# The hadronic contribution to the running
of the electromagnetic coupling
and the electroweak mixing angle

Marco Cè, ab111Current affiliation: Theoretical Physics department, CERN,
Geneva, Switzerland Teseo San José,00footnotemark: 0 abc Antoine Gérardin,d
Harvey B. Meyer,abc Kohtaroh Miura,abe Konstantin Ottnad,bc Andreas Risch,bc
Jonas Wilhelm,bc Hartmut Wittigabc
aHelmholtz-Institut Mainz, Johannes Gutenberg-Universität Mainz, Germany
bPRISMA+ Cluster of Excellence, Johannes Gutenberg-Universität Mainz, Germany
cInstitut für Kernphysik, Johannes Gutenberg-Universität Mainz, Germany
dJohn von Neumann-Institut für Computing (NIC), DESY Zeuthen, Germany
eKobayashi-Maskawa Institute for the Origin of Particles and the Universe,
Nagoya University, Japan
E-mail Speakers<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract:

The electromagnetic coupling $\alpha$ and the electroweak mixing angle
$\thetaW$ are parameters of the Standard Model (SM) that enter precision SM
tests and play a fundamental rôle in beyond SM physics searches. Their values
are energy dependent, and non-perturbative hadronic contributions are the main
source of uncertainty to the theoretical knowledge of the running with energy.
We present a lattice study of the leading hadronic contribution to the running
of $\alpha$ and $\sIIW$. The former is related to the hadronic vacuum
polarization (HVP) function of electromagnetic currents, and the latter to the
HVP mixing of the electromagnetic current with the vector part of the weak
neutral currents. We use the time-momentum representation (TMR) method to
compute the HVP on the lattice, estimating both connected and disconnected
contributions on $\Nf=2+1$ non-perturbatively $\order{a}$-improved Wilson
fermions ensembles from the Coordinated Lattice Simulations (CLS) initiative.
The use of different lattice spacings and quark masses allows us to reliably
extrapolate the results to the physical point.

MITP/19-064

DESY 19-179

π i m e^#1 N_f SU Δ_had Δα_had θ_W sin^2 sin^2^θ_W ¯Π τ_int x_0^cut x_0i
t_0^sym $\overline{\hbox{ms}}$

SM

    Standard Model
QFT

    quantum field theory
QCD

    quantum chromodynamics
HVP

    hadronic vacuum polarization
TMR

    time-momentum representation
CCS

    Lorentz-covariant coordinate space
FFT

    fast Fourier transform
CLS

    Coordinated Lattice Simulations
BC

    boundary condition
OBC

    open boundary condition
GS

    Gounaris-Sakurai
S/N

    signal-to-noise ratio

Marco Cè and Teseo San José

##  1 The running of the electromagnetic coupling

The predictions of the SM of particle physics are today tested to a high
degree of precision in experiments that span a vast range of energy scales,
from atomic physics to high-energy colliders. The connection between these
energy scales is encoded in the _running_ with energy of the strength of
interactions. For instance, the fine-structure constant
$\alpha=1/$137.035\,999\,139(31)$$ [1] is known from low-energy experiments to
better than a part per billion. However, its effective value for physics
around the $Z$ pole is $\hat{\alpha}^{(5)}(M_{Z})=1/$127.955(10)$$ [1], a
$7\text{\,}\mathrm{\char 37\relax}$ larger value. In the _on-shell_ scheme,
the running of $\alpha$ at a given time-like momentum transfer $q^{2}$ is
described by

$\alpha(q^{2})=\frac{\alpha}{1-\Delta\alpha(q^{2})},$ (1)

in terms of the $\Delta\alpha(q^{2})$ function. While the lepton contribution
to $\Delta\alpha(q^{2})$ can be computed in perturbation theory, the estimate
of the quark contribution at low energies requires non-perturbative
calculations of hadronic physics. Conventionally, the hadronic contribution
$\Dalphahad(q^{2})$ is related through the optical theorem to the $R$-ratio,
_i.e._ the total cross-section $e^{+}e^{-}\to\text{hadrons}$ over
$e^{+}e^{-}\to\mu^{+}\mu^{-}$, which is estimated using a compilation of
experimental data. This results in
$\Dalphahad^{(5)}(M_{Z}^{2})=$0.027\,64(7)$$ [1] constituting the main
contribution to the uncertainty on $\alpha(M_{Z}^{2})$. In an alternative
approach [2], the Adler function $D(Q^{2})$ at space-like $Q^{2}=-q^{2}$ is
estimated from the same data and used to compute the running up to
$Q^{2}\approx$2\text{\,}{\mathrm{GeV}}^{2}$$. Around this $Q^{2}$ and above,
$D(Q^{2})$ can be computed reliably in perturbative quantum chromodynamics
(QCD). Crucially, $\Dalphahad(-Q^{2})$ at space-like momenta $Q^{2}>0$ is
accessible to non-perturbative lattice techniques defined in Euclidean space-
time [3, 4, 5]. It is given by

$\Dalphahad(-Q^{2})=4\cpi\alpha\SPi^{\gamma\gamma}(Q^{2}),\qquad\SPi^{\gamma\gamma}(Q^{2})=[\Pi^{\gamma\gamma}(Q^{2})-\Pi^{\gamma\gamma}(0)],$
(2)

where $\SPi(Q^{2})$ is the subtracted HVP function111 In this work we denote
with $\SPi(Q^{2})$ with $Q^{2}>0$ the HVP function at _space-like_ momenta.

$(Q_{\mu}Q_{\nu}-\delta_{\mu\nu}Q^{2})\Pi^{\gamma\gamma}(Q^{2})=\Pi_{\mu\nu}^{\gamma\gamma}(Q^{2})=\int\differential[4]{x}\e{\I
Q\cdot x}\expectationvalue{j_{\mu}^{\gamma}(x)j_{\nu}^{\gamma}(0)}$ (3)

of the electromagnetic current

$j_{\mu}^{\gamma}=\frac{2}{3}\bar{u}\gamma_{\mu}u-\frac{1}{3}\bar{d}\gamma_{\mu}d-\frac{1}{3}\bar{s}\gamma_{\mu}s+\frac{2}{3}\bar{c}\gamma_{\mu}c.$
(4)

Lattice QCD can thus provide an estimate that does not depend on experimental
$R$-ratio data and cross-check the phenomenological estimate. This is of great
interest in the context of global SM fits, where $\Dalphahad(M_{Z}^{2})$ is an
input. Indeed, the best-fit result for the Higgs mass excluding kinematic
constraints is $M_{H}=$90^{+17}_{-16}\text{\,}\mathrm{GeV}$$, $1.9\sigma$
below the measured value, and a shift of $\pm${10}^{-4}$$ in
$\Dalphahad^{(3)}($4\text{\,}{\mathrm{GeV}}^{2}$)$ corresponds to a shift of
$\mp$4.5\text{\,}\mathrm{GeV}$$ in $M_{H}$ [1]. This is connected to the
tension between the SM and experimental determinations of the anomalous
magnetic moment of the muon $(g-2)_{\mu}$ (see Ref. [6] for a review), because
a solution of the $(g-2)_{\mu}$ puzzle that involves an increase of the SM
estimate of the leading hadronic contribution $a_{\mu}^{\mathrm{HLO}}$ has to
avoid a correlated increase of $\Dalphahad$ [7].

