[a]Wen-Chen Chang

# QCD effects in lepton angular distributions of Drell-Yan/$Z$ production and
jet discrimination

Randall Evan McClellan  Jen-Chieh Peng  Oleg Teryaev

###### Abstract

We present a comparison of data of lepton angular distributions of Drell-
Yan/$Z$ production with the fixed-order pQCD calculations by which the
baseline of pQCD effects is illustrated. As for the $Z$ production, we predict
that $A_{0}$ and $A_{2}$ for $Z$ plus single gluon-jet events are very
different from that of $Z$ plus single quark-jet events, allowing a new
experimental tool for checking various algorithms which attempt to
discriminate quark jets from gluon jets. Using an intuitive geometric
approach, we show that the violation of the Lam-Tung relation, appearing at
large transverse-momentum region, is attributed to the presence of a non-
coplanarity effect. This interpretation is consistent with the appearance of
violation beyond LO-QCD effect.

##  1 Introduction

Measuring lepton angular distributions of Drell-Yan (D-Y) process [1] provides
a powerful tool to explore the reaction mechanisms and the parton
distributions of colliding hadrons. For example, the Lam-Tung (L-T) relation
[2] has been proposed as a benchmark of the perturbative QCD (pQCD) effect in
D-Y process. Violations of L-T relation were observed in the measurements of
D-Y production by the fixed-target experiments as well as $\gamma^{*}$/$Z$
production by the CMS [3] and ATLAS [4] experiments at LHC. It is important to
understand the origin of these violations.

It is found that the violation of the L-T relation seen in CMS and ATLAS data
in the region of transverse momentum ($q_{T}$) greater than 5 GeV could be
well described taking into account NNLO pQCD effect [5]. On the other hand,
the agreement is not as good in a similar comparison [6, 7] for the fixed-
target data of NA10 [8], E615 [9] and E866 [10] at $q_{T}<3$ GeV. Transverse-
momentum dependent Boer-Mulders functions [11], correlating the nucleon spin
with the intrinsic transverse momentum of partons, have been suggested to
account for a violation of the L-T relation observed at small $q_{T}$ in the
fixed-target experiments.

In this proceedings, we show that the $q_{T}$ dependence of the angular
distribution coefficients, as well as the violation of the Lam-Tung violation,
could be obtained if the angular distribution coefficients were analyzed as a
function of the number of accompanying jets in $Z$-boson production measured
by the CMS and ATLAS Collaborations [12]. Furthermore we compare the data of
dilepton angular parameters $\lambda$, $\mu$, $\nu$ and the L-T violation
quantity $1-\lambda-2\nu$ measured by E615 [9] with the fixed-order pQCD
calculations. Finally we interpret some notable features of pQCD results using
the geometric model [13, 14, 15]. More results and greater details can be
found in Ref. [7, 12].

##  2 Lepton angular distributions of $Z$ production and jet discrimination

The lepton angular distribution in the $Z$ rest frame can be expressed as [3,
4]

$\displaystyle\frac{d\sigma}{d\Omega}$ $\displaystyle\propto$
$\displaystyle(1+\cos^{2}\theta)+\frac{A_{0}}{2}(1-3\cos^{2}\theta)+A_{1}\sin
2\theta\cos\phi+\frac{A_{2}}{2}\sin^{2}\theta\cos 2\phi$ (1) $\displaystyle+$
$\displaystyle A_{3}\sin\theta\cos\phi+A_{4}\cos\theta+A_{5}\sin^{2}\theta\sin
2\phi+A_{6}\sin 2\theta\sin\phi+A_{7}\sin\theta\sin\phi,$

where $\theta$ and $\phi$ are the polar and azimuthal angles of leptons in the
rest frame of $Z$. Since the intrinsic transverse momenta of the annihilating
quark and antiquark is neglected in the original Drell-Yan model, the angular
distribution is simply $1+\cos^{2}\theta$ and all angular distribution
coefficients, $A_{i}$, vanish. For a non-zero dilepton transverse momentum,
$q_{T}$, these coefficients can deviate from zero due to QCD effects. However,
it was predicted that the coefficients $A_{0}$ and $A_{2}$ should remain
identical, $A_{0}=A_{2}$, i.e. the Lam-Tung relation [2].

Figure 1: Comparison between the CMS data [3] of $A_{0}$ and $A_{0}-A_{2}$ for
$Z$ production from $p-p$ collisions with fixed-order pQCD calculations.
Curves correspond to calculations described in the text.

