- Email: [email protected]
Solar neutrinos and leptonic CP violation
Physics Letters B 468 Ž1999. 256–260
Solar neutrinos and leptonic CP violation Hisakazu Minakata b
a,b,1
, Shinji Watanabe
c,2
a Department of Physics, Tokyo Metropolitan UniÕersity, 1-1 Minami-Osawa, Hachioji, Tokyo 192-0397, Japan Research Center for Cosmic Neutrinos, Institute for Cosmic Ray Research, UniÕersity of Tokyo, Tanashi, Tokyo 188-8502, Japan c Department of Physics, Waseda UniÕersity, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan
Received 6 July 1999; received in revised form 20 August 1999; accepted 18 October 1999 Editor: H. Georgi
Abstract We examine the possibility of detecting effects of leptonic CP violation by precise measurement of the solar neutrinos within the framework of standard electroweak theory minimally extended to include neutrino masses and mixing. We prove a ‘‘no-go theorem’’ which states that effects of CP violating phase disappear in the ne survival probability to the leading order in electroweak interaction. The effects due to the next-to-leading order correction is estimated to be extremely small, effectively closing the door to the possibility we intended to pursue. q 1999 Published by Elsevier Science B.V. All rights reserved.
Exciting discovery of neutrino oscillation in atmospheric neutrino observation w1x strongly suggests that neutrinos are massive. Then, it is likely that the leptonic CKM w2x matrix, which is now called w3x as the Maki–Nakagawa–Sakata ŽMNS. w4x matrix, exists so that the nature admits CP violation in the lepton sector. The CP violation in the lepton sector has been the topics of interests because of various reasons; on the one hand it may inherit a key to understanding the lepton-quark correspondence, and on the other hand it may provide us with an intriguing mechanism for generating baryon asymmetry in our universe w5x. It is then important to explore the methods for measuring leptonic CP violating phase. Neutrino os-
1 2
E-mail: [email protected] E-mail: [email protected]
cillation has been known to be one of the promising ways of measuring effects of CP violation w6x. Its effect may be detectable if the flavor mixing angles are not small. Recently, detailed investigations have been done to investigate how to measure CP violation in long-baseline neutrino oscillation experiments, with particular emphasis on how to discriminate matter effect contamination from the genuine effect of the Kobayashi–Maskawa phase w7–11x. We consider in this paper an alternative possibility of measuring CP violation by precise measurement of the solar neutrino flux, assuming that neutrinos interact as dictated by the standard electroweak theory. Unfortunately, we will end up with the ‘‘nogo theorem’’. That is, we will show that the effects of CP violating phase vanish in any observables in solar neutrino experiments to leading order in the electroweak interaction. While the ‘‘theorem’’ is not quite a theorem because its validity is limited to
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 1 2 2 4 - 1
leading order, it effectively closes the door to any practical attempts to measure CP violating effects in solar neutrino experiments because the higher order effects are so small. Or, in other word, if a single effect of CP violating phase is detected in a solar neutrino experiment, then it would imply the existence of the neutrino interactions beyond that of the standard model. The goal of our argument in the first half of this paper is to show that the effect of CP-violating phase vanishes to first order in electroweak interactions in the survival probability P Ž ne ™ ne .. Notice that the solar neutrino observation is inherently a disappearance experiment; there is no way to detect appearance events in charged current interactions because the energy is well below the thresholds, and the neutral current interaction cannot distinguish between nm and nt . Toward the goal let us start by recalling the well known results for vacuum neutrino oscillation. There exists a general theorem that states that the measure for CP violation vanishes in the disappearance probability, i.e. P Ž na ™ na . y P Ž na ™ na . s 0
Ž 1.
