= Two-flavour vacuum survival model
{title2=$P_{\mu\mu}$}
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 <probability>
$$
P_{\mu\mu}=1-\sin^2(2\theta)\sin^2\!\left(1.27\frac{\Delta m^2}{\mathrm{eV}^2}\frac{L}{\mathrm{km}}\frac{\mathrm{GeV}}{E}\right),\qquad \Delta m^2=m_2^2-m_1^2.
$$
Here $L$ is path <length> and $E$ is <neutrino> <energy>; the numerical coefficient incorporates unit conversion.
The paper states the corresponding conversion formula.<Evidence for oscillation of atmospheric neutrinos by Y. Fukuda and others>[PDF p. 2, Eq. (1)]{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 is
$$
L(c)=-Rc+\sqrt{(R+h)^2-R^2(1-c^2)}.
$$
Thus $L(+1)=h$ 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 $\Delta m^2=2.5\times10^{-3}\,\mathrm{eV}^2$. At $E=1\,\mathrm{GeV}$, $L=20\,\mathrm{km}$ gives $P_{\mu\mu}\simeq0.9960$, while $L=500\,\mathrm{km}$ gives $P_{\mu\mu}\simeq0.000279$. The first minimum is at $L/E\simeq494.7\,\mathrm{km/GeV}$. 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. Define
$$
U=\begin{pmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{pmatrix},\qquad
S=U\begin{pmatrix}1&0\\0&e^{-2i\varphi}\end{pmatrix}U^T,
\qquad \varphi=1.27\frac{\Delta m^2}{\mathrm{eV}^2}\frac{L}{\mathrm{km}}\frac{\mathrm{GeV}}{E}.
$$
Then $S_{11}=\cos^2\theta+\sin^2\theta e^{-2i\varphi}$. Taking its squared <absolute value> yields the displayed survival formula because $1-\cos(2\varphi)=2\sin^2\varphi$. <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>.
Back to article page
