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From Eq. (3), we see that, apart from the factor of g/v2 out front and kinematical factors, the
amplitude for W+ -+ + + vi or for f- + W+ - vi is just Uri, while that for vi - f- + W+ is just
Up,. Writing out the mixing matrix explicitly, it is
vi v2 v3
e U,1 Ue2 Ue3
U = P U 1 Up2 U/'3 (4)
T UTi UT2 UT3]
From Eq. (3), we see that, for example, the e (top) row of U tells us what linear combination of the
neutrino mass eigenstates v1, v2, and v3 couples to an e and a W. Similarly, the vi (first) column of U
tells us what linear combination of the charged lepton mass eigenstates e, p, and T couples to a v, and a
W. Similarly for the other rows and columns.
2.2 Probability of Neutrino Oscillation in Vacuum
Let us now find the probability P(v, - v; L, E) that a neutrino born as a v, - a neutrino of flavor a
- will then behave like a v, - a neutrino of flavor 3 - after traveling through vacuum for a distance
L with energy E. The conventional derivation of this probability may be found in many places in the
literature [1]. Rather than reproduce that derivation, or present the one we gave in the lectures (which
may be found in Ref. [2]), here we present a new approach [3]. We apply this approach to neutrinos
produced in the pion decay 7 - p + v, which we view in the pion rest frame, as in Fig. 1. To illustrate
(tK, x ) (tv, xv)
e
ei
-i , . 1 * i ei7iv
m - 1_
2
Fig. 1: The pion-rest-frame view of a pion decay followed by interactions of the daughter neutrino and muon. The
amplitudes shown for the various parts of this scenario are explained in the text.
how the approach works, we consider a scenario in which the neutrino, having been created in the 7
decay at a spacetime point (0, 0), then interacts at the spacetime point (t", x") whose pion-rest-frame
coordinates are t" and x". The interaction is via W exchange with a target in a neutrino detector, and, for
illustration, we suppose that the charged lepton into which the neutrino is converted by this interaction is
an electron. Of course, the interaction will also produce a recoil X. As part of the same overall scenario,
the muon created together with the neutrino in the 7 decay interacts at the spacetime point (t , X )
whose pion-rest-frame coordinates are t1' and xN'. We imagine that the interaction is with matter that
surrounds the 7 decay region. The full scenario is pictured in Fig. 1.
We shall find the amplitude for the entire scenario, including the 7 decay and the v and y inter-
actions [4]. To do this, we shall use the fact that if a particle has mass m and width F, adding up to a
complex mass A = m - iF/2, then the amplitude for this particle to propagate for a proper time T in
its own rest frame is exp(-iAT). (If this propagation is through a distance x during a time t in some3
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Kayser, Boris. Neutrino Oscillation Physics, report, June 1, 2012; Batavia, Illinois. (https://digital.library.unt.edu/ark:/67531/metadc837195/m1/3/: accessed July 16, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.