



Neutrino oscillations in matter of varying density

Paul M. Fishbane*
Physics Dept. and Institute for Nuclear and Particle Physics,
Univ. of Virginia, Charlottesville, VA 22904

Peter Kaus**
Aspen Center for Physics, Aspen, CO 81611



We consider two-family neutrino oscillations in a medium of
continuously-varying density as a limit of the process in a series
of constant-density layers. We construct analytic expressions
for the conversion amplitude at high energies within a medium
with a density profile that is piecewise linear. We compare
some cases to understand the type of effects that depend on the
order of the material traversed by a neutrino beam.



I. Transition amplitudes for structured matter.

The problem of neutrino oscillations in matter is of obvious importance, and it is interesting to
see to what extent and in what manner analytic solution is possible. In this note we discuss a
general approach to the problem which allows us to solve in a new way the case of neutrino
passage through matter whose density varies linearly with distance.
Under the assumption that a two-channel approximation to neutrino mixing holds, the
amplitude Aj for passage of a neutrino beam of energy E through a medium of constant electron
density, whose properties (density Nj and thickness xj) we label by j, is a 2  2 matrix whose
indices label flavors. It is given by the expression

j
A = cosf + i sinf cos q s - i f q s (1.1)
j j (2 j) sin sin
z j (2 j) x

where the work [1] of Mikheyev and Smirnov and of Wolfenstein [MSW] showed that the
effect of the matter is summarized by

* e-mail address pmf2r@virginia.edu
** e-mail address pkaus@futureone.com


1


2 2 2
m m = m q - x + q
k (cos2 k )2 2
sin 2
2
m
k
j j = x (1.2)
k 4 k
E
2
sin 2q
2
q q sin 2q =
k k (cos2q -x + q
k )2 2
sin 2
and
2EV
k
x with V = 2G N . (1.3)
k 2
m k F k
The mass parameter m2 = m 2 2
2 - m1 is positive. We recover the vacuum result, Cabibbo
angle q, for V = 0.
Let us now consider the elements of the amplitude A123.. = A1A2...An for passage
through a series of layers labeled sequentially from n to 1. These can be found either by trace
techniques or by the following direct technique: We extract a factor of cosfj from each factor
Aj. Then the amplitude takes the form

n
12Ln
A = cosf cosf Lcosf b
(1.4)
1 2 n j
j 1
=
where r
b = 1+sr B (1.5)
j j
and r
B = i tan f (-sin2q ,0,cos2q (1.6)
j j j j )
To find the ordered product over the bj, we require products of the form
( r r r
sr B sr B sr
L B , and these can be found recursively using the m = 2 result
1 )( 2 ) ( m ) r r r r r r
(srB srB = B B i+sr B B .
1 )( 2 ) 1 2 ( 1 2)
(In the recursive development, it is helpful that the vector B has no y-component.) The general
form for our product is thereby found to be
( m
r r r r r r
n
s B s B L s B = i tanf

1 )( 2 ) ( m) j
j 1=
(1.7)
s
cos q - q +L+ q -s q - q +L + q modd
z (2 2 2 sin 2 2 2
1 2 m ) x ( 1 2 m )
 co
s
(2q -2q +L- 2q +is sin 2q - 2q +L+ 2q meven
1 2 m ) y ( 1 2 m )
In turn the transition matrix elements of interest are





2


n

1+ itanf cos2q + i f f q - q
j j ( )2 tan tan cos
j k (2 2
j k )
j=1 j<k

n

L
12 n
A = cosf + i tanf tanf tanf cos 2q -2q + 2q +K (1.8a)
11 j ( )3 j k l ( j k l )
j 1= j<k <
l
(
+ i)n tanf tanf Ltanf cos 2q -2q +L- -1 n 2q
1 2 n ( 1 2 ( ) n )

and
n

-itanf sin2q + -i f f q - q
j j ( )2 tan tan sin
j k (2 2
j k )
j 1
= j k
<

n

L
12 n
A = cosf + i
- tanf tanf tanf sin 2q -2q +2q +K (1.8b)
12 j ( )3 j k l ( j k l )
j 1= j<k <
l
(
+ -i)n tanf tanf Ltanf sin 2q -2q +L- -1 n 2q
1 2 n ( 1 2 ( ) n)

One can immediately check that |A11|2 + |A12|2 = 1.

