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# Electroweak Precision Measurements of a Nearly-Degenerate $Z^{\prime}$-$Z$
System

Dayun Qiu  Yi-Lei Tang  School of Physics, Sun Yat-Sen University, Guangzhou
510275, China

###### Abstract

In this paper, we discuss the possibility to probe a nearly-degenerate
$Z^{\prime}$-$Z$ system by analyzing the $Z$-lineshape at an electron-positron
collider. Compared with the usual $Z^{\prime}$ in the literature well
separated with the standard mocel (SM) $Z$ boson in mass, the nearly-
degenerate $Z^{\prime}$-$Z$ mixing affects the observed effective “oblique
parameters” $\tilde{S}$, $\tilde{T}$, $\tilde{U}$, and the effective deviation
of “number of neutrino species” $\delta\tilde{N}_{\nu}$ in a more complicated
way and cannot be simply computed perturbatively up to a particular order.
Aiming at solving this problem, we write down a general simplified effective
Lagrangian and enumerate some parameter spaces corresponding to some typical
models, and suggest a method to extract the constraints by looking into the
line-shape of the $Z$-like resonance at an electron-positron collider.

##  I Introduction

The standard model (SM) of particle physics can be extended with an additional
gauge group, thus accommodates various exotic vector bosons. The
electromagnetic neutral $Z^{\prime}$ boson, accompanied with an extra
$U(1)^{\prime}$ gauge symmetry is the simplest selectionFayet:1980ad ;
Fayet:1980rr ; Okun:1982xi ; Fayet:1990wx . $Z$-$Z^{\prime}$
mixingHoldom:1985ag ; Galison:1983pa ; Foot:1991kb  can arise due to the
absence of any unbroken symmetry below the electroweak scale, affecting
various phenomenologies for different searching proposals.

Straightforward “bump” searches are effective for a $Z^{\prime}$ with a
measurable interaction with the SM particles (For a review, see Ref.
Leike:1998wr ). In the literature, TevatronD0:2010kuq ; CDF:2011nxq ;
CDF:2006uji ; CDF:2005hun ; CDF:2011pih ; CDF:2008ieg ; CDF:2010zwq  and the
ATLAS and CMS collaborations at the LHCATLAS:2019erb ; CMS:2021ctt ;
CMS:2019buh ; LHCb:2020ysn ; ATLAS:2020zms ; CMS:2018rmh ; ATLAS:2019fgd ;
CMS:2021xuz ; ATLAS:2021suo  published constraints for such kind of
$Z^{\prime}$ boson with its mass well-separated with the SM-$Z$ boson. In
contrast with this “direct” strategy, an “oblique” way is to look into the
“oblique parameters”, e.g., Peskin-Takeuchi parameters $S$, $T$, and
$U$Peskin:1990zt ; Peskin:1991sw , and even $W$, $X$, $Y$, $Z$Barbieri:2004qk
, etc., to observe the hints imprinted on the electroweak precision
measurement parameters from an off-shell $Z^{\prime}$Holdom:1990xp ;
Babu:1997st  which might be veiled beneath its faint coupling with, or low
decay ratio into the visible SM particles during a straightforward searching
process. In the literature, most of the results utilizing the LEP data
actually follow this routeAltarelli:1990wt ; Richard:2003vc ; Peskin:2001rw .
The recent $W$ boson mass data published by the CDF collaborationCDF:2022hxs
and its deviation from the SM predicted value give rise to the possibility of
the exisitence of an exotic vector boson contributing to the electroweak
precision measurement valuesStrumia:2022qkt ; Asadi:2022xiy ; Fan:2022yly ;
Gu:2022htv ; Lu:2022bgw ; deBlas:2022hdk ; Zhang:2022nnh ; Zeng:2022lkk ;
Thomas:2022gib ; Cheng:2022aau ; Cai:2022cti ; Alguero:2022est ; Du:2022fqv ;
Harigaya:2023uhg .

Theoretical predictions on both “direct” and “oblique” ways to find a
$Z^{\prime}$ are based upon the perturbative expansions up to a particular
order. This is sufficient for the case with a significant mass difference
between $Z^{\prime}$ and the SM-$Z$, and the mixing angle is usually extremely
small. Yet in the literature, it seems that discussions about a nearly
degenerate $Z^{\prime}$-$Z$ system are rare.

Inspired by the famous $K^{0}$-$\overline{K^{0}}$ systemDonoghue:1992dd ;
Workman:2022ynf , we have realized that despite the Hermitian squared mass
elements, the non-Hermitian widths might also play important roles in finding
the “mass eigen-states” of two mixing states. In the case when two vector
bosons mix, diagonalizing a mass-squared matrix including the non-Hermitian
width contributions is equivalent to a resummation over all possible “string
diagrams” including the imaginary contributions from all possible one-loop
self-energy diagrams. Then these two “mass eigen-states” both contribute to
the line-shape of a $Z$-boson-like object at an electron-positron collider,
which might not be naively considered as a simple composition of two
resonances. Compared with a sheer SM-$Z$ line-shape, such an object might
appear as a distorted “resonance” to affect the electroweak precision
measurement results extracted from its appearance.

In this paper, we try to compute the electroweak precision measurement
distortions induced by a $\hat{Z}^{\prime}$ field which is nearly degenerate
with the SM $\hat{Z}$ field through diagonalizing their mass-squared matrix
including their widths. In order to compare our results with the famaliar
parameters, we define observables $\tilde{S}$, $\tilde{T}$, $\tilde{U}$ and
$\delta\tilde{N}_{\nu}$, corresponding to the well-known Peskin-Takeuchi
parameters $S$, $T$, $U$, and the deviation of the neutrino species $\delta
N_{\nu}$. Unlike $S$, $T$ and $U$, our $\tilde{S}$, $\tilde{T}$, $\tilde{U}$
only reflect the change of the line-shape, and is unable to be attributed to
some particular effective operator contributions. Since we are not able to
find the complete original LEP data published or extractedALEPH:2005ab , a
constraint on the $Z^{\prime}$-model parameter space is non-viable. As a
compromise, we adopt a set of events simulated in the conditions similar to
the LEP environments dubbed “pseudo-LEP” data to predict the sensitivity if
the real original LEP data are utilized.

Besides LEP, proposed future leptonic colliders, e.g., ILCILC:2013jhg ,
CEPCCEPCStudyGroup:2018ghi , FCC-eeFCC:2018evy , have been proposed with
extremely large integrated luminosities. Usually at least a calibration around
the $Z$-scale should be proceeded, at the same time electroweak precision
measurement data are updated by the way. For an example, the CEPC takes the
potential to produce $\sim 10^{11}$-$10^{12}$ $Z$-bosons, which can be
regarded as a super $Z$-factoryZheng:2018fqv  to significantly improve the
sensitivity of the oblique parameters. Simulating such a large “pseudo-CEPC”
data set is beyond our current computational resources, however its
sensitivities can still be estimated by utilizing the pseudo-LEP results.

This paper is organized as follows. In Sec. II, we introduce the effective
Lagrangian for an exotic vector boson $\hat{Z}^{\prime}$. Other basic concepts
are elaborated. Then simulation details and settings are illustrated in Sec.
III. In Sec. IV, the numerical results are presented and described in three
scenarios. Finally, Sec. V summarize this paper.

##  II Effective Lagrangian

In this paper, we rely on a simplified general effective Lagrangian introduced
in Ref. Cai:2022cti , which is rewritten as

$\displaystyle\mathcal{L}$ $\displaystyle\supset$
$\displaystyle-\frac{1}{4}\hat{Z}^{\prime}_{\mu\nu}\hat{Z}^{\prime\mu\nu}+\frac{1}{2}m_{\hat{Z}^{\prime}}^{2}\hat{Z}^{\prime}_{\mu}\hat{Z}^{\prime\mu}-\frac{\epsilon_{B}}{2}\hat{Z}^{\prime}_{\mu\nu}B^{\mu\nu}-\frac{1}{2\Lambda_{W}^{2}}\hat{Z}^{\prime}_{\mu\nu}W^{a\mu\nu}H^{\dagger}\sigma^{a}H$
(1) $\displaystyle-$
$\displaystyle\frac{1}{2\Lambda_{BW}^{2}}B_{\mu\nu}W^{a\mu\nu}H^{\dagger}\sigma^{a}H-\frac{1}{4\Lambda_{WW}^{4}}W^{a\mu\nu}H^{\dagger}\sigma^{a}HW^{b}_{\mu\nu}H^{\dagger}\sigma^{b}H$
$\displaystyle+$
$\displaystyle\frac{1}{\Lambda_{HD}^{2}}(H^{\dagger}D_{\mu}H)^{\dagger}(H^{\dagger}D^{\mu}H)+\hat{Z}^{\prime\mu}\left[i\lambda_{HZ^{\prime}}(D_{\mu}H)^{\dagger}H+\text{H.c.}\right].$

where $\hat{Z}^{\prime\mu}$ is the exotic neutral vector boson
$\hat{Z}^{\prime}$, and
$\hat{Z}^{\prime}_{\mu\nu}\equiv\partial_{\mu}\hat{Z}^{\prime}_{\nu}-\partial_{\nu}\hat{Z}^{\prime}_{\mu}$.
$H$ indicates the SM Higgs doublet, $W^{a\mu}$ and $B^{\mu}$ are the
$SU(2)_{L}$ and $U(1)_{Y}$ gauge fields respectively, and
$D_{\mu}=\partial_{\mu}-i\hat{g}^{\prime}_{Y}B_{\mu}/2-i\hat{g}\sigma^{a}W^{a}_{\mu}/2$,
where $\hat{g}^{\prime}$ and $\hat{g}$ are the “original” coupling constants.
$\epsilon_{B}$, $\Lambda_{W}$, $\Lambda_{BW}$, $\Lambda_{WW}$, $\Lambda_{HD}$,
$\lambda_{HZ^{\prime}}$ are the corresponding constants. Here the mass term
$m_{\hat{Z}^{\prime}}^{2}\hat{Z}^{\prime}_{\mu}\hat{Z}^{\prime\mu}$ is put by
hand, and might originate from an exotic Higgs carrying a $U(1)^{\prime}$
charge corresponding with the $Z^{\prime}$, or from the Stueckelberg
mechanisms.

