# Probing the vector charge of Sagittarius A* with pulsar timing

Zexin Hu  Lijing Shao,11footnotetext: Corresponding author. Rui Xu  Dicong
Liang  and Zhan-Feng Mai

###### Abstract

Timing a pulsar orbiting around Sagittarius A* (Sgr A*) can provide us with a
unique opportunity of testing gravity theories. We investigate the
detectability of a vector charge carried by the Sgr A* black hole (BH) in the
bumblebee gravity model with simulated future pulsar timing observations. The
spacetime of a bumblebee BH introduces characteristic changes to the orbital
dynamics of the pulsar and the light propagation of radio signals. Assuming a
timing precision of 1 ms, our simulation shows that a 5-yr observation of a
pulsar with an orbital period $P_{b}\sim 0.5\,{\rm yr}$ and an orbital
eccentricity $e\sim 0.8$ can probe a vector charge-to-mass ratio as small as
$Q/M\sim 10^{-3}$, which is much more stringent than the current constraint
from the Event Horizon Telescope (EHT) observations, and comparable to the
prospective constraint from extreme mass-ratio inspirals with the Laser
Interferometer Space Antenna (LISA).

##  1 Introduction

Due to the extreme rotation stability, pulsars are regarded as one of the best
probes to test gravity theories [1, 2, 3, 4]. With the observational data of
16 years, the Double Pulsar system provides the most precise test of general
relativity (GR) in a system containing strongly self-gravitating bodies [5].
It also has been anticipated that timing a pulsar orbiting closely around the
supermassive black hole (SMBH) residing in our Galactic Center (GC), the so-
called Sagittarius A* (Sgr A*), can provide us with a unique opportunity of
studying the black hole (BH) physics as well as the environment around the GC
[6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16].

From both theoretical models and observational evidences of the stellar
population around Sgr A*, we expect the existence of a large number of neutron
stars in the GC [17, 18, 19]. The discovery of the magnetar in this region
also strongly suggests the existence of normal pulsars as magnetars are
thought to be rare pulsars [20]. However, due to the large dispersion measures
and highly turbulent interstellar medium in the GC [21], no normal pulsar has
been found yet within the inner parsec even in those searches performed at
high radio frequencies [22, 23, 24, 25]. Nevertheless, future observations
with the next-generation facilities, such as the Square Kilometre Array (SKA)
and the next-generation Very Large Array (ngVLA), are expected to find a
number of pulsars in this region. We also expect to time these pulsars well,
which will open a new avenue of testing gravity theories [6, 7, 10, 26, 14,
27].

In this work we focus on the possibility of using a pulsar-SMBH system to
constrain the vector charge carried by the Sgr A* in a specific non-minimally
coupled vector-tensor theory, the so-called bumblebee gravity model. The
action of this theory reads [28]

$S=\int{\rm
d}^{4}x\sqrt{-g}\left(\frac{1}{2\kappa}R+\frac{\xi}{2\kappa}B^{\mu}B^{\nu}R_{\mu\nu}-\frac{1}{4}B^{\mu\nu}B_{\mu\nu}-V\right)+S_{m}\,,$
(1.1)

where $S_{m}$ is the action for conventional matter, $\kappa=8\pi G$,
$B_{\mu\nu}\equiv D_{\mu}B_{\nu}-D_{\nu}B_{\mu}$ is the field strength of the
vector field $B_{\mu}$ with $D_{\mu}$ being the covariant derivative, $\xi$ is
the coupling constant of the non-minimal coupling between the bumblebee vector
and the Ricci tensor $R_{\mu\nu}$, and $V$ is the potential for the vector
field $B_{\mu}$. The bumblebee gravity model has been widely studied in the
literature as an example for spontaneous violation of the Lorentz symmetry or
for being a simple extension of the Einstein-Maxwell theory [28, 29, 30, 31].
While observations in the Solar System set stringent constraints on the
bumblebee gravity in the weak-field region [32, 31], recent studies using the
BH shadows observed by the Event Horizon Telescope (EHT) leave a large
parameter space in the strong-field region untested [31, 33]. Considering the
high precision of pulsar timing experiments, the main purpose of this work is
to develop a timing model for pulsar-SMBH systems in the bumblebee gravity
theory and to investigate the potential of constraining the bumblebee gravity
with future timing observations.

The paper is organized as follows. In Sec. 2, we briefly overview the
numerical BH solutions in the bumblebee gravity theory and derive a leading
order solution expanded in terms of the bumblebee charge of the BH. We
construct two pulsar timing models in Sec. 3, which include calculations for
the beyond-GR light propagation and orbital motion in the BH spacetime. The
simulation results of possible constraints on the bumblebee charge of Sgr A*
are shown in Sec. 4 and we have some final discussion in Sec. 5.

In the remaining part of this paper, we use the geometrized unit system where
$G=c=1$, except when traditional units are written out explicitly. The sign
convention of the metric is $(-,+,+,+)$.

##  2 Bumblebee BHs

The field equations in the bumblebee gravity are obtained by taking variations
of the metric field $g_{\mu\nu}$ and the bumblebee field $B^{\mu}$ in Eq.
(1.1). Detailed calculation can be found in Ref. [31]. For convenience, we
briefly review the construction of the static spherical BH solutions here. To
focus on the static spherical BH solutions, we adopt the metric ansatz

${\rm d}s^{2}=-e^{2\nu}{\rm d}t^{2}+e^{2\mu}{\rm d}r^{2}+r^{2}\left({\rm
d}\theta^{2}+\sin^{2}\theta{\rm d}\phi^{2}\right)\,,$ (2.1)

as well as the assumption that $B_{\mu}=(b_{t},0,0,0)$, where $\nu$, $\mu$,
and $b_{t}$ are functions of $r$ only. Following the assumptions in Ref. [31],
we ignore the contribution from the potential $V$ which, in the setting of
spontaneous Lorentz symmetry breaking, merely plays a role of vacuum energy
after giving the background field values of $B_{\mu}$. Then the vacuum field
equations can be simplified to ordinary differential equations for $\nu$,
$\mu$, and $b_{t}$ [31, 34],

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
e^{2\mu}-2r\nu^{\prime}-1-r^{2}e^{-2\nu}\left(\frac{\kappa}{2}b_{t}^{\prime
2}-\xi b_{t}b_{t}^{\prime}\nu^{\prime}+\xi b_{t}^{2}\nu^{\prime 2}\right)\,,$
(2.2) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\mu^{\prime}\nu^{\prime}+\frac{1}{r}\mu^{\prime}-\frac{1}{r}\nu^{\prime}-\nu^{\prime\prime}-\nu^{\prime
2}+e^{-2\nu}\left(\frac{\kappa}{2}b_{t}^{\prime 2}-\xi
b_{t}b_{t}^{\prime}\nu^{\prime}+\xi b_{t}^{2}\nu^{\prime 2}\right)\,,$ (2.3)
$\displaystyle 0$ $\displaystyle=$ $\displaystyle
b_{t}^{\prime\prime}-b_{t}^{\prime}\left(\mu^{\prime}+\nu^{\prime}-\frac{2}{r}\right)+\frac{\xi}{\kappa}b_{t}\left(\mu^{\prime}\nu^{\prime}-\frac{2}{r}\nu^{\prime}-\nu^{\prime\prime}-\nu^{\prime
2}\right)\,,$ (2.4)

where the prime denotes the derivative with respect to $r$.

The general assumption for $B_{\mu}$ with spherical symmetry should contain a
radial component $b_{r}$. However, as discussed by Xu  _et al._ [31], the BH
solutions with a vanishing $b_{r}$ can recover the Reissner-Nordström (RN)
solution when $\xi$ tends to zero, while the solutions with a nonzero $b_{r}$
only exist when $\xi\neq 0$. It means that the solutions with a nontrivial
$b_{r}$ are extraordinary solutions caused by the nonminimal coupling [34].
For simplicity, we only consider the ordinary solutions that are connected to
the RN solution in the present work.

###  2.1 Numerical solutions

A general analytical solution of Eqs. (2.2–2.4) has not been found yet, thus
we can only obtain the BH solutions numerically. Following Refs. [31, 34], one
first eliminates $\mu$ with Eq. (2.2) to obtain two second-order ordinary
differential equations for $\nu$ and $b_{t}$. Then one can integrate the
system outward from the horizon $r=r_{h}$. The initial conditions that ensure
a BH solution have been carefully discussed in Ref. [31], which read

$\displaystyle e^{2\nu}$ $\displaystyle=$ $\displaystyle
N_{11}(r-r_{h})+N_{12}(r-r_{h})^{2}+\cdots\,,$ (2.5) $\displaystyle b_{t}$
$\displaystyle=$ $\displaystyle L_{11}(r-r_{h})+L_{12}(r-r_{h})^{2}+\cdots\,,$
(2.6)

where $N_{11},L_{11}$ are free parameters and

$\displaystyle N_{12}$ $\displaystyle=$
$\displaystyle-\frac{N_{11}}{r_{h}}+\left(\frac{\kappa}{2}-\frac{\xi}{4}\right)L_{11}^{2}+\left(\frac{\kappa\xi}{4}-\frac{\xi^{2}}{4}+\frac{\xi^{3}}{16\kappa}\right)\frac{L_{11}^{4}r_{h}}{N_{11}}\,,$
(2.7) $\displaystyle L_{12}$ $\displaystyle=$
$\displaystyle-\frac{L_{11}}{r_{h}}+\left(\frac{\xi}{4}-\frac{\xi^{2}}{8\kappa}\right)\frac{L_{11}^{3}}{N_{11}}+\left(\frac{\kappa\xi}{4}-\frac{\xi^{2}}{4}+\frac{\xi^{3}}{16\kappa}\right)\frac{L_{11}^{5}r_{h}}{N_{11}^{2}}\,.$
(2.8)

After integrating outward with given $r_{h}$, $N_{11}$, and $L_{11}$, one
needs to redefine the time coordinate such that $g_{tt}=-e^{2\nu}$ goes to
$-1$ as $r\rightarrow\infty$. Thus only two of these three parameters are
independent, which are equivalent to the mass $M$ and bumblebee charge $Q$ of
the BH that are defined with the asymptotic behavior of $\nu$ and $b_{t}$ at
spacial infinity [31, 34]

$\displaystyle M$
$\displaystyle\equiv-\lim_{r\rightarrow\infty}r^{2}\mu^{\prime}=\lim_{r\rightarrow\infty}r^{2}\nu^{\prime}\,,$
(2.9) $\displaystyle Q$
$\displaystyle\equiv-\sqrt{\frac{\kappa}{2}}\lim_{r\rightarrow\infty}r^{2}b_{t}^{\prime}\,.$
(2.10)

The factor $\sqrt{\kappa/2}$ in the definition of $Q$ is chosen so that it can
recover the RN solution when $\xi=0$. With this factor, $Q$ then can be
interpreted as an extension of the electric charge. The bumblebee charge $Q$
can be positive or negative depending on the sign of $b_{t}$. As one can see
from Eqs. (2.2–2.3) that the metric functions only depend on $b_{t}$ in
quadratic form, we only consider $Q\geq 0$ in this paper.

There are two special cases for the BH solutions in the bumblebee gravity [31,
34]. One is $\xi=0$, in which the theory reduces to the Einstein-Maxwell
theory so that the BH solution recovers the RN solution. The other case is
$\xi=2\kappa$. In this configuration, the BH solution is described by a
Schwarzchild metric accompanied by a nonvanishing bumblebee field. Similar
solutions exist in many other vector-tensor theories and scalar-tensor
theories [35, 36, 37]. In this case, one cannot distinguish the charged BH in
the bumblebee theory from a Schwarzchild BH in GR with only the background BH
metric considered [38].

###  2.2 Leading order solutions in $Q/M$

Although current observations with EHT and thermodynamic consideration of the
bumblebee BHs only set loose constraints on $Q/M$ for Sgr A* even for the case
that $\xi$ deviates from $2\kappa$ significantly [31, 33, 39], it turns out
that the prospective measurement precision of $Q/M$ using pulsar timing can be
as small as $Q/M\sim\mathcal{O}(10^{-2})$ for $|\xi/2\kappa-1|\gtrsim 1$ as we
will find out later. Moreover, from Eqs. (2.2–2.3) one can see that the
influence of the bumblebee field $b_{t}$ only appears in quadratic form, which
means that the next-to-leading order effects caused by the bumblebee charge
will be $\mathcal{O}(10^{-4})$ smaller than the leading order effects in the
cases that we are interested in. Thus a BH solution accurate to the leading
order of $Q/M$ can be used in constructing our timing model for a pulsar
around Sgr A* which is equipped with a bumblebee charge. Here we present such
an analytic approximate solution as follows

$\displaystyle b_{t}$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2\kappa}}\left(1-\frac{2M}{r}\right)\frac{Q}{M}+\mathcal{O}\left(\frac{Q}{M}\right)^{3}\,,$
(2.11) $\displaystyle g_{tt}$ $\displaystyle=$
$\displaystyle-1+\frac{2M}{r}-\left(1-\frac{\xi}{2\kappa}\right)\frac{Q^{2}}{r^{2}}+\mathcal{O}\left(\frac{Q}{M}\right)^{4}\,,$
(2.12) $\displaystyle g_{rr}$ $\displaystyle=$
$\displaystyle\left[1-\frac{2M}{r}+\left(1-\frac{\xi}{2\kappa}\right)\frac{Q^{2}}{r^{2}}\right]^{-1}+\mathcal{O}\left(\frac{Q}{M}\right)^{4}\,,$
(2.13)

and the detailed derivation can be found in Appendix A. The radius of the
event horizon $r_{h}$ can also be determined to the same order,

$\frac{r_{h}}{M}=1+\sqrt{1-\left(1-\frac{\xi}{2\kappa}\right)\left(\frac{Q}{M}\right)^{2}}+\mathcal{O}\left(\frac{Q}{M}\right)^{4}\,,$
(2.14)

where the retainment of the square root shows the recovery of the RN solution
for $\xi=0$ explicitly.

An important observation from the leading order solution is that, in the
leading order metric functions, $Q$ and $\xi$ always appear as a combined
factor $(1-\xi/2\kappa)Q^{2}$, which means that we cannot distinguish a
charged BH in the bumblebee theory from a RN BH in GR when $Q$ is small and
$\xi/2\kappa<1$ by measuring the background metric only. Such a degeneracy can
be broken by considering higher order terms as shown in Appendix A, or
probably by metric perturbations.

