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On the question of neutrino spin precession in a magnetic field
Physics Letters B 294 (1992) 243-247 North-Holland
PHYSICS LETTERS B
On the question of neutrino spin precession in a magnetic field M . M . G u z z o a,b a n d J. BeUandi a " Departamento de Raios Cbsmicos e Cronologia, IFGW,, Unicamp, 13081 Campinas, SP, Brazil b Instituto de Fisica Te6rica, UNESP, Rua Pamplona 145, 01405 S~o Paulo, SP, Brazil
Received 19 May 1992
Using the Feynman procedure of ordered exponential operators we solve the evolution equations for a two-neutrino system considering arbitrarily varying matter density and magnetic field along the neutrino trajectory. We show that a large geometrical phase velocity suppresses PL--d"PRtransitions unless some stationary trajectory is found along the neutrino path. Concerning the solar neutrino case, if we admit the standard solar model matter distribution, no such trajectory can be found.
1. Neutrino left-right transitions have been evoked as a possible mechanism to explain the observation o f a smaller-than-expected solar neutrino flux [ 1 ]. This possibility relies on the fact that existing solar neutrino detectors are sensible only to left-handed neutrinos and, therefore, transitions from left-handed to right-handed neutrinos imply a reduction of the measured solar neutrino flux. A possible way of generating left-right transitions arises when we assume that neutrinos have a nonvanishing magnetic m o m e n t and interact with a magnetic field. In particular, solar neutrinos could interact with the magnetic field o f the Sun [ 2 ]. To be effective for the solar neutrino data, such a mechanism requires - over a region as large as the convective zone o f the Sun, L - 1 0 ~° c m - a strong magnetic field, B ~ l0 3 G, if we consider a value for the neutrino magnetic m o m e n t close to the present experimental b o u n d s , / ~ < 10-1o #a (/~a is the Bohr magneton). Some remarks have been made concerning this possibility of generating neutrino left-right transitions. It was proposed by Vidal and Wudka [ 3 ] that a fast rotating magnetic field in a transverse plane (with respect to the neutrino path) could produce a sizeable reduction o f the left-handed solar neutrino flux even if#~ ~- 10-13 #B. A rotation o f the magnetic field introduces a geometrical or topological phase [ 4 ], determined by the phase velocity o f the rotating ~r Financial support from CNPq, Brazil.
field. This geometrical phase is to be compared with the usual dynamical one,/~B. Nevertheless, Smirnov showed in ref. [ 5 ] that this result does not apply for the case where the intensity of the magnetic field as well as the density o f matter interacting with the neutrinos are constants. Here we want to give a more general analysis of the problem o f neutrino left-right transitions. Employing the Feynman procedure o f ordered exponential operators [6,7] we solve a system of two coupled evolution equations for the neutrinos, taking into account the general case where the magnetic field B (t) a n d / o r the effective density of matter that interacts with neutrinos n e f f ( t ) vary in an arbitrary way along the neutrino trajectory. Without assuming any particular functional form for the parameters B ( t ) and n elf(t), we come to the following conclusions. The effect of neutrino interactions with matter in the presence o f a magnetic field (when the neutrino magnetic m o m e n t is non-vanishing) is just that neutrinos will effectively interact with a new magnetic field which coincides with the old one rotated by an angle proportional to the coupling of neutrinos with matter. Furthermore, the rotation o f the magnetic field in a transverse plane in the relevant case pointed out by Vidal and Wudka [3], i.e., the case where the geometrical phase is much larger than the dynamical one, does not lead to an appreciable neutrino left to fight transition for any behaviour of the magnetic field and of the density o f matter, unless some stationary phase
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
243
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is found along the neutrino path. For the case of solar neutrinos, if we admit the standard solar model matter distribution [ 8], we do not find any stationary trajectory along the neutrino path. Therefore large phase velocity does not provide any relevant contribution to a solution of the solar neutrino problem. 2. Transitions from left/]L to right 1/Rneutrinos are described by a system of two coupled evolution equations which can be written in a very general form as d
Go(t, to) - E x p [
- i A (t, t0)tr3 ]
= e x p [ - iA (t, to)t73]
(5)
with
A(t,
t') = .i a ( t " )
dt" ,
(6)
we can decompose the expansional in eq. (3) obtaining
_%)°,,,,
(1)
where q~(t) is a column matrix • (t) = (UR(t), VL(t)), 27(0 is a 2 X 2 matrix defined by the equation above where a (t) and b (t) are two general time dependent functions. The diagonal elements in eq. ( 1 ) are just the PR and PL level splitting which can receive contributions from the mass difference [AmE/E, where Am2=/T/E(pR)--mE(/JL) and E is the neutrino energy ] as well as from neutrino interactions with matter [v/2 Gvneff(t), where GF is the Fermi constant]. The non-diagonal entries can receive contributions proportional to the coupling of neutrinos with a magnetic field via the neutrino magnetic moment [#~B(t) ]. We want to solve the system of evolution equations ( 1 ) in a time interval to ~
• (t) =G(t, to)~(to) • The 2X2 matrix G(t, to) in eq.