in vacuum. Hereafter, the Greek indices label the flavor eigenstates; a s e, m ,t for three-generations. This comes from the CPT invariance which implies that P Ž na ™ nb . s P Ž nb ™ na .. The lepton flavor mixing is described by the mixing matrix Ua i that relates the flavor eigenstate na and mass eigenstate n i as na s Ua i n i , where i is the index for specifying mass eigenstates. In the framework of three flavor mixing the feature implied by the general theorem has a simple realization; the effect of CP violation shows up in the vacuum neutrino oscillation probabilities in the particular way as P Ž nb ™ na . y P Ž nb ™ na . s 4 Jba sin
ž
D m2 L 2E
/
,
the unique fermion rephasing invariant measure for CP violation. In fact, the CP-violating piece in P Ž nb ™ na . is identical with that of P Ž nb ™ na . apart from the sign, and is given by a half of the righthand-side of Ž2.. By unitarity the disappearance probability can be written in terms of the appearance probabilities as 1 y P Ž ne ™ ne . s P Ž ne ™ nm . q P Ž ne ™ nt .. The CP-violating pieces in the right-hand-side of this equation cancel owing to the cyclic property of Jba . This is all about how the CP violating effect vanishes in the survival probabilities P Ž na ™ na . in vacuum neutrino oscillation. The situation changes completely when the matter effect w13x is taken into account. Since the matter is in general neither CPT nor CP symmetric there is no general argument which enforces that CP violation effect disappears in neutrino oscillation. Moreover, the CP violation which shows up in neutrino conversion probabilities contains both the fake matter and the genuine effects due to the Kobayashi–Maskawa phase. Therefore, it is difficult to make general statement on how CP violating effects come in. To attack the problem we write down the evolution equation of three flavor neutrinos in matter which is valid to leading order in electroweak interaction:
°
m12r2 E ne nm s U 0 i dx nt 0 d
~
¢
aŽ x . q 0 0
0 0 0
0
0
m22r2 E
0
0
m23r2 E
0 0 0
Uq
¶n •n
ß
e
m
,
Ž 4.
nt
where aŽ x . s '2 G F Ne Ž x . indicates the index of refraction with G F and Ne Ž x . being the Fermi constant and the electron number density, respectively. 3
Ž 2. for a / b and Jba stands for the leptonic Jarlskog factor w12x, Jba s Im Ua 1Ua)2 Ub)1 Ub 2 ,
Ž 3.
3 We note, in passing, that the extra phases which appear for Majorana neutrinos do not give any effects in neutrino evolution in vacuum and in matter because they are multiplied from right to U and therefore drop out in Ž4..
We take a particular parametrization of the mixing matrix w14x. U s e i l 7 u 23 Gd e i l 5 u 13 e i l 2 u 12 ,
Ž 5.
where l i are SUŽ3. Gell–Mann’s matrix and G contains the CP violating phase 1 Gd s 0 0
0 1 0
where T is defined in Ž7. and its explicit form in our parametrization Ž5. of the mixing matrix reads
0 0 . eid
Ž 6.
We rewrite the evolution Eq. Ž4. in terms of the new basis defined w15x by
n˜a s w eyi l 5 u 13 G y1 eyi l 7 u 23 x ab nb ' Ž T t . a b nb .
c13
0
T s ys23 s13 e
id
yc23 s13 e
id
c 23 ys23
s13 s23 c13 e i d . c 23 c13 e
Ž 10 .
id
It is then evident that the ne survival amplitude ² ne N ne :, and hence the probability, does not contain the CP violating phase: 2 ² 2 ² ² ne N ne : s c13 n˜e N n˜e : q s13 n˜t N n˜t :
Ž 7. q c13 s13 ² n˜e N n˜t : q ² n˜t N n˜e : .
ž
Ž 11 .
/
It reads
°
n˜e m12 d 1 il u n˜m s i e 2 12 0 dx 2E n˜t 0
~
¢
2 c13 qa Ž x . 0 c13 s13
0
0
m22
0
0
m 23
0 0 0
c13 s13 0 2 s13
eyi l 2 u 12
¶ •
ß
n˜e n˜m . n˜t
Ž 8. The CP phase d disappears from the equation. It is due to the specific way that the matter effect comes in; aŽ x . only appears in Ž1.1. element in the Hamiltonian matrix and therefore the matter matrix diag Ž a,0,0. is invariant under rotation in 2 y 3 space by e i l1 u 23 . Then the rotation by the phase matrix G does nothing. It is clear from Ž8. that any transition amplitudes computed with n˜a basis is independent of the CP violating phase. Of course, it does not immediately imply that the CP violating phase d disappears in the physical transition amplitude ² nb N na :. The latter is related with the transition amplitude defined with n˜a basis as ² nb N na : s Tag Tbd) ² n˜d N n˜g : ,
Ž 9.