After $H$ acquires the vacuum expectation value (VEV),

$\displaystyle H=\begin{pmatrix}i\phi^{+}\\\
\frac{\hat{v}+h+i\phi^{0}}{\sqrt{2}}\end{pmatrix},$ (2)

where $\hat{v}\approx 246\rm{GeV}$, we therefore acquire the effective
kinematic mixing terms

$\mathcal{L}_{\mathrm{eff}}\supset-\frac{\epsilon_{B}}{2}\hat{Z}^{\prime}_{\mu\nu}B^{\mu\nu}-\frac{\epsilon_{W}}{2}\hat{Z}^{\prime}_{\mu\nu}W^{3\mu\nu}-\frac{\epsilon_{BW}}{2}B_{\mu\nu}W^{3\mu\nu}-\frac{\epsilon_{WW}}{4}W^{3}_{\mu\nu}W^{3\mu\nu}.$
(3)

where
$\epsilon_{W}\equiv-\hat{v}^{2}/(2\Lambda_{W}^{2}),~{}\epsilon_{BW}\equiv-\hat{v}^{2}/(2\Lambda_{BW}^{2})$,
and $\epsilon_{WW}\equiv\hat{v}^{4}/(4\Lambda_{WW}^{4})$. Then, the mass terms
as well as the kinematic terms can be written in the form of matrices,

$\displaystyle\mathcal{L}_{mass}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\hat{Z}^{\prime}_{\mu},&B_{\mu},&W^{3}_{\mu}\end{pmatrix}\mathcal{M}_{V}^{2}\begin{pmatrix}\hat{Z}^{\prime\mu}\\\
B^{\mu}\\\ W^{3\mu}\end{pmatrix},$ (4) $\displaystyle\mathcal{M}_{V}^{2}$
$\displaystyle=$
$\displaystyle\begin{pmatrix}m_{\hat{Z}^{\prime}}^{2}&\hat{g}^{\prime}\delta
m^{2}&-\hat{g}\delta m^{2}\\\ \hat{g}^{\prime}\delta
m^{2}&\frac{\hat{g}^{\prime 2}}{4}(\hat{v}^{2}+\delta
v^{2})&-\frac{\hat{g}^{\prime}\hat{g}}{4}(\hat{v}^{2}+\delta v^{2})\\\
-\hat{g}\delta m^{2}&-\frac{\hat{g}^{\prime}\hat{g}}{4}(\hat{v}^{2}+\delta
v^{2})&\frac{\hat{g}^{2}}{4}(\hat{v}^{2}+\delta v^{2})\end{pmatrix}.$ (5)

and

$\displaystyle\mathcal{L}_{\mathrm{kin}}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\begin{pmatrix}\hat{Z}^{\prime}_{\mu\nu},&B_{\mu\nu},&W^{3}_{\mu\nu}\end{pmatrix}\mathcal{K}_{V}\begin{pmatrix}\hat{Z}^{\prime\mu\nu}\\\
B^{\mu\nu}\\\ W^{3\mu\nu}\end{pmatrix},$ (6) $\displaystyle\mathcal{K}_{V}$
$\displaystyle=$ $\displaystyle\begin{pmatrix}1&\epsilon_{B}&\epsilon_{W}\\\
\epsilon_{B}&1&\epsilon_{BW}\\\
\epsilon_{W}&\epsilon_{BW}&1+\epsilon_{WW}\end{pmatrix},$ (7)

where $\delta m^{2}\equiv-\lambda_{HZ^{\prime}}\hat{v}^{2}/2$ and $\delta
v^{2}\equiv\hat{v}^{4}/(4\Lambda_{HD}^{2})$. Ref. Cai:2022cti  aims at
diagonalizing (5) and (7), and perturbatively expand the results in the case
that $m_{Z^{\prime}}$ is well-separated with the $m_{Z}$.

The contributions from the self-energy diagrams are usually calculated through
two ways, order-by-order perturbatively, or resummed to correct the mass term
of every propagator. Near the resonance of each s-channel propagator, the
imaginary part of its self-energy diagrams must be resummed prior to any other
process and contributes to the Breit-Wigner form of the propagator by adding
up an imaginary part in the denominator. Other contributions may be taken into
account perturbatively order by order later, and behave like sub-leading
corrections in quantities.

Figure 1: Cross terms between $Z$ and $Z^{\prime}$ particles.

Besides the self-energy of each particle, the self-energy among different
types of particles, or the “cross terms” may also give rise to the imaginary
parts, as shown in Fig. 1 as an example. This is not a problem if $Z$ and
$Z^{\prime}$ are well separated in mass spectrum, and its contributions are
suppressed by a factor $\frac{1}{m_{Z^{\prime}}^{2}-m_{Z}^{2}}$ so they can be
taken into account order-by-order perturbatively. However, when $Z$ and
$Z^{\prime}$ are nearly degenerate so $m_{Z^{\prime}}^{2}\approx m_{Z}^{2}$,
such a suppression becomes non-viable.

Denote $c_{XY}\equiv\rm{Im}\left\\{\Pi_{X\leftrightarrow Y}(p^{2}\approx
m_{X,Y}^{2})\right\\}$, where $\Pi_{X\leftrightarrow Y}(p^{2})$ is the
$g_{\mu\nu}$ coefficient of the self-energy of a particle $X$ transforming
into $Y$, one has to correct each element of (5) with an additional $ic_{XY}$
term before the diagonalizion processes. In fact, as we will illustrate in
Appendix A, in order to keep the photon massless, we utilize the matrix below

$\displaystyle\mathcal{M}_{V}^{2~{}\prime}$ $\displaystyle=$
$\displaystyle\mathcal{M}_{V}^{2}+i\left(C_{X\leftrightarrow
Y}\right)_{3\times 3}$ (8) $\displaystyle=$
$\displaystyle\begin{pmatrix}m_{\hat{Z}^{\prime}}^{2}+ic_{\hat{Z}^{\prime}\hat{Z}^{\prime}}&\hat{g}^{\prime}(\delta
m^{2}-\frac{ic_{\hat{Z}^{\prime}\hat{Z}}}{\sqrt{\hat{g}^{\prime
2}+\hat{g}^{2}}})&-\hat{g}(\delta
m^{2}-\frac{ic_{\hat{Z}^{\prime}\hat{Z}}}{\sqrt{\hat{g}^{\prime
2}+\hat{g}^{2}}})\\\ \hat{g}^{\prime}(\delta
m^{2}-\frac{ic_{\hat{Z}^{\prime}\hat{Z}}}{\sqrt{\hat{g}^{\prime
2}+\hat{g}^{2}}})&\frac{\hat{g}^{\prime 2}}{4}(\hat{v}^{2}+\delta
v^{2}+\frac{4ic_{\hat{Z}\hat{Z}}}{\hat{g}^{\prime
2}+\hat{g}^{2}})&-\frac{\hat{g}^{\prime}\hat{g}}{4}(\hat{v}^{2}+\delta
v^{2}+\frac{4ic_{\hat{Z}\hat{Z}}}{\hat{g}^{\prime 2}+\hat{g}^{2}})\\\
-\hat{g}(\delta
m^{2}-\frac{ic_{\hat{Z}^{\prime}\hat{Z}}}{\sqrt{\hat{g}^{\prime
2}+\hat{g}^{2}}})&-\frac{\hat{g}^{\prime}\hat{g}}{4}(\hat{v}^{2}+\delta
v^{2}+\frac{4ic_{\hat{Z}\hat{Z}}}{\hat{g}^{\prime
2}+\hat{g}^{2}})&\frac{\hat{g}^{2}}{4}(\hat{v}^{2}+\delta
v^{2}+\frac{4ic_{\hat{Z}\hat{Z}}}{\hat{g}^{\prime
2}+\hat{g}^{2}})\end{pmatrix},$

Now the mass terms are no longer hermitian, and this can be understood by
adding some non-conjugate corrections into the Lagrangian, or the Hamiltonian,
just as what happens in a $K^{0}$-$\overline{K^{0}}$ systemDonoghue:1992dd ;
Workman:2022ynf . We also have to note that the “hatted” $\hat{Z}$ denotes the
“SM-$\hat{Z}$” when all the mixing terms are switched off, that is to say,

$\displaystyle\hat{Z}^{\mu}=\frac{1}{\sqrt{\hat{g}^{2}+\hat{g}^{\prime
2}}}(-\hat{g}^{\prime}B^{\mu}+\hat{g}W^{3\mu}).$ (9)

In the following of this paper, the hatless “$Z$” usually appears within the
symbols associated with the aspect of the experimentalists who might be
unaware that their observed resonance can accommodate exotic contributions.
“$Z$” also appears within the definitions from the literature or simulating
tools. In this paper, the hatless “$Z^{\prime}$” is also associated with a
general reference of “$Z^{\prime}$-model” or “$Z^{\prime}$-$Z$ system”, while
the hatted “$\hat{Z^{\prime}}$” particularly refers to the $\hat{Z^{\prime}}$
field that we introduce in (1).

Then we can follow Ref. Cai:2022cti  to diagonalize the kinetic matrix (7)
beforehand:

$\displaystyle V_{C}=V_{1}V_{2}V_{3},\quad
V_{C}^{\mathrm{T}}\mathcal{K}_{V}V_{C}=I_{3\times 3},$ (10)

where

$\displaystyle V_{1}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}1&-\epsilon_{B}&-\epsilon_{W}\\\ 0&1&0\\\
0&0&1\end{pmatrix},~{}V_{2}=\begin{pmatrix}1&0&0\\\
0&1&\frac{-\epsilon_{BW}+\epsilon_{B}\epsilon_{W}}{1-\epsilon_{B}^{2}}\\\
0&0&1\end{pmatrix},$ $\displaystyle V_{3}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}1&0&0\\\
0&\frac{1}{\sqrt{1-\epsilon_{B}^{2}}}&0\\\
0&0&\sqrt{\frac{1-\epsilon_{B}^{2}}{1+\epsilon_{WW}-\epsilon_{B}^{2}-\epsilon_{W}^{2}-\epsilon_{BW}^{2}-\epsilon_{B}^{2}\epsilon_{WW}+2\epsilon_{B}\epsilon_{W}\epsilon_{BW}}}\end{pmatrix},$
(11)

and then the mass-squared matrix becomes

$\displaystyle(V_{1}V_{2}V_{3})^{\mathrm{T}}\mathcal{M}_{V}^{2~{}\prime}V_{1}V_{2}V_{3},$
(12)

and diagonalizing this matrix gives

$V=V_{1}V_{2}V_{3}V_{\mathrm{SM}}V_{\mathrm{f}},\quad
V^{\mathrm{T}}\mathcal{M}_{V}^{2~{}\prime}V=\mathrm{diag}(m^{2}_{1}-\mathrm{i}m_{1}\Gamma_{1},\,m_{2}^{2}-\mathrm{i}m_{2}\Gamma_{2},\,0),$
(13)

where $V_{\mathrm{SM}}$ is the familiar EW rotation matrix

$V_{\text{SM}}=\begin{pmatrix}1&0&0\\\
0&-\frac{\hat{g}^{\prime}}{\sqrt{\hat{g}^{\prime
2}+\hat{g}^{2}}}&\frac{\hat{g}}{\sqrt{\hat{g}^{\prime 2}+\hat{g}^{2}}}\\\
0&\frac{\hat{g}}{\sqrt{\hat{g}^{\prime
2}+\hat{g}^{2}}}&\frac{\hat{g}^{\prime}}{\sqrt{\hat{g}^{\prime
2}+\hat{g}^{2}}}\end{pmatrix}.$ (14)

$m_{1}$, $\Gamma_{1}$, $m_{2}$, $\Gamma_{2}$ are the “masses” and “widths” of
two “mass eigenstates” of the $Z^{\prime}$-$Z$ system. Since their masses are
nearly degenerate, and their mixing angle might be large, they might
altogether form a single SM-$Z$-like object, and which of them is identified
to be the $Z$\- or $Z^{\prime}$ is unessential.

We have to note that although (8) is no longer hermitian, one can still verify
that it is symmetric, that is,
$\mathcal{M}_{V}^{2~{}\prime}=(\mathcal{M}_{V}^{2~{}\prime})^{\rm{T}}$. This
guarantees the existence of $V_{\mathrm{f}}$ in (13) with the condition
$V_{\mathrm{f}}^{\mathrm{T}}V_{\mathrm{f}}=V_{\mathrm{f}}V_{\mathrm{f}}^{\mathrm{T}}=I$.
However, the elements of $V_{\mathrm{f}}$ can be complex, which are weird to
be understood as the “mixing terms” among real vector fields, since the mixed
“eigenstate fields” are no longer real numbers. This indicates that the usual
perceptions of the “mixing fields” are non-viable and should be replaced with
the concept of the resummmed propagators as described in Appendix A.

