[a]Federico Vazzoler

# Probing the Weak Mixing Angle at high energies at the LHC

Simone Amoroso  Ekaterina Lipka  Clara Lavinia Del Pio  Mauro Chiesa  Fulvio
Piccinini  Alessandro Vicini

###### Abstract

The electroweak mixing angle is a fundamental parameter of the theory of
electroweak interactions. Its value has been measured precisely at the Z–pole
at colliders. In this contribution, we propose to exploit measurements of
Neutral–Current Drell–Yan production at the Large Hadron Collider at large
invariant dilepton masses to determine the energy scale dependence (running)
of the electroweak mixing angle in the $\overline{\text{MS}}$ renormalisation
scheme, $\sin^{2}\theta_{W,\ell}^{\overline{\text{MS}}}(\mu)$. Such a
measurement can be used to confirm the Standard Model predictions for the
$\overline{\text{MS}}$ running at TeV scales, and to set model–independent
constraints on new states with electroweak quantum numbers. To this end, we
make use of a dedicated implementation of
$\sin^{2}\theta_{W,\ell}^{\overline{\text{MS}}}(\mu)$ in the POWHEG-BOX-V2
Monte Carlo event generator, which we use to explore the potential of future
analyses using the data of the LHC Run 3 and High–Luminosity.

##  1 Introduction

The electroweak mixing angle $\sin^{2}\theta_{W}$ is one of the fundamental
parameters of the Standard Model (SM). In the electroweak (EW) theory it can
be defined as

$\sin^{2}\theta_{W}\equiv\frac{g_{1}^{2}}{g_{1}^{2}+g_{2}^{2}}=1-\frac{\text{m}_{\text{W}}^{2}}{\text{m}_{\text{Z}}^{2}},$
(1)

where $g_{1}$ and $g_{2}$ are the $U(1)_{Y}$ and $SU(2)_{L}$ gauge couplings,
$\text{m}_{\text{W}}$ and $\text{m}_{\text{Z}}$ represent the physical masses
of the massive gauge bosons, and the last equality is assumed to all orders in
perturbation theory in the on–shell renormalization scheme. In other
renormalization schemes, the definition of Eq. 1 is modified by radiative
corrections. In particular, in the modified minimal–subtraction
($\overline{\text{MS}}$) renormalisation scheme, $\sin^{2}\theta_{W}$ becomes
a scale–dependent quantity (running).

At hadron colliders, $\sin^{2}\theta_{W}$ is measured by studying the angular
distribution of the lepton pairs produced in neutral current Drell–Yan (NCDY)
processes
$\text{q}\overline{\text{q}}\to\text{Z}/\gamma^{*}\to\ell^{+}\ell^{-}$. The
next–to–leading (NLO) order QCD prediction of the differential cross section
for the NCDY process can be expressed as a function of the angular
distribution of the produced leptons, their invariant mass,
$\text{m}_{\ell\ell}$, and rapidity, $y_{\ell\ell}$, as [1]

$\frac{\text{d}\sigma}{\text{d}\text{m}_{\ell\ell}\text{d}y_{\ell\ell}\text{d}\cos\theta_{CS}}=\frac{\text{d}\sigma_{\text{unpol}}}{\text{d}\text{m}_{\ell\ell}\text{d}y_{\ell\ell}}\biggr{[}(1+\cos^{2}\theta_{CS})+\frac{A_{0}}{2}(1-3\cos^{2}\theta_{CS})+A_{4}\cos\theta_{CS}\biggr{]},$
(2)

where $\theta_{CS}$ is the angle between the incoming and outgoing fermions in
the Collins–Soper reference frame [2]. Because of the vector– and axial–vector
couplings of the Z boson, Eq. 2 shows an asymmetry in the number of events
expected between the forward ($\cos\theta_{CS}>0$) and backward
($\cos\theta_{CS}<0$) directions ($A_{\text{FB}}$). The value of the EW mixing
angle at the scale of the Z–boson mass $\text{m}_{\text{Z}}$ has been
extracted with a precision at the sub–per cent level [3] by studying the
$A_{\text{FB}}$ asymmetry, while its value has never been tested at high
energies, i.e. above $\text{m}_{\text{Z}}$.

