# Characterizing and Tuning Exceptional Points Using Newton Polygons

Rimika Jaiswal  Undergraduate Programme, Indian Institute of Science,
Bangalore 560012, India  Ayan Banerjee  Solid State and Structural Chemistry
Unit, Indian Institute of Science, Bangalore 560012, India  Awadhesh Narayan
<EMAIL_ADDRESS>Solid State and Structural Chemistry Unit, Indian
Institute of Science, Bangalore 560012, India

###### Abstract

The study of non-Hermitian degeneracies – called exceptional points – has
become an exciting frontier at the crossroads of optics, photonics, acoustics,
and quantum physics. Here, we introduce the Newton polygon method as a general
algebraic framework for characterizing and tuning exceptional points. These
polygons were first described by Isaac Newton in 1676 and are conventionally
used in algebraic geometry, with deep roots in various topics in modern
mathematics. We have found their surprising connection to non-Hermitian
physics. We propose and illustrate how the Newton polygon method can enable
the prediction of higher-order exceptional points, using a recently
experimentally realized optical system. Using the paradigmatic Hatano-Nelson
model, we demonstrate how our Newton Polygon method can be used to predict the
presence of the non-Hermitian skin effect. As further application of our
framework, we show the presence of tunable exceptional points of various
orders in $PT$-symmetric one-dimensional models. We further extend our method
to study exceptional points in higher number of variables and demonstrate that
it can reveal rich anisotropic behaviour around such degeneracies. Our work
provides an analytic recipe to understand and tune exceptional physics.

Energy non-conserving and dissipative systems are described by non-Hermitian
Hamiltonians moiseyev2011non . Unlike their Hermitian counterparts, they are
not always diagonalizable and can become defective at some unique points in
their parameter space – called exceptional points (EPs) – where both the
eigenvalues and the eigenvectors coalesce Heiss_2012 ; kato2013perturbation .
Around such an EP, the complex eigenvalues lie on self-intersecting Riemann
sheets. This means that upon encircling an EP once, the system does not return
to its initial state, but to a different state on another Riemann sheet,
manifesting in Berry phases and topological charges heiss2016circling ;
el2018non ; alvarez2018topological ; ozdemir2019parity ; science_review ;
ghatak2019new ; ashida2020non ; bergholtz2021exceptional .

While the notion of EPs was known theoretically for several decades, their
controllable realization has only been possible recently. This has led to
enormous interest and, by now, EPs are ubiquitous in acoustic shi2016accessing
; acoustic2 , optical science_review , photonic yin2013 ; zhang2016observation
; ozawa2019topological ; feng2014single , mechanical
mechanical_PhysRevB.100.054109 , and condensed matter systems midya2018non ;
zhang2017 ; ferro_PhysRevLett.121.197201 ; cmpnew_PhysRevB.98.035141 ;
cmpnew_PhysRevB.99.121101 . They also appear in the study of atomic and
molecular physics molecular_PhysRevLett.103.123003 , electronics
electronic_Stehmann_2004 , superconductivity
superconductivity_PhysRevLett.99.167003 ; San-Jose2016 ; Avila2019 , and
quantum phase transitions phasetrans_PhysRevLett.99.100601 . They can lead to
a variety of intriguing phenomena such as uni-directional invisibility
lin2011unidirectional ; peng2014parity , double refraction makris2008beam ,
laser mode selectivity feng2014single ; hodaei2014parity , non-reciprocal
energy transfer xu2016topological , non-Hermitian skin effects alvarez2018non
; yao2018edge ; kunst2018biorthogonal ; yokomizo2019non ; lee2019anatomy ;
borgnia2020non ; okuma2020topological ; Helbig2020 ; Weidemann311  and
interesting quantum dynamics of exciton-polaritons gao2018chiral ;
gao2015observation , to name just a few.

While early studies focused on second order EPs (where only two eigenvectors
coalesce), very recently, the focus has shifted to higher order EPs, where
more than two eigenvectors coalesce xiao2019anisotropic ; xiao2020exceptional
; hepth_mandal2021symmetry . Apart from interesting fundamental physics, they
show promise for several fascinating applications
nature_EnhancedSensitivity_Hodaei2017 ; wiersig2014enhancing . Higher order
EPs and their unconventional phase transitions have been experimentally
realized in various acoustic and photonic systems
nature_EnhancedSensitivity_Hodaei2017 ; science_nexus ;
ferrohep_PhysRevB.101.144414 ; hep_exp .

Here, we introduce a new algebraic framework for characterizing such non-
Hermitian degeneracies using Newton polygons. These polygons were first
described by Isaac Newton, in 1676, in his letters to Oldenburg and Leibniz
brieskorn2012plane . They are conventionally used in algebraic geometry to
prove the closure of fields algebraic  and are intimately connected to Puiseux
series – a generalization of the usual power series to negative and fractional
exponents edwards2005essays ; np_notes . Furthermore, Newton polygons have
deep connections to various topics in mathematics, including homotopy theory,
braid groups, knot theory and algebraic number theory brieskorn2012plane . We
develop the Newton polygon method to study EPs and illustrate its utility in
predicting higher order EPs in experimentally realized systems, as well as
predicting the non-Hermitian skin effect. We also present parity and time
reversal ($PT$) symmetric one-dimensional models to demonstrate how this
method can provide an elegant way of tuning different system parameters to
obtain a higher order EP, or to choose from a spectrum of EPs of various
orders. The Newton polygon method can also be naturally extended to higher
number of variables. Using such an extension we show rich anisotropic
behaviour around such EPs. We hope that our results stimulate further
exploration of non-Hermitian degeneracies and their applications.

## The Newton Polygon Method

We consider a system at an EP described by the Hamiltonian
$H_{0}(t_{1},t_{2},...)$, where $t_{1},t_{2},...$ are system-dependent
parameters. If a perturbation of the form $\epsilon H_{1}(t_{1},t_{2},...)$ is
now added, one can write the eigenvalues of the perturbed Hamiltonian
$H(\epsilon)=H_{0}+\epsilon H_{1}$ as a Puiseux series in $\epsilon$.

$\omega(\epsilon)=\alpha_{1}\epsilon^{1/N}+\alpha_{2}\epsilon^{2/N}+...$ (1)

Note that, to leading order, they have the form $\omega\sim\epsilon^{1/N}$
where $N$ is the order of the EP (see Supplementary Information supplement
for a detailed discussion). The Newton polygon method gives us an algorithmic
way of determining the order of an EP by evaluating the power of the leading
order term in the eigenvalue expansion starting from the characteristic
equation. This can be done though the following steps:

  1. 1.

