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boson with its target in the detector, this neutrino turns into a charged lepton. Then, as far as we know,
this charged lepton will always be of the same flavor as the neutrino. Thus, it will be of the same flavor
as the charged lepton with which the neutrino was born.
Now imagine that we send a neutrino on a long journey, say from your present location straight
downward to a detector on the opposite side of the Earth. Suppose that this neutrino is created in the
pion decay r - Virtual W - p + v/, so that at birth it is a v/. Imagine that this neutrino interacts via
W exchange in the distant detector, turning into a charged lepton. If neutrinos have masses and leptons
mix, then this charged lepton need not be a p, but could be, say, a T. Since it is only a VT that can turn
into a T, the appearance of this T would imply that during its journey, our neutrino has evolved from a
v/ into a VT, or at least into a neutrino with a nonzero vT component. The last 14 years have brought us
compelling evidence that such changes of neutrino flavor actually occur. As we shall see, the probability
of flavor change in vacuum has an oscillatory character, so flavor change is commonly referred to as
neutrino oscillation.
That neutrinos have masses means that there is some spectrum of neutrino mass eigenstates vi,
whose masses mi we would like to determine. That leptons mix means that the neutrinos of definite
flavor, ve, v/, and VT, are not the mass eigenstates vi. Instead, the neutrino state Iv,) of flavor a, which
is the neutrino state that is created in leptonic W decay together with the charged lepton of the same
flavor, is a quantum superposition
Va) = ZUsvi) (1)
of the mass eigenstates Ivi). (From now on, a v with a Greek subscript such as a or # will denote
a neutrino of definite flavor, while one with a Latin subscript such as i or j will denote a neutrino of
definite mass.) In the superposition of Eq. (1), the coefficients Uri are (complex conjugates of the)
elements of the leptonic mixing matrix U - the leptonic analogue of the quark mixing matrix. Now,
there are at least 3 neutrinos v, of definite flavor, and they must be orthogonal to one another, or a
neutrino of one flavor, interacting via W exchange, would sometimes turn into a charged lepton of a
different flavor. Out of these 3 orthogonal vi, we can form 3 orthogonal linear combinations that will be
neutrino mass eigenstates vi. (The mass eigenstates must be orthogonal because they are eigenstates of a
Hermitean operator, the Hamiltonian.) For all we know, there are more than 3 neutrino mass eigenstates.
However, if there are only 3, then U is a 3 x 3 matrix, and, being the matrix that transforms the states
of one quntum basis into those of another, it is unitary. The matrix U is sometimes referred to as the
Maki-Nakagawa-Sakata (MNS) matrix, or as the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix,
to honor several pioneering contributors to the physics of mixing and oscillation.
Mixing is readily incorporated into the SM description of the coupling of the leptons to the W.
For this coupling, we have in the SM Lagrangian density the term
Geuw = - 9 1 (TL7 <VLaW +vL <LyaW7) (2)
Here, g is the semi-weak coupling constant, v, is the neutrino of flavor a as before, and f,, is the charged
lepton of flavor a. That is, fe = e, i?, = p, and ET T T. The subscript L denotes a left-handed chiral
projection: ELa = [(1 - -5) /2] , and similarly for PL,. Note from Eq. (2) that, in conformity with the
rule quoted earlier, the neutrino of flavor a couples only to the charged lepton of the same flavor. To
explicitly incorporate mixing into the E v W coupling, we insert Eq. (1) into Eq. (2), so that the latter
becomes
,Cuw = - v 7 (TL7 U ivLiW- + ULi2UcEfL.WA) (3)
i=1,2,3
Here, as before, vi is a neutrino mass eigenstate, and we have taken into account the fact that the field
operator which absorbs the state 2g U* Ivy) of Eq. (1) is not Zg U,*vi, but 2, Univi.2
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Kayser, Boris. Neutrino Oscillation Physics, report, June 1, 2012; Batavia, Illinois. (https://digital.library.unt.edu/ark:/67531/metadc837195/m1/2/: accessed July 16, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.