##  III Details of the event generation and the extraction of the observables

The standard way to extract the electroweak precision data is to compare the
line-shape of the $Z$-resonance with the parameterized functions considering
the Breit-Wigner propagators, initial-state radiation (ISR) effects, and the
momentum distribution of the beams (See section 55 of “Reviews, Tables &
Plots” in Ref. Workman:2022ynf  for a review, and for the references therein).
The photon mediated s-channel diagrams and all the t- and u-channel
contributions with the interference effects should also be considered. After
finding out the most fitted parameterized function, $m_{Z}$, $\Gamma_{Z}$,
$R_{e,\mu,\tau}$, $A_{\rm FB}^{0e,\mu,\tau}$, and $N_{\nu}$, which are the
mass of the $Z$-boson, the width of the $Z$-boson, the ratios of the
$Z\rightarrow\rm{hadrons}$ over $Z\rightarrow e/\mu/\tau$ branching ratios,
the forward-backward asymmetry parameters, and the effective number of the
active neutrinos, respectively, are extracted for further comparison with the
SM predicted values.

In this paper, we alternatively adopt what we called the “SM templates” to
replace the role of the parameterized functions. These are the line-shape data
acquired from the event generator based upon the dubbed “pseudo-SM” model
file, which is a modified variation of the default SM model file provided by
FeynRulesAlloul:2013bka , added with four additional parameters $m_{Z}$,
$s^{2}_{l}$, $\Gamma_{Z}$ and $\xi$ as the input value. Here $s^{2}_{l}$ is
the effective Weinberg angle affecting the weak coupling constants,
independent with the $s^{2}_{W}$ associated with the ratio of the $W$ and $Z$
bosons. $\Gamma_{Z}$ can also be assigned with an arbitrary value, which might
not equal to the SM-predicted one. $\xi$ appears in

$\displaystyle B_{\mu}$ $\displaystyle=$ $\displaystyle-s_{W}\xi
Z_{\mu}+c_{W}A_{\mu},$ $\displaystyle W^{3}_{\mu}$ $\displaystyle=$
$\displaystyle c_{W}\xi Z_{\mu}+s_{W}A_{\mu},$ (15)

modifying the definitions of the $B_{\mu}$ and $W^{3}_{\mu}$ in the model
file. This parameter aims at rescaling the height of the resonance while
keeping the shape of it intact.

The purpose to utilize the event generator to generate the “pseudo-SM
templates” is to elude the complicated comparison between our generator
results and the SM values with intricated loop-level contribution considered
under various renormalization schemes. Since the real LEP data are absent, it
is also reasonable to compare our simulated $Z^{\prime}$-model events with the
SM templates generated by exactly the same event generator, eliminating the
errors from the theoretical uncertainties automatically.

To compare the continuous line-shape curves, one should sample a set of
discrete $\sqrt{s}$’s, which are defined as the invariant mass, or the total
energy of the colliding $e^{+}e^{-}$ system. LEP published some of the early
detailed data at each of the sampled $\sqrt{s}$, while later as the integrated
luminosity accumulates, only the fitted results of the electroweak precision
measurements were published. With these incomplete information recorded in
Ref. LEP:2000pgt , we adopt
$\sqrt{s}=[88.2,89.2,90.2,91.2,92.2,93.2,94.2]\rm{GeV}$ as our samples as a
reference, and the integrated luminosity is set 300 $\mathrm{pb}^{-1}$ for
each $\sqrt{s}$ of the $Z^{\prime}$-model events. Since in Ref. LEP:2000pgt ,
the total luminosity is 60 $\mathrm{pb}^{-1}$, we therefore multiply a
$\sqrt{5}$ when evaluating the statistical uncertainties. We call our data
with such settings the “pseudo-LEP” results, and believe that these can
characterize the main features of the LEP in the case of incomplete data.

For the pseudo-SM templates, we generate $10^{7}$ events for each
$\sqrt{s}=[88.2$, $89.2$, $90.2$, $91.2$, $92.2$, $93.2$, $94.2]\rm{GeV}$
sample with various input values of $m_{Z}$, $s_{l}^{2}$, $\Gamma_{Z}$, and
$\xi$. Then we use the polynomial

$\displaystyle\sigma_{t}^{\text{PSM}}(m_{Z},\Gamma_{Z},s_{l}^{2},\xi,\sqrt{s})=a_{0t}(\sqrt{s})+(m_{Z}-m_{Z0})^{2}a_{1t}(\sqrt{s})+(s_{l}^{2}-s_{l0}^{2})^{2}a_{2t}(\sqrt{s})+$
(16)
$\displaystyle(\Gamma_{Z}-\Gamma_{Z0})^{2}a_{3t}^{F/B}(\sqrt{s})+(\xi-\xi_{0})^{2}a_{4t}(\sqrt{s})+(m_{Z}-m_{Z0})(s_{l}^{2}-s_{l0}^{2})a_{5t}(\sqrt{s})+$
$\displaystyle(m_{Z}-m_{Z0})(\Gamma_{Z}-\Gamma_{Z0})a_{6t}(\sqrt{s})+(m_{Z}-m_{Z0})(\xi-\xi_{0})a_{7t}(\sqrt{s})+(s_{l}^{2}-s_{l0}^{2})(\Gamma_{Z}-\Gamma_{Z0})a_{8t}(\sqrt{s})$
$\displaystyle+(s_{l}^{2}-s_{l0}^{2})(\xi-\xi_{0})a_{9t}(\sqrt{s})+(\Gamma_{Z}-\Gamma_{Z0})(\xi-\xi_{0})a_{10t}(\sqrt{s})+(m_{Z}-m_{Z0})a_{11t}(\sqrt{s})+$
$\displaystyle(s_{l}^{2}-s_{l0}^{2})a_{12t}(\sqrt{s})+(\Gamma_{Z}-\Gamma_{Z0})a_{13t}(\sqrt{s})+(\xi-\xi_{0})a_{14t}(\sqrt{s}),$

to fit the pseudo-SM template events by quadratic fitting for each
$\sqrt{s}=[88.2$, $89.2$, $90.2$, $91.2$, $92.2$, $93.2$, $94.2]\rm{GeV}$ in
each final product channel
$t\in\\{(e^{+}e^{-})_{\text{F}/\text{B}}$,$(\mu^{+}\mu^{-})_{\text{F}/\text{B}}$,$\rm{hadrons}\\}$.
Here the subscript of the “$(e^{+}e^{-})_{\text{F}}$” or
“$(e^{+}e^{-})_{\text{B}}$” denotes the “forward” or “backward” directions
that the positive charged product outcomes parallels or anti-parallels to the
incoming positron beam. The $a_{(0-14)t}$ are the factors to be determined by
the fitting processes. The charge asymmetry of the quarks also affect the
charge unbalance of the final hadrons, however analyzing such an asymmetry is
beyond our ability of simulation, so we evade taking this into account.

In principle the $e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ channels should also
be considered. $\tau$ might decay into muons, electrons, or hadrons to fake
the corresponding channels. The leptonic decay channels can be distinguished
by the additional missing energy/momentum criterion during the event
selections, and the hadronic decay channels might be problematic. At a lepton
collider, the hadronic $\tau$-decay products can be well discriminated from
the hadronic-jet eventsGrunewald:1995ny . The efficiency of the
“$\tau$-tagging” techniques at a hadronic collider seems to have been
significantly improved during these yearsBagliesi:2007qx ; CMS:2018jrd , and
it is reasonable to expect that such a technology might in turn contribute to
the future lepton collider programs. However, all these algorithms are too
complicated for a fast simulation in this paper, and with the consideration of
the technology improvement in the future, we just assume that all the $\tau$
products can be well-discriminated, so we neglect them in this paper.

In this work, we apply WHIZARD Kilian:2007gr ; Moretti:2001zz ;
Christensen:2010wz  as our event generator. LHAPDF6 Buckley:2014ana ;
Andersen:2014efa , PYTHIA6 Bierlich:2022pfr , FastJet Cacciari:2005hq ;
Cacciari:2011ma  and DELPHES deFavereau:2013fsa ; Selvaggi:2014mya ;
Mertens:2015kba  are connected for the detective-level data. In DELPHES, we
utilize the CEPC card regardless our purpose for a LEP-like prediction, and
not so many differences are expected. The beams structure is chosen to be the
Gaussian distribution. The WHIZARD parameter $\sigma$ of both the
“gaussian_spread1” and “gaussian_spread2” are set to be 250/90000. ISR is also
switched on. During the event generation, the cuts are set $\mathrm{E}\geq
100$ MeV, all $\mathrm{abs}(\cos(\mathrm{Theta}))\geq 0.8$ and all
$\mathrm{M2}\geq 1$ $\mathrm{GeV}^{2}$ for final products, where “E”, “Theta”
and “M2” are energy of the particle in the argument, absolute polar angle in
the lab frame and invariant mass squared of the particle in the argument,
respectively. Then transverse momentum $\mathrm{P_{T}}>7$ GeV and
pseudorapidity $\eta<2.4$ are set for event selection.

We then compare the line-shape cross sections in the $Z^{\prime}$-model for
each parameter point with the pseudo-SM template cross sections fitted in (16)
to find the best fitted $(m_{Z}^{*},s_{l}^{*2},\Gamma_{Z}^{*},\xi^{*})$ by
minimizing the $\chi_{F}^{2}$ defined as

$\chi_{F}^{2}(m_{Z},\Gamma_{Z},s_{l}^{2},\xi)=\sum_{t,\sqrt{s}}\frac{(\sigma_{t}^{Z^{\prime}}(\sqrt{s})-\sigma_{t}^{\mathrm{PSM}}(m_{Z},\Gamma_{Z},s_{l}^{2},\xi,\sqrt{s}))^{2}}{(\Delta_{t}^{Z^{\prime}}(\sqrt{s}))^{2}+(\Delta_{i}^{\mathrm{PSM}}(m_{Z},\Gamma_{Z},s_{l}^{2},\xi,\sqrt{s}))^{2}},$
(17)

where $\sqrt{s}=[88.2$, $89.2$, $90.2$, $91.2$, $92.2$, $93.2$,
$94.2]\rm{GeV}$,
$\sigma_{t}^{\mathrm{PSM}}(m_{Z},\Gamma_{Z},s_{l}^{2},\xi,\sqrt{s})$ is
defined in (16). $\sigma_{t}^{Z^{\prime}}(\sqrt{s})$ are line-shape sample
cross sections for the $Z^{\prime}$-model with a particular set of parameters.
$\Delta_{i}^{Z^{\prime}}$ and
$\Delta_{i}^{\mathrm{PSM}}(m_{Z},\Gamma_{Z},s_{l}^{2},\xi,\sqrt{s})$ are the
statistical uncertainties of the pseudo-SM template cross sections and the
$Z^{\prime}$-model cross sections respectively. The statistical uncertainty
$\Delta_{\rm X}$ for each cross section $\sigma_{\rm X}$ at channel $X$ is
evaluated by

$\displaystyle\Delta_{\rm X}=\frac{1}{\sqrt{n_{\rm X}}}\sigma_{\rm X},$ (18)

where $n_{\rm X}$ is the number of events passing through all selection
criterion. The total $\chi_{F}^{2}$ includes seven points of center-of-mass
energy, three channels, and the leptonic channels are separated into forward-
back parts, so the total degree of freedom is counted to be 35.

