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\begin{document}

\title{Changing Neutrinos}
\date{09. Januar 2009}
\author{Steffen Klemer}
\maketitle

\begin{abstract}
Neutrino physics gained a huge momentum in the last couple of years. What once
evolved as a simple idea to save energy conservation has turned into a
theoretical and experimental very interesting subject. In this paper I
will present some basic ideas of the neutrino oscillation phenomenon.
\end{abstract}

\section{Neutrinos}
\subsection{The Invention}

In 1930 Pauli proposed in his very famous letter starting with "Dear
radioactives,..." a particle \emph{Neutron} that is produced in $\beta$-decay and does
not interact. With such a brave hypothesis it was finally possible to
explain the continuous energy spectrum in $\beta$-decay which should be
a single peak if it was a two body decay and energy+momentum
conservation hold. Later he commented his \emph{invention} of this
ghost with "I've done something terrible by proposing a particle that
cannot be detected - something no theorist should ever do." After the
discovery of the neutron by Chadwick in 1932 Pauli's particle was
renamed by Fermi to its present-days name.

While Pauli guessed a mass between the electron and the proton it soon
became widely accepted that it should be massless. Direct searches
have ruled out masses over 1eV which is well below the electron mass
of 0.5MeV, the lightest known massive particle.

\subsection{The Standard Model}

	The Standard Model (SM) of particle physics arises from a
	SU(3)$\times$SU(2)$\times$U(1) local gauge symmetry. I will concentrate on
	the electro-weak sector, the SU(2)$\times$U(1) part. It is constituted by
	3 charged Leptons $e$, $\mu$, $\tau$ and the corresponding neutral
	neutrinos $\nu_e$, $\nu_\mu$, $\nu_\tau$. To each particle belongs an anti-
	particle, differing in charge and parity. Between charged particles the
	electromagnetic force is carried by the photon and between all 6 particles
	the weak force by the charged $W$- and the neutral $Z$-Bosons. The
	representation of the particles in the theory is a Vector-Axial Vector
	doublet in agreement with experiments.


\subsection{Mass in the Dirac equation}

	I will show that the mass couples left and right handed particles.
	
	Therefore I use the Feynman slash notation: $A\!\!\!/ := \gamma^\mu A_\mu$ with $\gamma^\mu;\;\mu=0..3$
	the Dirac Gamma-Matrices and $\gamma^5:=\I \gamma^0\gamma^1\gamma^2\gamma^3$. $\gamma^5$ is hermitean,
	$(\gamma^5)^2=\mathbbm{1}$. I conclude that $\gamma^5$ is a nice operator with
	eigenvalues $EW=\pm 1$ and it is possible to diagonalize it so one gets in the \emph{chiral}
	representation $\gamma^5=diag(1,1,-1,-1)$. The eigenfunctions are
	\begin{align*}
		\gamma^5 \psi_R &= + \psi_R \text{ right handed spinor} \\
		\gamma^5 \psi_L &= -\psi_L \text{ left handed spinor}
	\end{align*}
	One can always decompose a generic spinor: $\psi=\psi_R+\psi_L$. Now one defines the chiral projection operators
	with the following properties:
	\begin{align*}
		P_{R,L} &= \frac{1\pm \gamma^5}{2} \\
		\psi_{R,L} &= P_{R,L} \psi \\
		P^\dagger_{R,L} &=  P_{R,L} \\
		P_{R,L}\gamma^0 &= \gamma^0 P_{L,R}
	\end{align*}
	
	I will use this in the Dirac Lagrangian:
	\begin{align*}
		\mathscr{L} &= \bar{\psi}(x) (\I \overleftrightarrow{\partial\!\!\!/}-m)\psi(x) \\
		\text{with } \bar{\psi} &= \psi^\dagger \gamma^0 \\
		\text{and } \overleftrightarrow{\partial}_\mu &= \frac{\overrightarrow{\partial}_\mu-\overleftarrow{\partial}_\mu}{2} \\
		\Rightarrow\;\; \mathscr{L} &= (\bar\psi_R + \bar\psi_L)(\I \overleftrightarrow{\partial\!\!\!/} - m) (\psi_R+\psi_L)
	\end{align*}
	
