# Relativistic Breit-Wigner

Hello,

Relativistic Breit-Wigner pdf is not pre-defined in RooFit, so I have used the RooGeneric class to define it and use to fit the dikaon m(KK) mass distribution. The fit is working and is giving me some results also but I’m not sure if I have defined the pdf correctly or not. Please find the macro attached below.rel_bw.C (6.2 KB)

“Phimass” in the script is the variable in the ntuple which I’m trying to fit.

Hi @RISHABH_RATURI1, thanks for posting in the ROOT forum!

I can’t judge if you are doing the right thing for your physics analysis, because I’m not familiar with the details of this di-kaon mass distribution. But on the programming side, it appears to me that you implemented the relativistic Breit-Wigner distribution correctly. I would only change 2 things:

1. The approximations of `2*sqrt(2)` and `pi` are not nice instead you can do:
``````RooFormulaVar k1(
"k1",
"(2*TMath::Sqrt2()*mn*sigma1*gamma1)/(TMath::Pi()*sqrt(mn*mn+gamma1))",
RooArgList(mn, sigma1, gamma1));
``````
Same for `k2`.
2. I would rename `sigma1` and `sigma2` to `decayWidth1` and `decayWidth2` or something like that. The `sigma` makes me think of a resolution parameter, but you use it to name the decay width \Gamma in the relativistic BW formula, which is confusing in my opinion.

On the physics analysis side, I still have some doubts if this is a meaningful fit. The Breit Wigner has a clear physics meaning: it describes the distribution of the mass of the di-kaon system that come from the decay of a phi resonance. The parameter \Gamma is the decay width of the phi meson. So what’s the point of summing two Breit-Wigners with different \Gamma_i in your fit? Isn’t there only one decay width from one phi meson?

I’m just careful, because in your script you also describe `mn` as the `"common means for Crystal Balls"`, while you use it for the Breit-Wigners. So it appears to me you just replaced Breit-Wigners with crystal balls at some point, which might not make sense. However, Crystall balls are used to model resolution effects with exponential tails from initial/final state radiation, and it makes sense to add more than one for the combination of different resolution effects.

But a Breit-Wigner is physics motivated and you should only have one per resonance. So that’s a bit alarming in your fit, I would make sure with your collaborators if what you are doing is correct. What is often done, however, is the combination of a Breit-Wigner for the decay width with Gaussians or Crystal balls for detector/resolution effects. For example the convolution of a non-relativistic Breit-Wigner (aka. Cauchy distribution) with a Gaussian is so common that we even have a class for it in RooFit, the RooVoigtian. Maybe what you want to ultimately do is something like that, but with a relativistic Breit-Wigner? And are you sure you can’t just use the non-relativistic one?

Cheers,
Jonas

1 Like

Hello @jonas

I apologise for the late reply and thanks a lot for such a detailed response, it cleared many of my doubts.

I made the changes which you suggested on 2*sqrt(2) and pi approximations as well as renamed sigma variables to decaywidth which is absolutely right as that parameter is the decay width in the relativistic Breit-Wigner distribution.

As you pointed out, we use double crystal balls for the resolution effects on either side of the distribution. Similarly, I was trying to use two relativistic Breit-Wigner thinking that it may give better fitting results than by using just one. I will now use only one relativistic BW as the kaons are coming from one phi mesons only.

About “common means for crystal balls”, I just edited my old script which used double crystal balls for fitting the mass distributions and forgot to change the name of the variable which led to some confusion.

Actually, I tried non-relativistic Breit-Wigner distribution only first to fit the dikaon mass distribution and then tried relativistic BW just to compare the two pdfs. Thanks again for suggesting using the RooVoigtian class which is physics motivated and appropriate for the analysis, I’ll use it also and check with the relativistic and non-relativistic BW.

Regards
Rishabh

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