##  3 Pulsar timing model

As studied in Refs. [6, 8, 9, 16], timing a pulsar in a close orbit around Sgr
A* with future instruments like SKA or ngVLA can provide us with a unique
opportunity to measure BH properties to a high precision. A generic numerical
model of pulsar-SMBH systems was developed in Ref. [16]. To consider the
possibility of constraining the bumblebee charge of Sgr A* with pulsar timing,
the first step is to construct a timing model that consistently includes the
effects caused by the bumblebee charge. In pulsar timing, the timing model
connects a pulsar’s proper rotation with the observed times of arrival (TOAs)
at radio telescopes by modeling various physical effects taking part in the
travel history of the radio signal from the pulsar to the Earth [40, 41, 4].
Ignoring the aberration effect [40], the pulse arrival time $t^{\rm TOA}$ in
the barycenter of the Solar System is related to the pulsar’s proper time $T$
via

$t^{\rm TOA}=T+\Delta_{\rm E}+\Delta_{\rm prop}\,,$ (3.1)

where the Einstein delay $\Delta_{\rm E}=t-T$ translates the proper time of
the pulsar to the coordinate time $t$, and $\Delta_{\rm prop}$ accounts for
the propagation effects. The rotation of the pulsar is usually described by a
simple rotation model [42]

$N=N_{0}+\nu_{p}T+\frac{1}{2}\dot{\nu}_{p}T^{2}+\cdots\,,$ (3.2)

where $N$ is the rotation number and $\nu_{p}$, $\dot{\nu}_{p}$ are the
pulsar’s spin and spin down rate respectively. Combining these two equations,
one can calculate the TOA for each pulse after knowing the propagation of
light and pulsar’s orbital motion.

In Sec. 3.1 and Sec. 3.2, we develop the timing model that fully uses a
numerical method to account for all the possible effects caused by the
bumblebee charge of the BH. We also provide a timing model in Sec. 3.3 based
on the leading order solution introduced above.

###  3.1 Light propagation

The propagation time delay $\Delta_{\rm prop}$ in Eq. (3.1) represents the
time delay during the pulse signal travelling from the pulsar to the Earth.
Here we focus on the additional time delay caused by the bumblebee charge and
ignore the various effects caused by the interstellar medium, Galactic
acceleration, as well as the translation between the Solar System barycenter
and the local coordinate time on the Earth [41, 42]. The distance between Sgr
A* and the Solar System is treated as infinity, which means we also ignore the
proper motion of Sgr A* [43]. These simplifications are adopted as we are only
aiming to estimate the measurability of the bumblebee charge carried by the
Sgr A*. While for timing models used in real observations one should take into
account all these effects depending on the real timing precision.

In the spherical spacetime represented by Eq. (2.1), the light-like geodesic
equation can be rewritten with the help of its first integrals as

$\displaystyle\left(\frac{{\rm d}r}{{\rm d}t}\right)^{2}$
$\displaystyle=e^{2\nu-2\mu}\left(1-\frac{b^{2}e^{2\nu}}{r^{2}}\right)\,,$
(3.3) $\displaystyle\frac{{\rm d}\phi}{{\rm d}t}$
$\displaystyle=\frac{be^{2\nu}}{r^{2}}\,,$ (3.4)

where $b$ is the impact parameter and we have set the orbit in the equatorial
plane because of the spherical symmetry. For a pulsar at a given position,
solving the light propagation time is the so-called emitter-observer problem,
which in general dose not have an analytic solution even in the spherical
spacetime [44, 45]. With the metric solved numerically in Sec. 2, one can
first solve the impact parameter $b$ with shooting method from the equation

$\Delta\phi=\int_{r\rightarrow\infty}\frac{{\rm d}\phi}{{\rm d}r}{\rm d}r\,,$
(3.5)

where $\Delta\phi$ is the angle between the pulsar and the Earth as seen from
Sgr A*. The range of $\Delta\phi$ is considered to be between $0$ and $\pi$,
which corresponds to the direct propagation paths. The light paths with
$\Delta\phi>\pi$ bend so strongly such that they in general result in images
too faint to be observed [46]. One should also notice that the integration in
Eq. (3.5) splits into two pieces with ${\rm d}r/{\rm d}\phi$ taking different
signs if the path has a periastron at ${\rm d}r/{\rm d}\phi=0$.

Figure 1:  The difference in the light propagation time $\Delta
t=\big{|}\Delta_{\rm prop}^{\rm Bum}-\Delta_{\rm prop}^{\rm GR}\big{|}$
between GR and the bumblebee gravity for $\xi=\kappa$ and $Q=M$. The bottom
horizontal axis $r$ is the distance between the pulsar and Sgr A*, and the top
horizontal axis $P_{b}$ is the corresponding period of the pulsar on a
circular orbit. The vertical axis $\Delta\phi$ is the angle between the pulsar
and the Earth as seen from Sgr A*. The red line corresponds to an eccentric
Newtonian orbit with orbital parameters shown in Eq. (3.8).

One can directly integrate Eqs. (3.3–3.4) to obtain the light propagation time
$\Delta_{\rm prop}$ after getting the impact parameter $b$. In Fig. 1, we plot
the time difference $\Delta t=\big{|}\Delta_{\rm prop}^{\rm Bum}-\Delta_{\rm
prop}^{\rm GR}\big{|}$ between GR and the bumblebee gravity for $\xi=\kappa$
and $Q=M$ as an example. The red line in the figure illustrates a pulsar orbit
with an orbital period $P_{b}=0.5\,{\rm yr}$ and an orbital eccentricity
$e=0.8$. Although this example is somewhat unrealistic for its extremely large
$Q$, we illustrate it in order to show the deviation from GR clearly. For a
pulsar with orbital period $P_{b}\sim 0.1\,{\rm yr}$ around Sgr A*, which has
a mass $M\sim 4.3\times 10^{6}\,M_{\odot}$, the typical time difference is
$\sim 20\,{\rm ms}$, which may be detectable in future observations as the
expected timing precision for such a pulsar can reach $0.1$–$1$ ms [6]. The
time difference can be even larger if the pulsar is in an orbit with its
inclination angle close to $\pi/2$ or has a large orbital eccentricity.
However, this time difference is always proportional $(Q/M)^{2}$ at the
leading order. For $Q/M\lesssim 0.1$, the time difference is about or less
than $1\,{\rm ms}$ for pulsars in the interested parameter space, which are
hard to be directly detected in the near future. Nevertheless, one still
should at least include its leading order effect in the timing model as the
residuals caused by this effect might be close to the timing precision,
similar to the spirit in the treatment of those next-to-leading order effects
in the Double Pulsar [5].

###  3.2 Orbital motion

As the mass ratio of the pulsar and the Sgr A* BH is smaller than $10^{-6}$,
we will treat the pulsar as a test particle in the BH’s spacetime for the
moment. With this assumption, we also neglect the gravitational radiation that
causes the decay of orbital period, usually encoded in the $\dot{P}_{b}$
parameter. Therefore, the equations of motion for the pulsar read as

$\displaystyle\left(\frac{{\rm d}r}{{\rm d}t}\right)^{2}$
$\displaystyle=e^{2\nu-2\mu}\left(1-\frac{l^{2}}{\epsilon^{2}}\frac{e^{2\nu}}{r^{2}}-\frac{e^{2\nu}}{\epsilon^{2}}\right)\,,$
(3.6) $\displaystyle\frac{{\rm d}\phi}{{\rm d}t}$
$\displaystyle=\frac{l}{\epsilon}\frac{e^{2\nu}}{r^{2}}\,,\ \ \frac{{\rm
d}T}{{\rm d}t}=\frac{1}{\epsilon}e^{2\nu}\,,$ (3.7)

where $\epsilon$ is the specific energy constant, and $l$ is the specific
angular momentum constant. By numerically integrating the above equations, one
can obtain the orbital motion of the pulsar, as well as the Einstein delay
$\Delta_{\rm E}$.

As a fiducial case, we study a pulsar with the following orbital parameters,

$\displaystyle P_{b}=0.5\,{\rm yr}\,,\quad\quad e=0.8\,,\quad\quad
i=\frac{1}{5}\pi\,,\quad\quad\omega=\frac{5}{7}\pi\,,\quad\quad\sigma=-\frac{1}{3}\pi\,,$
(3.8)

where $P_{b}$ is the orbital period, $e$ is the orbital eccentricity, $i$ is
the inclination angle, $\omega$ is the longitude of the periastron at $t=0$,
and $\sigma$ is the initial orbital phase of the pulsar. As we have ignored
the proper motion of Sgr A*, the longitude of the ascending node, $\Omega$,
will not affect the TOAs and we choose to fix $\Omega=0$ [47]. The orbital
parameters of the pulsar is understood in a Keplerian manner at $t=0$. For
example, the semi-major axis $a$ of the orbit is defined through
$a=(MP_{b}^{2}/4\pi^{2})^{1/3}$, and the pericenter and apocenter distances
are given by $r_{\rm min}=a(1-e)$ and $r_{\rm max}=a(1+e)$ respectively. The
relation between the parameters $(\epsilon,l)$ and $(P_{b},e)$ is determined
by the assumption that $r_{\rm min}$ and $r_{\rm max}$ are the two roots that
result in ${\rm d}r/{\rm d}t=0$ in Eqs. (3.6–3.7).

Figure 2:  Five-year pulsar orbits in GR (orange dashed line) and in the
bumblebee gravity with $\xi=\kappa$ and $Q=M$ (blue solid line). The pulsar’s
orbital parameters are given in Eq. (3.8). The black dot denotes the position
of Sgr A* and the blue star denotes the initial location of the pulsar.

Due to the assumed spherical symmetry of the BH spacetime, the inclination
angle $i$ is a constant and the motion of the pulsar is in a fixed plane.
Figure 2 gives an illustration of the pulsar’s orbits in the orbital plane for
GR and the bumblebee gravity with $\xi=\kappa$ and $Q=M$. The total time span
$t_{\rm total}=5\,{\rm yr}$ is used, and one can see a secularly increasing
deviation between these two orbits. This secular effect corresponds to an
additional periastron advance caused by the bumblebee charge. At the leading
order of $Q/M$, it reads [31]

$k^{Q}=-\frac{1}{6}\left(1-\frac{\xi}{2\kappa}\right)\left(\frac{Q}{M}\right)^{2}k^{\rm
1PN}\,,$ (3.9)

where $k^{\rm 1PN}=3M/a(1-e^{2})$ is the periastron advance in GR at the first
post-Newtonian (PN) level [40]. The leading order effect caused by the
bumblebee charge is at the 1 PN level as expected in the parameterized PN
(PPN) framework where the PPN parameter $\beta\neq 1$ for a charged BH in the
bumblebee gravity [31]. Compared to its GR counterpart $k^{\rm 1PN}$, $k^{Q}$
is suppressed by a factor of $(Q/M)^{2}$ and thus becomes much smaller when
$Q/M\lesssim 0.1$. However, depending on the value of $Q/M$, this additional
periastron advance might be still comparable to the effects caused by the spin
of Sgr A*, which is numerically at the 1.5 PN level [48, 6]. Thus it is
important in the timing model. Besides the secular effect, we also include all
the periodic effects caused by the bumblebee charge by resorting to numerical
calculations, although they are in general smaller than the leading order
secular effect.

###  3.3 Leading order effects in $Q/M$

Combining the results of orbital motion and light propagation, one obtains the
numerical timing model for the pulsar-SMBH system in the bumblebee gravity.
However, as we discussed before, for $Q/M\lesssim 0.1$, it might be
unnecessary to calculate all these effects numerically. A leading order
approximation should work for the purpose of forecasting the measurability of
the bumblebee charge while higher order corrections can be added in real
timing models depending on the timing accuracy of the real data.

To construct a timing model with leading order effects in $Q/M$, we adopt the
expansion in $Q/M$ as well as the usual PN expansion [49], which describes the
pulsar’s motion in the harmonic coordinate. The relation between the radial
position $R$ in the harmonic coordinate and $r$ in Eq. (2.1) is described by
[31]

$R^{\prime\prime}+\left(\nu^{\prime}-\mu^{\prime}+\frac{2}{r}\right)R^{\prime}-\frac{2e^{2\mu}}{r^{2}}R=0\,.$
(3.10)

With the solution derived in Appendix A, one can obtain the relation between
$R$ and $r$ as

$R=r-M+\frac{\xi}{2\kappa}\left(1-\frac{\xi}{2\kappa}\right)^{2}\frac{M^{2}}{2r}\left(\frac{Q}{M}\right)^{4}+\mathcal{O}\left(\frac{Q}{M}\right)^{6}\,.$
(3.11)

The leading order correction caused by the bumblebee charge in this relation
is at the order of $(Q/M)^{4}$, which will not enter the later calculations as
the leading order effects in light propagation and orbital motion are both at
the $(Q/M)^{2}$ order.

Solving Eqs. (3.3–3.5) to the leading order of $Q/M$ and combining the
coordinate transformation in Eq. (3.11), one can obtain $\Delta_{\rm prop}$ in
the harmonic coordinate

$\Delta_{\rm prop}=\Delta_{\rm R}+\Delta_{\rm S}+\Delta_{\rm Q}\,,$ (3.12)

where $\Delta_{\rm R}$ and $\Delta_{\rm S}$ are the so-called Römer delay and
Shapiro delay respectively [40], given by

$\displaystyle\Delta_{\rm R}$ $\displaystyle=-R\cos{\Delta\phi}\,,\ $ (3.13)
$\displaystyle\Delta_{\rm S}$ $\displaystyle=-2M\ln(R+R\cos\Delta\phi)\,,$
(3.14)

and $\Delta_{\rm Q}$ is the additional light propagation delay caused by the
bumblebee charge $Q$ at the leading order

$\Delta_{\rm
Q}=-\frac{Q^{2}}{4R}\left(1-\frac{\xi}{2\kappa}\right)\left(\frac{3\Delta\phi}{\sin\Delta\phi}+\cos\Delta\phi\right)\,.$
(3.15)

Similarly to the leading order Shapiro delay, the leading order expression of
$\Delta_{\rm Q}$ diverges at the point $\Delta\phi=\pi$, as the leading order
calculation is effectively done by integrating along a straight line that will
cross the BH when $\Delta\phi=\pi$ [44]. Here we only include the leading
order Shapiro delay, which is at 1 PN order. As including higher-order terms
is usually expected to further break the degeneracies between parameters in
such a problem, the treatment here is rather conservative. It is sufficient
for our purpose here and we leave the problem of constructing a more delicate
timing model for future studies.

The orbital motion of a pulsar with leading order corrections of $Q/M$ can be
calculated analytically using the so-called osculating orbital elements [50]
or a similar procedure described by Damour and Deruelle [49]. However, those
methods are hard to extend when considering higher order corrections or
combining other possible effects such as the BH spin. Simple analytic formula
does not exist in general cases. Therefore, we adopt the expansion form of the
equations of motion in the harmonic coordinate

$\ddot{\bm{R}}=-\frac{M}{R^{2}}\hat{\bm{n}}-\frac{M}{R^{2}}\left[\left(-\frac{4M}{R}+v^{2}\right)\hat{\bm{n}}-4\dot{R}\,\bm{v}\right]+\left(1-\frac{\xi}{2\kappa}\right)\frac{Q^{2}}{R^{3}}\hat{\bm{n}}+\cdots\,,$
(3.16)

where $\bm{R}$ points from the central BH to the pulsar, the dots denote time
derivatives, $\hat{\bm{n}}\equiv\bm{R}/R$, $\bm{v}\equiv\dot{\bm{R}}$, and
$v\equiv|\bm{v}|$. We only keep the 1 PN term and the leading order correction
in $Q/M$ in this expansion for similar reasons that have been argued before.