The main goal of this work is to investigate the sensitivity of the Large
Hadron Collider (LHC) data to the energy scale dependence of the EW mixing
angle in the $\overline{\text{MS}}$ scheme above $\text{m}_{\text{Z}}$. To do
so, an $\overline{\text{MS}}$ definition of the EW mixing angle
$\sin^{2}\theta_{W,\ell}^{\overline{\text{MS}}}(\mu)$ [4], where $\mu$ is
chosen as the invariant mass of the final state leptons
$\mu=\text{m}_{\ell\ell}$, is considered. This quantity is related to the
coupling–based definition in Eq. 1, and it features a well–defined subtraction
of the singular terms which arises in dimensional regularisation [4]. To
extract the expected sensitivity on
$\sin^{2}\theta_{W,\ell}^{\overline{\text{MS}}}(\mu)$, predictions for the
NCDY differential cross section of Eq. 2 are obtained by generating
$\text{p}\text{p}\to\ell^{+}\ell^{-}$ Monte Carlo events at a centre–of–mass
energy of $13~{}\text{TeV}$ in several $\text{m}_{\ell\ell}$ bins. We show
that the LHC Run 3 data could be already sensitive to the running of
$\sin^{2}\theta_{W,\ell}^{\overline{\text{MS}}}(\mu)$ at the per cent–level
and even stronger sensitivity can be expected in the High–Luminosity LHC
(HL–LHC).

##  2 Simulation setup

In the simulations used in this work, the running of
$\sin^{2}\theta_{W,\ell}^{\overline{\text{MS}}}(\mu)$ is accounted for by
using a hybrid EW scheme with $\overline{\text{MS}}$ input parameters
$(\alpha_{\text{EM}}^{\overline{\text{MS}}}(\mu),\sin^{2}\theta_{W,\ell}^{\overline{\text{MS}}}(\mu))$
and the Z boson mass, $\text{m}_{\text{Z}}$, renormalised in the on–shell
scheme. The renormalisation scale $\mu$ is taken as the lepton–pair invariant
mass, $\mu=\text{m}_{\ell\ell}$. This scheme has been introduced into an
upgraded version [5] of the Z_ew-BMNNPV code [6], within the Monte Carlo event
generator framework POWHEG-BOX-V2 [7].

For each $\text{m}_{\ell\ell}$ mass bin, the nominal predictions are obtained
by initialising the EW parameters at their expected values at the Z boson pole
corresponding to $\mu=\text{m}_{\text{Z}}=91.1876\,\mathrm{GeV}$,
$\alpha_{\text{EM}}^{\overline{\text{MS}}}(\mu)=0.007297353$, and
$\sin^{2}\theta_{W,\ell}^{\overline{\text{MS}}}(\mu)=0.23122$, and evolving
those to the specific energy scale considered. Additional predictions,
labelled as templates, are generated by shifting the expected input value of
$\sin^{2}\theta_{W,\ell}^{\overline{\text{MS}}}(\mu)$ up and down by a factor
of $0.01$. The values of $\text{m}_{\ell\ell}$ and
$\alpha_{\text{EM}}^{\overline{\text{MS}}}(\mu)$ are modified accordingly to
match their $\overline{\text{MS}}$ values at the corresponding energy scale.
All the predictions are obtained at LO EW and NLO+PS QCD by using the
NNPDF31_nnlo_as_0118_hessian PDF set [9], while the factorisation and
renormalisation scales are chosen to be equal to $\text{m}_{\ell\ell}$. Events
are showered with pythia8 [8] without including the effect of QED radiation
thus only born–level leptons are considered.