A second connection to $(g-2)_{\mu}$ comes from the high-precision measurement
of $\alpha(-Q^{2})$ at space-like $Q^{2}$ in $t$-channel scattering proposed
by the MUonE collaboration [8, 9]. Isolating the hadronic contribution
$\Dalphahad(-Q^{2})$, an independent determination of $a_{\mu}^{\mathrm{HLO}}$
is obtained from [10]

$a_{\mu}^{\mathrm{HLO}}=\frac{\alpha}{\cpi}\int_{0}^{1}\differential{x}(1-x)\Delta\alpha_{\mathrm{had}}(t(x)),\qquad
t(x)=\frac{x^{2}m_{\mu}^{2}}{x-1}\leq 0.$ (5)

The proposed experiment is limited to $x<0.932$, corresponding to
$Q^{2}\lesssim$0.14\text{\,}{\mathrm{GeV}}^{2}$$, that leaves out
$13\text{\,}\mathrm{\char 37\relax}$ of the $a_{\mu}^{\mathrm{HLO}}$ integral.
Lattice data can complement the experiment for
$Q^{2}\gtrsim$0.14\text{\,}{\mathrm{GeV}}^{2}$$ [11].

##  2 The running of the electroweak mixing angle

As a second quantity, we consider the electroweak mixing angle or Weinberg
angle $\thetaW$, that is, the parameter of the Standard Model of particle
physics that parametrizes the mixing between electromagnetic and weak
interactions

$\sIIW=\frac{g^{\prime 2}}{g^{2}+g^{\prime 2}},\qquad
e=g\sin\thetaW=g^{\prime}\cos\thetaW,$ (6)

where $g$ and $g^{\prime}$ are the $\mathrm{SU}(2)_{L}$ and
$\mathrm{U}(1)_{Y}$ couplings, respectively. Beyond tree level, as the
couplings themselves, its precise value is scheme and energy dependent. In a
given scheme, the mixing angle is measured to sub-permille precision at
energies around $M_{Z}$, _e.g._  _on-shell_
$\sIIW=1-M_{W}^{2}/M_{Z}^{2}=$0.223\,43(7)$$ or in the -scheme
$\sIIWh(M_{Z})=$0.231\,22(7)$$ [1]. Conversely, the mixing angle in the
Thomson limit, $\absolutevalue*{q^{2}}\ll m_{e}^{2}$, can be defined in a
scheme-independent way [12, 13]. Its experimental value is less well known,
but upcoming experiments target up to a $0.15\text{\,}\mathrm{\char 37\relax}$
precision at a momentum transfer of
$4.5\text{\times}{10}^{-3}\text{\,}{\mathrm{GeV}}^{2}$ [14, 15]. A more
precise value, $\sIIWh(0)=$0.238\,68(5)(2)$$ [16], is obtained computing the
running in the -scheme from $M_{Z}$ to low energies. The first error is the
uncertainty on the $Z$-pole value, while the second error is the total
theoretical uncertainty on the running and it is dominated by the non-
perturbative hadronic contribution.

The energy dependence of $\sIIW$ in the on-shell scheme can be written as

$\sIIW(q^{2})=\sIIW\left[1+\Delta\sIIW(q^{2})\right],$ (7)

where $\sIIW$ is the value in the low-energy limit. Similarly to Eq (2), the
leading hadronic contribution to the running at space-like $Q^{2}=-q^{2}$ is
given by [17, 2]

$\Dhad\sin^{2}\thetaW(-Q^{2})=-\frac{e^{2}}{\sIIW}\SPi^{Z\gamma}(Q^{2}),$ (8)

where $\SPi^{Z\gamma}(Q^{2})$ is HVP mixing of the electromagnetic current
$j_{\mu}^{\gamma}$ and the vector part of the neutral weak current
$j_{\mu}^{Z}$

$j_{\mu}^{Z}\big{|}_{\mathrm{vector}}=j_{\mu}^{T_{3}}\big{|}_{\mathrm{vector}}-\sIIW
j_{\mu}^{\gamma},\qquad
j_{\mu}^{T_{3}}\big{|}_{\mathrm{vector}}=\frac{1}{4}\bar{u}\gamma_{\mu}u-\frac{1}{4}\bar{d}\gamma_{\mu}d-\frac{1}{4}\bar{s}\gamma_{\mu}s+\frac{1}{4}\bar{c}\gamma_{\mu}c.$
(9)

As in the electromagnetic case, $\SPi^{Z\gamma}(Q^{2})$ is directly accessible
to lattice computations [3, 4, 18, 19].

##  3 The TMR method

$\displaystyle 0$$\displaystyle 2$$\displaystyle 4$$\displaystyle
6$$\displaystyle 8$$\displaystyle 10$$\displaystyle x_{0}$
[$\mathrm{fm}$]$\displaystyle 0.0$$\displaystyle 0.2$$\displaystyle
0.4$$\displaystyle 0.6$$\displaystyle 0.8$$\displaystyle 1.0$$\displaystyle
K(x_{0})/x_{0}^{3}$ [${\mathrm{fm}}^{-1}$]$\displaystyle
a_{\mu}^{\textrm{HLO}}$$\displaystyle\bar{\Pi}($0.1\text{\,}{\mathrm{GeV}}^{2}$)$$\displaystyle\bar{\Pi}($0.5\text{\,}{\mathrm{GeV}}^{2}$)$$\displaystyle\bar{\Pi}($1.0\text{\,}{\mathrm{GeV}}^{2}$)$$\displaystyle
0$$\displaystyle 1$$\displaystyle 2$$\displaystyle 3$$\displaystyle
4$$\displaystyle 5$$\displaystyle x_{0}$ [$\mathrm{fm}$]$\displaystyle
0.0$$\displaystyle 0.2$$\displaystyle 0.4$$\displaystyle 0.6$$\displaystyle
0.8$$\displaystyle 1.0$$\displaystyle
K(x_{0})G(x_{0})/\int\differential{x_{0}}K(x_{0})G(x_{0})$
[${\mathrm{fm}}^{-1}$]$\displaystyle
a_{\mu}^{\textrm{HLO}}$$\displaystyle\bar{\Pi}($0.1\text{\,}{\mathrm{GeV}}^{2}$)$$\displaystyle\bar{\Pi}($1.0\text{\,}{\mathrm{GeV}}^{2}$)$

Figure 1: Left: the kernel $K(x_{0},Q^{2})$ of the TMR integral in Eq. (10)
divided by $x_{0}^{3}$ for different values of $Q^{2}$, compared to the kernel
for $a_{\mu}^{\mathrm{HLO}}$ [20, 21] (blue line), as a function of time
$x_{0}$. Right: contribution of $G(x_{0})K(x_{0},Q^{2})$ to the TMR integral
normalized to the value of the integral, comparing different kernels
$K(x_{0})$. The light coloured lines are drawn using a model for the
Euclidean-time correlator $G(x_{0})$ [20], that is also used for the integral,
while the data points with error bars are obtained using actual lattice
correlator data at the physical pion mass.