(2) (2) is the time evo-
lution operator
G(t, to)=Exp(-i i Z(t') dt' ) ,
(3)
where Exp indicates an expansional defined as a sum of multiple ordered integrals [ 6,7 ]. The 2 × 2 _r(t) matrix can be expanded in the basis of Pauli matrices. Introducing the lowering and raising operators, a+ and tr_, we can write
+b(t)tr+ +b*(t)a_.
G( t, to)=Go( t, to) ×Ex~-i)dt'Go(t,t') to
× [b(t' )tr+ +b*(t'
) a ]Gr-l(t,
t' ) ) .
(7)
Finally, using the commutation relations for the Pauli matrices, we obtain the very general result
G( t, to)=Go( t, to) X E x p ( - i f dt'
{exp[i.2A(t,t')lb(t')a+
to
+exp[-i.2A(t, t')]
b*(t')a_}).
(8)
The integrand in eq. (8) can be thought as a rotation of the non-diagonal element b ( t ' ) in the complex plane by an angle 2A(t, t' ). If we define a "rotated" element b' (t') =exp[i.2A(t,t' ) ] b(t' ), we can write eq. (8) again as
G( t, to)=Go( t, to) t
to
244
Since Pauli matrices are non-commuting operators, we should be careful when dealing with the expansional form, eq. (3). Defining
t'
i ~ ~(t) =~r(t)~(t)
_r(t) =a(t)cr3
12 November1992
(4)
×Exp(-ifdt'[b'(t')a++b'*(t')a_]).
(9)
to
Eq. (9) [together with eq. (2)] gives all the necessary information on the time evolution of a two-neutrino system described by a general evolution matrix Z'(t), with boundary conditions defined by • (to).
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PHYSICS LETTERSB
3. Let us discuss now some specific examples related to left-right transitions of neutrinos which interact with a magnetic field as well as with matter in a standard way. Assuming a non-vanishing neutrino magnetic moment, the neutrino evolution matrix ( 1 ) can be written, up to an irrelevant overall phase, as
[5] Z(t)=(l[V(t)-O(t)] /t~ IB(t) I
#~lB(t)l
-½[V(t)-O(t)]
)
' (10)
where ~(t) is the phase velocity describing the rotation of the magnetic field B(t)= IB(t) I exp[iq~(t) ] and V(t) is the URand//L level splitting
V( t ) =x/~ GFn elf(t) -- AmE/EE .
( 11 )
For constant parameters n e~, IBI and ~ we have the case analysed by Smirnov [ 5 ] and it is very easy to reproduce his results applying the expansional formalism presented here. In this case, the integration appearing in the definition of G (t, to), eq. (3), is trivial and G(t, to) can be written as a simple exponential
G( t, t o ) = e x p [ - i ( t - t o ) Z ]
information about the behaviour of the solar magnetic field. We will consider the general evolution matrix 27(t), given by eq. (10). We concentrate on the processes where a left-handed neutrino transforms into a righthanded one; these processes are related to the nondiagonal elements of the time evolution operator G (t, to). From the very general form of G(t, to), eq. (8), it is easy to see that it is not necessary to take Go(t, to) into account since it is diagonal and leads to just an overall phase in the relevant transitions. Since neutrinos are relativistic particles, the time dependent parameters n eft(t) and B (t) vary quickly along the neutrino trajectory (note that neff(t) - according to the solar standard model [ 8 ] - passes from values like O(100) NA/Cm 3, where NA is Avogadro's number, in the central regions of the Sun, to zero, on its surface, while B(t) can present strong variations from values like O( 103-5) G to zero in the convective zone due to solar internal activity). Consequently, neutrino spin precession is a rapid transition process. In processes like this, the expansional form appearing in eq. (8) can be approximated by [ 9 ] 1 - i i {b(t') exp[i.2A(t, t' )] tr+
=cos[ ( t - t o ) t o ] - i sin [ ( t - to)O0] 27,
to
(12)
+b*(t') exp[-i.2A(t,t')] a_} d t ' .