The relation between the pulsar’s proper time and the coordinate time has an
expansion form as

$\frac{{\rm d}T}{{\rm
d}t}=1-\frac{M}{R}-\frac{v^{2}}{2}+\frac{Q^{2}}{2R^{2}}\left(1-\frac{\xi}{2\kappa}\right)+\cdots\,.$
(3.17)

We integrate this equation to obtain the Einstein delay $\Delta_{\rm E}$.

Figure 3:  Various time delays corresponding to the pulsar orbit shown in Fig.
2 in the bumblebee gravity.

After dropping the constant term, one finally has

$t^{\rm TOA}=T+\Delta_{\rm E}+\Delta_{\rm R}+\Delta_{\rm S}+\Delta_{\rm Q}\,.$
(3.18)

In Fig. 3, we show these time delays for the orbit illustrated before with
$\xi=\kappa$ and $Q=M$. The dominant modification from the bumblebee charge is
in the Römer delay at the 1 PN level, due to the additional periastron advance
in the bumblebee gravity. The additional propagation time caused by the
bumblebee charge has a similar behavior as the Shapiro delay but with an
opposite sign, and it is much smaller than the Shapiro delay.

##  4 Parameter estimation

As a standard procedure for parameter estimation in the pulsar timing process
[40, 51], we use the covariance matrix,

$C_{\alpha\beta}=\left(\frac{\partial^{2}\mathcal{L}}{\partial\Theta^{\alpha}\partial\Theta^{\beta}}\right)^{-1}\,,$
(4.1)

to estimate the measurement uncertainties of the parameters
$\big{\\{}\Theta^{\mu}\big{\\}}$ in the timing model. The log-likelihood
function $\mathcal{L}$ is defined via [40]

$\displaystyle\mathcal{L}$ $\displaystyle=-\ln P\big{(}\Theta\big{|}t^{\rm
TOA}\big{)}\,,\ $ (4.2) $\displaystyle P\big{(}\Theta\big{|}t^{\rm
TOA}\big{)}$
$\displaystyle\propto\exp\left(-\frac{1}{2\nu_{p}^{2}}\sum_{i=1}^{N_{\rm
TOA}}\frac{\left[N^{(i)}(\Theta)-N^{(i)}(\bar{\Theta})\right]^{2}}{\sigma_{\rm
TOA}^{2}}\right)\,,$ (4.3)

where $N_{\rm TOA}$ is the total number of observed TOAs, $\sigma_{\rm TOA}$
is the timing precision, $N^{(i)}(\Theta)$ is the pulsar rotation number of
the $i$-th TOA calculated from the timing model with parameters
$\big{\\{}\Theta^{\mu}\big{\\}}$, and $\bar{\Theta}$ represents the true
values of those parameters. We list all the model parameters here

$\big{\\{}\Theta^{\mu}\big{\\}}=\Big{\\{}N_{0},\nu_{p},\dot{\nu}_{p},P_{b},e,i,\omega,\sigma,M,(Q/M)^{2}\Big{\\}}\,.$
(4.4)

As the dependence on $Q$ only appears in the quadratic form, we choose
$(Q/M)^{2}$ as the model parameter. Here we treat the coupling constant $\xi$
as a given parameter that describes the gravity model and do not estimate it
simultaneously. However, one would have the probability to constrain $\xi$ and
$Q$ simultaneously with high enough timing precision so that the next to
leading order effects can be observed. We assume the pulsar spin frequency
$\nu_{p}=2\,{\rm Hz}$, corresponding to a normal pulsar, and take the timing
precision $\sigma_{\rm TOA}$ to be 1 ms in our simulations, which is
achievable for future observations [6, 10]. The total observation time is
assumed to be 5 years, and TOAs are extracted weekly and uniformly in time,
which gives $N_{\rm TOA}=260$.

Figure 4:  Constraints on $Q/M$ as functions of $\xi$. We compare the results
of a simple estimation and the two timing models (see text for more details).
The pulsar orbital parameters are given in Eq. (3.8).

We have constructed a fully numerical timing model and a semi-analytic timing
model based on the leading order approximation in Sec. 3. A comparison of the
estimation results of the two timing models are shown in Fig. 4. For a pulsar
orbit with orbital parameters shown in Eq. (3.8), we present the constraint on
$Q/M$ as a function of $\xi$ obtained with the two timing models. In the
simulation, we assume that GR is correct, which means that the true value of
$Q$ is set to be zero. The constraint on $Q/M$ should be understood as an
upper limit corresponding to the square root of the uncertainty of
$(Q/M)^{2}$. Intuitively, one can estimate upper limits on $Q/M$ with $|k^{\rm
Q}|\lesssim\delta k$, where $\delta k$ represents the measurement precision of
the periastron advance. For the assumed pulsar-Sgr A* system, $\delta k/k$ can
be measured to a precision of $\sim 10^{-7}$ [6]. However, this kind of simple
estimation is in general too optimistic because the degeneracy between the
periastron advance caused by the BH mass $M$ and the bumblebee charge $Q$
leads to larger uncertainties for $M$ and $Q$ when doing a fully global
determination of them. Nevertheless, we still plot this simple estimation in
Fig. 4 as a rough estimate. From this figure, one can see that the constraints
from the leading order model are slightly larger than the constraints from the
numerical model, with a factor of about 1.5. This is reasonable as we have
ignored all higher order effects of $Q/M$ in the leading order timing formula.
Therefore, in the following discussion we only present the estimation results
obtained from the leading order timing model. One may expect that the results
are similar if the fully numerical model were adopted. One also notices that
the constraints on $Q/M$ are weak when $\xi$ is around $2\kappa$ as the BH
metric reduces to the Schwarzchild solution when $\xi=2\kappa$ [31].

Figure 5:  Constraints on $Q/M$ as functions of the pulsar orbital period
$P_{b}$ for $\xi=\kappa$, $\xi=0$ and $\xi=-\kappa$. Other orbital parameters
are shown in Eq. (3.8). Figure 6:  Constraints on $Q/M$ as functions of the
pulsar orbital eccentricity $e$ for $\xi=\kappa$, $\xi=0$ and $\xi=-\kappa$.
Other orbital parameters are shown in Eq. (3.8).

In order to investigate more pulsar-Sgr A* systems than the prototype system
in Fig. 4, we vary the orbital parameters. Figure 5 shows the constraints on
$Q/M$ as functions of the pulsar orbital period $P_{b}$ for $\xi=\kappa$,
$\xi=0$ and $\xi=-\kappa$. We can see an approximate symmetry with respect to
$\xi=2\kappa$ from the leading order metric solution. Therefore, with the
constraints on $Q/M$ being almost symmetric, we only show the $\xi<2\kappa$
cases. The upper limits for $Q/M$ decrease when $P_{b}$ becomes smaller as
both the secular and periodic effects become larger when the pulsar gets
closer to the SMBH. We also investigate the influence of the orbital
eccentricity and the results are shown in Fig. 6. Compared to the orbital
period, a higher orbital eccentricity is relatively more important for
constraining the bumblebee charge. This is because that the largest effect
caused by the bumblebee charge is the additional periastron advance, which
becomes degenerate with the orbital period when the orbital eccentricity is
small. For a pulsar with an orbital period of 0.5 yr and an orbital
eccentricity of 0.8, one can expect to constrain $Q/M$ to
$\sim\mathcal{O}(10^{-3}\mbox{--}10^{-2})$ when $|\xi/2\kappa-1|\gtrsim 1$.

##  5 Discussion

We have explored pulsar-SMBH systems in the bumblebee gravity and discussed
possible constraints on the bumblebee charge of Sgr A* by timing a pulsar
orbiting around it. We first constructed a timing model based on the fully
numerical calculations of the BH metric, light propagation, and pulsar orbital
motion in the bumblebee gravity. Considering the future timing precision for
such GC pulsars and current constraints on the bumblebee charge of Sgr A* [31,
33], we also constructed a timing model only including the leading order
effects based on the expansion solution of the metric that is more efficient
and flexible than the fully numerical approach. Our simulation shows that for
a pulsar with an orbital period $P_{b}\sim 0.5\,{\rm yr}$, an orbital
eccentricity $e\sim 0.8$, a 5-yr observation with weekly recorded TOAs, and a
timing precision of 1 ms, one can constrain the bumblebee charge of Sgr A* to
be $\sim\mathcal{O}(10^{-3}\mbox{--}10^{-2})$ for $|\xi/2\kappa-1|\gtrsim 1$,
which is much better than the current constraints from the EHT observation
[33], and at the same order of magnitude as the expected constraint from the
extreme-mass-ratio inspiral observation with the Laser Interferometer Space
Antenna (LISA) [34].

The limitation of the timing model built here is that we have only considered
the static spherical BHs in the bumblebee gravity. For spinning BHs, the spin-
orbit coupling and the quadruple moment of the BH both have detectable effects
in the pulsar timing observation [48, 6, 8, 16]. Also, the complex environment
around Sgr A* perturbs the system [52, 16]. The periastron advance caused by
these effects is degenerate with that caused by the bumblebee charge, which
brings larger uncertainties in the measurement even though one may separate
these effects by considering their periodic effects or combining other
observations. Nevertheless, spinning BHs with bumblebee charges might
introduce additional features in TOAs that can be used to break the
degeneracy. To make a more reliable estimation, one should extend the timing
model to include, for example, the spin of Sgr A*. Although complete solutions
of arbitrarily rotating BHs in the bumblebee gravity have not been obtained
yet, it is still possible to extend the leading order timing model by adding
the spin-orbit coupling term in the equations of motion using some specific
slowly-rotating BH solutions obtained by Ding and Chen [53]. We leave these
kinds of extension for future studies.

##  Appendix A An analytic solution in orders of $Q/M$

To obtain a solution of Eqs. (2.2–2.4) arranged in the order of $Q/M$, one can
start with the expansion

$\displaystyle b_{t}$ $\displaystyle=$ $\displaystyle
b_{t0}\epsilon+b_{t1}\epsilon^{3}+\mathcal{O}\left(\epsilon^{5}\right)\,,$
(A.1) $\displaystyle g_{tt}$ $\displaystyle=$ $\displaystyle
g_{tt0}+g_{tt1}\epsilon^{2}+g_{tt2}\epsilon^{4}+\mathcal{O}\left(\epsilon^{6}\right)\,,$
(A.2) $\displaystyle g_{rr}$ $\displaystyle=$ $\displaystyle
g_{rr0}+g_{rr1}\epsilon^{2}+g_{rr2}\epsilon^{4}+\mathcal{O}\left(\epsilon^{6}\right)\,,$
(A.3) $\displaystyle r_{h}$ $\displaystyle=$ $\displaystyle
r_{h0}+r_{h1}\epsilon^{2}+r_{h2}\epsilon^{4}+\mathcal{O}\left(\epsilon^{6}\right)\,,$
(A.4)

where $b_{t0},b_{t1},\cdots$ are functions of $r$ and $\epsilon=Q/M$ is a
notation to denote the different orders of $Q/M$. Substituting these equations
into the field equations, one can obtain the equations for each order. The
expansion for the radius of event horizon $r_{h}$ is used when applying the
boundary conditions.

For a BH with mass $M$ and bumblebee charge $Q$, we choose the zeroth order
solution to be a Schwarzchild solution with mass $M$, i.e.,

$\displaystyle g_{tt0}$ $\displaystyle=-1+\frac{2M}{r}\,,$ (A.5)
$\displaystyle g_{rr0}$ $\displaystyle=\left(1-\frac{2M}{r}\right)^{-1}\,,$
(A.6) $\displaystyle r_{h0}$ $\displaystyle=2M\,.$ (A.7)

Different choices of the zeroth order solution can change the higher order
terms but the entire series refer to the same exact solution.

From Eq. (2.2), we obtain the ordinary differential equation for $b_{t0}$,
which reads as

$b_{t0}^{\prime\prime}+\frac{2}{r}b_{t0}^{\prime}=0\,.$ (A.8)

The general solution of this equation, $b_{t0}=c_{1}+c_{2}/r$, has two
integration constants, which can be determined by the two boundary conditions

$\displaystyle b_{t}$ $\displaystyle=0\,,$ $\displaystyle\mbox{when}\quad
r=r_{h}$ (A.9) $\displaystyle-r^{2}b_{t}^{\prime}$
$\displaystyle\rightarrow\sqrt{2/\kappa}\,Q\,,$ $\displaystyle\mbox{when}\quad
r\rightarrow\infty$ (A.10)

Note that the boundary conditions should also be satisfied order by order.
Thus one can obtain

$b_{t0}=-\frac{1}{\sqrt{2\kappa}}\left(1-\frac{2M}{r}\right)\,.$ (A.11)

Then from Eqs. (2.3–2.4) one can derive the ordinary differential equation for
$g_{tt1}$, which reads as

$g_{tt1}^{\prime\prime}+\frac{2}{r}g_{tt1}^{\prime}+\frac{2M^{2}}{r^{4}}\left(1-\frac{\xi}{2\kappa}\right)=0\,.$
(A.12)

The general solution is

$g_{tt1}=c_{1}+\frac{c_{2}}{r}-\frac{M^{2}}{r^{2}}\left(1-\frac{\xi}{2\kappa}\right)\,.$
(A.13)

The boundary conditions for $g_{tt1}$ come from the requirements that when
$r\rightarrow\infty$, $g_{tt}\rightarrow-1$ and $r^{2}\nu^{\prime}\rightarrow
M$. So one finds $c_{1}=c_{2}=0$. After working out $g_{tt1}$, $g_{rr1}$ and
$r_{h1}$ can be determined directly from Eq. (2.2) together with
$g_{tt}(r_{h})=0$ respectively.