Recovery of the vacuum result. A useful check on Eqs. (1.8) follows from the assumption
that the parameter xj is small compared to cos2q for all j. We can thus approximate qj q as
well as m 2j m2. Assuming that the n slabs have equal width, we in addition have tanfj
tan(ffull/n), where ffull = m2X/(4E) is the angle appropriate to passage of a total length X
through vacuum. Thus when there are an odd number of angles,
2qi - 2qj +...+ 2qk 2q,
and when there are an even number of angles
2qi - 2qj +...- 2qk 0.
Thus, with f ffull/n, the 11 element of the amplitude is
n(n - )
1
1+
(itanf )2 +
K
2
A cosn f
=
11
n n - n -
cos2q n(itanf) ( )1( 2) (itanf)3
+ + +
K
3!

1
= { inf -in
e
+ e f + cos2 inf -in
q e
- e f
} (1.9a)
2
= cosf + i cos2q sinf .
full full
We have here used
1 ( n n - n -
1+ x)n -(1-x)n ( )1( 2) 3
= nx + x +
K
2 3!
1 ( n n -
1 x)n (1 x)n ( )1 2

+ + - =1+ x +K
2 2!
Similarly,




3


n n - n -
A cosn f sin2q n
( i tanf ) ( )1( 2) (itanf)3
= - + +K
12
3!

n 1 n n
= - + - -
cos f sin2q (1 i tanf ) (1 i tanf )
2
(1.9b)
1
= - sin2 inf -in
q e + e f
2
= -isin2q sinf .
full
The expressions (1.9) match the vacuum result for a single width X, Eq. (1.1) with the angles
replaced by their vacuum values.

A Given ordering and its reverse. One of the important features of oscillations within matter
is that the amplitude for transitions depends on the order of the density of the layers through
which a beam passes. The amplitude for passage through a single layer, Eq. (1.1), is symmetric,
A T
j = Aj . Therefore
2
L 1 = ( 12L )T
n n
A A , (1.10)
where we recall that the sequence of superscripts matches the layer order. That means in
particular that the diagonal (survival) elements are equal, e.g.,
n 2
L 1 12 n
A = A L (1.11)
11 11
Unitarity in the two channel problem then gives us equal probabilities for the off-diagonal
(conversion) elements for a given order and its reverse,
2 2
n 2
L 1 12Ln
A = A (1.12)
12 12

This proof fails for the three-channel conversion problem [2].
To learn the relation between the off-diagonal elements of the amplitudes themselves,
we note that A12...n has the simple form
L a b
12 n
A = ,
(1.13)
b * a *
-

To see this, one can for example use a recursive proof: Equation (1.1) shows it is true for one
layer, and explicit calculation shows that it is also true for two layers. Then one calculates
L L a b g d ag - bd * ad + bg *
12 (n+1) 12 n n+1
A = A A = = ,

b * a * d * g * b *g a *d * b *d a *g *
- - - - - +

and this has the requisite property.
We can now put Eqs. (1.10) and (1.13) together to show that
a
L -b *
n 21
A = . (1.14)
b a *

Thus in particular
n 2
L 1 = -( 12 n
A A L * . (1.15)
12 12 )
These results have been verified for the particular profiles we study below.