The computation of $\SPi^{\gamma\gamma}$ and $\SPi^{Z\gamma}$ as functions of
$Q^{2}$ is similar to that of $a_{\mu}^{\mathrm{HLO}}$, the leading-order HVP
contribution to $(g-2)_{\mu}$, and, as in that case, different methods are
available, such as the four-momentum _hybrid method_ , the time moments, or
the TMR method [20, 22]. In this study, we employ the TMR method to compute
the subtracted HVP function

$\SPi(Q^{2})=\int_{0}^{\infty}\differential{x_{0}}G(x_{0})\left[x_{0}^{2}-\frac{4}{Q^{2}}\sin[2](\frac{Qx_{0}}{2})\right],\qquad
G(x_{0})=-\frac{1}{3}\int\differential[3]{x}\sum_{k=1}^{3}\expectationvalue{j_{k}(x)j_{k}(0)},$
(10)

where we have to integrate over Euclidean time the product of the zero-
momentum-projected correlator, $G^{\gamma\gamma}(x_{0})$ or
$G^{Z\gamma}(x_{0})$, times a $Q^{2}$-dependent kernel
$K(x_{0},Q^{2})=x_{0}^{2}-(4/Q^{2})\sin^{2}(Qx_{0}/2)$. This allows us, in
principle, to input any value of $Q^{2}$ in the kernel. The properties of the
kernel significantly influence the systematics of the integral: On the one
hand, a shorter-range kernel puts a larger weight on the correlator at short
times. Since the correlator on the lattice is sampled at a spacing $a$, one
needs $Q^{2}\ll(\cpi/a)^{2}$ in order to avoid large cut-off effects. On the
other hand, a longer-range kernel weights relatively more the long-time
behaviour of the correlator, which is noisier and susceptible to finite-volume
effects. Different kernels are compared in the left plot of Figure 1,
including the one used to compute $a_{\mu}^{\mathrm{HLO}}$, Eq. (84) from Ref.
[20]. In the right plot we show the corresponding relative contribution to the
integral against the time variable. The HVP function $\SPi$ at
$Q^{2}=$0.1\text{\,}{\mathrm{GeV}}^{2}$$ receives the larger contribution
around $1\text{\,}\mathrm{fm}$, as $a_{\mu}^{\mathrm{HLO}}$ does, but at long
times the contribution to the former is smaller. When lattice data at physical
pion masses is used for $G(x_{0})$, this results in a smaller statistical
error on $\SPi($0.1\text{\,}{\mathrm{GeV}}^{2}$)$ than on
$a_{\mu}^{\mathrm{HLO}}$. In the case of
$\SPi($1\text{\,}{\mathrm{GeV}}^{2}$)$, the correlator tail has a negligible
contribution to the statistical error.

##  4 Lattice setup

We perform the computation on the $\Nf=2+1$ set of ensembles from the CLS
initiative [23], with tree-level Lüscher-Weisz gauge action and non-
perturbatively $\order{a}$-improved Wilson fermions. The list of ensembles
employed in this work is in Table 1. We use four lattice spacings, $u$ and $d$
quark masses are degenerate, thus we have exact isospin symmetry, and the
pseudoscalar meson masses span from
$M_{\pi}=M_{K}\approx$415\text{\,}\mathrm{MeV}$$ at the $\SU(3)$-symmetric
point to the physical ones along a trajectory on which the sum of the bare
$u$, $d$ and $s$ quark masses is kept constant. We set the scale using
$(8t_{0}^{\mathrm{phys}})^{1/2}=$0.415(4)(2)\text{\,}\mathrm{fm}$$ [24, 25].

Table 1: List of ensembles from the CLS initiative employed, with approximate lattice spacings, sizes and pion and kaon masses. A * denotes the ensembles for which the disconnected contribution is available. All ensembles have open boundary conditions in time, except those denoted with § that have periodic boundary conditions in time. Values of $\tnotsym$ and $a$ are from Ref. [25].  | $T/a$ | $L/a$ | $\tnotsym/a^{2}$ |  $a$ [$\mathrm{fm}$] |  $L$ [$\mathrm{fm}$] |  $M_{\pi}$, $M_{K}$ [$\mathrm{MeV}$] | $M_{\pi}L$
---|---|---|---|---|---|---|---
H101 | $96$ | $32$ | $2.860$ | $0.086$ | $2.8$ | $415$ | $5.8$
H102 | $96$ | $32$ |  |  | $2.8$ | $355$ | $440$ | $5.0$
H105* | $96$ | $32$ |  |  | $2.8$ | $280$ | $460$ | $3.9$
N101 | $128$ | $48$ |  |  | $4.1$ | $280$ | $460$ | $5.8$
C101* | $96$ | $48$ |  |  | $4.1$ | $220$ | $470$ | $4.6$
B450§ | $64$ | $32$ | $3.659$ | $0.076$ | $2.4$ | $415$ | $5.1$
S400 | $128$ | $32$ |  |  | $2.4$ | $350$ | $440$ | $4.3$
N401* | $128$ | $48$ |  |  | $3.7$ | $285$ | $460$ | $5.3$
H200 | $96$ | $32$ | $5.164$ | $0.064$ | $2.1$ | $420$ | $4.4$
N202 | $128$ | $48$ |  |  | $3.1$ | $410$ | $6.4$
N203* | $128$ | $48$ |  |  | $3.1$ | $345$ | $440$ | $5.4$
N200* | $128$ | $48$ |  |  | $3.1$ | $285$ | $465$ | $4.4$
D200* | $128$ | $64$ |  |  | $4.1$ | $200$ | $480$ | $4.2$
E250*§ | $192$ | $96$ |  |  | $6.2$ | $130$ | $490$ | $4.1$
N300 | $128$ | $48$ | $8.595$ | $0.050$ | $2.4$ | $420$ | $5.1$
N302* | $128$ | $48$ |  |  | $2.4$ | $345$ | $460$ | $4.2$
J303 | $192$ | $64$ |  |  | $3.2$ | $260$ | $475$ | $4.2$

###  4.1 Flavour decomposition and renormalization

The correlators are computed on the ensembles in Table 1 as described in Ref.
[26], to which we refer to for the unexplained notation. At the sink, we
employ both the local and conserved discretizations of the vector current