Repeating the above procedure, one can obtain the solution for a BH with mass
$M$ and bumblebee charge $Q$ in orders of $Q/M$. The result up to the order of
$(Q/M)^{6}$ is listed here

$\displaystyle b_{t}$
$\displaystyle=-\frac{1}{\sqrt{2\kappa}}\left(1-\frac{2M}{r}\right)\frac{Q}{M}$
$\displaystyle\ \ \ \
-\frac{1}{4}\frac{1}{\sqrt{2\kappa}}\left(1-\frac{\xi}{2\kappa}\right)\left[\left(1-\frac{\xi}{2\kappa}\right)+\frac{4M^{2}}{r^{2}}\frac{\xi}{2\kappa}\right]\left(\frac{Q}{M}\right)^{3}+\mathcal{O}\left(\frac{Q}{M}\right)^{5}\,,$
(A.14) $\displaystyle g_{tt}$
$\displaystyle=-1+\frac{2M}{r}-\left(1-\frac{\xi}{2\kappa}\right)\frac{Q^{2}}{r^{2}}$
$\displaystyle\ \ \ \
-\frac{\xi}{2\kappa}\left(1-\frac{\xi}{2\kappa}\right)^{2}\left(1-\frac{M}{r}\right)\frac{M^{2}}{r^{2}}\left(\frac{Q}{M}\right)^{4}+\mathcal{O}\left(\frac{Q}{M}\right)^{6}\,,$
(A.15) $\displaystyle g_{rr}$
$\displaystyle=\left[1-\frac{2M}{r}+\left(1-\frac{\xi}{2\kappa}\right)\frac{Q^{2}}{r^{2}}\right.$
$\displaystyle\ \ \ \
+\left.\frac{\xi}{2\kappa}\left(1-\frac{\xi}{2\kappa}\right)^{2}\left(2-\frac{3M}{r}\right)\frac{M^{2}}{r^{2}}\left(\frac{Q}{M}\right)^{4}\right]^{-1}+\mathcal{O}\left(\frac{Q}{M}\right)^{6}\,,$
(A.16) $\displaystyle\frac{r_{h}}{M}$
$\displaystyle=1+\sqrt{1-\left(1-\frac{\xi}{2\kappa}\right)\left(\frac{Q}{M}\right)^{2}}-\frac{1}{4}\frac{\xi}{2\kappa}\left(1-\frac{\xi}{2\kappa}\right)^{2}\left(\frac{Q}{M}\right)^{4}+\mathcal{O}\left(\frac{Q}{M}\right)^{6}\,.$
(A.17)

The forms of $g_{rr}$ and $r_{h}$ are chosen such that the RN solution and the
Schwarzschild metric can be recovered by the leading order terms at $\xi=0$
and $\xi=2\kappa$ respectively.

As discussed in Sec. 2, the charged BHs in the bumblebee gravity are
indistinguishable with RN BHs in GR at the leading order of $Q/M$. However,
this degeneracy can be broken by considering higher order terms as the
combination of $\xi$ and $Q$ changes in the next-to-leading order
contributions.

## Acknowledgments

This work was supported by the National SKA Program of China (2020SKA0120300),
the National Natural Science Foundation of China (11975027, 11991053,
12247128, 11721303), the China Postdoctoral Science Foundation (2021TQ0018),
the Max Planck Partner Group Program funded by the Max Planck Society, and the
High-Performance Computing Platform of Peking University.

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uses the input submitted to the Strategy process to map the landscape of Higgs
physics at future colliders.

  * •

The “convenors” in the PPG who are connected to this specific topic (Beate
Heinemann and Keith Ellis) and the ECFA chair will be included as ex-officio
observers.

##  Appendix B Theoretical Cross Sections and Partial Width Uncertainties

Table 18:  Cross sections for the main production channels expected for Higgs
boson production at the different types of colliders (as defined in Table 1).

|  Cross Section $\sigma$ [pb] |
---|---|---
$pp$ collider | Total | $ggH$ | VBF | $WH$ | $ZH$ | $t\bar{t}H$ | $tH$ | $ggHH$
LHC (13 TeV) | 56 | 48.6 | 3.77 | 1.36 | 0.88 | 0.510 | 0.074 | 0.031
HL-LHC | 62 | 54.7 | 4.26 | 1.50 | 0.99 | 0.613 | 0.090 | 0.037
HE-LHC | 168 | 147 | 11.8 | 3.40 | 2.47 | 2.86 | 0.418 | 0.140
FCChh | 936 | 802 | 69 | 15.7 | 11.4 | 32.1 | 4.70 | 1.22

|  Cross Section $\sigma$ [fb]
---|---
$e^{+}e^{-}$ collider ($\mathcal{P}_{e^{-}}/\mathcal{P}_{e^{+}}$) | Total | VBF | $ZH$ | $t\bar{t}H$ | $ZHH$
|  | CC/NC |  |  | (CC VBF)
CEPC | 199 | 6.19/0.28 | 192.6 |  |
FCC-ee | 199 | 6.19/0.28 | 192.6 |  |
ILC250 (-80/30) | 313 | 15.4/0.70 | 297 |  |
ILC500 (-80/30) | 262 | 158/7.8 | 96 | 0.41 | 0.2
CLIC380 (0/0) | 160 | 40/7.4 | 113 | – | 0.029 (0.0020)
CLIC1500(0/0) | 329 | 290/30 | 7.5 | 1.3 | 0.082 (0.207)
CLIC3000(0/0) | 532 | 480/49 | 2 | 0.48 | 0.037 (0.77)
CLIC380 (-80/0) | 209 | 68/8.7 | 133 | – | 0.034 (0.0024)
CLIC1500(-80/0) | 574 | 528/35 | 8.8 | 1.70 | 0.97 (0.37)
CLIC3000(-80/0) | 921 | 860/57 | 2.4 | 0.61 | 0.043 (1.38)
CLIC380 (+80/0) | 112 | 13/6.0 | 93 | – | 0.024 (0.0016)
CLIC1500(+80/0) | 91 | 59/24 | 6.2 | 0.89 | 0.068 (0.045)
CLIC3000(+80/0) | 138 | 96/40 | 1.7 | 0.34 | 0.30 (1.56)

|  Cross Section $\sigma$ [fb] |
---|---|---
$e^{-}p$ collider ($\mathcal{P}_{e^{-}}$) | Total | VBF | $tH$ | $HH$
|  | CC/NC |  | (CC VBF)
LHeC (0) | 130 | 110/20 | 0.07 | 0.01
HE-LHeC (0) | 247 | 206/41 | 0.37 | 0.04
FCCeh (0) | 674 | 547/127 | 4.2 | 0.26
LHeC (-80) | 221 | 197/24 | 0.12 | 0.02
HE-LHeC (-80) | 420 | 372/48 | 0.67 | 0.07
FCCeh (-80) | 1189 | 1040/149 | 7.6 | 0.47

Table 19:  Partial decay widths for the Higgs boson to specific final states and the uncertainties in their calculation [97]. The uncertainties arise either from intrinsic limitations in the theoretical calculation (ThIntr) and parametric uncertainties (ThPar). The parametric uncertainties are due to the finite precision on the quark masses, ThPar($m_{q}$), on the strong coupling constant, ThPar($\alpha_{s}$), and on the Higgs boson mass, ThPar($M_{H}$). The columns labelled "partial width" and "current uncertainty" and refer to the current precision [97], while the predictions for the future are taken from ref. [131]. For the future uncertainties, the parametric uncertainties assume a precision of $\delta m_{b}=13$ MeV, $\delta m_{c}=7$ MeV, $\delta m_{t}=50$ MeV, $\delta\alpha_{s}=0.0002$ and $\delta M_{H}=10$ MeV.  Decay | Partial width | current unc. $\Delta\Gamma/\Gamma$ [%] | future unc. $\Delta\Gamma/\Gamma$ [%]
---|---|---|---
| [keV] | ThIntr | ThPar($m_{q}$) | ThPar($\alpha_{s}$) | ThPar($m_{\mathrm{H}}\,$) | ThIntr | ThPar($m_{q}$) | ThPar($\alpha_{s}$) | ThPar($m_{\mathrm{H}}\,$)
$H\to b\bar{b}$ | 2379 | $<0.4$ | $1.4$ | $0.4$ | $-$ | $0.2$ | $0.6$ | $<0.1$ | $-$
$H\to\tau^{+}\tau^{-}$ | 256 | $<0.3$ | $-$ | $-$ | $-$ | $<0.1$ | $-$ | $-$ | $-$
$H\to c\bar{c}$ | 118 | $<0.4$ | $4.0$ | $0.4$ | $-$ | $0.2$ | $1.0$ | $<0.1$ | $-$
$H\to\mu^{+}\mu^{-}$ | 0.89 | $<0.3$ | $-$ | $-$ | $-$ | $<0.1$ | $-$ | $-$ | $-$
$H\to W^{+}W^{-}$ | 883 | $0.5$ | $-$ | $-$ | $2.6$ | $0.4$ | $-$ | $-$ | $0.1$
$H\to gg$ | 335 | $3.2$ | $<0.2$ | $3.7$ | $-$ | $1.0$ | $-$ | $0.5$ | $-$
$H\to ZZ$ | 108 | $0.5$ | $-$ | $-$ | $3.0$ | $0.3$ | $-$ | $-$ | $0.1$
$H\to\gamma\gamma$ | 9.3 | $<1.0$ | $<0.2$ | $-$ | $-$ | $<1.0$ | $-$ | $-$ | $-$
$H\to Z\gamma$ | 6.3 | $5.0$ | $-$ | $-$ | 2.1 | $1.0$ | $-$ | $-$ | $0.1$

##  Appendix C Inputs

In this section we report some information relative to the inputs to the
strategy process. Figure 13 shows the start date and extent of the runs of
proposed future projects, using the earliest start time provided in the
submitted documentation.

Figure 13:  Sketch of timeline of various collider projects starting at the
"earliest start time" stated in the respective documents. For FCC-eh/hh this
figure assumes that it is not preceeded by FCC-ee. If it comes after FCC-ee it
would start in the early 2060s. Only scenarios as submitted to the European
Strategy by Dec. 2018 are displayed. Additional scenarios considered in this
report (e.g. Giga-Z at ILC/CLIC, ILC at 1 TeV, LE-FCC) are not shown.

###  C.1 Inputs for Higgs studies

The uncertainties on inputs for all the colliders used in our analysis are
listed in Tables 20-26. In all cases the relative uncertainty on the
measurement is given corresponding to a Gaussian $1$-$\sigma$ uncertainty.

