MadGraph5_aMC@NLO

Longitudinally polarised W bosons

Asked by Baptiste Ravina

Hi, I'm investigating the production of longitudinally polarised W bosons at sqrt(s)=13 TeV with MG 3.4.2 and the LO sm UFO. In particular, I'm interested in applications of the Goldstone boson equivalence theorem (GBET), which states that at high enough energies (E_W>>M_W) the probability of emission of a longitudinally polarised W boson becomes equivalent to that of a Goldstone boson. Therefore I would expect it to vanish for light quarks in the u d~ > w+ and c s~ > w+ processes, since the yukawas would be 0.

I run three independent event generations:
1. generate p p > w+ --> xsec = 8.36e+04 +- 77.94 pb
2. generate p p > w+{T} --> xsec = 8.36e+04 +- 77.94 pb
3. generate p p > w+{0} --> error

in run 3. I get the following error message:
"Survey return zero cross section.
   Typical reasons are the following:
   1) A massive s-channel particle has a width set to zero.
   2) The pdf are zero for at least one of the initial state particles
      or you are using maxjetflavor=4 for initial state b:s.
   3) The cuts are too strong.
   Please check/correct your param_card and/or your run_card."

I don't think any of these reasons apply here, so am I really just seeing a vanishing cross section for the production of a longitudinally polarised W boson? I would expect to get *some* W bosons produced close to at rest, and therefore by the GBET a small but non-zero cross section...

Interestingly, if I now generate ttW at LO (exactly the same diagrams, with an extra gluon->ttbar emission from an ISR quark), I get
1. generate p p > t t~ w+ --> xsec = 0.2352 +- 0.0001772 pb
2. generate p p > t t~ w+{T} --> xsec = 0.1848 +- 0.0001151 pb
3. generate p p > t t~ w+{0} --> xsec = 0.0502 +- 3.573e-05 pb

and now there is a non-vanishing longitudinal component. The cross sections from 2. and 3. add up to that from 1., so I presume there is negligible interference. Looking at the distribution of the energy of the W boson, the longitudinally polarised sample 3. does peak at lower values and decreases initially faster than the transversely polarised one, but never fully vanishes, even at very high energies.

Am I doing something wrong here, or perhaps misunderstanding the consequences of the GBET?

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Olivier Mattelaer (olivier-mattelaer) said : 		#1

If you check the log of the computation for your syntax:
generate p p > w+{0}

You will get this information:
        1000 points passed the cut but all returned zero
 therefore considering this contribution as zero
 Deleting file events.lhe

> I would expect to get *some* W bosons produced close to at rest.

This statement is obviously frame dependent. For that process, we can only do the computation in one frame... the one where the W is exactly at rest.

For your second example, You also need to be careful about the lorentz frame. And I guess be sure that your plot is in the same frame in which you evaluate your polarization. Now with the syntax that you use your W is always onshell and it always exists one frame where it is exactly at rest which makes me think that the GBET is not suppose to be valid as well.

Cheers,

Olivier

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