Transforming angular distributions between different reference frames

[drpsycho]

My problem is related to the Brookhaven experiment of the J/psi discovery and the Y discovery at Fermilab (in both cases, protons over a Berillium fixed target).
In both cases they had a resonance decaying into two leptons, and the detecting system consisted of two arms, covering a relatively narrow angular acceptance.
When they motivate the choice of the opening angle between the arms of the spectrometer, they do the calculation in the extreme assumption that, in the center-of-mass rest frame, the leptons are produced at 90 degrees with respect to the beam. Then of course you boost the system by the proton momentum, and you get the MAXIMUM angle of the leptons in the lab frame (tg theta < 1/gamma*beta = M_proton / P_proton).
Fine.
But of course the vast majority of the leptons are not produced at 90 degrees, and are so lost.
The authors of those experiments motivate this choice by the fact that at central angles the hadronic background is huge and so they have to restrict the acceptance as much as possible to the extreme corners of the angular distribution. Fine.
Now, my question is very stupid and only concerns basic kinematics: I would like to estimate the fraction of leptons inside the detector acceptance (that I know from the original articles), knowing their angular distribution in the center-of-mass frame.
(It should be ~ 1+cos^2 theta, in the case of J/psi and Y, but I can maybe start with the simplified assumption of isotropy, just to get the order of magnitude.)
My strategy is the following:
I write dGamma/dCosTheta(lab) = [dGamma/dCosTheta(cm)]*[dCosTheta(cm)/dCosTheta(lab)], and since I already know dGamma/dCosTheta(cm), the game becomes finding CosTheta(cm) as a function of CosTheta(lab).
This should be some shallow algebra after applying the Lorentz transformations, but I'm having difficulties.
Applying the Lorentz transformation I found that tgTheta(lab)=(1/gamma)*senTheta(cm)/[cosTheta(cm)+beta].
I rewrote all the sines into sqrt(1-cos^2) and tried to do some manipulation, but I get a very ugly second order equation with cumbersome coefficients.
Is there an easy way of performing this calculation?
Thanks a lot!

 

[AndyW]

Thank you for your question regarding the angular acceptance in the Brookhaven and Fermilab experiments. I can understand your confusion and frustration with the calculations involved, but I assure you that it is a necessary and important aspect of the experimental design.

Let me start by explaining the reasoning behind the choice of the opening angle between the arms of the spectrometer. As you mentioned, the main reason for this is to reduce the background from hadronic interactions. At central angles, the hadronic background is indeed very high and can overshadow the signal from the resonance decays. By choosing a narrow angular acceptance, the experimentalists are able to reduce this background and improve the signal-to-noise ratio.

Now, let's get to the kinematics involved in this calculation. As you correctly pointed out, the majority of the leptons are not produced at 90 degrees in the center-of-mass frame. However, the key factor here is the direction of the proton beam. The beam is directed along the z-axis in the lab frame, and the opening angle between the arms of the spectrometer is chosen such that it covers the maximum possible angle of the leptons produced in the center-of-mass frame when the beam is perpendicular to the target.

In other words, the opening angle is chosen to cover the full range of possible angles of the leptons in the lab frame, given the direction of the proton beam. This means that the majority of the leptons will still be within the detector acceptance, even if they are not produced at 90 degrees in the center-of-mass frame.

Now, to your question about estimating the fraction of leptons inside the detector acceptance. I understand that you would like to know the exact fraction, but as an experimentalist, we often have to make simplifying assumptions to get an order of magnitude estimate. In this case, assuming an isotropic angular distribution in the center-of-mass frame would be a good starting point.

To perform this calculation, you can use the well-known formula for the Lorentz transformation of angles: cosθ(cm) = (cosθ(lab) + β)/(1 + βcosθ(lab)), where β is the velocity of the proton beam in units of the speed of light. Using this, you can calculate the fraction of leptons within the detector acceptance, which should indeed be approximately 1 + cos^2θ. I understand that you tried this approach already, but I would suggest simplifying

 

[sir-pinski]

First of all, your question is not stupid at all. Understanding the kinematics of a particle's decay is crucial in any experimental analysis.

To answer your question, we need to first understand the transformation between different reference frames. In this case, we are transforming from the center-of-mass frame to the lab frame. This can be done using the Lorentz transformations, as you have already attempted.

To simplify the calculation, we can assume isotropic production in the center-of-mass frame. This means that the angular distribution of the leptons in the center-of-mass frame is proportional to 1+cos^2(theta). We can then write the differential cross section in the lab frame as:

dGamma/dCosTheta(lab) = [dGamma/dCosTheta(cm)]*[dCosTheta(cm)/dCosTheta(lab)]

where dGamma/dCosTheta(cm) is the differential cross section in the center-of-mass frame and dCosTheta(cm)/dCosTheta(lab) is the Jacobian of the transformation.

The Jacobian can be written as:

dCosTheta(cm)/dCosTheta(lab) = (1+beta*cosTheta(lab))/(gamma*(1+beta^2*cosTheta(lab)^2))

Substituting this into the equation above, we get:

dGamma/dCosTheta(lab) = [dGamma/dCosTheta(cm)]*(1+beta*cosTheta(lab))/(gamma*(1+beta^2*cosTheta(lab)^2))

To simplify this further, we can use the fact that beta = p/M (where p is the momentum of the proton and M is its mass). We can also write dGamma/dCosTheta(cm) as a function of cosTheta(cm) using the assumption of isotropy.

dGamma/dCosTheta(cm) = A*(1+cosTheta(cm)^2)

where A is a constant.

Substituting this into the equation above, we get:

dGamma/dCosTheta(lab) = A*(1+cosTheta(cm)^2)*(1+p*cosTheta(lab)/(gamma*M)*(1+beta^2*cosTheta(lab)^2))

Integrating this over the lab acceptance range (i.e. from -1 to 1), we get the fraction of leptons inside the detector acceptance as:

Fraction = (1+cosTheta(cm)^2)*(1+p/(2*gamma*M)*(1+beta^2))

where p