(.O
where to=x/(½ V - ½~)2+ (/~ IBI )2 .
(13)
The amplitude of the probability of transitions from left- to right-handed neutrinos is given by the upper non-diagonal element of the matrix G(t, to) and can be written as A (VL""*//R, t) = - i g , [B]
12 November 1992
sin[ ( t - t o ) m ] O)
Again, the VL--'gR transition is given by the upper nondiagonal element of the matrix (15), i.e., terms accompanying a+. This transition can be written as A (/.'L---*UR; t) t
= - i .J b(t') exp[i.2A(t, t') ] d t ' .
(16)
to
(14)
The probability that is derived from this amplitude coincides with the one obtained by Smirnov [ 5 ], and his conclusions straightforwardly apply here. Let us now analyse the more general case where the behaviour of the magnetic field a n d / o r of the density of particles along the neutrino path is not specified. This is an important case for the Sun. Even if theoretical calculations based on the standard solar model [ 8 ] give us some idea about the matter distribution inside the Sun [ 8 ], it is very difficult to obtain some
(15)
The values for b(t' ) and A(t, t' ) in the case we are analysing can be read from eqs. ( 1 ), (6) and ( 10 ):
b(t' )=lt~lB(t' )[
(17)
and
A(t, t ' ) = ½ i [V(t")-O(t")] dt".
(18)
t'
On the basis of the result (15) we shall analyse next some physically interesting limit cases. In order to 245
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PHYSICS LETTERS B
avoid irrelevant complications, we will consider the phase velocity 0 (t) as a constant. Thus
A(t, t')=½[7(t, t')-(t, t')6] ,
(19)
where
7(t, t') = _i V(t") dt" .
(20)
t'
If the absolute value of the magnetic field IB (t) I is a regular function of time, but its phase velocity can be considered extremely large (i.e, in the limit where 6--,oo), the Riemann-Lebesgue lemmas [10] guarantee that the integration appearing in eq. ( 15 ) is such that
12 November 1992
So, in the case where 2=2o, the slope of the function V(2) must present the same sign as 6. Eq. (24) is a generalization of Smirnov's resonance condition which was derived in ref. [5] in a naive way. Note that, differently from what was discussed by Smirnov, to have eq. (24) satisfied, a fixed direction of the magnetic field is not necessary. If there exist a stationary phase, we will find for the left to right amplitude [ 11 ]
A(VL---}VR;t ) " - - i
I
b(2o)
t
Xexp(i f d/t [ V(/t) - 6] + , i ~ ) ,
(26)
Ao
and the probability of a VL--'VRtransition is negligibly small. Note that this result is very general since in the limit case under analysis the time evolution operator given by eq. (8) goes to a diagonal form and all the relevant contributions to a left-right neutrino transition are negligible. If ~ is large (but q~-'~-ov) this integration does not vanish in the case where there exists a stationary phase. In this situation we can employ the stationary phase method [ 11 ] to analyse the behaviour of the following integration:
and the probability for a left-handed neutrino interacting with a magnetic field to be transformed into a ri.ght-handed neutrino will be given by 2zt[b (2o) ]2/ I V(2o) I. Thus, in the case where there exists a stationary phase along the neutrino trajectory, the VL~ VRtransition probability strongly depends on the absolute value of the magnetic field and on the matter distribution along the neutrino path [see eqs. ( 11 ) and (17)]. If the matter distribution allows more than one stationary trajectory, the total VL--'VRamplitude will be given by the sum of the contributions from each of them:
A(PL-'-}IIR; t)
A(PL-'}PR; t) _~
lim A(VL-'-}VR; t ) = lim o ( l ~ = o , ~oo ~oo
(21)
(22)
to
where
7(t,2) 6 - (t-2).
