





Neutrino propagation in matter and CP violation



Paul M. Fishbane*
Physics Dept. and Institute for Nuclear and Particle Physics,
Univ. of Virginia, Charlottesville, VA 22904-4714

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





We point out that the dependence on the order of the matter
through which neutrinos pass can provide a window into
CP violation in the neutrino sector. This allows for study of
CP in the neutrino sector without the necessity of making a
comparison between the behavior of neutrinos and that of
antineutrinos.



arXiv: v2 29 Dec 2000






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


1


The presence of neutrino oscillations [1-2] leads to the question of possible CP violation
in a (fully-coupled) three-family neutrino sector [3]. We suggest here a probe of CP that
uses an effect of the modification of neutrino evolution within matter1 [4-6]. The early
two-family work of Refs. 4 and 5 arrives at a clear analytic result for propagation of
neutrinos within a medium of constant density. Our method is also clearly illustrated in
the two-family case. After development of our method we illustrate the three-family case
with some numerical calculations.
Neutrino propagation in material differs from propagation in vacuum because, in
contrast to  and , any e component in the neutrino beam can scatter from electrons in
the material through charged current interactions. Consider neutrino passage through a set
of N constant density material layers that we label 1, 2,..., N. These could be materials of
finite width or could be a set of labels that represent an (arbitrarily good) approximation
to a material of continuously varying density. Because of multi-family transitions, the
order of the materials matters. With n neutrino families, the n  n amplitude A12...N is
(t) = A12...N(t)(0). (1)
where are the flavor states and t is the total time (or interchangably the total distance).
The multiple-layer amplitude is the ordered product of the amplitudes for passage
through the respective single layers,
A12...N = A1A2...AN (2)
While Eq. (2) is general, to proceed further we want to start with two families, for
which we know the single-layer amplitude Aj is symmetric (see below). Then the
amplitude for passage through the "reversed" material2, in which the order of layers is
inverted, is
T
reverse N 21 N 2 1
A = A = A A A
= ( 1 2 N
A A A
)
(3)
= ( T T
12 N
A ) =( direct
A ) .
To see the consequences of this, recall that for two families with mass eigenvalues
m 2 2 2
i and a single (positive) mass difference factor m2 = m2 - m1 , and a Cabibbo matrix
describing the flavor states in terms of the energy eigenstates in the usual way, the single-
layer amplitude [4, 5] takes the general form

. (4)
* *

-
(In fact is pure imaginary, but that is unimportant here.) We can then show by
recurrence that the N layer amplitude has the structure

direct 12 N
A = A = . (7)
* *

-
This follows because the amplitude for N + 1 layers is





1 In other work [3], the effects of matter imitate CP violation and it is necessary to distinguish the effects of
matter from those of the presence of a CP-violating phase.
2 We generally refer to quantities for the order 12...N as the "direct" case, and quantities for the order
N...21 as the "reverse" case.


2



12 N 1
+ 12 N N 1
A A A +
= = * * * *

- -
.
* *
- +
= ,

* * * * * *

- - -
and this indeed has the structure of (7). It remains only to show by direct calculation
starting from (4) that a two layer amplitude has the structure of (7), and this is easily
done. This result can alternatively be derived in a continuum limit, for which the
amplitude has an ordered exponential form.
The combination of Eqs. (3) and (7) then tells us that
*

-
reverse
A = . (8)
*

Comparison between Eqs. (7) and (8) reveals that the probabilities Pij |Aij|2 satisfy
reverse direct
P = P ,
ij ij

although only the diagonal elements of the amplitudes are equal.
The presence of three families complicates the situation. Even if we could
establish that Eq. (3) were valid, it would give only a useful relation for the diagonal
matrix elements of the direct and reverse amplitudes; assuming the validity of (3) we find
reverse direct
A = A . (9)
jj jj

With this statement alone and unitarity, one can deduce the fact that for two families the
off-diagonal probabilites are the same; for example,
reverse = 1 reverse
- =1 direct direct
P P - P = P .
12 11 11 12

For three families the off-diagonal probabilities for the direct and reverse probabilities
would not generally be equal. Rather, for example,
reverse reverse
+ =1 reverse
- =1 direct direct direct
P P P - P = P + P .
12 13 11 11 12 13

