Kinematic coverage for dijet measurements

[gstagnit]

Hi all, I'm producing some plots for my thesis @stefanoforte . In particular kinematic coverage plots for jet measurements. I'm using the plot_xq2 function of validphys. While I think things are working fine for single jet measurements, dijet data points in the (x,Q^2) plane are completely off.

I don't know if you are aware of this already, I'm opening this issue to share my thoughts with you, with the hope this will be useful (I guess that at some point you would like to produce kinematic plots for NNPDF4.0, which will include jet measurements).

At the moment, given the three kinematic variables [k1,k2,k3] associated to each data point, the JETXQ2MapMixin function (implemented here:

nnpdf/validphys2/src/validphys/plotoptions/kintransforms.py

Line 103 in 330a53c

class JETXQ2MapMixin:

) returns:

 (**) x = k2/k3 * (exp(k1) + exp(-k1)), Q2 = k2^2

This function is used for any jet measurement. Studying the LO kinematics, I came up with the following conclusions:

• INCLUSIVE JETS: [k1, k2, k3] = [|y|, pt, sqrt(s)] with pt= jet transverse momentum and y= jet rapidity. Given pt and |y|, the minimum and maximum momentum fractions read:

x_min = 2 pt/sqrt(s) exp(-|y|), x_max = 2 pt/sqrt(s) exp(+|y|)

Thus (**) returns (x_min + x_max)/2, and this should be fine.

• DIJETS (CMS_2JET_7TEV): [(k1, k2, k3] = [|ymax|=max(|y1|,|y2|), m12, sqrt(s)] with m12 invariant mass of the dijet system. Given that, at LO, the following relation holds:

x1 = m12/sqrt(s) exp(+|yb|), x2 = m12/sqrt(s) exp(-|yb|) 

with |yb|=|y1+y2|/2 (as in DY), I would say that the min and max x are given by the previous relations with |yb|=|ymax| i.e.

x_min = m12/sqrt(s) exp(-|ymax|), x_max = m12/sqrt(s) exp(+|ymax|)

Thus the average value should be given by (**)/2.

• DIJETS (ATLAS_2JET_7TEV): [(k1, k2, k3] = [|y*|=|y1-y2|/2, m12, sqrt(s)]. Here I think that the situation is similar to eq.(7.33) of pinkbook, with tanh(y*) = cos(theta*). You see that parton luminosities factorize from matrix elements, and x1*x2*s = m12^2. Then I think that max and min x are simply given by:

x_min = m12^2/sqrt(s), x_max = 1

In this case (**) is completely off, and the mean would be something like

x = 1/2 * (1 + m12^2/sqrt(s))

• DIJETS 3D (CMS_2JET_3D_8TEV): [(k1, k2, k3] = [|y*|=|y1-y2|/2, ptavg=(pt1+pt2)/2, |yb|=|y1+y2|/2]. In this case we know exactly the two momentum fractions:

x1 = 2 pt/sqrt(s) * cosh(|y*|) * exp(+|yb|), x2 = 2 pt/sqrt(s) * cosh(|y*|) * exp(-|yb|)

However, we need also sqrt(s), not present among the kinematic variables. So for now, I don't know how to represent these points on the (x,Q^2) plane.

[stefanoforte]

Dear Giovanni, x_min = 2 pt/sqrt(s) exp(-|y|), x_max = 2 pt/sqrt(s) exp(+|y|) Thus (**) returns (x_min + x_max)/2, and this should be fine. it is as good a definition as any other given that what one plots is somewhat arbitrary, but I'd much rather do what we did in the past, namely plot xmin and xmax (so two point for each jet datapoint) given that the plot is supposed to show the coverage of the kinematic plane. In this case (**) is completely off, and the mean would be something like x = 1/2 * (1 + m12^2/sqrt(s)) As far as I can tell, the dijet functions have never been used - at least I never saw a kinematic plane plot fo dijets, so it could well be that the functions coded into validphys are bugged Let;s discuss this at the next PC However, we need also sqrt(s), not present among the kinematic variables. So for now, I don't know how to represent these points on the (x,Q^2) plane. I am sorry, but our dijet data are 7 or 8 TeV, so sqrt s is 7 or 8 Tev. No? Cheers Stefano — You are receiving this because you were mentioned. Reply to this email directly, view it on GitHub, or unsubscribe.*

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[Zaharid]

cc @enocera @rabah-khalek

[enocera]

@gstagnit Thanks for pointing out this issue. I'd say that in the case of 2D dijet distributions the issue can be solved by defining the correct kinematic transformations. My apologies for having overlooked them when I implemented the data sets. For the 3D dijets, the issue is more subtle, in that we can currently store only 3 kinematic variables; ND differential distributions, with N>2, require more than three kinematic variables, of course. As mentioned long ago by @Zaharid, in principle we should modify the data layout to include an arbitrary number of kinematic variables. I'm not sure that I'd like to embark in such a work right now.

[gstagnit]

@enocera @stefanoforte I've tried to implement the transformations discussed above in validphys (see #758)

it is as good a definition as any other given that what one plots is
somewhat arbitrary, but I'd much rather do what we did in the past,
namely plot xmin and xmax (so two point for each jet datapoint) given
that the plot is supposed to show the coverage of the kinematic plane.

In my implementation I'm plotting both xmin and xmin, both for jets and dijets (with the transformations I wrote above --- I hope they are correct). I tried to test them, but I found that in several cases max(xmin,xmax) > 1, and so in the plot it is clipped by validphys. I need to recheck the transformations.

I am sorry, but our dijet data are 7 or 8 TeV, so sqrt s is 7 or 8
Tev. No?

Exactly, we know sqrt(s), as the only dijet 3D measurement we have up to now is a CMS measurement at 8 TeV. However, as Emanuele was pointing out, the clean way to implement a kinematic transformation with sqrt(s) is to have a fourth field in the kinematic variables.
(In order to produce the plot for my thesis I will harcode that)