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CP properties of Majorana neutrinos and double beta decay
Volume 107B, number 1,2
PHYSICS LETTERS
3 December 1981
CP PROPERTIES OF MAJORANA NEUTRINOS AND DOUBLE BETA DECAY
Lincoln WOLFENSTEIN
Carnegie-Mellon University, Pittsburgh, PA 15213, USA Received 8 September 1981
It is pointed out that in theories with CP invaxiance and two (or more) massive Majorana neutrinos the relative sign of their CP eigenvalues is significant. For opposite values o f the CP eigenvalue the contributions of the two neutrinos to neutrinoless double beta decay tend to cancel, so that published bounds on the mass of a Majorana electron neutrino do not apply. In the case o f the Zee model o f the Majorana neutrino mass matrix, the cancellation is complete and the double beta decay rate vanishes identically.
If we assume CP invariance, Majorana neutrinos should be CP eigenstates. In many recent papers [1,2] it is implicitly assumed that if there are several Majorana neutrinos that all have even CP eigenvalues. We wish to point out in this note interesting changes in analyses that occur when there exist two massive Majorana neutrinos with opposite CP eigenvalues. In particular, limits on the masses of Majorana neutrinos deduced from double beta decay [3,4] are no longer valid. We have previously discussed [5] a model due to Zee [6] in which there are two almost-degenerate massive Majorana neutrinos; it turns out that these have opposite CP eigenvalues and so this model serves to exemplify the point we are making. The Majorana neutrino mass matrix is written -~?m = ~abM - ~. (. .f.f. '. -. v c ° ~ + V -cR b P L a ) ' where
(1)
PLa is a chiral neutrino field with flavor a and
lecRa = (ce)12La(Ce)- 1 We assume that we are starting in a basis in which the charged leptons have already been diagonalized. The matrix Mab must be symmetric by construction. We assume CP invariance with the consequence that Mab (as well as the charged couplings) are real. The matrix Mab may be diagonalized by an orthogonal matrix O, T ab Opa M abObn = 77pmp6pn'
(2)
where mp is a positive real number and ~p = +1 .We can then put "~qm in diagonal form by defining the Majorana field
Xn = ~a (OnaPLa + ~nOnaVRa)'
-Z? m = ~n mn~n×n .
(3)
It follows that the CP eigenvalues of the Majorana fields are CP Xn (CP)- 1 = r?n Xn"
(4)
The factor r?n enters in eq. (3) so that the Majorana particles all have positive mass. An alternative formalism has been used in a number of papers [1,2]. These start from the general case in which Mab is symmetric but not necessarily real, thus allowing CP non-invariance. It is then always possible to diagonalize Mab using a unitary matrix U, U op T =ran8 np . ab U n aM .ao
(5)
Note the presence of U T, not U -1 in eq. (5), so that by an appropriate choice of the phase of Una it is always possible to make rn n positive. For the cases we have considered with Mab real this corresponds to (a)
r/n = +1,
Una = Ona,
(6) 77
Volume 107B, number 1,2
(b)
r/n = - 1 ,
Una = i
PHYSICS LETTERS
Ona.
(6 cont'd)
Because the masses all come out positive, the Majorana fields are all chosen to have positive CP. However, those for which 77n = - 1 now have pure imaginary couplings for the weak interactions coupled to flavor. Thus they are discussed as if they were CP-violating interactions and are even referred to as maximally CPviolating. Since, however, this CP violation can simply be removed by redefining CP as in eq. (4), there can be no observable CP violation * 1 For any single Majorana neutrino there is no significance in whether it is CP-even or CP-odd. This follows from the presence within our theoretical framework of an overall lepton number superselection rule. However, the relative CP value of two neutrinos is significant. This can be seen in the process v 2 -+ u 1 + % If CP(P2) = CP(v 1) we obtain magnetic dipole radiation whereas ff CP(v2) = -CP(v 1) we obtain electric dipole radiation [7]. Schechter and Valle [8], who assume all Maj orana neutrinos have the same CP property, include the case of electric dipole radiation as a CP-violating case. O ~violation occurs, however, only if E1 and M1 radiation occur simultaneously. The case of neutrinoless double beta decay has been discussed by Doi et al. [1]. Using the notation of eq. (5) they find the rate is proportional to rccl~U n
U
ne ne
m 12
n[ '
(7)
Thus, in the case that Une is complex, they note that different virtual Majorana neutrinos may partially cancel. In the case of interest to us, this translates using eq. (6) to
pcx ( ? r l n O n e O n e m n ) 2.
