- Email: [email protected]
T′ predictions of PMNS and CKM angles
Physics Letters B 679 (2009) 347–349
Contents lists available at ScienceDirect
Physics Letters B www.elsevier.com/locate/physletb
T predictions of PMNS and CKM angles Paul H. Frampton ∗ , Shinya Matsuzaki Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599-3255, United States
a r t i c l e
i n f o
Article history: Received 8 March 2009 Received in revised form 9 July 2009 Accepted 3 August 2009 Available online 7 August 2009 Editor: T. Yanagida
a b s t r a c t Generalizing a previous model to accommodate the third quark family and CP violation, we present a T model which predicts tribimaximal neutrino (PMNS) mixings while the central predictions for quark mixings are | V td / V ts | = 0.245 and | V ub / V cb | = 0.237 with a predicted CP violating KM phase δKM = 65.8◦ . All these are acceptably close to experiment, including the KM phase for which the allowed values are 63◦ < δKM < 72◦ , and depend only on use of symmetry T × Z 2 to define the model and no additional parameters. © 2009 Elsevier B.V. All rights reserved.
The justly celebrated standard model (SM) of particle phenomenology rests as one of the great accomplishments of theory. Nevertheless, in the extended form which can accommodate neutrino masses, it has 28 free parameters none of which has been calculated or predicted. Notable are quark and lepton masses (12, hereafter SM masses) and mixing angles and phases (10, hereafter SM mixings) which relate mass eigenstates for quarks and leptons to states in the Lagrangian. Except for a hint about the neutrino hierarchy, we find no new insight about SM masses.1 With regard to SM mixings we report at least limited progress towards understanding 6 out of 10 phenomenological parameters by the study of T (binary tetrahedral group) as a flavor symmetry commuting with the SM theory. If even one of these explanations survives we feel it will be useful and important. In recent work, the present authors, together with Kephart [1], presented a simplified model based on T flavor symmetry. The principal simplification was that the CKM mixing angles2 involving the third quark family were taken to vanish Θ23 = Θ13 = 0. In terms of the scalar field content, all scalar fields are taken to be doublets under electroweak SU (2) with vacuum values which underly the symmetry breaking. Great simplification was originally achieved by the device of restricting scalar fields to irreducible representations of T which are singlets and triplets only, without any T doublets. There was a good reason for this because the admission of T -doublet scalars enormously complicates the √ symmetry breaking. This enabled the isolation of the Cabibbo angle Θ12 and to a reasonable prediction thereof, namely [1] tan 2Θ12 = ( 2 )/3. Within the same simplified model, in a subsequent paper with Eby [2], the departure of Θ12 from this T prediction was used to make predictions for the departure of the neutrino PMNS angles θi j from their tribimaximal values [3]. Also in that model [4], we suggested a smoking-gun T prediction for leptonic decay of the standard model Higgs scalar. Other related works are [5–15]. In the present Letter, we examine the addition of T -doublet scalars. As anticipated in [1], this allows more possibilities of T symmetry breaking and permits non-zero values for Θ23 , Θ13 and δKM . We present an explicit ( T × Z 2 ) model which leads to consistent results for PMNS and KM angles as well as testable predictions. To understand the incorporation of T -doublet scalars and to make the present article self-contained, it is necessary to review the simplified model employed in [1,2,4] in which T -doublet scalars were deliberately excluded in order to isolate the Cabibbo angle Θ12 . We here adopt the global symmetry ( T × Z 2 ).
* 1 2
Corresponding author. E-mail addresses: [email protected] (P.H. Frampton), [email protected] (S. Matsuzaki). The result in an earlier paper [1] involving the quark mass squared ratio (md2 /m2s ) has only small corrections from the considerations of the present Letter. Note that here upper case Θi j refer to quarks (CKM) and lower case θi j will refer to neutrinos (PMNS).