Then the best-fitted $(m_{Z}^{*},s_{l}^{*2},\Gamma_{Z}^{*},\xi^{*})$ will be
converted into effective oblique parameters $\tilde{S}$, $\tilde{T}$,
$\tilde{U}$, and $\delta\tilde{N}_{\nu}$, which are the effective Peskin-
Takeuchi oblique parameters and the the effective deviation of the species of
neutrinos respectively. Ref. Peskin:1990zt ; Peskin:1991sw  derived the $S$,
$T$, $U$ expressions depending on $m_{W},s_{l},\Gamma_{Z}$. Here we reverse
these formulas as the definitions of the effective $\tilde{S}$, $\tilde{T}$,
$\tilde{U}$ so that the constraints of $S$, $T$, $U$ in the literature can be
straightforwardly cast here. One subtle thing is that in the usual electroweak
precision measurement discussions, $m_{Z}$ is the input parameter, so $\delta
m_{W}$ is given by

$\delta m_{W}=-\frac{\alpha
m_{W}^{\mathrm{SM}}}{4(c_{\mathrm{W}}^{2}-s_{\mathrm{W}}^{2})}(S-2c^{2}_{\mathrm{W}}T-\frac{c_{\mathrm{W}}^{2}-s_{\mathrm{W}}^{2}}{2s^{2}_{\mathrm{W}}}U),$
(19)

where $\delta m_{W}=m_{W}-m_{W}^{\mathrm{SM}}$. However, in our pseudo model,
$m_{Z}$ is no longer the “measured” $Z$ boson mass. The correction on $m_{W}$
is equivalent to a correction on $m_{Z}$ towards the opposite direction, so
according to $m_{W}^{\mathrm{SM}}=m_{Z}c_{\mathrm{W}}$,
$m_{W}=m_{Z}^{\mathrm{SM}}c_{\mathrm{W}}$ and Ref. Ciuchini:2013pca ;
Fan:2014vta , we define $\tilde{S}$, $\tilde{T}$, $\tilde{U}$, as well as the
deviation of the neutrino species number $\delta\tilde{N}_{\nu}$ by solving
the following equations.

$\displaystyle m_{Z}^{*}-m_{Z}^{\mathrm{SM}}$ $\displaystyle=$
$\displaystyle\frac{\alpha
m_{Z}^{\mathrm{SM}}}{4(c_{\mathrm{W}}^{2}-s_{\mathrm{W}}^{2})}(\tilde{S}-2c^{2}_{\mathrm{W}}\tilde{T}-\frac{c_{\mathrm{W}}^{2}-s_{\mathrm{W}}^{2}}{2s^{2}_{\mathrm{W}}}\tilde{U}),$
$\displaystyle s_{l}^{*2}-(s_{l}^{2})^{\mathrm{SM}}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{4(c_{\mathrm{W}}^{2}-s_{\mathrm{W}}^{2})}(\tilde{S}-4s_{\mathrm{W}}^{2}c_{\mathrm{W}}^{2}\tilde{T}),$
$\displaystyle\Gamma_{Z}^{*}-(\Gamma_{Z})^{\mathrm{SM}}$ $\displaystyle=$
$\displaystyle\frac{\alpha^{2}m_{Z}^{\mathrm{SM}}}{72s^{2}_{\mathrm{W}}c^{2}_{\mathrm{W}}(c_{\mathrm{W}}^{2}-s_{\mathrm{W}}^{2})}(-10(3-8s^{2}_{\mathrm{W}})\tilde{S}+(63-126s^{2}_{\mathrm{W}}-40s^{4}_{\mathrm{W}})\tilde{T}),$
$\displaystyle\delta\tilde{N}_{\nu}$ $\displaystyle=$
$\displaystyle\frac{\Gamma_{Z}^{*}-(\Gamma_{Z})^{\mathrm{pSM}}}{(\Gamma_{\nu\nu})^{\mathrm{pSM}}},$
(20)

where $m_{Z}^{\text{SM}}$ is the $\hat{Z}$-mass encoded in the model file of
the $Z^{\prime}$-model file if all the vector boson’s mixing terms are
switched off. $(s_{l}^{2})^{\mathrm{SM}}$ and $(\Gamma_{Z})^{\mathrm{SM}}$ are
also the calculated values of $s_{l}^{2}$ and $\Gamma_{Z}$ in this case.
$(\Gamma_{Z})^{\mathrm{pSM}}$ and $(\Gamma_{\nu\nu})^{\mathrm{pSM}}$ are
calculated according to the pseudo-SM model files with the best fitted
$m_{Z}^{*}$, $s_{l}^{*2}$ and $\xi^{*}$ as the input parameter. Solving (20),
we obtain

$\displaystyle\tilde{S}$ $\displaystyle=367.29\delta
s_{l}^{2}+37.92\delta\Gamma_{Z},$ (21) $\displaystyle\tilde{T}$
$\displaystyle=132.35\delta s_{l}^{2}+52.94\delta\Gamma_{Z},$
$\displaystyle\tilde{U}$ $\displaystyle=-2.62\delta m_{Z}+144.31\delta
s_{l}^{2}-37.92\delta\Gamma_{Z},$

where $\delta m_{Z}=m_{Z}^{*}-m_{Z}^{\mathrm{SM}}$, $\delta
s_{l}^{2}=s_{l}^{*2}-(s_{l}^{2})^{\mathrm{SM}}$ and
$\delta\Gamma_{Z}=\Gamma_{Z}^{*}-(\Gamma_{Z})^{\mathrm{SM}}$.

| Result | Correlation
---|---|---
$S$ | $0.06\pm 0.1$ | $1.00$ |  |
$T$ | $0.11\pm 0.12$ | $0.90$ | $1.00$ |
$U$ | $-0.02\pm 0.09$ | $-0.57$ | $-0.82$ | $1.00$
Table 1: Global fit results of the oblique parameters $S$, $T$, and $U$
adopted from Ref. Lu:2022bgw .

The $\chi^{2}$ comparing the $\tilde{S}$, $\tilde{T}$ and $\tilde{U}$ with the
global fitted data is defined by

$\displaystyle\chi^{2}=\begin{pmatrix}\tilde{S}-S_{0},&\tilde{T}-T_{0},&\tilde{U}-U_{0}\end{pmatrix}C^{-1}\begin{pmatrix}\tilde{S}-S_{0}\\\
\tilde{T}-T_{0}\\\ \tilde{U}-U_{0}\end{pmatrix},$ (22)

where $S_{0}$, $T_{0}$ and $U_{0}$ are the best global-fitted values, $C^{-1}$
represents the inverse of the covariance matrix of $S_{0}$, $T_{0}$ and
$U_{0}$. In this paper, the $S_{0}$, $T_{0}$, $U_{0}$ and $C^{-1}$ are adopted
from Tab. 1.

In this paper, based upon some typical models (See an enumeration of the
$Z^{\prime}$-models in Ref. Langacker:2008yv , and the references therein), we
are going to show our calculated $\tilde{S}$, $\tilde{T}$, $\tilde{U}$,
$\delta\tilde{N}_{\nu}$, as well as the estimated $e^{+}e^{-}$ collider
sensitivity in three scenarios. They are

  * •

Scenario I: $\hat{Z}^{\prime}$ only couples to invisible particles.

This scenario is inspired from the dark matter model associated with a
$Z^{\prime}$, in which $Z^{\prime}$ plays a crucial role connecting the
visible world with the dark sector. $Z^{\prime}$ might couple with the dark
matter, and the dark matter particles decayed from it are invisible at a
collider. Besides the dark matter, $Z^{\prime}$ might also couple with the
sterile neutrino, which might also be invisible if they are long-lived enough
to decay outside the detector.

This scenario is accomplished by straightforwardly assigning a
$c_{\hat{Z}^{\prime}\hat{Z}^{\prime}}$ value in (8) for convenience, rather
than introducing some invisible fields for the $\hat{Z}^{\prime}$ to decay
into them.

  * •

Scenario II: $\hat{Z}^{\prime}$ couples with the SM fermions universally among
all three generations.

Charged under the $U(1)^{\prime}$ gauge symmetry universally among all three
generations, the coupling constants are stringently constrained by the off-
shell $\hat{Z}^{\prime}$ mediated processes, leading to a particularly small
$c_{\hat{Z}^{\prime}\hat{Z}^{\prime}}$ to give a narrow but sharp valley-like
structure imposed on the resonance. The initial momentum distribution in the
beams and the ISR effect smear this structure to give a relatively “smooth”
curve. Since the LEP collaborations only published the final electroweak
precision measurement results, and they are acquired by comparing the
experimental data with a trivial resonance structure, we only follow this
principle to compare our pseudo-LEP results with the pseudo-SM templates to
find the best-fitted electroweak observables, regardless of the non-standard
resonance shape of the pseodu-LEP data.

Here we define the coupling constants as

$\displaystyle
g_{u_{R}}\hat{Z}^{\prime}_{\mu}\bar{u}_{R}\gamma^{\mu}u_{R}+g_{u_{L}}\hat{Z}^{\prime}_{\mu}\bar{u}_{L}\gamma^{\mu}u_{L}$
$\displaystyle=$
$\displaystyle\hat{Z}^{\prime}_{\mu}\bar{u}\gamma^{\mu}(\frac{g_{u_{L}}+g_{u_{R}}}{2}-\frac{g_{u_{L}}-g_{u_{R}}}{2}\gamma^{5})u,$
$\displaystyle
g_{d_{R}}\hat{Z}^{\prime}_{\mu}\bar{d}_{R}\gamma^{\mu}d_{R}+g_{d_{L}}\hat{Z}^{\prime}_{\mu}\bar{d}_{L}\gamma^{\mu}d_{L}$
$\displaystyle=$
$\displaystyle\hat{Z}^{\prime}_{\mu}\bar{d}\gamma^{\mu}(\frac{g_{d_{L}}+g_{d_{R}}}{2}-\frac{g_{d_{L}}-g_{d_{R}}}{2}\gamma^{5})u,$
$\displaystyle
g_{l_{R}}\hat{Z}^{\prime}_{\mu}\bar{l}_{R}\gamma^{\mu}l_{R}+g_{l_{L}}\hat{Z}^{\prime}_{\mu}\bar{l}_{L}\gamma^{\mu}l_{L}$
$\displaystyle=$
$\displaystyle\hat{Z}^{\prime}_{\mu}\bar{l}\gamma^{\mu}(\frac{g_{l_{L}}+g_{l_{R}}}{2}-\frac{g_{l_{L}}-g_{l_{R}}}{2}\gamma^{5})u,$
$\displaystyle
g_{\nu_{L}}\hat{Z}^{\prime}_{\mu}\bar{\nu}_{L}\gamma^{\mu}\nu_{L}$
$\displaystyle=$
$\displaystyle\hat{Z}^{\prime}_{\mu}\bar{\nu}(\frac{g_{\nu_{L}}}{2}-\frac{g_{\nu_{L}}}{2}\gamma^{5})\nu.$
(23)

  * •

Scenario III: $\hat{Z}^{\prime}$ couples with the SM fermions depending on
their generations.

For some particular models (Ref. Salvioni:2009jp  enumerated such models, and
references can be found therein), a particular generation of particles might
be charged under the $U(1)^{\prime}$ group, or two generations of particles
takes a particular combination of the $U(1)^{\prime}$ charge (For an example,
the $U(1)_{L_{e}-L_{\mu}}$ modelsHe:1990pn ; He:1991qd ; Ma:2001md ;
Foot:1994vd ; Baek:2001kca ). This not only affects the shape of the
$Z$-resonance, but also breaks the universality of the $l^{+}l^{-}$ branching
ratios, where $l=e,\mu,\tau$. In this paper, we only discuss the $e$-$\mu$
asymmetry, and utilize the

$\frac{R_{e}}{R_{\mu}}$ (24)

where

$R_{l}\equiv\frac{\Gamma_{\mathrm{had}}}{\Gamma_{l^{+}l^{-}}}$ (25)

to observe such an asymmetry.

