



On equations for neutrino propagation in matter

Paul M. Fishbane*
Physics Dept. and Institute for Nuclear and Particle Physics,
Univ. of Virginia, Charlottesville, VA 22903

Stephen G. Gasiorowicz**
Physics Dept. and Institute of Theoretical Physics,
University of Minnesota. Minneapolis, MN 55455




We study the dynamical equations for two-family neutrino
oscillations in a medium of continuously-varying density.
We can find explicit solutions to these equations in terms of
series of nested integrals. These solutions can serve as a
basis for numerical calculation of these processes or for
further study of their analytical properties.








arXiv: v2 29 Dec 2000






* email address pmf2r@virginia.edu
** email address gasior@umn.edu


1


I. Introduction

The presence of neutrino oscillations [1-2] has renewed interest in the question of
oscillations within matter [3-5]. The early work ("MSW") of Refs. 3 and 4 solved the
problem of propagation within a medium of constant density, and it is possible to treat
nonconstant density by numerical means. It is nevertheless always useful to think about
an analytic approach [6] in order to develop insight and understanding. Since the case we
study here is that of a two-channel problem, some aspects of the methods we describe are
also applicable to spins in varying magnetic fields and other two-channel order-
dependent problems.
For convenience we recall here the MSW results [3, 4], constant density We
assume a two-channel approximation to neutrino mixing and give the amplitude T(t) for a
neutrino beam of energy E passing through a medium of constant electron density Ne
given some initial neutrino flavor mixture (0), namely
(t) = T(t)(0). (1.1)
(The time t can be interchangably viewed as the thickness x of the medium.) Here the


two-vector weak state (i.e., flavor basis) is ( ) e
t = . The transition amplitude T is

T = cos + i sin cos (2 ) - i sinsin (2 (1.2)
z ) x

where the primed variables contain the effect of the matter:

2
2EA
2 2 2 2
m m = m cos 2 - + sin 2

2
m
2
m
= t (1.3)
4E
2
sin 2
2
sin 2 = 2
2EA 2
cos 2
- + sin 2
2
m
with

A = 2G N . (1.4)
F e
The mass parameter m2 = m 2 2 2
2 - m1 , where mi is the ith mass eigenvalue, is positive. We
recover the vacuum result, Cabibbo angle , for A = 0.
This introduction sets some notation and recalls well-known results. Below we
shall be concerned with variable density.

II. Passage through a medium of variable density

The Schrdinger equation for a two-family weak state (t) propagating through a
medium of electron density Ne is
d
i = (H +W , (2.1)
F )
dt




2


where, using the approximation mi << E and after pieces in the hamiltonian proportional
to the unit matrix are removedthey lead only to overall phases, as we describe further
belowwe have

2
m -cos 2 sin 2
H = (2.2)
F
4E sin 2
cos 2
and
2G N 0 A 0
F e
W = (2.3)
0 0 0 0

cos sin
Here H =
F is related to the mass eigenstates by the Cabibbo matrix V ,
-sin cos
2
1 m 0

1
H = V V
VH V . (2.4)
F 2
2 D
E
0 m

2
A solution to Eq. (2.1) will give us the amplitude T(t), through Eq. (1.1).
It is convenient to approach the dynamical equation (2.1) by starting in the mass
basis, using the transformed states
V . We do so by multiplying Eq. (2.1) by
V ,
giving
d
i = (
H +V WV . (2.5)
D )
dt
Next we remove the factor HD by defining a new function by
-iHDt
e .
Starting from Eq. (2.5) it is easy to see that this function obeys the equation
d
iH t -
D iHDt
i = e V WVe . (2.6)
dt
In this expression, only the mass difference enters into the exponentials. To see this,
define
2 2 2
m + m m
1 2
E E + and
- . (2.7)
4E 4E
In terms of these quantities, HD reads
H = E +
D 3

The portion of HD proportional to the unit matrix commutes through the quantities in Eq.
(2.6) and cancels, leaving
d
i
-

