# Soft particle

in HEP, what exactly is the definition of soft particle?

also, why are branching fractions $$\Gamma$$ in GeV instead of a unitless ratio?

## Answers and Replies

Staff Emeritus
There is no exact definition of "soft". What is the cutoff of height to be considered tall?

Gamma is a width or a partial width, not a branching fraction.

What exactly is a soft particle? Is it just a particle with low pT?

Gamma is the width of what? why is it in GeVs?

Hello,

What exactly is a soft particle? Is it just a particle with low pT?

depends on the context. But it can be "soft Pt particle" !

Gamma is the width of what? why is it in GeVs?

Gamma can be the width of the resonance of an unstable particle (like a Z boson).
Within the scheme c=h=1, you can choose arbritraly to express lengths, times, energies with a single unit (like GeV).

"Soft Particle" would be a good name for a band.

blechman
What exactly is a soft particle? Is it just a particle with low pT?

"soft" means that it has momentum below a certain cutoff. For experimentalists, these are particles that do not have enough energy/momentum to set off a detector. For a theorist, we usually refer to soft particles as those that have the "minimum" energy/momentum. For example: when a theorist says "soft photon" (s)he is referring to a photon that has zero energy and momentum (since they're massless, this can happen).

Gamma is the width of what? why is it in GeVs?

Due to the Heisenberg uncertainty principle, particles do not have a fixed mass but have a "mean" mass (what people quote as "the mass") and an "uncertainty" $\Delta m$. It is this uncertainty that is $\Gamma$. It has units of mass (and therefore energy when c=1), and by the uncertainty principle, the lifetime of the particle is $\hbar/\Gamma$. Physically, when you make a mass measurement you don't get a spike at "the mass" but a sort-of bell-curve centered at the mass (technically it's a Cauchy distribution in most cases), and $\Gamma$ is the width of the bump half way down ("Full Width at Half Max").

The branching ratio is the "partial width" of the particle decaying to one final state, divided by the total width (sum of all the partial widths = total lifetime). This is dimensionless and represents the probability of a particle decaying into a *particular* final state.

Physically, when you make a mass measurement you don't get a spike at "the mass" but a sort-of bell-curve centered at the mass (technically it's a Cauchy distribution in most cases)

Could you expound on this Cauchy distribution business? I would have expect that any more than 1 single reading would cause the distribution to gain a finite variance?

blechman
typically, when you "measure" a particle, you are *actually* measuring the decay products of said particle. When you plot the number of events as a function of the total energy of the decay products, you do not get a single spike but a distribution of energy that (assuming the decay did not happen too close to threshold) reproduces a Cauchy (or Breit-Wigner) distribution:

$$\frac{d\sigma}{dE}\sim\frac{1}{(E-E_0)^2+\Gamma^2/4}$$

The center of the distribution (E_0) is interpreted as the mass of the resonance, and Gamma is the FWHM, representing the inverse-lifetime of the resonance.

This is all explained quite well in most QM textbooks. See Sakurai's "Modern QM" for example. In fact, Jackson's "Intro to E&M" text also talks about it, as does Landau-Lif****z, since this result comes from wave mechanics and thus there is an analogy in classical E&M.

Hope that helps!

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blechman
... as does Landau-Lif****z, since this result ...

Any moderators out there: this is the second time I couldn't put the poor Russian's name on a post! Maybe you should make an exception for that in your code...

Just making an observation.

typically, when you "measure" a particle, you are *actually* measuring the decay products of said particle. When you plot the number of events as a function of the total energy of the decay products, you do not get a single spike but a distribution of energy that (assuming the decay did not happen too close to threshold) reproduces a Cauchy (or Breit-Wigner) distribution:

$$\frac{d\sigma}{dE}\sim\frac{1}{(E-E_0)^2+\Gamma^2/4}$$

The center of the distribution (E_0) is interpreted as the mass of the resonance, and Gamma is the FWHM, representing the inverse-lifetime of the resonance.

This is all explained quite well in most QM textbooks. See Sakurai's "Modern QM" for example. In fact, Jackson's "Intro to E&M" text also talks about it, as does Landau-Lif****z, since this result comes from wave mechanics and thus there is an analogy in classical E&M.

Hope that helps!

Aha -- that makes a lot of sense. I'm familiar with the maths, but never really made the connection between Breit-Wigner and experimental data... Thank you for taking the time to explain.