One billion nominal events are generated, separately for the electron and muon
decay channel of the Z boson, and divided in 6 $\text{m}_{\ell\ell}$ and 6
$|y_{\ell\ell}|$ bins separately for $\cos\theta_{CS}>0$ and
$\cos\theta_{CS}<0$. The boundaries in $\text{m}_{\ell\ell}$ are at $116$,
$150$, $200$, $300$, $500$, $1500$ and $5000\,\mathrm{GeV}$, while in
$|y_{\ell\ell}|$ are at $0.0$, $0.4$, $0.8$, $1.2$, $1.6$, $2.0$ and $2.5$. In
order to reproduce a realistic measurement scenario, the following kinematical
selection criteria, inspired by the recent CMS work [11], are imposed to the
leading (sub–leading) leptons: $p_{T}^{\ell}>40\,(30)\,\text{GeV}$ and
$|\eta^{\ell}|<2.5$. Detector efficiencies and resolutions are emulated by
passing each event through a dedicated Rivet [12] routine. The identification
and reconstruction efficiencies and energy–smearing functions used for the
electrons and muons are inspired by those derived by ATLAS during the LHC Run
2 data take [13, 14].

##  3 Analysis strategy

The expected sensitivity to the running of the EW mixing angle is obtained
with the xFitter analysis tool [10]. The templates obtained in Sec. 2, are
fitted to the nominal differential cross section
$\text{d}\sigma/\text{d}|y_{\ell\ell}|\text{d}\text{m}_{\ell\ell}$,
simultaneously for forward ($\cos\theta_{CS}>0$) and backward
($\cos\theta_{CS}<0$) events of the nominal sample. The expected sensitivity
on the EW mixing angle in each $\text{m}_{\ell\ell}$ bin is obtained by
minimising the function

$\chi^{2}(\beta_{\mathrm{exp}},\beta_{\mathrm{th}})=\sum_{i=1}^{N_{\mathrm{data}}}\frac{(\sigma_{i}^{\mathrm{exp}}+\sum_{j}\Gamma_{ij}^{\mathrm{exp}}\beta_{j,\mathrm{exp}}-\sigma_{i}^{\mathrm{th}}-\sum_{k}\Gamma_{ik}^{\mathrm{th}}\beta_{k,\mathrm{th}})^{2}}{\Delta_{i}^{2}}+\sum_{j}\beta_{j,\mathrm{exp}}^{2}+\sum_{k}\beta_{k,\mathrm{th}}^{2},$
(3)

where $\beta_{\text{th.}}$ are the expected theoretical uncertainties and
$\beta_{\text{exp.}}$ the expected experimental uncertainties. For each
$\text{m}_{\ell\ell}$ bin, the post–fit value
$\delta\sin^{2}\theta_{W,\ell}^{\overline{\text{MS}}}(\mu)^{\text{post--fit}}$
is identified as the expected sensitivity on the value of the EW mixing angle
at a particular value of $\mu$.

###  3.1 Measurement uncertainties

Several sources of uncertainties are considered in the fitting procedure. The
expected statistical uncertainties in the data are obtained from the
reconstructed–level number of events in each
$(\text{m}_{\ell\ell},|y_{\ell\ell}|)$ bin. An expected luminosity of
$300\,\text{fb}^{-1}$ ($3000\,\text{fb}^{-1}$) is used for the Run 3 (HL–LHC)
scenarios. The systematic uncertainties are inspired by an $8~{}\mathrm{TeV}$
ATLAS measurement of high–mass Drell–Yan cross sections [15], and are
extrapolated to match the $(\text{m}_{\ell\ell},|y_{\ell\ell}|)$ binning used
in this work by assuming a reduction factor of $0.5$ ($0.25$) for the Run 3
(HL–LHC) scenarios. For what concerns the estimated uncertainty in the
luminosity measurement, a $1\%$ uncertainty is considered for both the Run 3
and HL–LHC scenarios. Theoretical uncertainties originating from the
uncertainties in the PDFs, are modelled using aMCFast grids [16, 17],
convoluted with the NNPDF31_nnlo_as_0118_hessian PDF set, and included as
nuisance parameters in Eq. 3.