(23)
The condition for having a stationary phase leads us to
~h(t, 20) ~ 0 ~ V(;to)
02
246
(27)
1 0 ~--~ V(2o) < 0 .
Finally, we would like to point out that, for the solar neutrino case, if we admit the matter distribution usually accepted to fit the predictions of the solar standard model [ 8 ], i.e., an approximately monotonically decreasing exponential function (in the radial direction from the center to the surface of the Sun), then it will not be possible to satisfy simultaneously conditions (24) and (25) in the large ~ case we are analysing and there will be no stationary phase in the problem.
(24)
and 02----~h(t, 2o) = -
.
Ao
= --i .i b(2 exp[i6h(t, 2) ] d2,
h(t, 2 ) -
~ A(PL-'-}PR; 2 0 )
(25)
4. Using the Feynman procedure of ordered exponential operators we have solved the evolution equations for a two-neutrino system which interacts with matter and magnetic field without assuming any particular behaviour for the matter density nor for the
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PHYSICS LETTERS B
magnetic field along the neutrino trajectory. From the very general result ofeq. (8), we have concluded that all physical contributions to the non-diagonal entries o f the general neutrino evolution matrix 27(t) will generate an effective rotation o f the magnetic field interacting with the neutrinos. The simplest contributions to these entries come from interactions o f the neutrino with matter as well as from the neutrino mass splitting. Note that, since the absolute value o f the magnetic field does not change in a rotation, the necessary values for the neutrino magnetic m o m e n t and the magnetic field obtained in ref. [ 2 ] for an effective reduction o f the measurable solar neutrino flux, are still the same, independently o f whether the neutrino interacts with matter in the way discussed in the present paper. Furthermore, we have observed that a quick rotating magnetic field B ( t ) (or, equivalently, a large phase velocity ~ for the magnetic field) along the neutrino path, generates negligibly small probabilities of neutrino spin precession, unless some stationary phase can be found. In this case, the PL---~1/R transition probability will strongly depend on the matter distribution along the neutrino path. I f we admit the standard solar model distribution of matter inside the
12 November 1992
Sun, we will not find any stationary phase satisfied in the large ~ case. Therefore, the conclusions presented in ref. [ 3 ] by Vidal and Wudka are not valid not only in the case where the relevant parameters B(t) and neff(t) are constants, as discussed by Smirnov in ref. [ 5 ], but also in the general case where these parameters vary arbitrarily.
References [1 ] R. Davis et al., in: Proc. XXI Intern. Cosmic ray Conf. ( Adelaide, Australia, 1990) Vol. 7, p. 155. [2] M. Volishin, M.I. Vysotsky and L. Okun, Yad. Fiz. 44 (1986) 677 [Soy. J. Nucl. Phys. 44 (1986) 440]; A. Cisneros, Astrophys. Space Sci. 10 ( 1971 ) 87. [3] J. Vidal and J. Wudka, Phys. Lett. B 249 (1990) 473. [4] M. Berry, Proc. R. Soc. A 362 (1984) 45. [5] A.Yu. Smirnov, Phys. Left. B 260 (1991) 161. [6] R.P. Feynman, Phys. Rev. 84 ( 1951 ) 108; I. Fujikawa, Prog. Theor. Phys. 7 (1952) 433. [7] J. Bellandi et al., J. Phys. A 25 (1992) 877. [ 8 ] J. Bahcall and R.K. Ulrich, Rev. Mod. Phys. 60 (1988) 298; S. Turck-Chi~ze et al., Astrophys. J. 335 ( 1988) 425. [9] A. Messiah, Mrcanique Quantique, Vol. II (Dunod, Paris, 1964). [10]E.T. Whittaker and G.N. Watson, A course of modern analysis (Cambridge U.P., Cambridge, 1969). [11] A. Erdrlyi, Asymptotic expansions (Dover, New York, 1956).