Given a characteristic equation
$p(\omega,\epsilon)=\det{H-\omega\mathbb{I}}=0$, write $p(\omega,\epsilon)$ in
the form $\sum_{m,n}a_{mn}(t_{1},t_{2},...)\omega^{m}\epsilon^{n}$.

  2. 2.

For each term of the form $a_{mn}\omega^{m}\epsilon^{n}$ in the polynomial,
plot a point $(m,n)$ in $\mathbb{R}^{2}$. The smallest convex shape that
contains all the points plotted is called the Newton polygon.

  3. 3.

Select a segment of the Newton polygon such that all plotted points are either
on, above or to the right of it. The negative of the slope of this line-
segment gives us the lowest order dependence of $\omega$ on $\epsilon$.

### Predicting higher-order EP

To illustrate our method, we consider a recently realized optical system,
consisting of three coupled resonators, that exhibits a higher order EP and an
unprecedented sensitivity to changes in the environment
nature_EnhancedSensitivity_Hodaei2017 . The system can be described by a
remarkably simple non-Hermitian Hamiltonian

$H(\epsilon)=\begin{bmatrix}ig+\epsilon&\kappa&0\\\ \kappa&0&\kappa\\\
0&\kappa&-ig\end{bmatrix},$ (2)

where $g$ accounts for gain and loss, $\kappa$ is the coupling between the
resonators and $\epsilon$ is the external perturbation. The characteristic
equation, $p(\omega,\epsilon)=0$, reads

$-\omega^{3}+\omega^{2}\epsilon+(2\kappa^{2}-g^{2})\
\omega+ig\omega\epsilon-\kappa^{2}\epsilon=0.$ (3)

Figure 1: Newton polygon method enabled prediction of higher order exceptional
points. Left panel shows the Newton polygon for the characteristic polynomial
of $H(\epsilon)$. The line-segment which has all points either on, above, or
to the right of it (shown in blue) has a slope of $-1$ implying that
$\omega\sim\epsilon^{1}$, or, no exceptional behaviour. However, if the term
corresponding to the point $(1,0)$ can be made to vanish (which occurs for
$g=\sqrt{2}\kappa$), we can expect to observe exceptional behaviour. Right
panel shows the Newton polygon for this case. Here, a slope of $-1/3$ implies
$\omega\sim\epsilon^{1/3}$, or the presence of a third order exceptional
point. Here $n(\omega$) and $n(\epsilon)$ denote the exponents of $\omega$ and
$\epsilon$, respectively.

Figure 1 shows the Newton polygon for $p(\omega,\epsilon)$, where each point
on the graph corresponds to a term in the characteristic equation. The line-
segment that contains all points on, above or to the right of it is shown in
blue. This line has a slope of $-1$ implying that the lowest order dependence
of the eigenvalues on $\epsilon$ has the form $\omega\sim\epsilon$. Notice,
however, that if we set $g=\sqrt{2}\kappa$, the coefficient of $\omega$
vanishes and the point $(1,0)$ is no longer present in the Newton polygon. The
slope of the desired line-segment is now $-1/3$ which, in turn, means that
$\omega\sim\epsilon^{1/3}$, or, we have a third order EP (EP3). The Newton
polygon method could thus predict the presence of an EP3 for
$g=\sqrt{2}\kappa$. This is indeed what has been found in the experiments by
Hodaei et al. nature_EnhancedSensitivity_Hodaei2017 .

Here, we have illustrated the use of the Newton polygon method to evaluate the
degree of an EP. In addition, it also provides an algebraic way of evaluating
the expansion of the eigenvalues beyond just the leading order, including the
coefficients of the terms at various orders. We present a detailed discussion
in the Methods sections, with worked out examples in Supplementary Information
supplement .

Figure 2: Newton polygon enabled prediction of non-Hermitian skin effect. Left
panel shows the Newton polygon for the characteristic polynomial of
$H_{N}(\epsilon)$ [Eq. 5]. The right panel shows the Newton Polygon for the
case of $t=\gamma/2$ wherein the coefficients of all the point corresponding
to $\omega^{M}$ vanish, other than when $M=0,N$. The line-segment which has
all points either on, above, or to the right of it thus has a slope of $-1/N$,
which indicates the presence of an $N$-th order EP and correspondingly the
occurrence of non-Hermitian skin effect.

### Predicting non-Hermitian skin effect

We now use the paradigmatic Hatano-Nelson model to show that our Newton
Polygon method can predict the presence of skin effect, which is a remarkable
characteristic of non-Hermitian systems wherein a macroscopic number of
eigenstates accumulate at the edge alvarez2018non ; yao2018edge . The non-
Hermitian skin effect is indicative of the presence of higher-order EPs as per
the anomalous bulk-boundary correspondence. The Hamiltonian for the Hatano-
Nelson model reads HN_PhysRevLett.77.570

$H=\sum_{m=1}^{N}J_{R}c^{\dagger}_{m+1}c_{m}+J_{L}c^{\dagger}_{m}c_{m+1},$ (4)

where $J_{R/L}=t\pm\gamma/2$ are the right and left hopping amplitudes. If we
now add a perturbation, $\epsilon$, coupling the first and the last sites, the
Hamiltonian matrix takes the form

$H_{N}(\epsilon)=\begin{bmatrix}0&t-\gamma/2&0&\cdots&\epsilon\\\
t+\gamma/2&0&t-\gamma/2&\cdots&0\\\ 0&t+\gamma/2&0&\cdots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\ \end{bmatrix}_{N\times N}.$ (5)

The characteristic equation, in turn, is

$p(\omega,\epsilon)=(t+\gamma/2)\epsilon+\sum_{M=N,N-2,\cdots}z_{M}[t^{2}-(\gamma/2)^{2}]^{\frac{N-M}{2}}\omega^{M},$
(6)

where each $z_{M}$ is a constant, $z_{M}\in\mathbb{Z}$. The Newton polygon for
$p(\omega,\epsilon)$ is shown in the left panel of Figure 2. We note that
remarkably at $t=\gamma/2$, the coefficients of all the terms in
$p(\omega,\epsilon)$ vanish other than the $\epsilon^{1}$ and the $\omega^{N}$
terms. The Newton polygon for this case is plotted in the right panel in
Figure 2. The slope of the relevant line-segment is $1/N$. This implies the
presence of an $N$-th order EP and correspondingly the presence of non-
Hermitian skin effect at $t=\gamma/2$, which is physically the condition for
unidirectional hopping ($J_{R}=\gamma$, $J_{L}=0$). Thus, our Newton polygon
method can elegantly predict the occurrence of non-Hermitian skin effect.

