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Solar neutrinos and gravity
UCLEAR PHYSICS
PROCEEDINGS SUPPLEMENTS ELSEVIER
Nuclear Physics B (Proc. Suppl.) 100 (2001) 65-67
www.elsevier.nl/locateJnpe
Solar neutrinos and gravity T.K. K u o Physics D e p a r t m e n t , P u r d u e University, W. Lafayette, IN 47907, U S A We review t h e possibility t h a t t h e solar n e u t r i n o p r o b l e m can be explained by n e u t r i n o s violating t h e equivalence principle. It is found t h a t such a scenario can be ruled o u t w h e n one takes into a c c o u n t d a t a from high energy accelerator n e u t r i n o e x p e r i m e n t s . Neutrino oscillations have been invoked as a very likely cause for the observed deficit of solar neutrinos. Usually, the mismatch between mass and weak interaction eigenstates is used to analyze oscillation effects. However, there are other, albeit less likely mechanisms which can also lead to oscillations. In this talk I will give a brief review of the status of using violation of equivalence principle (VEP) as such a possible mechanism. [1,2] When a neutrino propagates in a gravitational field, in the presence of a small and constant potential (¢), the energym o m e n t u m relation is approximately given by
summarized [3] in the equation:
z~-~
vg
+5
(vj)
Ej = (1 -
27j¢)P.
(i)
In general relativity, 7j = 1. However, if there is a violation of the equivalence principle, then one can entertain the possibility that 7i # 73,
i # j.
(2)
This means that for two neutrinos, say vl and Vl, after they propagate in the potential over a distance L, a phase difference is developed Am = 2ATCEL.
(3)
This then gives rise to neutrino oscillations from VEP. The formalism for a quantitative analysis is similar to that of the usual neutrino oscillation problem. For two flavors, it can be
+E[¢[A7
cos 0
= Jt~
sin
20M
1
0
-e(0
(-cos2OG sin20v
sin cos
20M ) 20M
sin20G ) ( r e ) cos20c
}
u~
"
(4) Here, besides the familiar terms in the usual MSW analyses, the VEP term is included. 0a represents the mismatch between weak and gravitational eigenstates. From this equation we can immediately extract some salient features of V E P effects. . Neutrino oscillation can occur even if AM 2 = 0. In fact, for simplicity, one usually concentrates on V E P effects by assuming A M 2 = 0. On the other hand, matter effects, which are important in the usual analysis with A M 2 # 0, will be kept and combined with the VEP contribution. . The physics of V E P oscillation is entirely similar to ordinary oscillation phenomena. The equivalence of the two is established by the substitution
Am2/4E ~
0920-5632/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII S0920-5632(01 )01412-8
EI¢IAT.
(5)
T.K. Kuo/Nuclear Physics B (Proc. Suppl.) 100 (2001) 65-67
66
For a given E (or a narrow range of E), one can infer the relevant parameter I¢1A7 from the corresponding information on A m 2. .
Note the different E-dependence in Arn2/4E versus EI¢IA'~. If an experiment covers a wide range of energies, the E-dependence can distinguish oscillation effects due to the mass or VEP.
4. The parameter which enters into the problem is the product [ClAy. To obtain any information about Ay, which measures the genuine V E P effect, we must know 1¢1. Unfortunately, there is considerable uncertainties here. Estimates range from 6 x 10 -1° (earth on earth) to 3 x 10 -5 (super cluster on solar system). A common practice is to use I¢1 ~ 1 0 - 5 " We turn now to a detailed analysis of VEP effects in the solar neutrino problem. We will concentrate on the MSW-like analysis, [3,4,5] substituting the usual A m 2 effect by that of VEP. We will not consider the "just-so" solution [6] with V E P effects. Just like the familiar plots of allowed regions in A m 2 vs sin 2 20 M for the solar neutrino analysis, the existent d a t a constrain the parameters 1¢IA and sin 2 20G to similarly shaped regions. The result [5] using av£ilable data, is presented in Fig. 1. We can understand qualitatively the main features of this plot. For solar neutrinos, the energy is roughly in the range E -,~ 1 - 10MeV. Also, the MSW effect in the usual analysis gives the relevant range of Arn 2 ~ 1 0 - 4 - - 10-7eV 2. According to Eq.(5), this translates into a range I¢1A3, --~ 10-118_ 10-2t, as shown in the figure. Also, the shape of the allowed regions correspond to the usual MSW regions with the proper substitution of 1¢IA~/for A m 2. The E-dependence of VEP effects favors high energy neutrinos. This means that accelerator neutrino experiments can put strong constraints on the VEP parameters. In fact, for the C C F R experiment, the neutrino energy is in the range E ,,~ 30-600GeV,
while the neutrino path length is L ,-, lkrn. This is equivalent to the solar neutrino problem with Arn '~ <_ leV 2. Or, in the V E P sector, I¢IA~/ _< 10 -2a. Since no oscillators were observed, this sets a very strong bound, I¢]A'~ > 10 -2a, A detailed analysis, including three flavor effects, gives a similar conclusion and is depicted by the dotted line in Fig. 1. Thus, when accelerator neutrino d a t a are taken into account, the parameters allowed by the solar neutrino d a t a are ruled out at a high level of confidence level. In summary, while the V E P hypothesis can be made to be compatible with the solar neutrino data, the same is not true if one includes the accelerator results. It should be mentioned that. atmospheric neutrino d a t a [8] also exclude the V E P possibility strongly.