247
PHYSICS LETTERS B
On the question of neutrino spin precession in a magnetic field M . M . G u z z o a,b a n d J. BeUandi a " Departamento de Raios Cbsmicos e Cronologia, IFGW,, Unicamp, 13081 Campinas, SP, Brazil b Instituto de Fisica Te6rica, UNESP, Rua Pamplona 145, 01405 S~o Paulo, SP, Brazil
Received 19 May 1992
Using the Feynman procedure of ordered exponential operators we solve the evolution equations for a two-neutrino system considering arbitrarily varying matter density and magnetic field along the neutrino trajectory. We show that a large geometrical phase velocity suppresses PL--d"PRtransitions unless some stationary trajectory is found along the neutrino path. Concerning the solar neutrino case, if we admit the standard solar model matter distribution, no such trajectory can be found.
1. Neutrino left-right transitions have been evoked as a possible mechanism to explain the observation o f a smaller-than-expected solar neutrino flux [ 1 ]. This possibility relies on the fact that existing solar neutrino detectors are sensible only to left-handed neutrinos and, therefore, transitions from left-handed to right-handed neutrinos imply a reduction of the measured solar neutrino flux. A possible way of generating left-right transitions arises when we assume that neutrinos have a nonvanishing magnetic m o m e n t and interact with a magnetic field. In particular, solar neutrinos could interact with the magnetic field o f the Sun [ 2 ]. To be effective for the solar neutrino data, such a mechanism requires - over a region as large as the convective zone o f the Sun, L - 1 0 ~° c m - a strong magnetic field, B ~ l0 3 G, if we consider a value for the neutrino magnetic m o m e n t close to the present experimental b o u n d s , / ~ < 10-1o #a (/~a is the Bohr magneton). Some remarks have been made concerning this possibility of generating neutrino left-right transitions. It was proposed by Vidal and Wudka [ 3 ] that a fast rotating magnetic field in a transverse plane (with respect to the neutrino path) could produce a sizeable reduction o f the left-handed solar neutrino flux even if#~ ~- 10-13 #B. A rotation o f the magnetic field introduces a geometrical or topological phase [ 4 ], determined by the phase velocity o f the rotating ~r Financial support from CNPq, Brazil.
field. This geometrical phase is to be compared with the usual dynamical one,/~B. Nevertheless, Smirnov showed in ref. [ 5 ] that this result does not apply for the case where the intensity of the magnetic field as well as the density o f matter interacting with the neutrinos are constants. Here we want to give a more general analysis of the problem o f neutrino left-right transitions. Employing the Feynman procedure o f ordered exponential operators [6,7] we solve a system of two coupled evolution equations for the neutrinos, taking into account the general case where the magnetic field B (t) a n d / o r the effective density of matter that interacts with neutrinos n e f f ( t ) vary in an arbitrary way along the neutrino trajectory. Without assuming any particular functional form for the parameters B ( t ) and n elf(t), we come to the following conclusions. The effect of neutrino interactions with matter in the presence o f a magnetic field (when the neutrino magnetic m o m e n t is non-vanishing) is just that neutrinos will effectively interact with a new magnetic field which coincides with the old one rotated by an angle proportional to the coupling of neutrinos with matter. Furthermore, the rotation o f the magnetic field in a transverse plane in the relevant case pointed out by Vidal and Wudka [3], i.e., the case where the geometrical phase is much larger than the dynamical one, does not lead to an appreciable neutrino left to fight transition for any behaviour of the magnetic field and of the density o f matter, unless some stationary phase
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
243
Volume 294, number 2
PHYSICS LETTERSB
is found along the neutrino path. For the case of solar neutrinos, if we admit the standard solar model matter distribution [ 8], we do not find any stationary trajectory along the neutrino path. Therefore large phase velocity does not provide any relevant contribution to a solution of the solar neutrino problem. 2. Transitions from left/]L to right 1/Rneutrinos are described by a system of two coupled evolution equations which can be written in a very general form as d
Go(t, to) - E x p [
- i A (t, t0)tr3 ]
= e x p [ - iA (t, to)t73]
(5)
with
A(t,
t') = .i a ( t " )
dt" ,
(6)
we can decompose the expansional in eq. (3) obtaining
_%)°,,,,
(1)
where q~(t) is a column matrix • (t) = (UR(t), VL(t)), 27(0 is a 2 X 2 matrix defined by the equation above where a (t) and b (t) are two general time dependent functions. The diagonal elements in eq. ( 1 ) are just the PR and PL level splitting which can receive contributions from the mass difference [AmE/E, where Am2=/T/E(pR)--mE(/JL) and E is the neutrino energy ] as well as from neutrino interactions with matter [v/2 Gvneff(t), where GF is the Fermi constant]. The non-diagonal entries can receive contributions proportional to the coupling of neutrinos with a magnetic field via the neutrino magnetic moment [#~B(t) ]. We want to solve the system of evolution equations ( 1 ) in a time interval to ~
• (t) =G(t, to)~(to) • The 2X2 matrix G(t, to) in eq.