0370-2693/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2009.08.001
348
P.H. Frampton, S. Matsuzaki / Physics Letters B 679 (2009) 347–349
Left-handed quark doublets (t , b) L , (c , d) L , (u , d) L are assigned under this as
t b
QL L ⎫ c ⎪ ⎪ s L ⎬ QL ⎪ u ⎪ ⎭ d
(11 , +1),
(21 , +1),
(1)
L
and the six right-handed quarks as
(11 , +1),
tR bR cR uR sR
(12 , −1),
CR
(23 , −1),
SR
(22 , +1).
dR
(2)
The leptons are assigned as
⎫
ντ ⎪ ⎪ ⎪ τ− L ⎪ ⎪ ⎪ ⎪ ⎬ νμ L (3, +1) − μ ⎪ L L⎪ ⎪ ⎪ ⎪ νe ⎪ ⎪ e−
L
τ R− (11 , −1) μ−R (12 , −1)
(1 )
N R (11 , +1), (2 )
N R (12 , +1),
−
e R (13 , −1)
⎭
(3)
(3 )
N R (13 , +1).
Next we turn to the symmetry breaking and the necessary scalar sector with its own potential3 and Yukawa coupling to the fermions, leptons and quarks. For scalar fields we keep those used previously namely the two T triplets and two T singlets
H 3 (3, +1);
H 3 (3, −1);
H 11 (11 , +1);
H 13 (13 , −1)
(4)
which led to the simplified model discussed previously with CKM angles Θ23 = Θ13 = 0. That model was used to derive a formula for the Cabibbo angle [1], to predict corrections [2] to the tribimaximal values [3] of PMNS neutrino angles, and to make a prediction for Higgs boson decay [4]. We now introduce one T doublet scalar in an explicit model. Non-vanishing Θ23 and Θ13 will be induced by symmetry breaking due to the addition the T doublet scalar
H 23 (23 , +1).
(5) 4
The field in Eq. (5) allows a new Yukawa coupling
Y QS Q L S R H 23 + h.c.
(6)
∗ is real and we accommodate CP violation by the use of complex T Clebsch–Gordan coefficients. Y sb
where Y sb = The vacuum expectation value (VEV) for H 23 is taken with the alignment
H 23 = V 23 (1, 1)
(7)
while as in [1] the other VEVs include
H 3 = V (1, −2, 1).
(8)
The hermitian squared mass matrix D ≡ D D for the charge (−1/3) quarks is then †
⎛
mb2
√ ⎜ 1 √ Y Y V V 23 (1 − 2 2ω−2 ) D=⎜ ⎝ 6 S QS √ √1 Y S Y QS V V 2 (ω−2 + 2 ) 3
√1 Y S Y QS V V 2 (1 − 2 3 6
√
2ω 2 )
−
√
2 (Y S V )2 3
+
√
2 (Y S V )2 3
−
3(Y S V )2
6
√1 Y S Y QS V V 2 (ω2 3 6 √
2)
⎞ ⎟ ⎟. ⎠
(9)
2
(Y S V )
3 The scalar potential will not be examined explicitly. We assume that it has enough parameters to accommodate the required VEVs in a finite neighborhood of parameter values. 4 We list the Yukawa couplings already discussed for the T -triplet and T -singlet scalars:
LY =
1 2
M 1 N R N R + M 23 N R N R + Y 1 L L N R H 3 + Y 2 L L N R H 3 + Y 3 L L N R H 3 + Y τ L L τ R H 3 + Y μ L L μ R H 3 + Y e L L e R H 3 (1)
(1)
(2)
(3)
(1)
(2)
(3)
+ Y t {Q L }11 {t R }11 H 11 + Y b ({Q L }11 {b R }12 H 13 ) + Y C ({ Q L }21 {C R }23 H 3 ) + Y S { Q L }21 {S R }22 H 3 + h.c.
P.H. Frampton, S. Matsuzaki / Physics Letters B 679 (2009) 347–349
349
In Eq. (9) the 2 × 2 sub-matrix √ for the first two families coincides with the result discussed earlier [1] and hence the successful Cabibbo angle formula tan 2Θ12 = ( 3 )/2 is preserved. Note that in this model the mass matrix for the charge +2/3 quarks is diagonal5 so the CKM mixing matrix arises purely from diagonalization of D in Eq. (9). The presence of the complex T Clebsch–Gordan coefficients [16] in Eq. (9) which will lead to non-zero KM CP violating phase. For mb2 the experimental value 17.6 GeV2 [17] although the CKM angles and phase do not depend on this overall normalization. Actually our results depend only on assuming that the ratio (Y QS V 23 /Y S V ) is much smaller than one. In this case we find the results for the ratios of CKM matrix elements
| V td / V ts | = 0.245
(10)
| V ub / V cb | = 0.237.