We have checked that this conclusion is not specific to the particular parametrization of the MNS matrix, as it should not. Notice that the same statement does not apply to the nm disappearance amplitude: ² nm N nm : 2 2 ² 2 ² 2 2 ² s s23 s13 n˜e N n˜e : q c 23 n˜m N n˜m : q s23 c13 n˜t N n˜t :
y c13 s23 s13 e i d ² n˜m N n˜e : q eyi d ² n˜e N n˜m :
/
q c 23 s13 c13 e i d ² n˜m N n˜t : q eyi d ² n˜t N n˜m :
/
ž
ž
2 y s23 c13 s13 ² n˜e N n˜t : q ² n˜t N n˜e : .
ž
/
Ž 12 .
One clearly sees the d dependence in the survival amplitude of m neutrinos in Ž12.. Notices also that it’s dependence is not necessarily cos d because ² n˜b Ž x . N n˜a Ž 0 . : / ² n˜a Ž x . N n˜b Ž 0 . :
Ž 13 .
owing to the fact that aŽ0. / aŽ x . in general. If Ž13. holds there exists a CP-odd observable P Ž nm ™ nm .
y P Ž nm ™ nm . which is nonvanishing due to the CPviolating phase. We now turn to the problem of CP violating effect due to the next-to-leading order correction of the standard electroweak interaction. It was noticed by Botella, Lim, and Marciano w16x that it gives rise to a small splitting between the indices of refraction of muon and tau neutrinos; the matter effect matrix in Ž4., diag.Ža,0,0., becomes diag.Ža,0,b. in an appropriate phase convention of the neutrino wave function. With a suitable redefinition of the Fermi constant G F aŽ x . is still given by the same form, aŽ x . s '2 G F Ne Ž x ., and the ratio bra is computed to be w16x bŽ x . aŽ x .
3a sy
mt
2p sin2u W
2
ž / mW
2ln
mt
q
mW
4 P Ž ne ™ ne . s c13 Ž cv2 c122 q sv2 s122 . q s134
4E q
Ž 14 .
for an isoscalar medium where mt and mW denote the masses of tau lepton and W boson, respectively. With the next-to-leading order effect the Hamiltonian matrix in the evolution Eq. Ž8. in n˜ basis has the additional term
b Ž x . c 23 s23 s13 e i d 2 y c 23 c13 s13
yi d
c 23 s23 s13 e 2 s23
y c 23 s23 c13 eyi d
Ž a y bc232 . c132 s132
2 = c13 Ž cv2 c122 q sv2 s122 . y s132
4E q
DM2
2 bc13 Ž c122 y s122 .
2 =cv sv c 23 s23 c13 s13 cos d ,
sin2 v s
Bs
Ds 2 2 c 23 s13
DM2
Ž 16 .
2 where D M 2 s D m223 , D m13 as defined above. The angle v in Ž16. is the value of an angle at the solar core;
5 6
, 5.02 = 10y5
generation neutrinos. We give here only the result, leaving the detail to Ref. w17x.
B
(B q 2
2 D m12
2E 2 D m12
2 y c 23 c13 s13
2E
1 4
D2
,
c12 s12 q bc23 s23 s13 eyi d ,
Ž c122 y s122 . y ac132 y b Ž c232 s132 y s232 . , Ž 17 .
y c 23 s23 c13 e i d . 2 2 c 23 c13
Ž 15 . Clearly the n˜ evolution is affected by the CP violating phase. Its effect is however small, giving rise to the small correction term in the flavor conversion probability which is of the order of bra , 5 = 10y5 . If the neutrinos have hierarchy in their masses in 2 such a way that the D m12 relevant for the solar neutrino conversion is much smaller than that of the 2 atmospheric neutrino anomaly, i.e., D m13 , D m223 ' 2 2 D M 4 D m12 then CP violation due to higher order effects are suppressed even further. To give a feeling we compute the next-to-leading order correction to the ne survival probability under the adiabatic approximation. We also restrict ourselves to the leading order perturbative correction of the effects of third
which parametrizes matter mass eigenstate in 2 = 2 submatrix. Notice that the CP violating effect comes out in a form very reminiscent of the Jarlskog factor, 2 s13 sin d , apart from modification J s c12 s12 c 23 s23 c13 by the matter effect. As clearly seen in Ž16., the CP violation effect is suppressed, in addition to the possible smallness of angle factors, by two small ratios, b DM2
;
2 b D m12
a DM2
, 5 = Ž 10y8 y 10y7 . ,
Ž 18 .