(2) (2) is the time evo-
lution operator
G(t, to)=Exp(-i i Z(t') dt' ) ,
(3)
where Exp indicates an expansional defined as a sum of multiple ordered integrals [ 6,7 ]. The 2 × 2 _r(t) matrix can be expanded in the basis of Pauli matrices. Introducing the lowering and raising operators, a+ and tr_, we can write
+b(t)tr+ +b*(t)a_.
G( t, to)=Go( t, to) ×Ex~-i)dt'Go(t,t') to
× [b(t' )tr+ +b*(t'
) a ]Gr-l(t,
t' ) ) .
(7)
Finally, using the commutation relations for the Pauli matrices, we obtain the very general result
G( t, to)=Go( t, to) X E x p ( - i f dt'
{exp[i.2A(t,t')lb(t')a+
to
+exp[-i.2A(t, t')]
b*(t')a_}).
(8)
The integrand in eq. (8) can be thought as a rotation of the non-diagonal element b ( t ' ) in the complex plane by an angle 2A(t, t' ). If we define a "rotated" element b' (t') =exp[i.2A(t,t' ) ] b(t' ), we can write eq. (8) again as
G( t, to)=Go( t, to) t
to
244
Since Pauli matrices are non-commuting operators, we should be careful when dealing with the expansional form, eq. (3). Defining
t'
i ~ ~(t) =~r(t)~(t)
_r(t) =a(t)cr3
12 November1992
(4)
×Exp(-ifdt'[b'(t')a++b'*(t')a_]).
(9)
to
Eq. (9) [together with eq. (2)] gives all the necessary information on the time evolution of a two-neutrino system described by a general evolution matrix Z'(t), with boundary conditions defined by • (to).
Volume 294, number 2
PHYSICS LETTERSB
3. Let us discuss now some specific examples related to left-right transitions of neutrinos which interact with a magnetic field as well as with matter in a standard way. Assuming a non-vanishing neutrino magnetic moment, the neutrino evolution matrix ( 1 ) can be written, up to an irrelevant overall phase, as
[5] Z(t)=(l[V(t)-O(t)] /t~ IB(t) I
#~lB(t)l
-½[V(t)-O(t)]
)
' (10)
where ~(t) is the phase velocity describing the rotation of the magnetic field B(t)= IB(t) I exp[iq~(t) ] and V(t) is the URand//L level splitting
V( t ) =x/~ GFn elf(t) -- AmE/EE .
( 11 )
For constant parameters n e~, IBI and ~ we have the case analysed by Smirnov [ 5 ] and it is very easy to reproduce his results applying the expansional formalism presented here. In this case, the integration appearing in the definition of G (t, to), eq. (3), is trivial and G(t, to) can be written as a simple exponential
G( t, t o ) = e x p [ - i ( t - t o ) Z ]
information about the behaviour of the solar magnetic field. We will consider the general evolution matrix 27(t), given by eq. (10). We concentrate on the processes where a left-handed neutrino transforms into a righthanded one; these processes are related to the nondiagonal elements of the time evolution operator G (t, to). From the very general form of G(t, to), eq. (8), it is easy to see that it is not necessary to take Go(t, to) into account since it is diagonal and leads to just an overall phase in the relevant transitions. Since neutrinos are relativistic particles, the time dependent parameters n eft(t) and B (t) vary quickly along the neutrino trajectory (note that neff(t) - according to the solar standard model [ 8 ] - passes from values like O(100) NA/Cm 3, where NA is Avogadro's number, in the central regions of the Sun, to zero, on its surface, while B(t) can present strong variations from values like O( 103-5) G to zero in the convective zone due to solar internal activity). Consequently, neutrino spin precession is a rapid transition process. In processes like this, the expansional form appearing in eq. (8) can be approximated by [ 9 ] 1 - i i {b(t') exp[i.2A(t, t' )] tr+
=cos[ ( t - t o ) t o ] - i sin [ ( t - to)O0] 27,
to
(12)
+b*(t') exp[-i.2A(t,t')] a_} d t ' .
