Fox-Wolfram moments in Higgs physics

Geometric correlations between jets as part of hard processes or in addition to hard processes are key ingredients to many LHC analyses. Fox-Wolfram moments systematically describe these correlations in terms of spherical harmonics. These moments, computed either from the tagging jets or from all jets in each event, can significantly improve Higgs searches in weak boson fusion. Applications of Fox-Wolfram moments in LHC analyses obviously surpass jets as analysis objects, as well as Higgs searches in terms of analyses.

Figure 1

Fox-Wolfram moments for $\ell =2$, 17 with the weight factors $W_{ij}^s$ (left), $W_{ij}^z$ (center), and $W_{ij}^T$ (right) for WBF $H$ signal (solid line/gray), $Z+2$ jets (short dashed line/light gray), and $t\overline{t}+1$ jet (long dashed line/black). All jets entering the moments pass the basic acceptance cuts of Eqs. (5, 6).

Figure 2

Fox-Wolfram moments $H_{\ell }^{p,T}$ for two jets as a function of ${\Omega }_{12}$ and $r=r_{p,T}$. We show the analytic result in our toy model [Eq. (8)].

Figure 3

Normalized distributions of Fox-Wolfram moments computed only from the two tagging jets for $\ell =2–5$, 8, 9, 12, and 19 with weight factors $W_{ij}^p$ (left) and $W_{ij}^T$ (right) for WBF $H$ signal (solid line/gray), $Z+2$ jets (short dashed line/light gray), and $t\overline{t}+1$ jet (long dashed line/black). All events pass the cuts in Eqs. (5, 6).

Figure 4

Top: correlations of $H_3^T$ with $H_3^p$ and $H_4^T$ computed from the two tagging jets alone for WBF $H$ signal (square), $Z+2$ jets (cross), and $t\overline{t}+1$ jet (triangle). Bottom: Normalized distributions for $H_7^T$ and $H_8^T$ computed from the two tagging jets alone and after requiring $H_3^T<0.3$ for WBF $H$ signal (solid line/gray), $Z+2$ jets (short dashed line/light gray), and $t\overline{t}+1$ jet (long dashed line/black). All event in this figure pass the basic cuts in Eqs. (5, 6), but not the final $\Delta y_{jj}$ requirement.

Figure 5

Normalized distributions of Fox-Wolfram moments computed only from the two tagging jets for $\ell =2–5$, 8, 9, 12, and 19 with a weight factor $W_{ij}^T$ for WBF $H+1$ jet signal (solid line/gray), $Z+2$ jets (short dashed line/light gray), and $t\overline{t}+1$ jet (long dashed line/black). All events pass the full set of QCD cuts in Eqs. (5, 6, 7).

Figure 6

Normalized distributions on Fox-Wolfram moments computed from all available jets for $\ell =2–5$, 8, 9, 12, and 19 with weight factors $W_{ij}^p$ (left) and $W_{ij}^T$ (right) for WBF $H+1$ jet signal (solid line/gray), $Z+2$ jets (short dashed line/light gray), and $t\overline{t}+1$ jet (long dashed line/black). All events pass the cuts in Eqs. (5, 6).

Figure 7

Normalized distributions of Fox-Wolfram moments computed from all jets for $\ell =2–5$, 8, 9, 12, and 19 with a weight factor $W_{ij}^T$ for WBF $H+1$ jet signal (solid line/gray), $Z+2$ jets (short dashed line/light gray), and $t\overline{t}+1$ jet (long dashed line/black). All events pass the full set of QCD cuts in Eqs. (5, 6, 7).

Figure 8

Normalized distributions of Fox-Wolfram moments computed from all jets for $\ell =2–5$, 8, 9, 12, and 19 with weight factor $W_{ij}^1$ for WBF $H+1$ jet signal (solid line/gray), $Z+2$ jets (short dashed line/light gray), and $t\overline{t}+1$ jet (long dashed line/black). All events pass the full set of QCD cuts in Eqs. (5, 6, 7).