	This is pretty complicated so we look at some individual parts:
	\begin{align*}
		\bar\psi_R &= \overline{P_R\psi} = (P_R\psi)^\dagger\gamma^0 = \psi^\dagger P_R\gamma^0 \\
		&= \psi^\dagger \gamma^0 P_L =\bar\psi P_L \\
			\Rightarrow\;\;	\I \bar\psi_R \overleftrightarrow{\partial\!\!\!/} \psi_L &= \I \psi P_L \overleftrightarrow{\partial\!\!\!/} P_L \psi = \I \bar\psi \overleftrightarrow{\partial\!\!\!/} P_R P_L \psi \\
			 &= 0 = \I \bar\psi_L \overleftrightarrow{\partial\!\!\!/} \psi_R \\
	m\bar\psi_R\psi_R &= m\bar\psi P_L P_R \psi \\
	&= 0 = m\bar\psi_L\psi_L \\
	\Rightarrow\;\; \mathscr{L} &= \underbrace{\bar\psi_R \I\overleftrightarrow{\partial\!\!\!/} \psi_R + \bar\psi_L \I\overleftrightarrow{\partial\!\!\!/} \psi_L}_{\text{individual kinetic terms}}\\
	 &\;\; - \underbrace{m(\bar\psi_R\psi_L + \bar\psi_L\psi_R)}_{\text{coupled mass term}}
	\end{align*}
	Now we can write the equations of motion:
	\begin{align*}
		\I \partial\!\!\!/ \psi_R &= m\psi_L \\
		\I \partial\!\!\!/ \psi_L &= m\psi_R
	\end{align*}

	One often combines left and right handed wave functions in the chiral
	representation $\psi=(\psi_R, -\psi_L)$. For $m=0$ the left and right handed
	solution decouple. It turns out that right handed Leptons in general don't
	couple to the weak interaction and as neutrinos are neutral the right handed
	neutrinos can't be measured. So we only use the left handed of the chiral
	components in the SM. The anti-neutrino is consequently right handed. As a
	note besides: for $m=0$ chirality equals helicity, the generalization of the
	angular momentum.
	
\subsection{Higgs mechanism}

	In the SM the masses of the charged Leptons are generated by a symmetry
	break of the vacuum (and therefore all its excitations - our particles). In
	the SM we get two neutrino contributions to the currents. One is a neutrino -
	neutrino interaction in the neutral weak Noether current. The other one is a
	coupling to a lepton of the same flavor:
	
	\begin{align*}
		j^\varrho_L &= 2 \sum_{\alpha=e,\mu,\tau} \bar\nu_{\alpha,L} \gamma^\varrho l_\alpha;\;\;\varrho=0..3 \\
	\Rightarrow\;\; L_\alpha &= \int \D^3x j^0_\alpha (x) \text{ as the \emph{charge}}
	\end{align*}
	We can interpret the \emph{charge} as the flavor lepton number. As we have
	the U(1)-symmetry, it is conserved as well as the total lepton number $L=L_e+L_\mu+L_\tau$.

\section{Mixing}

	Now we consider massive neutrinos. One possible way to give them masses is
	adding another Higgs-Term to the Lagrangian. The problem that arises is,
	that the flavor eigenstates don't necessarily have to be mass eigenstates.
	But one after the other. First the important new term in the Lagrangian (there are others):
	\begin{align*}
		\mathscr{L}_{H,L} &= \ldots \bar{\vec{\nu}}_L Y' \vec{\nu}_R\ldots + \text{ Herm. conj.}
	\end{align*}
	
