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Neglect radiation, take $E_{\tau }=M_Z/2\simeq 45.6\mathrm{GeV}$, $m_{\tau }\simeq 1.777\mathrm{GeV}$, and treat a fixed hadronic invariant mass $m_h=1.5\mathrm{GeV}$ as a single recoil system in ${\tau }^{-}\to h^{-}{\nu }_{\tau }$. With negligible neutrino mass,Thus $E_h^{\min ,\max }=\gamma (E_h^{\ast }\mp \beta p_h^{\ast })$, numerically approximately $32.50$ to $45.60\mathrm{GeV}$. In the ultrarelativistic limit the range is $E_{\tau }{m}_h^2/m_{\tau }^2\le E_h\le E_{\tau }$. A spread of hadronic invariant masses broadens these endpoints; an average mass alone is not a strict endpoint specification for the whole sample.

Let $P_{\tau }=+1$ denote a positive-helicity ${\tau }^{-}$, and measure $\vartheta$ from its laboratory flight direction in its rest frame. For a spin-zero hadronic recoil, the left-helicity tau neutrino and angular momentum conservation giveFor example a tau lepton with spin along $+z$ can emit a left-helicity neutrino along $-z$, carrying spin along $+z$, with the spin-zero recoil along $+z$. A positive-helicity tau lepton then gives a harder visible recoil. If $x_h=E_h/E_{\tau }$, $r=m_h^2/m_{\tau }^2$ and $\beta \simeq 1$, the more general tau hadronic polarimetry formula isFor spin-zero recoil ${\alpha }_h=1$. For a spin-one system produced by the V-A interaction, the longitudinal and transverse weights are $m_{\tau }^2:2m_h^2$, with opposite angular slopes; summing gives ${\alpha }_h=(m_{\tau }^2-2m_h^2)/(m_{\tau }^2+2m_h^2)$. At $m_h=1.5\mathrm{GeV}$ this is about $-0.175$, so the negative-helicity tau lepton has the higher mean total hadronic energy, about $39.43$ versus $38.67\mathrm{GeV}$, within that spin-one model. ^[ref]

The supplied mean invariant mass does not specify the recoil's spin composition or its helicity amplitudes. Consequently the last sign cannot be universal: positive helicity is harder for a spin-zero recoil, negative helicity is harder for the specified $1.5\mathrm{GeV}$ spin-one recoil, and a mixture requires its measured spectral weights. Polarization of an intermediate vector or axial-vector meson also changes the division of energy among its daughter hadrons; that information is not fixed by the total jet invariant mass alone.

The Goldhaber experiment inferred neutrino helicity from the circular polarization of a nuclear gamma ray. A spin-zero metastable europium-152 nucleus captures an inner-shell electron, producing a spin-one excited samarium-152 daughter and an electron neutrino; the daughter promptly emits a roughly $960\mathrm{keV}$ gamma ray to its spin-zero ground state. For a nearly stationary initial atom, the daughter recoils opposite the neutrino. Selecting photons emitted along that recoil relates the daughter spin projection, the photon helicity and the neutrino helicity.

The selection used resonant absorption and scattering in samarium. Nuclear recoil normally shifts an emitted gamma ray below the absorption resonance; the daughter recoil from electron capture supplies a compensating forward Doppler shift. Resonant scattering therefore preferentially selects forward photons, fixing their direction relative to the otherwise invisible neutrino. A samarium scatterer and scintillation counting identify this resonant component. Transmission through magnetized iron, whose Compton scattering depends on the relative electron and photon spins, then analyzes the circular polarization. Reversing the magnetization reverses the analyzing asymmetry and helps control instrumental offsets.

The selected photon and neutrino normalized helicities have the same sign: their momenta are opposite, while the daughter angular-momentum accounting gives opposite spin projections along the recoil axis. The observed negative photon helicity therefore establishes predominantly left-helicity neutrinos, consistent with the V-A interaction. It measures a helicity correlation, not a tiny neutrino mass or a literal observation of the neutral particle's trajectory.^[ref]

The local Fermi interaction gives a three-body leptonic decay width of the format tree level with negligible neutrino masses and universal weak coupling constant values. The fifth power follows from the dimension of $G_F$ and the three-body phase space; $f$ results from integrating the V-A interaction matrix element over the reduced phase space when the charged daughter has nonzero mass. It is more than a purely geometric volume correction because the spin-dependent matrix element is included.

As $z\to 0$, $f(z)\to 1$; increasing daughter mass reduces the available energy and suppresses the rate. The largest ratio is $m_{\mu }/m_{\tau }$, so the $\tau \to \mu$ correction is visibly below one. The electron is extremely light relative to either parent. Using the lepton masses givesThese reproduce the supplied factors to their stated accuracy. Radiative and finite-W boson corrections are separate small corrections when a precision lepton universality test is performed.