##  IV Numerical Results

The effective Lagrangian (1) takes six parameters, $m_{Z^{\prime}}$,
$\epsilon_{B}$, $\Lambda_{W}$, $\Lambda_{BW}$, $\Lambda_{WW}$, $\lambda_{BD}$,
which are not convenient for a relatively intuitive presentation.
Equivalently, $\epsilon_{B}$, $\epsilon_{W}$, $\epsilon_{BW}$, $\delta v^{2}$,
$\delta m^{2}$ and $m_{\hat{Z}^{\prime}}$ appeared in (8) and (11) and can be
treated as the free parameters for further discussions. Among them,
$\epsilon_{B}$ and $\epsilon_{W}$ and $m_{\hat{Z}^{\prime}}$ are the most
important. In fact, the $\delta v^{2}$, $\epsilon_{BW}$ have nothing to do
with the $Z^{\prime}$ sector, so their contributions are perturbatively
calculable by previous algorithms and we do not discuss them. $\delta m^{2}$
can also give rise to non-perturbative mixings, however in this paper, we
focus on the kinematic mixing effects.

In the following of this section, we plot the $\tilde{S}$, $\tilde{T}$,
$\tilde{U}$ and $\delta\tilde{N}_{\nu}$ for Scenario I-III on the continuous
$\epsilon_{B}$-$\epsilon_{W}$ plain with some discrete $\delta
m_{Z^{\prime}}=m_{Z^{\prime}}-m_{Z}^{\rm SM}$ values adopted. For Scenario III
when universality among generations is broken,
$\frac{\tilde{R}_{e}}{\tilde{R}_{\mu}}-1$ is also plotted. Here the definition
of $m_{Z}^{\rm SM}$ is the same as in (20).

The Monte-Carlo algorithm that we utilize inevitably introduces statistical
fluctuations in our results, thus smoothness is lost in the plotted figures.
We use a second-order polynomial of $\epsilon_{B}$ and $\epsilon_{W}$ to fit
each of the fluctuated $\tilde{S}$, $\tilde{T}$, $\tilde{U}$,
$\delta\tilde{N}_{\nu}$ and $\frac{\tilde{R}_{e}}{\tilde{R}_{\mu}}-1$ by least
square method to smooth the results. The constant terms of the second-order
polynomials are set zero since all new physics corrections are destined to
vanish at the $\epsilon_{B}=\epsilon_{W}=0$ origin. The similar results from
several independent runs with different random seeds verify the reliability of
this fitting algorithm, so in this paper, we show the fitted results in the
figures.

In principle, we should compare our results with the global fitted results of
all $\tilde{S}$, $\tilde{T}$, $\tilde{U}$, $\delta\tilde{N}_{\nu}$, and
$\frac{\tilde{R}_{e}}{\tilde{R}_{\mu}}-1$ parameters. Unfortunately, in the
literature $\tilde{S}$, $\tilde{T}$, $\tilde{U}$ are fitted with the
assumption of universality and $\delta\tilde{N}_{\nu}=0$.
$\delta\tilde{N}_{\nu}$ is also fitted with the assumption that the visible
$Z$-propagator is not distorted, that is to say, $\tilde{S}=0$, $\tilde{T}=0$,
$\tilde{U}=0$. A complete global fitting including all these parameters is far
beyond our target, and remember that we are only able to show readers the
“sensitivity” of a LEP-like electron-positron collider without performing a
real fitting process due to the lack of the published data, so in this paper,
we still show the corresponding “STU-$(1,2)$-$\sigma$”,
“$\delta\tilde{N}_{\nu}$-$(1,2)$-$\sigma$”, “$R_{e}/R_{\mu}$-$(1,2)$-$\sigma$”
contours in each of the figures. The prefixes “STU-”,
“$\delta\tilde{N}_{\nu}$-”, “$R_{e}/R_{\mu}$-” indicate that the $1$-$\sigma$
and $2$-$\sigma$ fitted results originate from the global fitted oblique
parameter in Tab. 1, indirect $N_{\nu}$ results from Ref. Workman:2022ynf ,
$R_{e}$ and $R_{\mu}$ data with their uncertainties adopted from Ref.
Workman:2022ynf . However we should note that these contours only characterize
the sensitivity of the collider on this model, and should not be regarded as
real constraints.

Besides, people are more interested about the sensitivity of some proposed
future colliders. According to Ref. CEPCStudyGroup:2018ghi , the CEPC is
expected to significantly improve the uncertainties of the electroweak
precision measurements. With the expected sensitivities published in Ref.
CEPCStudyGroup:2018ghi  around the $Z$-pole, we also plot the “CEPC-
ST-$(1,2)$-$\sigma$”, “CEPC-$\delta\tilde{N}_{\nu}$-$(1,2)$-$\sigma$”,
“CEPC-$R_{e}/R_{\mu}$-$(1,2)$-$\sigma$” contours within one plot for each of
the $\epsilon_{B}$ and $\epsilon_{W}$ plain. Note that Ref.
CEPCStudyGroup:2018ghi  only shows the expected $S$-$T$ results in Figure
11.18, so we are forced to give up the $U$. Although in principle
$U$-sensitivity can be extracted from the Tab. 11.16 of Ref.
CEPCStudyGroup:2018ghi , however the complete covariance matrix is missing for
a complete fitting. Since we only target at the collider’s potential
sensitivity, neglecting the $U$ is expected not to affect the final results
significantly. Another subtle thing is that Ref. CEPCStudyGroup:2018ghi  only
gives $R_{\mu}$, which is insufficient to estimate the $R_{e}/R_{\mu}$
uncertainties. In fact, at a lepton collider, the $e^{+}e^{-}\rightarrow
e^{+}e^{-}$ channel data are slightly less precise than the corresponding
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$ data because of the less accuracy of the
electron/positron trajectory measurements. It is also reasonable to assume
that the CEPC $R_{e}$ uncertainty decreases synchronously with the $R_{\mu}$
uncertainty as the integrated luminosity increases, so the $R_{e}$ uncertainty
at CEPC can be reasonably estimated by assuming that the ratio of the $R_{e}$
and $R_{\mu}$ uncertainties should remain similar to the LEP results recorded
in Ref. CEPCStudyGroup:2018ghi .

Now we show our results for the three scenarios respectively.

###  IV.1 Scenario I: $\hat{Z}^{\prime}$ couples with invisible matters

In this scenario, the width $\Gamma_{\hat{Z}^{\prime}}$ that
$\hat{Z}^{\prime}$ only decays to invisible matter is regarded as a input
parameter, so $c_{\hat{Z}^{\prime}\hat{Z}^{\prime}}$ and $c_{\hat{Z}\hat{Z}}$
are given by

$c_{\hat{Z}^{\prime}\hat{Z}^{\prime}}=-m_{\hat{Z}^{\prime}}\Gamma_{\hat{Z}^{\prime}},\quad
c_{\hat{Z}\hat{Z}}=-m_{Z}^{\mathrm{SM}}\Gamma_{Z}^{\mathrm{SM}},$ (26)

Since the $\hat{Z}^{\prime}$ has no coupling to SM fermions, there is no self
energy diagram of $\hat{Z}^{\prime}$-$\hat{Z}$. Therefore we have

$c_{\hat{Z}^{\prime}\hat{Z}}=0.$ (27)

If the $\Gamma_{\hat{Z}^{\prime}}$ is close to $\Gamma_{Z}^{\mathrm{SM}}$, and
the $m_{\hat{Z}^{\prime}}$ is also close to $m_{Z}^{\mathrm{SM}}$, maximum
mixings between $Z^{\prime}$-$Z$ might arise, however the overlapped and
interfered peaks still look like a single $Z$-pole.

For convenience we define

$\lambda_{Z^{\prime}}=\frac{\Gamma_{\hat{Z}^{\prime}}}{\Gamma_{Z}^{\mathrm{SM}}},$
(28)

which is the ratio of the widths of the two “interaction-eigenstates”. We have
tried several combinations of parameters like $\delta m_{Z^{\prime}}=0,-0.1$
GeV, and $\lambda_{Z^{\prime}}=0.5,0.9$, and found that the results are quite
similar if $\hat{Z}^{\prime}$ and $\hat{Z}$ are nearly-degenerate. Therefore,
we choose to plot the results when $\lambda_{Z^{\prime}}=0.9$ and $\delta
m_{Z^{\prime}}=-0.1$ GeV in Fig. 2 as a paradigm.

Figure 2: Plots for $\delta m_{Z^{\prime}}=-0.1$ GeV,
$\lambda_{Z^{\prime}}=0.9$ and $\delta
v^{2}=\epsilon_{WW}=\epsilon_{BW}=\delta m^{2}=0$ on the
$\epsilon_{B}$-$\epsilon_{W}$ plain. The lower right panel shows the estimated
CEPC sensitivity. The remaining four panels display the heat maps of
$\tilde{S}$, $\tilde{T}$, $\tilde{U}$ and $\delta\tilde{N}_{\nu}$, and plot
the 1-$\sigma$ and 2-$\sigma$ contours on the $\epsilon_{B}$-$\epsilon_{W}$
plain. Current $N_{\nu}$ data is extracted from Ref. Workman:2022ynf .

###  IV.2 Scenario II: $\hat{Z}^{\prime}$ couples with the SM fermions
universally among all three generations

In this scenario, $\hat{Z}^{\prime}$ couples directly with the SM quarks and
leptons. General definitions of the coupling constants have been listed in
(23). To compute the $\text{Im}\Pi_{\hat{Z}^{\prime}\hat{Z}}$, the fermions’
couplings with the $\hat{Z}_{\mu}$ fields are required.

$\displaystyle\mathcal{L}\supset\hat{Z}_{\mu}\bar{\psi}^{f}\gamma^{\mu}(V_{\hat{Z}}^{f}-A_{\hat{Z}}^{f}\gamma^{5})\psi^{f},\quad
f=u,d,l,\nu$ (29)

where according to the SM,

$\displaystyle
V_{\hat{Z}}^{u}=\frac{1}{4c_{\mathrm{W}}}-\frac{2s^{2}_{\mathrm{W}}}{3c_{\mathrm{W}}},\quad
A_{\hat{Z}}^{u}=\frac{1}{4c_{\mathrm{W}}},\quad
V_{\hat{Z}}^{d}=-\frac{1}{4c_{\mathrm{W}}}+\frac{s^{2}_{\mathrm{W}}}{3c_{\mathrm{W}}},\quad
A_{\hat{Z}}^{d}=-\frac{1}{4c_{\mathrm{W}}},$ $\displaystyle
V_{\hat{Z}}^{l}=-\frac{1}{4c_{\mathrm{W}}}+\frac{s^{2}_{\mathrm{W}}}{c_{\mathrm{W}}},\quad
A_{\hat{Z}}^{l}=-\frac{1}{4c_{\mathrm{W}}},\quad
V_{\hat{Z}}^{\nu}=\frac{1}{4c_{\mathrm{W}}},\quad
A_{\hat{Z}}^{\nu}=\frac{1}{4c_{\mathrm{W}}}.$ (30)

Denote $V_{\hat{Z}^{\prime}}^{f}\equiv\frac{g_{f_{L}}+g_{f_{R}}}{2}$ and
$A_{\hat{Z}^{\prime}}^{f}\equiv\frac{g_{f_{L}}-g_{f_{R}}}{2}$, under the
approximation that the decay products are massless, $c_{\hat{Z}\hat{Z}}$,
$c_{\hat{Z}^{\prime}\hat{Z}}$ and $c_{\hat{Z}^{\prime}\hat{Z}^{\prime}}$ are
calculated to be

$\displaystyle c_{\hat{Z}\hat{Z}}$ $\displaystyle=$ $\displaystyle-
m_{Z}^{\mathrm{SM}}\Gamma_{Z}^{\mathrm{SM}},$ $\displaystyle
c_{\hat{Z}^{\prime}\hat{Z}^{\prime}}$ $\displaystyle=$
$\displaystyle-\frac{(m_{Z}^{\mathrm{SM}})^{2}}{12\pi}\sum_{f\neq
t}(V^{f}_{\hat{Z}^{\prime}}V^{f}_{\hat{Z}^{\prime}}+A^{f}_{\hat{Z}^{\prime}}A^{f}_{\hat{Z}^{\prime}}),$
$\displaystyle c_{\hat{Z}^{\prime}\hat{Z}}$ $\displaystyle=$
$\displaystyle-\frac{(m_{Z}^{\mathrm{SM}})^{2}}{12\pi}\sum_{f\neq
t}(V^{f}_{\hat{Z}^{\prime}}V^{f}_{\hat{Z}}+A^{f}_{\hat{Z}^{\prime}}A^{f}_{\hat{Z}}).$
(31)

In this scenario, the direct coupling constants between $\hat{Z}^{\prime}$ and
the SM particles should be stringently constrained. Therefore the
$c_{\hat{Z}^{\prime}\hat{Z}^{\prime}}$ and $c_{\hat{Z}^{\prime}\hat{Z}}$
become much smaller than $c_{\hat{Z}\hat{Z}}$. This actually suppresses the
mixing angle and in this case, the $\hat{Z}^{\prime}$-like object cleaves a
deep but narrow valley within the resonance induced by the $Z$-like object,
which will later be smeared by the ISR and beam momentum distribution effects,
just as we have plotted in Fig. 3 as an example.

