Monday, December 17, 2007

Three Neutrino Families

[Source: The ALEPH Collaboration et al., Precision Electroweak Measurements on the Z Resonance, Physics Reports 427 (2006) 257; arXiv: hep-ex/0509008v3, Fig. 1.13.]

In my car, parts of the glove compartment start to make a rattling noise at a specific engine speed. That's a typical example of a resonance: A mechanical system that can sustain vibrations of a certain frequency, called the natural frequency, will start to vibrate if it is driven by an external excitation with a frequency that closely matches the natural frequency. If there is damping in the vibrating system, the response to the external excitation is smaller than without damping, but on the other hand, response sets in already for larger mismatches between driving and natural frequency than without damping.

In a very similar way, the annihilation of electrons and positrons in a particle collider can produce a new particle if the centre-of-mass energy of the electron-positron pair matches the energy corresponding to the mass of the particle. Most particles produced in this way - for example, the J/Ψ and Υ mesons, or the electrically neutral Z boson, the massive partner of the photon in the electroweak theory, - are unstable and will decay on a very short time scale into other particles. For example, the Z particle can decay in pairs of charged leptons (electron and positron, muon and antimuon, tau and antitau), in neutrino-antineutrino pairs (one pair for each charged lepton pair), or in quark-antiquark pairs, which will end up in some hadrons. These processes can be represented by the diagrams

or or .

As a result, the Z boson, with a mass of 91.2 GeV/c², decays with a half-life of about 10-25 seconds into any of these particle pairs. But this means, in the language of resonances, that the Z is quite strongly damped, and that it can be produced also if there is a certain mismatch between the mass of the Z and the centre-of mass energy of the electron-positron pair.

This is shown in the green curve in the figure, which fits the data measured at different experiments at the now dismantled Large Electron Positron Collider (LEP) at CERN. The curve shows the total cross-section σhad for the production of hadrons as a function of the center of mass energy Ecm. This cross-section has a clear maximum at the energy corresponding to the mass of the Z, but this is not a sharp peak - the bump is quite broad, and has a certain width Γ of about 2.5 GeV. This width is related to the lifetime τ of the Z by the relation Γ·τ = ħ, where ħ is Planck's constant divided by 2π.

But there is more information hidden in the resonance curve of the Z: From all the different decay products, charged lepton pairs, such as muon-antimuon pairs, and hadrons can be measured in detectors, but the neutrino-antineutrino pairs are elusive. How can one be sure that they are there, and can one say anything about their number?

The exciting point is that one can see that they are there, and that there are three different families of neutrinos: the electron-, muon-, and tau neutrino with their respective antineutrinos.

In fact, while not directly detectable, the neutrinos contribute to the damping of the Z boson, and hence, to the height and width of the resonance peak. For example, one can calculate how the resonance curve should look like if there were only two neutrino families, or four neutrino families: The Z would be less, or more damped than for three families, and the peak should be higher and sharper, or flatter and wider, respectively. The corresponding curves are shown in the plot, labelled as "2ν" and "4ν", respectively.

Data fit perfectly well the resonance curve for three neutrino families. Taking into account the caveat that a neutrino has to couple to the Z (it should not be "sterile"), and that its mass should be below half the mass of the Z (the Z should be able to decay into the particle-antiparticle pair) for the method to work, these LEP data show that there are three neutrino families - 2.9840 ± 0.0082, to be precise...

All the details from The ALEPH Collaboration, the DELPHI Collaboration, the L3 Collaboration, the OPAL Collaboration, the SLD Collaboration, the LEP Electroweak Working Group, the SLD electroweak, heavy flavour groups in Precision Electroweak Measurements on the Z Resonance, Physics Reports 427 (2006) 257; arXiv: hep-ex/0509008v3

OPAL Events at LEP1 is a beautiful collection of event displays that shows how the different processes that are involved in the creation and decay of the Z boson look like in the particle detector.

The form of the resonance peak is known as Breit-Wigner distribution. The width Γ is the full width at half maximum (FWHM).

This post is part of our 2007 advent calendar A Plottl A Day.


  1. Could there be fourth neutrino type that was at least as massive as a Z0? Presumably then the Z0 would be unable to decay into it and so it would not show up on this plot?

  2. Hi Steve:

    In principle, yes. As we've written

    its mass should be below half the mass of the Z (the Z should be able to decay into the particle-antiparticle pair) for the method to work

    Such a very heavy neutrino might mess up some other data though, astrophysics/cosmology wise. I am not exactly sure about the constraints from this.



  3. But wouldn't a fourth neutrino type imply an entire fourth generation of quarks/leptons/etc, which one assumes would create all kinds of difficult consequences?

    Or is there some way we could possibly posit a fourth neutrino family, without having to add an entire new row onto that lepton periodic table?

  4. Hi Coin:

    Well, if they are sufficiently heavy, I'd think it was possible. Besides this people like to play around with the mentioned 'sterile' neutrinos that don't carry flavor and thus wouldn't be bound to charged leptons. These could even be lighter than 45 GeV since the Z wouldn't decay into them. E.g. some people have played around with 'hot' dark matter made of sterile neutrinos I think somewhere in the MeV range. I can't say though I find these speculations neither appealing nor plausible. Best,


  5. Hi Stefan and Bee,

    Your 'resonance' wiki reference with the helical bouncing spring reminds me of the likely helical trajectories for QM particles such as neutrinos and GR celestial bodies.

    See this example from Cleon Teunissen, Nederlands
    'Nonsymmetric velocity time dilation' [bottom of web page] Picture 7. Animation
    A straight worldline and a helical worldline.

  6. Hi Bee and Stefan, thanks for this interesting post. A question: you say

    "But this means, in the language of resonances, that the Z is quite strongly damped..."

    I don't understand what the word "this" refers to, i.e., what aspect of this decay process can be thought of as corresponding to a strong damping, in the language of resonances. Could you elaborate?

  7. Hi mono,

    the Z has many different "decay channels", i.e. options to decay in various other particles. As a result, the Z decays rapidly, and its energy is "dissipated" - it will distributed over the decay products.

    In a vibrating system, damping means dissipation of the energy which has been applied to the system to set it into vibration. The stronger the damping, the faster the dissipation of the energy, and the sooner the vibrations comes to a stop.

    So, the Z with its many efficient decay channels and fast decay is analogous to a strongly damped resonating system - including the characteristic feature of a broadened peak.

    Maybe it helps to compare the Z peak to the J/Psi or Upsilon states - these states have a longer lifetime, and as a consequence, they correspond to much sharper lines - as for a resonance with small damping.

    Hope that helps, Stefan


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