11institutetext: CIEMAT, Madrid, Spain 22institutetext: LPNHE, IN2P3/CNRS,
Paris France, and University of Geneva, Switzerland 33institutetext: Niels
Bohr Institute, Copenhagen, Denmark 44institutetext: CERN, EP Department,
Geneva, Switzerland

# The Z lineshape challenge: ppm and keV measurements

Juan Alcaraz Maestre  11 Alain Blondel  22 Mogens Dam  33 Patrick Janot  44

((Submitted to EPJ+ special issue: A future Higgs and Electroweak factory
(FCC): Challenges towards discovery, Focus on FCC-ee) )

###### Abstract

The FCC-ee offers powerful opportunities for direct or indirect evidence for
physics beyond the Standard Model, via a combination of high precision
measurements and searches for forbidden and rare processes and feebly coupled
particles. A key element of FCC-ee physics program is the measurement of the Z
lineshape from a total of $5\times 10^{12}$ Z bosons and a beam-energy
calibration with relative uncertainty of $10^{-6}$. With this exceptionally
large event sample, five orders of magnitude larger than that accumulated
during the whole LEP1 operation at the Z pole, the defining parameters -
$m_{\rm Z}$, $\Gamma_{\rm Z}$, $N_{\nu}$, $\sin^{2}\theta_{\rm W}^{\rm eff}$,
$\alpha_{\rm S}(m_{\rm Z}^{2})$, and $\alpha_{\rm QED}(m^{2}_{\rm Z})$ \- can
be extracted with a leap in accuracy of up to two orders of magnitude with
respect to the current state of the art. The ultimate goal that experimental
and theory systematic errors match the statistical accuracy (4 keV on the Z
mass and width, $3\times 10^{-6}$ on $\sin^{2}\theta_{\rm W}^{\rm eff}$, a
relative $3\times 10^{-5}$ on $\alpha_{\rm QED}$, and less than 0.0001 on
$\alpha_{\rm S}$) leads to highly demanding requirements on collider
operation, beam instrumentation, detector design, computing facilities,
theoretical calculations, and Monte Carlo event generators. Such precise
measurements also call for innovative analysis methods, which require a joint
effort and understanding between theorists, experimenters, and accelerator
teams.

###### pacs:

PACS-keydescribing text of that key and PACS-keydescribing text of that key

††offprints:

##  1 Introduction

With an integrated luminosity of $150\,\mathrm{ab}^{-1}$ collected in $\approx
4$ years of running at centre-of-mass energies between 88 and 94 GeV, FCC-ee
Benedikt:2651299  offers a unique opportunity to perform ultra-precise
electroweak measurements of the Z resonance. More than $5\times 10^{12}$ Z
decays, constituting the so-called TeraZ scenario, will be available for
study. The statistical power of this sample is complemented by an
extraordinary precision in the knowledge of the collision energy, of $\approx
100\,\mathrm{keV}$ R0polblondel . Robust procedures to monitor other relevant
beam collision parameters and the relative uncertainties between the energy
points in Z lineshape scans Blondel:2019jmp  are also an integral part of the
physics program.

The increase in luminosity with respect to past LEP experiments translates
into an increase of more than two orders of magnitude in statistical
sensitivity. With such large potential improvement over previous measurements,
a key question is how much the associated systematic uncertainties can be
reduced, and ultimately match the statistical uncertainties. Challenges arise
both at the theoretical and experimental levels. New theoretical paths will
have to be pursued in order to provide most precise predictions for
experimental observables Blondel:2018mad , and detailed experimental studies
will have to be performed to optimize accelerator and detector designs.
Exploring new analysis strategies and observables to simultaneously reduce
both theoretical and experimental uncertainties will be another essential
component of the challenge.

An initial review of the FCC-ee potential regarding ultra-precise measurements
at the Z pole can be found in Ref. Abada:2019lih . In the following we focus
on some elements of the challenge that we consider relevant for success. The
physics implications of the proposed program are expected to be deep.
Regarding universal deviations, the new estimates of the oblique parameters
$S$ and $T$ will constitute an early probe for Higgs compositeness or new
interactions possibly occurring at the deca-TeV scale Blondel:2021ema ;
Strategy:2019vxc , i.e. one order of magnitude above currently explored
scales. Deviations from universality related with these scales will be
searched for through the precise measurement of the couplings of each fermion
flavour to the Z. Separate access to the left- and right-handed components of
the couplings will be available at FCC-ee even in the absence of polarised
beams Blondel:2019yqr , as described below.