Let us turn now to the question of three families, including the possibility that the
mixing matrix is complex (CP in the mixing matrix). We begin by recalling the amplitude
in vacuo. With the CKM matrix V connecting the flavor eigenstate and energy
eigenstate according to the = V, one can easily show that
-i 1
E t
e 0 0

A(t) -iE2t
= V 0 e 0 V

-i 3
0 0 E t
e

(Given the freedom to remove an overall phase, it is possible to use, for example, Ei E
+ m 2
i /(2E), and pull out factors so that the central matrix gives only dependence on the
two mass difference factors m 2 2 2 2 2 2
21 m2 - m1 and m31 m3 - m1 of the problem.)
Only if there is no CP violation, so that V is real, is A symmetric. When matter is
involved, and ignoring terms proportional to the unit matrix, the problem comes down to
diagonalizing the matrix
0 0 0 V
0 0
cc
1
2
H V 0 m 0 V + 0 0 0 . (10)
flavor 21
2E
2
0 0 m 0 0 0
31


3


This means finding a unitary matrix U such that
0 0
1


UH U = 0 0 (11)
flavor 2

0 0
3
The time evolution equation is then solved in the diagonal basis, and after transformation
back to the flavor basis we have
-i1t
e 0 0

A(t) -i2t
= U 0 e 0 U U D
(t)U (12)
-i3
0 0 t
e

(We remark here that if the consequences of a nonzero are not to be lost, U cannot be
found by using approximations based on a presumed neutrino mass hierarchy. Doing so
leads to a decoupling in which the oscillation proceeds essentially through two-family
steps, and this removes any effect of CP violation [7]. Similarly, if any m 2
ij = 0, then
can simply be set to 0.)
At this point we can understand the effect of a CP-violating phase in V. If = 0,
then V is real, and hence so is U, and A is symmetric. Equation (3) holds for the three
channel case, and the respective diagonal elements of the direct and reverse processes are
equal to one another. If, however, is non-zero, then V is no longer real, and neither is U.
More specifically, we can generally write our primitive amplitude (12) with an index j for
the material (including its widtht tj). Then for passage through a sequence of layers
we have
(A )T
reverse = (( T

U D t U U D t U U D t U
N N ( N ) N )( ( ) )(
2 2 2 2 1 1 ( 1 ) 1 ))
* * *
= (
U ) *
D (-t ) *
U (
U ) *
D (-t ) *
U (
U D -t U
N ) *
N ( N ) *
1 1 1 1 2 2 2 2 N


= ( direct
A (t t-
i i ))* .
In other words, the more general form of Eq. (3) is
T
*
reverse
A = (( direct
A (t -t )) ) =( direct
A t -t (13)
i i ( i i ))

In particular, the diagonal matrix elements of the direct and reverse processes are no
longer equal, and this provides a way to test for the presence of a nonzero value of .
Of the diagonal processes available to us, we cannot use the process e e. That
is because, given the fact that the charged current occurs in the electron sector, there is no
CP violation visible in e e to leading order in the weak interaction [8]. We have
verified the argument with, among others, direct numerical calculation with the
parameters described below: the process e e remains independent of material order
and indeed independent of .
The process   does allow us to test the effect described here. To illustrate
we use a set of numbers that correspond to the so-called large-angle solar MSW scenario
- -
[9], namely m 2 2 3 2 5
32 m31 = 3.5  10 eV2, m21 = 5  10 eV2, sin13 = 0.10, sin23 =
0.71, sin12 = 0.53, with two layers, layer 1 corresponding to a material with density 8
-
gm/cm3 (V 13
cc = 2.4  10 eV) and layer 2 being that of the vacuum (or, for our purposes,