2E 2 assuming that D m12 s 10y6 y 10y5 eV 2 and D M 2 y3 2 s 10 eV . We argue that the double suppression of the CP phase effect is not the artifact of the adiabatic ap-
proximation. Since b carries the dimension of mass, it must be compensated by D m2rE. The D m2 has to be D M 2 because the CP violation must vanish in an effective two-flavor limit D M 2 ™ `. By the similar argument one can show that the CP violating term in P Ž nm ™ nm . obtains a mild suppression factor ; aErD M 2 , 0.1 in the earth crust. In this paper we have proven that the effect of CP violating phase disappears from the survival probability P Ž ne ™ ne . within the framework of the standard model of electroweak interactions with minimal extension of including neutrino masses and mixing. Several concluding remarks are in order: Ž1. We note that our ‘‘no-go theorem’’ does not come out as a consequence of general principles, such as the CPT invariance in the case of vacuum neutrino oscillation, but results due to the particular way that the matter effect comes in into the problem. Nonetheless, it appears to be the first general statement on how the CP violating phase enters into the neutrino oscillation probabilities in matter. Ž2. Unfortunately, our result implies that it is practically impossible to detect the effect of CP violating phase in solar neutrino measurements even with highest attainable accuracies. 4 Ž3. Our ‘‘no-go theorem’’ and hence the similar conclusion does apply to long-baseline electron Žanti-. neutrino disappearance experiments. Muon Žanti-. neutrino disappearance can be a hope. Ž4. We emphasize that our result does not mean all negative; it implies that super high-statistics solar neutrino experiments in the future will provide a clean laboratory for precise determination of the lepton mixing angles without any uncertainties due to the leptonic Kobayashi–Maskawa phase. Ž5. It is interesting to examine how our ‘‘theorem’’ would be invalidated by introduction of new neutrino properties which may exist in less conservative extension of the standard model.
4 Of course, even if the effect turns out to be reasonably large, we would have to worry about next on how to extract the CP violating effects from the solar neutrino data in which no information of anti-neutrino oscillation probability is involved.
Acknowledgements The research of HM is partly supported by the Grant-in-Aid for Scientific Research in Priority Areas No. 11127213, and by the Grant-in-Aid for International Scientific Research No. 09045036, Inter-University Cooperative Research, Ministry of Education, Science, Sports and Culture.
References w1x T. Kajita, Talk at 18th International Conference on Neutrino Physics and Astrophysics ŽNeutrino’98., June 4–9, 1998, Takayama, Japan, hep-exr9810001, to appear in Proceedings; Y. Fukuda et al., SuperKamiokande Collaboration, Phys. Rev. Lett. 81 Ž1998. 1562. w2x N. Cabbibo, Phys. Rev. Lett. 10 Ž1963. 531; M. Kobayashi, T. Maskawa, Prog. Theor. Phys. 49 Ž1973. 652. w3x P. Ramond, Opening talk at 18th International Conference on Neutrino Physics and Astrophysics ŽNeutrino’98., June 4–9, 1998, Takayama, Japan, hep-phr9809401, to appear in Proceedings. w4x Z. Maki, M. Nakagawa, S. Sakata, Prog. Theor. Phys. 28 Ž1962. 247. w5x M. Fukugita, T. Yanagida, Phys. Lett. B 174 Ž1986. 45. w6x N. Cabbibo, Phys. Lett. B 72 Ž1978. 333; V. Barger, K. Wisnant, R.J.N. Phillips, Phys. Rev. Lett. 45 Ž1980. 2084; S. Pakvasa, in: L. Durand, L.G. Pondrom ŽEds.., Proc. XXth International Conference on High Energy Physics, AIP Conf. Proc. No. 68, vol. 2, AIP, New York, 1981, p. 1164. w7x J. Arafune, J. Sato, Phys. Rev. D 55 Ž1997. 1653; J. Arafune, M. Koike, J. Sato, Phys. Rev. D 56 Ž1997. 3093. w8x H. Minakata, H. Nunokawa, Phys. Lett. B 413 Ž1997. 369; Phys. Rev. D 57 Ž1998. 4403. w9x M. Tanimoto, Phys. Rev. D 55 Ž1997. 322; Prog. Theor. Phys. 97 Ž1997. 901. w10x S.M. Bilenky, C. Giunti, W. Grimus, Phys. Rev. D 58 Ž1998. 033001. w11x A. De Rujula, M.B. Gavela, P. Hernandez, Nucl. Phys. B 547 Ž1999. 21. w12x C. Jarlskog, Phys. Rev. Lett. 55 Ž1985. 1039. w13x L. Wolfenstein, Phys. Rev. D 17 Ž1978. 2369. w14x L.-L. Chau, W.-Y. Keung, Phys. Rev. Lett. 53 Ž1984. 1802; T.K. Kuo, J. Pantaleone, Phys. Lett. B 198 Ž1987. 406; S. Toshev, Phys. Lett. B 226 Ž1989. 335. w15x T.K. Kuo, J. Pantaleone, Phys. Rev. D 35 Ž1987. 3432. w16x F.J. Botella, C.-S. Lim, W.J. Marciano, Phys. Rev. D 35 Ž1987. 896. w17x S. Watanabe, H. Minakata, in preparation.