Table 20: Inputs used for CEPC and FCC-ee projections. All uncertainties are given as fractional 68% CL intervals and are taken to be symmetric. The upper limits are given at 68% CL. A dash indicates the absence of a projection for the corresponding channel. | FCC-ee240 | FCC-ee365 | CEPC
---|---|---|---
$\delta\sigma_{ZH}$ | 0.005 | 0.009 | 0.005
$\delta\mu_{ZH,bb}$ | 0.003 | 0.005 | 0.0031
$\delta\mu_{ZH,cc}$ | 0.022 | 0.065 | 0.033
$\delta\mu_{ZH,gg}$ | 0.019 | 0.035 | 0.013
$\delta\mu_{ZH,WW}$ | 0.012 | 0.026 | 0.0098
$\delta\mu_{ZH,ZZ}$ | 0.044 | 0.12 | 0.051
$\delta\mu_{ZH,\tau\tau}$ | 0.009 | 0.018 | 0.0082
$\delta\mu_{ZH,\gamma\gamma}$ | 0.09 | 0.18 | 0.068
$\delta\mu_{ZH,\mu\mu}$ | 0.19 | 0.40 | 0.17
$\delta\mu_{ZH,Z\gamma}$ | $-$ | $-$ | 0.16
$\delta\mu_{\nu\nu H,bb}$ | 0.031 | 0.009 | 0.030
$\delta\mu_{\nu\nu H,cc}$ | $-$ | 0.10 | $-$
$\delta\mu_{\nu\nu H,gg}$ | $-$ | 0.045 | $-$
$\delta\mu_{\nu\nu H,ZZ}$ | $-$ | 0.10 | $-$
$\delta\mu_{\nu\nu H,\tau\tau}$ | $-$ | 0.08 | $-$
$\delta\mu_{\nu\nu H,\gamma\gamma}$ | $-$ | 0.22 | $-$
$\rm{BR}_{\rm inv}$ |  $<$0.0015 |  $<$0.003 |  $<$0.0015
Table 21: Inputs used for ILC projections at the 250 and 350 GeV energy stages
and two polarisations. All uncertainties are given as fractional 68% CL
intervals and are taken to be symmetric. The upper limits are given at 68% CL.
ILC250
---
Polarization: |  $e^{-}$: -80% $e^{+}$: +30% |  $e^{-}$: +80% $e^{+}$: -30%
$\delta\sigma_{ZH}/\sigma_{ZH}$ | 0.011 | 0.011
$\delta\mu_{ZH,bb}$ | 0.0072 | 0.0072
$\delta\mu_{ZH,cc}$ | 0.044 | 0.044
$\delta\mu_{ZH,gg}$ | 0.037 | 0.037
$\delta\mu_{ZH,ZZ}$ | 0.095 | 0.095
$\delta\mu_{ZH,WW}$ | 0.024 | 0.024
$\delta\mu_{ZH,\tau\tau}$ | 0.017 | 0.017
$\delta\mu_{ZH,\gamma\gamma}$ | 0.18 | 0.18
$\delta\mu_{ZH,\mu\mu}$ | 0.38 | 0.38
$\delta\mu_{\nu\nu H,bb}$ | 0.043 | 0.17
$\rm{BR}_{\rm inv}$ |  $<$0.0027 |  $<$0.0021
|  |
ILC350
Polarization: |  $e^{-}$: -80% $e^{+}$: +30% |  $e^{-}$: +80% $e^{+}$: -30%
$\delta\sigma_{ZH}/\sigma_{ZH}$ | 0.025 | 0.042
$\delta\mu_{ZH,bb}$ | 0.021 | 0.036
$\delta\mu_{ZH,cc}$ | 0.15 | 0.26
$\delta\mu_{ZH,gg}$ | 0.11 | 0.20
$\delta\mu_{ZH,ZZ}$ | 0.34 | 0.59
$\delta\mu_{ZH,WW}$ | 0.076 | 0.13
$\delta\mu_{ZH,\tau\tau}$ | 0.054 | 0.094
$\delta\mu_{ZH,\gamma\gamma}$ | 0.53 | 0.92
$\delta\mu_{ZH,\mu\mu}$ | 1.2 | 2.1
$\delta\mu_{\nu\nu H,bb}$ | 0.025 | 0.18
$\delta\mu_{\nu\nu H,cc}$ | 0.26 | 1.9
$\delta\mu_{\nu\nu H,gg}$ | 0.10 | 0.75
$\delta\mu_{\nu\nu H,ZZ}$ | 0.27 | 1.9
$\delta\mu_{\nu\nu H,WW}$ | 0.078 | 0.57
$\delta\mu_{\nu\nu H,\tau\tau}$ | 0.22 | 1.6
$\delta\mu_{\nu\nu H,\gamma\gamma}$ | 0.61 | 4.2
$\delta\mu_{\nu\nu H,\mu\mu}$ | 2.2 | 16
$\rm{BR}_{\rm inv}$ |  $<$0.0096 |  $<$0.015
Table 22: Inputs used for ILC projections at the 500 and 1000 GeV energy
stages and two polarisations. All uncertainties are given as fractional 68% CL
intervals and are taken to be symmetric. The upper limits are given at 68% CL.
ILC500
---
Polarization: |  $e^{-}$: -80% $e^{+}$: +30% |  $e^{-}$: +80% $e^{+}$: -30%
$\delta\sigma_{ZH}/\sigma_{ZH}$ | 0.017 | 0.017
$\delta\mu_{ZH,bb}$ | 0.010 | 0.010
$\delta\mu_{ZH,cc}$ | 0.071 | 0.071
$\delta\mu_{ZH,gg}$ | 0.059 | 0.059
$\delta\mu_{ZH,ZZ}$ | 0.14 | 0.14
$\delta\mu_{ZH,WW}$ | 0.030 | 0.030
$\delta\mu_{ZH,\tau\tau}$ | 0.024 | 0.024
$\delta\mu_{ZH,\gamma\gamma}$ | 0.19 | 0.19
$\delta\mu_{ZH,\mu\mu}$ | 0.47 | 0.47
$\delta\mu_{\nu\nu H,bb}$ | 0.0041 | 0.015
$\delta\mu_{\nu\nu H,cc}$ | 0.035 | 0.14
$\delta\mu_{\nu\nu H,gg}$ | 0.023 | 0.095
$\delta\mu_{\nu\nu H,ZZ}$ | 0.047 | 0.19
$\delta\mu_{\nu\nu H,WW}$ | 0.014 | 0.055
$\delta\mu_{\nu\nu H,\tau\tau}$ | 0.039 | 0.16
$\delta\mu_{\nu\nu H,\gamma\gamma}$ | 0.11 | 0.43
$\delta\mu_{\nu\nu H,\mu\mu}$ | 0.4 | 1.7
$\delta\mu_{ttH,bb}$ | 0.20 | 0.20
$\rm{BR}_{\rm inv}$ |  $<$0.0069 |  $<$0.0050
Direct constraint on Higgs self-interaction |  |
$\delta\kappa_{3}$ | 0.27
|  |
ILC1000
Polarization: |  $e^{-}$: -80% $e^{+}$: +20% |  $e^{-}$: +80% $e^{+}$: -20%
$\delta\mu_{\nu\nu H,bb}$ | 0.0032 | 0.010
$\delta\mu_{\nu\nu H,cc}$ | 0.017 | 0.064
$\delta\mu_{\nu\nu H,gg}$ | 0.013 | 0.047
$\delta\mu_{\nu\nu H,ZZ}$ | 0.023 | 0.084
$\delta\mu_{\nu\nu H,WW}$ | 0.0091 | 0.033
$\delta\mu_{\nu\nu H,\tau\tau}$ | 0.017 | 0.064
$\delta\mu_{\nu\nu H,\gamma\gamma}$ | 0.048 | 0.17
$\delta\mu_{\nu\nu H,\mu\mu}$ | 0.17 | 0.64
$\delta\mu_{ttH,bb}$ | 0.045 | 0.045
Direct constraint on Higgs self-interaction |  |
$\delta\kappa_{3}$ | 0.10
Table 23: Inputs used for CLIC projections at the three energy stages and two
polarisations. All uncertainties are given as fractional 68% CL intervals and
are taken to be symmetric. The upper limits are given at 68% CL. CLIC380
---
Polarization: |  $e^{-}$: -80% $e^{+}$: 0% |  $e^{-}$: +80% $e^{+}$: 0%
$\delta\sigma_{ZH,Z\to ll}/\sigma_{ZH,Z\to ll}$ | 0.036 | 0.041
$\delta\sigma_{ZH,Z\to qq}/\sigma_{ZH,Z\to qq}$ | 0.017 | 0.020
$\delta\mu_{ZH,bb}$ | 0.0081 | 0.0092
$\delta\mu_{ZH,cc}$ | 0.13 | 0.15
$\delta\mu_{ZH,gg}$ | 0.057 | 0.065
$\delta\mu_{ZH,WW}$ | 0.051 | 0.057
$\delta\mu_{ZH,\tau\tau}$ | 0.059 | 0.066
$\delta\mu_{\nu\nu H,bb}$ | 0.014 | 0.041
$\delta\mu_{\nu\nu H,cc}$ | 0.19 | 0.57
$\delta\mu_{\nu\nu H,gg}$ | 0.076 | 0.23
$\rm{BR}_{\rm inv}$ |  $<$0.0027 |  $<$0.003
|  |
CLIC1500
Polarization: |  $e^{-}$: -80% $e^{+}$: 0% |  $e^{-}$: +80% $e^{+}$: 0%
$\delta\mu_{ZH,bb}$ | 0.028 | 0.062
$\delta\mu_{\nu\nu H,bb}$ | 0.0025 | 0.015
$\delta\mu_{\nu\nu H,cc}$ | 0.039 | 0.24
$\delta\mu_{\nu\nu H,gg}$ | 0.033 | 0.20
$\delta\mu_{\nu\nu H,WW}$ | 0.0067 | 0.04
$\delta\mu_{\nu\nu H,ZZ}$ | 0.036 | 0.22
$\delta\mu_{\nu\nu H,\gamma\gamma}$ | 0.1 | 0.6
$\delta\mu_{\nu\nu H,Z\gamma}$ | 0.28 | 1.7
$\delta\mu_{\nu\nu H,\tau\tau}$ | 0.028 | 0.17
$\delta\mu_{\nu\nu H,\mu\mu}$ | 0.24 | 1.5
$\delta\mu_{eeH,bb}$ | 0.015 | 0.033
$\delta\mu_{ttH,bb}$ | 0.056 | 0.15
|  |
CLIC3000
Polarization: |  $e^{-}$: -80% $e^{+}$: 0% |  $e^{-}$: +80% $e^{+}$: 0%
$\delta\mu_{ZH,bb}$ | 0.045 | 0.10
$\delta\mu_{\nu\nu H,bb}$ | 0.0017 | 0.01
$\delta\mu_{\nu\nu H,cc}$ | 0.037 | 0.22
$\delta\mu_{\nu\nu H,gg}$ | 0.023 | 0.14
$\delta\mu_{\nu\nu H,WW}$ | 0.0033 | 0.02
$\delta\mu_{\nu\nu H,ZZ}$ | 0.021 | 0.13
$\delta\mu_{\nu\nu H,\gamma\gamma}$ | 0.05 | 0.3
$\delta\mu_{\nu\nu H,Z\gamma}$ | 0.16 | 0.95
$\delta\mu_{\nu\nu H,\tau\tau}$ | 0.023 | 0.14
$\delta\mu_{\nu\nu H,\mu\mu}$ | 0.13 | 0.8
$\delta\mu_{eeH,bb}$ | 0.016 | 0.036
Direct constraint on Higgs self-interaction |  |
$\delta\kappa_{3}$ | 0.11
Table 24: Inputs used for LHeC and FCC-eh projections. All uncertainties are given as fractional 68% CL intervals and are taken to be symmetric. Observable | LHeC | FCC-eh
---|---|---
$\delta\mu_{WBF,bb}$ | 0.008 | 0.0025
$\delta\mu_{WBF,cc}$ | 0.071 | 0.022
$\delta\mu_{WBF,gg}$ | 0.058 | 0.018
$\delta\mu_{ZBF,bb}$ | 0.023 | 0.0065
$\delta\mu_{WBF,WW}$ | 0.062 | 0.019
$\delta\mu_{WBF,ZZ}$ | 0.120 | 0.038
$\delta\mu_{WBF,\tau\tau}$ | 0.052 | 0.016
$\delta\mu_{WBF,\gamma\gamma}$ | 0.15 | 0.046
$\delta\mu_{ZBF,cc}$ | 0.200 | 0.058
$\delta\mu_{ZBF,gg}$ | 0.160 | 0.047
$\delta\mu_{ZBF,WW}$ | 0.170 | 0.050
$\delta\mu_{ZBF,ZZ}$ | 0.350 | 0.100
$\delta\mu_{ZBF,\tau\tau}$ | 0.15 | 0.042
$\delta\mu_{ZBF,\gamma\gamma}$ | 0.42 | 0.120
Table 25: Left) Inputs used for FCC-hh. All uncertainties are given as fractional 68% CL intervals and are taken to be symmetric. Right) Extra inputs used in the $\kappa$ fit studies.  | FCC-hh
---
$\delta\mu_{ggF,4\mu}$ | 0.019 |
$\delta\mu_{ggF,\gamma\gamma}$ | 0.015 |
$\delta\mu_{ggF,Z\gamma}$ | 0.016 |
$\delta\mu_{ggF,\mu\mu}$ | 0.012 |
$\delta(\rm{BR}_{\mu\mu}/\rm{BR}_{4\mu})$ | 0.013 |
$\delta(\rm{BR}_{\gamma\gamma}/\rm{BR}_{2e2\mu})$ | 0.008 |
$\delta(\rm{BR}_{\gamma\gamma}/\rm{BR}_{\mu\mu})$ | 0.014 |
$\delta(\rm{BR}_{\mu\mu\gamma}/\rm{BR}_{\gamma\gamma})$ | 0.018 |
$\delta(\sigma_{ttH}^{bb}/\sigma_{ttZ}^{bb})$ | 0.019 |
Invisible decays |  |
$\rm{BR}_{\rm inv}$ |  $<$0.00013 |
Direct constraint on Higgs self-interaction |  |
$\delta\kappa_{3}$ | 0.05 |
| FCC-hh
---
(Extra inputs used in $\kappa$ fits)
$\delta(\sigma_{WH}^{H\to\gamma\gamma}/\sigma_{WZ}^{Z\to e^{+}e^{-}})$ | 0.014
$\delta(\sigma_{WH}^{H\to\tau\tau}/\sigma_{WZ}^{Z\to\tau\tau})$ | 0.016
$\delta(\sigma_{WH}^{H\to bb}/\sigma_{WZ}^{Z\to bb})$ | 0.011
$\delta(\sigma_{WH}^{H\to WW}/\sigma_{WH}^{H\to\gamma\gamma})$ | 0.015
Table 26: Inputs used for a low-energy FCC-hh running at 37.5 TeV (LE-FCC).
All uncertainties are given as fractional 68% CL intervals and are taken to be
symmetric.  LE-FCC
---
$\delta(\rm{BR}_{\mu\mu}/\rm{BR}_{4\mu})$ | 0.029 |
$\delta(\rm{BR}_{\gamma\gamma}/\rm{BR}_{2e2\mu})$ | 0.015 |
$\delta(\rm{BR}_{\gamma\gamma}/\rm{BR}_{\mu\mu})$ | 0.028 |
$\delta(\rm{BR}_{\mu\mu\gamma}/\rm{BR}_{\gamma\gamma})$ | 0.06 |
$\delta(\sigma_{ttH}^{bb}/\sigma_{ttZ}^{bb})$ | 0.04-0.06 |
Direct constraint on Higgs self-interaction |  |
$\delta\kappa_{3}$ | 0.15 |

###  C.2 Inputs for electroweak precision observables

The uncertainties on several electroweak precision observables related to the
properties (masses and couplings) of the electroweak vector bosons are
presented in Table 27. We also report the expected uncertainties on the top-
quark mass, which enters in the analysis as an input of the global electroweak
fit.

For the extraction of $m_{\textrm{top}}$ from a $t\bar{t}$ scan threshold at
$e^{+}e^{-}$ colliders the current theoretical uncertainty is $\sim 40$ MeV
[132]. As it was done for the other intrinsic theory uncertainties, this is
expected to improve in the future and is neglected in the baseline fits. We
therefore use a common statistical uncertainty of $\sim 20$ MeV for all lepton
colliders running at the $t\bar{t}$ threshold.

For the circular colliders, the asymmetries $A_{f}$ and partial-width ratios
$R_{f}$ are not given for $\sqrt{s}\gg 90$ GeV as the statistical precision is
much higher at the $Z$ pole, and the $Z$-pole run is part of the default
programme. For the linear colliders a Giga-$Z$ run is not part of the running
plan submitted to the EPPSU, but it is used in some of the results presented
in this report to illustrate the impact of such a run in the EW and Higgs
programmes. For the ILC and CLIC Giga-$Z$ option, an integrated luminosity of
100 fb-1 is assumed, and polarisations as stated in Table 1. Note that the
asymmetry parameters $A_{f}$ can be extracted in different ways depending on
the access of polarization of the initial and/or final states. For instance,
at linear colliders with polarized beams $A_{b}$ can be directly extracted
from a left-right forward-backward asymmetry. Without polarized beams,
circular colliders can access that quantify via a forward-backward asymmetry
$A_{FB}^{b}=\frac{3}{4}A_{e}A_{b}$, but it requires to know $A_{e}$. On the
other hand, both circular and linear colliders could access $A_{e}$ and
$A_{\tau}$ separately, measuring the polarization of the final states in
$e^{+}e^{-}\to\tau^{+}\tau^{-}$. We refer to the discussion in Section 3.4.1
for the assumptions adopted in the treatment of systematic uncertainties for
the heavy flavor observables included the fits.

For ILC all values are taken from Ref. [4]. For CLIC all values are taken from
Refs. [11, 5]. For CEPC they are taken from either Ref. [2] or from Ref.
[133]. For FCC-ee they are taken from Refs. [134, 79, 135, 136].