4


High energy limit. Let us call xmin = 2EVmin/m2 the minimum value taken on by the
parameter xj as j runs from 1 to n. Then we can study a high energy limit,
xmin >> cos2q, sin2q (1.16)
In this limit
sin2q sin2q
2q =1 (1.17)
j x x
j min
and, because m 2j xj m2 in this limit, we also have
1
f f x (1.18)
j full j
n
For arbitrary n this could be large or small; however, because we are ultimately interested in
large n and because for many situations this quantity is small in any case, we shall also assume,
as part of the definition of the high energy limit, that fj is small for all j and retain only first order
terms in fj. In particular the prefactor of the product over the cosfj in Eqs. (1.9) is unity.
The effect of our limit is most easily seen in the 12 element of the transition amplitude A,
Eq. (1.8b). Let us refer to the m-tuple sum in the curly brackets on the right of Eq. (1.8b) as
Tm, so that A12 is a sum over these sums.
Generally Tm is a function of the potential V; however for m = 1, which is the one term
in A12 that is order independent, the potential dependence cancels,
n
T = i
- tan f sin2q = i
- f sin2q
. (1.19)
1 j j full
j 1
=
This leading term essentially reproduces the vacuum result, independent of n. Both the potential
and the order dependence are present in the m = 2 term,
T (V ) = - tanf tanf sin 2q - 2q

2 j k ( j k )
j k
<
2 (1.20)
1 2E
- f sin2q V x -V x
full 2 ( k )
( j )
n m
j<k
This expression sets the pattern for the general term,
T (V ) = - tanf Ltanf sin 2q -2q +K- 1 m
- 2q
m j1 jm ( j1 j2 ( ) jm )
j < j <K j
1 2 m (1.21)
m m-1
1 2E
i f sin2q
- S (V ),
full 2 m
n m

where the multiple sum Sm is defined by
1 1
m 1
S (V ) = V z V
L z
- +K- - . (1.22)
m ( 1
1 ) ( m )
< <K V z V z V z
1
j 2
j m
j ( 1) ( 2) ( ) ( m)
In the following section, we consider these sums for specific potentials.
The continuum limit of the sums of Eq. (1.22) can also be found in the usual manner.
With the scaling variables zk jk/n, the large n limit of Sm(V) is
1 m
z z2
m 1 1 m 1
S V n dz dz L dzV z V
L z
- +K- -
m ( ) 1
m m 1
- 1 ( 1) ( m )
V z V z V z
0 0 0 ( 1 ) ( 2 ) ( ) ( m)

5


(1.23)


II. Linear density profile

Here we consider a simple linear profile V1 given by
V1(xj) = Vmin + VjX/n. (2.1)
The linear case has in fact been solved in other ways. In work [3, 4] on passage of neutrinos
through layers of constant density matter, a linear density profile was used to interpolate the
layers, and in so doing it was noticed that a formally identical problem had been solved much
earlier in the context of atomic physics [5, 6]. This work was more thoroughly recalled and
refined by Petcov [7]. More recently, another approach has produced a solution to the case of
linear matter for an arbitrary number of channels [8]. In addition there is a body of work based
on various approximations[9]. What we present here differs considerably in technique from the
exact work cited and the high energy approximation that applies to the examples below
complements the approximate work.
For the profile of Eq. (2.1) the double sum term, Eq. (1.22), takes the form
n k-1
X X X n n +1 n -1
S V = V k - j = V k - j = V
(2.2)
2 ( 1 ) ( ) ( ) ( )( )
n j<k n k 1= j 1= n 6
This behaves as n2 at large n, and since there is an additional factor n-2 in T2, Eq. (1.20), T2
itself has a finite large n limit, namely
( ) X n (n + )
1 (n - ) 2
1 1 2E
T V = V
- f
sin2q
2 1 full 2
n 6 n m (2.3)
1 2

- V X
f sin2q.
n 12 full
The pattern is repeated for the general term1. We have
m 1
V X -