(8)
3 December 1981
values. As a result, the limits on the electron neutrino Majorana mass from double beta decay do not apply in this case. For example, the limit ofm u < 15 eV from double beta decay [4] is not contradictory with an observed m v of 30 eV in beta decay [9] if the betadecay electron is a mixture of two Majorana mass eigenstates with a small mass difference. It may be noted that the case 0 = n/4, m 1 = m2, r/= - 1 corresponds to the limit in which the two Majoranas merge to form a Dirac neutrino. Clearly the double beta dec~iy must vanish in this limit. Zee [6] has suggested a model for obtaining massive Majorana neutrinos associated with a AL = 2 coupling of a massive charged scalar boson. For a grand unified theory such as SU(5) in which there are no righthanded neutrinos, this is essentially a unique method of obtaining a sizeable neutrino mass [10]. The version of this model we have discussed [5] has the feature of yielding Mab with zero diagonal values. As a result, after the diagonalization via eq. (2) the eigenvalues m n cannot all have the same sign and therefore some of the Majorana fields must have opposite CP eigenvalues. In this model it turns out that the cancellation in eq. (8) is complete. This is easy to see by noting, using eq. (2), that
Me e = ~ OTnr7 m 8 0 = ~n r l n m n 0 n2e . np e n n np pe Thus, the decay rate F from eq. (8) is directly proportional to M2e . Therefore, in models such as the Zee model, in which the diagonal Majorana mass of ue vanishes, the decay rate F vanishes identically. In such models the mass of the electron neutrino results only from off-diagonal terms in the original mass matrix. Of course, other possible contributions to neutrinoless double beta decay such as those due to right-handed currents are not affected by this discussion.
In the case of two flavors this becomes p cc (m 1 cos20 + r/m2 sin20) 2, where r/= r/lr/2. Thus, in the absence of CP violation, if two neutrinos contribute to double beta decay their effects tend to cancel ff they have opposite CP eigen-
This research was carried out at the Aspen Center for Physics. I wish to thank G. Karl, P. Herczeg, L.F. Li, and P.B. Pal for useful discussions. This work was supported in part by the US Department of Energy
References ¢1 In the thought experiment discussed by Schechter and VaUe [2] one can see by comparing their eqs. (5) and (6) that when their CP-violating phase 0 = n/2, the amplitudes A g b and A a g become equal.
78
[1] M. Doi et al., Phys. Lett. 102B (1981) 323. [2] S.M. Bilenky et al., Phys. Lett. 94B (1981) 495; J. Schechter and J.W.F. VaUe, Phys. Rev. D23 (1981) 1666.
Volume 107B, number 1,2
PHYSICS LETTERS
[3] P. Rosen, in: Neutrino mass Mini-Conf., Eds. V. Barger and D. Cline, U. of Wisconsin report 186 (1980) 180; M. Doi et al., Phys. Lett. 103B (1981) 219. [4] W.C. Haxton et at., Phys. Rev. Lett. 47 (1981) 153. [5] L. Wolfenstein, Nucl. Phys. B175 (1980) 93. [6] A. Zee, Phys. Lett. 93B (1980) 389.
3 December 1981
[7] P.B. Pal and L. Wolfenstein, Carnegie-Mellon preprint (1981). [8] J. Schechter and J.W.E. Valle, Syracuse preprint COO3533-192(SU4217-192) (1981). [9] V.A. Lubimov et al., Phys. Lett. 94B (1980) 266. [10] J. Nieves, Nucl. Phys. B, to be published.