##  2 General considerations for a successful TeraZ program

Regarding cross-section measurements, relative statistical uncertainties are
expected in the $1/\sqrt{N}=10^{-5}$-$10^{-6}$ range, where $N$ is the number
of events selected in the decay channel under study. Similar statistical
uncertainties, of the order of $\scriptstyle{\sqrt{(1-\mathcal{A}^{2})/N}}$,
are expected on the absolute value of measured asymmetries $\mathcal{A}$.

At the theoretical front, and focusing on cross-section measurements, a
limiting factor is the precision of the theoretical predictions used in
luminosity measurements. Current studies using Bhabha scattering at low angle
consider a relative accuracy of order $10^{-4}$ as a realistic target
Jadach:2018jjo . The $\mathrm{e^{+}e^{-}}\to\gamma\gamma$ process has also
been suggested as an alternative channel to consolidate a $10^{-4}$ precision
measurement, owing to the almost negligible size of the theoretical
uncertainties of hadronic origin for this process CarloniCalame:2019dom .
Cross-section ratio and asymmetry measurements are not affected by luminosity
uncertainties. Besides the calculation of missing higher orders in these
observables, reaching precisions of order $10^{-5}$ or better will probably
require a deep change of philosophy in the measurement of lineshape
parameters: inclusion of non-factorizable terms; interference effects between
initial and final state radiation; redefinition of electroweak parameters at
the amplitude level; fits of complete differential distributions using
dedicated weighted Monte Carlo calculations, etc. Blondel:2018mad  For the
discussion below, and in order to better illustrate the main challenges, we
nevertheless assume a simplified LEP-like strategy in the measurement of
electroweak parameters ALEPH:2005ab .

Focusing on experimental aspects, a typical limiting factor for cross-section
measurements is the systematic uncertainty on the acceptance determination. A
$10^{-5}$ uncertainty, even in processes presenting a relatively smooth
behaviour of the angular distributions, implies a knowledge of the positions
of the edges of sub-detectors at the $10\,\mu\mathrm{m}$ level over distances
of the order of a meter. A first consequence is that detectors should be as
homogeneous as possible. Such a precision is a realistic target given current
tracking accuracy, but it demands dedicated efforts in terms of metrology,
alignment, monitoring and designs able to ensure the stability of large
detector volumes as a function of time. The challenge is even bigger for
detectors located at very low polar angles and measuring differential cross
sections with a $d\sigma/d\theta\propto 1/\sin\theta$ behaviour. For instance,
a luminosity monitor located at 1 m of the interaction point with an inner
radius of $\approx 65$ mm demands a $1\,\mu\mathrm{m}$ ($1\,\mu\mathrm{rad}$)
precision in positioning, in order to reach $10^{-4}$ uncertainties
Benedikt:2651299 . Other requirements imposed by acceptance systematics are
the uniformity in the detector response, redundant particle identification
capabilities, beam stability and a detailed monitoring of the beam geometry
conditions at the interaction point.

##  3 Z lineshape determination and ppm/keV precision observables

At FCC-ee Z lineshape scans are expected to provide a measurement of
$m_{\mathup{{{Z}}}}$ with unprecedented precision, $\delta
m_{\mathup{{{Z}}}}\approx 0.1\,\mathrm{MeV}$, i.e., $\approx 20$ times better
than the present precision from LEP. The uncertainty is fully dominated by the
uncertainty on the collision energy, which can be determined with $\approx
100\,\mathrm{keV}$ precision using resonant depolarisation of the transversely
polarised beams Blondel:2019jmp . This method was already used at LEP and will
be significantly improved at FCC-ee. There will be very frequent in-situ
calibrations using non-colliding pilot bunches, simultaneous with nominal
collisions. Nevertheless, detailed studies of the differences between
colliding and pilot bunches will be necessary. Also, the beam energy spread
and the energy asymmetries between the two beams should be monitored via
analysis of the longitudinal boosts of $\mathrm{e^{+}e^{-}}\to\mu^{+}\mu^{-}$
events Blondel:2019jmp .

