and inverted hierarchies. For the cases already studied in literature the corresponding reference are given in Ref. [5]. The entries denoted with the symbol represent situations that do not satisfy the inequality for any value of mlight . χ, ξ 1,1 1,2 1,3 2,1 2,2 2,3 3,1 3,2 3,3
A–NH 0.010 0.006 0.019 0.004 0.011 0.023
A–IH 0.044 0.046 0.011 0.026 0.046 0.061
B–NH 0.008 0.008 0.030 0.006 0.023 0.007 0.004 0.004 0.029
B – IH 0.036 0.027 0.005 0.007 0.008 0.008 0.008 0.021 0.031
Table 1: Minimal values for mee in eV for the cases (A) and (B) for normal and inverse hierarchies. The means that the triangular inequality can not be valid.
We observe that most of the values can be tested in future neutrinoless double beta decay experiments. 1 Notice that there are three inequality of the type |x | ≤ i |xj | + |xk | obtained permuting the three index i, j and k, but only one gives constraints on the lightest neutrino mass.
χ, ξ 1,1 1,2 1,3 2,1 2,2 2,3 3,1 3,2 3,3
C– NH 0.006 0.000 0.017 0.000 0.011
C –IH 0.029 0.014 0.014 0.031 0.019
D –NH .005 .004 .018 .003 .005 .018
D– IH .008 .026 .025 .007 .015 .026 .007 .016
Table 2: Minimal values for mee in eV for the cases (C) and (D).
Acknowledgments This work was supported by the Spanish MICINN under grants FPA2008-00319/FPA, FPA2011-22975 by Prometeo/2009/091 (Generalitat Valenciana), DFG grant WI 2639/4-1 and by a Juan de la Cierva contract. References [1] S. M. Boucenna, S. Morisi, M. Tortola and J. W. F. Valle, Phys. Rev. D 86, 051301 (2012). [2] S. F. King, C. Luhn and A. J. Stuart, Nucl. Phys. B 867, 203 (2013). [3] G. Altarelli, F. Feruglio and L. Merlo, JHEP 0905, 020 (2009). [4] L. J. Hall, H. Murayama and N. Weiner, Phys. Rev. Lett. 84, 2572 (2000). [5] L. Dorame, D. Meloni, S. Morisi, E. Peinado and J. W. F. Valle, Nucl. Phys. B 861, 259 (2012). [6] J. Barry and W. Rodejohann, Nucl. Phys. B 842, 33 (2011). [7] M. C. Gonzalez-Garcia, M. Maltoni and J. Salvado, JHEP 1004, 056 (2010). [8] G. L. Fogli, E. Lisi, A. Marrone, A. Palazzo and A. M. Rotunno, Phys. Rev. D 84, 053007 (2011). [9] T. Schwetz, M. Tortola and J. W. F. Valle, New J. Phys. 13, 109401 (2011). [10] M. C. Gonzalez-Garcia, M. Maltoni, J. Salvado and T. Schwetz, JHEP 1212, 123 (2012). [11] G. L. Fogli, E. Lisi, A. Marrone, D. Montanino, A. Palazzo and A. M. Rotunno, Phys. Rev. D 86, 013012 (2012). [12] D. V. Forero, M. Tortola and J. W. F. Valle, Phys. Rev. D 86, 073012 (2012).