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- 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.

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.