	The raise of right handed neutrinos is clear - the fields don't decouple
	anymore. $\vec{\nu} = (\nu_e,\nu_\mu,\nu_\tau)$. $Y$ is the Yukawa coupling, a
	3$\times$3 complex matrix. It is possible to diagonalize any complex,
	quadratic matrix by a biunitary transformation (see \cite[sec 4.1]{chung})
	with two unitary matrices $V_L,V_R$: $Y = V_L^\dagger Y' V_R$. One defines
	the massive neutrino array as:
	\begin{align*}
		\vec{n}_{L,R} = V_{L,R}^\dagger \vec{\nu}'_{L,R} = (\nu_{1_{L,R}},\nu_{2_{L,R}},\nu_{3_{L,R}})
	\end{align*}
	For $\vec{n}_{L,R}$ the mass term is diagonal but the current becomes:
	\begin{align*}
		j^\varrho = 2 \bar{\vec{\nu}}'_L \gamma^\varrho \vec{l}_L = 2 \bar{\vec{n}}_L V^{\nu\dagger}_L \gamma^\varrho \vec{l}_L
	\end{align*}
	
	So the mass eigenstates and the flavor eigenstates are connected via a
	unitary 3 $\times$ 3 matrix. Now one asks how many parameters one needs for
	this matrix. Naively that are 3 angles and 6 phases. But 5 phases have no
	physical meaning as they vanish in all physical terms in $\mathscr{L}$.
	So we are left with 3 angles and one phase that one usually writes as
	(with $c_{ij}=\cos(\theta_{ij}),s_{ij}=\sin(\theta_{ij})$)
	\begin{align*}
		U =
		\begin{pmatrix}
		1 & 0 & 0 \\
		0 & c_{23} & s_{23} \\
		0 & -s_{23} & c_{23}
		\end{pmatrix}
		\begin{pmatrix}
		c_{13} & 0 & s_{13}e^\delta \\
		0 & 1 & 0 \\
		-s_{13}e^\delta & 0 & c_{13}
		\end{pmatrix}
		\begin{pmatrix}
		c_{12} & s_{12} & 0 \\
		-s_{12} & c_{12} & 0 \\
		0 & 0 & 1
		\end{pmatrix}
	\end{align*}		

	In bra-ket notation this reads
	\begin{align*}
		\ket{\nu_\alpha} &= \sum_{k=1}^3 U^\dagger_{\alpha k} \ket{\nu_k} \quad\quad\quad
		\ket{\nu_k} = \sum_{\alpha=e,\mu,\tau} U_{\alpha k} \ket{\nu_\alpha}
	\end{align*}
	or in other words: flavor neutrinos are constituted by a mixing of 3 different
	massive neutrinos and the other way round.

\section{Oscillations}

	Now it is time to look at the propagation of flavor neutrinos. We can
	imagine an electron neutrino created in a fusion process in the sun or an
	anti-electron-neutrino from a beta decay in a reactor.
	
	The massive neutrinos are eigenstates of the Hamiltonian $\mathscr{H}\ket{\nu_k} = E_k \ket{\nu_k}$ with $E_k^2=\vec{p}^2+m_k^2$. So if we are interested in the time evolution we need Schrödinger's equation:
	\begin{align*}
		\I \partial_t\ket{\nu_k(t)} &= \mathscr{H}\ket{\nu_k(t)} \\
		\Rightarrow\;\; \ket{\nu_k(t)} &= e^{-\I E_kt} \ket{\nu_k}
	\end{align*}
	Now we are interested in flavor eigenstates:
	\begin{align*}
		\ket{\nu_\alpha(t)} &= \sum_k U^\dagger_{\alpha k}e^{-\I E_kt} \ket{\nu_k}
	\end{align*}
	For an ultra relativistic neutrino\footnote{Experiments usually only see neutrinos above a certain threshold of some MeV. This justifies this assumption pretty well.} we get:
	
	\begin{align*}
	E_k &\approx E + m_k^2/2E &\text{where } E=|\vec{p}| \\
	\Rightarrow\; E_k - E_j &\approx \Delta m^2_{kj}/2E &\text{with } \Delta m^2_{kj} = m_k^2-m_j^2
	\end{align*}
	
