E1,e1 close to muon pair production threshold

Dear Camilla,
thanks for your question.
ad 1) what do you mean by SSS correction? The process e1, e1 => e2, E2 has exactly 2 Feynman diagrams (in the model SM), namely s-channel photon and Z exchange.
ad 2) this depends whether you want matrix element photons or shower photons. For ME photons, just add more processes with additional photons, which you could even do in the same process definiton with the notation
process proc = E1, e1 => (E2, e2) + (E2, e2, A) + (E2, e2, A, A) + ....
If you want shower photons (keeping the main cross section), use either the Pythia6 or Pythia8 interfaces. At the moment, there is not (yet) an automatized method to generate an inclusive sample with an arbitrary number of photons.
ad 3)
Note that WHIZARD as a default does not apply any cuts, so the radiation of photons is not infrared safe without adding a cut statement. In your setting above, both muons and electrons are massive, so there are no true collinear singularities, but only soft singularities. You should demand a photon minimum energy for the integration, e.g.
cuts = all E > 500 MeV [A]
Hope this helps.
Cheers,
 JRR (Juergen Reuter)

Dear Juergen,
thank you for replying so fast! I have some more to ask...

1) I mean the Sommerfeld–Schwinger–Sakharov (SSS) threshold Coulomb resummation factor, that is relevant at threshold, cf. eg S.J. Brodsky, R.F. Lebed, PRL 102 (2009) 213401, http://de.arxiv.org/abs/0904.2225, lately introduced also in Geant4 as explained here: https://geant4-userdoc.web.cern.ch/UsersGuides/PhysicsReferenceManual/BackupVersions/V10.6c/html/electromagnetic/electron_incident/AnnihiToMuPair.html.

2) Perfect, I'll try with process proc = E1, e1 => (E2, e2) + (E2, e2, A) + (E2, e2, A, A) + .... and cuts = all E > 500 MeV [A]. I'm trying to get the NLO so this should be the best way.

3)I'm trying to understand if it makes sense to have a lower cross section when I ask for one additional photon...I have some doubts about the way I ask for the cross section actually. If I write
 -------------------------------
 process proc = E1, e1 => E2, e2, A
 compile
 beams_momentum = 45.000 GeV, 0. GeV
 beams = E1, e1
 integrate(proc)
 simulate (proc) {
 $sample="45GeV"
 sample_format=short
 n_events=100000
 }
------------------------------
at the beginning of the proc.log file i read
-------------------------
Process [scattering]: 'proc'
   Run ID = ''
   Library name = 'default_lib'
   Process index = 1
   Process components:
     1: 'proc_i1': e+, e- => mu+, mu-, A [omega]
------------------------------------------------------------------------
###############################################################################
   Integral = 3.9819953787E+08
   Error = 2.4012825726E+07
-------------------------------------------

but at the end of it:
------------------------------------
[undefined] num_id(proc) = [unknown integer]
[undefined] integral(proc) = [unknown real]
[undefined] error(proc) = [unknown real]
----------------------------

moreover I'm getting a different result everytime I run, am I doing anything wrong? Is the cross section value influenced by the number of events?

Thank you very much,
ciao

Dear Camila,
no, this Sommerfeld–Schwinger–Sakharov (SSS) threshold Coulomb resummation factor is not implemented in Whizard for this case. There is something similar for the top threshold in e+e- annihilation, but there also a special physics model is used solely for that purpose. Something similar would be necessary here. Clearly, this is not something which could be done immediately and straightforwardly. Regarding your third point: absolutely should you expect the emission of additional _hard_ quanta to result in lower cross sections, parameterically lower by alpha/pi, so 2 orders of magnitude. This you don't observe because the alpha suppression is compensated by large logarithms which are generated by soft or collinear emission of massless quanta like the photon. For collinear emission the true expansion parameter would be alpha/pi * log(s/me**2), for soft emissions something like
alpha/pi * log(s/Eres**2) where Eres is the lower photon resolution of your detector (or a theoretical cut-off).
Cheers,
    JRR

Dear Camila,

adding to Jürgen's comments, some general statements. I apologize if this evident anyway:

you may ask for the cross section as an inclusive quantity with an undefined/arbitrary number of radiated photons. This is a physics question but not supported in Whizard.

Whizard computes cross sections as exclusive quantities with a definite initial and final state. If there are massless particles involved (such as the photons), you have to provide cuts such that the exclusive cross section is finite. If the chosen cuts do not guarantee finiteness,, the program will not warn you but nevertheless give you numbers, but they should be unstable and in any case, they are wrong. If the cuts are sufficient, the result may fluctuate still due to statistics and phase-space structure. Such fluctuations can be reduced by increasing the number of calls or the number of iterations in the integration pass.

In perturbation theory, the cross section with a photon depends on cuts, but it has to be less than the cross section without photons, as Jürgen explains. If this is not the case, you have probably chosen the cuts such that the result is not well-defined in perturbation theory. You could employ resummation methods that fix this problem, but this is beyond the capabilities of Whizard.

You may ask for the cross sections with zero and one (+ two, ..) photons to be added by the program. This can be done using the syntax that Jürgen quotes. You have to specify the cuts such that they apply to all processes in this list. The result will be exactly this sum, no resummation or virtual corrections are included.

Best,
Wolfgang

I forgot to mention that in the event-generation step, you can switch on a shower algorithm (Pythia) that adds a meaningful number of photons to each event. This is important and useful, but it does not modify the cross section.

(For the initial state, Whizard actually supports resumming photon emission in some approximation [ISR] but not for the final state. By default, this is switched off though.)

Thank you very much for your replies.

Regarding the SSS factor, I am a bit puzzled by the results that I get for the cross section by running
--------------------------
 process proc = E1, e1 => E2, e2
 compile
 beams_momentum = 45.000 GeV, 0. GeV
--------------------------------
which returns 511 nb.

That appears to be super close to what predicted by the analytic LO computation with SSS as reported here:
https://geant4-userdoc.web.cern.ch/UsersGuides/PhysicsReferenceManual/BackupVersions/V10.6c/html/electromagnetic/electron_incident/AnnihiToMuPair.html
If the SSS factor is not included I would have expected a lower result, corresponding to the red curve in the plots at the above link.
What is then the reason for the cross section returned by Whizard to be the equal to the one of the black curve (with the SSS factor)?
Thank you, ciao

Thank you very much for your help!
ciao, Camilla

Thanks Juergen Reuter, that solved my question.