Table 27: Uncertainty on several observables related to the properties of the electroweak vector bosons. We also list the uncertainty on the top mass. For dimensionful quantities the absolute uncertainty is given, while relative errors are listed for dimensionless quantities. A few comments on some particular numbers are in order: a) For hadron colliders the top mass is not the pole mass. b) For the top mass all lepton colliders require a dedicated top threshold scan to achieve the uncertainty given here. (For ILC the quoted value comes from a dedicated run at 350 GeV.) c) From direct reconstruction in the $ZH$ run 2-3 MeV can be achieved [2]. d) In a 4-year dedicated run 2 MeV can be achieved by ILC [137]. e) From $\tau$ polarization measurements. f) At circular colliders, for $A_{b}$ and $A_{c}$ previous measurement uncertainties were dominated by the physics modelling [138] and the systematic uncertainty arising from this was only estimated by FCC-ee [135]. When these systematics are set to zero in the measurements of $A_{FB}^{b}$ and $A_{FB}^{c}$ the uncertainty in $A_{b}$ and $A_{c}$ is controlled by the statistical errors plus the uncertainty on $A_{e}$. This is the setup used for the baseline fits. See discussion in Section 3.4.1 for details. g) $R_{\nu}\equiv\Gamma_{Z\to{\rm inv}}/\Gamma_{Z\to{\rm had}}$ and $R_{\rm inv}\equiv\Gamma_{Z\to{\rm inv}}/\Gamma_{Z\to\ell\ell}$.  Quantity | Current | HL-LHC | FCC-ee | CEPC | ILC | CLIC
---|---|---|---|---|---|---
|  |  |  |  | Giga-Z | 250 GeV | Giga-Z | 380 GeV
$\delta m_{\textrm{top}}$ [MeV] |  $\sim$500 a) |  $\sim$400 a) | 20 b) | $-$ | $-$ | 17 b) | $-$ | 20-22 b)
$\delta M_{Z}$ [MeV] | 2.1 | $-$ | $0.1$ | $0.5$ | $-$ | $-$ | $-$ | $-$
$\delta\Gamma_{Z}$ [MeV] | 2.3 | $-$ | $0.1$ | $0.5$ | $1$ | $-$ | $1$ | $-$
$\delta\Gamma_{Z\to{\rm had}}$ [MeV] | 2.0 | $-$ | $-$ | $-$ | $0.7$ | $-$ | $0.7$ | $-$
$\delta\sigma_{{\rm had}}^{0}$ [pb] | 37 | $-$ | $4$ | $5$ | $-$ | $-$ | $-$ | $-$
$\delta M_{W}$ [MeV] | 12 | 7 | 0.7 | 1.0 (2-3) c) | $-$ | 2.4 d) | $-$ | 2.5
$\delta\Gamma_{W}$ [MeV] | 42 | $-$ | 1.5 | 3 | $-$ | $-$ | $-$ | $-$
$\delta{\rm BR}_{W\to e\nu}$[$10^{-4}$] | 150 | $-$ | 3 | 3 | $-$ | 4.2 | $-$ | 11
$\delta{\rm BR}_{W\to\mu\nu}$[$10^{-4}$] | 140 | $-$ | 3 | 3 | $-$ | 4.1 | $-$ | 11
$\delta{\rm BR}_{W\to\tau\nu}$[$10^{-4}$] | 190 | $-$ | 4 | 4 | $-$ | 5.2 | $-$ | 11
$\delta{\rm BR}_{W\to{\rm had}}$[$10^{-4}$] | 40 | $-$ | 1 | 1 | $-$ | $-$ | $-$ | $-$
$\delta A_{e}$ [$10^{-4}$] | 140 | $-$ | 1.1 e) | 3.2 e) | 5.1 | 10 | 10 | 42
$\delta A_{\mu}$ [$10^{-4}$] | 1060 | $-$ | $-$ | $-$ | 5.4 | 54 | 13 | 270
$\delta A_{\tau}$ [$10^{-4}$] | 300 | $-$ | 3.1 e) | 5.2 e) | 5.4 | 57 | 17 | 370
$\delta A_{b}$ [$10^{-4}$] | 220 | $-$ | $-$ | $-$ | 5.1 | 6.4 | 9.9 | 40
$\delta A_{c}$ [$10^{-4}$] | 400 | $-$ | $-$ | $-$ | 5.8 | 21 | 10 | 30
$\delta A^{\mu}_{\rm FB}$ [$10^{-4}$] | 770 | $-$ | 0.54 | 4.6 | $-$ | $-$ | $-$ | $-$
$\delta A^{b}_{\rm FB}$ [$10^{-4}$] | 160 | $-$ | 30 f) | 10 f) | $-$ | $-$ | $-$ | $-$
$\delta A^{c}_{\rm FB}$ [$10^{-4}$] | 500 | $-$ | 80 f) | 30 f) | $-$ | $-$ | $-$ | $-$
$\delta R_{e}$ [$10^{-4}$] | 24 | $-$ | 3 | 2.4 | 5.4 | 11 | 4.2 | 27
$\delta R_{\mu}$ [$10^{-4}$] | 16 | $-$ | 0.5 | 1 | 2.8 | 11 | 2.2 | 27
$\delta R_{\tau}$ [$10^{-4}$] | 22 | $-$ | 1 | 1.5 | 4.5 | 12 | 4.3 | 60
$\delta R_{b}$ [$10^{-4}$] | 31 | $-$ | 2 | 2 | 7 | 11 | 7 | 18
$\delta R_{c}$ [$10^{-4}$] | 170 | $-$ | 10 | 10 | 30 | 50 | 23 | 56
$\delta R_{\nu}$ [$10^{-3}$] g) | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 9.4
$\delta R_{\rm inv}$ [$10^{-3}$] g) | $-$ | $-$ | 0.27 | 0.5 | $-$ | $-$ | $-$ | $-$

##  Appendix D Correlation matrices

The correlations of three of the lepton collider kappa-3 fits, discussed in
Section 3.2, are shown in Figure 14.

Figure 14:  Correlation seen in the kappa-3 scenario fit for three future
colliders as discussed in section 3.2. Top: ILC250. Center: FCC365. Bottom:
CLIC3000.

##  Appendix E Additional Kappa scenario fits

This appendix contains additional kappa scenarios to complement the results
shown in section 3.2. Tables 28 and 29 present the results of the different
colliders in the kappa-2 scenario, in which BSM decays are allowed and future
colliders are considered independently and not fitted together with the HL-LHC
prospects.

Table 28:  Expected relative precision (%) of the $\kappa$ parameters in the kappa-2 (standalone) scenario described in section 2 for the HL-LHC,LHeC, and HE-LHC. A bound on $|\kappa_{V}|\leq 1$ is applied since no direct access to the Higgs width is possible, thus the uncertainty on $\kappa_{W}$ and $\kappa_{Z}$ is one-sided. For the remaining kappa parameters one standard deviation is provided in $\pm$. The corresponding 95%CL upper limit on $\mathrm{BR}_{\mathrm{inv}}$ is also given. In this scenario $\mathrm{BR}_{\mathrm{unt}}$ is a floating parameter in the fit, to propagate the effect of an assumed uncertain total width on the measurement of the other $\kappa_{i}$. Based on this constraint the reported values on $\mathrm{BR}_{\mathrm{unt}}$ are inferred. Cases in which a particular parameter has been fixed to the SM value due to lack of sensitivity are shown with a dash ($-$). In the case of $\kappa_{t}$ sensitivity at the LHeC, note that the framework relies as input on $\mu_{ttH}$, and does not take into consideration $\mu_{tH}$. A star ($\star$) indicates the cases in which a parameter has been left free in the fit due to lack of input in the reference documentation. The integrated luminosity and running conditions considered for each collider in this comparison are described in Table 1.  kappa-2 scenario | HL-LHC | LHeC | HE-LHC (S2) | HE-LHC (S2’)
---|---|---|---|---
$1\geq\kappa_{W}>$ (68%) | 0.985 | 0.994 | 0.988 | 0.992
$1\geq\kappa_{Z}>$ (68%) | 0.987 | 0.988 | 0.990 | 0.990
$\kappa_{g}$ [$\%$] | $\pm 2.$ | $\pm 3.9$ | $\pm 1.6$ | $\pm 1.1$
$\kappa_{\gamma}$ [$\%$] | $\pm 1.6$ | $\pm 7.8$ | $\pm 1.3$ | $\pm 0.96$
$\kappa_{Z\gamma}$ [$\%$] | $\pm 10.$ | $-$ | $\pm 5.6$ | $\pm 3.8$
$\kappa_{c}$ [$\%$] | $-$ | $\pm 4.3$ | $-$ | $-$
$\kappa_{t}$ [$\%$] | $\pm 3.2$ | $-$ | $\pm 2.6$ | $\pm 1.6$
$\kappa_{b}$ [$\%$] | $\pm 2.5$ | $\pm 2.3$ | $\pm 2.0$ | $\pm 1.5$
$\kappa_{\mu}$ [$\%$] | $\pm 4.4$ | $-$ | $\pm 2.3$ | $\pm 1.5$
$\kappa_{\tau}$ [$\%$] | $\pm 1.6$ | $\pm 3.6$ | $\pm 1.2$ | $\pm 0.85$
$\mathrm{BR}_{\mathrm{inv}}$ ($<$%, 95% CL) | $1.9$ | $2.2$ | $3.2\star$ | $2.4\star$
$\mathrm{BR}_{\mathrm{unt}}$ ($<$%, 95% CL) | inferred using constraint $|\kappa_{V}|\leq 1$
| $4.$ | $2.2$ | $3.2$ | 2.1
Table 29:  Expected relative precision (%) of the $\kappa$ parameters in the kappa-2 (standalone collider) scenario described in section 2 for future accelerators beyond the LHC era. The corresponding 95%CL upper limits on $\mathrm{BR}_{\mathrm{unt}}$ and $\mathrm{BR}_{\mathrm{inv}}$ and the derived constraint on the Higgs width (in %) are also given. Cases in which a particular parameter has been fixed to the SM value due to lack of sensitivity are shown with a dash (-). An asterisk ($\ast$) indicates the cases in which a parameter has been left free in the fit due to lack of input in the reference documentation. The integrated luminosity and running conditions considered for each collider in this comparison are described in Table 1. FCC-ee/eh/hh corresponds to the combined performance of FCC-ee240+FCC-ee365, FCC-eh and FCC-hh. kappa-2 scenario |  ILC250 |  ILC500 |  ILC1000 |  CLIC380 |  CLIC1500 |  CLIC3000 | CEPC |  FCC-ee240 |  FCC-ee365 | FCC-ee/eh/hh
---|---|---|---|---|---|---|---|---|---|---
$\kappa_{W}$ [$\%$] | 1.8 | 0.31 | 0.26 | 0.86 | 0.39 | 0.38 | 1.3 | 1.3 | 0.44 | 0.2
$\kappa_{Z}$ [$\%$] | 0.3 | 0.24 | 0.24 | 0.5 | 0.39 | 0.39 | 0.19 | 0.21 | 0.18 | 0.17
$\kappa_{g}$ [$\%$] | 2.3 | 0.98 | 0.67 | 2.5 | 1.3 | 0.96 | 1.5 | 1.7 | 1.0 | 0.52
$\kappa_{\gamma}$ [$\%$] | 6.8 | 3.5 | 1.9 | 88.$\ast$ | 5. | 2.3 | 3.7 | 4.8 | 3.9 | 0.32
$\kappa_{Z\gamma}$ [$\%$] |  87.$\ast$ |  75.$\ast$ |  74.$\ast$ |  110.$\ast$ | 15. | 7. | 8.2 |  71.$\ast$ |  66.$\ast$ | 0.71
$\kappa_{c}$ [$\%$] | 2.5 | 1.3 | 0.91 | 4.4 | 1.9 | 1.4 | 2.2 | 1.8 | 1.3 | 0.96
$\kappa_{t}$ [$\%$] | - | 6.9 | 1.6 | - | - | 2.7 | - | - | - | 1.0
$\kappa_{b}$ [$\%$] | 1.8 | 0.6 | 0.5 | 1.9 | 0.6 | 0.52 | 1.3 | 1.3 | 0.69 | 0.48
$\kappa_{\mu}$ [$\%$] | 15. | 9.4 | 6.3 |  290.$\ast$ | 13. | 5.9 | 9. | 10. | 8.9 | 0.43
$\kappa_{\tau}$ [%] | 1.9 | 0.72 | 0.58 | 3.1 | 1.3 | 0.95 | 1.4 | 1.4 | 0.74 | 0.49
$\mathrm{BR}_{\mathrm{inv}}$ ($<$%, 95% CL) | 0.26 | 0.23 | 0.23 | 0.65 | 0.65 | 0.65 | 0.28 | 0.22 | 0.19 | 0.024
$\mathrm{BR}_{\mathrm{unt}}$ ($<$%, 95% CL) | 1.8 | 1.4 | 1.3 | 2.7 | 2.4 | 2.4 | 1.1 | 1.2 | 1.1 | 1.0

##  Appendix F Additional Comparisons

In this section additional potential scenarios for accelerators are compared.
The inputs for these were mostly provided after the European Strategy meeting
in Granada.

Table 30 and Fig. 15 show the results of the kappa-0-HL fit for various FCC
scenarios. This fit is a replica of the already described kappa-0 one, which
does not allow any BSM decay of the Higgs, but incorporating the HL-LHC
information in a combined fit for completeness. With 4 instead of 2 IPs the
uncertainties reduce by a factor of up to 1.4 due to the increased statistics.
With the FCC-hh only, the uncertainties all increase by factors of 2-5. When
omitting FCC-eh, the uncertainty on $\kappa_{W}$ increases by a factor of two
and that on $\kappa_{b}$ increases by 20%, the others are mostly unaffected.
When omitting FCC-ee, most uncertainties increase by about 20% and that on
$\kappa_{Z}$ increases by more than a factor of two.

Table 30:  Results of kappa-0-HL fit for various scenarios of the FCC. In all cases the FCC data are combined with HL-LHC. The "4 IP" option considers 4 experiments instead of the 2 experiments considered in the CDR. For the FCC-hh scenario constraints on the $b$, $\tau$ and $W$ couplings come from measurements of ratios of $WH$ to $WZ$ production with the $H$ and $Z$ decaying to $b$-quarks or $\tau$ leptons, see Ref. [139]. kappa-0-HL |  HL+FCC-ee240 | HL+FCC-ee | HL+FCC-ee (4 IP) | HL+FCC-ee/hh | HL+FCC-eh/hh | HL+FCC-hh | HL+FCC-ee/eh/hh
---|---|---|---|---|---|---|---
$\kappa_{W}[\%]$ | $0.86$ | $0.38$ | $0.23$ | $0.27$ | $0.17$ | $0.39$ | $0.14$
$\kappa_{Z}[\%]$ | $0.15$ | $0.14$ | $0.094$ | $0.13$ | $0.27$ | $0.63$ | $0.12$
$\kappa_{g}[\%]$ | $1.1$ | $0.88$ | $0.59$ | $0.55$ | $0.56$ | $0.74$ | $0.46$
$\kappa_{\gamma}[\%]$ | $1.3$ | $1.2$ | $1.1$ | $0.29$ | $0.32$ | $0.56$ | $0.28$
$\kappa_{Z\gamma}[\%]$ | $10.$ | $10.$ | $10.$ | $0.7$ | $0.71$ | $0.89$ | $0.68$
$\kappa_{c}[\%]$ | $1.5$ | $1.3$ | $0.88$ | $1.2$ | $1.2$ | $-$ | $0.94$
$\kappa_{t}[\%]$ | $3.1$ | $3.1$ | $3.1$ | $0.95$ | $0.95$ | $0.99$ | $0.95$
$\kappa_{b}[\%]$ | $0.94$ | $0.59$ | $0.44$ | $0.5$ | $0.52$ | $0.99$ | $0.41$
$\kappa_{\mu}[\%]$ | $4.$ | $3.9$ | $3.3$ | $0.41$ | $0.45$ | $0.68$ | $0.41$
$\kappa_{\tau}[\%]$ | $0.9$ | $0.61$ | $0.39$ | $0.49$ | $0.63$ | $0.9$ | $0.42$
$\Gamma_{H}[\%]$ | $1.6$ | $0.87$ | $0.55$ | $0.67$ | $0.61$ | $1.3$ | $0.44$
---
Figure 15:  Comparison of the different FCC scenarios in the kappa-0-HL
scenario (similar to kappa-0 in that it does not allow any BSM decay, but
including HL-LHC data).

Another interesting questions is what uncertainties are obtained when
combining any of the lower energy stages of $e^{+}e^{-}$ colliders with the
FCC-hh. This is shown in Table 31 for the kappa-0-HL fit and in Table 32 for
the EFT fit. The results for the various 1st-stage $e^{+}e^{-}$ colliders are
comparable within a factor of about two for the Higgs couplings in the kappa-
fits. Figure 16 shows this comparison graphically. For the EFT framework, the
differences are a bit larger, in particular for the aTGC values, see Table 32.