### Application to $PT$-symmetric 4-site model

Figure 3: Tunable higher order exceptional points in the $PT$-symmetric model
via Newton polygons. (a) Illustration of the 4-site $PT$-symmetric model which
has the Hamiltonian $H_{4}$. (b) The Newton polygon for $H_{4}(\epsilon)$. The
red and blue surfaces shown in (c) correspond to the regions in the parameter
space where the red and blue points in the Newton polygon vanish. This
predicts that one would get an EP2 if the system parameters fall on the red
surface, while one would get an EP4 if they fall on the black curve (which
depicts the intersection of the two surfaces). Numerical verification of these
predictions are shown in middle and right panels for EP2 and EP4,
respectively. (d) The numerically obtained band structure near the exceptional
point EP2 (green) closely matches the predicted behaviour of $\epsilon^{1/2}$
(red). Inset shows the plot on a logarithmic scale. (f) The numerical band
structure near the exceptional point EP4 (green) closely matches the predicted
behaviour of $\epsilon^{1/4}$ (red). Panels (e) and (g) show the behaviour of
the four bands as one encircles the exceptional points once (choosing the
contour $\epsilon=0.05\exp(i\theta)$). As expected, two of the bands swap for
the middle panel (e) while the four bands undergo a cyclic permutation for the
right panel (g). Parameter values for middle panel: $\delta=2,p=\sqrt{2},q=2$.
Parameter values for right panel: $\delta=3,p=\sqrt{3},q=2$.

We next consider a 4-site $PT$-symmetric system as shown schematically in
Figure 3(a). We consider a general form consistent with the $PT$-symmetry. Due
to parity symmetry, two hopping parameters $p$ and $q$ are sufficient to
describe the couplings between the four sites. The balanced gain and loss for
the outer and inner sites are given by $\delta$ and $\gamma$ respectively. We
will show that when such a system is perturbed, one can get an EP2, or an EP4
depending on the tuning of various parameters. As we shall demonstrate below,
the Newton polygon method elegantly predicts the required tuning. The
Hamiltonian for the 4-site system can be written as

$H_{4}=\begin{bmatrix}i\delta&p&0&0\\\ p&i\gamma&q&0\\\ 0&q&-i\gamma&p\\\
0&0&p&-i\delta\\\ \end{bmatrix}.$ (7)

As the overall energy scale does not affect the behaviour, we shall set
$\gamma=1$ from hereon.

If the system is perturbed, say by slightly varying one of the couplings by
$\epsilon$, the perturbed Hamiltonian reads

$H_{4}(\epsilon)=\begin{bmatrix}i\delta&p+\epsilon&0&0\\\ p+\epsilon&i&q&0\\\
0&q&-i&p\\\ 0&0&p&-i\delta\\\ \end{bmatrix}.$ (8)

We present the Newton polygon of the characteristic polynomial in Figure 3(b).
Observe that if the point at $(0,0)$ is absent, we would get an EP2. The part
of the three-dimensional parameter space which shows a second order EP is thus
the surface where the coefficient of $\omega^{0}\epsilon^{0}$ vanishes, which
is shown in red in Figure 3(c). Now, if the point $(2,0)$ was also absent, we
would get an EP4. The parameter space for which this vertex vanishes is shown
in blue in Figure 3(c). The locus of EP4 is then the curve formed by the
intersection of these two surface [dashed line in Figure 3(c)]. Note that this
system cannot give us an EP3 because of the $PT$-symmetry. Our Newton polygon
framework, thus, enables the identification and selection of higher order EPs
in a straightforward manner.

We verify these predictions from our method in several ways. First, we show
that close to the EPs, our bands scale exactly as $\omega\sim\epsilon^{1/N}$,
where $N$ is the order of the EP as determined from our approach [Figure 3(d)
and (f)]. The Newton polygon method also allows calculating the coefficients
of the Puiseux expansions. For example, we obtain
$\omega=\sqrt{\frac{2p(\delta+p^{2})}{2p^{2}+q^{2}-\delta^{2}-1}}\epsilon^{1/2}$
to first order for EP2, which has an excellent match to the numerical fit.
Derivations of such analytical expressions are presented in the Supplementary
Information supplement . As a second check, we numerically show that if we
encircle the EPs, the bands swap among themselves as expected. Upon encircling
an EPN, $N$ eigenvalues undergo a cyclic perturbation among themselves
supplement . For an EP2, two of the bands swap [Figure 3 (e)], while for the
EP4, all four bands undergo a cyclic permutation [Figure 3 (g)].

We remark here that Zhang et al. have studied a model for supersymmetric
arrays showing an EP4 supersymmetric_PhysRevA.101.033820 . Their model is a
special case of ours with $p=\sqrt{3}$, $\delta=3$, and $q=2$.

The 4-site model we presented here can show tunable second and fourth order
EPs. Analogously, we have devised a 5-site $PT$-symmetric model. Using the
Newton polygon framework, we have shown that it can exhibit tunable third and
fifth order EPs supplement . Such $PT$-symmetric models have been
experimentally realized in magnetic multi-layers
PTmagnetic_PhysRevB.101.144414 , waveguides Zhong_2016 ; RevModPhys.88.035002
, topoelectric circuits PTelectric_PhysRevLett.126.215302 , and photonic
lattices El-Ganainy2018 ; Feng2017 ; nanophotonics_review ;
anotherRev_10.1093/ptep/ptaa094 . The Newton polygon method can serve as a
useful tool for tuning to EPs in these experimental platforms.

### Extension to higher number of variables

Figure 4: Determination of anisotropic behaviour around an exceptional point
using Newton polygons. For most directions in the $\epsilon$-$\lambda$ plane
making an angle $\phi$ with the $\epsilon$-axis, the Newton polygon has the
form in (i.a) implying $\omega\sim\epsilon^{1/3}$. However, the $(0,1)$ term
can vanish for $\phi=-\pi/4$ giving us the Newton polygon (ii.a) which gives
$\omega\sim\epsilon^{1/2}$ instead. The Newton Polygon method thus predicts
second order behaviour along $\phi=-\pi/4$, but third order behaviour along
all other directions. Numerical verification of these predictions are shown in
(b) and (c). Note that the numerical band structures along a chosen direction
(in green) closely match the predicted behaviours (in red) along that
direction. Panels (i.c) and (ii.c) show the behaviour of the phase rigidity,
$r$. The phase rigidity scales as $\epsilon^{2/3}$ for $\phi\neq-\pi/4$ as is
characteristic for an EP3, while it scales as $\epsilon^{1}$ for $\phi=-\pi/4$
confirming an EP2. For plots in the left panel, $\phi=\pi/6$ was chosen as a
representative direction.