Figure 1 shows the CMS data for $A_{0}$ and $A_{0}-A_{2}$. Pronounced $q_{T}$
dependence of $A_{0}$ is observed and the Lam-Tung relation, $A_{0}-A_{2}=0$,
is clearly violated. There are two NLO QCD subprocesses for $Z$ production:
$q\bar{q}\rightarrow Zg$ annihilation process, and $qg\rightarrow Zq$ quark
Compton scattering process. In the Collins-Soper frame [16], the NLO pQCD
predictions of $A_{0}$ and $A_{2}$ as a function of $q_{T}$ of $Z$ for these
two processes are $A_{0}=A_{2}=q^{2}_{T}/(M_{Z}^{2}+q^{2}_{T})$ [17] and
$A_{0}=A_{2}=5q^{2}_{T}/(M_{Z}^{2}+5q^{2}_{T})$ [18, 19], respectively. The
dotted and dashed curves in Fig. 1(a) correspond to these NLO expressions.

As the $q\bar{q}$ and $qg$ processes contribute to the $pp\to ZX$ reaction
incoherently, the observed $q_{T}$ dependence of $A_{0}$ reflects the combined
effect of these two contributions. A best-fit to the CMS $A_{0}$ data, shown
as the solid line in Fig. 1(a), gives a mixture of 58.5$\pm$1.6% $qg$ and
41.5$\pm$1.6% $q\bar{q}$ processes. For $pp$ collisions at the LHC, the $qg$
process is expected to be more important than the $q\bar{q}$ process, in
agreement with the best-fit result. For $Z$ plus single-jet events, Fig. 1(a)
shows that there is remarkable difference in the $q_{T}$ dependence for
$A_{0}$ between the $q\bar{q}$ annihilation process and the $qg$ Compton
process. Since it is a high-$p_{T}$ gluon (quark) jet associated with the
$q\bar{q}(qg)$ process at the $\alpha_{s}$ level, one could first utilize the
existing algorithms for quark (gluon) jet identification to separate the
$q\bar{q}$ annihilation events from the $qg$ Compton events and investigate
the angular distribution of individual event samples. Their angular
distribution coefficients for $Z$ plus single jet data would also provide a
powerful tool for testing various algorithms designed to distinguish quark
jets from gluon jets.

For the $Z$ plus multi-jet data, the L-T relation is expected to be violated
at a higher level than that of the inclusive production data. Exclusion of the
$Z$ plus single-jet events satisfying the L-T relation, would enhance the
violation of the L-T relation. We have carried out pQCD calculations using
DYNNLO [20, 21] to demonstrate this effect. Figure 1(b) compares the DYNNLO
calculations with the CMS $A_{0}-A_{2}$ data. The black band corresponds to
the NNLO calculation including contributions from the events of $Z$ with
single jet and two jets. The blue band singles out the contributions to
$A_{0}-A_{2}$ from $Z$ plus two jets only, showing that the violation of the
Lam-Tung relation is indeed amplified for the multi-jet events. This can be
readily tested with the data collected at the LHC.

##  3 Lepton angular distributions of Drell-Yan production in fixed-target
experiments

In the rest frame of the virtual photon in the D-Y process, another expression
for the lepton angular distributions commonly used by the fixed-target
experiments is given as [2]

$\frac{d\sigma}{d\Omega}\propto 1+\lambda\cos^{2}\theta+\mu\sin
2\theta\cos\phi+\frac{\nu}{2}\sin^{2}\theta\cos 2\phi,$ (2)

where $\theta$ and $\phi$ refer to the polar and azimuthal angles of leptons.
The $\lambda,\mu,\nu$ are related to $A_{0},A_{1},A_{2}$ in Eq. (1) via

$\displaystyle\lambda=\frac{2-3A_{0}}{2+A_{0}};~{}~{}~{}\mu=\frac{2A_{1}}{2+A_{0}};~{}~{}~{}\nu=\frac{2A_{2}}{2+A_{0}}.$
(3)

Equation (3) shows that the L-T relation, $1-\lambda-2\nu=0$, is equivalent to
$A_{0}=A_{2}$.

Figure 2: Comparison of NLO (red points) and NNLO (blue points) fixed-order
pQCD calculations with the E615 $\pi^{-}+W$ D-Y data at 252 GeV [9] (black
points) for $\lambda$, $\mu$, $\nu$ and $1-\lambda-2\nu$.

In Fig. 2, we compare the results of $\lambda$, $\mu$, $\nu$, and the L-T
violation, $1-\lambda-2\nu$, from the fixed-order pQCD calculations with
252-GeV $\pi^{-}+W$ data from E615 experiment [9]. The angular parameters are
evaluated as a function of $q_{T}$ in the Collins-Soper frame. Overall, the
calculated $\lambda$, $\mu$ and $\nu$ exhibit distinct $q_{T}$ dependencies.

At $q_{T}\rightarrow 0$, $\lambda$, $\mu$ and $\nu$ approach the values
predicted by the collinear parton model: $\lambda=1$ and $\mu=\nu=0$. As
$q_{T}$ increases, Fig. 2 shows that $\lambda$ decreases toward its
large-$q_{T}$ limit of $-1/3$ while $\nu$ increases toward $2/3$, for both
$q\bar{q}$ and $qG$ processes [18, 19]. The $q_{T}$ dependence of $\mu$ is
relatively mild compared to $\lambda$ and $\nu$. This is understood as a
result of some cancellation effect, to be discussed in Sec. 4.