3t i 3t
i = e V WVe . (2.8)
dt
We next write out in 2  2 form the quantity multiplying on the right side of Eq.
(2.8), using in particular the identity
ia
e 0

ia 3
e = cos a + i sin a = . (2.9)
3
0 -ia
e

We find





3


2
+ e
i
t -i

t 1 1 cos 2 i t sin 2
3 3
e V WVe = A
2
2 - it
e sin 2
1- cos 2
(2.10)
2
1 cos 2 i t
e sin 2
A -
2
2 it
e sin 2
- cos 2
In the last step we have taken out the piece proportional to the unit matrix; again, it
contributes only an overall phase to . (This can be seen in a variety of ways. For
t
i
example, one can define a new function exp + A
(t)dt and show that the
2
0
equation for is identical to that for but with the last form in Eq. (2.10) on the right
multiplying .) We can finally transform away the term in Eq. (2.10) proportional to 3.
We define the new function by
i
exp I
(t)cos2 0

i 2
exp + I t cos 2 =
,(2.11)
3 ( )
2 i
0 exp
- I
(t)cos2
2
where
t

I (t) A
(t)dt. (2.12)
0

The function obeys the simpler equation
1
2
0 i t

i Ae sin 2 i
d -
3 I cos 2 3 I cos 2
2
2 2
i = e e
dt 1
-2i t
Ae sin 2 0
2
1
i(2t+ I cos 2 )
0 Ae sin 2

2
= (2.13)
1 -i
(2 t
+I cos2 )
Ae sin 2 0
2
P(t) .
The last line defines the matrix P(t), in terms of which we can write our formal solution
to this equation. We remark in particular, for later use, that the 21 element P21 equals the
complex conjugate of the 12 element P12. We will below express the solution to Eq.
(2.13) in terms of P12 and P12*.
The form that P(t) takes is easy to understand. The central matrix in the first line
i
 I cos2
3
of Eq. (2.13) is a linear combination of 2
1 and 2, while the external factors e
take the form of a rotation about the 3-axis. Their effect is then to rotate the combination
of 1 and 2 to give a different combination that lies at a different anglethat takes the
form of the original combination but with the phase shifted. This is indeed what happens,
as the explicit calculations that give us P(t) show.




4


Solution of the equation for
. It is clear from the matrix structure of Eq. (2.13) that the
commutator [P(t), P(t)] 0, so that the solution to Eq. (2.13) involves an ordering.
Formally, the solution is
t

(t ) = exp i
- P
(s)ds (0) M (t) (0), (2.14a)

0
where the time ordering operator
ensures that the matrices P in the expression are
ordered so that P with a later argument stands to the left of P with an earlier argument.
We may write
n t t t
i
-
M (t) ( )
= ds ds ds
P s P s P s
,
1 2 n ( 1) ( 2 ) ( n )
= n
n 0 ! 0 0 0

and it is a standard exercise to show that this is equivalent to
t
1
s n
s 1
-
n
M (t) = (-i) P
(s ds P s ds P s ds
(2.14b)
1 ) 1 ( 2) 2 ( n ) n
n=0 0 o 0

Finally we can separate out the explicit matrix elements of M by using the matrix
structure of P, namely P(t) = Px(t)1 + Py(t)2. This leads immediately to
t t

M =1- P
(t) *P t
dt dt
+ (2.15a)
11 12 12 ( )
0 0
t t t t

M = -i P
(t)dt+i P
(t) *
P t
P t dt
dt dt -
12 12 12 12 ( ) 12 ( )
(2.15b)
0 0 0
0
M = M (2.15c)
*
21 12
1
P 2 1
P 2

*
M = M (2.15d)
22 11

Alternatively, direct differentiation of the group of Eqs. (2.15) shows that it satisfies the
dM
necessary condition = -iPM .
dt
The solution given by Eqs. (2.15) is essentially a power series in A. The order
dependence of the result is contained in the fact that the integrals in M are nested. We can
write the transition amplitude T in terms of M by undoing our series of transformations,
leaving
1
-i t
+ I (t)cos2
2
e 0
T (t ) = V M
(t)
V (2.16)
1
i t
+ I (t)cos2
2
0 e

Generally speaking the nested integrals in M are complicated, even for the case of
constant density. (Of course numerical integration is always possible.)

