# Is resonance total width conserved in different frames?

Since we have to perform a relativistic transformation to transform the resonance peak energy from the lab frame to the resonances rest frame, I would find it surprising that we could use the total width of the resonance peak in the lab frame directly to calculate the resonance lifetime in its rest frame (i.e. just use $$\Gamma_{lab} = \frac{\hbar}{\tau}$$).

The question is equivalent to 'can we use the lab frame lifetime to get the rest frame lifetime?'

The answer is yes, if you know the velocity you can just lorentz transform the time

• I thought this initially, but I think in the lab frame the resonance width is actually dominated by kinetic energy transfer (not just 'mass energy transfer') due to the scatter kinematics – Alex Gower May 13 at 20:16
• Does your answer assume we plot the graph against mass energy in the lab frame? – Alex Gower May 14 at 9:07

The resonance energy width is not conserved between frames. However it is also not just as simple as the resonance width is always inversely proportional to the resonance particle lifetime in that frame.

A resonance particle is unstable and therefore exists as a superposition of many 'mass energy' eigenstates. In the CM frame, there is no kinetic energy of the resonance particle so the change in energy always corresponds to a change in mass energy and therefore the resonance width in the CM frame is purely due to the superposition of energies of an unstable particle.

However in the lab frame (e.g. for a projectile-target scatter topology), the resonance width plotted on a graph of projectile energy would not give the same width as in the CM frame, and also not give a thinner resonance width due to the increased lifetime of the resonance particle in this frame (where $$\Delta E \Delta t \sim \hbar$$). This is because if you vary the energy of the projectile by a given amount, it won't just vary the 'mass energy' of the resonance particle in the lab frame, but it will also vary the kinetic energy too. Therefore you will need to vary the projectile energy by a greater amount in order to explore the same range of 'mass energy' of the unstable resonant particle.

Therefore the width of the resonance in the lab frame will often be dominated by the scatter kinematics and not just the superposition of 'mass energies' of the resonant particle.

This all assumes the graph is plotted with respect to the projectile energy not the resonance mass energy like below.

• I do not think so. Kinetic energy is a part of the four vector, the other three components are momentum. The invariant mass is invariant to all Lorentz transformations, and it is the invariant mass that is plotted to get the width. The components of the resonance (or any sum of four vectors) should have the same invariant mass whenever they are Lorentz transformed. – anna v May 14 at 4:45
• I think there might be two different definitions of 'width' being used. There is the width of the spike in the graph and the width of the particle (which is defined in the rest frame of the particle anyway) – Toby Peterken May 14 at 9:04
• I have given an example picture for my definition – Alex Gower May 14 at 9:05