(11)
and
For the CP violating phase δKM
δKM = 65.8◦ is in the experimentally allowed range 63◦
(12)
< δKM
< 72◦ .
More technical details will be provided in [18]. Note that once the off-diagonal third-family elements in Eq. (9) are taken as much smaller than the elements involved in the Cabibbo angle, the two KM angles and the CP phase are predicted without further free parameters so this vindicates the hope expressed in [1]. In summary, we have reported progress in understanding the PMNS and CKM mixing angles by exploiting the binary tetrahedral group (T ) as a global discrete flavor symmetry commuting with the local gauge symmetry SU (3) × SU (2) × U (1) of the standard model of particle phenomenology. The results are encouraging to pursue this direction of study. Acknowledgement This work was supported in part by the US Department of Energy under Grant No. DE-FG02-06ER41418. References [1] [2] [3] [4] [5] [6] [7]
[8]
[9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
5
P.H. Frampton, T.W. Kephart, S. Matsuzaki, Phys. Rev. D 78 (2008) 073004, arXiv:0807.4713 [hep-ph]. D.A. Eby, P.H. Frampton, S. Matsuzaki, Phys. Lett. B 671 (2009) 386, arXiv:0810.4899 [hep-ph]. P.F. Harrison, D.H. Perkins, W.G. Scott, Phys. Lett. B 530 (2002) 167, hep-ph/0202074. P.H. Frampton, S. Matsuzaki, arXiv:0810.1029 [hep-ph], Mod. Phys. Lett. A (2009), in press. P.H. Frampton, S. Matsuzaki, Mod. Phys. Lett. A 24 (2009) 429, arXiv:0807.4785 [hep-ph]. E. Ma, G. Rajasekaran, Phys. Rev. D 64 (2001) 113012, hep-ph/0106291; K.S. Babu, E. Ma, J.W.F. Valle, Phys. Lett. B 552 (2003) 207, hep-ph/0206292. E. Ma, Mod. Phys. Lett. A 20 (2005) 2601, hep-ph/0508099; E. Ma, Phys. Lett. B 632 (2006) 352, hep-ph/0508231; B. Adhikary, B. Brahmachari, A. Ghosal, E. Ma, M.K. Parida, Phys. Lett. B 638 (2006) 345, hep-ph/0603059; E. Ma, Phys. Rev. D 73 (2006) 057304; E. Ma, H. Sawanaka, M. Tanimoto, Phys. Lett. B 641 (2006) 301, hep-ph/0606103; E. Ma, Mod. Phys. Lett. A 21 (2006) 1917, hep-ph/0607056. G. Altarelli, F. Feruglio, Nucl. Phys. B 720 (2005) 64, hep-ph/0504165; G. Altarelli, F. Feruglio, Nucl. Phys. B 742 (2006) 215, hep-ph/0512103; G. Altarelli, F. Feruglio, Y. Lin, Nucl. Phys. B 775 (2007) 31, hep-ph/0610165. P.H. Frampton, T.W. Kephart, Int. J. Mod. Phys. A 10 (1995) 4689, JHEP 0709 (2007) 110, arXiv:0706.1186 [hep-ph]. F. Feruglio, C. Hagedorn, Y. Lin, L. Merlo, Nucl. Phys. B 775 (2007) 120, hep-ph/0702194. M.-C. Chen, K.T. Mahanthappa, Phys. Lett. B 652 (2007) 34, arXiv:0705.0714. P.F. Harrison, W.G. Scott, Phys. Lett. B 628 (2005) 93, hep-ph/0508012. P.H. Frampton, S.L. Glashow, Phys. Lett. B 461 (1999) 95, hep-ph/9906375. P.H. Frampton, S.L. Glashow, D. Marfatia, Phys. Lett. B 536 (2002) 79, hep-ph/0201008. P.H. Frampton, S.L. Glashow, T. Yanagida, Phys. Lett. B 548 (2002) 119, hep-ph/0208157. M.-C. Chen, K.T. Mahanthappa, arXiv:0904.1721 [hep-ph]. Particle Data Group, Phys. Lett. 667 (2008) 1. D.A. Eby, P.H. Frampton, S. Matsuzaki, arXiv:0907.3425 [hep-ph].