The Newton polygon approach can be naturally extended to study exceptional
behaviour for higher number of variables. Remarkably, it has been recently
shown that if a system at an EP can be perturbed in two different ways such
that $H(\epsilon,\lambda)=H_{0}+\epsilon H_{1}+\lambda H_{2}$, then it
possible to observe different exceptional behaviours along different
directions in the $\epsilon$-$\lambda$ plane science_nexus ;
exp_aniso_PhysRevLett.121.085702 . The Newton polygon framework can predict
such anisotropic variations, as we show next. We consider the Hamiltonian
given in Equation 2 and add a second perturbation, $\lambda$, to obtain

$H(\epsilon,\lambda)=\begin{bmatrix}ig+\epsilon&\kappa&0\\\ \kappa&0&\kappa\\\
0&\kappa&-ig+\lambda\end{bmatrix}.$ (9)

We can set $g=\sqrt{2}\kappa$ to obtain exceptional behaviour, as discussed
earlier. If we pick any direction in the $\epsilon$-$\lambda$ plane making an
angle $\phi$ with the $\epsilon$-axis, then along that direction
$\lambda=\epsilon\tan{\phi}$. We can now use the Newton polygon method to
determine, in one go, the order of the EPs along all directions (i.e., for all
values of $\phi$). The characteristic equation reads (for $\kappa=1$)

$\displaystyle-\omega^{3}+(\tan\phi+1)\ \omega^{2}\epsilon-\tan\phi\
\omega\epsilon^{2}+$ $\displaystyle i\sqrt{2}(\tan\phi-1)\
\omega\epsilon-(\tan\phi+1)\ \epsilon=0.$ (10)

The corresponding Newton polygon is shown in Figure 4(a). Notice that
$\omega\sim\epsilon^{1/3}$ along all directions unless $\tan\phi+1=0$ for
which the point $(0,1)$ vanishes from the Newton polygon. In this case we
instead obtain $\omega\sim\epsilon^{1/2}$. The Newton polygon approach thus
predicts the presence of an anisotropic EP, which shows second order behaviour
along $\phi=-\pi/4$, but third order behaviour along all other directions. We
note that a similar model along two special directions has been studied before
signatures , although a complete picture along all possible directions was
lacking. Our method provides a unified way of obtaining the exceptional
properties along all directions.

We again numerically verify the predictions from the Newton polygon method in
multiple ways (see Figure 4). We first verify our analytical expressions for
the eigenvalues by fitting them to the numerical band structure. Next, we
study the variation of the phase rigidity, $r$, with the perturbation.
Physically, phase rigidity is a measure of the bi-orthogonality of the
eigenfunctions, and helps identify EPs muller2008exceptional ;
exp_aniso_PhysRevLett.121.085702 ; xiao2019anisotropic . We present further
details in Methods. Here, we find that the phase rigidity vanishes at the
location of the EP at $\epsilon=0$ [Figure 4 (c)]. Importantly, we observe
that the phase rigidity scales as $\epsilon^{2/3}$ for $\phi\neq-\pi/4$, while
it scales as $\epsilon^{1}$ for $\phi=-\pi/4$, thus confirming our predictions
of the anisotropic nature of the EP.

## Discussions

We put forward a new algebraic framework using Newton polygons for
characterizing EPs. Using several examples, we illustrated how the Newton
polygon method enables prediction and selection of higher order EPs. Using the
celebrated Hatano-Nelson model, we showed how this method allows prediction of
the non-Hermitian skin effect. We also proposed an extension to higher number
of variables and used it to reveal rich anisotropic behaviour around such non-
Hermitian degeneracies.

Looking ahead, our analytical approach could be useful for tuning to EPs in
different experimental platforms, whether it be for enhanced performance of
sensors or exploring unconventional phase transitions, especially as the
dimensionality and complexity of the non-Hermitian Hamiltonians in question
increases. Newton polygons play a natural role in homotopy theory, braid
groups, knot theory and algebraic number theory brieskorn2012plane ;
edwards2005essays . Our work lays the foundation for exploring such
connections in the context of non-Hermitian physics. With the recent growing
interest in knotted and linked exceptional nodal systems
carlstrom2018exceptional ; yang2019non ; carlstrom2019knotted ; yang2020jones
; hu2021knots ; zhang2021tidal ; wang2020simulating , it would be worthwhile
to find a Newton polygon based characterization of such systems. In
conclusion, we hope that our results inspire further exploration of EPs and
their applications.

## Methods

### Computing the coefficients and higher-order terms

Our Newton polygon approach also provides a straightforward algorithmic method
to evaluate the expansion of the eigenvalues beyond just the leading order. We
describe here the steps for finding the coefficients and higher order terms in
the Puiseux series expansion of eigenvalues.

Any solution of the characteristic equation $p(\omega,\epsilon)=0$ has the
form

$\omega=c_{1}\epsilon^{\gamma_{1}}+c_{2}\epsilon^{\gamma_{1}+\gamma_{2}}+c_{3}\epsilon^{\gamma_{1}+\gamma_{2}+\gamma_{3}}+....$
(11)

The steps for computing $\gamma_{1}$ were described earlier. Once $\gamma_{1}$
has been determined, we can write
$\omega=\epsilon^{\gamma_{1}}(c_{1}+\omega_{1})$ where
$\omega_{1}=c_{2}\epsilon^{\gamma_{2}}+c_{3}\epsilon^{\gamma_{2}+\gamma_{3}}+...$.

The steps to evaluate the other unknowns in the expansion are as follows.

  1. 1.

Collect the lowest order terms in $\epsilon$ in the polynomial
$p(\epsilon^{\gamma_{1}}(c_{1}+\omega_{1}),\epsilon)$. They must cancel each
other as $p(\omega,\epsilon)=0$. The first coefficient $c_{1}$ can be
extracted from this requirement.

  2. 2.

To get the next order term, find the polynomial
$p_{1}(\omega,\epsilon)=\epsilon^{-\beta}p(\epsilon^{\gamma_{1}}(c_{1}+\omega_{1}),\epsilon)$
where $\beta$ is the $y$-intercept of the line segment whose slope gave us
$-\gamma_{1}$.

  3. 3.

Next, calculate the Newton polygon for $p_{1}(\omega,\epsilon)$ and repeat the
steps to find $\gamma_{2}$ and $c_{2}$, and so on.