III. Second order equation

Equation (2.13) represents two coupled first order differential equations. These equations
can be rewritten as a single uncoupled second order equation, and the second order





5


equation lends itself to a variety of treatments beyond the formal solution we have
already expressed. In terms of the explicit components of , Eq. (2.13) reads
0

1 1 2
= = , (3.1)


0

2 2 1

where
-iP12 and -iP21 = -iP12*. (3.2)

One more derivative of, say, the upper component gives = + =
+ , or
1 2 2 1 1


- - = 0 (3.3)
1 1 1

Using the explicit expressions for and , we find that
2
A

1
= + i(2 + Acos2 ) and
= - Asin 2
.
A 2
Thus we have finally the second order equation
2
A
1
- + i 2 + Acos 2 + Asin 2 = 0 (3.4)
1 1 ( ) 1 1
A 2
It is worthwhile noting that the quantity I, Eq. (2.12), does not appear in this equation.
We may take the required two boundary conditions to be
1(0) and 2(0) = (0) / (0) .
1

1 d t
1 ( )
Once we find the solution for =
1(t), we have t . We also remark here
2 ( ) (t) dt
that we have verified that our formal solution to the equations for , Eq. (2.15), satisfies
Eq. (3.4).

Recovery of constant density case. For the MSW case (constant A A0), reviewed in
Section I, Eq. (3.4) takes the form
2
-ib + c = 0 (3.5)
1 0 1 0 1

where
1
b = 2 + A cos 2 and c = A sin 2 . (3.6)
o 0 0 0
2
If in addition we define

2 2
b + 4c , (3.7)
0 0

then the solution to this equation is
it it
t = B exp
+ b +C exp
- + b (3.8a)
1 ( ) 1 [ 0 ] 1 [ 0 ]
2 2
it it
t = B exp
- b +C exp
- - b , (3.8b)
2 ( ) 2 [ 0 ] 2 [ 0 ]
2 2
where





6


1 1
B = 0 - b - c 0
1 1 ( )[ 0 ] 0 2 ( )

2
1 1
C = 0 + b + c 0
1 1 ( )[ 0 ] 0 2 ( )

2

1 1
B = 0 + b - c 0
2 2 ( )[ 0 ] 0 1 ( )

2
1 1
C = 0 - b + c 0
2 2 ( )[ 0 ] 0 1 ( )

2
This solution allows us to identify M(t) through Eq. (2.14) and hence the
transition amplitude T through Eq. (2.16) adapted to constant A, namely
it
- ( it
2+ -
0
A cos 2 ) 0
b
2 2
e 0 e 0

T (t )
= V MV = V MV
(3.9)
it ( it
2+ A cos 2
0 ) 0
b
2 2
0 e 0 e

The result of the exercise is
( -b ) it/2
e + ( + b ) -it/2
e 2 it -it
- -
1 c ( / 2 / 2
e e
0 0 0 )

T = V V (3.10)
2 -2c
( it/2 -it/2
e - e ) ( +b ) it/2
e + ( -b ) -it/2
e
0 0 0
Then specific calculation shows, for example,
2i
T = T = sin 2 sin t / 2 . (3.11)
12 ( e ) ( )

This can be put into the canonical form of Ref. [4], -i(sinm)(sin2m), where the subscript
2
m
m indicates the propagation is in material, and where eff
= t , if we identify
m 4E
2 sin 2 sin 2 sin 2
sin 2 = = = (3.12a)
m 2 2
4EA 2EA 2EA 2
1- cos 2 + cos 2
- + sin 2
2 2 2
m m m
and

2 2
4EA 2EA 2EA
2 2 2 2
m = 2E = m 1- cos 2 + = m cos 2 - + sin 2 .(3.12b)
eff
2 2 2
m m m
Indeed, in terms of these new variables the full transition amplitude is
T = cos + (i sin cos 2 ) - (i sin sin 2 . (3.13)
m m m z m m ) x

This is the full canonical form described in Section I for propagation in a medium of
constant density.

