Physical meaning of alternative weights

Asked by Leonhard Reichenbach

I played around a bit with O'Mega because I would like to generate the alternative weights, which I want to use for my templates to do a triple gauge coupling analysis, on the fly at analysis time. I was able to do that and reproduce the values produced by the corresponding alternative simulation setups in Whizard, however, I am left with some questions.

First, some context.
I have the following (shortened) Sindarin:

model = SM_ac

alias u_q = u:U
alias d_q = d:D
alias lep = e1:E1
alias nu = n1:N1

process ww = e1, E1 => lep, nu, u_q, d_q

n_events = 10
sqrts = 250

sample_format = lcio
?write_raw = false

# ILD tracking acceptance cut 84 mrad
cuts = all abs (cos (Theta)) < cos (84 mrad) [lep]

compile
integrate (ww)
simulate (ww) {
    alt_setup = {
    g1z = 1.001
    ka = 1.000
    la = 0.000
    kz = 1.0 - (ka - 1.0) * sw**2/cw**2 + (g1z - 1.0)
    lz = la
    }
}

This generates an amplitude from O'Mega with two flavor combinations in the description:
! flavor combinations:
!
! 1: e- e+ -> e+ nue ubar d
! 2: e- e+ -> e- nuebar u dbar

I figure out that I can reproduce the sqme value stored in my lcio file by doing the following calculation (omitting the necessary setup to get this running in c++ here):
double sqme(int flv)
{
    double res = 0.0:

    for (int i = 1; i <= 64; i++) {
        res += __opr_ww_i1_MOD_color_sum(flv, i);
    }
    // average over the 4 input helicities
    return res / 4.0;
}

Then, per event, the sqme from whizard is equal to sqme(1) + sqme(2), so equal to the sum of the two flavors.
The same holds for the calculation of sqme_alt1, and weight_alt1 = sqme_alt1 / sqme.

Now my confusion begins. Why do we have to add the two flavor combinations?
I can understand that one could do the Monte Carlo integration with the sum of the two and then only decide on the type of the actual event by throwing a weighted coin afterwards. But once the type of the event is decided, does the sum still carry any meaning? In particular, if we re-weight an event by the sum of this, and sqme1 and sqme2 scale differently, isn't it just wrong to do this?

As an example for one specific event (which according to the MC history is flavor combination 2) I get:

sqme: 5.796556004566848e-05, sqme1: 1.4072181689328032e-06, sqme2: 5.655834187673568e-05
sqme_alt1: 5.800501132090408e-05, sqme1_alt1: 1.404883357738779e-06, sqme2_alt1: 5.66001279631653e-05
weight: 1.0, weight_alt1: 1.0006805986333678
sqme_alt1/sqme: 1.000680598534794
sqme1_alt1/sqme1: 0.9983408321143303
sqme2_alt1/sqme2: 1.0007388138520872

I.e. with weight 1.0006805986333678 but naively I would think it should be 1.0007388138520872 as that would be the scaling for the final state of this event.

Question information

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English Edit question
Status:
Solved
For:
WHIZARD Edit question
Assignee:
Juergen Reuter Edit question
Solved by:
Juergen Reuter
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Revision history for this message
Juergen Reuter (j.r.reuter) said :
#1

Hi Leonhard,
let me start commenting on this, but very likely this is not yet a final answer. First remark: there is no need for you to sum over particles and antiparticles, this is a model assumption if you like. What you are effectively doing is summing over the charge-conjugate decay channels of W+ and W-. If there are SMEFT operators or couplings that to violate CP symmetries like e.g. ones with the dual field strength, then indeed they might be affected differently. To cross check you could do a run and on-the-fly analysis by keeping processes and event files. Secondly, if I understand correctly, your confusion comes from the fact that the flavor sum is always an incoherent and inclusive one: so Whizard assumes that full sample of flavor combination cannot be experimentally resolved into single flavor combinations, so the ratio is taken with respect to the full flavor sample.
Cheers,
    JRR

Revision history for this message
Leonhard Reichenbach (zehvogel) said :
#2

Hi Juergen,

thanks for the fast reply!
I have not fully digested it yet, but regarding the last point:

    Whizard assumes that full sample of flavor combination cannot be experimentally resolved into single flavor combinations, so the ratio is taken with respect to the full flavor sample.

I would understand that for the hadronic part of the final state, but here this would mean Whizard assumes that I cannot experimentally distinguish between an electron and a positron in the final state?

Revision history for this message
Best Juergen Reuter (j.r.reuter) said :
#3

Well, that explanation might have been misleading. But if you sum over them this means that their cross sections are added and they will be mixed in the event samples according to the probability sigma(flv_i) / sigma(tot). What you can do or not do in reconstruction depends on you, but the event sample is completely inclusive. This also means that alternative weights are always calculated with respect to the total cross section (summed over all flavor combinations). If you want flavor-specific reweighting factors you need to keep the processes exclusive in e.g. lepton flavors (or lepton charges).

Revision history for this message
Leonhard Reichenbach (zehvogel) said :
#4

Ah, now I understand. I naively thought that this e1:E1 etc. flavor sum is more like a syntactic sugar and less me explicitly requesting Whizard to treat the particles as indistinguishable. (I also hoped it gives O'Mega a chance to simplify the calculation and to only need one compiled amplitude). I will just do things exclusive in the lepton flavors then.

Thanks!

Maybe this whole part could use a bit more coverage in the documentation, but I am also not an MC expert so maybe this behavior is obvious to others (or I missed the relevant parts) ;)

Revision history for this message
Thorsten Ohl (thomega) said :
#5

> I also hoped it gives O'Mega a chance to simplify the calculation

That hope was not unjustified. Common pieces of the amplitudes are shared among processes.

> to only need one compiled amplitude

That's also correct. You can access the individual amplitudes in O'Mega, but you can't unmix the event sample.