Solar neutrinos and leptonic CP violation Hisakazu Minakata b
a,b,1
, Shinji Watanabe
c,2
a Department of Physics, Tokyo Metropolitan UniÕersity, 1-1 Minami-Osawa, Hachioji, Tokyo 192-0397, Japan Research Center for Cosmic Neutrinos, Institute for Cosmic Ray Research, UniÕersity of Tokyo, Tanashi, Tokyo 188-8502, Japan c Department of Physics, Waseda UniÕersity, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan
Received 6 July 1999; received in revised form 20 August 1999; accepted 18 October 1999 Editor: H. Georgi
Abstract We examine the possibility of detecting effects of leptonic CP violation by precise measurement of the solar neutrinos within the framework of standard electroweak theory minimally extended to include neutrino masses and mixing. We prove a ‘‘no-go theorem’’ which states that effects of CP violating phase disappear in the ne survival probability to the leading order in electroweak interaction. The effects due to the next-to-leading order correction is estimated to be extremely small, effectively closing the door to the possibility we intended to pursue. q 1999 Published by Elsevier Science B.V. All rights reserved.
Exciting discovery of neutrino oscillation in atmospheric neutrino observation w1x strongly suggests that neutrinos are massive. Then, it is likely that the leptonic CKM w2x matrix, which is now called w3x as the Maki–Nakagawa–Sakata ŽMNS. w4x matrix, exists so that the nature admits CP violation in the lepton sector. The CP violation in the lepton sector has been the topics of interests because of various reasons; on the one hand it may inherit a key to understanding the lepton-quark correspondence, and on the other hand it may provide us with an intriguing mechanism for generating baryon asymmetry in our universe w5x. It is then important to explore the methods for measuring leptonic CP violating phase. Neutrino os-
1 2
E-mail: [email protected] E-mail: [email protected]
cillation has been known to be one of the promising ways of measuring effects of CP violation w6x. Its effect may be detectable if the flavor mixing angles are not small. Recently, detailed investigations have been done to investigate how to measure CP violation in long-baseline neutrino oscillation experiments, with particular emphasis on how to discriminate matter effect contamination from the genuine effect of the Kobayashi–Maskawa phase w7–11x. We consider in this paper an alternative possibility of measuring CP violation by precise measurement of the solar neutrino flux, assuming that neutrinos interact as dictated by the standard electroweak theory. Unfortunately, we will end up with the ‘‘nogo theorem’’. That is, we will show that the effects of CP violating phase vanish in any observables in solar neutrino experiments to leading order in the electroweak interaction. While the ‘‘theorem’’ is not quite a theorem because its validity is limited to
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 1 2 2 4 - 1
leading order, it effectively closes the door to any practical attempts to measure CP violating effects in solar neutrino experiments because the higher order effects are so small. Or, in other word, if a single effect of CP violating phase is detected in a solar neutrino experiment, then it would imply the existence of the neutrino interactions beyond that of the standard model. The goal of our argument in the first half of this paper is to show that the effect of CP-violating phase vanishes to first order in electroweak interactions in the survival probability P Ž ne ™ ne .. Notice that the solar neutrino observation is inherently a disappearance experiment; there is no way to detect appearance events in charged current interactions because the energy is well below the thresholds, and the neutral current interaction cannot distinguish between nm and nt . Toward the goal let us start by recalling the well known results for vacuum neutrino oscillation. There exists a general theorem that states that the measure for CP violation vanishes in the disappearance probability, i.e. P Ž na ™ na . y P Ž na ™ na . s 0
Ž 1.