S (V
j L j - j j L j +K- -1 m j L j (2.4)
m 1 ) ( 2 m 1 3 m ( ) 1 m-1 )
n 1
j < j2 <K jm
This multiple sum is explicitly calculable for any finite n but is not very enlightening. The large n
behavior is simpler, and we give here the first results for the first few values of m:
S (V ) V X
n
n ( ) 2 1/3!
2 1 ( )
S (V ) V X
n
n ( )2 3 7/5!
3 1 ( )
S (V ) V X
n
n ( )3 4 27/7!
4 1 ( )
(2.5)
S (V ) V X
n
n ( )4 5 321/9!
5 1 ( )
S (V ) V X
n
n ( )5 6 2265/11!
6 1 ( )
S (V ) V X
n
n ( )6 7 37575/13!
7 1 ( )

1 We have ignored the terms containing Vmin because they are nonleading as n becomes large. In other
words, the expression of Eq. (2.4) is already a large-n approximation.


6


These results can be found either with the large n behavior of the multiple sums of Eq. (1.22) or,
more simply, with the multiple integrals of the large n form Eq. (1.23).
We first remark that the factors of n are those necessary to make the result finite, since
Tm contains an additional factor of n-m. The sequence in the denominator can be identified [10]
as follows:
m 1
- m 1
2 2 -
m V X
a m -1 V X
a (m - )
1
T = i
- f q = T -i (2.6)
m ( ) ( )
2 (2m ) sin2 ,
full 1
1 ! 2
- (2m-
)1!
where a(m) obeys the two-term recurrence relation
a(m + 1) = a(m) + 2m(2m + 1)a(m - 1), with a(0) = a(1) = 1. (2.7)
The quantity a(m) is an expansion coefficient in several elementary functions, including some
combinations of inverse trigonmetric functions and algebraic functions. Perhaps the most
interesting relation is
y
2
2k (
a k) 1 2
y = exp y / 2 -u
e du

= k + y
k 2 1 !
0 ( ) ( )
0

1 p
= exp( 2
y / 2) erf ( y)
y 2
1 k
y +
= exp( y / 2) 2 1
2 (- )1k ,
y = k + k
k 2 1 !
0 ( )
a relation that can be applied to our case with the substitution
2
V X

y = exp ( -ip / 4) . (2.8)
2
The result is
1 p
A (V ) = T exp( 2
y / 2 erf y (2.9)
12 1 1 ) ( )
y 2


III. Comparison to other density profiles

It is instructive to compare the results of the previous section with two other density profiles of
the same total thickness X, each representing a rearrangement of the matter that composes the
profile V1, i. e., each having the same integral of V over x from 0 to X. Specifically, we consider
a profile peaked at the center,
X n
2V j j =1,
K
n 2
V x = V + (3.1)
2 ( j ) min n X n
V
X -2V j - j = +1,Kn

2
n 2
and a constant profile,
V3(xj) = Vmin + V X/2. (3.2)
We again work at our high energy regime.


7


The profile V2 can be treated by many of the same techniques that we used for V1, even
if the algebra is rather more complicated. We find for the double sum term, Eq. (1.22) with m =
2,
1
S V = V
x -V x = - VXn

. (3.3)
2 ( 2 ) 2 ( k ) 2
( j )
j k
< 2
This result should be compared to the corresponding one for S2(V1), Eq, (2.2), which is
proportional to n2 at large n. The term T2(V2) vanishes in the large-n limit. The cancellation is a
consequence of the symmetry of the matter distribution about X/2 (as will be confirmed below
for the V3 case); indeed it is generally true that Sm(V2) = 0 for even values of m in the large-n
limit. For odd values of m, however, there is a nonzero limit. We have worked through the first
few odd-m expressions for Sm(V2) for finite n . These are not simple, even in the large-n limit,
unless we set Vmin to 0, which we do to make the expressions clear2. Then the large-n results
for all m are
0
m even