(.O
where to=x/(½ V - ½~)2+ (/~ IBI )2 .
(13)
The amplitude of the probability of transitions from left- to right-handed neutrinos is given by the upper non-diagonal element of the matrix G(t, to) and can be written as A (VL""*//R, t) = - i g , [B]
12 November 1992
sin[ ( t - t o ) m ] O)
Again, the VL--'gR transition is given by the upper nondiagonal element of the matrix (15), i.e., terms accompanying a+. This transition can be written as A (/.'L---*UR; t) t
= - i .J b(t') exp[i.2A(t, t') ] d t ' .
(16)
to
(14)
The probability that is derived from this amplitude coincides with the one obtained by Smirnov [ 5 ], and his conclusions straightforwardly apply here. Let us now analyse the more general case where the behaviour of the magnetic field a n d / o r of the density of particles along the neutrino path is not specified. This is an important case for the Sun. Even if theoretical calculations based on the standard solar model [ 8 ] give us some idea about the matter distribution inside the Sun [ 8 ], it is very difficult to obtain some
(15)
The values for b(t' ) and A(t, t' ) in the case we are analysing can be read from eqs. ( 1 ), (6) and ( 10 ):
b(t' )=lt~lB(t' )[
(17)
and
A(t, t ' ) = ½ i [V(t")-O(t")] dt".
(18)
t'
On the basis of the result (15) we shall analyse next some physically interesting limit cases. In order to 245
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PHYSICS LETTERS B
avoid irrelevant complications, we will consider the phase velocity 0 (t) as a constant. Thus
A(t, t')=½[7(t, t')-(t, t')6] ,
(19)
where
7(t, t') = _i V(t") dt" .
(20)
t'
If the absolute value of the magnetic field IB (t) I is a regular function of time, but its phase velocity can be considered extremely large (i.e, in the limit where 6--,oo), the Riemann-Lebesgue lemmas [10] guarantee that the integration appearing in eq. ( 15 ) is such that
12 November 1992
So, in the case where 2=2o, the slope of the function V(2) must present the same sign as 6. Eq. (24) is a generalization of Smirnov's resonance condition which was derived in ref. [5] in a naive way. Note that, differently from what was discussed by Smirnov, to have eq. (24) satisfied, a fixed direction of the magnetic field is not necessary. If there exist a stationary phase, we will find for the left to right amplitude [ 11 ]
A(VL---}VR;t ) " - - i
I
b(2o)
t
Xexp(i f d/t [ V(/t) - 6] + , i ~ ) ,
(26)
Ao
and the probability of a VL--'VRtransition is negligibly small. Note that this result is very general since in the limit case under analysis the time evolution operator given by eq. (8) goes to a diagonal form and all the relevant contributions to a left-right neutrino transition are negligible. If ~ is large (but q~-'~-ov) this integration does not vanish in the case where there exists a stationary phase. In this situation we can employ the stationary phase method [ 11 ] to analyse the behaviour of the following integration:
and the probability for a left-handed neutrino interacting with a magnetic field to be transformed into a ri.ght-handed neutrino will be given by 2zt[b (2o) ]2/ I V(2o) I. Thus, in the case where there exists a stationary phase along the neutrino trajectory, the VL~ VRtransition probability strongly depends on the absolute value of the magnetic field and on the matter distribution along the neutrino path [see eqs. ( 11 ) and (17)]. If the matter distribution allows more than one stationary trajectory, the total VL--'VRamplitude will be given by the sum of the contributions from each of them:
A(PL-'-}IIR; t)
A(PL-'}PR; t) _~
lim A(VL-'-}VR; t ) = lim o ( l ~ = o , ~oo ~oo
(21)
(22)
to
where
7(t,2) 6 - (t-2).