Adiabatic Expansion. If the factor in Eq. (3.4) that contains the derivative of A is small
compared to the other factors, one can make a systematic adiabatic expansion [6zz] in
terms of it about the 0th order (MSW) answer. To do so, it is useful to recast the solution
technique somewhat. We shall first take the starting point of the neutrino beam, at t = 0,
to specify the constant background level of the material density factor, i.e., A(0) = A0. We



7


leave the boundary conditions 1(0) and 2(0) unspecified for the moment but remark that
using Eq. (2.13) the boundary condition for 2(0) gives us alternatively a condition for the
d i
derivative of 1 = - = -
1 at t = 0: (0) iP 0 0 A sin 2 0 .
12 ( ) 2 ( ) 0 ( ) 2 ( )
dt 2
We see from our earlier solution of the constant density case (Eq. 3.8) that the t-
dependence is contained in a pair of phases. An alternative way to derive these phases in
the constant density case is through a solution ansatz of the schematic form
1 = R0 exp(iS0(t)), (3.14)
where R0 is constant and where S0(t = 0) = 0. The real and imaginary parts of Eq. (3.5)
lead to the following equations for S0(t):
2
dS dS
0 0 2
- b - c = 0
0 0
dt dt
(3.15)
2
d S0 = 0.
2
dt
Together with the vanishing of S0 at t = 0, these imply that
S0(t) = t (3.16)
where
2 2
- b - c = 0 (3.17)
0 0

The solution of the quadratic equation gives
1
=  +
 (b ) 2 2
, where
b 4c . (3.18)
0 0 0
2
Thus, as indeed Eq. (3.8) shows, a better ansatz for the solution is
i t i
+ t
t = R e + R e - (3.19)
1 ( ) + -

The boundary conditions for 1(t) give us immediately R+ + R- = 1(0) and
1
R + = -
+ R
+ - A
- 0 sin 2 ; in turn these last two relations determine
0 2 ( )
2
0 b 0 + A 0 sin 2
1 ( ) 0 1 ( ) 0 2 ( )
R =
 * . (3.20)
2 2
Having reviewed the 0th order (constant density) problem, we go on to include
time (distance) dependence in the material density. We accordingly write the input
density as
A(t) = A0(1 + f1(t)), (3.21)
where f1(t) << 1 for all t in the problem and f1(t 0) = 0. We extend our ansatz for the
solution to the form
(t) = R +
+ (t ) iS (t )
e + R
+ - (t ) iS (t )
e - , (3.22a)
1 -
where R = +
 (t ) R (1  t (3.22b)
1 ( ))
S = +
 (t ) t  t (3.22c)
1 ( )
 
The quantities with subscript "1" are all small; moreover, 1 (0) = 1 (0) = 0. We also set
b(t) = b0 + b1(t), where b1(t) = A0 f1(t) cos2 (3.23a)
c(t) = c0 + c1(t), where c1(t) = (A0 f1(t) sin2)/2. (3.23b)




8


We now insert our ansatz into Eq. (3.5). The spirit of the adiabatic expansion is to keep
only first order terms in quantities with the subscript "1." In addition, we insist that the
coefficients of exp(iS(t)) vanish separately.
The real and imaginary parts of the coefficients of exp(iS) give respectively
2
d  d 
1 1
* + K f = 0 (3.24a)
2  1
dt dt
2
d  d  df
1 1 1
 - = 0, (3.24b)
2 
dt dt dt
where
K A0(cos2 + c0sin2) = 2(-2  )/2. (3.25)

Equations (3.24) contain only derivatives of the functions we seek, so they are in
fact two coupled first order equations for the functions

 d   d 
1 1
v and
u . (3.26)
1 1
dt dt
 
To the equations for v1 and u1 we add boundary conditions that follow from d1/dt = 0,
 
namely v1 (0) = 0 = u1 (0). As we shall see, these boundary conditions guarantee that 1
and 1 remain small (i.e., O(f1)).  
We decouple the two equations for v1 and u1 by taking one more derivative of,

say, Eq. (3.24a), giving a single second order equation for u1 :
2
d u  df
1 2 1
+ u = 2
, (3.27)
2 1 
dt dt
where for the coefficient of df1/dt we have used  - K = 2. This equation is solved
by a standard Green's function G(t - t) that satisfies
2
d G (t - t)
2
+ G t - t = t - t
2 ( ) ( )
dt
with, as causality suggests, G(x) = 0 for x < 0. The Green's function required is
1
G (x) = (t )sint
. (3.28)
In terms of this function we have
df t
u t = a sint + a cost + 2 dt G
t - t

1 ( ) s c  ( ) 1( )
dt
-
(3.29)