in vacuum. Hereafter, the Greek indices label the flavor eigenstates; a s e, m ,t for three-generations. This comes from the CPT invariance which implies that P Ž na ™ nb . s P Ž nb ™ na .. The lepton flavor mixing is described by the mixing matrix Ua i that relates the flavor eigenstate na and mass eigenstate n i as na s Ua i n i , where i is the index for specifying mass eigenstates. In the framework of three flavor mixing the feature implied by the general theorem has a simple realization; the effect of CP violation shows up in the vacuum neutrino oscillation probabilities in the particular way as P Ž nb ™ na . y P Ž nb ™ na . s 4 Jba sin
ž
D m2 L 2E
/
,
the unique fermion rephasing invariant measure for CP violation. In fact, the CP-violating piece in P Ž nb ™ na . is identical with that of P Ž nb ™ na . apart from the sign, and is given by a half of the righthand-side of Ž2.. By unitarity the disappearance probability can be written in terms of the appearance probabilities as 1 y P Ž ne ™ ne . s P Ž ne ™ nm . q P Ž ne ™ nt .. The CP-violating pieces in the right-hand-side of this equation cancel owing to the cyclic property of Jba . This is all about how the CP violating effect vanishes in the survival probabilities P Ž na ™ na . in vacuum neutrino oscillation. The situation changes completely when the matter effect w13x is taken into account. Since the matter is in general neither CPT nor CP symmetric there is no general argument which enforces that CP violation effect disappears in neutrino oscillation. Moreover, the CP violation which shows up in neutrino conversion probabilities contains both the fake matter and the genuine effects due to the Kobayashi–Maskawa phase. Therefore, it is difficult to make general statement on how CP violating effects come in. To attack the problem we write down the evolution equation of three flavor neutrinos in matter which is valid to leading order in electroweak interaction:
°
m12r2 E ne nm s U 0 i dx nt 0 d
~
¢
aŽ x . q 0 0
0 0 0
0
0
m22r2 E
0
0
m23r2 E
0 0 0
Uq
¶n •n
ß
e
m
,
Ž 4.
nt
where aŽ x . s '2 G F Ne Ž x . indicates the index of refraction with G F and Ne Ž x . being the Fermi constant and the electron number density, respectively. 3
Ž 2. for a / b and Jba stands for the leptonic Jarlskog factor w12x, Jba s Im Ua 1Ua)2 Ub)1 Ub 2 ,
Ž 3.
3 We note, in passing, that the extra phases which appear for Majorana neutrinos do not give any effects in neutrino evolution in vacuum and in matter because they are multiplied from right to U and therefore drop out in Ž4..
We take a particular parametrization of the mixing matrix w14x. U s e i l 7 u 23 Gd e i l 5 u 13 e i l 2 u 12 ,
Ž 5.
where l i are SUŽ3. Gell–Mann’s matrix and G contains the CP violating phase 1 Gd s 0 0
0 1 0
where T is defined in Ž7. and its explicit form in our parametrization Ž5. of the mixing matrix reads
0 0 . eid
Ž 6.