79
PHYSICS LETTERS
3 December 1981
CP PROPERTIES OF MAJORANA NEUTRINOS AND DOUBLE BETA DECAY
Lincoln WOLFENSTEIN
Carnegie-Mellon University, Pittsburgh, PA 15213, USA Received 8 September 1981
It is pointed out that in theories with CP invaxiance and two (or more) massive Majorana neutrinos the relative sign of their CP eigenvalues is significant. For opposite values o f the CP eigenvalue the contributions of the two neutrinos to neutrinoless double beta decay tend to cancel, so that published bounds on the mass of a Majorana electron neutrino do not apply. In the case o f the Zee model o f the Majorana neutrino mass matrix, the cancellation is complete and the double beta decay rate vanishes identically.
If we assume CP invariance, Majorana neutrinos should be CP eigenstates. In many recent papers [1,2] it is implicitly assumed that if there are several Majorana neutrinos that all have even CP eigenvalues. We wish to point out in this note interesting changes in analyses that occur when there exist two massive Majorana neutrinos with opposite CP eigenvalues. In particular, limits on the masses of Majorana neutrinos deduced from double beta decay [3,4] are no longer valid. We have previously discussed [5] a model due to Zee [6] in which there are two almost-degenerate massive Majorana neutrinos; it turns out that these have opposite CP eigenvalues and so this model serves to exemplify the point we are making. The Majorana neutrino mass matrix is written -~?m = ~abM - ~. (. .f.f. '. -. v c ° ~ + V -cR b P L a ) ' where
(1)
PLa is a chiral neutrino field with flavor a and
lecRa = (ce)12La(Ce)- 1 We assume that we are starting in a basis in which the charged leptons have already been diagonalized. The matrix Mab must be symmetric by construction. We assume CP invariance with the consequence that Mab (as well as the charged couplings) are real. The matrix Mab may be diagonalized by an orthogonal matrix O, T ab Opa M abObn = 77pmp6pn'
(2)
where mp is a positive real number and ~p = +1 .We can then put "~qm in diagonal form by defining the Majorana field
Xn = ~a (OnaPLa + ~nOnaVRa)'
-Z? m = ~n mn~n×n .
(3)
It follows that the CP eigenvalues of the Majorana fields are CP Xn (CP)- 1 = r?n Xn"
(4)
The factor r?n enters in eq. (3) so that the Majorana particles all have positive mass. An alternative formalism has been used in a number of papers [1,2]. These start from the general case in which Mab is symmetric but not necessarily real, thus allowing CP non-invariance. It is then always possible to diagonalize Mab using a unitary matrix U, U op T =ran8 np . ab U n aM .ao
(5)
Note the presence of U T, not U -1 in eq. (5), so that by an appropriate choice of the phase of Una it is always possible to make rn n positive. For the cases we have considered with Mab real this corresponds to (a)
r/n = +1,
Una = Ona,
(6) 77
Volume 107B, number 1,2
(b)
r/n = - 1 ,
Una = i
PHYSICS LETTERS
Ona.
(6 cont'd)
Because the masses all come out positive, the Majorana fields are all chosen to have positive CP. However, those for which 77n = - 1 now have pure imaginary couplings for the weak interactions coupled to flavor. Thus they are discussed as if they were CP-violating interactions and are even referred to as maximally CPviolating. Since, however, this CP violation can simply be removed by redefining CP as in eq. (4), there can be no observable CP violation * 1 For any single Majorana neutrino there is no significance in whether it is CP-even or CP-odd. This follows from the presence within our theoretical framework of an overall lepton number superselection rule. However, the relative CP value of two neutrinos is significant. This can be seen in the process v 2 -+ u 1 + % If CP(P2) = CP(v 1) we obtain magnetic dipole radiation whereas ff CP(v2) = -CP(v 1) we obtain electric dipole radiation [7]. Schechter and Valle [8], who assume all Maj orana neutrinos have the same CP property, include the case of electric dipole radiation as a CP-violating case. O ~violation occurs, however, only if E1 and M1 radiation occur simultaneously. The case of neutrinoless double beta decay has been discussed by Doi et al. [1]. Using the notation of eq. (5) they find the rate is proportional to rccl~U n
U
ne ne
m 12
n[ '
(7)
Thus, in the case that Une is complex, they note that different virtual Majorana neutrinos may partially cancel. In the case of interest to us, this translates using eq. (6) to
pcx ( ? r l n O n e O n e m n ) 2.