  1 t df t
1 ( )
= a sint + a cost + 2 dt sin -
 t t

s c ( ( ))
dt
0

(The lower limit reflects the fact that f1 is identically zero for negative argument.) Once
 
we have u1 we can get v1 from Eq. (3.24a):
1 du
 1
v =  + K f
1  1

dt

(3.30)
K
=  a cost - a sint 
+ f t
s c 1 ( )





9


   
At this point we can get as and ac from the boundary conditions for v1 (0) and u1 (0),
 
which determine as = ac = 0. (It is worth noting that since these quantities are not
proportional to f1, the only way for them to be small is to be zero.) In turn, we have
finally


 2 t t
df t
t 
= sin t - t dtd
t
(3.31)
1 ( ) ( ( )) 1( )
dt
0 0

 1 t
=  K f t dt
(3.32)
1  1 ( )
0

The phase function S(t) takes on a suggetive form if we use the identity K =  -
(b0  ), in which case we can write
t 2 t

S = + *
 (t )  (1 f t dt  f t dt
. (3.33)
1 ( )) 1 ( )

0 0

The first term integrates the material density.

An example: We take a linear variation, f1 = qt, together with the condition that
the beam is pure  at t = 0 (which translates into 1(0) = -sin and 2(0) = cos). Then
we have immediately
1 2 1
S = +
 (t ) 2  2
t qt * qt
 (3.34a)
2 2
and
2 1
R = + -
 (t ) R 1 q  t sin
 t
. (3.34b)
2


In Figs. 1 and 2 we plot some probabilities associated with the resulting amplitude for
some representative values of the parameters.

Acknowledgements
We would like to thank the Aspen Center for Physics, where much of this work was
done. PMF would also like to thank Dominique Schiff and the members of the LPTHE at
Universit de Paris-Sud for their generous hospitality. This work is supported in part by
the U.S. Department of Energy under grant number DE-FG02 -97ER41027.

References

1. Super-Kamiokande Collaboration, Y. Fukuda et al., Phys. Lett. B, 433 (1998) 9; ibid.,
436 (1998) 33; Phys. Rev. Lett. 81 (1998) 1562; ibid., 82 (1999) 1810; ibid., 82 (1999)
2430; E. Kearns, TAUP97, The 5th International Workshop on Topics in Astroparticle
and Underground Physics, Nucl. Phys. Proc. Suppl. 70 (1999) 315; A. Habig for the
Super-Kamiokande Collaboration, ; K. Scholberg for the Super-
Kamiokande Collaboration,  to appear in the Proceedings of the 8th
International Workshop on Neutrino Telescopes (Venice, Italy, 1999); G. L. Fogli, E.
Lisi, A. Marrone, and G. Scioscia,  to appear in the Proceedings of WIN
'99, 17th Annual Workshop on Weak Interactions and Neutrinos (Cape Town, South
Africa, 1999); Kamiokande Collaboration, K. S. Hirata et al., Phys. Lett. B280, 146



10


(1992); Y. Fukuda et al., Phys. Lett. B335 (1994) 237; IMB collaboration, R. Becker-
Szendy et al., Nucl. Phys. B (Proc. Suppl.) 38 (1995) 331; Soudan-2 collaboration, W.
W. M. Allison et al., Phys. Lett. B391 (1997) 491; Kamiokande Collaboration, S.
Hatekeyama et al., Phys. Rev. Lett. 81 (1998) 2016; MACRO Collaboration, M.
Ambrosio et al., Phys. Lett. B 434 (1998) 451; CHOOZ Collaboration, M. Apollonio et
al., Phys. Lett. B420 (1998) 397.

2. J. N. Bahcall and M. H. Pinsonneault, Rev. Mod. Phys. 67 (1995) 781; J. N. Bahcall, S.
Basu, and M. H. Pinsonneault, Phys. Lett. B 433 (1998) 1; J. N. Bahcall, P. I. Krastev,
and A. Yu. Smirnov, Phys. Rev. D58 (1998) 096016; B. T. Cleveland et al., Nucl. Phys.
B (Proc. Suppl.) 38 (1995) 47; Kamiokande Collaboration, Y. Fukuda et al., Phys. Rev.
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(1996) 384; SAGE Collaboration, J. N. Abdurashitov et al., Phys. Rev. Lett. 77 (1996)
4708; Liquid Scintillator Neutrino Detector (LSND) Collaboration, C. Athanassopoulos
et al., Phys. Rev. Lett. 75 (1996) 2650; ibid., 77 (1996) 3082; Phys. Rev. C58 (1998)
2489; Phys. Rev. Lett. 81 (1998)1774; G. L. Fogli, E. Lisi, A. Marrone, and G. Scioscia,
Phys. Rev. D59 (1999) 033001. See also G. L. Fogli, E. Lisi, and A. Marrone, Phys. Rev.
D57 (1998) 5893 and references therein.