We rewrite the evolution Eq. Ž4. in terms of the new basis defined w15x by
n˜a s w eyi l 5 u 13 G y1 eyi l 7 u 23 x ab nb ' Ž T t . a b nb .
c13
0
T s ys23 s13 e
id
yc23 s13 e
id
c 23 ys23
s13 s23 c13 e i d . c 23 c13 e
Ž 10 .
id
It is then evident that the ne survival amplitude ² ne N ne :, and hence the probability, does not contain the CP violating phase: 2 ² 2 ² ² ne N ne : s c13 n˜e N n˜e : q s13 n˜t N n˜t :
Ž 7. q c13 s13 ² n˜e N n˜t : q ² n˜t N n˜e : .
ž
Ž 11 .
/
It reads
°
n˜e m12 d 1 il u n˜m s i e 2 12 0 dx 2E n˜t 0
~
¢
2 c13 qa Ž x . 0 c13 s13
0
0
m22
0
0
m 23
0 0 0
c13 s13 0 2 s13
eyi l 2 u 12
¶ •
ß
n˜e n˜m . n˜t
Ž 8. The CP phase d disappears from the equation. It is due to the specific way that the matter effect comes in; aŽ x . only appears in Ž1.1. element in the Hamiltonian matrix and therefore the matter matrix diag Ž a,0,0. is invariant under rotation in 2 y 3 space by e i l1 u 23 . Then the rotation by the phase matrix G does nothing. It is clear from Ž8. that any transition amplitudes computed with n˜a basis is independent of the CP violating phase. Of course, it does not immediately imply that the CP violating phase d disappears in the physical transition amplitude ² nb N na :. The latter is related with the transition amplitude defined with n˜a basis as ² nb N na : s Tag Tbd) ² n˜d N n˜g : ,
Ž 9.
We have checked that this conclusion is not specific to the particular parametrization of the MNS matrix, as it should not. Notice that the same statement does not apply to the nm disappearance amplitude: ² nm N nm : 2 2 ² 2 ² 2 2 ² s s23 s13 n˜e N n˜e : q c 23 n˜m N n˜m : q s23 c13 n˜t N n˜t :
y c13 s23 s13 e i d ² n˜m N n˜e : q eyi d ² n˜e N n˜m :
/
q c 23 s13 c13 e i d ² n˜m N n˜t : q eyi d ² n˜t N n˜m :
/
ž
ž
2 y s23 c13 s13 ² n˜e N n˜t : q ² n˜t N n˜e : .
ž
/
Ž 12 .
One clearly sees the d dependence in the survival amplitude of m neutrinos in Ž12.. Notices also that it’s dependence is not necessarily cos d because ² n˜b Ž x . N n˜a Ž 0 . : / ² n˜a Ž x . N n˜b Ž 0 . :
Ž 13 .
owing to the fact that aŽ0. / aŽ x . in general. If Ž13. holds there exists a CP-odd observable P Ž nm ™ nm .
y P Ž nm ™ nm . which is nonvanishing due to the CPviolating phase. We now turn to the problem of CP violating effect due to the next-to-leading order correction of the standard electroweak interaction. It was noticed by Botella, Lim, and Marciano w16x that it gives rise to a small splitting between the indices of refraction of muon and tau neutrinos; the matter effect matrix in Ž4., diag.Ža,0,0., becomes diag.Ža,0,b. in an appropriate phase convention of the neutrino wave function. With a suitable redefinition of the Fermi constant G F aŽ x . is still given by the same form, aŽ x . s '2 G F Ne Ž x ., and the ratio bra is computed to be w16x bŽ x . aŽ x .
3a sy
mt
2p sin2u W
2
ž / mW
2ln
mt
q
mW
4 P Ž ne ™ ne . s c13 Ž cv2 c122 q sv2 s122 . q s134
4E q
Ž 14 .
for an isoscalar medium where mt and mW denote the masses of tau lepton and W boson, respectively. With the next-to-leading order effect the Hamiltonian matrix in the evolution Eq. Ž8. in n˜ basis has the additional term
b Ž x . c 23 s23 s13 e i d 2 y c 23 c13 s13
yi d
c 23 s23 s13 e 2 s23
y c 23 s23 c13 eyi d
Ž a y bc232 . c132 s132
2 = c13 Ž cv2 c122 q sv2 s122 . y s132
4E q
DM2
2 bc13 Ž c122 y s122 .
