MSW-A Possible Solution to the Solar-Neutrino Problem

MSW
|(tt) [a 1 (tt)e iM 1 t cos a 2 (tt)e iM 2 t sin ] | e
[a 1 (tt)e iM 1 t sin a 2 (tt)e iM 2 t cos ] | . (8)
We next consider the neutrino as a superposition of flavor states:
|(t) a e (t) | e a (t) | . (9)
Because only electron neutrinos interact via charged currents, the two flavor states
have different forward-scattering amplitudes, and each sees a different effective refractive
index in matter. We assume that the change in the probability amplitudes a e (t) and
a (t) during an infinitesimal time t can be expressed as a simple phase shift that is
proportional to the refractive index:
a e (tt) a e (t) exp [ip (n nc n cc 1)t] a e (t)exp [i( )t , and (10a)
a (tt) a (t) exp [ip (n nc 1)t] a (t)exp [i t] , (10b)
where (2N e /p)f nc and 2G F N e . The latter relation is the matter
oscillation term. We have also used x t. The neutrino state, therefore, evolves as
|(tt) ae (t)ei( )t |e a (t)e i t | ae (t)eit |e a (t) | , (11)
where again we have dropped the overall phase factor of exp(it) because it does not
affect the final result. Equations (8) and (11) are expressions for |(t t). Equating
the coefficients of | e and | results in a set of coupled equations:
a 1 (tt)e iM 1 t cos a 2 (tt)e iM 2 t sin a e (t)e i( )t , (12a)
a 1 (tt)e iM 1 t sin a 2 (tt)e iM 2 t cos a (t)e i t . (12b)
Both sides of Equations (12a) and (12b) are expanded to first order in t,
[a 1 (t) a˙ 1 (t)t ia 1 (t)M 1 t]cos [a 2 (t) a˙ 2 (t)t ia 2 (t)M 2 t]sin a e (t)(1it) (13a)
[a 1 (t) a˙ 1 (t)t ia 1 (t)M 1 t]sin [a 2 (t) a˙ 2 (t)t ia 2 (t)M 2 t]cos a (t) , (13b)
where a dot indicates the time derivative. Equations (4c) and (4d) are used to express
a 1 (t), . a 1 (t), a 2 (t), and . a 2 (t) in terms of a e (t), a e (t), a (t), and a (t). Following more algebraic
operations,
i .a˙ e (t) [M 1 cos 2 M 2 sin 2 ]a e (t) (M 2 M 1 )cos sin a (t) , (14a)
a˙ (t) (M 2 M 1 )cos sin a e (t) [M 1 sin 2 M 2 cos 2 ]a (t) . (14b)
These expressions can be cast in a Schrödinger-like equation for a column matrix A
consisting of the probability amplitudes a e (t) and a (t):
dA
i HA ,
dt
(15)
where H and A M1cos ae (t)
a (t)
2 M2sin2 (M2 M1 )cos sin
(M2 M1 )cos sin M1sin2 M2cos2 62 Los Alamos Science Number 25 1997
The eigenvalues of the matrix H are given by
1,2 . (16)
Equation (15) can then be solved:
e
A(t) I H A(0) , (17)
where I is the identity matrix. At time t 0, the beam consists only of electron
neutrinos. Thus, ae (0) 1, and a (0) 0 so that
i 2 t ei 1 t
2e
2
1
i 1 t
1ei 2 t
2
1
2 (M 2 M 1 ) 2 1 2 2(M 2 M 1 cos ) 2
2
2
ae (t) (M1cos2 + M2sin2 e
) ,(18a)
i 2 t ei 1 t
2e
2
1
i 1 t
1ei 2 t
2
1
e
a (t) (M2 M1 )cos sin . (18b)
i 2 t ei 1 t
2
1
The probability of detecting a muon neutrino after a time t is given by
P MSW ( e → ) (t) 2 a e (t) e a (t) 2 a (t) 2 (19)
so that
(M2 M1 )
PMSW (e → )
2(1 cos(
2 1 )t)
2cos2 sin2
(
2
1 ) 2
sin2 t . (20)
By substituting in the expressions for
1 ,
2 , M1 , M2 , we have
(2 1 ) 2 (M2 M1 ) 2 sin2 (M ( )
2 M1 ) 2 1
2
2
2 cos2 M2
M1 2sin22
( )
2 1 2
(21a)
m
(M2 M1 ) . (21b)
2
m
2E
2
2p
Recalling that x t, and 2G F N e ,we arrive at the MSW probability for
an electron neutrino to oscillate into a muon neutrino:
PMSW (e → ) sin2 sin xW
, (22)
22
W2 W2 sin22 2G F Ne cos 22 2E
, (23)
m2 where is the in vacuo oscillation length,
2E = 2
m2 , (24)
and m 2 m 2 2 m1 2 is required to be nonzero. If the numerical values of the
hidden factors of h _ and c are included, the expression for the oscillation length
becomes E/1.27m 2 . ■
Number 25 1997 Los Alamos Science
MSW