Figure 3: $Z$ line-shape predicted by an example model that $\hat{Z}^{\prime}$
couples with the SM fermions. The left panel shows the results when the
electron/positron is “bare”, while the right panel considers the ISR and beam
momentum distribution effects.

It is actually impossible for a practical lepton collider to scan every
$\sqrt{s}$ in a sufficient resolution to depict such inconspicuous structure,
so what we can do is still adopt the scattered $\sqrt{s}=[88.2$, $89.2$,
$90.2$, $91.2$, $92.2$, $93.2$, $94.2]\rm{GeV}$ samples to extract the
$\tilde{S}$, $\tilde{T}$, $\tilde{U}$, $\delta\tilde{N}_{\nu}$ parameters.

In this paper, we only consider two cases, one is that the $\hat{Z}^{\prime}$
only couples universally with all the leptons, and the other is that the
$\hat{Z}^{\prime}$ couple with leptons and quarks with the ratio of the
coupling constants to be $3:1$, inspired by the $U(1)_{B-L}$ model. For the
first case, since we find out that the chirality of the interaction terms
defined in (23) does not affect the results to a significant extent, we only
adopt the vector-type interaction pattern $g_{l_{L}}=g_{l_{R}}$, and show an
example when $g_{l_{L}}=g_{l_{R}}=0.001$ in Fig. 4. Inspired by the
$U(1)_{B-L}$ models, we plot the Fig. 5 as an example when
$g_{l_{L}}=g_{l_{R}}=-3g_{u_{L}}=-3g_{u_{R}}=-3g_{d_{L}}=-3g_{d_{R}}=0.0015$.
One can again evaluate the LEP sensitivity as well as the estimated CEPC
sensitivity on the $\epsilon_{B}$-$\epsilon_{W}$ plain.

Figure 4: Plots for $\hat{m}_{Z^{\prime}}=90.2$ GeV,
$g_{l_{L}}=g_{l_{R}}=0.001$ and $\delta
v^{2}=\epsilon_{WW}=\epsilon_{BW}=\delta
m^{2}=g_{u_{L}}=g_{u_{R}}=g_{d_{L}}=g_{d_{R}}=g_{\nu_{L}}=0$ on the
$\epsilon_{B}$-$\epsilon_{W}$ plain. The other symbols are the same as in Fig.
2.

Figure 5: Plots for $\hat{m}_{Z^{\prime}}=92.2$ GeV,
$g_{u_{L}}=g_{u_{R}}=g_{d_{L}}=g_{d_{R}}=-0.0005$,
$g_{l_{L}}=g_{l_{R}}=g_{\nu_{L}}=0.0015$ and $\delta
v^{2}=\epsilon_{WW}=\epsilon_{BW}=\delta m^{2}=0$ on the
$\epsilon_{B}$-$\epsilon_{W}$ plain. The other symbols are the same as in Fig.
2

###  IV.3 Scenario III: $\hat{Z}^{\prime}$ couples with the SM fermions
depending on the generations

The $Z^{\prime}$ couplings to fermions might be generation-dependent, in which
case coefficients in (23) need to be changed into the generation-dependent
version. Constrained by our current calculation ability and inspired by some
eminent models, we only consider the case that $\hat{Z}^{\prime}$ only couples
with $\mu$ and/or $\tau$. The interaction terms are parametrized by

$\mathcal{L}_{\mathrm{eff}}\supset\hat{Z}^{\prime}_{\mu}\bar{\mu}\gamma^{\mu}(\frac{g_{\mu_{L}}+g_{\mu_{R}}}{2}-\frac{g_{\mu_{L}}-g_{\mu_{R}}}{2})\mu+\hat{Z}^{\prime}_{\mu}\bar{\tau}\gamma^{\mu}(\frac{g_{\tau_{L}}+g_{\tau_{R}}}{2}-\frac{g_{\tau_{L}}-g_{\tau_{R}}}{2})\tau.$
(32)

Since $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$ might be affected by (32),
firstly, we delete the all the $\mu^{+}\mu^{-}$ terms in (17) to find the
best-fitted point in our pseudo-SM template, then again $\tilde{S}$,
$\tilde{T}$, $\tilde{U}$, $\delta\tilde{N}_{\nu}$ are extracted.

To translate the definition in (24) into the expression of our simulated data,
we adopt

$\frac{R_{e}}{R_{\mu}}=\frac{\sigma^{\mathrm{NP}}_{\mu^{+}\mu^{-}}(\sqrt{s}=m_{Z})}{\sigma^{\mathrm{NP}}_{se^{+}e^{-}}(\sqrt{s}=m_{Z})},$
(33)

where subscript “$s$” in $\sigma^{\mathrm{NP}}_{se^{+}e^{-}}(m_{Z})$ indicates
the $s$-channel cross section the $e^{+}e^{-}\rightarrow e^{+}e^{-}$ channel.
Certainly, $\sigma^{\mathrm{NP}}_{se^{+}e^{-}}(m_{Z})$ is impossible to
acquire straightforwardly due to the contamination from the $t$-,
$u$-channels. Fortunately, in the pseudo-SM template, the universality is
preserved, and the $\sigma^{\mathrm{PSM}}_{\mu^{+}\mu^{-}}$ does not include
the $t$-, $u$-channel contributions. Therefore, the
$\sigma^{\mathrm{PSM}}_{\mu^{+}\mu^{-}}(\sqrt{s}=91.2~{}\mathrm{GeV})$ adopted
from the pseudo-SM template can be utilized in place of
$\sigma^{\mathrm{NP}}_{se^{+}e^{-}}(m_{Z})$, and we also take
$\sqrt{s}=91.2\rm{GeV}$ to replace the $m_{Z}$, so

$\frac{R_{e}}{R_{\mu}}=\frac{\sigma^{\mathrm{NP}}_{\mu^{+}\mu^{-}}(\sqrt{s}=91.2~{}\mathrm{GeV})}{\sigma^{\mathrm{PSM}}_{\mu^{+}\mu^{-}}(\sqrt{s}=91.2~{}\mathrm{GeV})}.$
(34)

We will extract the current $\frac{R_{e}}{R_{\mu}}$ data from Ref.
Workman:2022ynf , and plot the $1$-$\sigma$ and $2$-$\sigma$ contour in the
$\epsilon_{B}$-$\epsilon_{W}$ plain. The CEPC sensitivity is extracted from
Ref. CEPCStudyGroup:2018ghi . Notice that according to Ref.
CEPCStudyGroup:2018ghi , the sensitivity of “$R_{l}$” denoted there is
actually $R_{\mu}$. Usually the detector’s sensitivities on electron/positrons
are a bit lower than those on muons, and it is reasonable to expect the
uncertainties of $R_{e}$ and $R_{\mu}$ at the same collider will synchronously
decrease as the integrated luminosity rises. Therefore, we estimate the
expected uncertainty of the $R_{e}$ at CEPC by multiplying the expected
uncertainty of the $R_{\mu}$ by the same factor of the ratio of the
uncertainties of $R_{e}$ and $R_{\mu}$ recorded in Ref. Workman:2022ynf .
Therefore, the expected uncertainty of $\frac{R_{e}}{R_{\mu}}$ at CEPC can be
extracted.

Inspired by the models that $Z^{\prime}$ couples with a particular generation
of leptons, we show an example that the $\hat{Z}^{\prime}$ only couples with
the second family of the leptons in Fig. 6. Inspired by the
$U(1)_{L_{e}-L_{\mu}}$ modelsHe:1990pn ; He:1991qd ; Ma:2001md ; Foot:1994vd ;
Baek:2001kca , we show an example in Fig. 7.

|  |
---|---|---
|  |
Figure 6: Plots for $\delta m_{Z^{\prime}}=0.1$ GeV, $g_{\mu_{L}}=g_{\mu_{R}}=0.001$ and $g_{\tau_{L}}=g_{\tau_{R}}=\delta v^{2}=\epsilon_{WW}=\epsilon_{BW}=\delta m^{2}=0$ on the $\epsilon_{B}$-$\epsilon_{W}$ plain. Compared with the Fig. 2, $\frac{R_{e}}{R_{\mu}}-1$ is additionally plotted. |  |
---|---|---
|  |
Figure 7: Plots for $\delta m_{Z^{\prime}}=0.1$ GeV,
$g_{\mu_{L}}=g_{\mu_{R}}=0.001$, $g_{\tau_{L}}=g_{\tau_{R}}=-0.001$ and
$\delta v^{2}=\epsilon_{WW}=\epsilon_{BW}=\delta m^{2}=0$ on the
$\epsilon_{B}$-$\epsilon_{W}$ plain. The other symbols are the same as in Fig.
6

##  V Summary

In a nearly-degenerate $Z^{\prime}$-$Z$ system, not only the widths, or
equivalently, the imaginary part of the self-energy diagram of each fields,
but also the indispensable “cross terms”, that is to say, the imaginary part
of the self-energy diagram connecting two different fields play important
roles in calculating the line-shape observables. After diagonalizing the “mass
matrix” with the imaginary contributions included, sometimes “SM-like $Z$” and
“$Z^{\prime}$” cannot be well discriminated and they overlap coherently to
form a single resonance-like object which might be recognized as a single
particle.

Relying on the effective field theory model in which many of the $Z^{\prime}$
models can be accommodated, we simulate the shape of this resonance-like
object, and follow the usual literature to utilize “oblique parameters”
$\tilde{S}$, $\tilde{T}$, $\tilde{U}$, and “neutrino species deviation”
$\delta\tilde{N}_{\nu}$ to describe the shape of it. Comparing the results
with the current data mainly contributed by the LEP, one can estimate the
sensitivity of the LEP at the $\epsilon_{B}$ and $\epsilon_{W}$ plain if the
LEP data can be reanalyzed. As a paradigm of the future high-luminosity lepton
colliders, the predicted sensitivity of this model at the CEPC is also
evaluated. Besides, we also estimate the $R_{e}/R_{\mu}-1$ of the non-
universality models, and compare them with the LEP and CEPC sensitivities.

###### Acknowledgements.

We thank to Chengfeng Cai, Zhao-Huan Yu, and Hong-Hao Zhang for helpful
discussions. This work is supported in part by the National Natural Science
Foundation of China under Grants No. 12005312, the Guangzhou Science and
Technology Program under Grant No.202201011556, and the Sun Yat-Sen University
Science Foundation.