### Diagnosing exceptional points using phase rigidity

The phase rigidity, $r_{\alpha}$, is defined as eleuch2016clustering ;
alvarez2018topological

$r_{\alpha}=\dfrac{\langle\psi_{\alpha}|\phi_{\alpha}\rangle}{\langle\phi_{\alpha}|\phi_{\alpha}\rangle},$
(12)

where $\psi_{\alpha}$ and $\phi_{\alpha}$ are the normalized biorthogonal left
and right eigenvectors of a state $\alpha$. The phase rigidity quantitatively
measures the eigenfunctions’ bi-orthogonality. At an EP the states coalesce
and thus $|r_{\alpha}|$ vanishes. The phase rigidity is also experimentally
measurable. It enables defining a critical exponent around an EP in the
parameter space zhang2020high ; tang2020exceptional . For example, around an
EP3, the scaling exponents are given by $(N-1)/N$ and $(N-1)/2$, where $N=3$
is the order of the EP.
We numerically evaluate phase rigidity for the bivariable model with
directional anisotropy, which shows that the phase rigidity scales as
$\epsilon^{2/3}$ and $\epsilon^{1}$ for $\phi\neq-\pi/4$ and $\phi=-\pi/4$,
respectively [see Fig. 3 (c)].

##  I Acknowledgements

We acknowledge Madhusudan Manjunath (Department of Mathematics, Indian
Institute of Technology, Bombay) and Chinmaya Kausik (Department of
Mathematics, University of Michigan) for illuminating discussions. R.J. thanks
the Kishore Vaigyanik Protsahan Yojana (KVPY) for a fellowship. A.B. is
supported by the Prime Minister’s Research Fellowship (PMRF). A.N.
acknowledges support from the startup grant of the Indian Institute of Science
(SG/MHRD-19-0001) and DST-SERB (project number SRG/2020/000153).

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#  Theory Requirements and Possibilities for the FCC-ee and other Future High
Energy and Precision Frontier Lepton Colliders111Input to the European
Strategy Particle Physics 2018-2020

Alain Blondel (Université de Genéve)  Ayres Freitas (University of Pittsburgh)
Janusz Gluza222Corresponding author<EMAIL_ADDRESS>and Tord Riemann
(U. Silesia)
Sven Heinemeyer (IFT/IFCA CSIC Madrid/Santander  ECI/UAM/CSIC Madrid)
Stanisław Jadach (IFJ PAN Kraków)  Patrick Janot (CERN)

(18 December 2018)

###### Abstract

The future lepton colliders proposed for the High Energy and Precision
Frontier set stringent demands on theory. The most ambitious, broad-reaching
and demanding project is the FCC-ee. We consider here the present status and
requirements on precision calculations, possible ways forward and novel
methods, to match the experimental accuracies expected at the FCC-ee. We
conclude that the challenge can be tackled by a distributed collaborative
effort in academic institutions around the world, provided sufficient support,
which is estimated to about 500 man-years over the next 20 years.

##  1 Shaping the future of particle physics theory with FCC-ee

The FCC project proposes to build a sequence of colliders in a common
infrastructure close to CERN. It is a unique opportunity to step up in our
understanding the physical world at the smallest scales. It is the most
versatile, feasible high energy collider project for the next few decades.
Two, or even four, interaction points on the 100 km facility (important for
setting up independent experimental analyses) along with at least three
different types of collisions ($ee$ as a first step, $hh$ and $eh$ in a second
life, and may be more) will challenge the physics community with a very broad
program. Fine details of the Standard Model theory of particle interactions
will be revealed; even more importantly, the widest possible spectrum of new
theoretical concepts in the fundamental laws of physics will be explored. This
quest will take several forms: i) exquisite precision measurements, ii) search
for tiny violations of conservation laws, or iii) discovery of extremely rare
phenomena forbidden in the Standard Model. Details can be found in [1].

The following three characteristics make the FCC multi-task and multi-purpose
circular collider project special in its theoretical challenges. They call for
a proper strategy for theoretical computations to back the analysis and
interpretation of FCC-ee data.

  1. 1.

FCC-ee: the first stage of the most powerful high energy discovery tool. The
FCC-ee will provide a set of ground-breaking measurements of a large number of
new-physics sensitive observables, with improvement by one to two orders of
magnitude in experimental precision. This will require the corresponding
improvement in calculation ability with respect to the present baseline of
perturbative Quantum Field Theory (pQFT). An important feature of FCC-ee will
be to improve the precision of input parameters of the SM, in particular
$m_{Z},m_{t}$ and $\alpha_{QCD}(m_{Z})$; it will also offer a unique
possibility to measure directly $\alpha_{QED}(m_{Z})$, without the use of low
energy hadronic data. Theoretical calculations will have to keep pace with
these improvements. More precise knowledge of electroweak, Higgs and flavour
physics will open the way to accurate investigations of the most burning
questions in particle physics: like the baryon asymmetry of the Universe, the
nature of Dark Matter, the origin of neutrino masses, to name a few. See FCC
CDR vol. 1 [1] for more extensive discussions.

  2. 2.

FCC-ee: an interdisciplinary challenge. An interdisciplinary character of the
FCC-ee project is manifest in the accelerator itself and the experiments where
cutting edge technologies, equipment, facilities and software must be
developed, often in the cooperation with the industry.
Presently available techniques of pQFT are unable to match the experimental
precision of the future FCC data for almost all observables. The needs are
well defined, especially in the case of the most demanding FCC-ee Tera-Z case
[2, 3]. Future theoretical computations for FCC-ee physics will require a
considerable synergy of pQFT with advanced and explorative mathematics in
order to develop new computational methods to deal with much higher level of
complexity of the anticipated pQFT calculations. Development of the new
sophisticated (Monte Carlo) programs for direct simulation of the scattering
processes at FCC-ee will provide another important synergy with the
corresponding branch of the computing science. This connection between
advanced pQFT and advanced exploratory mathematics and computing has played a
fertile role in the development of both branches of science for a long time.
New challenges in pQFT will induce significant progress in the computer
algebra systems, algebraic geometry – new cutting edge algorithms for
analytical and numerical methods will be developed, to the profit of several
branches in advanced mathematics and computing science.

  3. 3.

FCC-ee: highly demanding collective educational, scientific and social
networking. The enormous diversity of the physics topics that will be explored
at several stages on the FCC constitutes a challenge both to theory and
experiment parts of the grand project, concerning a proper organization of the
work and the choice of dedicated teams. The project needs skillful and
versatile researchers, in all stages of their scientific career. Well prepared
educational and research theoretical programs must be set up, including
Initial Training Networks, ERC and national grants. A world-wide collaborative
effort of theoretical physicists and contributors from related relevant
disciplines will be carried out mainly in the framework of the academic
environment. An initial estimate of the efforts related to development of new
theory calculation techniques and tools for FCC experiments points to about
500 person years, corresponding to about 25 full-time-equivalent TH experts
throughout the design, construction and operation phases of the FCC-ee
programme.

