### Guest post: Why Flavour Tastes Funny

26May08

Today is a guest post from my officemate, Tracey Li, who is the organiser of the IPPP Neutrino Journal Club and is currently working on long baseline neutrino phenomenology. The post below is on the very interesting topic of why it isn’t possible to quantise weak eigenstates.

A couple of weeks ago, in between drawing penguins and hunting down elusive faculty members, Flip asked me to write a piece about the GSI anomaly, arXiv:0801.2079 [1]. This nucl-ex paper was chosen as the paper for the IPPP’s first neutrino journal club meeting some time ago [2]. The discussion strayed to topics including quantum coherence, the definition of weak/ flavour eigenstates, and the problem with quantising them. Similarly, my attention’s strayed from “GSI anomaly” to these topics, so I’m not actually going to be writing about the anomaly at all, but rather about the associated question: Why it isn’t possible to quantise weak eigenstates?

Particle mixing is known to occur between quarks, mesons, and neutrinos. It occurs when the mass eigenstates (eigenstates of the free Hamiltonian) of a particle do not coincide with the interaction eigenstates.

If you’re familiar with basic quantum field theory and the process of quantising a free field, you’ll know that the field $\phi$ and its conjugate $\pi$ satisfy the equal-time (anti-)commutation relations, as do the creation and destruction operators for each type of particle.

So what’s the problem with weak eigenstates? The operators and fields for each of the mass eigenstates, $\mid\psi_{i}\rangle$, satisfy these required (anti-)commutation relations. The interaction states, $latex\mid\chi_{a}\rangle$, however, are linear combinations of mass eigenstates:

$\mid\chi_{a}\rangle = \sum_{i} U_{ai}\mid\psi_{i}\rangle$,

where $U_{ij}$ are elements of the appropriate unitary mixing matrix. Each operator in the Fourier expansion of a field comes with a coefficient containing factors of $E, m$, (often hidden in the normalisation of the spinors) and each mass eigenstate has a well-defined energy, $E_{i}$, and mass, $m_{i}$. When taking a linear combination of these mass states to define flavour states, the flavour operators are then defined as including factors of $U_{ai}, E_{i}$ and $m_{i}$. Because one cannot extract a common factor of $E$, the (anti-)commutation relations no longer hold.

I was going to try to avoid putting in any equations at all, but I suspect that last paragraph is otherwise totally incomprehensible to anyone other than myself. To clarify I’ll stick a couple of lines in, ignoring all but the relevant factors just so I can illustrate the point:

For the mass eigenstates
$\psi_{i}: [A_{i}^{mass}(p),A_{j}^{mass\dagger}(q)] \sim (m_{i}m_{j}/E_{i}E_{j})^{1/2}) \delta_{ij}\delta^{3}(p-q)$

Now for the flavour states
$\chi_{a}: [A_{a}^{weak}(p),A_{b}^{weak\dagger}(q)] \sim \delta^{3}(p-q)\sum_{i}U_{ai}U_{ib}^{\ast}m_{i}/E_{i}$

Even though the matrix U is unitary, the kinematic factors spoil the diagonality of the flavour basis (Phys. Rev. D 45, 2414). As an interesting (and potentially time-comsuming) aside, one can consult Phys. Rev. 107, 307, referenced in the above paper, to follow through the quantisation of a Majorana field, for which the upper two components of a four-component spinor are related to the lower two components via charge conjugation. [3] The result looks a little unfamiliar because it’s not often that you see a field expansion explicitly containing helicities ($h = \pm 1$) in the operator coefficients, and the sum is over helicities rather than spins, but it’s nice
to see how it can be done.

Anyhow, the conclusion is that because there are no well-behaved (anti-)commutation relations, one cannot construct a Fock space of weak states. So plane wave descriptions of these states do not exist as they do not have definite energies and momenta. That’s the quantum field theoretical justification. From a quantum mechanical point of view, flavour states are a superposition of definite-energy (stationary) mass states, so the flavour states are non-stationary states, hence we are led to the same conclusion. We must instead use wave-packets to describe flavour states.

For my own favourite particle, the neutrino, this leads to the point that energy-momentum conservation does not hold in neutrino oscillations. This is rather shocking – is nothing
sacred?! However, upon closer investigation this should have been obvious since wave-packets have a spread of energies and momenta. The conservation laws for particle states hold only within the extent of these uncertainties (see Giunti’s paper 0801.0653 for a nice overview). Bilenky (hep-ph/0605228) presents this using QFT, showing that flavour states are not invariant under time translation, hence energy non-conservation [4].

On the topic of energy conservation, Giunti’s paper also contains a nice detailed explanation of why physical neutrino states do not correspond to the standard expression for $\mid\chi_{a}\rangle$ given above, because one has to take into account the kinematics of the interaction which produced the neutrino. Recommended reading if you enjoy these sort of discussions! And if you do and can shed any more light on anything I’ve touched on, do please share your thoughts…

Hopefully normal service from Mr. Tomato himself will be resumed shortly!

Postscript

[Editor’s remark:] Giunti’s paper Phys. Rev. D 45, 2414 mentioned above provides an alternate process-dependent definition of neutrino flavour eigenstates.

Notes

[1] Just swallow hard when you see the “nucl-ex” label – we did! You can skip to the discussion section for the physics. There has been a recent debate about this phenomenon can be explained by neutrino mixing: 0801.1465, 0801.4639, 0804.1099, 0805.0431, 0805.0435.

[2] The neutrino club is one of Flip’s legacies to us. Others include an office pet (Ed: I’ll probably write a small post about this some time), awareness of the existence of pi(e) day (Ed: the British system of writing day-month means Pi day isn’t well known here), another welcome on-line distraction (this blog), and the very sage advice that one should always be extremely careful when writing said blog because you’ll be surprised at who reads it.

[3] I love reading old papers like this from the 50’s and 60’s, although notation and undefined conventions can often prove to be a battle. Did hermitian conjugates not exist in 1957?! (Ed: I hate reading old papers from the 50s and 60s… Did LaTeX not exist in 1957?!)