$\displaystyle j_{\mu}^{\mathrm{l}}(x)=\bar{q}(x)\gamma_{\mu}q(x),$ (11a)
$\displaystyle
j_{\mu}^{\mathrm{c}}(x)=\frac{1}{2}\left[\bar{q}(x+a\hat{\mu})(1+\gamma_{\mu})U^{\dagger}_{\mu}(x)q(x)-\bar{q}(x)(1-\gamma_{\mu})U_{\mu}(x)q(x+a\hat{\mu})\right],$
(11b)

while only the local current is used at the source. The currents are non-
perturbatively $\order{a}$-improved and renormalized. To this purpose, we
introduce a flavour $\SU(3)$ decomposition of the current. For the local
discretization, we have [27]

$\displaystyle
j_{\mu,R}^{3,\mathrm{l}}=Z_{V}\left(1+3\bar{b}_{V}am_{q}^{\mathrm{av}}+b_{V}am_{q,\ell}\right)j_{\mu}^{3,\mathrm{il}},$
(12a) $\displaystyle\begin{pmatrix}j_{\mu}^{8}\\\
j_{\mu}^{0}\end{pmatrix}_{R}^{\mathrm{l}}=Z_{V}\begin{pmatrix}1+3\bar{b}_{V}am_{q}^{\mathrm{av}}+b_{V}\frac{a(m_{q,\ell}+2m_{q,s})}{3}&\left(\frac{b_{V}}{3}+f_{V}\right)\frac{2a(m_{q,\ell}-m_{q,s})}{\sqrt{3}}\\\
r_{V}d_{V}\frac{a(m_{q,\ell}-m_{q,s})}{\sqrt{3}}&r_{V}+r_{V}(3\bar{d}_{V}+d_{V})am_{q}^{\mathrm{av}}\end{pmatrix}\begin{pmatrix}j_{\mu}^{8}\\\
j_{\mu}^{0}\end{pmatrix}^{\mathrm{il}},$ (12b)

where the improved non-singlet and singlet local currents are

$j_{\mu}^{a,\mathrm{il}}=j_{\mu}^{a,\mathrm{l}}+ac_{V}^{\mathrm{l}}\tilde{\partial}_{\nu}T_{\nu\mu}^{a},\qquad
j_{\mu}^{0,\mathrm{il}}=j_{\mu}^{0,\mathrm{l}}+a\bar{c}_{V}^{\mathrm{l}}\tilde{\partial}_{\nu}T_{\nu\mu}^{0},$
(13)

and the breaking of flavour $\SU(3)$ symmetry introduces a mixing between the
singlet and non-singlet $I=0$ components. For the conserved discretization, no
renormalization is needed

$j_{\mu,R}^{a,\mathrm{c}}=j_{\mu}^{a,\mathrm{c}}+ac_{V}^{\mathrm{c}}\tilde{\partial}_{\nu}T_{\nu\mu}^{a},\qquad
j_{\mu,R}^{0,\mathrm{c}}=j_{\mu}^{0,\mathrm{c}}+a\bar{c}_{V}^{\mathrm{c}}\tilde{\partial}_{\nu}T_{\nu\mu}^{0}.$
(14)

We use the renormalization and improvement coefficients determined non-
perturbatively in Ref. [28]. Since the coefficients to renormalize the singlet
local current are unknown, we use only the conserved vector current for the
singlet component. This implies that we can only put the $Z$ current at the
sink, where the conserved discretization is available. Moreover, we set
$f_{V}=0$ and
$\bar{c}_{V}^{\mathrm{c},\mathrm{l}}=c_{V}^{\mathrm{c},\mathrm{l}}$, which is
valid up to $\order{g_{0}^{5}}$ and introduces a negligible error.

The charm contribution is also computed in the quenched approximation, with
the charm quark mass tuned using the experimental $D_{s}$ meson mass and the
local current renormalized computing the mass-dependent $Z_{V}^{c}$ [26].
Including the latter contribution, the $\gamma\gamma$ and $Z\gamma$ bare
correlators are

$\displaystyle
G_{\mu\nu}^{\gamma\gamma}(x)=G_{\mu\nu}^{33}(x)+\frac{1}{3}G_{\mu\nu}^{88}(x)+\frac{4}{9}C_{\mu\nu}^{c,c}(x),$
(15a) $\displaystyle
G_{\mu\nu}^{Z\gamma}(x)=\left(\frac{1}{2}-\sIIW\right)G_{\mu\nu}^{\gamma\gamma}(x)-\frac{1}{6\sqrt{3}}G_{\mu\nu}^{08}(x)-\frac{1}{18}C_{\mu\nu}^{c,c}(x),$
(15b)

where the flavour $\SU(3)$ contributions are defined as222In the usual lattice
notation, $G_{\mathrm{conn}}^{\ell}=2G^{33}$ and
$G_{\mathrm{conn}}^{s}=3G_{\mathrm{conn}}^{88}-G^{33}$. Moreover,
$G^{08}_{\mathrm{conn}}=\sqrt{3}(G^{33}-G_{\mathrm{conn}}^{88})/2$.

$\displaystyle G_{\mu\nu}^{33}(x)=\frac{1}{2}C^{\ell,\ell}_{\mu\nu}(x),$ (16a)
$\displaystyle
G_{\mu\nu}^{88}(x)=\frac{1}{6}\left[C^{\ell,\ell}_{\mu\nu}(x)+2C^{s,s}_{\mu\nu}(x)+2D^{\ell-s,\ell-s}_{\mu\nu}(x)\right],$
(16b) $\displaystyle
G_{\mu\nu}^{08}(x)=\frac{1}{2\sqrt{3}}\left[C^{\ell,\ell}_{\mu\nu}(x)-C^{s,s}_{\mu\nu}(x)+D^{2\ell+s,\ell-s}_{\mu\nu}(x)\right],$
(16c)

and the connected and disconnected Wick’s contractions are

$\displaystyle
C^{f_{1},f_{2}}_{\mu\nu}(x)=-\expectationvalue{\Tr{D_{f_{1}}^{-1}(x,0)\gamma_{\mu}D_{f_{2}}^{-1}(0,x)\gamma_{\nu}}},$
(17a) $\displaystyle
D^{f_{1},f_{2}}_{\mu\nu}(x)=\expectationvalue{\Tr{D_{f_{1}}^{-1}(x,x)\gamma_{\mu}}\Tr{D_{f_{2}}^{-1}(0,0)\gamma_{\nu}}}.$
(17b)

###  4.2 Autocorrelation study

Due to the update procedure of Monte Carlo simulations different
configurations within one chain are not independent. In order to give a
reliable estimate of the observable’s uncertainty we need to take these
autocorrelations into account. For this work we have used the $\Gamma$-method
[29, 30] to estimate the autocorrelation time, $\tauint$ and, from it, an
appropriate bin size to obtain statistically independent samples. Since
autocorrelations are observable-dependent we have computed $\tauint$ of
$\Dalphahad$ for the light and strange flavours on each ensemble at different
energies. Extra caution needs to be taken regarding the tail of the
correlator, which shows the signal-to-noise ratio problem (see Section 4.4).
To obtain a good estimate of $\tauint$ we only take into account the
correlator until a certain maximum value of $x_{0}$. Repeating the process for
different time cuts, we know where the noise of the tail starts to dominate
and the value of $\tauint$. We choose a bin size $B=2\tauint$ for each
ensemble to bin the correlator data and obtain a distribution of independent
bootstrap samples to carry out the main analysis.