##  2 Precision theoretical issues and objectives at the FCC-ee

FCC-ee defines a very clean experimental set-up compared to hadron colliders,
but the demands in precision are much higher. Initial state QED radiation is
well understood, leading to the possibility of very precise measurements
requiring leap-jumps in the precision of theoretical computations for Standard
Model phenomena and so-called electroweak pseudo-observables (EWPOs). Such are
partial widths and couplings of Z and W, forward-backward and polarization
asymmetries, peak cross sections at the Z resonance position, effective
electroweak mixing angle [4, 2], and additional EWPOs in the WW, ZH and
$t\bar{t}$ processes. EWPOs encapsulate experimental data after extraction of
well known and controllable QED and QCD effects, in a model-independent
manner. They provide a convenient bridge between real data and the predictions
of the SM (or SM plus New Physics). Contrary to raw experimental data (like
differential cross sections), EWPOs are well suited for archiving and long
term exploitation. In particular archived EWPOs can be exploited over long
periods of time for comparisons with steadily improving theoretical
calculations of the SM predictions, and for validations of the New Physics
models beyond the SM. They are also useful for comparison and combination of
results from different experiments.

For extracting the interesting physics from data in form of EWPOs it will be
mandatory to develop a library of new Monte Carlo (MC) event generators
implementing both QED and EW/QCD higher order effects. They will have to be
developed for all processes at FCC-ee in all four stages, Tera-Z, WW, HZ and
$t\bar{t}$. Well tested legacy MC programs of the LEP era can be used for
benchmarking purposes, but the development of new event generation techniques
and programs, should be done in parallel with a campaign of advances in the
theoretical physics calculation methods.

Fitting EWPOs to data near Z-resonance was possible at LEP using non-MC semi-
analytical calculations of QED effects implemented in ZFITTER and TOPAZ0
programs, and MC generators such as KORALZ. However, already at LEP the
extraction of W mass from WW process was done by means of a direct fitting of
the output of the MC programs (with W mass as a parameter) to WW cross section
and final four fermion distributions, without any use of analytical formulas,
with all QED effects handled by the MC [5]. For FCC-ee data analysis, due to
the rise of non-factorisable QED effects above the experimental errors, the
latter way of using MC programs might become the standard for fitting EWPOs to
data, even at the Tera-Z stage. New MC event generators will have to provide
built-in provisions for efficient direct fitting of EWPOs to data, which are
not present in the LEP legacy MCs. Section C3 of [2] describes possible forms
of future EWPOs at FCC-ee experiments and specification of a new required MC
software is given. It is underlined there that due to non-factorisable QED
contributions, the factorization of the multiphoton QED effects will have to
be done at the amplitude level. Additional quantities available in tau and
heavy flavour physics will reach the $10^{-5}$ precision and are likely to
need similar attention.

The state of art and the open issues for these analysis in the context of FCC-
ee near $Z$-boson resonance has been reported in [2]. As FCC will start from
$e^{+}e^{-}$ Tera-Z option and it will bring the highest luminosities and
stringiest experimental results, we quote below the main goals and tasks,
adapted from the executive summary given in [2].

  1. 1.

In order to meet the experimental precision of the FCC-ee Tera-Z for EWPOs,
3-loop and partial 4-loop calculations of the $Zf{\bar{f}}$-vertex will be
needed. The leading 3-loop corrections are
${\cal{O}}(\alpha\alpha_{s}^{2}),{\cal{O}}(N_{f}\alpha^{2}\alpha_{s}),{\cal{O}}(N_{f}^{2}\alpha^{3})$,
where $\alpha$ denotes an electroweak loop, and $N_{f}^{n}$ denotes $n$ or
more closed internal fermion loops. Sub-leading corrections may also be
needed, depending on insights gained from computing the leading terms.

  2. 2.

Full 2-loop corrections to the $Zf{\bar{f}}$-vertex have been completed
recently. The principal techniques for the electroweak loop calculations have
been and will likely be numerical, due to the large number of scales involved.
A few digits of internal precision will be sufficient at the 3-loop level.
Several additional exploratory strategies, methods and tools have been also
identified and discussed.

  3. 3.

The $Zf{\bar{f}}$-vertex corrections are embedded in a structure describing
the hard scattering process $e^{+}e^{-}\to f{\bar{f}}$, based on matrix
elements in form of Laurent series around the $Z$ pole. Here, additional non-
trivial contributions like 2-loop weak box diagrams must be implemented
properly.

  4. 4.

The experience gained in the high precision calculation for physics near Z
pole will provide a strong basis for more advanced treatment of four fermion
processes in the W mass and width measurements. In spite of the fact that the
experimental precision is less demanding, (permil precision rather than a few
per-million), more precise SM calculations will be required for the Higgs and
top physics, as well.

  5. 5.

Numerically dominant effects due multi-photon emission constitute another key
problem; their complexity will be often comparable to that of the electroweak
loop calculations. The methodology of joint treatment of electroweak and QCD
loop corrections with the photonic corrections is essentially at hand. In
practice, however, construction of new Monte Carlo programs, which can handle
efficiently the above problems, will need special efforts [6].

The elementary techniques for higher order perturbative SM corrections are
basically understood, but their use in practical calculations at the level of
computer programs will be highly nontrivial and will require considerable
effort. The understanding of all sources of possible theoretical uncertainties
will be fundamental for success of the FCC-ee data analysis.

| $\delta\Gamma_{Z}\;[\rm MeV]$ | $\delta R_{l}\;[10^{-4}]$ | $\delta R_{b}\;[10^{-5}]$ | $\delta\sin^{2,l}_{eff}\theta\;[10^{-6}]$
---|---|---|---|---
Present EWPO theoretical uncertainties
EXP-2018 | 2.3 | $250$ | $66$ | $160$
TH-2018 | 0.4 | 60 | $10$ | $45$
EWPO theoretical uncertainties when FCC-ee will start
EXP-FCC-ee | 0.1 | 10 | $2\div 6$ | $6$
TH-FCC-ee | $0.07$ | $7$ | $3$ | $7$
Table 1: Comparison for selected precision observables of present experimental
measurements (EXP-2018), current theory errors (TH-2018), FCC-ee precision
goals at the end of the Tera-Z run (EXP-FCC-ee) and rough estimates of the
theory errors assuming that electroweak 3-loop corrections and the dominant
4-loop EW-QCD corrections are available at the start of FCC-ee (TH-FCC-ee).
Based on discussion in [2].

Table 1 shows the current experimental and theoretical errors (EXP-2018,
TH-2018) for some basic Z-physics EWPOs, and the prospective measurement
errors at FCC-ee (EXP-FCC-ee) together with the corresponding estimate for
theoretical uncertainties after the leading 3/4-loop results become available
(TH-FCC-ee). The entry TH-2018 takes into account recent completion of the
2-loop electroweak calculations [7, 8], so the error estimate comes solely
from an estimate of magnitudes of missing 3-loop and 4-loop EW and mixed EW-
QCD corrections. The estimated TH-FCC-ee error stems from remaining 4-loop and
higher effects. These are rather difficult to estimate presently, however, a
rough conservative upper bound on them has been provided in [2]. They are
denoted in Tab. 1 as TH-FCC-ee. As we can see, the uncertainties of the TH-
FCC-ee scenario are comparable to the corresponding EXP-FCC-ee experimental
errors and will not limit the FCC-ee physics goals.