The ordering of Weyl components depends on the supplied basis transformation. In this paper use $u_W=U{u}_D=(\xi ,\eta {)}^{\mathsf{T}}$. Applying the printed $U$ to the printed Dirac matrices givesThus $\xi$ is right chiral and $\eta$ left chiral in this ordering. The free massless Weyl equations areThey describe spin-one-half massless fields in the two inequivalent chiral spinor representations of the Lorentz group. Their rotation generators agree, but their boost generators have opposite signs; spatial parity interchanges the two. A positive-energy particle in the $\xi$ field has helicity $+1/2$, and one in the $\eta$ field has helicity $-1/2$. Their antiparticles have the opposite helicities, so a Weyl field is not a field containing only one helicity for both particles and antiparticles.

Neutrinos provide the observed ultrarelativistic approximation to a left-chiral particle and right-helicity antiparticle. However, neutrino oscillation establishes nonzero neutrino mass differences, so physical neutrinos are not all exactly massless Weyl particles. Chiral Weyl fields occur in the Standard Model, but an isolated exactly massless elementary Weyl particle is not established by neutrino observations. Whether neutrino mass eigenstates are Dirac or Majorana remains a separate question.^[ref]

A mass eigenstate $|{\nu }_i⟩$ has a definite neutrino mass $m_i$ and acquires a definite propagation phase at fixed momentum. An interaction eigenstate is the neutrino flavor selected by a weak charged current: ${\nu }_e$ is produced or detected together with an electron, and ${\nu }_{\mu }$ together with a muon. The two bases are related by a unitary matrix, here the real rotation matrix $U$.

Inverting the supplied transformation gives $|{\nu }_e⟩=c|{\nu }_1⟩+s|{\nu }_2⟩$ and $|{\nu }_{\mu }⟩=-s|{\nu }_1⟩+c|{\nu }_2⟩$, where $c=\cos \theta$, $s=\sin \theta$. If $\theta =0$, these bases coincide: each produced neutrino flavor propagates as one mass eigenstate, and there is no neutrino oscillation, even if the two masses differ. Conversely, equal masses also prevent vacuum neutrino oscillation, even when a nonzero basis-rotation angle is written down.

For a deliberately simplified neutrino oscillation calculation, take two neutrino flavors related to two mass eigenstates by a real rotation matrix with angle $\theta$. A quantum superposition acquires different phases during propagation; projecting back onto the initial neutrino flavor gives the probabilityHere $L$ is path length and $E$ is neutrino energy; the numerical coefficient incorporates unit conversion.

The paper states the corresponding conversion formula.^[ref]

The path dependence has a geometrical origin. In an original spherical-Earth model, let $R$ be the radius and $h$ a fixed production height, neglecting detector depth. Let $\Theta$ be the zenith distance of the direction in the sky from which the neutrino arrives, so $c=\cos \Theta =+1$ means downward propagation. The backward ray from detector to production obeys $R^2+L^2+2RLc=(R+h{)}^2$. Its positive solution isThus $L(+1)=h$ and $L(-1)=2R+h$. The calculation explains why opposite directions test very different baselines, rather than attributing every observed direction to the same travel distance. Actual production heights and the angle between the outgoing charged lepton and incoming neutrino smear that relationship.

For an original chosen example, set $\theta =\pi /4$ and $\Delta m^2=2.5\times 10^{-3}{\mathrm{eV}}^2$. At $E=1\mathrm{GeV}$, $L=20\mathrm{km}$ gives $P_{\mu \mu }\simeq 0.9960$, while $L=500\mathrm{km}$ gives $P_{\mu \mu }\simeq 0.000279$. The first minimum is at $L/E\simeq 494.7km/GeV$. These are stipulated inputs, not a fit to the 1998 data. A broad distribution of true $L/E$, together with energy and directional uncertainty, smooths the sharp minima.

A direct amplitude check makes the mechanism explicit. DefineThen $S_{11}={\cos }^2\theta +{\sin }^2\theta e^{-2i\phi }$. Taking its squared absolute value yields the displayed survival formula because $1-\cos (2\phi )=2{\sin }^2\phi$. Unitarity gives $|S_{11}{|}^2+|S_{21}{|}^2=1$: disappearance from one channel becomes conversion to the other within this closed model.

This illustration does not describe all solar neutrino propagation. Coherent forward scattering in matter changes the effective propagation operator, and three active neutrino flavors require a larger mixing description. Even in the simplified model, neutrino oscillation measures a squared-mass difference; it does not determine the absolute neutrino mass.