	This means, the energy and that's why the propagation depends on the mass. The
	interesting property is the transition probability between two flavor states:
	\begin{align*}
		P_{\nu_\alpha\rightarrow \nu_\beta}(t) &= |\langle \nu_\beta|\nu_\alpha(t)\rangle |^2 \\
		&= \sum_{k,j} U^\dagger_{\alpha k} U_{\beta k}U_{\alpha j} U^\dagger_{\beta j} e^{- \frac{\I \Delta m^2_{kj}t}{2E}}
	\end{align*}
	or with $t=L$ for the distance to the source $L$ as we have ultra-relativistic neutrinos:
	\begin{align*}
		P_{\nu_\alpha\rightarrow \nu_\beta}(L) &= \sum_{k,j} U^\dagger_{\alpha k} U_{\beta k}U_{\alpha j} U^\dagger_{\beta j} e^{- \frac{\I \Delta m^2_{kj}L}{2E}}
	\end{align*}
	
	What does it mean? As $U$ is a constant of nature we just have to tune $E/L$
	to determine the mass difference as the frequency. In the case of a diagonal
	$U$ (mass eigenstates equal flavor eigenstates) we would get
	$\delta_{\alpha\beta}$, otherwise a real oscillation pattern. And in fact from
	quite a dozen experiments we now know that\cite{pdb}
	\begin{align*}
	\Delta m^2_{12}&=8(3)\cdot 10^{-5} \mathrm{eV} \\
	 \Delta m^2_{23}&=2.4(5)\cdot 10^{-3} \mathrm{eV}	\\
	 \sin^22\theta_{12} &= 0.86^{+0.03}_{-0.04}  \\
	 \sin^22\theta_{23} &> 0.92 \\
	 \sin^22\theta_{13} &< 0.19
	\end{align*}
	
	This (standard) approach of explaining neutrino oscillation has some drawbacks:
	\begin{enumerate}
		\item We assume the same $\vec{p}$ for all $\nu_k$. This is not true but
		even in more correct calculations $P$ stays the same.
		\item As (at least 2) neutrinos do have a mass, $v\neq c$. Also we need
		wave packets
		to describe the propagation in a consistent way. Packets of different
		mass-eigenstates have slightly different group velocities which leads to a
		separation of the packets with time (and distance). But this effect is
		very, very small.
	\end{enumerate}
	
	We see a similar behavior for quarks (the famous CKM-Matrix) but not for charged leptons. All 3 are exactly the same except for mass. So the only way to
	distinguish them is their mass.

	The following picture is taken from the KamLAND experiment and shows the 
	oscillation pattern of anti electron neutrinos of different nuclear reactors
	each in a distant of around 200km.

\includegraphics[width=0.5\textwidth]{sinus.png}

\section{Finally}

	We've seen an explanation for the neutrino oscillation. This is nowadays in
	a pretty good agreement with experiments. Some elements of the matrix are
	still missing (like the phase) as well as the absolute mass of one of
	the neutrinos.
	
	What can we do with this knowledge? We can use neutrinos to study the sun,
	supernovae, observe the creation of plutonium in reactors, communicate over
	big distances, measure the CP-violation or build a neutrino beam of death.
	

\begin{thebibliography}{[AaaXX]}
\bibitem[Giu07]{chung} Giunti, C; Kim, C.: 
  \emph{Fundamentals of Neutrino Physics and Astrophysics}
  Oxford University Press, 2007
\bibitem[Gom08]{Cadenas}  Gomez-Cadenas, J-J.:
	\emph{Neutrino Physics} (Lecture at CERN Summer Student Program 2008)
Video+Slides: \url{http://indico.cern.ch/conferenceDisplay.py?confId=34693}, 2009-01-04
\bibitem[Les03]{lesch} Lesch, H.: \emph{Wo sind die Neutrinos}, \href{http://www.br-online.de/br-alpha/alpha-centauri/alpha-centauri-neutrinos-2003-ID1208269738527.xml}{http://www.br-online.de/br-alpha/alpha-centauri/}, 2009-01-04
\bibitem[PDB08]{pdb}C. Amsler et al. (Particle Data Group), Physics Letters B667, 1 (2008)
\end{thebibliography}

You can find this document (with real hyperlinks) and it's source on \url{http://www.noch-mehr-davon.de/vortr.shtml}.

\end{document}