Table 31:  Results for the kappa-0-HL fit for FCC-hh combined with any of the four $e^{+}e^{-}$ colliders proposed. kappa-0-HL |  HL+ILC250+FCC-hh |  HL+CLIC380+FCC-hh | HL+CEPC+FCC-hh |  HL+FCC-ee365/hh
---|---|---|---|---
$\kappa_{W}[\%]$ | $0.37$ | $0.36$ | $0.35$ | $0.27$
$\kappa_{Z}[\%]$ | $0.19$ | $0.26$ | $0.12$ | $0.13$
$\kappa_{g}[\%]$ | $0.65$ | $0.69$ | $0.55$ | $0.55$
$\kappa_{\gamma}[\%]$ | $0.31$ | $0.34$ | $0.29$ | $0.29$
$\kappa_{Z\gamma}[\%]$ | $0.71$ | $0.74$ | $0.69$ | $0.7$
$\kappa_{c}[\%]$ | $1.8$ | $3.8$ | $1.8$ | $1.2$
$\kappa_{t}[\%]$ | $0.96$ | $0.96$ | $0.95$ | $0.95$
$\kappa_{b}[\%]$ | $0.63$ | $0.68$ | $0.52$ | $0.5$
$\kappa_{\mu}[\%]$ | $0.43$ | $0.47$ | $0.41$ | $0.41$
$\kappa_{\tau}[\%]$ | $0.61$ | $0.78$ | $0.52$ | $0.49$
$\Gamma_{H}[\%]$ | $0.90$ | $0.98$ | $0.74$ | $0.67$
Figure 16:  Combination of the different future ee colliders with FCC-hh and HL-LHC, in an extension of the kappa-0-HL scenario. Note that ILC250 and CLIC380 (first stages) are shown in comparison with CEPC (240) and FCC-ee365.  Table 32:  Results for the global EFT fit for FCC-hh combined with any of the four $e^{+}e^{-}$ colliders proposed, also shown in Table 31.  SMEFTND |  HL+ILC250+FCC-hh |  HL+CLIC380+FCC-hh | HL+CEPC+FCC-hh |  HL+FCC-ee365/hh
---|---|---|---|---
$g_{HZZ}^{\mathrm{eff}}[\%]$ | $0.35$ | $0.46$ | $0.38$ | $0.21$
$g_{HWW}^{\mathrm{eff}}[\%]$ | $0.36$ | $0.46$ | $0.36$ | $0.21$
$g_{H\gamma\gamma}^{\mathrm{eff}}[\%]$ | $0.47$ | $0.55$ | $0.48$ | $0.38$
$g_{HZ\gamma}^{\mathrm{eff}}[\%]$ | $0.78$ | $0.83$ | $0.76$ | $0.72$
$g_{Hgg}^{\mathrm{eff}}[\%]$ | $0.73$ | $0.88$ | $0.54$ | $0.56$
$g_{Htt}[\%]$ | $3.1$ | $2.2$ | $3.1$ | $1.7$
$g_{Hcc}[\%]$ | $1.8$ | $3.9$ | $1.8$ | $1.2$
$g_{Hbb}[\%]$ | $0.75$ | $0.95$ | $0.58$ | $0.51$
$g_{H\tau\tau}[\%]$ | $0.78$ | $1.2$ | $0.61$ | $0.54$
$g_{H\mu\mu}[\%]$ | $0.54$ | $0.61$ | $0.53$ | $0.46$
$\delta g_{1Z}[\times 10^{2}]$ | $0.078$ | $0.04$ | $0.08$ | $0.028$
$\delta\kappa_{\gamma}[\times 10^{2}]$ | $0.12$ | $0.079$ | $0.089$ | $0.048$
$\lambda_{Z}[\times 10^{2}]$ | $0.042$ | $0.043$ | $0.1$ | $0.047$

The ILC and CLIC documents submitted to the European Strategy did not contain
any explicit analysis related to the $Z$ boson properties. In the meantime, it
was explored what can be done either using radiative $Z$ boson events during
the standard running or by a dedicated running at $\sqrt{s}\approx M_{Z}$ for
a period of 1-3 years, collecting 100 fb-1. The results shown in the main body
of this paper now include the radiative return analysis. Here, we present the
additional improvement which can be made when a dedicated $Z$ running period
is considered. Table 33 shows the effective Higgs boson couplings with and
without Giga-Z running for the ILC and CLIC. It is seen that for ILC250 a
Giga-Z running improves the $H$ couplings to vector bosons by about 30%, and
for other couplings the improvement is much smaller. For ILC500 and CLIC380
the impact of dedicated Giga-Z running is low, except for the precision on the
TGC parameter $\delta\kappa_{\gamma}$.

Table 33:  Comparison of the effective Higgs coupling sensitivities for the ILC and CLIC with and without a dedicated running at $\sqrt{s}\approx M_{Z}$. SMEFTND |  HL+ILC250 |  HL+ILC250(GigaZ) |  HL+ILC500 |  HL+ILC500(GigaZ) |  HL+CLIC380 |  HL+CLIC380(GigaZ)
---|---|---|---|---|---|---
$g_{HZZ}^{\mathrm{eff}}[\%]$ | $0.39$ | $0.31$ | $0.22$ | $0.19$ | $0.5$ | $0.47$
$g_{HWW}^{\mathrm{eff}}[\%]$ | $0.41$ | $0.32$ | $0.22$ | $0.2$ | $0.5$ | $0.47$
$g_{H\gamma\gamma}^{\mathrm{eff}}[\%]$ | $1.3$ | $1.3$ | $1.2$ | $1.2$ | $1.4$ | $1.4$
$g_{HZ\gamma}^{\mathrm{eff}}[\%]$ | $9.6$ | $9.4$ | $6.8$ | $6.7$ | $9.7$ | $9.5$
$g_{Hgg}^{\mathrm{eff}}[\%]$ | $1.1$ | $1.1$ | $0.79$ | $0.79$ | $1.3$ | $1.3$
$g_{Htt}[\%]$ | $3.2$ | $3.1$ | $2.9$ | $2.8$ | $3.2$ | $3.2$
$g_{Hcc}[\%]$ | $1.8$ | $1.8$ | $1.2$ | $1.2$ | $4.$ | $4.$
$g_{Hbb}[\%]$ | $0.78$ | $0.74$ | $0.52$ | $0.51$ | $0.99$ | $0.98$
$g_{H\tau\tau}[\%]$ | $0.81$ | $0.78$ | $0.59$ | $0.58$ | $1.3$ | $1.3$
$g_{H\mu\mu}[\%]$ | $4.1$ | $4.1$ | $3.9$ | $3.9$ | $4.4$ | $4.3$
$\delta g_{1Z}[\times 10^{2}]$ | $0.091$ | $0.071$ | $0.047$ | $0.031$ | $0.045$ | $0.044$
$\delta\kappa_{\gamma}[\times 10^{2}]$ | $0.12$ | $0.087$ | $0.076$ | $0.047$ | $0.079$ | $0.066$
$\lambda_{Z}[\times 10^{2}]$ | $0.042$ | $0.042$ | $0.021$ | $0.021$ | $0.043$ | $0.043$

Table 34 shows the impact of the Giga-Z running on the precision on the
effective couplings of the $Z$ boson to fermions. In many cases, the impact is
significant, improving the precision by up to a factor of $\sim 4$. Also shown
are the results expected for CEPC and FCC-ee. In most cases, CEPC and FCC-ee
achieve the highest precision. A notable exception is the top quark coupling
which is best constrained by the ILC500.

Table 34:  Comparison of the effective $Z$-boson coupling sensitivities for the ILC and CLIC with and w/o a dedicated running at $\sqrt{s}\approx M_{Z}$. Also shown are the values for CEPC and FCC-ee In all cases, the combination with HL-LHC is shown but the sensitivity is dominated by the $e^{+}e^{-}$ collider.  SMEFTND |  HL+ILC250 |  HL+ILC250(GigaZ) |  HL+ILC500 |  HL+ILC500(GigaZ) |  HL+CLIC380 |  HL+CLIC380(GigaZ) | HL+CEPC | HL+FCC-ee
---|---|---|---|---|---|---|---|---
$g_{L}^{\nu_{e}}[\%]$ | $0.082$ | $0.058$ | $0.048$ | $0.032$ | $0.027$ | $0.024$ | $0.032$ | $0.028$
$g_{L}^{\nu_{\mu}}[\%]$ | $0.11$ | $0.075$ | $0.088$ | $0.064$ | $0.16$ | $0.11$ | $0.036$ | $0.03$
$g_{L}^{\nu_{\tau}}[\%]$ | $0.12$ | $0.079$ | $0.095$ | $0.071$ | $0.16$ | $0.11$ | $0.038$ | $0.034$
$g_{L}^{e}[\%]$ | $0.048$ | $0.025$ | $0.037$ | $0.023$ | $0.035$ | $0.021$ | $0.014$ | $0.0073$
$g_{R}^{e}[\%]$ | $0.055$ | $0.028$ | $0.041$ | $0.025$ | $0.047$ | $0.025$ | $0.016$ | $0.0089$
$g_{L}^{\mu}[\%]$ | $0.072$ | $0.023$ | $0.07$ | $0.022$ | $0.17$ | $0.022$ | $0.031$ | $0.007$
$g_{R}^{\mu}[\%]$ | $0.09$ | $0.025$ | $0.087$ | $0.024$ | $0.24$ | $0.026$ | $0.047$ | $0.0092$
$g_{L}^{\tau}[\%]$ | $0.076$ | $0.027$ | $0.073$ | $0.026$ | $0.17$ | $0.029$ | $0.016$ | $0.0076$
$g_{R}^{\tau}[\%]$ | $0.094$ | $0.031$ | $0.091$ | $0.03$ | $0.24$ | $0.034$ | $0.018$ | $0.0094$
$g_{L}^{u=c}[\%]$ | $0.24$ | $0.13$ | $0.23$ | $0.12$ | $0.26$ | $0.11$ | $0.051$ | $0.05$
$g_{R}^{u=c}[\%]$ | $0.35$ | $0.15$ | $0.35$ | $0.15$ | $0.43$ | $0.16$ | $0.08$ | $0.066$
$g_{L}^{t}[\%]$ | $11.$ | $11.$ | $0.84$ | $0.84$ | $2.4$ | $2.3$ | $11.$ | $1.5$
$g_{R}^{t}[\%]$ | $-$ | $-$ | $2.$ | $2.$ | $6.$ | $6.$ | $-.$ | $3.5$
$g_{L}^{d=s}[\%]$ | $0.23$ | $0.14$ | $0.21$ | $0.13$ | $0.25$ | $0.14$ | $0.056$ | $0.051$
$g_{R}^{d=s}[\%]$ | $4.8$ | $3.2$ | $3.9$ | $2.5$ | $5.9$ | $3.9$ | $1.4$ | $1.1$
$g_{L}^{b}[\%]$ | $0.071$ | $0.034$ | $0.068$ | $0.033$ | $0.13$ | $0.041$ | $0.017$ | $0.011$
$g_{R}^{b}[\%]$ | $0.51$ | $0.31$ | $0.51$ | $0.32$ | $3.$ | $0.75$ | $0.29$ | $0.088$

It is also interesting to compare the highest energy options closely. This is
done in Table 35. In all cases, it is assumed that the colliders also include
a Giga-Z run of 1-3 years [4, 11].

Table 35:  Effective Higgs couplings precision for the EFT fit for a selection of colliders at high energy. For the linear colliders it is assumed that 100 fb-1 of dedicated running on the Z-pole, corresponding to 1-3 years of data taking, are part of the programme. SMEFTND |  HL+ILC500 |  HL+ILC1000 |  HL+CLIC1500 |  HL+CLIC3000 | HL+FCC-ee/hh | HL+FCC-ee/eh/hh
---|---|---|---|---|---|---
$g_{HZZ}^{\mathrm{eff}}[\%]$ | $0.19$ | $0.15$ | $0.2$ | $0.16$ | $0.21$ | $0.13$
$g_{HWW}^{\mathrm{eff}}[\%]$ | $0.2$ | $0.16$ | $0.18$ | $0.15$ | $0.21$ | $0.13$
$g_{H\gamma\gamma}^{\mathrm{eff}}[\%]$ | $1.2$ | $1.1$ | $1.3$ | $1.1$ | $0.38$ | $0.34$
$g_{HZ\gamma}^{\mathrm{eff}}[\%]$ | $6.7$ | $6.6$ | $4.5$ | $3.6$ | $0.72$ | $0.7$
$g_{Hgg}^{\mathrm{eff}}[\%]$ | $0.79$ | $0.55$ | $0.97$ | $0.75$ | $0.56$ | $0.49$
$g_{Htt}[\%]$ | $2.8$ | $1.5$ | $2.2$ | $2.1$ | $1.7$ | $1.7$
$g_{Hcc}[\%]$ | $1.2$ | $0.88$ | $1.8$ | $1.4$ | $1.2$ | $0.95$
$g_{Hbb}[\%]$ | $0.51$ | $0.43$ | $0.44$ | $0.37$ | $0.51$ | $0.44$
$g_{H\tau\tau}[\%]$ | $0.58$ | $0.49$ | $0.92$ | $0.74$ | $0.54$ | $0.46$
$g_{H\mu\mu}[\%]$ | $3.9$ | $3.5$ | $4.1$ | $3.4$ | $0.46$ | $0.42$
$\delta g_{1Z}[\times 10^{2}]$ | $0.031$ | $0.03$ | $0.012$ | $0.0099$ | $0.028$ | $0.018$
$\delta\kappa_{\gamma}[\times 10^{2}]$ | $0.047$ | $0.044$ | $0.022$ | $0.018$ | $0.048$ | $0.047$
$\lambda_{Z}[\times 10^{2}]$ | $0.021$ | $0.014$ | $0.0053$ | $0.0018$ | $0.047$ | $0.045$

After the Granada meeting, it was also studied what could be achieved with a
hadron-hadron collider with $\sqrt{s}=37.5$ TeV and ${\cal L}=15$ ab-1, in
conjunction with one of the $e^{+}e^{-}$ colliders [10]. This is shown in
Table. 36 compared to the nominal FCC-hh in combination with the various
$e^{+}e^{-}$ colliders. For most coupling parameters the sensitivity of the
$37.5$ TeV collider is degraded by about a factor $1.5-2$ w.r.t. the 100 TeV
collider, except for $Z\gamma$ where it is a factor of 5. For $\kappa_{Z}$ and
$\kappa_{c}$ there is no difference as both are very much dominated by the
lepton collider sensitivity.