This uses the approximation that the electron mass is me = 0; cf. Ref. [1].
Contents lists available at ScienceDirect
Physics Letters B www.elsevier.com/locate/physletb
T predictions of PMNS and CKM angles Paul H. Frampton ∗ , Shinya Matsuzaki Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599-3255, United States
a r t i c l e
i n f o
Article history: Received 8 March 2009 Received in revised form 9 July 2009 Accepted 3 August 2009 Available online 7 August 2009 Editor: T. Yanagida
a b s t r a c t Generalizing a previous model to accommodate the third quark family and CP violation, we present a T model which predicts tribimaximal neutrino (PMNS) mixings while the central predictions for quark mixings are | V td / V ts | = 0.245 and | V ub / V cb | = 0.237 with a predicted CP violating KM phase δKM = 65.8◦ . All these are acceptably close to experiment, including the KM phase for which the allowed values are 63◦ < δKM < 72◦ , and depend only on use of symmetry T × Z 2 to define the model and no additional parameters. © 2009 Elsevier B.V. All rights reserved.
The justly celebrated standard model (SM) of particle phenomenology rests as one of the great accomplishments of theory. Nevertheless, in the extended form which can accommodate neutrino masses, it has 28 free parameters none of which has been calculated or predicted. Notable are quark and lepton masses (12, hereafter SM masses) and mixing angles and phases (10, hereafter SM mixings) which relate mass eigenstates for quarks and leptons to states in the Lagrangian. Except for a hint about the neutrino hierarchy, we find no new insight about SM masses.1 With regard to SM mixings we report at least limited progress towards understanding 6 out of 10 phenomenological parameters by the study of T (binary tetrahedral group) as a flavor symmetry commuting with the SM theory. If even one of these explanations survives we feel it will be useful and important. In recent work, the present authors, together with Kephart [1], presented a simplified model based on T flavor symmetry. The principal simplification was that the CKM mixing angles2 involving the third quark family were taken to vanish Θ23 = Θ13 = 0. In terms of the scalar field content, all scalar fields are taken to be doublets under electroweak SU (2) with vacuum values which underly the symmetry breaking. Great simplification was originally achieved by the device of restricting scalar fields to irreducible representations of T which are singlets and triplets only, without any T doublets. There was a good reason for this because the admission of T -doublet scalars enormously complicates the √ symmetry breaking. This enabled the isolation of the Cabibbo angle Θ12 and to a reasonable prediction thereof, namely [1] tan 2Θ12 = ( 2 )/3. Within the same simplified model, in a subsequent paper with Eby [2], the departure of Θ12 from this T prediction was used to make predictions for the departure of the neutrino PMNS angles θi j from their tribimaximal values [3]. Also in that model [4], we suggested a smoking-gun T prediction for leptonic decay of the standard model Higgs scalar. Other related works are [5–15]. In the present Letter, we examine the addition of T -doublet scalars. As anticipated in [1], this allows more possibilities of T symmetry breaking and permits non-zero values for Θ23 , Θ13 and δKM . We present an explicit ( T × Z 2 ) model which leads to consistent results for PMNS and KM angles as well as testable predictions. To understand the incorporation of T -doublet scalars and to make the present article self-contained, it is necessary to review the simplified model employed in [1,2,4] in which T -doublet scalars were deliberately excluded in order to isolate the Cabibbo angle Θ12 . We here adopt the global symmetry ( T × Z 2 ).
* 1 2
Corresponding author. E-mail addresses: [email protected] (P.H. Frampton), [email protected] (S. Matsuzaki). The result in an earlier paper [1] involving the quark mass squared ratio (md2 /m2s ) has only small corrections from the considerations of the present Letter. Note that here upper case Θi j refer to quarks (CKM) and lower case θi j will refer to neutrinos (PMNS).
