ISR recoil kinematics broken

[Juergen Reuter]

In comparing photon energy distributions for e+e- -> mu+mu- with ISR and recoil on between WHIZARD 1.95 and WHIZARD 2.2.8 there is a discrepancy in the energy peak of the radiative return to the Z peak. Most likely the kinematics in WHIZARD 2 for the onshell projection of the splitting kinematics is broken. Besides that there is no longer the possibility to set the onshell mode between energy conservation and momentum conservation for all relevant structure functions as in WHIZARD 1.

[Juergen Reuter]

Did some first cross check: the ISR photons are really all on the mass shell, as you can see in the plot attached. I coud reproduce the strange "shoulder" around 104 GeV as you could see from the PDF attached.

[Juergen Reuter]

WHIZARD with ISR, reproduced shoulder, checked for masslessness of photons.

[Juergen Reuter]

Corresponding SINDARIN file.

[Juergen Reuter]

KEEP_MOMENTUM version

[Juergen Reuter]

Changed settings to KEEP_MOMENTUM, and - magically - the shoulder has vanished. (Raising the electron mass to 1 GeV doesn't change anything). Plot attached for KEEP_MOMENTUM.

[Juergen Reuter]

Following info on the status: first of all, I repaired the momentum distribution for the recoil which now has the form per leg as in WHIZARD1. Particularly, the sign of cos(theta) was never set, and the ultrasoft photons led to the infamous peak for cos(theta) = 0. Then: for massless particles, the two options in WHIZARD1 (keep_momentum and keep_energy) led to the same behaviour in the case of massless emitters. The crucial observation is that for the new keep_energy option in WHIZARD2, the system in the end (final state plus the two recoil photons) has a non-vanishing momentum (pT and pL), such that the derivation of e.g. the recoil in the photon energy from the dilepton pair gets a shoulder from cos(theta) part of the distribution close - but not exactly equal - to 1. This can be derived analytically when taking into account the on-shell projection with keep_energy. This option was never there in this form in WHIZARD1 and should be taken with utmost care, and I pledge for making keep_momentum the default, which comes closest to the WHIZARD1 case and doesn't show the shoulder. The shoulder is gone both in the collinear case as well as in the non-linear case for keep_energy when cutting on cos(theta) > 0.90 or better 0.95. WK, comments?

[kilian]

@JRR: First, thanks for the archaeological survey!

I accept the fact that the projection would have deserved extensive testing at the time it was implemented. I should have implemented the precise W1 formula first, before adding 'improved' versions of it. However, is there a genuine obstacle against implementing the original version(s) of Pt generation? If I read this correctly, the new versions are not the same as any of the W1 versions? Probably re-implementing the latter would require special-casing the both the splitting and on-shell procedures, and might have some restrictions on the allowed external masses.

If the new version with keep_momentum gives reasonable results, then it may well become the default. Any other variants should remain there as alternatives, of course, so we can reproduce both W1 and the misleading W2 distributions.

[Juergen Reuter]

BACN and JRR have discussed this further: W1 collected all recoil momenta from all emitted particles, then subtracted this from the beam momentum in a way that seems to be the same for a single recoil structure function and massless emitters. Then a Lorentz boost of the modified beam particle momentum to the 3-axis was constructed to calculate the true sqrt(shat), which was then used to construct incoming momenta of the correct energy-mass relation. They were then - together with all outgoing particles - boosted by the aforementioned Lorentz trafo.

This seems difficult in the modular and factorized approach in W2. BACN and JRR are in favour of letting keep_momentum be the standard and adapt the functional tests accordingly. This is only numerical noise for hepmc_8 (rescan with ISR on), pythia_3/4 (shower in e+e- with ISR), tauola_1 (incl. ISR). ewa_1/2/3 and ep_3 give completely different integrals, but we all know that EWA is not a good approximation anyways. We are going to commit this, assuming that there are no objections.

Methods for pT generation in 1->2 splitting might be improved during CF's thesis.

[Juergen Reuter]

My proposal is to close this ticket for now, and communicate to the LC generator group that with the next alpha tarball both their E_gamma distribution will be correct as will be their cos(theta) distribution of their ISR recoil samples.

[kilian]

OK, all those kinematics considerations must have been done by myself ages ago - so it's gone from memory. I apologize for the complete lack of useful documentation. For practical purposes, if there is an algorithm that works to sufficient accuracy in the relevant observables, that's all we can hope for anyway.

I agree with closing this now. Is it possible to sum this up in some LaTeX lines with the essentials, including plots - for our own reference only, of course? Just in case we're facing the problem again at a later time.

[Juergen Reuter]

Agreement was that this problem is closed for now, and all that is left - as usual - is documentation. This will be done in a special part of the general section on the physics of structure functions and lepton collider ISR in the manual to be done. Closing this for now.