3. S. P. Mikheyev and A. Yu. Smirnov, Sov. J. Nucl. Phys. 42, 913 (1985).

4. L. Wolfenstein, Phys. Rev. D17, 2369 (1978).

5. The first paper to point out that a two-layer case is relevant to passage and conversion
of neutrinos within the Earth is S. T. Petcov, Phys. Lett. B 434 (1998) 321. There is much
subsequent work, including (and not necessarily in chronological order) M.V. Chizhov
and S. T. Petcov, Phys. Rev. Lett. 83 (1999) 1096; M.V. Chizhov and S. T. Petcov, hep-
; M.V. Chizhov, ; E. Kh. Akhmedov, Nucl. Phys. B538, 25
(1999); E. Kh. Akhmedov, A. Dighe, P. Lipari, and A. Yu. Smirnov, Nucl Phys. B542, 3
(1999); M. V. Chizhov, M. Maris, and S. T. Petcov, ; E. Kh. Akhmedov,
; E. Kh. Akhmedov, Pramana 54, 47 (2000); E. Kh. Akhmedov and A.
Yu. Smirnov, ; M. V. Chizhov and S. T. Petcov, ; I.
Mocioiu and R. Shrock, Phys. Rev. D62 (2000) 053017; P. M. Fishbane, Phys. Rev. D62
(2000) 093009; M. Freund and T. Ohlsson, Mod. Phys. Lett. A 15 (2000) 867; T. Ohlsson
and H. Snellman, Phys. Lett. B 474 (2000) 153 and 480 (2000) 419(E).

6. This is of course an approach that we are not the first to take. Among many possible
citations, and beyond those of references 3 and 4, we single out S. T. Petcov, Phys. Lett.
B 191 (1987) 299; T. Ohlsson and H. Snellman, J. Math. Phys. 41 (2000) 2768; P. Osland
and T. T. Wu, Phys. Rev, D62 (2000) 013008; H. Lehmann, P. Osland, and T. T. Wu,


7. For other work on the "adiabatic" treatment of neutrino oscillation amplitudes, see ref.
3; S. Toshov, Phys. Lett. B 185 (1987) 177; S. T. Petcov and S. Toshov, Phys. Lett. B
187 (1987) 120; P. Langacker et al., Nucl Phys. B282 (1987) 589. The approximation in





11


these citations is not, we believe, the one we have developed here. We shall discuss this
matter elsewhere.

Acknowledgements

We would like to thank the Aspen Center for Physics, where much of this work was
done. PMF would also like to thank Dominique Schiff and the members of the LPTHE at
Universit de Paris-Sud for their hospitality. This work is supported in part by the U.S.
Department of Energy under grant number DE-FG02-97ER41027.


Figure Captions

Figure 1. Probability, as calculated in the adiabatic approximation described in the text,
of e as a function of time from production as a pure  at t = 0, in a medium with density
- -
factor A 1 13
0(1 + qt), where A0 = 6  109 cm = 10 eV corresponds to Earth-like density.
We assume the primary mixing angle is = 0.7 and that the difference of the square of
- -
the neutrino masses is 5  10 6 eV2. The factor = -7.9  10 14 eV, a value for which the
neutrino energy lies around the MSW resonance value, an energy of roughly 20 MeV.
- -
The horizontal exis is in units of 1014 eV 1; note that for q = -2  10 15 eV, qt = -0.2 at t
-
= 1014 eV 1.

-
Figure 2. Same as Fig. 1, but with the factor = -7.5  10 15 eV, a value for which the
neutrino energy lies roughly ten times higher than that corresponding to the MSW
resonance value.





12



Figure 1







Figure 2





13



