Dependency of phase space generator to differential distributions

[minits]

TL;DR

Hello! I calculated the cross section of a scattering process myself and with a program to compare with. I then made histograms of differential distributions which were supposed to match, which they didn´t. I am now asking myself why they should match in the first place. Isn´t it totally dependent on the phase space generator one uses? (for both programs they are different)

I attatched an example plot where I created the histogram for the differential distribution with respect to the energy of the d-quark produced in the scattering process. My conception is that the phase space generator can "decide" how much of the available energy it assigns to the respective particles in the final state based on how it was programmed. So if you have two different phase space generators one could naturally assign different pairs of energy to the final state particles more ofter than the other or is this a misconception? Any suggestions are highly appreciated!

 

[mfb]

What is the process, what did you use for the two distributions?

 

[minits]

The process is u b < d t. Without using pdfs the distributions were basically the same Bit now with pdfs they are not anymore. I used mad analysis and a self written program (blue). Mine works in principle since I tested it without the pdfs

 

[RGevo]

The answers should be equivalent.

If you use the same set if inputs to both calculations, there should be a “correct” answer (up to numerical uncertainties of the numerical integration procedure).

If the integration without the pdf convolution agrees (i.e. you get the same exact answer for a single phase space point as your comparison generator) then likely there is an issue in your pdf sampling.

 

[minits]

Thanks! So I will have to look into this again. I was just wondering since I calculated the cross sections before with both programs and the results matched. Do you have any idea why that could be?

 

[SBoh]

There are several point from me.

1) Please provide information such as is this proton-proton collision, at what center-of-mass energy, at which level you produced histograms (matrix element level, after parton shower with specific particle status number, ...), at which perturbation order (LO in QCD, NLO in QCD ...), if you used exactly same PDF set. Currently, there should be too many guesses here. It will be difficult to you to get useful answer.

2) For me, the point that you got agreement in the cross section but difference in kinematic shape is difficult to understand too. Are they total cross sections using all possible Bjoken scale x values? Are both cross-sections calculated using the Monte-Carlo (MC) method?

3) If your distributions are from the matrix element level, I would check transverse momentum of d + t which should be zero.

 

[minits]

Thank you for your response! Regarding the points you mentioned:

1) it is a proton-proton collision at 1TeV center of mass energy. Histograms are on matrix element level the process is in LO and I used the same pdf set for both calculations

2) Yes, x may take all the values from 0 to 1 and both cross sections were calculated via MC methods

3) I checked the transverse momentum, it is zero.

 

[SBoh]

Okay great.

I am sorry but I have no idea what is the reason.
Regarding your answer, effect of all possible scenarios that I can think of should be propagated into cross section difference, where cross sections are calculated using MC method with error depending on number of generated events.

 

[minits]

Hi,

I have found the error. The problem is that I have been in the wrong reference frame.

 

[minits]

Maybe you could help me with solving this problem.
The four-momenta in the initial state have the following form:
$$p_1=(x_1\cdot\frac{\sqrt{s}}{2},0,0,x_1\cdot\frac{\sqrt{s}}{2})
p_2=(x_2\cdot\frac{\sqrt{s}}{2},0,0,-x_2\cdot\frac{\sqrt{s}}{2})$$

The routine that generates the final state momenta produces them in the CM-system. What should the Lorentz-boost look like to boost them in the right system? It must be a boost in the z-direction so I took a usual boost matrix for this direction. However, I am not confident with the v-vector that is part of the gamma factor. I said it should read
$$v=(x_1+x_2,0,0,x_1-x_2)$$
since adding up the $p_1$ and $p_2$ this would be the difference to the initial frame without the bjorken scalings but I am not very confident about it.

 

[victorspanabauer]

good luck an w.h.o. is the lucky 2

 

[RGevo]

I would recommend that you implement this boost into your code.

1) generate the momenta in the CoM frame as you are doing.
2) apply the resultant boost to both p1 and p2
3) print the boosted (Lab frame momenta) and compare them to what you expect (i.e. p1_Lab = x1 Ehadron (1, 0, 0, 1), p2_Lab )

they should be consistent to the numerical precision of the implementation. You can also construct the invariant mass of your partonic system to make sure it is indeed the correct (and invariant one) in the Lab frame.