0370-2693/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2009.08.001
348
P.H. Frampton, S. Matsuzaki / Physics Letters B 679 (2009) 347–349
Left-handed quark doublets (t , b) L , (c , d) L , (u , d) L are assigned under this as
t b
QL L ⎫ c ⎪ ⎪ s L ⎬ QL ⎪ u ⎪ ⎭ d
(11 , +1),
(21 , +1),
(1)
L
and the six right-handed quarks as
(11 , +1),
tR bR cR uR sR
(12 , −1),
CR
(23 , −1),
SR
(22 , +1).
dR
(2)
The leptons are assigned as
⎫
ντ ⎪ ⎪ ⎪ τ− L ⎪ ⎪ ⎪ ⎪ ⎬ νμ L (3, +1) − μ ⎪ L L⎪ ⎪ ⎪ ⎪ νe ⎪ ⎪ e−
L
τ R− (11 , −1) μ−R (12 , −1)
(1 )
N R (11 , +1), (2 )
N R (12 , +1),
−
e R (13 , −1)
⎭
(3)
(3 )
N R (13 , +1).
Next we turn to the symmetry breaking and the necessary scalar sector with its own potential3 and Yukawa coupling to the fermions, leptons and quarks. For scalar fields we keep those used previously namely the two T triplets and two T singlets
H 3 (3, +1);
H 3 (3, −1);
H 11 (11 , +1);
H 13 (13 , −1)
(4)
which led to the simplified model discussed previously with CKM angles Θ23 = Θ13 = 0. That model was used to derive a formula for the Cabibbo angle [1], to predict corrections [2] to the tribimaximal values [3] of PMNS neutrino angles, and to make a prediction for Higgs boson decay [4]. We now introduce one T doublet scalar in an explicit model. Non-vanishing Θ23 and Θ13 will be induced by symmetry breaking due to the addition the T doublet scalar
H 23 (23 , +1).
(5) 4
The field in Eq. (5) allows a new Yukawa coupling
Y QS Q L S R H 23 + h.c.
(6)
∗ is real and we accommodate CP violation by the use of complex T Clebsch–Gordan coefficients. Y sb
where Y sb = The vacuum expectation value (VEV) for H 23 is taken with the alignment
H 23 = V 23 (1, 1)
(7)
while as in [1] the other VEVs include
H 3 = V (1, −2, 1).
(8)
The hermitian squared mass matrix D ≡ D D for the charge (−1/3) quarks is then †
⎛
mb2
√ ⎜ 1 √ Y Y V V 23 (1 − 2 2ω−2 ) D=⎜ ⎝ 6 S QS √ √1 Y S Y QS V V 2 (ω−2 + 2 ) 3
√1 Y S Y QS V V 2 (1 − 2 3 6
√
2ω 2 )
−
√
2 (Y S V )2 3
+
√
2 (Y S V )2 3
−
3(Y S V )2
6
√1 Y S Y QS V V 2 (ω2 3 6 √
2)
⎞ ⎟ ⎟. ⎠
(9)
2
(Y S V )
3 The scalar potential will not be examined explicitly. We assume that it has enough parameters to accommodate the required VEVs in a finite neighborhood of parameter values. 4 We list the Yukawa couplings already discussed for the T -triplet and T -singlet scalars:
LY =
1 2
M 1 N R N R + M 23 N R N R + Y 1 L L N R H 3 + Y 2 L L N R H 3 + Y 3 L L N R H 3 + Y τ L L τ R H 3 + Y μ L L μ R H 3 + Y e L L e R H 3 (1)
(1)
(2)
(3)
(1)
(2)
(3)
+ Y t {Q L }11 {t R }11 H 11 + Y b ({Q L }11 {b R }12 H 13 ) + Y C ({ Q L }21 {C R }23 H 3 ) + Y S { Q L }21 {S R }22 H 3 + h.c.
P.H. Frampton, S. Matsuzaki / Physics Letters B 679 (2009) 347–349
349
In Eq. (9) the 2 × 2 sub-matrix √ for the first two families coincides with the result discussed earlier [1] and hence the successful Cabibbo angle formula tan 2Θ12 = ( 3 )/2 is preserved. Note that in this model the mass matrix for the charge +2/3 quarks is diagonal5 so the CKM mixing matrix arises purely from diagonalization of D in Eq. (9). The presence of the complex T Clebsch–Gordan coefficients [16] in Eq. (9) which will lead to non-zero KM CP violating phase. For mb2 the experimental value 17.6 GeV2 [17] although the CKM angles and phase do not depend on this overall normalization. Actually our results depend only on assuming that the ratio (Y QS V 23 /Y S V ) is much smaller than one. In this case we find the results for the ratios of CKM matrix elements
| V td / V ts | = 0.245
(10)
| V ub / V cb | = 0.237.
