WWA aTGC with VBFNLO

Table of Contents:

Graphs:

In collaboration with Fermilab and CERN, for a project utilizing VBFNLO, below are a few figures for the photon pT spectrum of the WWγ process with anomalous coupling. This particular process used VBFNLO's version 2.7.0 Beta 1 standard cuts for a proton-proton collision at a center-of-mass beam energy of 8 TeV. The WWγ decays leptonically, and this is the photon's differential cross section as a function of its pT. The authors have suggested using at least 4 iterations to optimize the integrations, but have currently reported a bug in the software for any process using more than one phase space (this process uses 3). The 2.7.0 Beta 1 version was used for this process because previous versions do not employ anomalous coupling for the WWγ leptonic decay.

Parabloid/Parabolic variation in dσ Ratio (aTGC : SM):

The above figure is for the ratio Anomalous Differential Cross-section : SM Differential Cross-section, with Δκ values on the y-axis (in percent deviation from SM), while λ is on the x-axis. The differential cross-sections used in the ratio correspond to a photon pT of approximately 50 GeV. The histogram has been fitted with a 4th degree polynomial, z(x,y) = (c1 + c2*x + c3*x^2 +c4*y + c5*y^2 + c6*x*y), as a red grid. Since the photon pT spectrum ranges from 20 - 500 GeV with bin sizes of 1 GeV, then there are 480 such plots for a parabloid (one for each pT bin).

The above figure is for the ratio Anomalous Differential Cross-section : SM Differential Cross-section, with Δκ values on the x-axis (in percent deviation from SM), while λ = 0. The differential cross-sections used in the ratio correspond to a photon pT of approximately 250 GeV. The histogram has been fitted with a 2nd degree polynomial (red line), and represents essentially plane cross-section at λ = 0 removed from the parabloid (such as the one plotted above).

The constant coefficients c1-c6 used in the parabloid fit to the differential photon cross-section ratio are actually dependent on the photon pT, e.c. c1(pT), etc. Below are a few figures that attempt to characterize the pT-dependence of these coefficients with 6th-degree polynomials. As can be seen in all the figures, the fit parameters are dominated by p0 and p1, corresponding to coefficients of 1 and x, respectively, in the 6th degree polynomial; however, without the higher order terms, the fit becomes noticeably worse under a logscale y-axis plot. Each figure's title stipulates which polynomial term the coefficient corresponds to.

It actually turns out that c1(pT) should be constant at c1 = 1. This is because in the parabloid equation: z = c1 + ...., where z is the differential cross-section ratio (aTGC:SM), means that when Δκ and λ are both equal to zero, the ratio should be unity (no aTGC). Although the following plots correspond to a parabloid fit with c1 taking values in the above plot, fixing c1 = 1 in the fit algorithm did not change the "nature" of the pT-dependence of the remaining coefficeints (c2-c6).

dσ Ratio (aTGC : SM) vs. Photon pT Functional Fit:

The following images are Ratio plots between the Standard Model cross section of the photon, in the WWγ fully leptonic decay as a function of pT, and the photon's cross section for various anomolous coupling processes of varying values in Δκ and λ. The y-axis is in a logarithmic scale in order to elaborate on the exponential nature of the Ratio plots, as well as to help differentiate between different plot lines. The bin sizes are 1 GeV, and the MC generation was done with 6 iterations and 2^26 points.

The above figure is for Δκ = 0.03 and λ = 0.03, fitted with a 2nd degree polynomial with χ2 = 20.

The above figure is for Δκ = 0.06 and λ = 0.06, fitted with a 2nd degree polynomial with χ2 = 93.

As can be seen in the above two figures, the 2nd degree polynomial fit seems to not follow along the MC data, especially at low pT. The following few figures demonstrate a few different fit methods to fix this issue for both small and large anomalous coupling.

The above figure is for Δκ = 0.03 and λ = 0, fitted with a 2nd degree polynomial with χ2 = 6.

The above figure is for Δκ = 0.03 and λ = 0, fitted with a 5th degree polynomial with χ2 = 6.

The above figure is for Δκ = 0.03 and λ = 0, fitted with a gaussian.

The above figure is for Δκ = 0.3 and λ = 0.3, fitted with a 2nd degree polynomial with χ2 = 37831.

The above figure is for Δκ = 0.3 and λ = 0.3, fitted with a 2nd degree polynomial with χ2 = 37831, except that the y-axis is not logscaled.

The above figure is for Δκ = 0.3 and λ = 0.3, fitted with a 5th degree polynomial with χ2 = 8607.

The above figure is for Δκ = 0.3 and λ = 0.3, fitted with a gaussian.

dσ Ratio (aTGC : SM) vs. Photon pT:

Below are figures demonstrating the difference the Ratio takes as the two coefficients are varied from their SM values. The logarithmic nature indicates that Δκ contributes the "elbow" appearance in the Ratio plots in the low pT range (i.e. seems to influence the low pT photon cross section)....

....while λ has a dramatic impact on the photon's cross section at higher pT....