2 =cv sv c 23 s23 c13 s13 cos d ,
sin2 v s
Bs
Ds 2 2 c 23 s13
DM2
Ž 16 .
2 where D M 2 s D m223 , D m13 as defined above. The angle v in Ž16. is the value of an angle at the solar core;
5 6
, 5.02 = 10y5
generation neutrinos. We give here only the result, leaving the detail to Ref. w17x.
B
(B q 2
2 D m12
2E 2 D m12
2 y c 23 c13 s13
2E
1 4
D2
,
c12 s12 q bc23 s23 s13 eyi d ,
Ž c122 y s122 . y ac132 y b Ž c232 s132 y s232 . , Ž 17 .
y c 23 s23 c13 e i d . 2 2 c 23 c13
Ž 15 . Clearly the n˜ evolution is affected by the CP violating phase. Its effect is however small, giving rise to the small correction term in the flavor conversion probability which is of the order of bra , 5 = 10y5 . If the neutrinos have hierarchy in their masses in 2 such a way that the D m12 relevant for the solar neutrino conversion is much smaller than that of the 2 atmospheric neutrino anomaly, i.e., D m13 , D m223 ' 2 2 D M 4 D m12 then CP violation due to higher order effects are suppressed even further. To give a feeling we compute the next-to-leading order correction to the ne survival probability under the adiabatic approximation. We also restrict ourselves to the leading order perturbative correction of the effects of third
which parametrizes matter mass eigenstate in 2 = 2 submatrix. Notice that the CP violating effect comes out in a form very reminiscent of the Jarlskog factor, 2 s13 sin d , apart from modification J s c12 s12 c 23 s23 c13 by the matter effect. As clearly seen in Ž16., the CP violation effect is suppressed, in addition to the possible smallness of angle factors, by two small ratios, b DM2
;
2 b D m12
a DM2
, 5 = Ž 10y8 y 10y7 . ,
Ž 18 .
2E 2 assuming that D m12 s 10y6 y 10y5 eV 2 and D M 2 y3 2 s 10 eV . We argue that the double suppression of the CP phase effect is not the artifact of the adiabatic ap-
proximation. Since b carries the dimension of mass, it must be compensated by D m2rE. The D m2 has to be D M 2 because the CP violation must vanish in an effective two-flavor limit D M 2 ™ `. By the similar argument one can show that the CP violating term in P Ž nm ™ nm . obtains a mild suppression factor ; aErD M 2 , 0.1 in the earth crust. In this paper we have proven that the effect of CP violating phase disappears from the survival probability P Ž ne ™ ne . within the framework of the standard model of electroweak interactions with minimal extension of including neutrino masses and mixing. Several concluding remarks are in order: Ž1. We note that our ‘‘no-go theorem’’ does not come out as a consequence of general principles, such as the CPT invariance in the case of vacuum neutrino oscillation, but results due to the particular way that the matter effect comes in into the problem. Nonetheless, it appears to be the first general statement on how the CP violating phase enters into the neutrino oscillation probabilities in matter. Ž2. Unfortunately, our result implies that it is practically impossible to detect the effect of CP violating phase in solar neutrino measurements even with highest attainable accuracies. 4 Ž3. Our ‘‘no-go theorem’’ and hence the similar conclusion does apply to long-baseline electron Žanti-. neutrino disappearance experiments. Muon Žanti-. neutrino disappearance can be a hope. Ž4. We emphasize that our result does not mean all negative; it implies that super high-statistics solar neutrino experiments in the future will provide a clean laboratory for precise determination of the lepton mixing angles without any uncertainties due to the leptonic Kobayashi–Maskawa phase. Ž5. It is interesting to examine how our ‘‘theorem’’ would be invalidated by introduction of new neutrino properties which may exist in less conservative extension of the standard model.
4 Of course, even if the effect turns out to be reasonably large, we would have to worry about next on how to extract the CP violating effects from the solar neutrino data in which no information of anti-neutrino oscillation probability is involved.
Acknowledgements The research of HM is partly supported by the Grant-in-Aid for Scientific Research in Priority Areas No. 11127213, and by the Grant-in-Aid for International Scientific Research No. 09045036, Inter-University Cooperative Research, Ministry of Education, Science, Sports and Culture.
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