(11)
and
For the CP violating phase δKM
δKM = 65.8◦ is in the experimentally allowed range 63◦
(12)
< δKM
< 72◦ .
More technical details will be provided in [18]. Note that once the off-diagonal third-family elements in Eq. (9) are taken as much smaller than the elements involved in the Cabibbo angle, the two KM angles and the CP phase are predicted without further free parameters so this vindicates the hope expressed in [1]. In summary, we have reported progress in understanding the PMNS and CKM mixing angles by exploiting the binary tetrahedral group (T ) as a global discrete flavor symmetry commuting with the local gauge symmetry SU (3) × SU (2) × U (1) of the standard model of particle phenomenology. The results are encouraging to pursue this direction of study. Acknowledgement This work was supported in part by the US Department of Energy under Grant No. DE-FG02-06ER41418. References [1] [2] [3] [4] [5] [6] [7]
[8]
[9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
5
P.H. Frampton, T.W. Kephart, S. Matsuzaki, Phys. Rev. D 78 (2008) 073004, arXiv:0807.4713 [hep-ph]. D.A. Eby, P.H. Frampton, S. Matsuzaki, Phys. Lett. B 671 (2009) 386, arXiv:0810.4899 [hep-ph]. P.F. Harrison, D.H. Perkins, W.G. Scott, Phys. Lett. B 530 (2002) 167, hep-ph/0202074. P.H. Frampton, S. Matsuzaki, arXiv:0810.1029 [hep-ph], Mod. Phys. Lett. A (2009), in press. P.H. Frampton, S. Matsuzaki, Mod. Phys. Lett. A 24 (2009) 429, arXiv:0807.4785 [hep-ph]. E. Ma, G. Rajasekaran, Phys. Rev. D 64 (2001) 113012, hep-ph/0106291; K.S. Babu, E. Ma, J.W.F. Valle, Phys. Lett. B 552 (2003) 207, hep-ph/0206292. E. Ma, Mod. Phys. Lett. A 20 (2005) 2601, hep-ph/0508099; E. Ma, Phys. Lett. B 632 (2006) 352, hep-ph/0508231; B. Adhikary, B. Brahmachari, A. Ghosal, E. Ma, M.K. Parida, Phys. Lett. B 638 (2006) 345, hep-ph/0603059; E. Ma, Phys. Rev. D 73 (2006) 057304; E. Ma, H. Sawanaka, M. Tanimoto, Phys. Lett. B 641 (2006) 301, hep-ph/0606103; E. Ma, Mod. Phys. Lett. A 21 (2006) 1917, hep-ph/0607056. G. Altarelli, F. Feruglio, Nucl. Phys. B 720 (2005) 64, hep-ph/0504165; G. Altarelli, F. Feruglio, Nucl. Phys. B 742 (2006) 215, hep-ph/0512103; G. Altarelli, F. Feruglio, Y. Lin, Nucl. Phys. B 775 (2007) 31, hep-ph/0610165. P.H. Frampton, T.W. Kephart, Int. J. Mod. Phys. A 10 (1995) 4689, JHEP 0709 (2007) 110, arXiv:0706.1186 [hep-ph]. F. Feruglio, C. Hagedorn, Y. Lin, L. Merlo, Nucl. Phys. B 775 (2007) 120, hep-ph/0702194. M.-C. Chen, K.T. Mahanthappa, Phys. Lett. B 652 (2007) 34, arXiv:0705.0714. P.F. Harrison, W.G. Scott, Phys. Lett. B 628 (2005) 93, hep-ph/0508012. P.H. Frampton, S.L. Glashow, Phys. Lett. B 461 (1999) 95, hep-ph/9906375. P.H. Frampton, S.L. Glashow, D. Marfatia, Phys. Lett. B 536 (2002) 79, hep-ph/0201008. P.H. Frampton, S.L. Glashow, T. Yanagida, Phys. Lett. B 548 (2002) 119, hep-ph/0208157. M.-C. Chen, K.T. Mahanthappa, arXiv:0904.1721 [hep-ph]. Particle Data Group, Phys. Lett. 667 (2008) 1. D.A. Eby, P.H. Frampton, S. Matsuzaki, arXiv:0907.3425 [hep-ph].
This uses the approximation that the electron mass is me = 0; cf. Ref. [1].