R2 wrongly dropped when interaction order are added

[celine degrande]

When I generate

g g > g z [QCD] with the attached model
everythning works fine but when I put restriction on the intaction order
even QCD=99 QED=1 NP=2, the R2 g g g z with NP=2 is removed and the process is no longer gauge invariant.

[Olivier Mattelaer (olivier-mattelaer)]

Additional information from Celine:

Dear all,

We have been testing processing related to axial anomalies for the SMEFT at NLO and in particular g g > z.
This process cancel when the Z is on-shell, so I am testing g g > ta+ ta- with heavy taus (177GeV) as the process vanishes for massless fermions. I use taus in the final state and not top to separate the coupling for the production of the Z (extra contribution from the top compared to the SM) and the decay to the final state. I have started with a simpler EFT model (SMEFTNLOaxial attached) in which I only have the operators that modify the quarks z vertices (additional contribution to the left and right couplings but no new lorentz structure) and the top gluon interaction (the chromomagnetic operator). We have checked the R2 by hands and they are correct

The MG results seems to have several problems :

the brs check work only for taus in the final state but no for top, there seems to be Lambda^-4 remaining term but the ampltitude itself should vanish not only the squared amplitude.

The squared amp does not match with the expected value. The loop is known (see Cen’s notes attached, eq.1.2, it has been checked by Andres independently) and I have added the extra factor from the z propagator and tau decay and take the square. The squared amp is proportional to ga_ta^2 mta^2/mz^4 aS^2 ga_top^2 where ga_ta and ga_top are the axial coupling of the tau and top to the z, I do not find the eta and mz scaling while the other one are fine. aS is obvious but there are no dependence on the vector couplings. I have replaced the loop by an operator with the same lorentz structure (part with the levi-civita between () in 1.2) and then I got a perfect agreement with this tree-level result. The ME go MG is much smaller (5orders) than my own computation so I do not think that we are dominated by numerical uncertainties after large cancellation.

I have created a new NLO model with the QCD part of the SM plus 4 independent axial couplings to z with the up(1st generation only)/ down (1st generation only)/ tau/top with different coupling order (NPu/NPd/NPta/NPT, the model is attached as axialanotest) to try to dig the issue but I do not get the check working then. However, the two matrix element are a factor 2.25 apart (mine being bigger than MG) for the up or down only or the two together. This factor is constant so the amplitude in this case has the right mta and mz dependence. When I do the top only on the other hand, this is not working, I do not get the right mta and mz dependence and in the large mta limit I do not recover the massless case (the two MG results are a factor 4 appart while mines have a 10% dif).

Sorry for the long mail but hopefully this will help you to find the problem.

Best,

Celine

[Olivier Mattelaer (olivier-mattelaer)]

Valentin, any comment on this?

[Valentin Hirschi (valentin-hirschi)]

Hi Celine/Olivier,

The model attached by Celine seems to pre-date your longer email, is it the latest one that needs to be tested?

In any case, I tried to run this in MG v3.1.1 (I had to convert the model to v3 for this) and I can confirm that I get:

```
MG5_aMC>check brs g g > g z QCD=99 QED=1 NP=2 [virt=QCD]
[...]
Gauge results:
Process [virt=QCD] matrix BRS ratio Result
g g > g z 1.9378359556e-03 4.2873465380e+00 2.2124403904e+03 Failed
Summary: 0/1 passed, 1/1 failed
Failed processes: g g > g z
```

while of course removing the NP contributions, i.e. going back to the pure SM case, it still works:

```
MG5_aMC>check brs g g > g z [virt=QCD]
[...]
Gauge results:
Process [virt=QCD] matrix BRS ratio Result
g g > g z 9.0039116013e-02 1.6706986246e-19 1.8555253523e-18 Passed
Summary: 1/1 passed, 0/1 failed
```

However, when I do:
```
MG5_aMC>import model SMEFTNLOWZtaugsctGonly
MG5_aMC>set acknowledged_v3.1_syntax True
MG5_aMC>generate g g > g z QCD=99 QED=1 NP=2 [virt=QCD]
MG5_aMC>output CelineTEST
MG5_aMC>launch -f
```

and then I inspect the code generated, i.e. I look into:
```
CelineTEST/SubProcesses/P0_gg_gz/helas_calls_ampb_1.f
```
I find the following:
```
[...]
C Counter-term amplitude(s) for loop diagram number 58
      CALL VVVV21_0(W(1,1),W(1,2),W(1,3),W(1,4),R2GC_127_18,AMPL(1,34))
[...]
```

where `R2GC_127_18` is the R2 coupling of g g g Z which has `NP=2` in its coupling orders.
So it seems to be correctly included in this case.

So @Celine, what makes you think that it is removed/absent from the process output generated by MadLoop?