i0.I~0"4
10-s
i0-2
lO-X
10-16
i0-16
lO-i~
i0-~ i0-18
lO-le lO-t9
ilO -~9
10-2o
10-2o
I0-21
Ii0-21 10-22
i0-22 i0-23
.............. .,~..~.~.~
.................... , ........
I0-~3
i0-24 i0-3
10-2
lO-t
1
i0-24 i0-25
sin 2 (2¢)
F I G 1. P a r a m e t e r region allowed by solar neutrino observations at 90% C.L. (solid curve). Also shown is the region excluded by a three-neutrino analysis of the C C F R accelerator experiment at 90% C.L. (dotted curve). The accelerator d a t a exclude the VEP explanation of the solar neutrino observations. REFERENCES
1. M. Gasperini, Phys. Rev. D 38, 2635 (1988); 39, 3606 (1989).
TK. Kuo /Nuclear Physics B (Proc. Suppl.) 100 (2001) 65-67
2. A. Halprin and C.N. Leung, Phys. Rev. Lett. 67, 1833 (1991). 3. A. Halprin, C.N. Leung and J. Pantaleone, Phys. Rev. D53, 5365 (1996); J. Bahcall, P. Kraztev and C.N. Leung, ibid 52, 1770 (1995). 4. S. Mansour and T.K. Kuo, Phys. Rev. D60, 097301 (1999). 5. J. Pantaleone, S. Mansour and T.K. Kuo, Phys. Rev. D61, 033011 (2000). 6. H. Nunokawa, these proceedings. 7. D. Naples et al., Phys. 031101 (1999).
Rev.
D59,
8. P. Lipari and M. Lusignoli, Phys. Rev. D60, 013003 (1999); G. Gofli, E. Lisi, A. Marrane and G. Scioscia, ibid, 60, 053006 (1999).
67
PROCEEDINGS SUPPLEMENTS ELSEVIER
Nuclear Physics B (Proc. Suppl.) 100 (2001) 65-67
www.elsevier.nl/locateJnpe
Solar neutrinos and gravity T.K. K u o Physics D e p a r t m e n t , P u r d u e University, W. Lafayette, IN 47907, U S A We review t h e possibility t h a t t h e solar n e u t r i n o p r o b l e m can be explained by n e u t r i n o s violating t h e equivalence principle. It is found t h a t such a scenario can be ruled o u t w h e n one takes into a c c o u n t d a t a from high energy accelerator n e u t r i n o e x p e r i m e n t s . Neutrino oscillations have been invoked as a very likely cause for the observed deficit of solar neutrinos. Usually, the mismatch between mass and weak interaction eigenstates is used to analyze oscillation effects. However, there are other, albeit less likely mechanisms which can also lead to oscillations. In this talk I will give a brief review of the status of using violation of equivalence principle (VEP) as such a possible mechanism. [1,2] When a neutrino propagates in a gravitational field, in the presence of a small and constant potential (¢), the energym o m e n t u m relation is approximately given by
summarized [3] in the equation:
z~-~
vg
+5
(vj)
Ej = (1 -
27j¢)P.
(i)
In general relativity, 7j = 1. However, if there is a violation of the equivalence principle, then one can entertain the possibility that 7i # 73,
i # j.
(2)
This means that for two neutrinos, say vl and Vl, after they propagate in the potential over a distance L, a phase difference is developed Am = 2ATCEL.
(3)
This then gives rise to neutrino oscillations from VEP. The formalism for a quantitative analysis is similar to that of the usual neutrino oscillation problem. For two flavors, it can be
+E[¢[A7
cos 0
= Jt~
sin
20M
1
0
-e(0
(-cos2OG sin20v
sin cos
20M ) 20M
sin20G ) ( r e ) cos20c
}
u~
"
(4) Here, besides the familiar terms in the usual MSW analyses, the VEP term is included. 0a represents the mismatch between weak and gravitational eigenstates. From this equation we can immediately extract some salient features of V E P effects. . Neutrino oscillation can occur even if AM 2 = 0. In fact, for simplicity, one usually concentrates on V E P effects by assuming A M 2 = 0. On the other hand, matter effects, which are important in the usual analysis with A M 2 # 0, will be kept and combined with the VEP contribution. . The physics of V E P oscillation is entirely similar to ordinary oscillation phenomena. The equivalence of the two is established by the substitution
Am2/4E ~
0920-5632/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII S0920-5632(01 )01412-8
EI¢IAT.
