Measuring the Legendre moments of the distribution of the $t \bar{t}$ production angle

University of Michigan, Ann Arbor, MI, USA
Michigan State University, East Lansing, MI, USA
CDF public note 10974: We measure the differential cross section for $t\bar{t}$ production as a function of the top production angle $\cos θ_t$, employing a projection onto the Legendre polynomials to characterize the shape of the cross section. We observe agreement with the standard model prediction for all except the 1st Legendre moment. The top forward-backward asymmetry is dominated by the anomalously large 1st Legendre moment.

We use a standard lepton+jets selection with only very minor modifications:

The differences from the standard lepton+jets are the $H_T$ cut, which rejects a large fraction of our Non-$W$ background while maintaining efficiency for top, and the addition of a 3 tight jets plus at least one loose jet sample. Other lepton+jets analyses do not cut on $H_T$ and require 4 or more tight jets.

We normalize our background with Method II. Our signal Monte Carlo is a sample of Powheg generated with the luminosity profile of the entire Run II.

The production angle, $\cos \theta_t$, is the angle between the proton momentum and the top quark momentum, as measured in the top-antitop center-of-mass frame. It is related to the top-antitop rapidity difference, $\Delta y$, used in previous $A_\text{FB}$ measurements.

Event yields

Table 1
CDF Run II Preliminary, 9.4/fb: Expected and observed event yields
$W$+HF 481 ± 178
$W$+LF 201 ± 72
Non-$W$ 207 ± 86
Single top 67 ± 6
Diboson 36 ± 4
$Z$+jets 34 ± 5
Total Background 1026 ± 210
$t\bar{t}$ (7.4 pb) 2750 ± 427
Total Prediction 3776 ± 476
Observed3864

Some validation plots

Most validation has already been accomplished in the previous $A_{FB}$ analysis1. We present just a few variables including the variable of interest, $\cos \theta_t$, and show that we are well modeled in our full signal sample, a side-band ("Anti-tag"), and several sub-regions of our signal sample.

Figures 1-10

Legendre polynomials

The Legendre polynomials $P_\ell (x)$ are a complete set of orthogonal polynomials. They frequently appear in electromagnetics and quantum mechanics, and are closely related to the theory of angular momentum in quantum mechanics.

Angular momentum

The 1959 paper of Jacob and Wick details the general theory of $2 \to 2$ scatter of particles with mass and spin. The angular dependence is entirely described by Legendre polynomials, and the Legendre moments are determined from a sum over total angular momentum states, incoming and outgoing helicities, and the Clebsch-Gordan coefficients. In the end, a diagram that proceeds with total angular momentum $J$, interfering with a diagram that proceeds with total angular momentum $J'$, contributes to the Legendre moments with $\left\lvert J - J' \right\rvert \leq \ell \leq J + J'$.

The only Standard Model diagram for $q\bar{q} \to t\bar{t}$ at leading order contains an intermediate state of a single gluon, so that diagram proceeds only with total angular momentum $J=1$ (the spin of the gluon). Thus, leading order Standard Model production of $t\bar{t}$ will have nonzero Legendre moments $a_0$ and $a_2$ ($a_1$ is zero because the final state is a parity eigenstate).

Beyond tree-level, additional diagrams with multi-particle intermediate states contribute. These diagrams proceed with any total angular momentum, and so contribute non-zero Legendre moments everywhere. We expect a roughly exponential decrease in the magnitude of the Legendre moments as $\ell$ increases due to the decreasing amplitude for initial states with large total angular momentum.

A model containing an axi-gluon can produce a non-zero $a_1$ at tree level because the final state is no longer a parity eigenstate. Models with a flavor-changing $Z'$ exchanged in the $t$-channel produce large non-zero moments everywhere because there is no single-particle intermediate state, similar to the NLO SM box diagram.

NLO SM prediction

We use a calculation by Bernreuther et al as our Standard Model benchmark. This calculation includes both QCD and electroweak effects on $t \bar{t}$ production at next-to-leading order, and has been performed with three different scale settings.

Figure 11

Other theoretical predictions

We also compare our data to several Monte Carlo calculations. These include an NLO SM calculation from POWHEG, a LO SM calculation from PYTHIA, and two new physics models from MadGraph. Octet A is a representative $s$-channel model, with an axigluon with a mass of 2 TeV. $Z'$ 200 is a $t$-channel model, containing a flavor-changing $Z'$ with a mass of 200 GeV.

