Timescale of virtual photon interaction in particle scattering

[Lux137]

TL;DR Summary

Why is the time-scale of the virtual photon exchange τ~1/Q in two particle scattering?

I'm studying the electron-proton deep inelastic scattering. In the notes that I'm studying from the author states that the time-scale for a virtual photon to interact with a proton is $τ\sim\frac{1}{Q}$, where $Q$ is the momentum transfer with $Q\gg M$, which is the mass of the proton. I assume by $Q$ he means the Minkowski norm of the 4-momentum transfer. And then he compares it with the time-scale of the quarks interaction in a proton $τ\sim\frac{1}{M}$. All this in a reference frame where the proton is at rest.
Later on delving more into time-scales he writed that the typical time of interaction of an electron with a virtual photon is also $\frac{1}{Q}$ which is to be compared with the time the electron emits and reabsorb its own virtual photons $\frac{1}{q}$

I'm having a very hard time justifying all these time-scale estimations by myself. I tried to apply the time evolution on a state with an electron and a photon, as eigenstates of the free hamiltonian $H_0$ (not the full $H=H_0+H_i$) in the interaction picture, in order to visualize what happens with virtual particles, and what i got is that the amplitudes for scattered states reach its max at $t\sim\frac{1}{ΔE}$ where $ΔE$ is the difference between the virtual photon energy and the energy it would have if it were on-shell. Which is still different from the statements above.

I can't seem to get past this and i feel like it's something silly that i should have already grasped, since in all the textbooks i have checked the topic of interaction time-scale is always barely mentioned or completely skipped. Does anyone know a derivation of these results or maybe some sources where i can find this topic discussed in detail?

Sorry for my bad english.

 

[LightHero]

The time-scale concepts you are discussing are related to the uncertainty principle. In essence, the uncertainty principle states that the more precisely we measure one physical quantity, the more uncertain we will be about another physical quantity. In particular, the position-momentum uncertainty principle states that the uncertainty in the position of an object multiplied by the uncertainty in its momentum must be greater than or equal to a certain constant (usually denoted by ħ). This means that if we know the momentum of an object very precisely, then we cannot know its position equally precisely, and vice versa.In the case of electron-proton deep inelastic scattering, this means that if the momentum transfer is very large (i.e. Q is much larger than the proton's mass M), then we can only be certain that the virtual photon interacted with the proton for a short amount of time. The time-scale for this interaction is approximately 1/Q. On the other hand, if the momentum transfer is small (i.e. Q is much smaller than the proton's mass M), then the time-scale for the interaction is approximately 1/M.Similarly, the time-scale for the interaction between an electron and a virtual photon is approximately 1/Q, while the time-scale for the electron to emit and reabsorb its own virtual photons is approximately 1/q.In order to understand these time-scales better, it may be helpful to look at some diagrams of the scattering process. You can find some diagrams in the following references: 1. S. D. Ellis, "Deep Inelastic Scattering", Cambridge University Press, 2001.2. P. E. Nishijima, "Quantum Mechanics of Particles and Fields", World Scientific, 1994.