The total Z width, $\Gamma_{\mathup{{{Z}}}}$, is directly connected with the
width of the Z lineshape. A statistical precision of $4\,\mathrm{keV}$ is
expected from a fit of the hadronic lineshape. The overall precision is
dominated by the so-called “point-to-point” uncertainties, which correspond to
systematic differences between their central $\sqrt{s}$ values that are not
100% positively correlated. The high statistics and the expected muon momentum
resolution of tracking detectors ($\delta\left(1/p_{\rm T}\right)\approx$ a
few $10^{-5}$ for $p=45$ GeV at normal incidence) allow a quantification of
these differences from the dimuon invariant mass distributions obtained at
each collision energy. An uncertainty of $\approx 25\,\mathrm{keV}$ was
obtained in preliminary studies Blondel:2019jmp , which translates into a
precision improvement of two orders of magnitude over current LEP results. re
The ratio between hadronic and leptonic cross sections, $R_{\ell}$, is an
essential observable for the extraction from the total width of the global and
individual leptonic and hadronic partial widths. The global leptonic partial
width is a direct test of new physics involving weak isospin violations, while
the individual partial widths constitute a powerful test of lepton coupling
universality in the neutral current. $R_{\ell}$ also provides one of the most
precise ways to measure $\alpha_{\mathrm{S}}(m^{2}_{\mathup{{{Z}}}})$ Gomez-
Ceballos:2013zzn ; Benedikt:2651299 . Its measurement is independent of
luminosity uncertainties, and therefore relative precisions below $10^{-4}$
can be contemplated. Improvements over LEP results by a factor ranging between
$20$ and $100$, i.e., $\delta R_{\ell}/R_{\ell}=(1-5)\times 10^{-5}$ are
expected. Leptonic and hadronic Z decays provide clean signatures at the Z
pole, and are only affected by limited backgrounds at small visible mass (two-
photon processes). According to LEP experience, the largest source of
uncertainty could be the acceptance of the leptonic channels. Even with a
rather hermetic detector and sufficiently redundant identification criteria,
the edges of the tracking acceptance and the interplay with beam position and
width parameters will have to be understood in detail. A precision in the
position of these edges at the level of of $10\,\mu\mathrm{m}$ might be
required.

A measurement of the relative invisible width $\Gamma_{\rm
inv}/\Gamma_{\ell}$, or equivalently of the number of neutrinos,
$N_{\nu}=\Gamma_{\rm inv}/\Gamma_{\nu}$, involves the measurement of the Z
peak cross section ALEPH:2005ab ; Voutsinas:2019hwu ; Janot:2019oyi . The
measurement is therefore limited by the precision in the measurement of the
luminosity, as commented before. A $10^{-4}$ precision represents already one
order of magnitude improvement with respect to LEP for a similar luminosity
detector coverage. The study of radiative recoil ratios like
$\sigma(\mathit{{\nu}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\mathit{{\mkern
4.0mu\overline{\mkern-4.0mu{\nu}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}\gamma)/\sigma(\ell^{+}\ell^{-}\gamma)$
above the Z pole is also being considered as an alternative at a similar level
of precision, dominated by statistical uncertainties Abada:2019lih .

##  4 Precise measurements of the electroweak fermion couplings to the Z and
of $\alpha(m^{2}_{\mathup{{{Z}}}})$

One of the main targets of the electroweak program is the precise measurement
of the chiral couplings to the Z for each individual fermion. The baseline
FCC-ee proposal relies on a direct measurement of the Z polarisation induced
by the beam particles with studies of the $\tau$ polarisation as a function of
the polar angle. This approach suppresses the need of longitudinal beam
polarisation, which would otherwise imply a non-negligible loss in luminosity.
The tau polarisation in Z decay is measured from the charged particle momentum
distribution in the semi-leptonic decays $\tau\to{\rm e}\nu_{\rm
e}\nu_{\tau}$, $\mu\nu_{\mu}\nu_{\tau}$ or in hadronic decays
$\tau\rightarrow{\rm h}\nu_{\tau}$ where $\rm h$ can be $\pi$, ${\rm K}$,
$\rho$, ${\rm K}^{\ast}$, ${\rm a}_{1}$, etc. Each channel has a different
polarisation analysis power and must therefore be analysed independently. A
clean separation between channels is also essential. The analysis of the
$\tau$ polarisation dependence on the $\mathrm{e^{+}e^{-}}\to\tau^{+}\tau^{-}$
scattering angle $\theta$ gives access to both the tau and electron chiral
coupling asymmetries ${\cal A}_{\tau}$ and ${\cal A}_{\rm e}$ independently

$P(\cos\theta)=-\frac{{\cal A}_{\tau}(1+\cos^{2}\theta)+2{\cal A}_{\rm
e}\cos\theta}{(1+\cos^{2}\theta)+2{\cal A}_{\rm e}{\cal A}_{\tau}\cos\theta},$
(1)