Figure 12

Find the moments of the data and BG $\cos \theta_t$ distributions

We start by evaluating the Legendre moments of the data and the background model. We also estimate uncertainties on these as covariance matrices, to account for the correlations among the measured moments. We will correct the difference between the data and the background moments ("Background subtracted") to the parton level. We also compare the Legendre series from the raw moments to a histogram of the raw data. The series curve follows the histogram very nicely, demonstrating that the Legendre moments are an equivalently faithful representation of our data.

Figure 13-14

Background moments

Figure 15-16

Legendre moments corrected to the parton level

We correct the background-subtracted data moments to the parton level and evaluate all of our systematic uncertainties. We intend to present moments up through $\ell = 8$ in a table, but only up through $\ell = 4$ in the principal plot of the analysis.

Figure 17
Table 2
CDF Run II Preliminary, 9.4/fb: The parton-level Legendre moments, including both the systematic and statistical uncertainty
Parton-level data NLO SM (PRD 86 034026)
$\ell$ $a_\ell \pm \mathrm{stat} \pm \mathrm{syst}$ $a_\ell$ (scale uncertainties)
$1$$ 0.40 \pm 0.09 \pm 0.08$${ 0.15}_{-0.033}^{+0.066}$
$2$$ 0.44 \pm 0.14 \pm 0.21$${ 0.28}_{-0.030}^{+0.053}$
$3$$ 0.11 \pm 0.20 \pm 0.08$${ 0.030}_{-0.007}^{+0.014}$
$4$$ 0.22 \pm 0.25 \pm 0.11$${0.035}_{-0.008}^{+0.016}$
$5$$ 0.11 \pm 0.32 \pm 0.07$${0.0048}_{-0.001}^{+0.002}$
$6$$ 0.24 \pm 0.39 \pm 0.12$${0.0060}_{-0.003}^{+0.002}$
$7$$-0.15 \pm 0.46 \pm 0.14$${-0.0028}_{-0.001}^{+0.001}$
$8$$ 0.16 \pm 0.56 \pm 0.33$${-0.0019}_{-0.0003}^{+0.0003}$

Systematic uncertainties

We consider many sources of systematic uncertainty. To evaluate the effect of a source of systematic uncertainty, we vary some unknown parameter, the re-do the background subtraction and parton-level correction, obtaining a varied set of parton level moments, $a_\ell^{varied}$. We compare the varied moments with the nominal moments, and obtain a covariance matrix (with 100% correlation in the effect on each moment) $$σ_{\ell m} = (a_\ell^{varied} - a_\ell^{nominal}) \cdot (a_m^{varied} - a_m^{nominal}).$$ In this manner, we obtain a covariance matrix describing the effect of each source of systematic uncertainty. We sum all of these and add them to the covariance matrix describing the statistical uncertainty, giving us a covariance matrix which fully describes the uncertainty on the measurement of the Legendre moments.

Figure 18

Covariance matrix

The covariance matrix may be inverted to form a $\chi^2$ statistic which measures goodness of fit to our measurement. We also show the correlation matrix, $\dfrac{cov_{ij}}{sqrt(cov_{ii} cov_{jj})}$.

