Infinite cross section in scattering

[gop]

Hi

For example in e-e- -> e-e- scattering (electron-electron scattering) the differential cross section goes to infinity as theta goes to zero. Consequently the cross section is infinite.
But how can we measure and interpret the cross section/differential cross section and interpret it as a probability/event rate if it yields infinite values?

This problem DOESN't exist in e-mu- -> e-mu- scattering. What is the fundamental difference between these two processes?

All computations are done to lowest order.

thx

 

[Bob_for_short]

... in e-e- -> e-e- scattering (electron-electron scattering) the differential cross section goes to infinity as theta goes to zero. Consequently the cross section is infinite. But how can we measure and interpret the cross section/differential cross section and interpret it as a probability/event rate if it yields infinite values?

It means all projectiles scatter to a non-zero angle: N = j*sigma.

This problem DOESN't exist in e-mu- -> e-mu- scattering. What is the fundamental difference between these two processes?

If you use the Coulomb potential, you should obtain the same result. Thus you used a different potential or made a mistake.

 

[Vanadium 50]

Electrons are identical particles, so you don't have the same diagrams as electrons and muons.

I don't recall the details of this calculation; I did it many. many years ago, but I am not surprised that a one-order calculation gives a nonsensical answer as Q^2 goes to 0; that's exactly the point at which higher orders become important.

 

[gop]

@Bob I'm not really sure I'm following your first point. As the cross section increases shouldn't the number of scattered particle increase (everything else being equal).

@Vanadium That sounds plausible I guess. I'm still not sure however why this isn't pointed out in the book. Maybe I overlooked something.

 

[nrqed]

Hi

For example in e-e- -> e-e- scattering (electron-electron scattering) the differential cross section goes to infinity as theta goes to zero. Consequently the cross section is infinite.
But how can we measure and interpret the cross section/differential cross section and interpret it as a probability/event rate if it yields infinite values?

This problem DOESN't exist in e-mu- -> e-mu- scattering. What is the fundamental difference between these two processes?

All computations are done to lowest order.

thx

Are you sure? Do you have the full expression for the tree level ee scattering (without neglecting the electron mass)? I know that most references neglect the electron mass in their calculation. However, in the limit theta goes to zero this approximation is not justified (since the electron mass is not negligible relative to the momentum transferred). It seems to me that there should be no tree level divergence as theta goes to zero.

 

[GreyBadger]

There's a nice discussion of this in Srednicki, Chapter 26: Infrared divergences. Indeed, as said above, the problem is due to the calculation being performed in the $$m\to 0$$ limit, and higher order corrections are needed to control this.

What is more formidable, calculating loop corrections or performing the calculation taking into account massive fermions? I don't know...

 

[Bob_for_short]

@Bob I'm not really sure I'm following your first point. As the cross section increases shouldn't the number of scattered particle increase (everything else being equal).

Yes, it should, as it follows from the cross section definition: the number N of scattered particles from a target is proportional to the particle flux j and the area of scattering surface σ: N = j⋅σ. Sigma is determined as in Classical Mechanics. In a Coulomb field the total cross section is infinite because whatever impact parameter R is, the particles are scattered to a non-zero angle due to long-range character of this potential. In this sense the cross section diverges due to infinite impact parameter: σ_total = π⋅R²→∞ when R→∞.

In practice the projectile beam is limited by diaphragms so the number of scattered particles is finite - instead of π⋅R² you have to use the beam cross section ( j(r) is not uniform but turns to zero starting from some D).

 

[nrqed]

There's a nice discussion of this in Srednicki, Chapter 26: Infrared divergences. Indeed, as said above, the problem is due to the calculation being performed in the $$m\to 0$$ limit, and higher order corrections are needed to control this.

But this problem arises in loop calculations. At tree level, there should be no infrared divergence due to the emission of soft photons.

 

[GreyBadger]

Sorry, you're correct. The chapter I'm thinking of is talking about controlling IR divergences in calculations to one loop.