$Z\rightarrow b\bar{b}$
---
| 1 loop |  2 loops |  3 loops
Number of topologies | 1 | 5 | 51
Number of diagrams | 15 | 1074 | 120472
Fermionic loops | 0 | ${150}$ | $17580$
Bosonic loops | 15 | ${924}$ | ${102892}$
QCD / EW | 1 / 14 | 98 / 1016 | ${10386/110086}$
Table 2: Numbers of relevant topologies and diagrams for $Z\to b\bar{b}$
decays. Adapted from [2].

An illustration of complexity of future perturbative calculations is provided
in Table 2, where the numbers of distinct topologies of diagrams to be
calculated and the numbers of diagrams and various categories at the 1-, 2-
and 3-loop order are shown for the most complicated $Z\to b\bar{b}$ decay.
According to Ref. [2], the calculation of fermionic 3-loop corrections, which
will be the largest part of them, are within reach, already with present
methods and tools.

Another important issue in electroweak fits are uncertainties from the
existing (often experimental) errors on SM input parameters, themselves EWPOs,
which affect the SM predictions of EWPOs when comparing them with experiment.

Well-defined subsets of experimental data (EWPOs) are treated as input SM
parameters, like [9, 10] for instance: Fermi constant (from muon decay),
$\alpha_{QED}(M_{Z})$ (from the dispersion integral of $e^{+}e^{-}$ data at
low energies), $\alpha_{s}(M_{Z})$ (from the measurement of $R_{\ell}$ and
$R_{b}$ at the Z peak), $M_{Z}$, $m_{t}$, $M_{H}$ and the masses of other
fermions – all of them with well defined experimental and theoretical errors.
Remaining EWPOs, like $\sin^{2}\theta^{eff}_{W}$, $m_{W}$, $\Gamma_{Z}$, and
more, can be measured either directly with an experimental error or can be
predicted using SM perturbative machinery, with parametric errors inherited
from the chosen subset of input parameters (EWPOs).

EWPO | Exp. direct error | Param. error | Main source | Theory uncert.
---|---|---|---|---
$\Gamma_{{Z}}$ [MeV] | 0.1 | 0.1 | $\delta\alpha_{{s}}$ | 0.07
$R_{b}$ [$10^{-5}$] | 6 | $1$ | $\delta\alpha_{{s}}$ | 3
$R_{\ell}$ [$10^{-3}$] | 1 | $1.3$ | $\delta\alpha_{{s}}$ | 0.7
$\sin^{2}\theta_{\rm eff}^{\ell}$ [$10^{-5}$] | 0.5 | 1 | $\delta(\Delta\alpha)$ | 0.7
$M_{{W}}$ [MeV] | $0.5$ | 0.6 | $\delta(\Delta\alpha)$ | 0.3
Table 3: Estimated experimental precision for the direct measurement of
several important EWPOs at FCC-ee [2] (column two) and experimental parametric
error (column three), with the main source shown in the forth column.
Important input parameter errors are $\delta(\Delta\alpha)=3\cdot 10^{-5}$,
$\delta\alpha_{{s}}=0.00015$ see FCC CDR, vol. 2 [1]. Last column shows
anticipated theory uncertainties at start of FCC-ee.

In Table. 3 the size and main sources of parametric errors connected with QED
and QCD running couplings for selected important EWPOs are shown. It is
important to note that e.g. $\delta\alpha_{{s}}$ is a common parametric error
on $\Gamma_{{Z}}$ and $R_{\ell}$, and will not affect e.g. the leptonic
partial width of the Z. These EWPOs serving as SM input parameters will be
determined very precisely at the FCC-ee. It should be noted that determination
of $\Delta\alpha$ with error at the level of $3\cdot 10^{-5}$ assumed in the
table is from present perspective a big challenge. Standard approach
exploiting low energy hadronic contributions to vacuum polarization will try
to reach such a precision. Another independent method aiming at the same
precision was proposed [11]. It relies on measuring at FCC-ee charge asymmetry
of the $e^{+}e^{-}\to\mu^{+}\mu^{-}$ process very precisely at $s^{1/2}\simeq
M_{Z}\pm 3.5$ GeV. Future projections of the theory uncertainty are also shown
in Table. 3. Both theoretical uncertainties and parametric errors are smaller
or comparable to the corresponding experimental direct errors.

To complete the picture of precision studies at the FCC-ee, the remaining FCC-
ee-W, FCC-ee-H and FCC-ee-t stages of the $e^{+}e^{-}$ collider should be also
considered.

Very precise determinations of $M_{W}$ at FCC-ee will rely on the precise
measurement of the cross section of the $e^{+}e^{-}\to W^{+}W^{-}$ process
near the threshold. A statistical precision of $0.04\%$ of this cross section
translates into 0.6 MeV experimental error on $M_{W}$, which is almost three
orders of magnitude better than at LEP. Theoretical legacy calculations from
the LEP era for this process with a precision tag of $\sim 0.2\%$ would yield
an unacceptable 3 MeV theory error for $M_{W}$ [12]. Therefore, improved
theoretical calculations are required for the generic $e^{+}e^{-}\to 4f$
process near the WW threshold with an improvement of one order of magnitude.
As advocated in [13], the most economical solution will be to combine ${\cal
O}(\alpha^{1})$ calculation for $e^{+}e^{-}\to 4f$ process with ${\cal
O}(\alpha^{2})$ calculation for the doubly-resonant $e^{+}e^{-}\to W^{+}W^{-}$
subprocess. The former calculation is already available [14]. The latter will
need to be developed; Inclusion of the resummed QED corrections will be
mandatory.

Present estimation for the above two-loop electroweak ${\cal O}(\alpha^{2})$
corrections based on the unstable-particle Effective Field Theory approach
translates into 1 MeV theory uncertainty of $M_{W}$ [15, 16]. Assuming that
the two-loop electroweak ${\cal O}(\alpha^{2})$ corrections are known, an
estimate of the missing three-loop electroweak ${\cal O}(\alpha^{3})$
corrections in such a hybrid calculation [13], shows that $\sim 0.02\%$ theory
precision for the threshold cross section could be achieved, which translates
into 0.3 MeV theory uncertainty of $M_{W}$ given in Tab. 3, comparable with
the FCC-ee experimental precision. On the other hand, $M_{W}$ mass will be
also extracted from the subsequent decays of W’s to four fermions, away from
the threshold. Most likely, the same progress in the theory calculations will
be mandatory for this measurement as well.

