Physics for Non-Physicists: Decay Widths

While cross sections tell us how strong the interactions are between different types of particles, there is another quantity – the decay width – which tells us how quickly the decay of an unstable particle is likely to happen.

As in particle collisions, decays proceed in a probabilistic manner. If we start with an unstable particle, the probability that it will have decayed by a time interval t later is

$P(t) = \left(\frac{1}{2}\right)^{t/\tau_{1/2}} = \exp\left(-\frac{t}{\tau}\right) = \exp\left(-\Gamma t\right)$

(though checking resets the clock for decays) where τ1/2 is the half-life, τ is the lifetime, and Γ is the decay width. You can easily work out the relationships between these three values. They are just three different ways to parameterize the same quantity.

Half-lives are the most commonly used value for a general audience because reducing a quantity by factors of 2 is easier to visualize than reducing by factors of e. In high energy physics, there are several reasons to use the decay width rather than a lifetime or half-life.

In calculations, the decay width is directly analogous to the cross section. It includes factors for the kinematics of the initial and final state and a squared matrix element (or amplitude). If a decay cannot happen, the decay width will end up being 0, which is more mathematically useful than the corresponding infinite lifetime.

When calculating decay widths, there can be many different channels. Each decay channel (final state) has its own width (a partial width), while the sum of the partial widths gives the total width. So, if there are a series of decay channels labeled i, the total width is

$\Gamma = \frac{1}{\tau} = \sum\limits_{i} \Gamma_i.$

The different types of decays all occur with the same lifetime. The ratio of a partial width to the total width tells gives us the fraction of all decays that go into the channel(s) described by the partial width. This ratio is known as the branching fraction.

The decay width has effects beyond just telling us the lifetime. It turns out that the decay width also forces us to modify the mass of the decaying particle in calculations by adding an imaginary part. One interesting effect of this is that it means that unstable particles do not have a well-defined mass. For example, if you create a Z boson in a collision and it then decays to particles that you measure, even if you perfectly measure the momenta of the decay products you won’t necessarily be able to reconstruct the Z mass of 91.2 GeV. Rather, you’ll see a distribution of masses centered near 91.2 GeV with a characteristic width equal to Γ for the Z boson. Typically, the production and subsequent decay of an unstable particle results in us seeing a peak in the invariant mass (center of mass energy) distribution of the final state particles that we measure.

This example for the Z boson was used to definitively determine that there can only be 3 flavors of Standard Model neutrinos. Collider detectors don’t actually find neutrino interactions directly. Rather, their branching ratios affect the shape of the Z peak, so by measuring the peak in another channel, the number of neutrinos could be extrapolated from the shape. The shape of the Z peak as measured by the LEP electron-positron collider at CERN (LEP was replaced by the LHC, which sits in the same tunnel) almost perfectly matched the expected result for three neutrino flavors.  From this result, we know that any additional neutrinos must either not interact with the Z (these would be “sterile” neutrinos if they don’t interact directly with anything in the standard model) or they must be heavy enough to prevent the Z from decaying to the new neutrinos and to prevent any hints of their existence from popping up in other measurements.

Hadrons that undergo strong decays, such as the Δ (delta) particles (spin 3/2 particles made of up and down quarks – like an excited state of a proton or neutron) are often called as “resonances” rather than “particles” because their lifetimes are so short that they cannot be feasibly measured. However, by having short lifetimes, they have large widths (clear resonant peaks) that can be measured, so we can indirectly find the lifetime by looking at an energy spectrum.

Finally, measurements of things like branching fractions provide a very nice way to test different models that might predict the values of these things. In some cases, the width can make a measurement more difficult. We are lucky that the particle we think is the Higgs has a fairly light mass of 125 GeV. At this mass, its width is of order 1 GeV, so the experiments are actually largely limited by the detector resolution. If the mass were more like 1 TeV or more, the width would expand dramatically, eventually being about as large as the mass. In this case, the peak, if you could still call it that for such a large width, would not be easy to find above background levels, and a discovery would be extremely difficult if not impossible.