(23)
The condition for having a stationary phase leads us to
~h(t, 20) ~ 0 ~ V(;to)
02
246
(27)
1 0 ~--~ V(2o) < 0 .
Finally, we would like to point out that, for the solar neutrino case, if we admit the matter distribution usually accepted to fit the predictions of the solar standard model [ 8 ], i.e., an approximately monotonically decreasing exponential function (in the radial direction from the center to the surface of the Sun), then it will not be possible to satisfy simultaneously conditions (24) and (25) in the large ~ case we are analysing and there will be no stationary phase in the problem.
(24)
and 02----~h(t, 2o) = -
.
Ao
= --i .i b(2 exp[i6h(t, 2) ] d2,
h(t, 2 ) -
~ A(PL-'-}PR; 2 0 )
(25)
4. Using the Feynman procedure of ordered exponential operators we have solved the evolution equations for a two-neutrino system which interacts with matter and magnetic field without assuming any particular behaviour for the matter density nor for the
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PHYSICS LETTERS B
magnetic field along the neutrino trajectory. From the very general result ofeq. (8), we have concluded that all physical contributions to the non-diagonal entries o f the general neutrino evolution matrix 27(t) will generate an effective rotation o f the magnetic field interacting with the neutrinos. The simplest contributions to these entries come from interactions o f the neutrino with matter as well as from the neutrino mass splitting. Note that, since the absolute value o f the magnetic field does not change in a rotation, the necessary values for the neutrino magnetic m o m e n t and the magnetic field obtained in ref. [ 2 ] for an effective reduction o f the measurable solar neutrino flux, are still the same, independently o f whether the neutrino interacts with matter in the way discussed in the present paper. Furthermore, we have observed that a quick rotating magnetic field B ( t ) (or, equivalently, a large phase velocity ~ for the magnetic field) along the neutrino path, generates negligibly small probabilities of neutrino spin precession, unless some stationary phase can be found. In this case, the PL---~1/R transition probability will strongly depend on the matter distribution along the neutrino path. I f we admit the standard solar model distribution of matter inside the
12 November 1992
Sun, we will not find any stationary phase satisfied in the large ~ case. Therefore, the conclusions presented in ref. [ 3 ] by Vidal and Wudka are not valid not only in the case where the relevant parameters B(t) and neff(t) are constants, as discussed by Smirnov in ref. [ 5 ], but also in the general case where these parameters vary arbitrarily.
References [1 ] R. Davis et al., in: Proc. XXI Intern. Cosmic ray Conf. ( Adelaide, Australia, 1990) Vol. 7, p. 155. [2] M. Volishin, M.I. Vysotsky and L. Okun, Yad. Fiz. 44 (1986) 677 [Soy. J. Nucl. Phys. 44 (1986) 440]; A. Cisneros, Astrophys. Space Sci. 10 ( 1971 ) 87. [3] J. Vidal and J. Wudka, Phys. Lett. B 249 (1990) 473. [4] M. Berry, Proc. R. Soc. A 362 (1984) 45. [5] A.Yu. Smirnov, Phys. Left. B 260 (1991) 161. [6] R.P. Feynman, Phys. Rev. 84 ( 1951 ) 108; I. Fujikawa, Prog. Theor. Phys. 7 (1952) 433. [7] J. Bellandi et al., J. Phys. A 25 (1992) 877. [ 8 ] J. Bahcall and R.K. Ulrich, Rev. Mod. Phys. 60 (1988) 298; S. Turck-Chi~ze et al., Astrophys. J. 335 ( 1988) 425. [9] A. Messiah, Mrcanique Quantique, Vol. II (Dunod, Paris, 1964). [10]E.T. Whittaker and G.N. Watson, A course of modern analysis (Cambridge U.P., Cambridge, 1969). [11] A. Erdrlyi, Asymptotic expansions (Dover, New York, 1956).
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