The previous expression is valid at Born-improved level and in the massless
lepton limit.
$\mathcal{A}_{\mathit{{f}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}}$
is the chiral asymmetry parameter of the fermion
$\mathit{{f}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$
in its coupling to the Z. The average $\tau$ polarisation,
$\mathcal{P}_{\mathup{{{\tau}}}}$, provides a direct measurement of
$\mathcal{A}_{\mathup{{{\tau}}}}$:
$\mathcal{P}_{\mathup{{{\tau}}}}=-\mathcal{A}_{\mathup{{{\tau}}}}$, whereas
the forward-backward polarisation asymmetry, ${A_{\rm FB}^{\rm
pol,\mathup{{{\tau}}}}}$, provides a direct measurement of
$\mathcal{A}_{\mathup{{{e}}}}$, the induced Z polarisation in the
$\mathrm{e^{+}e^{-}}$ collision: ${A_{\rm FB}^{\rm
pol,\mathup{{{\tau}}}}}=-\frac{3}{4}\mathcal{A}_{\mathup{{{e}}}}$. The
remaining systematic uncertainty on $\mathcal{A}_{\mathup{{{e}}}}$ at LEP was
originating from the limited knowledge of non-$\tau$ backgrounds
Heister:2001uh . At FCC-ee, huge control samples will be available to reduce
this component, but dedicated studies are still necessary to estimate the
ultimately reachable precision.

The forward-backward asymmetry in the
$\mathrm{e^{+}e^{-}}\to\mathit{{f}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\overline{\mathit{{f}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}}$
process, ${A_{\rm
FB}^{\mathit{{f}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}}}=\frac{3}{4}\mathcal{A}_{\mathup{{{e}}}}\mathcal{A}_{\mathit{{f}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}}$
gives access to the chiral couplings of the fermion
$\mathit{{f}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$
when combined with tau polarisation studies. Chiral couplings can be trivially
converted in measurements of vector and axial couplings or, alternatively, in
a Born-improved spirit, in measurements of the effective
$\mathrm{\overline{\rho}_{\mathit{{f}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}}}$
and
$\mathrm{\sin^{2}\theta_{W}^{\mathit{{f}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}},eff}}$
parameters for each individual fermion. Assuming universal deviations,
$\mathrm{\sin^{2}\theta_{W}^{\mathit{{f}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}},eff}}$
measurements become a measurement of the effective weak mixing angle
$\mathrm{\sin^{2}\theta_{W}^{eff}}$. The muon channel is particularly
promising in this respect, aiming for a precision approaching the ultimate
statistical sensitivity of $3\times 10^{-6}$ on ${A_{\rm FB}^{\mu}}$. At that
level of precision, further theoretical studies of QED corrections and in
particular of interference effects between initial and final-state radiation
will be mandatory. Detailed studies of the beam parameters and of the polar
angular resolution will be required as well Blondel:2019jmp ; Jadach:2018lwm .

The extreme FCC-ee precision requirements demand an equivalent level of
precision on the input parameters to theoretical predictions. One of these
parameters is $\alpha(m^{2}_{\mathup{{{Z}}}})$, the electromagnetic coupling
constant at the Z scale, which is the source of one of the dominant
uncertainties in present fits. In practice, $\alpha(m^{2}_{\mathup{{{Z}}}})$
can be treated as another parameter to be determined in the Z lineshape
running. This is supported by the study presented in Ref. Janot:2015gjr ,
which proposes a measurement of $\alpha(m^{2}_{\mathup{{{Z}}}})$ with a
$3\times 10^{-5}$ relative precision, largely uncorrelated with other Z
lineshape parameters. Most of the sensitivity lies in the linear dependence of
the ${A_{\rm FB}^{\mu}}$ asymmetry with respect to the
$\mathup{{{\gamma}}}-\mathup{{{Z}}}$ interference term around the Z peak. An
appropriate combination of measurements at the energy points with the largest
sensitivity, $\sqrt{s}=87.9$ and $94.3$ GeV, using one year of integrated
luminosity should provide the required precision. Still, reaching the aimed
precision will require the calculation of missing electroweak corrections of
higher order, as well as more detailed studies of initial-final state
interference effects.

##  5 Heavy quark precision measurements

The cross-section ratios and asymmetries $R_{\rm
b}\equiv\Gamma_{\mathup{{{b}}}}/\Gamma_{\rm had}$, $R_{\rm
c}\equiv\Gamma_{\mathup{{{c}}}}/\Gamma_{\rm had}$, ${A_{\rm FB}^{\rm b}}$, and
${A_{\rm FB}^{\rm c}}$ are expected to be measured at FCC-ee with more than
one order of magnitude better precision than at LEP/SLC. On top of the
increase in the number of collected events, the statistical precision should
be boosted with respect to LEP/SLC by the much higher b- or c-tagging
efficiencies and purities obtained with better detectors and more powerful
lifetime tagging techniques. Systematic uncertainties are expected to be
significantly reduced by the use of exclusive B decays and enriched control
samples in specific regions of phase space.