Table 36:  Expected relative precision (%) of the $\kappa$ parameters in the
kappa-0-HL scenario described in section 2 for future lepton colliders
combined with the HL-LHC and the FCC-hh37.5 (top part) and with HL-LHC and
FCC-hh (bottom part). No BSM width is allowed in the fit: both
$\mathrm{BR}_{\mathrm{unt}}$ and $\mathrm{BR}_{\mathrm{inv}}$ are set to 0.

kappa-0-HL scenario | HL-LHC+FCC-hh37.5+
---|---
ILC250 | CLIC380 | CEPC | FCC-ee365
$\kappa_{W}[$%$]$ | $0.94$ | $0.62$ | $0.81$ | $0.38$
$\kappa_{Z}[\%]$ | $0.21$ | $0.33$ | $0.13$ | $0.14$
$\kappa_{g}[\%]$ | $1.3$ | $1.3$ | $0.97$ | $0.87$
$\kappa_{\gamma}[\%]$ | $0.64$ | $0.68$ | $0.62$ | $0.62$
$\kappa_{Z\gamma}[\%]$ | $3.$ | $3.1$ | $2.8$ | $3.$
$\kappa_{c}[\%]$ | $1.9$ | $3.9$ | $1.9$ | $1.3$
$\kappa_{t}[\%]$ | $1.9$ | $1.9$ | $1.9$ | $1.9$
$\kappa_{b}[\%]$ | $0.99$ | $0.94$ | $0.81$ | $0.58$
$\kappa_{\mu}[\%]$ | $1.$ | $1.1$ | $1.$ | $1.$
$\kappa_{\tau}[\%]$ | $0.96$ | $1.2$ | $0.83$ | $0.6$

kappa-0-HL scenario | HL-LHC+FCC-hh+
---|---
ILC250 | CLIC380 | CEPC | FCC-ee365
$\kappa_{W}[\%]$ | $0.37$ | $0.36$ | $0.35$ | $0.27$
$\kappa_{Z}[\%]$ | $0.19$ | $0.26$ | $0.12$ | $0.13$
$\kappa_{g}[\%]$ | $0.65$ | $0.69$ | $0.55$ | $0.55$
$\kappa_{\gamma}[\%]$ | $0.31$ | $0.34$ | $0.29$ | $0.29$
$\kappa_{Z\gamma}[\%]$ | $0.71$ | $0.74$ | $0.69$ | $0.7$
$\kappa_{c}[\%]$ | $1.8$ | $3.8$ | $1.8$ | $1.2$
$\kappa_{t}[\%]$ | $0.96$ | $0.96$ | $0.95$ | $0.95$
$\kappa_{b}[\%]$ | $0.63$ | $0.68$ | $0.52$ | $0.5$
$\kappa_{\mu}[\%]$ | $0.43$ | $0.47$ | $0.41$ | $0.41$
$\kappa_{\tau}[\%]$ | $0.61$ | $0.78$ | $0.52$ | $0.49$
$\Gamma_{H}[\%]$ | $0.90$ | $0.98$ | $0.74$ | $0.67$

##  Appendix G Electroweak precision constraints on oblique parameters

In this section we will focus on the constraints on heavy new physics that can
be obtained by precise measurement of the on-shell $W$ and $Z$ properties. We
will focus on universal effects that can be fully encapsulated in the vector
boson propagators, with no direct correction to the interaction vertices with
fermions. This assumption allows on the one hand to limit the analysis to a
few parameters. One the other hand, it should be noted, motivated models, like
the minimal composite Higgs, often satisfy this assumption to a very good
approximation. When considering universal deviations from the SM one must
distinguish between the number of on-shell $Z$ and $W$ observables and the
total number of parameters, which corresponds to the total number of on- and
off-shell observables.

Under the assumption of universality, the relevant on-shell observables of $W$
and $Z$ physics reduce to three quantities: the relative normalization of
charged and neutral currents (or the $Z$’s axial coupling), and the two
relative differences among the three possible definitions of the Weinberg
angle (from $\alpha(M_{Z}),G_{F},M_{Z}$, from the $Z$’s vector coupling, and
from $M_{W}/M_{Z}$). These can be nicely encapsulated in the $\epsilon$’s of
Altarelli and Barbieri [140]:

$\displaystyle\varepsilon_{1}$ $\displaystyle=$ $\displaystyle\Delta\rho,$
$\displaystyle\varepsilon_{2}$ $\displaystyle=$
$\displaystyle\cos^{2}{\theta_{w}}\Delta\rho+\frac{\sin^{2}{\theta_{w}}}{\cos^{2}{\theta_{w}}-\sin^{2}{\theta_{w}}}\Delta
r_{W}-2s_{0}^{2}\Delta\kappa,$ $\displaystyle\varepsilon_{3}$ $\displaystyle=$
$\displaystyle\cos^{2}{\theta_{w}}\Delta\rho+(\cos^{2}{\theta_{w}}-\sin^{2}{\theta_{w}})\Delta\kappa.$
(36)

where we define the weak angle from
$\sin^{2}{\theta_{w}}\cos^{2}{\theta_{w}}\equiv\frac{\pi\alpha(M_{Z})}{\sqrt{2}G_{F}M_{Z}^{2}}$
and the $\Delta r_{W}$, $\Delta\rho$ and $\Delta\kappa$ parameters are defined
from the masses and effective vector and axial couplings of the electroweak
bosons :

$\displaystyle\frac{M_{W}^{2}}{M_{Z}^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{1+\sqrt{1-\frac{4\pi\alpha(M_{Z})}{\sqrt{2}G_{F}M_{Z}^{2}(1-\Delta
r_{W})}}},$ $\displaystyle g_{V}^{f}$ $\displaystyle=$
$\displaystyle\sqrt{1+\Delta\rho}\leavevmode\nobreak\
\\!(T_{3}^{f}-2Q_{f}(1+\Delta\kappa)\sin^{2}{\theta_{w}}),$ $\displaystyle
g_{A}^{f}$ $\displaystyle=$
$\displaystyle\sqrt{1+\Delta\rho}\leavevmode\nobreak\ \\!T_{3}^{f}.$ (37)

($T_{3}^{f}$ and $Q_{f}$ are weak isospin and charge of the corresponding
fermion.) Notice that $\epsilon_{2}$ relies on the measurement of the $W$ mass
while $\epsilon_{1,3}$ do not.

The number of parameters describing universal new physics in the $W,Z$ channel
on- and off-shell is instead four232323For a good part of the history of EW
precision tests, the community has mostly relied on a set of three quantities,
$S,T,U$. These are however inadequate in any realistic new physics scenario:
they are always either redundant or incomplete. Indeed in technicolor models,
it was understood that $U$ is negligible and the set is redundant in that case
[141, 142, 143, 144]. On the other hand $S,T,U$ are insufficient to describe
even the simplest sequential $Z^{\prime}$ models that fall into the class of
universal theories. That the relevant set should be consist of 4 quantities
was first realized in [145] in the context of linearly realized EW symmetry,
i.e. SMEFT. The generality of this counting, and therefore its validity also
in technicolor/HEFT scenarios, has been clarified in [48].. They correspond to
the leading effects in a derivative expansion of the vector boson self-
energies, $\Pi_{VV^{\prime}}(q^{2})$. More precisely they correspond to the
leading effects in each independent channel (with the channels characterized
by the relevant quantum numbers: electric charge, electroweak and custodial
symmetry). Considering the vector boson self-energies, using the constraints
from $U(1)_{EM}$ unbroken gauge invariance (massless photon), and subtracting
the quantities that play the role of SM inputs ($v$, $g$, $g^{\prime}$), one
is left with these four leading quantities

$\displaystyle\hat{S}$ $\displaystyle=$ $\displaystyle
g^{2}\Pi_{W_{3}B}^{\prime}(0),$ $\displaystyle\hat{T}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{M_{W}^{2}}\left(\Pi_{W_{3}W_{3}}(0)-\Pi_{W^{+}W^{-}}(0)\right),$
$\displaystyle W$ $\displaystyle=$
$\displaystyle\frac{1}{2}g^{2}M_{W}^{2}\Pi_{W_{3}W_{3}}^{\prime\prime}(0),$
$\displaystyle Y$ $\displaystyle=$
$\displaystyle\frac{1}{2}g^{\prime\leavevmode\nobreak\
2}M_{W}^{2}\Pi_{BB}^{\prime\prime}(0),$ (38)

which can be mapped to four linear combinations of operators in the SMEFT
Lagrangian. Finally, considering their effect on $W,Z$ on-shell propagation,
one finds, writing $\epsilon_{i}=\epsilon_{i,{\rm SM}}+\delta\epsilon_{i}$,

$\displaystyle\delta\varepsilon_{1}$ $\displaystyle=$
$\displaystyle\hat{T}-W-Y\tan^{2}\theta_{w},$
$\displaystyle\delta\varepsilon_{2}$ $\displaystyle=$ $\displaystyle-W,$
$\displaystyle\delta\varepsilon_{3}$ $\displaystyle=$
$\displaystyle\hat{S}-W-Y.$ (39)

Strongly coupled models come with a parametric enhancement of
$\hat{S},\hat{T}$ over $W,Y$, such that in the class of models, one can
simplify further the analysis of EW data and perform a two-dimensional fit.

The results of this fit setup are presented in Table 37 and Figure 17, for the
different future lepton-collider options, where the largest improvement in
terms of measurements of the EW precision observables (EWPO) is expected. In
the table and figures we also show the impact of the SM theory uncertainties
in the results. The results are presented assuming the projected future
improvements in SM theory calculations (Full ThIntr Unc.), neglecting the
intrinsic theory uncertainties associated to such calculations (No ThIntr
Unc.) and, finally, also assuming that parametric uncertainties become
subdominant (No ThPar+Intr Unc.). Since several of the SM EW inputs are to be
measured at the future collider under consideration, the latter scenario goes
beyond the physics potential of these machines. This scenario is presented
only to illustrate whether the precision of the measurements of such inputs
can become a limiting factor in terms of the reach of $\hat{S}$ and $\hat{T}$.
This seems to be the case for the circular colliders and, to a less extent,
the linear collider Giga-Z options.

Table 37:  Comparison of the sensitivity at 68% probability to new physics contributions to EWPO in the form of the oblique $S$ and $T$ parameters, under different assumptions for the SM theory uncertainties. We express the results in terms of the usually normalised parameters: $S=4\sin^{2}\theta_{w}\hat{S}/\alpha$ and $T=\hat{T}/\alpha$.  |  | HL-LHC |  HL+CLIC380 |  HL+CLIC380 (GigaZ) |  HL+ILC250 |  HL+ILC250 (GigaZ) | HL+CEPC | HL+FCC-ee
---|---|---|---|---|---|---|---|---
$S$ |  Full ThIntr Unc. | $0.053$ | $0.032$ | $0.013$ | $0.015$ | $0.012$ | $0.01$ | $0.0079$
|  No ThIntr Unc. | $0.053$ | $0.032$ | $0.011$ | $0.012$ | $0.009$ | $0.0068$ | $0.0038$
|  No ThPar+Intr Unc. | $0.052$ | $0.031$ | $0.0091$ | $0.011$ | $0.0067$ | $0.0031$ | $0.0013$
$T$ | Full ThIntr Unc. | $0.041$ | $0.023$ | $0.013$ | $0.015$ | $0.014$ | $0.0094$ | $0.0058$
| No ThIntr Unc. | $0.041$ | $0.023$ | $0.012$ | $0.014$ | $0.013$ | $0.0072$ | $0.0022$
| No ThPar+Intr Unc. | $0.039$ | $0.022$ | $0.01$ | $0.011$ | $0.0091$ | $0.0041$ | $0.0019$
|
---|---
Figure 17:  (Left) 2-$\sigma$ regions in the $S-T$ plane at the different
future colliders, combined with the HL-LHC (including also the LEP/SLD EWPO
programme). We express the results in terms of the usually normalised
parameters: $S=4\sin^{2}\theta_{w}\hat{S}/\alpha$ and $T=\hat{T}/\alpha$. The
results include the future projected parametric uncertainties in the SM
predictions of the different EWPO, but not the intrinsic ones. (Right) The
same illustrating the impact of neglecting such intrinsic theory errors. For
each project (including the Giga-Z option for linear colliders) the solid
regions show the results in the left panel, to be compared with the regions
bounded by the dashed lines, which include the full projected theory
uncertainty.

##  Appendix H Improvement with respect to HL-LHC

Figures 18 and 19 give a graphic comparison of the improvement with respect to
HL-LHC in the Kappa-3 and SMEFT-ND frameworks. This improvement is shown as
the ratio of the precision at the HL-LHC over the precision at the future
collider, with more darker colors corresponding to larger improvement factors.
The kappa-3 result shows large improvements, up to an order of magnitude, for
all future ee colliders for the measurement of the couplings to Z, W and b and
the limits on the invisible branching ratio, and an ’infinite’ improvement in
the case of the coupling to charm, immeasurable at the HL-LHC. Rare,
statistically dominated, couplings, as well as the coupling to the top quark
are shown to be markedly improved with respect to HL-LHC only with FCC-hh. The
more complete SMEFT-ND fit highlights more clearly the improvement in
precision, with improvements of the order of an order of magnitude in the
measurement of Z, W and b couplings for all future ee colliders. The aTGC
results show an even more dramatic improvement, with factors over 100 and 1000
for the last stages of the linear colliders.

Figure 18: Graphic comparison of the improvement with respect to HL-LHC in the
Kappa-3 framework.  Figure 19: Graphic comparison of the improvement with
respect to HL-LHC in the SMEFT-ND framework.

##  Appendix I Changes with respect to previous versions

The current version (v2) of this report includes extended studies with respect
to what was presented in the first version of the document of May 2019. Apart
from fixing typos and bugs, the following changes were implemented for the
different scenarios:

### HL-LHC

  * •

Fixed bug in kappa-0 fit that was accidentally imposing the constraint
$|\kappa_{V}|\leq 1$ in that fit.

  * •

Improved treatment of theory uncertainties, providing results closer to those
presented in Ref. [13].

### HE-LHC

  * •

Fixed bug in kappa-0 fit that was accidentally imposing the constraint
$|\kappa_{V}|\leq 1$.

  * •

Improved treatment of theory uncertainties, closer to the setup presented in
[13]. Results presented in some cases using both scenarios S2 and S2′ in [13].

  * •

Added correlation between theory uncertainties in the combination with HL-LHC.

### ILC

  * •

Added inputs from a run at centre-of-mass energy of 1 TeV [4].

  * •

Added the projections for EWPO obtained using radiative return events to the
$Z$ pole collected at $\sqrt{s}=250$ GeV [4].

  * •

Added inputs from a run at the $Z$ pole (Giga Z) [4].

### CLIC

  * •

Added the projections for EWPO obtained using radiative return events to the
$Z$ pole collected at $\sqrt{s}=380$ GeV [11].

  * •

Added inputs from a run at the $Z$ pole (Giga Z) [11].

  * •

Updated the effects of the parametric error associated to the Higgs mass
determination, assuming the HL-LHC measures $M_{H}$ with a precision of 20
MeV.

### CEPC

  * •

Included new projections for several EWPO not available at [2]. See table 27
(from [133]).

### FCC-ee

  * •

Fixed the precision of some of the asymmetries.

  * •

Added projections for a FCC-ee scenario with four interaction points.

### FCC-hh

  * •

Added projections for Higgs measurements in associated production of the Higgs
with a $W$ boson. (Only for $\kappa$ framework studies.)

  * •

Added results for FCC-hh in combination with the first stage as a Higgs
factory of ILC, CLIC and CEPC.

  * •

Added results for a low energy FCC-hh collider operating at 37.5 TeV (LE-FCC).
Results shown in combination with the Higgs factory option of ILC, CLIC, CEPC
and FCC-ee in appendix F

### Others

  * •

Added Appendix G with the contraints on oblique parameters from electroweak
precision measurements at the different future lepton colliders.