###  4.3 Finite-size corrections

In this work, we estimate finite-volume corrections using the same strategy
described in Ref. [26]. Namely, we compute the difference between the
infinite- and finite-volume $I=1$ correlator and apply it as a correction to
the lattice data. We use two distinct models to compute the difference,
depending on whether the correlator is considered at short or long times.

At long times, we use the $I=1$ correlator obtained from the time-like pion
form factor $F_{\pi}(\omega)$. In infinite volume, that is

$G^{33}(x_{0},\infty)=\int_{0}^{\infty}\differential{\omega}\omega^{2}\rho(\omega^{2})\e{-\omega
x_{0}},\qquad\rho(\omega^{2})=\frac{1}{48\cpi^{2}}\left(1-\frac{4M_{\pi}^{2}}{\omega^{2}}\right)^{\frac{3}{2}}\absolutevalue{F_{\pi}(\omega)}^{2},$
(18)

while the finite volume expression is computed from the Lüscher energies
$\omega_{n}$ and the Lellouch-Lüscher amplitudes,

$G^{33}(x_{0},L)=\sum_{n}\absolutevalue{A_{n}}^{2}\e{-\omega_{n}x_{0}},$ (19)

extracted applying the Lellouch-Lüscher formalism [31, 32, 33]. In this work,
we use the Gounaris-Sakurai (GS) parametrization [34] of $F_{\pi}(\omega)$,
that depends on two parameters, $m_{\rho}$ and $g_{\rho\pi\pi}$, which are
obtained either by fitting the correlator at long times or by a spectroscopic
analysis [26]. We emphasise that the model is used only to correct for the
relatively small finite-volume effect of the correlator, while no modelling of
the contribution from the correlator tail is assumed. In the future, we plan
to further reduce the model-dependence employing, where available, a full
lattice determination of $F_{\pi}(\omega)$ [35, 36] instead of the GS
parametrization.

The $F_{\pi}(\omega)$-based model provides a good spectral representation of
the correlator up to the three-pion threshold, thus we use it for the
correlator correction at times $x_{0}>\ti$, with
$\ti=(M_{\pi}L/4)^{2}/M_{\pi}$. At smaller times $x_{0}\leq\ti$ we compute the
difference between the infinite- and finite-volume correlator in scalar QED
(_i.e._ NLO PT) [22, 21]

$G^{33}(x_{0},\infty)-G^{33}(x_{0},L)=\frac{1}{3}\left(\int\frac{\differential[3]{\vec{k}}}{(2\cpi)^{3}}-\frac{1}{L^{3}}\sum_{\vec{k}}\right)\frac{\vec{k}^{2}+M_{\pi}^{2}}{\vec{k}^{2}}\e{-2x_{0}\sqrt{\vec{k}^{2}+M_{\pi}^{2}}}.$
(20)

This model omits the pion self-interaction and it is known to only account for
a fraction of the finite-volume correction to $\SPi(Q^{2})$ at $Q^{2}$ values
of $\order{${\mathrm{GeV}}^{2}$}$ [37]. However, comparing the correlator on
two pairs of ensembles in Table 1 with the same parameters except for
different physical volumes, we observe that the finite-volume correction at
times $x_{0}\leq\ti$ is smaller than our uncertainties. Therefore, we are
confident that this rather simple short-time description is sufficient at the
current level of precision, while recent developments [38] show a promising
path towards improving on it.

We can estimate the relative size of the finite-volume corrections. For
example, on ensembles D200, N302 and J303, the ratio of the correction over
the total $\Dalphahad$ at $Q^{2}=$1\text{\,}{\mathrm{GeV}}^{2}$$ amounts to
more than $1\text{\,}\mathrm{\char 37\relax}$. The contribution, although
small in absolute terms, is already bigger than the $0.5\text{\,}\mathrm{\char
37\relax}$ statistical uncertainty on these ensembles and therefore needs to
be included.

###  4.4 Signal-to-noise ratio

Both the correlator and its variance can be expressed, using the spectral
decomposition, as an infinite tower of exponentials, but with different decay
rates and amplitudes. In general, this leads to an exponential deterioration
of the signal with respect to the noise with Euclidean time, known as the
signal-to-noise ratio problem. As shown in Section 3, this problem has a
larger impact on the statistical error of $\SPi(Q^{2})$ at smaller $Q^{2}$.
Although it affects all flavour components, it is more acute for the light
correlator, whose contribution remains sizeable longer in time.

As described in Ref. [21], one possibility is to model the tail of the
correlator with a single exponential, _i.e._

$G(x_{0})=\begin{cases}\mathrm{data},&x_{0}<\tcut,\\\
A\e{-m_{V}x_{0}},&x_{0}\geq\tcut,\end{cases}$ (21)

where $A$ and $m_{V}$ are obtained fitting the correlator. In our case, due to
the high statistics available, $\tcut$ lies between $1.5\text{\,}\mathrm{fm}$
and $2.5\text{\,}\mathrm{fm}$, depending on the ensemble. However, there is no
guarantee that a single state describes the tail of the correlator. This is
particularly true for light ensembles in a large volume, where both the $\rho$
resonance and the tower of two-$\pi$ states are long range contributions.
Therefore, this method has its limitations, and we choose not to use it. The
results presented in the following sections are obtained using the correlator
data until the last time slice available. As explained in Section 3, the
contribution from the tail to $\SPi(Q^{2})$ is smaller than in the case of
$a_{\mu}^{\mathrm{HLO}}$.

In the future, we are considering improving the statistical error on
$\SPi(Q^{2})$ at small $Q^{2}$, especially on light-pion ensembles,
substituting the correlator at long times with a spectral reconstruction, and
controlling the systematics _e.g._ with the bounding method [39, 26]. In its
simplest incarnation, this relies on the fact that $G(x_{0})$ at
$x_{0}\geq\tcut$ is bounded by

$0\leq G(\tcut)\e{-E_{\mathrm{eff}}(\tcut)(x_{0}-\tcut)}\leq G(x_{0})\leq
G(\tcut)\e{-E_{0}(x_{0}-\tcut)},$ (22)

where
$E_{\mathrm{eff}}(x_{0})=-\differential\log(G(x_{0}))/\differential{x_{0}}$ is
the effective mass and $E_{0}$ the ground state energy, to obtain the $\tcut$
so both bounds yield the same result for the observable.