S (V ) m -
(3.4)
m n (V X )m 1
2 ( m-
) m odd
2 1 !!
This gives for the conversion amplitude
m-1
i 1
A (V ) 2
= T - V X (3.5)
12 2 1
= K m -
m 2 2 1 !!
1,3 ( )
where T1 is the amplitude for passage through a single layer of vacuum, Eq. (1.19). The sum on
1
2w 1
the right is a Lommel function [11] U ( ,
w 0) cos w
( 2
1 t
= - dt , and
1 )
p 2

2 0
p 1
A (V ) 2
= T U V X , 0
. (3.6)
12 2 1 2 1
V X 2

2
The Lommel function is associated with diffraction from edges.
The amplitude for V3 is simply given by the MSW result of Eq. (1.1), i.e.
A V = i
- sin2q sin f .
12 ( 3 ) 3
V 3
V
1
In the "high energy" limit we have f f x = V X and the conversion amplitude
3
V full 3
V 3
2
becomes
sin2q 1 f sin2q
full 1
A V i sin V X i sin V X
- = -
12 ( 3 ) 3 3
x 2 f x 2

3
V full 3
V

1 1
sin V X sin V X
(3.7)
3 3
2 2
T T
= = .
1 1
f x 1
full 3
V V X
3
2


2 Strictly speaking, this violates our high energy condition Eq. (1.16); however, because V is only non-zero
for an arbitrarily small range of x this should not be troublesome.


8


The complete high energy expressions of Eqs. (3.6) and (3.7) are not very useful as
grounds for comparison with the full result for V1, Eq. (2.9). It is more transparent to expand
each result for small values of V X2 (as well as for Vmin = 0). In that case,
2
1 i 7 1
A (V ) 2 2
T 1
- V X
- V X
+K (3.8a)
12 1 1
3!2 5!
2

2
1 1
A (V ) 2
T 1
- V X
+K (3.8b)
12 2 1
5!!
2

2
1 1
A (V ) 2
T 1
- V X
+K (3.8c)
12 3 1
24
2

From these the conversion probabilities are, to leading order in V X2,
2
4 1
P (V ) 2 2
T 1
- V X
+ K (3.9a)
12 1 1
45
2

2
2 1
P (V ) 2 2
T 1
- V X
+K (3.9b)
12 2 1
15
2

2
1 1
P (V ) 2 2
T 1
- V X
+K (3.9c)
12 3 1
12
2

The ratios of these probabilities is in principle subject to experimental test, although it is clear
that such tests would be very difficult.



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.


References

1. L. Wolfenstein, Phys. Rev. D17, 2369 (1978); S. P. Mikheyev and A. Yu. Smirnov, Sov. J.
Nucl. Phys. 42, 913 (1985).

2. P. M. Fishbane and P. Kaus, 8.

3. W. C. Haxton, Phys. Rev. Lett. 57 (1986) 1271.

4. S. J. Parke, Phys. Rev. Lett. 57 (1986) 1275.


9



5. L. D. Landau, Phys. Z. USSR 1 (1932) 426.

6. C. Zener, Proc. R. Soc. A 137 (1932) 696.

7. S. T. Petcov, Phys. Lett. B 191 (1987) 199.

8. H. Lehmann, P. Osland, and T. T. Wu, .

9. See for example T. K. Kuo and J. Pantaleone, Phys. Rev. Lett. 57 (1986) 1805; S. P.
Rosen and J. M. Gelb, Phys. Rev. D 34 (1986) 969; V. Barger, R. J. N. Phillips, and K.
Whisnant, Phys. Rev. D 34 (1986) 980.

10. P. S. Bruckman, Fib. Quarterly 10, 169 (1972). This identification was made using The
Encyclopedia of Integer Sequences (http://www.research.att.com/~njas/sequences/).

11. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge [Eng.]
The University Press; New York, The Macmillan Company, 1944; K. W. Knockenhauer, Ann.
der Physik und Chemie (2) XLI, 104 (1837).





10



