Dependencies of lepton angular distribution coefficients on the transverse momentum and rapidity of $Z$ bosons produced in $pp$ collisions at the LHC

High precision data of lepton angular distributions for ${\gamma }^{*}/Z$ production in $pp$ collisions at the LHC, covering broad ranges of dilepton transverse momenta ($q_T$) and rapidity ($y$), were recently reported. Strong $q_T$ dependencies were observed for several angular distribution coefficients, $A_i$, including $A_0-A_4$. Significant $y$ dependencies were also found for the coefficients $A_1$, $A_3$ and $A_4$, while $A_0$ and $A_2$ exhibit very weak rapidity dependence. Using an intuitive geometric picture, we show that the $q_T$ and $y$ dependencies of the angular distributions coefficients can be well described.

Figure 1

Definition of the Collins-Soper frame and various angles and planes in the rest frame of ${\gamma }^{*}/Z$. The hadron plane is formed by ${\overrightarrow{P}}_B$ and ${\overrightarrow{P}}_T$, the momentum vectors of the beam (B) and target (T) hadrons. The $\widehat{x}$ and $\widehat{z}$ axes of the Collins-Soper frame both lie in the hadron plane with the $\widehat{z}$ axis bisecting the ${\overrightarrow{P}}_B$ and $-{\overrightarrow{P}}_T$ vectors. The quark ($q$) and antiquark ($\overline{q}$) annihilate collinearly with equal momenta to form ${\gamma }^{*}/Z$, while the quark momentum vector ${\widehat{z}}^{\prime }$ and the $\widehat{z}$ axis form the quark plane. The polar and azimuthal angles of ${\widehat{z}}^{\prime }$ in the Collins-Soper frame are ${\theta }_1$ and ${\varphi }_1$. The $l^{-}$ and $l^{+}$ are emitted back-to-back with $\theta$ and $\varphi$ as the polar and azimuthal angles for $l^{-}$.

Figure 2

(a) Feynman diagram for $q-\overline{q}$ annihilation where a gluon is emitted from a quark in the beam hadron (B). (b) Momentum direction for $q$ and $\overline{q}$ in the C-S frame before and after gluon emission. The momentum direction of $q$ is now collinear with that of $\overline{q}$. (c) Feynman diagram for the case where a gluon is emitted from an antiquark in the target hadron (T). (d) Momentum direction for $q$ and $\overline{q}$ in the C-S frame before and after gluon emission for diagram (c).

Figure 3

(a) Feynman diagram for $q-\overline{q}$ annihilation where a gluon is emitted from an antiquark in the beam hadron (B). (b) Momentum direction for $q$ and $\overline{q}$ in the C-S frame before and after gluon emission. The momentum direction of $q$ is now collinear with that of $\overline{q}$. (c) Feynman diagram for the case where a gluon is emitted from a quark in the target hadron (T). (d) Momentum direction for $q$ and $\overline{q}$ in the C-S frame before and after gluon emission for diagram (c).

Figure 4

(a) Feynman diagram for $qG$ Compton process where a quark from the beam hadron annihilates with an antiquark from the splitting of a gluon in the target hadron. (b) Momentum direction of $q$, $\overline{q}$ and gluon in the C-S frame before and after gluon splitting. (c) Feynman diagram for $qG$ fusing into a quark which then emits a ${\gamma }^{*}/Z$. (d) Momentum direction of $q$, $\overline{q}$ and gluon before and after the $qG$ fusion.

Figure 5

The CMS data [25] on angular distribution coefficients $A_i$ versus $q_T$ for $|y|<1.0$.

Figure 6

Comparison between the CMS data [25] on $A_0$, $A_2$ and $A_0-A_2$ with calculations. Curves correspond to calculations described in the text.

Figure 7

Comparison between the CMS data [25] on $A_1$, $A_3$ and $A_4$ at $|y|<1.0$ with calculations. Curves correspond to calculations described in the text.

Figure 8

Comparison between the CMS data [25] on $A_0$ and $A_2$ at two rapidity regions with calculations. Curves correspond to calculations described in the text.

Figure 9

Comparison between the CMS data [25] on $A_1$, $A_3$ and $A_4$ at two rapidity regions with calculations. Curves correspond to calculations described in the text.