Second look at UEH issue ...

In my previous post, I showed that the difference between the particle and parton level jet yields from the full pythia simulation (which is used in the UEH correction to the cross section) is mainly due to the fact that the massless partons aquire a mass at the particle level, which comes into the full definition of the dijet mass. Below I look further at the effect the jet mass has on the dijet cross section as well as data / simu comparisons.


For the plots below, I am comparing the full dijet cross section to a massless expression given by:

Mass = Sqrt{2*Jet3_pt*Jet4_pt*[Cosh(Jet3_rapidity - Jet4_rapidity) - Cos(Jet3_phi - Jet4_phi)]}

There is some subtlety concerning whether to use the jet rapidity or pseudorapidity. Jets are created from several 4-vectors, and even in the massless case, the sum of the 4-vectors will aquire a non-zero 'mass'. So although pseudorapidity is the massless limit of rapidity, there will still be a slight difference between the two for jets. I did a quick spot check and the difference between the (no jet mass) dijet mass using rapidity or pseudorapidity is at the <= 1% .


Figure 1: Comparison between the full dijet mass spectra and the 'no jet mass' dijet mass spectra for the full data sample.

 

Figure 2: Data / Simulation comparison for the full dijet mass and the 'no jet mass' dijet mass.

As can bee seen, there is very little difference between the data / simu comparisons from the full and 'no jet mass' dijet mass definitions.


There are several pieces which go into calculating the cross section including the response matrix and the full pythia sample used for efficiency calculations. The plots below look at all of these in detail.


Figure 3: Comparison between the full and 'no jet mass' dijet mass definitions for the detector level dijets (left column) and the particle level dijets (right column). These spectra are the 1-D projections of the particle level response matrix. The corresponding plots for the parton level are here.

Figure 4: Comparisons for the particle and parton level dijets from the response matrix directly comparing the full and 'no jet mass' dijet mass definitions (top 4 panels) and the particle and parton levels (bottom 4 panels).

 

Figure 5: Same as figure 4, but for the full pythia sample used in the efficiency calculation.

 

The plots below show how the 'no jet mass' dijet mass definition affect the cross section.


Figure 6: Ratio of the dijet cross section using the full and 'no jet mass' definitions. The red points show the unfolding and efficiency while the blue points show the unfolding only with no efficiency correction

Figure 7: The (Data-Theory)/Theory ratios for the full and 'no jet mass' dijet cross sections with and without the corresponding UEH correction

Figure 8: Size of the UEH correction (particle-parton cross section divided by raw theory cross section) for the full and 'no jet mass' cases

The difference between the 'full mass' cross section and the 'massless' cross section is driven primarily by the individual jet masses which are sensitive to the fragmentation of nearly massless partons into massive final state particles. The UEH correction calculated from the 'full mass' cross section by taking the difference between the particle and parton level cross sections should take this effect, which is explicitly ignored in the 'massless' cross section, into account. Given this, I was somewhat supprised that in the comparisons with theory, the UEH corrected 'full mass' ratio (blue triangles in figure 7) didn't line up with the 'massless' cross section ratio (red triangles). 

To see why these points don't line up, I looked at the theory source code to determine how the dijet mass was calculated and found that the DSSV code uses the 'massless' formulation. When doing the UEH correction, I find the cross section at particle and parton levels and I use the 'full mass' formula to find the dijet mass at each level. Because I use the 'full mass' formula when finding the cross section and parton level, that cross section will be larger than what the theory code will predict, so when the particle - parton difference is taken, it is actually underestimating the difference between the theoretical calculation and the particle level cross section we present. To test this, I recalculated the UEH correction using the 'full mass' formula at the particle level, but now using the 'massless' formula at the parton level to better match what is being done in the theory calculation. The result can be seen in figure 9. 


Figure 9: Same as figure 7, but with the addition of the green triangles which show the (data-theory)/theory ratio after the theory cross section has been corrected for UEH by taking the particle level cross section extracted using the 'full mass' formula minus the parton level cross section using the 'massless' formula.

The above comparisons were made using a dijet mass formula which explicitly assumed the jet mass was zero (except for the use of rapidity, which I have found has a very small effect). However, as I mention above, the jet 4-momentum is constructed from the sum of several track/tower/particle 4-momenta and even if these constituent 4-momenta are massless, the sum will in general have a mass-like component because the magnitude of the vector sum of the individual 3-momenta will not equal the sum of the individual energies. The comparisons above do not capture this residual 'mass-like' component, but the full mass formula will:

M^2 = (Jet3_m)^2 + (Jet4_m)^2 + 2*Sqrt[(Jet3_m)^2 + (Jet3_pt)^2]*Sqrt[(Jet4_m)^2 + (Jet4_pt)^2]*[Cosh(Jet3_rapidity - Jet4_rapidity)] - 2*Jet3_pt*Jet4_pt*Cos(Jet3_phi - Jet4_phi)

I wanted to see how using the full formula, but assuming massless tracks/towers/particles would change the cross section. For the data, I did this by looping over all the tracks and towers in each jet and reconstructing their 4-momenta while specifically setting |p| = E (this assumes that the change between the pion mass and massless track assumption does not significantly influence the jet finding itself). Unfortunatelly, I could not do the same procedure for the particle and parton level jets because I don't save the individual particle momenta in my trees.    


Figure 10: Individual and dijet masses from a subset of the data assuming towers = massless and tracks = pion mass vs assuming all tracks and towers are massless. The full mass formula was used for the dijet mass

As can be seen in Figure 9, there is very little difference in the data between the pion mass and massless cases when using the full dijet mass formula. This is not too suprising given that only the tracks had a mass applied and that mass was relatively small. 

It would be very interesting to see this difference in the particle and parton levels of the simulation, because the jet-finding at these levels use the actual particle mass in the jet finding, and so the deviations between the massive and massless cases could be larger than in the actual data. Although I can't currently look at this in the pp simulation, I have been running some ep simulations for eRHIC studies and can look at massless vs massive particle 4-vectors at the particle level. Some quick properties of the ep simulation:

• Sqrt s = 141 GeV (20x250 GeV electrons on protons)
• Anti_kt Jet Finder with R=1.0
• Individual Jet pts >= 5.0, Delta phi > 120 degrees
• Subprocesses are 'Hard QCD', ie, resolved photon

Figure 11: Jet mass and dijet mass using massive and massless particle 4-vectors (particle level) from the ep simulation described above