##  Appendix A Resummation of the imaginary parts of the self-energy diagrams

Let us start from a group of real scalar particles for simplicity, e.g.,
$\phi_{i}$, $i=1,2,\dots,n$, with the mass matrix to be
$\mathcal{M}^{2}_{s}=\\{m^{2}_{ij}\\}$ where $m^{2}_{ij}=m^{2}_{ji}$ so
$\mathcal{M}_{s}^{2T}=\mathcal{M}_{s}^{2}$, and the mass terms being
$\mathcal{L}_{m}=-\frac{1}{2}\phi_{i}m^{2}_{ij}\phi_{j}$. The complete
propagator of these scalar particles can be written in the form of a matrix

$\displaystyle\frac{i}{p^{2}I_{n\times n}-\mathcal{M}^{2}_{s}}\equiv
i(p^{2}I_{n\times n}-\mathcal{M}^{2}_{s})^{-1}.$ (35)

The usual diagonalization processes to find the mass eigenstates of
$\phi_{i}$’s are equivalent to finding an orthogonal matrix $V$ to diagonalize
the propagator

$\displaystyle iV(p^{2}I_{n\times
n}-\mathcal{M}^{2}_{s})^{-1}V^{-1}=\text{diag}\left[\frac{i}{p^{2}-m_{1}^{2}},\frac{i}{p^{2}-m_{2}^{2}},\dots,\frac{i}{p^{2}-m_{n}^{2}}\right],$
(36)

where $V$ is a real matrix to satisfy $VV^{T}=V^{T}V=I$. Expand the left-hand
side of (36) according to $\mathcal{M}^{2}_{s}$, we acquire

$\displaystyle iV(p^{2}I_{n\times
n}-\mathcal{M}^{2}_{s})^{-1}V^{-1}=iV\frac{1}{p^{2}}\left(I-\sum_{t=1}^{n}(\frac{\mathcal{M}^{2}_{s}}{p^{2}})^{n}\right)V^{-1}$
(37) $\displaystyle=$ $\displaystyle
i\frac{1}{p^{2}}\left[I+\sum_{t=1}^{n}\left(\frac{V\mathcal{M}^{2}_{s}V^{-1}}{p^{2}}\right)^{n}\right]=i\left(p^{2}I_{n\times
n}-V\mathcal{M}_{s}^{2}V^{-1}\right)^{-1}.$

It is then clear to see that $\mathcal{M}_{s}^{2}$ and the propagator (36)
must be diagonalized at the same time.

Including all the imaginary parts of the $\phi_{i}$-$\phi_{j}$ self-energy
diagrams introduces a resummation

$\displaystyle\sum_{t=0}^{\infty}\frac{i}{p^{2}I_{n\times
n}-\mathcal{M}^{2}_{s}}\left[(-i)\text{Im}(\Sigma(p^{2}))\frac{i}{p^{2}I_{n\times
n}-\mathcal{M}^{2}_{s}}\right]^{t}=\frac{i}{p^{2}I_{n\times
n}-\mathcal{M}^{2}_{s}-\text{Im}(\Pi(p^{2}))},$ (38)

where $-i\Pi(p^{2})$ is a $n\times n$ symmetric matrix with each of its
element $-i\Pi_{jk}(p^{2})$ the (cross) self-energy diagram connecting
$\phi_{j}$ and $\phi_{k}$. Therefore, a complete diagonalization process
described in (37) should be replaced by

$\displaystyle iV(p^{2}I_{n\times
n}-\mathcal{M}^{2}_{s}-\text{Im}(\Pi(p^{2})))^{-1}V^{-1}=i\left\\{p^{2}I_{n\times
n}-V[\mathcal{M}_{s}^{2}+\text{Im}(\Pi(p^{2}))]V^{-1}\right\\}^{-1}.$ (39)

It is now clear that for the complete diagonalization processes including all
the width informations, one should in turn diagonalize
$\mathcal{M}_{s}^{2}+\text{Im}(\Pi(p^{2}))$ at each momentum $p$ instead of a
mere $\mathcal{M}_{s}^{2}$. Again, it is easy to verify that
$\mathcal{M}_{s}^{2}+\text{Im}(\Pi(p^{2}))$ is a complex symmetric diagram,
which guarantees the possibility of a successful diagonalization by a complex
orthogonal diagram $V=(V^{T})^{-1}$.

If all the scalars are nearly-degenerate around the mass $m_{d}$, a good
approximation can be $p^{2}=m_{d}^{2}$ to preserve the accuracy of the near-
shell performances of the propagators. This is a generalization of the Breig-
Wigner propagator for the single particle into a nearly-degenerate multiple
particle group.

Since we are discussing about the real scalars, and usually the complex
orthogonal $V$ contains not only real numbers, if we treat $V$ as the matrix
to recombine $\phi_{1,2,\dots,n}$ into “mass eigenstates”
$\phi^{\prime}_{j}=V_{jk}\phi_{k}$ as usual, then $\phi^{\prime}_{j}$ is
something “complex” but cannot be regarded as a “complex scalar field”.
Therefore, in this paper, we remind the reader that (39) cannot be understood
as a equivalence to diagonalizing the scalar fields, although we sometimes
still apply this less rigorous terminology for brevity.

For the $W^{3}$-$B$-$Z^{\prime}$ system, the propagators should be accompanied
with a Lorentz term

$\displaystyle\frac{ig_{\mu\nu}}{p^{2}I_{3\times 3}-\mathcal{M}^{2}_{V}}\equiv
ig_{\mu\nu}(p^{2}I_{3\times 3}-\mathcal{M}^{2}_{V})^{-1}.$ (40)

Here we adopt the Feynmann gauge. In principle Goldstone propagators should
also be considered. However, around the $Z$-pole, only light leptons and
quarks can be produced on-shell. Even the heaviest $b$ quark takes the
$m_{B}\sim 3$-$5$ GeV mass, so the Goldstone/Higgs contributions are
suppressed by a $\left(\frac{m_{b}}{v}\right)^{2}\ll 1$ factor, and can be
safely neglected.

Due to the Lorentz covariance, The one-loop self-energy diagram of the vector
bosons can be decomposed and parametrized by

$\displaystyle i\Pi^{\mu\nu}_{3\times 3}(p^{2})=i\Pi_{3\times
3}(p^{2})g^{\mu\nu}+i\Pi^{\prime}_{3\times 3}(p^{2})p^{\mu}p^{\nu}.$ (41)

Then we can follow (38) to resum the vector boson’s propagators. Notice that
all the masses of our initial and final state particles are ignorable since
they are extremely relativistic particles, so once a $p^{\mu}$ in (41)
appears, it will finally dots into the fermionic propagators of the external
legs. The Ward-Takahashi identity version in the broken phaseChanowitz:1985hj
transmute this into a Goldstone propagator, and its contributions are again
suppressed by the smallness of the Yukawa couplings. Therefore, we are able to
neglect the $p^{\mu}p^{\nu}$ terms in (41), and finally write down the
resummed propagators when only imaginary parts of the (41) are considered

$\displaystyle ig^{\mu\nu}\left[p^{2}I_{3\times
3}-\mathcal{M}^{2}_{v}-\text{Im}(\Pi_{3\times 3}(p^{2}))\right]^{-1}.$ (42)

In principle, $\text{Im}(\Pi_{3\times 3}(p^{2}))$ should be computed for each
$p$. However, practical event generators are not designed for such kind of
propagators. If we just fix $p^{2}\approx m_{Z}^{\text{SM}2}$ in (42), the
off-shell $s$-channel photon decay contributions are included, however the
softer $t$-channel photon propagators might also become massive. To avoid the
massive photon, we tentatively shut down the mixing terms, and diagonalize the

$\displaystyle\mathcal{M}_{V}^{\text{SM}2}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}m_{\hat{Z}^{\prime}}^{2}&0&0\\\
0&\frac{\hat{g}^{\prime
2}}{4}\hat{v}^{2}&-\frac{\hat{g}^{\prime}\hat{g}}{4}\hat{v}^{2}\\\
0&-\frac{\hat{g}^{\prime}\hat{g}}{4}\hat{v}^{2}&\frac{\hat{g}^{2}}{4}\hat{v}^{2}\end{pmatrix}.$
(43)

Shutting down the mixing terms does not affect the self-energy calculations up
to the one-loop order. Diagonalizing (43) requires $V_{\rm SM}$ defined in
(14). $V_{\rm SM}^{T}\mathcal{M}_{V}^{\text{SM}2}V_{\rm SM}$ gives

$\displaystyle\text{diag}[m_{\hat{Z}^{\prime}}^{2},m_{\hat{Z}}^{2},0],$ (44)

corresponding to the masses of the “$\hat{Z}^{\prime}$”, “SM-like $\hat{Z}$”
with the “hat” symbols to distinguish them from the true SM mass-eigenstates,
and the photon “eigenstate” masses respectively. Here
$m_{\hat{Z}^{\prime}}^{2}=\frac{\hat{g}^{2}+\hat{g}^{\prime
2}}{4}\hat{v}^{2}$. Actually, it is the $\hat{Z}^{\prime}$ and $\hat{Z}$ that
are nearly degenerate and might strongly mix together, and the photon lies far
from the $\hat{Z}^{\prime}$-$\hat{Z}$ mass spectrum, so we only have to
consider $\Pi_{\hat{Z}\hat{Z}}(p^{2})$,
$\Pi_{\hat{Z}^{\prime}\hat{Z}^{\prime}}(p^{2})$ and
$\Pi_{\hat{Z}^{\prime}\hat{Z}}(p^{2})$, whose contributions are not able to be
perturbatively expanded. With the definition
$c_{XY}\equiv\rm{Im}\left\\{\Pi_{X\leftrightarrow Y}(p^{2}\approx
m_{X,Y}^{2})\right\\}$, (44) becomes

$\displaystyle V_{\rm SM}^{T}\mathcal{M}_{V}^{\text{SM}2}V_{\rm
SM}+iC_{\hat{Z}/\hat{Z}^{\prime}}=\begin{pmatrix}m_{\hat{Z}^{\prime}}^{2}+ic_{\hat{Z}^{\prime}\hat{Z}^{\prime}}&ic_{\hat{Z}^{\prime}\hat{Z}}&0\\\
ic_{\hat{Z}^{\prime}\hat{Z}}&m_{\hat{Z}}^{2}+ic_{\hat{Z}\hat{Z}}&0\\\
0&0&0\end{pmatrix}.$ (45)

where $iC_{\hat{Z}/\hat{Z}^{\prime}}$ indicates the corrections from the
imaginary parts of the self-energy diagrams. Then we utilize $V_{\rm
SM}^{-1}=V_{\rm SM}^{T}$ to restore (45) into the form under the “interacting
eigenstate basis”,

$\displaystyle V_{\rm SM}(V_{\rm SM}^{T}\mathcal{M}_{V}^{\text{SM}2}V_{\rm
SM}+iC_{\hat{Z}/\hat{Z}^{\prime}})V_{\rm SM}^{T}$ (46) $\displaystyle=$
$\displaystyle\begin{pmatrix}m_{\hat{Z}^{\prime}}^{2}+ic_{\hat{Z}^{\prime}\hat{Z}^{\prime}}&-i\frac{\hat{g}^{\prime}}{\sqrt{\hat{g}^{\prime
2}+\hat{g}^{2}}}c_{\hat{Z}^{\prime}\hat{Z}}&i\frac{\hat{g}}{\sqrt{\hat{g}^{\prime
2}+\hat{g}^{2}}}c_{\hat{Z}^{\prime}\hat{Z}}\\\
-i\frac{\hat{g}^{\prime}}{\sqrt{\hat{g}^{\prime
2}+\hat{g}^{2}}}c_{\hat{Z}^{\prime}\hat{Z}}&\frac{\hat{g}^{\prime
2}}{4}\left(\hat{v}^{2}+\frac{4ic_{\hat{Z}\hat{Z}}}{\hat{g}^{\prime
2}+\hat{g}^{2}}\right)&-\frac{\hat{g}^{\prime}\hat{g}}{4}\left(\hat{v}^{2}+\frac{4ic_{\hat{Z}\hat{Z}}}{\hat{g}^{\prime
2}+\hat{g}^{2}}\right)\\\ i\frac{\hat{g}}{\sqrt{\hat{g}^{\prime
2}+\hat{g}^{2}}}c_{\hat{Z}^{\prime}\hat{Z}}&-\frac{\hat{g}^{\prime}\hat{g}}{4}\left(\hat{v}^{2}+\frac{4ic_{\hat{Z}\hat{Z}}}{\hat{g}^{\prime
2}+\hat{g}^{2}}\right)&\frac{\hat{g}^{2}}{4}\left(\hat{v}^{2}+\frac{4ic_{\hat{Z}\hat{Z}}}{\hat{g}^{\prime
2}+\hat{g}^{2}}\right)\end{pmatrix}.$

Supplement (46) with the mixing terms $\delta m^{2}$, $\delta v^{2}$ appeared
in (5), (8) is then acquired.

