Question

How is the decay width of a resonance is measured?

Answer 1

It's from spectroscopy. The measurement shows a strong peak (or line) in a particular place (measured in energy, one way or another, but often expressed in wavelength for light. It's still energy.)

Answer 2

Hello mate here is your answer.

I start with the decay law of Rutherford and Soddy which states that the rate of decay N˙(t)N˙(t) of an unstable particle (or nucleus) at a given time tt is proportional to the number of those particles which are present NN at that time tt.

dNdt=N˙(t)=∝N(t)⇒N˙=−λNdNdt=N˙(t)=∝N(t)⇒N˙=−λN

where λλ is the decay constant.

Now it is often the case that there is more than one decay mode and for each of the decay modes there is a corresponding decay constant so now one must write that the rate of decay depends on the sum of all the decay modes.

N˙all(t)=−λAN(t)−λBN(t)−λCN(t)−.....N˙all(t)=−λAN(t)−λBN(t)−λCN(t)−.....

where λA,λB,λCλA,λB,λC etc are the decay constants for the various decay modes.

From this you get N˙all=−λallNN˙all=−λallNwhere λall=λA+λB+λC+....λall=λA+λB+λC+....

So that decaying particle has a decay constant which is the sum of the decay constants for all of the possible modes of decay.
At the moment of decay the decaying particle chooses one particular mode of decay and the probability of such a decay is expressed as a branching fraction or branching ratio.

The average lifetime of an unstable particle ττ is related to the decay constant τ=1λτ=1λ.

Hope it helps you.