Neutrinos: in and out of the standard model Page: 3 of 16
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Using the same E formulation, we have that pj = E2 - m ~ E
P(v, -- vi)
and therefore
where A = 3m2L/4E is the kinematic phase, with Sm2 = m2 - mi. The disappearance
probability is given byP(v -- v )
1- P(v, -- v2) = 1- sin220 sin2A.
(5)
If we put the h's and c's into the appearance probability we find
vm2c4L
P(v, -> v2 ) = sin2 20 sin2 .
4hcE(6)
In the semi-classical limit, h -- 0, the oscillation length goes to zero and the oscillations
are averaged out. This is the same limit as letting 5m2 become large. This is precisely
what happens in the quark sector. In Fig. 1 we have shown the oscillation probability for
both fixed energy and a gaussian spread of 15% of the mean neutrino energy. Notice that
oscillations are observable only for a limited range of distance. At small distance the
simple flavor description is a good one. But at very large distance using the probability
description with mass eigenstates works well since the oscillations are averaged out. The
neutrino mass eigenstates are effectively incoherent. Thus, in terms of probabilities2At Production:
the fraction of v ) that is vi) is cos2 0
the fraction of v ) that is v2) is sin2 0During Propagation: flavor fractions in Ivi) and Iv2) remain unchanged
the fraction of vi) that is vN) is cos2 0
the fraction of vi) that is v,) is sin2 0
the fraction of v2) that is v ) is sin2 0
the fraction of v2) that is v,) is cos2 0Thus, in the v, beam, the fraction of vi is fi = cos2 6 and v2 is f2
dently of the neutrino energy, and the survival probability isP(vA -- v )
sin2 0, indepen-
fi cos20 + f2 sin2 0
(7)
= cos4 0 + sin4 6 = 1- sin220 Ksin2A),
since sin2 A) = 1/2. Notice that the full treatment given earlier is really only useful for
distances around (1/5 to 5 times, say) the oscillation length, LO = 4:rE/3m2. At small
distance, the oscillations haven't built up enough to be significant, whereas as at the
large distance the oscillations are average out.2 v is the neutrino produced in association with p+.
sin2 0 cos2 0 e-i#L/2E _ e-i# L/2E 2 = sin220 sin2
(4)
At Detection:
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Parke, Stephen. Neutrinos: in and out of the standard model, article, July 1, 2006; Batavia, Illinois. (https://digital.library.unt.edu/ark:/67531/metadc888181/m1/3/: accessed April 16, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.