4


air). We consider three energies, E1 = 100 MeV, E2 = 500 MeV (corresponding to
-
m2/(2E) V 4
cc in material 1 for m2 = 10 eV2), and E3 = 20 GeV (corresponding to a
possible neutrino factory energy [3]). In Figs. 1a-c we plot for each energy the difference
in probabilities for   for the direct and reverse process. In the range from = 0 to
= 1 rad, the dependence is essentially linear in the magnitude of the differences, with
no effect for = 0 to the largest effect for the largest . Aside from the height
magnification there is no additional -dependence, and we restrict the plots presented -
here to = 0.5. The horizontal axis is the thickness of the air layer in units of 1013 eV 1;
we have in Figs. 1a-c taken equal amounts of vacuum and material.
A well-known characteristic of the types of effects we are talking about is that
they are compressed at low energies and stretched out at higher energies, and this is
clearly visible in the three figures. We note the following: At E1, the effect tends to be
more symmetric about the horizontal axis. Because in any hypothetical experiment the
fine oscillations tend to be averaged over, the symmetry is a disadvantage. If one spreads
the horizontal axis over a much larger range, say from 0 to 100, then the envelope of the
fine oscillations, which occurs in a series of "lobes" as is most visible in Fig. 1b, moves
slowly to either side of the horizontal axis, but clearly E1 is not the most interesting case.
That is reserved for E2, where the lobes are clearly of one sign or another and the effect is
reasonably large. The case E3 also shows an average that is not symmetric about the
horizontal axis, but over a larger distance scale.
In Figure 2 we again consider the difference in probabilities for   for the
direct and reverse process, but this time there is ten times as much air as material (Fig.
2a) or ten times as much material as air (Fig. 2b); again the horizontal scale is the amount
of air in the same units as in Fig. 1. The effect remains large and systematic.
Finally we remark on an off-diagonal process,  e for definiteness. These
processes have order dependence even in the absence of a CP-violating phase . Figures
3a-b plot the probability difference for the direct and reverse processes for = 0 and =
1 respectively. These probabilities certainly differ from one another; however, the
standard for the detection of non-zero is different from the diagonal case.
Whether the phenomenon being described here is of eventual use will depend on
as-yet unknown features of the neutrino spectrum. Detectable oscillations over
managable distances would certainly render our results more relevant. It is nevertheless
interesting that there is at least in principle a way to look for CP-violation in the neutrino
sector without having to compare neutrino and antineutrino processes.

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.





5


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
(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.
Lett. 77 (1996) 1683; GALLEX Collaboration, W. Hampel et al., Phys. Lett. B388
(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. For a general review, see for example C. Albright et al., Physics at a Neutrino Factory,
Report to the Fermilab Directorate, . For work specifically relevant to CP
violation, see among other work H. Minakata and H. Nunokawa, Phys. Lett. B143 (1997)
369; J. Arafune and J. Sato, Phys. Rev. D55 (1997) 1653; M. Tanimoto, Phys. Rev. D55
(1997) 322 and Prog. Theor. Phys. 97 (1997) 901; J. Arafune, M. Koike, and J. Sato,
Phys. Rev. D56 (1997) 3093 and erratum ibid., D60 (1999) 119905; H. Minakata and H.
Nunokawa, Phys. Rev. D57 (1998) 4403; S. M. Bilenky, C. Giunti, and W. Grimus, Phys.
Rev. D58 (1998) 033001; A. De Rujula, M. B. Gavela, and P. Hernandez, Nucl. Phys.
B547 (1999) 21; K. Dick et al., Nucl. Phys. B562 (1999) 29; A. Gago, V. Pleitez, and R.
Zukanovich Funchal, Phys. Rev. D61 (2000) 016004; Z. Z. Xing, Phys. Lett. B 487
(2000) 327 and hep-ph/009294; H. Minakata and H. Nunokawa,  and
.

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





6


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

6. 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).

7. E. Kh. Akhmedov, Trieste Summer School in Particle Physics, 1999, .

8. H. Minakata and H. Nunokawa, Phys. Lett. B468 (1999) 256.

9. See for example V. Barger et al., Phys. Rev. D62 (2000) 013004.

Figure Captions

Figure 1. Probability differences for finding  as a function of time from production as a
pure  between the case where the neutrino beam passes through two different layers of
equal thickness in one order and the case where the beam passes through the layers in the
reverse order. The horizontal axis is the time for passage through (or thickness of) the air
layer. Parameter values as described in the text. (a) E = 100 MeV; (b) E = 500 MeV; (c)
E = 20 GeV.

Fig 2 As in Fig. 1, except the thickness of the air and the material layer are different. (a)
Air thickness 10 times that of material thickness; (b) air thickness 1/10 that of material
thickness.

Figure 3. Probability differences for finding e as a function of time from production as a
pure  between the case where the neutrino beam passes through two different layers of
equal thickness in one order and the case where the beam passes through the layers in the
reverse order. The horizontal axis is the time for passage through (or thickness of) the air
-
layer in units of 1013 eV 1. Parameter values as described in the text. (a) = 0; (b) = 1.










7























Figure 1



8



















Figure 2






9


















Figure 3






10



