﻿ How the Nature of a Neutrino would be Revealed by the Relationship between its Energy and Momentum
San José State University

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Thayer Watkins
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How the Nature of a Neutrino would be
Revealed by the Relationship between its
Energy and Momentum

The existence of the neutrino was deduced in the early 1930's by Wolfgang Pauli from an apparent lack of the conservation of momentum in beta decay nuclei. Pauli gave it the name neutron but Enrico Fermi called it the "little neutral one," the neutrino.

It was not so much as a quantitative tabulation of the momenta of the decay products as the angles of the decay products were not compatible with a zero initial momentum of the nuclei. This webpage is about how estimate of the deficits in energy and momentum would reveal the nature of the neutrino. That nature could be of three types:

• A particle traveling at a velocity far less than the speed of light in a vacuum
• A photon-like particle with zero rest mass traveling at the speed of light
• A particle traveling at velocity near the speed of light.

Let ΔE and ΔP be the energy and momentum deficits in a nuclear decay and hence equall to the energy and momentum of the neutrino produced by that decay.

• For a particle of mass m traveling at nonrelativistic velocity v

#### ΔP = mv and ΔE = ½mv² hence v = 2(ΔE/ΔP) and m = ½(ΔP)²/ΔE

• For a photon-like particle traveling at the speed of light

#### ΔE = cΔP where c is the speed of light and hence (ΔE/ΔP) should be equal to c

• For a particle traveling at a speed close to the speed of light

#### ΔE = mc² where m = m0/(1 −(v/c)²)½and ΔP = (∂E/∂v) = m0v/(1 −(v/c)²)3/2 = mv/(1 −(v/c)²) and thus ΔE/ΔP = c(1 −(v/c)²)/(v/c) and hence (v/c) can be found by solving a quadratic equation and thus m0 = (ΔE/c²)(1 −(v/c)²)½.

Thus estimates of the deficits in energy and momentum might establish the nature of the neutrino and give estimates of its mass and velocity. The empirical estimates of the velocities of neutrinos are so close to the speed of light that the differences are not statistically significance. The range of the estimates is from slightly above the speed of light to slightly below. The upper bound on the rest mass energy of the electron neutrino, based upon cosmological considerations, is 50 electron volts (eV). This about one thousandth of the mass of an electron or a half of a millionth of the mass of a proton.