##  5 Numerical results

$\displaystyle 0$$\displaystyle 1$$\displaystyle 2$$\displaystyle
3$$\displaystyle 4$$\displaystyle-0.004$$\displaystyle-0.002$$\displaystyle
0.000$$\displaystyle 0.020$$\displaystyle 0.040$$\displaystyle
0.060$$\displaystyle 0.080$$\displaystyle\bar{\Pi}(Q^{2})$N200$\displaystyle
M_{\pi}=$285\text{\,}\mathrm{MeV}$$$\displaystyle 33$$\displaystyle 88$
conn$\displaystyle 08$ conn$\displaystyle c$ conn$\displaystyle 88$
disc$\displaystyle 08$ disc$\displaystyle 0$$\displaystyle 1$$\displaystyle
2$$\displaystyle 3$$\displaystyle 4$$\displaystyle
t_{0}^{\mathrm{sym}}Q^{2}$D200$\displaystyle
M_{\pi}=$200\text{\,}\mathrm{MeV}$$$\displaystyle 0$$\displaystyle
1$$\displaystyle 2$$\displaystyle 3$$\displaystyle 4$E250$\displaystyle
M_{\pi}=$130\text{\,}\mathrm{MeV}$$$\displaystyle 0$$\displaystyle
2$$\displaystyle 4$$\displaystyle 6$$\displaystyle 8$$\displaystyle
0$$\displaystyle 2$$\displaystyle 4$$\displaystyle 6$$\displaystyle
8$$\displaystyle Q^{2}$ [${\mathrm{GeV}}^{2}$]$\displaystyle 0$$\displaystyle
2$$\displaystyle 4$$\displaystyle 6$$\displaystyle 8$

Figure 2: Running with energy $Q^{2}$ of different contributions to $\SPi(Q^{2})$ on three different ensembles at $a\approx$0.064\text{\,}\mathrm{fm}$$. The conserved, local discretization is shown and, when available, the local, local discretization in a lighter colour shade. The negative side of the vertical axis of the plot is inflated by a factor $10$ with respect to the positive side.  Table 2: Estimate of connected and disconnected contributions to $${10}^{5}$\cdot\SPi($1\text{\,}{\mathrm{GeV}}^{2}$)$ for the conserved-local (c.l.) and, when available, local-local (l.l.) discretizations, on three different ensembles at $a\approx$0.064\text{\,}\mathrm{fm}$$.  $\times${10}^{-5}$$ | $\SPi^{33}$ | $\SPi_{\mathrm{conn}}^{88}$ | $\SPi_{\mathrm{conn}}^{08}$ | $\SPi_{\mathrm{conn}}^{c}$ | $\SPi_{\mathrm{disc}}^{88}$ | $\SPi_{\mathrm{disc}}^{08}$
---|---|---|---|---|---|---
N200 c.l. | $3002(11)$ | $2537(5)$ | $393(6)$ | $266.2(5)$ | $-39(7)$ | $-110(29)$
l.l. | $2962(11)$ | $2497(5)$ |  | $228.4(5)$ | $-39(7)$ |
D200 c.l. | $3226(13)$ | $2526(5)$ | $600(8)$ | $270.3(6)$ | $-57(12)$ | $-120(36)$
l.l. | $3185(14)$ | $2485(5)$ |  | $232.5(5)$ | $-57(12)$ |
E250 c.l. | $3552(36)$ | $2594(12)$ | $826(21)$ | $271.8(40)$ | $-164(50)$ | $-301(120)$
l.l. | $3511(36)$ | $2553(12)$ |  | $233.7(34)$ | $-171(51)$ |

Figure 2 shows the running of different contributions to $\SPi(Q^{2})$,
defined through the correlators in Eq. (15), as a function of $Q^{2}$ on three
different lattices at the same lattice spacing with increasingly lighter
pions. As one moves away from the $\SU(3)$-symmetric point, the $\SPi^{33}$
contribution increases while the $\SPi_{\mathrm{conn}}^{88}$ contribution
decreases. The (quenched) charm contribution is also shown to be relatively
independent on the pion mass and linearly increasing in the range of $Q^{2}$
values. The negative disconnected contributions are also shown, enlarged by a
factor $10$. They are obtained cutting the TMR integration at
$\tcut\approx$2.5\text{\,}\mathrm{fm}$$, which results in a conservative
statistical error. Nevertheless, Table 2 shows that the statistical error on
the disconnected contribution is comparable to the connected one. The
exception is the E250 ensemble, with only a small number of disconnected loop
measurements being available at the current time. We will improve the
disconnected contribution estimate by implementing the bounding method for the
$I=0$ channel, in order to reduce the statistical error and correctly estimate
the integration tail systematics. It is worth noting that
$\SPi_{\mathrm{disc}}^{88}(Q^{2})$ is constant for
$Q^{2}\gtrsim$0.5\text{\,}{\mathrm{GeV}}^{2}$$, as predicted by perturbation
theory.

###  5.1 Extrapolation to the physical point

Table 3: Results extrapolated to the physical point for the different connected contributions to $\SPi$ for a range of $Q^{2}$ values. Only the connected contribution is included in $\Dalphahad$ and $\Dhad\sIIW$.  $Q^{2}$ [${\mathrm{GeV}}^{2}$] | $\SPi^{33}$ | $\SPi_{\mathrm{conn}}^{88}$ | $\SPi_{\mathrm{conn}}^{08}$ | $\Dalphahad$ | $\Dhad\sIIW$
---|---|---|---|---|---
$0.1$ | $0.007\,64(16)$ | $0.004\,56(8)$ | $0.002\,67(8)$ | $0.000\,841(16)$ | $-0.000\,821(15)$
$0.4$ | $0.020\,68(26)$ | $0.013\,54(19)$ | $0.006\,19(12)$ | $0.002\,310(29)$ | $-0.002\,300(29)$
$1.0$ | $0.032\,74(34)$ | $0.023\,40(27)$ | $0.008\,09(13)$ | $0.003\,718(38)$ | $-0.003\,770(40)$
$2.0$ | $0.042\,42(38)$ | $0.032\,26(33)$ | $0.008\,80(13)$ | $0.004\,876(45)$ | $-0.005\,012(47)$
$3.0$ | $0.048\,05(41)$ | $0.037\,68(36)$ | $0.008\,98(13)$ | $0.005\,558(48)$ | $-0.005\,752(50)$
$4.0$ | $0.052\,02(42)$ | $0.041\,57(38)$ | $0.009\,04(13)$ | $0.006\,041(50)$ | $-0.006\,278(53)$
$5.0$ | $0.055\,08(43)$ | $0.044\,61(39)$ | $0.009\,07(13)$ | $0.006\,415(51)$ | $-0.006\,687(54)$