(8)
3 December 1981
values. As a result, the limits on the electron neutrino Majorana mass from double beta decay do not apply in this case. For example, the limit ofm u < 15 eV from double beta decay [4] is not contradictory with an observed m v of 30 eV in beta decay [9] if the betadecay electron is a mixture of two Majorana mass eigenstates with a small mass difference. It may be noted that the case 0 = n/4, m 1 = m2, r/= - 1 corresponds to the limit in which the two Majoranas merge to form a Dirac neutrino. Clearly the double beta dec~iy must vanish in this limit. Zee [6] has suggested a model for obtaining massive Majorana neutrinos associated with a AL = 2 coupling of a massive charged scalar boson. For a grand unified theory such as SU(5) in which there are no righthanded neutrinos, this is essentially a unique method of obtaining a sizeable neutrino mass [10]. The version of this model we have discussed [5] has the feature of yielding Mab with zero diagonal values. As a result, after the diagonalization via eq. (2) the eigenvalues m n cannot all have the same sign and therefore some of the Majorana fields must have opposite CP eigenvalues. In this model it turns out that the cancellation in eq. (8) is complete. This is easy to see by noting, using eq. (2), that
Me e = ~ OTnr7 m 8 0 = ~n r l n m n 0 n2e . np e n n np pe Thus, the decay rate F from eq. (8) is directly proportional to M2e . Therefore, in models such as the Zee model, in which the diagonal Majorana mass of ue vanishes, the decay rate F vanishes identically. In such models the mass of the electron neutrino results only from off-diagonal terms in the original mass matrix. Of course, other possible contributions to neutrinoless double beta decay such as those due to right-handed currents are not affected by this discussion.
In the case of two flavors this becomes p cc (m 1 cos20 + r/m2 sin20) 2, where r/= r/lr/2. Thus, in the absence of CP violation, if two neutrinos contribute to double beta decay their effects tend to cancel ff they have opposite CP eigen-
This research was carried out at the Aspen Center for Physics. I wish to thank G. Karl, P. Herczeg, L.F. Li, and P.B. Pal for useful discussions. This work was supported in part by the US Department of Energy
References ¢1 In the thought experiment discussed by Schechter and VaUe [2] one can see by comparing their eqs. (5) and (6) that when their CP-violating phase 0 = n/2, the amplitudes A g b and A a g become equal.
78
[1] M. Doi et al., Phys. Lett. 102B (1981) 323. [2] S.M. Bilenky et al., Phys. Lett. 94B (1981) 495; J. Schechter and J.W.F. VaUe, Phys. Rev. D23 (1981) 1666.
Volume 107B, number 1,2
PHYSICS LETTERS
[3] P. Rosen, in: Neutrino mass Mini-Conf., Eds. V. Barger and D. Cline, U. of Wisconsin report 186 (1980) 180; M. Doi et al., Phys. Lett. 103B (1981) 219. [4] W.C. Haxton et at., Phys. Rev. Lett. 47 (1981) 153. [5] L. Wolfenstein, Nucl. Phys. B175 (1980) 93. [6] A. Zee, Phys. Lett. 93B (1980) 389.
3 December 1981
[7] P.B. Pal and L. Wolfenstein, Carnegie-Mellon preprint (1981). [8] J. Schechter and J.W.E. Valle, Syracuse preprint COO3533-192(SU4217-192) (1981). [9] V.A. Lubimov et al., Phys. Lett. 94B (1980) 266. [10] J. Nieves, Nucl. Phys. B, to be published.
79