# unitary bound of aa->W+ W- process

Asked by Ji-Chong Yang on 2019-12-15

Dear Olivier:

We are intereted in the unitary bound of a a -> W+ W- process.
We think the cross-section of a a -> W+ W- should not grow with beam energy otherwise the unitary bound is violated.
At tree level, there is no s-channel Feynman diagrams, so we woundering that how can the cross-section be supressed when the beam energy increase, and we do the following:

generate a a > w+ w-
launch
shower=OFF
detector=OFF
analysis=OFF
0
set lpp1=0
set lpp2=0
(We think this means do not use PDF)

and with ebeam1,2=1000,2000,3000,4000,5000,10000,20000,30000,50000
run the madgraph5, we obtain (pb):
93.19
94.29
94.62
95.14
95.87
100.8
121.5
155.5
262.4

My question is, it seems a a -> W+ W- does violate the unitary bound, is it because we did something wrong?

## Question information

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Solved by:
Olivier Mattelaer
Solved:
2019-12-20
Last query:
2019-12-20
2019-12-19
 Olivier Mattelaer (olivier-mattelaer) said on 2019-12-15: #1

Did you set all the width to zero for such computation.
The fix-width scheme breaks the gauge invariance of the Standard Model and therefore leads to violation of high-energy.

Another method to restore gauge invariance is to use the complex mass scheme but for such scheme to hold, you can not have any unstable particle in the final state.

Cheers,

Olivier

 Olivier Mattelaer (olivier-mattelaer) said on 2019-12-15: #2

Did you set all the width to zero for such computation.
The fix-width scheme breaks the gauge invariance of the Standard Model and therefore leads to violation of high-energy.

Another method to restore gauge invariance is to use the complex mass scheme but for such scheme to hold, you can not have any unstable particle in the final state.

Cheers,

Olivier

> On 15 Dec 2019, at 11:27, Ji-Chong Yang <email address hidden> wrote:
>
> New question #687151 on MadGraph5_aMC@NLO:
>
> Dear Oliver:
>
> We are intereted in the unitary bound of a a -> W+ W- process.
> We think the cross-section of a a -> W+ W- should not grow with beam energy otherwise the unitary bound is violated.
> At tree level, there is no s-channel Feynman diagrams, so we woundering that how can the cross-section be supressed when the beam energy increase, and we do the following:
>
> generate a a > w+ w-
> launch
> shower=OFF
> detector=OFF
> analysis=OFF
> 0
> set lpp1=0
> set lpp2=0
> (We think this means do not use PDF)
>
> and with ebeam1,2=1000,2000,3000,4000,5000,10000,20000,30000,50000
> run the madgraph5, we obtain (pb):
> 93.19
> 94.29
> 94.62
> 95.14
> 95.87
> 100.8
> 121.5
> 155.5
> 262.4
>
> My question is, it seems a a -> W+ W- does violate the unitary bound, is it because we did something wrong?
>
>
>
>
> --

 Ji-Chong Yang (nbalexis) said on 2019-12-16: #3

Dear Olivier

Thank you very much !
We did not set width to zero.
Did you mean in the case we have unstable particles in final states, we should set width to zero?

We then try to set all particle presented we find for
ebeam1,2=10000,20000,30000,50000
the results are (pb)
86.22
86.12
86.29
86.37
It seems still violate the unitary (we expected cross-section decrease as 1/s)
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?

Thank you!

 Olivier Mattelaer (olivier-mattelaer) said on 2019-12-19: #4

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

 Ji-Chong Yang (nbalexis) said on 2019-12-20: #5

Thanks Olivier Mattelaer, that solved my question.

 Ji-Chong Yang (nbalexis) said on 2019-12-20: #6

Deal Olivier!