Table 3
CDF Run II Preliminary, 9.4/fb: The covariance matrix, including both statistical and systematic uncertainties, describing the uncertainty on the measured Legendre moments
12345678
1 $ 1.47 \times 10^{-2}$ $-7.60 \times 10^{-4}$ $ 1.14 \times 10^{-2}$ $-2.95 \times 10^{-3}$ $ 6.86 \times 10^{-3}$ $-1.73 \times 10^{-3}$ $ 2.01 \times 10^{-3}$ $-7.34 \times 10^{-3}$
2 $-7.60 \times 10^{-4}$ $ 6.41 \times 10^{-2}$ $ 8.96 \times 10^{-3}$ $ 3.29 \times 10^{-2}$ $-3.70 \times 10^{-3}$ $ 4.12 \times 10^{-3}$ $ 1.05 \times 10^{-2}$ $-4.69 \times 10^{-2}$
3 $ 1.14 \times 10^{-2}$ $ 8.96 \times 10^{-3}$ $ 4.50 \times 10^{-2}$ $-8.18 \times 10^{-5}$ $ 2.72 \times 10^{-2}$ $-4.88 \times 10^{-3}$ $ 1.36 \times 10^{-2}$ $-1.33 \times 10^{-2}$
4 $-2.95 \times 10^{-3}$ $ 3.29 \times 10^{-2}$ $-8.18 \times 10^{-5}$ $ 7.72 \times 10^{-2}$ $-7.32 \times 10^{-4}$ $ 4.00 \times 10^{-2}$ $ 5.49 \times 10^{-3}$ $ 1.13 \times 10^{-2}$
5 $ 6.86 \times 10^{-3}$ $-3.70 \times 10^{-3}$ $ 2.72 \times 10^{-2}$ $-7.32 \times 10^{-4}$ $ 1.06 \times 10^{-1}$ $ 2.31 \times 10^{-3}$ $ 4.85 \times 10^{-2}$ $ 1.13 \times 10^{-2}$
6 $-1.73 \times 10^{-3}$ $ 4.12 \times 10^{-3}$ $-4.88 \times 10^{-3}$ $ 4.00 \times 10^{-2}$ $ 2.31 \times 10^{-3}$ $ 1.63 \times 10^{-1}$ $ 1.42 \times 10^{-2}$ $ 8.81 \times 10^{-2}$
7 $ 2.01 \times 10^{-3}$ $ 1.05 \times 10^{-2}$ $ 1.36 \times 10^{-2}$ $ 5.49 \times 10^{-3}$ $ 4.85 \times 10^{-2}$ $ 1.42 \times 10^{-2}$ $ 2.32 \times 10^{-1}$ $-5.45 \times 10^{-3}$
8 $-7.34 \times 10^{-3}$ $-4.69 \times 10^{-2}$ $-1.33 \times 10^{-2}$ $ 1.13 \times 10^{-2}$ $ 1.13 \times 10^{-2}$ $ 8.81 \times 10^{-2}$ $-5.45 \times 10^{-3}$ $ 4.17 \times 10^{-1}$
Figure 19

Significance of deviation

We subtract the NLO prediction from the measured moments, and divide by the uncertainty on the measured moments to obtain a significance. All of the moments except for $a_1$ are consistent with the predictions of the Standard Model. The first moment (linear term) deviates from the Standard Model by just over 2σ.

Figure 20

Contributions of the moments to the forward-backward asymmetry

The even moments have no contribution to the asymmetry, as the even Legendre polynomials are symmetric.

The odd moments all contribute to the asymmetry. We show the contribution of each moment to the total asymmetry, and the contribution of the linear terms ($a_1$) vs the contribution of the non-linear terms ($\ell > 1$).

Figure 21-22

Integrating the Legendre series

We take the measured Legendre moments, form the Legendre series, and integrate this series over the span of 10 bins in $\cos \theta_t$. This gives us a measurement of the differential cross section in a more traditional fashion than the shape analysis afforded by measuring the Legendre moments.

We also subtract the predicted cross section from the measured cross section, and plot a line corresponding to the anomalous first moment.

Figure 23-24
Table 4
CDF Run II Preliminary, 9.4/fb: Integral of the Legendre series over the bins, normalized to an integral of 1
Bin edges Cross section NLO (QCD+EWK)
$-1.0$ -- $-0.8$ $0.085 \pm 0.010$ ${0.106}_{-0.001}^{-0.002}$
$-0.8$ -- $-0.6$ $0.079 \pm 0.008$ ${0.096}_{-0.002}^{-0.004}$
$-0.6$ -- $-0.4$ $0.078 \pm 0.005$ ${0.090}_{-0.002}^{-0.004}$
$-0.4$ -- $-0.2$ $0.079 \pm 0.004$ ${0.087}_{-0.002}^{-0.003}$
$-0.2$ -- $ 0.0$ $0.084 \pm 0.002$ ${0.086}_{-0.001}^{-0.003}$
$ 0.0$ -- $ 0.2$ $0.091 \pm 0.002$ ${0.089}_{-0.001}^{-0.002}$
$ 0.2$ -- $ 0.4$ $0.101 \pm 0.004$ ${0.094}_{-0.000}^{-0.000}$
$ 0.4$ -- $ 0.6$ $0.114 \pm 0.005$ ${0.102}_{+0.001}^{+0.002}$
$ 0.6$ -- $ 0.8$ $0.132 \pm 0.008$ ${0.115}_{+0.003}^{+0.005}$
$ 0.8$ -- $ 1.0$ $0.158 \pm 0.010$ ${0.136}_{+0.006}^{+0.011}$

Assume values for some moments

Since all the moments with $\ell \geq 2$ are in agreement with the Standard Model prediction, we can improve our estimate of the first moment by making an additional assumption. We introduce the assumption that the NLO SM prediction (PRD 86 034026), with the scale uncertainties, is correct for the moments with $\ell \geq 2$, and combine this assumption with our measurement using the BLUE method. We obtain a new estimate of $a_1$ with an improvement in precision of approximately 10%.

Figure 25-26