At FCC-ee, $R_{\rm b}$ and $R_{\rm c}$ will be likely measured with double-
tagging techniques on both hemispheres of a
$\mathup{{{b}}}\overline{\mathup{{{b}}}}$ or a
$\mathup{{{c}}}\overline{\mathup{{{c}}}}$ event ALEPH:2005ab . This strategy
allows a measurement independent of the knowledge of the tagging efficiency in
the limit of negligible backgrounds and correlations between hemispheres. At
LEP, hemisphere correlations due to QCD effects (mostly hard gluon emission)
and primary vertex determination were dominant sources of uncertainty. Gluon
splitting increases the number of single tags in events with two light-flavour
jets, and also constitutes a significant source of correlated uncertainty
between experiments. The vertex precision of the new generation of detectors
should contribute to reduce vertex correlations significantly. Studies as a
function of the acoplanarity between b-tagged jets can help reducing QCD
correlations, and huge gluon splitting samples will become available for a
precise understanding of this source of uncertainty. Nevertheless, coming
studies are needed to quantify these improvements in more detail.

The bare forward-backward asymmetry of b quarks at the Z pole, ${A_{\rm
FB}^{0,{\rm b}}}$, is the electroweak observable that currently presents the
largest deviation with respect to the standard model expectation in current
fits ALEPH:2005ab  ($\approx 3\sigma$ pull). An order-of-magnitude improved
measurement at FCC-ee could thus become a clean signal of new physics if the
deviation in the central value is confirmed. The world-average measurement is
still dominated by statistical uncertainties ($\delta{A_{\rm FB}^{0,{\rm
b}}}({\rm stat.})=0.0016$), but is also affected by non-negligible systematic
uncertainties ($\delta{A_{\rm FB}^{0,{\rm b}}}({\rm syst.})=0.0007$). A
fraction of it can be reduced at FCC-ee through dedicated studies on high-
statistics control samples. A detailed analysis of the detector requirements
to maximise flavour identification capabilities is also mandatory. Exclusive B
decays can be exploited as well. For instance, about $10^{8}$ $\mathrm{B}^{+}$
decays, not affected by charm contamination or B-mixing effects, will be
available at FCC-ee Fabrizio-afbb . The expected improvements in flavour
tagging will be much more visible in the case of ${A_{\rm FB}^{0,{\rm c}}}$
measurements. In addition, the availability of exclusive, high purity, and
large statistics D decay samples should provide a significant improvement in
terms of precision compared with LEP measurements ALEPH:2005ab . There, most
measurements were performed with inclusive or pseudo-inclusive techniques on
samples with significant b-quark contamination. Dedicated studies are
certainly necessary.

An irreducible source of uncertainty in the current estimate of ${A_{\rm
FB}^{0,{\rm b}}}$, fully correlated among experiments, is the presence of a
QCD correction factor, of order $1-\alpha_{\mathrm{S}}/\pi$, that accounts for
the shift between the experimentally observed asymmetry and ${A_{\rm
FB}^{0,{\rm b}}}$ Djouadi-afbb ; Altarelli:1992fs ; ep-98 . The main role of
the correction is to absorb the angular distortions due to final-state QCD
radiation in the $\mathrm{Z}\to\mathrm{b\overline{b}(g)}$ decay. Recent re-
evaluations of that uncertainty dEnterria:2018jsx  based on modern parton
shower tunes seem to be consistent with the initial estimates. New strategies
to reduce or constrain experimentally the size of these uncertainties are
being developed, for an initial target of $\delta{A_{\rm FB}^{0,{\rm
b}}}\approx 0.0001$ AlcarazMaestre:2020fmp .

##  6 Outlook

Besides theory requirements, discussed in more detail in Freitas:2019bre ;
Heinemeyer:2021rgq , careful experimental studies with realistic detector
descriptions are necessary to estimate the ultimate precision for FCC-ee
electroweak measurements at the Z pole. In the particular case of cross
section measurements, aiming for a precision of $\approx 10^{-4}$ is already
imposing severe constraints on the design and tolerances of luminosity
monitors. Regarding final states involving heavy flavours (tau, bottom,
charm), an improvement of one order of magnitude with respect to previous
LEP/SLC measurements may imply new constraints on the detector design and the
development of new, more powerful tagging techniques. Let us note that the
huge available statistics at the Z peak suggests that exclusive decay
identification should be explored in more depth, as a complementary path to
reduce systematic uncertainties in some particular cases. Addressing all these
challenges is a critical step for success and, as such, one of the main
objectives of the present electroweak physics program at FCC-ee.

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