$\displaystyle 0.0$$\displaystyle 0.1$$\displaystyle 0.2$$\displaystyle
0.3$$\displaystyle 0.4$$\displaystyle 0.5$$\displaystyle 0.6$$\displaystyle
0.7$$\displaystyle 0.8$$\displaystyle\phi_{2}=8t_{0}M_{\pi}^{2}$$\displaystyle
0.022$$\displaystyle 0.024$$\displaystyle 0.026$$\displaystyle
0.028$$\displaystyle 0.030$$\displaystyle 0.032$$\displaystyle
0.034$$\displaystyle 0.036$$\displaystyle
0.038$$\displaystyle\bar{\Pi}_{\textrm{conn}}^{88}(Q^{2})$
$\displaystyle\bar{\Pi}^{33}(Q^{2})$140200240280315340370395420$\displaystyle
M_{\pi}$
[$\mathrm{MeV}$]$0.086\text{\,}\mathrm{fm}$$0.076\text{\,}\mathrm{fm}$$0.064\text{\,}\mathrm{fm}$$0.050\text{\,}\mathrm{fm}$

Figure 3: Combined extrapolation of $\SPi^{33}(Q^{2})$ and
$\SPi_{\mathrm{conn}}^{88}(Q^{2})$ at $Q^{2}=$1\text{\,}{\mathrm{GeV}}^{2}$$
to the physical point. Filled symbols denote the conserved, local
discretization, while open symbols denote the local, local one.

Four different lattice spacings and several quark masses, including the
physical ones, allow us to extrapolate our results to the physical point. In
these proceedings, we consider only the extrapolation of the light and strange
connected contributions, which is obtained with a combined fit of the two
discretizations of $\SPi^{33}(Q^{2})$ and $\SPi_{\mathrm{conn}}^{88}(Q^{2})$.
At a given fixed $Q^{2}$, each HVP function contribution in the combined fit
is modelled with

$\SPi(a^{2}/t_{0},\phi_{2},\phi_{4})=\SPi^{\mathrm{sym}}+\delta_{2}a^{2}/t_{0}+\delta_{3}(a^{2}/t_{0})^{3/2}+\beta_{21}a^{2}(\phi_{2}-\phi_{2}^{\mathrm{sym}})/t_{0}\\\
+\gamma_{1}(\phi_{2}-\phi_{2}^{\mathrm{sym}})+\gamma_{2}(\log\phi_{2}-\log\phi_{2}^{\mathrm{sym}})+\gamma_{4}(\phi_{2}-\phi_{2}^{\mathrm{sym}})^{2}+\eta_{1}(\phi_{4}-\phi_{4}^{\mathrm{sym}}),$
(23)

where $\phi_{2}=8\tnotsym M_{\pi}^{2}$ and
$\phi_{4}=8\tnotsym(M_{K}^{2}+M_{\pi}^{2}/2)$ are proxies for the $M_{\pi}$
and $M_{K}$ dependence. For the continuum limit extrapolation, $a^{2}$ and
$a^{3}$ terms are included with different coefficients for the two different
discretizations. To fit the dependence on the meson masses, we considered
different models. In the fit shown in Figure 3, the $\SPi^{33}(Q^{2})$
interpolation includes the $\gamma_{1}^{33}$ term $\sim M_{\pi}^{2}$ and the
$\gamma_{2}^{33}$ term $\sim\log M_{\pi}^{2}$, while the
$\SPi_{\mathrm{conn}}^{88}(Q^{2})$ interpolation includes an independent
$\gamma_{1}^{88}$ term and a $\gamma_{4}^{88}$ term $\sim M_{\pi}^{4}$, while
we constrain $\gamma_{2}^{88}=\gamma_{2}^{33}/3$. The latter term is included
to match the diverging behaviour of $\SPi^{33}(Q^{2})$ with $M_{\pi}\to 0$,
that is present in $\SPi_{\mathrm{conn}}^{88}(Q^{2})$, with a factor of one
third, due to the missing disconnected contribution [26].

The combined fit includes the uncertainties in the determination of the pion
and kaon masses, including the correlation with the HVP function
determination, that are obtained from a dedicated computation. For every
ensemble $l$, we build a residue vector

$\nu_{l}=\begin{pmatrix}\phi_{2}^{l}\\\
\SPi_{\mathrm{cl}}^{33}(a^{2}/\tnotsym,\phi_{2}^{l},\phi_{4}^{l})\\\
\SPi_{\mathrm{ll}}^{33}(a^{2}/\tnotsym,\phi_{2}^{l},\phi_{4}^{l})\\\
\phi_{4}^{l}\\\
\SPi_{\mathrm{cl}}^{88}(a^{2}/\tnotsym,\phi_{2}^{l},\phi_{4}^{l})\\\
\SPi_{\mathrm{ll}}^{88}(a^{2}/\tnotsym,\phi_{2}^{l},\phi_{4}^{l})\end{pmatrix}-\begin{pmatrix}8\tnotsym
M_{\pi}^{2}\\\ \SPi_{\mathrm{cl}}^{33}\\\ \SPi_{\mathrm{ll}}^{33}\\\
8\tnotsym(M_{K}^{2}+M_{\pi}^{2}/2)\\\ \SPi_{\mathrm{cl}}^{88}\\\
\SPi_{\mathrm{ll}}^{88}\end{pmatrix}_{\mathrm{data}}$ (24)

and we minimize the
$\chi^{2}=\sum_{l\in\\{\mathrm{ensembles}\\}}\nu_{l}^{\mathsf{T}}C^{-1}_{l}\nu_{l}$,
with the necessary adaptations for $\SU(3)$-symmetric ensembles where
$\SPi^{33}$ and $\SPi^{88}$, and $M_{\pi}$ and $M_{K}$, are not independent.

This choice leads to an acceptable fit with an additional cut on the physical
lattice volume that excludes lattices with $L<$2.5\text{\,}\mathrm{fm}$$. At
$Q^{2}=$1\text{\,}{\mathrm{GeV}}^{2}$$, we have
$\chi^{2}/\mathrm{dof}=52.13/38=1.37$, which corresponds to a $p$-values of
$0.0631$, and parameters

$\begin{gathered}\SPi^{\mathrm{sym}}=$0.025\,16(29)$,\mkern
12.0mu\phi_{2}^{\mathrm{sym}}=2\phi_{4}^{\mathrm{sym}}/3=$0.754(5)$,\mkern
12.0mu\eta_{1}=$-0.011(4)$,\\\ \delta_{2}^{\mathrm{cl}}=$0.0126(31)$,\mkern
12.0mu\delta_{2}^{\mathrm{ll}}=$0.0106(31)$,\mkern
12.0mu\delta_{3}^{\mathrm{cl}}=$-0.005(4)$,\mkern
12.0mu\delta_{3}^{\mathrm{ll}}=$-0.006(4)$,\mkern
12.0mu\beta_{21}=$0.0004(23)$,\\\ \gamma_{1}^{33}=$-0.0017(9)$,\mkern
12.0mu\gamma_{1}^{88}=$0.003\,63(33)$,\mkern
12.0mu\gamma_{2}^{33}=\gamma_{2}^{88}/3=$-0.002\,91(28)$,\mkern
12.0mu\gamma_{4}^{88}=$0.0026(5)$.\end{gathered}$ (25)

