Neutrino Oscillations in Matter. Refractive Index
Introduction to Neutrino Physics
Experiment: a plate between polarizers
One frequency, two polarizations. Along the axes of the plate the refractive indices are \(n_1\neq n_2\), and the waves acquire a phase difference \(\Delta\varphi=(n_1-n_2)kL\).
\(P=\sin^2 2\theta\,\sin^2\dfrac{\Delta\varphi}{2}\), where \(\theta\) is the angle between the polarizer and the axis of the plate; here \(\theta=45^\circ\).
Fresnel rings
A layer of thickness \(dz\) in the plane \(z=0\), scatterer concentration \(\mathcal N\). The observation point \(P\) lies on the axis \(z\) behind the layer.
A scatterer at a distance \(b\) from the axis sends to \(P\) the wave \(f(\vartheta)\,e^{ikr}/r\), where \(r=\sqrt{z^2+b^2}\), \(\cos\vartheta=z/r\).
The points of the layer with the same \(r\) form a ring: a sphere of radius \(r\) centered at \(P\) intersects the plane of the layer along a circle.
Fresnel zones: \(r_m=z+m\lambda/2\). For neighboring rings the path to \(P\) differs by \(\lambda/2\), and the contributions arrive in antiphase.
Why is the forward amplitude important?
A ring of radius \(b\) has area \(2\pi b\,db\), and \(b\,db=r\,dr\). The factor \(r\) cancels the \(1/r\) of the spherical wave, and the sum over the layer is written exactly:
\[ d\psi_{\rm sc}=\mathcal N\,dz\int\!2\pi b\,db\,f(\vartheta)\frac{e^{ikr}}{r} =2\pi\mathcal N\,dz\int_z^\infty\! f(\vartheta)\,e^{ikr}\,dr . \]
The beam is finite and the wave was switched on at some time: the distant rings are damped by the factor \(e^{-\varepsilon r}\). We integrate by parts, the lower limit \(r=z\) is the center of the layer:
\[ \int_z^\infty f\,e^{ikr}\,dr\;\simeq\;\frac{i}{k}\,f(0)\,e^{ikz} \quad\Longrightarrow\quad d\psi_{\rm sc}=i\,\frac{2\pi\mathcal N}{k}\,f(0)\,dz\,e^{ikz}. \]
The remainder is small as long as \(f(\vartheta)\) hardly changes over the first zone: \(z\gg ka^2\), where \(a\) is the size of the scatterer.
The contribution comes from the region of small angles, which is why \(f(0)\) appears. The factor \(i\) — a rotation by \(90^\circ\) — is given by the geometry of the plane layer. The phase of the layer’s contribution is equal to \(90^\circ+\arg f(0)\). Together with the incident wave behind the layer
\[ \begin{aligned} \psi&=e^{ikz}\,[1+i\,d\varphi],\\ d\varphi&=\frac{2\pi\mathcal N}{k}\,f(0)\,dz . \end{aligned} \]
Many layers: homogeneous matter
Let us add free propagation through \(dz\) to the result for a thin layer (waves \(e^{ikz-i\omega t}\)):
\[ \psi(z+dz)=e^{ikdz} \left[1+i\,\frac{2\pi\mathcal N}{k}f(0)\,dz\right]\psi(z). \]
Let us divide the thickness \(D\) into \(M\) layers of \(dz=D/M\) each:
\[ e^{ikD}\left(1+i\,\frac{2\pi\mathcal N f(0)}{k}\frac{D}{M}\right)^M \xrightarrow[M\to\infty]{} e^{ikD}\exp\!\left(i\,\frac{2\pi\mathcal N f(0)}{k}D\right). \]
\[ \psi(z)=e^{ik_{\rm med}z}\psi(0), \qquad k_{\rm med}=k+\frac{2\pi\mathcal N}{k}f(0), \]
\[ k_{\rm med}=nk:\qquad \boxed{n=1+\frac{2\pi\mathcal N}{k^2}f(0)}. \]
The exponential has accounted for all orders in \(\mathcal N D\), while \(n-1\) was found in the first order in the concentration of independent scatterers; the backscattered wave is neglected (\(|n-1|\ll1\)). In dense glass an atom is also acted upon by the field of neighboring atoms (the Lorentz–Lorenz formula), but the meaning of \(n\) is the same.
The nature of the first refractive index
of a neutrino
If a neutrino of any flavor scatters elastically through zero angle in its interaction with the particles of the medium via the exchange of a \(Z\) boson, then:
the state of the medium does not change, and
the neutrino keeps its direction.
The forward-scattered wave adds coherently to the wave that passed through without scattering.
A contribution to the refractive index common to the active flavors appears.
\(\alpha=e,\mu,\tau\), \(\quad f=e^-,p,n\).
Nature of the second
refractive index of the neutrino
An ordinary (normal) medium consists of atomic nuclei and electrons. It contains no \(\mu\), \(\tau\).
An electron neutrino \(\nu_e\) can scatter elastically off an electron through the exchange of a \(W^+\) boson, which changes the refractive index \(\nu_e\).
Therefore the matter contributions \(\delta n_e\) and \(\delta n_x\) are different.
Matter changes both the refractive indices and the eigenstates of the vacuum “crystal”.
Neutrino state
A neutrino with a given energy \(E\) travels along \(z\). We use \(\hbar=c=1\), and drop the common factor \(e^{-iEt}\).
We consider two flavors: \(\nu_e\) and \(\nu_x\) (\(\nu_\mu\) or \(\nu_\tau\) or a combination of them).
The state and its flavor amplitudes:
\[ |\psi\rangle=\psi_e|\nu_e\rangle+\psi_x|\nu_x\rangle, \qquad \psi_\alpha=\langle\nu_\alpha|\psi\rangle. \]
- The same vector can be expanded in the mass states \(|\nu_1\rangle,|\nu_2\rangle\):
\[ |\psi\rangle=a_1|\nu_1\rangle+a_2|\nu_2\rangle, \qquad a_i=\langle\nu_i|\psi\rangle. \]
Diagram of the bases in the state space. The amplitudes are in general complex.
New axes — new eigenstates
The eigenstates of the full medium satisfy the equation
\[ \hat n_{\mathrm{mat}}|\nu_{\mathrm{mat},i}\rangle =n_{\mathrm{mat},i}|\nu_{\mathrm{mat},i}\rangle. \]
Hence the spectral representation has the familiar form:
\[ \boxed{\hat n_{\mathrm{mat}}=\sum_i n_{\mathrm{mat},i} |\nu_{\mathrm{mat},i}\rangle\langle\nu_{\mathrm{mat},i}|.} \]
- \(n_1,n_2\) are the vacuum eigen-indices.
- \(\delta n_e,\delta n_x\) are the flavor contributions of matter.
- \(n_{\mathrm{mat},1},n_{\mathrm{mat},2}\) are the eigen-indices of the full medium.