In case of FCC-ee-H, $M_{H}$ will be obtained from $e^{+}e^{-}\to HZ$ process
with the precision better of order 10 MeV [1]. Theory uncertainties (mainly
due to final-state radiation effects) will be subdominant.

The anticipated experimental error for $m_{t}$ measurement at FCC-ee-t [2] is
$\delta m_{{t}}\sim{\cal O}(20)\,\,\mathrm{MeV}$. On the theory side, there
are several uncertainty sources: _(i)_ The perturbative uncertainty for the
calculation of the threshold shape with higher order QCD corrections, _(ii)_
the threshold mass definition translated into the $\overline{\text{MS}}$
scheme _(iii)_ precision of $\alpha_{{s}}$. Combining these three error
sources, a theory uncertainty close to experimental and less than $50$ MeV for
$m_{{t}}$ appears feasible. In addition a very accurate determination of the
efficiency of experimental acceptances and selection cuts is needed. This task
will require the inclusion of higher-order corrections and re-summation
results in a Monte-Carlo event generator. NLO QCD corrections for off-shell
$t\bar{t}$ production, and matching between these contributions are
complementing previous semi-analytic results.

An initial workshop "Precision EW and QCD calculations for the FCC studies:
methods and tools" was held at CERN in January 2018 [3] to start the process
of preparing long-term research for precision calculations needed in the FCC-
ee Tera-Z physics. This activity will continue in 2019 [17]; this follow-up
workshop will focus on establishing solid plans for precision calculations the
FCC-ee project connected with W, H and top quark physics.

##  3 The need for development of methods and refined tools in theoretical
calculations

An estimation of theory errors has always to be taken with a grain of salt. As
was shown recently in [8] they can differ from the actual calculations even by
factors 3-5. There is no other way than to perform the concrete calculation of
higher loop effects, and, in view of the huge complexity, ideally with at
least two independent methods and/or teams. In fact, we will need the leading
electroweak 3-loop and QCD 4-loop contributions in order not to limit the
interpretation of the Z resonance shape, and there are similar demands from
the other FCC-ee modes. There is no closed form theory for perturbation theory
calculations of Feynman integrals beyond one loop. For this reason, numerical
integration methods are the most promising, if not the only, avenues for
addressing that challenges. Analytical techniques are expected to be important
in many respects, but numerical integration methods have advantages for
increasing number of masses and momentum scales. Fortunately, there is an
impressive progress in recent years in this direction [2]. The renormalization
and treatment of ultra-violet divergences is basically understood. There are
only two numerical methods known to allow a systematic treatment of infra-red
divergences. In 2014 the only advanced automatic numerical two-loop method was
one of them, sector decomposition SD. However, the corresponding software was
not capable to work out the complete set of Feynman integrals for the electro-
weak bosonic two-loop corrections to the $Z$-boson decay with the desired high
precision (up to 8 digits per integral). The task could be completed
succesfully with a substantial development of the other numerical approach,
based on Mellin-Barnes MB representations of Feynman integrals [7, 8]. These
calculations are challenging due to the numerical role of particle masses
$M_{Z},M_{W},m_{t},M_{H}$, leading to (i) a huge number of contributions,
ranging from tens to hundreds of thousands of diagrams (at 3-loops), and (ii)
the occurrence of up to four dimensionless parameters in Minkowskian
kinematics (at $s=M_{Z}^{2}$) with intricate threshold and on-shell effects
where contour deformation fails. To tackle more loops or legs, merging both
the MB- and SD-methods in numerical calculations is a key for final success.

A well founded expectation is that these two methods, combined with more
specialized techniques for specific topologies, will allow to execute the next
steps of the necessary multi-loop calculations. There are also many
opportunities for improvement in other approaches and software packages; see
[2] for a year-2018 overview. There is a lively activity in many areas of
multi-loop calculations, and several promising (semi-) analytical methods are
also under development [2]. For instance a treatment of multi-scale integrals
beyond multiple polylogarithms, direct numerical calculations of Feynman
integrals in $d=4$, or calculations based on unitarity methods, all of them
with first examples of two-loop numerical evaluations.

Concerning the numerically important QED corrections, further refinements of
the factorization to infinite order of the multi-photon soft real and virtual
contributions, and their re-summation, using Monte Carlo methods have to be
pursued in order to meet the accuracy needs. Again infra-red problems appear
to be crucial: The inclusion of collinear (mass) singularities must be
improved in the existing schemes. Techniques for disentangling QED and EW
corrections for two and more loops in the framework of soft photon
factorization/resummation are in principle available, but their practical
implementation will require more work. A central issue is the implementation
of the higher loop terms in their proper form as Laurent series around the
Z-boson pole, characterized by mass and width, in the scattering energy, which
is mandated by analyticity, gauge-invariance and unitarity.

##  4 Summary

FCC-ee, a circular collider with extremely high statistics and high energy
resolution, will provide the possibility to test the Standard Model with its
fine quantum electroweak effects with a precision far beyond the current state
of the art. Significant future theory effort will be needed for both for
parametric and theoretical calculational errors to match the experimental
accuracy of FCC-ee physics program. No potential showstoppers are foreseen [1,
2]. It will be important that adequate theory funding will be available to
ensure that theory uncertainties are reduced to the desired level. The
combined advances in experimental and theoretical techniques will yield
unprecedented sensitivity to very weakly coupled or very massive new physics,
or to forbidden violations of symmetries, at FCC-ee.

##  5 Acknowledgments

The authors acknowledge the contributions of all participants in the FCC-ee
working groups, especially the participants of the workshop on theory
calculations [3].

A.B. gratefully acknowledges CERN hospitality. The work of T.R. was supported
in part by an Alexander von Humboldt Polish Honorary Research Fellowship. The
work of J.G. was supported by the Polish National Science Centre (NCN) under
the Grant Agreement 2017/25/B/ST2/01987. The work of A.F. was supported in
part by the National Science Foundation under grant no. PHY-1519175. The work
of S.H. was supported in part by the MEINCOP (Spain) under contract
FPA2016-78022-P, in part by the Spanish Agencia Estatal de Investigación
(AEI), in part by the EU Fondo Europeo de Desarrollo Regional (FEDER) through
the project FPA2016-78645-P, in part by the “Spanish Red Consolider MultiDark”
FPA2017-90566-REDC, and in part by the AEI through the grant IFT Centro de
Excelencia Severo Ochoa SEV-2016-0597. The work of S.J. was partly supported
by the Polish National Science Center grant 2016/23/B/ST2/03927.

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