The extrapolation to $a=0$ and to physical $\pi^{0}$ and $K^{0}$ masses gives
$\SPi^{33}=$0.032\,74(34)$$ and $\SPi_{\mathrm{conn}}^{88}=$0.023\,40(27)$$.
Moreover,
$\SPi_{\mathrm{conn}}^{08}=\sqrt{3}(\SPi^{33}-\SPi_{\mathrm{conn}}^{88})/2=$0.008\,09(13)$$.
Applying Eqs (2) and (6) allows us to compute the leading hadronic connected
contribution from three-flavour QCD to the running of $\alpha$ and $\sIIW$ at
$Q^{2}=$1\text{\,}{\mathrm{GeV}}^{2}$$

$\Dalphahad(-$1\text{\,}{\mathrm{GeV}}^{2}$)=$0.003\,718(38)$,\qquad\Dhad\sIIW(-$1\text{\,}{\mathrm{GeV}}^{2}$)=$-0.003\,770(40)$.$
(26)

###  5.2 The running with energy

$\displaystyle 0$$\displaystyle 1$$\displaystyle 2$$\displaystyle
3$$\displaystyle 4$$\displaystyle 5$$\displaystyle Q^{2}$
[${\mathrm{GeV}}^{2}$]$\displaystyle 0.000$$\displaystyle 0.001$$\displaystyle
0.002$$\displaystyle 0.003$$\displaystyle 0.004$$\displaystyle
0.005$$\displaystyle 0.006$$\displaystyle 0.007$$\displaystyle
0.008$$\displaystyle\Delta\alpha_{\mathrm{had}}(-Q^{2})$this work,
preliminaryBorsanyi _et al._ 2018, connected only$\displaystyle
0$$\displaystyle 1$$\displaystyle 2$$\displaystyle 3$$\displaystyle
4$$\displaystyle 5$$\displaystyle Q^{2}$
[${\mathrm{GeV}}^{2}$]$\displaystyle-0.008$$\displaystyle-0.007$$\displaystyle-0.006$$\displaystyle-0.005$$\displaystyle-0.004$$\displaystyle-0.003$$\displaystyle-0.002$$\displaystyle-0.001$$\displaystyle
0.000$$\displaystyle\Delta_{\mathrm{had}}\sin^{2}\theta_{\mathrm{W}}(-Q^{2})$

Figure 4: Leading hadronic (connected) contribution from three-flavour QCD to
the running of the electromagnetic coupling $\alpha$ (left) and the
electroweak mixing angle $\sIIW$ (right) as a function of the space-like
momentum transfer $Q^{2}$. The blue line (with error band) is the
extrapolation of our lattice data, while the orange points are constructed
from the HVP values from Table S3 of the supplemental material of Ref. [5].

Performing the fit described in Section 5.1 for different values of $Q^{2}$
results in a mild dependence of the $\chi^{2}$ on $Q^{2}$ up to around
$2\text{\,}{\mathrm{GeV}}^{2}$. At higher $Q^{2}$ discretization effects
become the dominant systematics and the quality of the fit, using the current
model, starts to deteriorate. Results extrapolated to the physical point for a
number of $Q^{2}$ values are given in Table 3, and in Figure 4 we plot the HVP
contribution to the running of $\alpha$ and $\sIIW$. In both plots, we compare
our preliminary result to the corresponding contribution from the percent-
level lattice determination of Ref. [5], given in the supplemental material
for five $Q^{2}$ values. The agreement is good, with a small tension only at
the smallest $Q^{2}=$1\text{\,}{\mathrm{GeV}}^{2}$$.

##  6 Conclusions and outlook

We computed on the lattice the leading hadronic contribution to the running of
the electromagnetic coupling $\alpha$ and of the electroweak mixing angle
$\thetaW$. After extrapolating to the physical point, the statistical error on
the connected contribution amounts to $\approx$4\text{\times}{10}^{-5}$$
which, for both quantities, is $1\text{\,}\mathrm{\char 37\relax}$ of the
contribution at $Q^{2}=$1\text{\,}{\mathrm{GeV}}^{2}$$. We also presented
results for the disconnected contribution, not yet extrapolated at the
physical point, which show that the statistical error is also around
$1\text{\,}\mathrm{\char 37\relax}$ of the total contribution. We included
finite-volume corrections, that are instrumental for obtaining a reliable
physical-point extrapolation. The precision of our results is comparable to
the phenomenological estimate, which is around $0.7\text{\,}\mathrm{\char
37\relax}$ and $1\text{\,}\mathrm{\char 37\relax}$ for the two quantities
respectively [40]. In particular, flavour contributions are naturally singled
out on the lattice. Thus, our results can be combined with the
phenomenological analysis of the electroweak mixing angle to reduce its
flavour separation systematics.

We are currently working on increasing the statistics of disconnected loops at
physical meson masses. Subsequently, we will extrapolate the disconnected and
the quenched charm contribution to the physical point. Moreover, a full
assessment of systematic errors is missing from the results in Table 3. We
plan to assess the systematics introduced by the extrapolation by varying the
choice of the fit function, and, in the future, adding a new ensemble at the
fine lattice spacing $a\approx$0.050\text{\,}\mathrm{fm}$$ with a lighter pion
mass of $M_{\pi}\approx$175\text{\,}\mathrm{MeV}$$. We plan, as described in
Section 4.4, to improve the treatment of the long-time tail of the lighter
ensembles with the bounding method. The scale setting introduces an error that
can be estimated as explained in Section B.2 of Ref. [21]. Also, a small
systematics from the mistuning of the charm hopping parameter needs to be
added. Finally, to compare with the physical world, isospin-breaking effects
from non-degenerate $u$ and $d$ quark masses and QED are to be included [41,
42].

Acknowledgements: We thank Jens Erler, Fred Jegerlehner and Daniel Mohler for
valuable discussions. Calculations for this project have been performed on the
HPC clusters “clover” and “himster2” at Helmoltz-Institut Mainz and “Mogon II”
at JGU Mainz, on the BG/Q system “JUQUEEN” at Jülich Supercomputing Center
(JSC), and on “Hazel Hen” at Höchstleistungsrechenzentrum Stuttgart (HLRS).
The authors gratefully acknowledge the support of the Gauss Centre for
Supercomputing and the John von Neumann-Institut für Computing (NIC) for
project HMZ21 at JSC and project GCS-HQCD at HLRS. Our programs use the
deflated SAP + GCR solver from the openQCD package, as well as the QDP++
library. We are grateful to our colleagues in the CLS initiative for sharing
ensembles.

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  *[SM]: Standard Model
  *[HVP]: hadronic vacuum polarization
  *[TMR]: time-momentum representation
  *[CLS]: Coordinated Lattice Simulations
  *[QCD]: quantum chromodynamics
  *[GS]: Gounaris-Sakurai