(5)
T.K. Kuo/Nuclear Physics B (Proc. Suppl.) 100 (2001) 65-67
66
For a given E (or a narrow range of E), one can infer the relevant parameter I¢1A7 from the corresponding information on A m 2. .
Note the different E-dependence in Arn2/4E versus EI¢IA'~. If an experiment covers a wide range of energies, the E-dependence can distinguish oscillation effects due to the mass or VEP.
4. The parameter which enters into the problem is the product [ClAy. To obtain any information about Ay, which measures the genuine V E P effect, we must know 1¢1. Unfortunately, there is considerable uncertainties here. Estimates range from 6 x 10 -1° (earth on earth) to 3 x 10 -5 (super cluster on solar system). A common practice is to use I¢1 ~ 1 0 - 5 " We turn now to a detailed analysis of VEP effects in the solar neutrino problem. We will concentrate on the MSW-like analysis, [3,4,5] substituting the usual A m 2 effect by that of VEP. We will not consider the "just-so" solution [6] with V E P effects. Just like the familiar plots of allowed regions in A m 2 vs sin 2 20 M for the solar neutrino analysis, the existent d a t a constrain the parameters 1¢IA and sin 2 20G to similarly shaped regions. The result [5] using av£ilable data, is presented in Fig. 1. We can understand qualitatively the main features of this plot. For solar neutrinos, the energy is roughly in the range E -,~ 1 - 10MeV. Also, the MSW effect in the usual analysis gives the relevant range of Arn 2 ~ 1 0 - 4 - - 10-7eV 2. According to Eq.(5), this translates into a range I¢1A3, --~ 10-118_ 10-2t, as shown in the figure. Also, the shape of the allowed regions correspond to the usual MSW regions with the proper substitution of 1¢IA~/for A m 2. The E-dependence of VEP effects favors high energy neutrinos. This means that accelerator neutrino experiments can put strong constraints on the VEP parameters. In fact, for the C C F R experiment, the neutrino energy is in the range E ,,~ 30-600GeV,
while the neutrino path length is L ,-, lkrn. This is equivalent to the solar neutrino problem with Arn '~ <_ leV 2. Or, in the V E P sector, I¢IA~/ _< 10 -2a. Since no oscillators were observed, this sets a very strong bound, I¢]A'~ > 10 -2a, A detailed analysis, including three flavor effects, gives a similar conclusion and is depicted by the dotted line in Fig. 1. Thus, when accelerator neutrino d a t a are taken into account, the parameters allowed by the solar neutrino d a t a are ruled out at a high level of confidence level. In summary, while the V E P hypothesis can be made to be compatible with the solar neutrino data, the same is not true if one includes the accelerator results. It should be mentioned that. atmospheric neutrino d a t a [8] also exclude the V E P possibility strongly.
i0.I~0"4
10-s
i0-2
lO-X
10-16
i0-16
lO-i~
i0-~ i0-18
lO-le lO-t9
ilO -~9
10-2o
10-2o
I0-21
Ii0-21 10-22
i0-22 i0-23
.............. .,~..~.~.~
.................... , ........
I0-~3
i0-24 i0-3
10-2
lO-t
1
i0-24 i0-25
sin 2 (2¢)
F I G 1. P a r a m e t e r region allowed by solar neutrino observations at 90% C.L. (solid curve). Also shown is the region excluded by a three-neutrino analysis of the C C F R accelerator experiment at 90% C.L. (dotted curve). The accelerator d a t a exclude the VEP explanation of the solar neutrino observations. REFERENCES
1. M. Gasperini, Phys. Rev. D 38, 2635 (1988); 39, 3606 (1989).
TK. Kuo /Nuclear Physics B (Proc. Suppl.) 100 (2001) 65-67
2. A. Halprin and C.N. Leung, Phys. Rev. Lett. 67, 1833 (1991). 3. A. Halprin, C.N. Leung and J. Pantaleone, Phys. Rev. D53, 5365 (1996); J. Bahcall, P. Kraztev and C.N. Leung, ibid 52, 1770 (1995). 4. S. Mansour and T.K. Kuo, Phys. Rev. D60, 097301 (1999). 5. J. Pantaleone, S. Mansour and T.K. Kuo, Phys. Rev. D61, 033011 (2000). 6. H. Nunokawa, these proceedings. 7. D. Naples et al., Phys. 031101 (1999).
Rev.
D59,
8. P. Lipari and M. Lusignoli, Phys. Rev. D60, 013003 (1999); G. Gofli, E. Lisi, A. Marrane and G. Scioscia, ibid, 60, 053006 (1999).
67