##  4 Results and summary

The expected sensitivity on the running of the EW mixing angle
$\delta\sin^{2}\theta_{W,\ell}^{\overline{\text{MS}}}(\mu)^{\text{post--
fit}}$, defined in Sec. 3, is presented in Tab. 1. It ranges from about $1\%$
($0.5\%$) to $7\%$ ($3\%$) when considering the LHC Run 3 (HL–LHC) scenario.
Due to the reduced expected statistical and systematic uncertainties,
$\delta\sin^{2}\theta_{W,\ell}^{\overline{\text{MS}}}(\mu)^{\text{post--fit}}$
is lower in the HL–LHC scenario with respect to the Run 3 one. By repeating
the fit without including the PDF uncertainties, the sensitivity in the last
$\text{m}_{\ell\ell}$ bin decreases to about $6\%$ ($2\%$) indicating a
non–negligible impact of the PDF uncertainty.

|  | Run 3 scenario | HL–LHC scenario
---|---|---|---
$\text{m}_{\ell\ell}\,[\text{GeV}]$ | $\sin^{2}\theta_{W,\ell}^{\overline{\text{MS}}}(\mu)$ | $\delta\sin^{2}\theta_{W,\ell}^{\overline{\text{MS}}}(\mu)^{\text{post--fit}}$ | $[\%]$ | $\delta\sin^{2}\theta_{W,\ell}^{\overline{\text{MS}}}(\mu)^{\text{post--fit}}$ | $[\%]$
133 | 0.2334 | 0.0021 | 0.9 | 0.0013 | 0.5
175 | 0.2347 | 0.0027 | 1.2 | 0.0017 | 0.7
250 | 0.2364 | 0.0034 | 1.4 | 0.0021 | 0.9
400 | 0.2386 | 0.0043 | 1.8 | 0.0029 | 1.2
1000 | 0.2340 | 0.0055 | 2.2 | 0.0039 | 1.6
3250 | 0.2486 | 0.0168 | 6.8 | 0.0077 | 3.1
Table 1: The theoretically predicted value of the EW mixing angle in the
$\overline{\text{MS}}$ renormalisation scheme
$\sin^{2}\theta_{W,\ell}^{\overline{\text{MS}}}(\mu)$ and the expected
sensitivity
$\delta\sin^{2}\theta_{W,\ell}^{\overline{\text{MS}}}(\mu)^{\text{post--fit}}$
obtained in this work. The results are given as a function of the invariant
mass of the final state leptons $\text{m}_{\ell\ell}$, separately for the Run
3 and HL–LHC scenarios.

The expected sensitivities at high energies, obtained in the HL–LHC scenario,
are compared to the available measurements at and below the Z boson pole in
Fig 1. All the measurements included in Fig 1 have been translated to the
$\overline{\text{MS}}$ renormalisation scheme [18], while the curve represents
the predicted scale dependence of the EW mixing angle in the
$\overline{\text{MS}}$ scheme obtained by including higher orders [4]. This
work shows that the HL–LHC will be able to probe the running of
$\sin^{2}\theta_{W}$ at the TeV scales and to set model–independent
constraints on new states with electroweak quantum numbers.

Figure 1: The running of the EW mixing angle compared to the available
measurements at low energy (red points), obtained in atomic parity violation,
neutrino–and polarised electron scattering experiments and to the measured
value at $\text{m}_{\text{Z}}$ (violet point) obtained at collider
experiments. The expectations for the HL–LHC reaches obtained in this work are
shown with black points. The total expected uncertainties are represented as
black error bars while the superimposed red error bars are obtained by
repeating the fit without including the PDFs uncertainties (figure updated
from [18]).

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