Comparing the results of the NLO with the NNLO calculations,
$\lambda{\rm(NNLO)}$ is smaller than $\lambda\rm{(NLO)}$ while $\mu$ and $\nu$
are very similar for NLO and NNLO. The amount of L-T violation,
$1-\lambda-2\nu$, is zero in the NLO calculation, and nonzero and positive in
the NNLO calculation. As seen in Fig. 2, the pQCD predicts a sizable magnitude
for $\nu$, comparable to the data. Such pQCD effect should be included in the
determination of nonperturbative Boer-Mulders effect from the data of $\nu$.

##  4 Geometric model

As introduced above, both CMS and E615 data of lepton angular distributions
for $Z$ and D-Y production can be reasonably well described by the NLO and
NNLO pQCD calculations. It is interesting to see that various salient features
of pQCD calculations could be understood using a geometric approach developed
in Refs. [13, 14].

In the Collins-Soper $\gamma^{*}/Z$ rest frame, the hadron plane, the quark
plane, and the lepton plane of collision geometry are defined [13, 14]. A pair
of collinear $q$ and $\bar{q}$ with equal momenta annihilate into a
$\gamma^{*}/Z$. The momentum unit vector of $q$ is defined as
$\hat{z}^{\prime}$, and the quark plane is formed by the $\hat{z}^{\prime}$
and the $\hat{z}$ axes of Collins-Soper frame. The angular coefficients
$A_{i}$ in Eq. (3) can be expressed in term of $\theta_{1}$ and $\phi_{1}$ as
follows:

$\displaystyle
A_{0}=\langle\sin^{2}\theta_{1}\rangle,~{}~{}~{}A_{1}=\frac{1}{2}\langle\sin
2\theta_{1}\cos\phi_{1}\rangle,~{}~{}~{}A_{2}=\langle\sin^{2}\theta_{1}\cos
2\phi_{1}\rangle,$ (4)

where the $\theta_{1}$ and $\phi_{1}$ are the polar and azimuthal angles of
the natural quark axis $\hat{z}^{\prime}$ of the quark plane in the Collins-
Soper frame.

Up to NLO ($\mathcal{O}(\alpha_{S})$) in pQCD, the quark plane coincides with
the hadron plane and $\phi_{1}=0$. Therefore $A_{0}=A_{2}$ or
$1-\lambda-2\nu=0$, i.e., the L-T relation is satisfied. Higher order pQCD
processes allow the quark plane to deviate from the hadron plane, i.e.,
$\phi_{1}\neq 0$. This acoplanarity effect leads to a violation of the L-T
relation. For a nonzero $\phi_{1}$, Eq. (4) shows that $A_{2}<A_{0}$.
Therefore, when the L-T relation is violated, $A_{0}$ must be greater than
$A_{2}$ or, equivalently, $1-\lambda-2\nu>0$. This expectation of
$1-\lambda-2\nu>0$ in this geometric approach agrees with the results of NNLO
pQCD calculations shown in Fig. 2. The geometric approach offers a simple and
intuitive interpretation for this result.

Furthermore the sign of $\mu$ could be either positive or negative, depending
on which parton and hadron the gluon is emitted from [14, 7]. Hence, one
expects some cancellation effects for $\mu$ among contributions from various
processes. Each process is weighted by the corresponding parton density
distributions. At mid-rapidity, the momentum fraction carried by the beam
parton is comparable to that of the target parton. Therefore, the weighting
factors for various processes are of similar magnitude and the cancellation
effect could be very significant, resulting in a small value of $\mu$.

##  5 Summary

We have presented a comparison of the measurements of the angular parameters
$A_{0}$ and $A_{0}-A_{2}$ of the $Z$ production from the CMS experiment as
well as $\lambda$, $\mu$, $\nu$ and $1-\lambda-2\nu$ of the D-Y process from
the fixed-target E615 experiment with the corresponding results from the NLO
and NNLO pQCD calculations. Qualitatively the transverse momentum dependence
of measured angular parameters could be described by pQCD. The L-T violation
part $A_{0}-A_{2}$ or $1-\lambda-2\nu$ remains zero in the NLO pQCD
calculation and turns positive in NNLO pQCD. The measurement of $A_{0}$ and
$A_{2}$ coefficients in $Z$ plus single-jet or multi-jet events would provide
valuable insights on the origin of the violation of the L-T relation and could
be used an an index in discriminating the intrinsic property of high-$q_{T}$
jets.

Within the geometric picture, the occurrence of acoplanarity between the quark
plane and the hadron plane ($\phi_{1}\neq 0$), for the pQCD processes beyond
NLO offers an interpretation of a violation of the L-T relation. The predicted
positive value of $1-\lambda-2\nu$, or $A_{0}>A_{2}$ when $\phi_{1}$ is
nonzero, is consistent with the NNLO pQCD results.

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