unitary bound of aa->W+ W- process

Hi,

>Did you mean in the case we have unstable particles in final states, we should set width to zero?

Well it really depends of what you are doing obviously. Note that for final states particles, the width has no impact (since final states particles are onshell and assumed to be assymptotically free). The real impact of setting the width to zero is on T/S channel propagator.

The width is NOT a LO effect, this is actually an all order effect. The issue is that including such width at LO breaks gauge invariance and can therefore leads to spurious term that can dominates the amplitude (and give rise to unphysical effect).
In your case, the "easiest" is to kill such effect by setting all the width to zero but in general you need to use the complex mass scheme (but in that case you have to decay the W)

 > Maybe I should try process a a > W+ W- > l+ l- vl vl~ with MadSpin, so we don't have unstable particles in final states?

MadSpin will not correct the cross-section, since it will simply scale it according to the narrow-width approximation (which does not hold if you hope to restore gauge invariance within madspin)
So I would do
a a > e+ mu- ve vm~
You can not use the syntax
a a > W+ W- > l+ l- vl vl~
since that one also breaks gauge invariance

Now indeed your cross-section is now at fixed. The question is what are you going to break first:
unitarity or perturbativity?
If you look at the rapidity plot, you will see that your "W" are extremelly forward. Since you have your W in the massless limit, you have some colinear singularity, so you expect to have huge contribution for higher order effect.
Part of such contribution should be in principle include in the running of the coupling. However we do not have the running of the weak coupling implemented and therefore if you want to include the running effect (which will decrease the cross-section) you have to edit the param_card accordingly.

Now If you think that this is a bug in MG5, I would advise to cross-check with another tool to see if they give the same prediction.
I have check with older version of the code (up to 2012) and found the same behaviour. I was not able to check up to version 1.0.0 of the code (2011) since they are a bug in aloha screwing such process before version 1.2.1.

I have also checked with the (very old) sm_v4 model which fully bypass the the ALOHA code for the evaluation of the matrix-element and instead used the original HELAS subroutine, I also get the exact same answer. Now I have not taken the time to do the computation by hand to see if I would get the decrease of energy and/or compare with another code (but the switch of method on how the matrix-element is evaluated).

For final check, I have also run the build-in check of MG5aMC on any matrix-element:

INFO: Note That all width have been set to zero for those checks
Lorentz invariance results:
Process Min element Max element Relative diff. Result
a a > w+ w- 3.3977080137e-01 3.3977080137e-01 2.9408080922e-15 Passed
Summary: 1/1 passed, 0/1 failed
Gauge results:
Process matrix BRS ratio Result
a a > w+ w- 5.2435646483e-01 5.9733537250e-30 1.1391780450e-29 Passed
Summary: 1/1 passed, 0/1 failed
Gauge results (switching between Unitary/Feynman):
Process Unitary Feynman Relative diff. Result
a a > w+ w- 3.8307185727e-01 3.8307185727e-01 9.9988275366e-15 Passed
Summary: 1/1 passed, 0/1 failed
Process permutation results:
Process Min element Max element Relative diff. Result
a a > w+ w- 5.4750055491e-01 5.4750055491e-01 6.0834076678e-16 Passed
Summary: 1/1 passed, 0/1 failed

I also run them in the complex-mass scheme (they fail without decay as expected):
INFO: Note that Complex mass scheme gives gauge/lorentz invariant
results only for stable particles in final states.

Lorentz invariance results:
Process Min element Max element Relative diff. Result
a a > e+ ve mu- v9.5367102400e-13 9.5367102400e-13 1.0799707362e-14 Passed
Summary: 1/1 passed, 0/1 failed
Gauge results:
Process matrix BRS ratio Result
a a > e+ ve mu- vm~ 2.1816484058e-13 3.3355387606e-40 1.5289075690e-27 Passed
Summary: 1/1 passed, 0/1 failed
Gauge results (switching between Unitary/Feynman):
Process Unitary Feynman Relative diff. Result
a a > e+ ve mu- vm~ 9.5780173303e-13 9.5780173303e-13 4.4277600261e-15 Passed
Summary: 1/1 passed, 0/1 failed
Process permutation results:
Process Min element Max element Relative diff. Result
a a > e+ ve mu- vm~ 1.4155007927e-13 1.4155007927e-13 7.4901339660e-15 Passed
Summary: 1/1 passed, 0/1 failed

And finally without the complex-mass-scheme:

INFO: Note That all width have been set to zero for those checks
Lorentz invariance results:
Process Min element Max element Relative diff. Result
a a > e+ ve mu- v4.1209069576e-13 4.1209069576e-13 8.2084735142e-15 Passed
Summary: 1/1 passed, 0/1 failed
Gauge results:
Process matrix BRS ratio Result
a a > e+ ve mu- vm~ 8.9813519049e-12 7.0940734756e-43 7.8986699895e-32 Passed
Summary: 1/1 passed, 0/1 failed
Gauge results (switching between Unitary/Feynman):
Process Unitary Feynman Relative diff. Result
a a > e+ ve mu- vm~ 1.2942400765e-13 1.2942400765e-13 1.7554080176e-15 Passed
Summary: 1/1 passed, 0/1 failed
Process permutation results:
Process Min element Max element Relative diff. Result
a a > e+ ve mu- vm~ 2.0910380701e-13 2.0910380701e-13 5.7946833573e-15 Passed
Summary: 1/1 passed, 0/1 failed

Cheers,

Olivier