Figure 8: (Cross) self-energy terms connecting among $\tilde{Z}^{\prime}$,
$\tilde{Z}$ and photon fields.

As we have mentioned, we include the imaginary parts of the diagrams in the
first panel of Fig. 8, while neglecting the rest of them with the photon
external leg(s) involved. Since the mass of the photon is far from the nearly-
denegerate $Z^{\prime}$-$Z$ system, one find it possible to manipulate these
photon-involved terms perturbatively in (45) and (46), in contrast with the
non-perturbative $\hat{Z}$-$\hat{Z}^{\prime}$ mixing terms. Neglecting the
photon-involved terms gives the error of $\sim\left[\frac{{\rm
Im}\left(\Pi_{\gamma\hat{Z}^{(\prime)}}\right)}{m_{\hat{Z}}^{2}}\right]^{2}$,
which can be ignored in our current stage of theoretical discussions.

In the practical event simulation processes that we have performed, we at
first shut down all the mixing terms to calculate the width of the SM-Z boson
by the event generator to extract the $c_{\hat{Z}\hat{Z}}$ from it. Then, we
compute $c_{\hat{Z}^{\prime}\hat{Z}}$ and
$c_{\hat{Z}^{\prime}\hat{Z}^{\prime}}$ by hand relying on different model
setups. After diagonalizing (8) by our own programs, we acquire the “masses”
and “widths” of the “mass eigenstates”, as well as the rotated “coupling
constants” to be input into the event generator for further simulations, which
is equivalent to diagonalizing the propagator matrix (42) to calculate the
amplitudes.

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# On some theoretical limitations of Generative Adversarial Networks

Oriol, Benoît
<EMAIL_ADDRESS>
École Polytechnique
Palaiseau, 91120, France
Miot, Alexandre
<EMAIL_ADDRESS>
Société Générale
La Défense, 92800, France

###### Abstract

Generative Adversarial Networks have become a core technique in Machine
Learning to generate unknown distributions from data samples. They have been
used in a wide range of context without paying much attention to the possible
theoretical limitations of those models. Indeed, because of the universal
approximation properties of Neural Networks, it is a general assumption that
GANs can generate any probability distribution. Recently, people began to
question this assumption and this article is in line with this thinking. We
provide a new result based on extreme value theory showing that GANs can’t
generate heavy tailed distributions. The full proof of this result is given.

##  1 Introduction

The universal approximation property of neural networks (see [8] and [4])
might make us assume that GANs can simulate any distribution from a Gaussian
prior. However, neural networks, as functions are by design almost everywhere
differentiable functions with bounded derivatives to limit exploding gradients
phenomenons (see [10]). By Rademacher (see [7] for a proof) and mean value
theorems, this is nearly equivalent to say that neural networks functions are
Lipschitz continuous. This fact basically sets the limitations of GANs to
express any probability distribution given a Gaussian prior. The are numerous
definitions of the concept of ”fat”, ”longed” or ”heavy” tailed distributions.
They are usually not equivalent but all convey a sense of having a larger
probability of being ”big” compared to a Gaussian or Exponential distribution.
Here we focus on two possible ways to define the concept. One, similarly to
[12], is focussing on finite samples and relies on classical concentration
inequalities. The other is asymptotic and uses Extreme Value Theory to prove a
new theorem in the continuity of the theoretical work of Huster et al. in [9]
and the experimental approach of [6].

Notations, in the following, we make use of the following notations:

  * -

$f$ is a Lipschitz function $\mathbb{R}^{n}\mapsto\mathbb{R}$,
$||f||_{l}=\sup_{x,y\in\mathbb{R}^{n},x\neq y}\frac{||f(x)-f(y)||}{||x-y||}$
its semi-norm,

  * -

$\gamma_{n}$ the Gaussian measure on $\mathbb{R}^{n}$,

  * -

$d$ the Euclidean distance in $\mathbb{R}^{n}$ i.e. $d(x,y)=||y-x||$,

  * -

for any set $S\subset\mathbb{R}^{n}$ the $\epsilon$ neighbourhood
$S_{\epsilon}=\\{x\in\mathbb{R}^{n}\text{ such that }d(x,S)<\epsilon\\}$,
where $\epsilon>0$,

  * -

$\bar{A}$ the complement of a subset $A\subset\mathbb{R}^{n}$,

  * -

$M$ the median of a mapping $\mathbb{R}^{n}\mapsto\mathbb{R}$ for the
$\gamma_{n}$ measure i.e. $\gamma_{n}(\\{f\geq M\\})\geq\frac{1}{2}$ and
$\gamma_{n}(\\{f\leq M\\})\geq\frac{1}{2}$,

  * -

$X$ a standard Gaussian random variable in $\mathbb{R}^{n}$,

  * -

$\bar{\Psi}=\frac{1}{\sqrt{2\pi}}\int_{x}^{+\infty}e^{-u^{2}/2}du$ the
Gaussian tail function.

##  2 Limitations through sub-gaussianity

In this section we prove that given a Lipschitz function and a Gaussian prior
$X$, $f(X)$ is sub-gaussian: a GAN with a Gaussian prior can only generate
sub-gaussian distributions.

######  Definition 1.

A real valued random variable $Y$ is said to be sub-gaussian if it satisfies
one of the two following equivalent properties :

$\Updownarrow\begin{array}[]{l}\exists K\in\mathbb{R},\forall t\geq
0,P(\left|Y\right|\geq t)\leq 2e^{-\frac{t^{2}}{K^{2}}}\\\ \exists
K^{\prime}\in\mathbb{R},\forall p\geq 1,||X||_{L^{p}}\leq
K^{\prime}\sqrt{p}\text{ .}\end{array}$

For a proof see [15].

If $G=f(X)$, where $f$ is Lipschitz, by Lipschitz continuity and
$\mathbb{E}(X)=0$

$\forall x\geq 0\quad\mathbb{P}(|G-f(0)|\geq
x)\leq\mathbb{P}\left(|X|\geq\frac{x}{||f||_{l}}\right)\text{ .}$

In particular if $X$ is one dimensional then using a standard upper bound of
the gaussian tail function $G$ will be sub-gaussian as a sum of two
independant sub-gaussian functions, considering $f(0)$ as a constant random
variable.

If $X$ was $n$ dimensional then,

$\mathbb{P}\left(|X|\geq\frac{x}{||f||_{l}}\right)=\mathbb{P}\left(|X|^{2}\geq\frac{x^{2}}{||f||_{l}^{2}}\right)\text{
,}$

and $|X|^{2}$ following a $\chi^{2}$ distribution with $n$ degrees of freedom
the sub-gaussianity of $G$ would seem to be dependent on the dimension of $X$.
Yet, this is not the case as stated by the following remarkable result:

######  Theorem 1 (Gaussian concentration theorem [14], [13] and [3]).

Let $X$ be a standard gaussian random variable on $\mathbb{R}^{n}$ and $f$ a
Lipschitz function then $f(X)-\mathbb{E}(f(X))$ is sub-gaussian. More
precisely,

$\forall\epsilon>0,\mathbb{P}(|f(X)-\mathbb{E}(f(X))|\geq\epsilon)\leq
2e^{-2\epsilon^{2}/||f||_{l}^{2}}\text{ .}$

So in particular, a GAN with a Gaussian prior will not be able to generate any
realistic samples even if trained on a ”fat tailed” distribution. This is not
the first time that concentration of measure gives some strong theoretical
limits to machine learning methods see [5] or [11] for a more recent paper.
Limitations of GANs has also been explored from a different perspective in
[16].

This theorem is also true when we replace the mean $\mathbb{E}(f(X))$ by a
median of $f(X)$. The original proof of the theorem can be found in [14]. The
proof is quite technical, a more accessible one can be found in [2] . To get a
sense of the concentration of measure phenomenon we provide here a simple
proof with the median.

The gaussian concentration theorem is a consequence of:

######  Theorem 2 (Gaussian Isoperimetric Theorem [3]).

Let’s $A$ be a Borel set in $\mathbb{R}^{n}$ and
$H=\\{x\in\mathbb{R}^{n}\text{ such that }x_{1}<a\\}$ with $a\in\mathbb{R}$
such that $\gamma_{n}(A)=\gamma_{n}(H)$ then

$\forall\epsilon\geq
0\;\gamma_{n}(A_{\epsilon})\geq\gamma_{n}(H_{\epsilon})\text{ .}$

It is easily seen that $\gamma_{n}(H_{r})=\Psi(a+r)$ where $\Psi$ is the
cumulative distributive function of the one dimensional standard Gaussian
distribution.
It is not obvious at first sight what is the link between this theorem and the
Gaussian concentration theorem for Lipschitz functions. The link is made
defining the following ‘isoperimetric function’ for $a\in[0,1]$ and
$\epsilon>0$

$\displaystyle\eta_{a}(\epsilon)$ $\displaystyle=\sup_{A\text{ borel set
}\subset\mathbb{R}^{n}}\\{\gamma_{n}(\bar{A_{\epsilon}})\;|\;\gamma_{n}(A)\geq
a\\}$ $\displaystyle=1-\inf_{A\text{ borel set
}\subset\mathbb{R}^{n}}\\{\gamma_{n}(A_{\epsilon})\;|\;\gamma_{n}(A)\geq
a\\}\text{ .}$

######  Lemma 1.

Let $f\,:\,\mathbb{R}^{n}\mapsto\mathbb{R}$ be a Lipschitz function and $M$ a
median for the Gaussian measure, then

$\forall\epsilon>0\quad\gamma_{n}(|f-M|>\epsilon)\leq\eta_{\frac{1}{2}}\left(\frac{\epsilon}{||f||_{l}}\right)\text{
.}$

###### Proof.

Let $A=\\{f\leq M\\}$, $\epsilon>0$ and $x\in A_{\epsilon}$ then

$\exists y\in A\text{ such that }d(x,A)\leq d(x,y)<\epsilon$

so, $f$ being Lipschitz and $y\in A$

$|f(x)-f(y)|\leq||f||_{l}\;\epsilon\text{ .}$

So, $f(x)\leq M+||f||_{l}\;\epsilon$ i.e. $\\{f\geq
M+||f||_{l}\;\epsilon\\}\subset\bar{A_{\epsilon}}$. Changing
$\epsilon\rightarrow\frac{\epsilon}{||f||_{l}}$ we have proved that for any
Lipschitz function $f$ of median $M$

$\forall\epsilon>0\quad\gamma_{n}(\\{f>M+\epsilon\\})\leq\eta_{\frac{1}{2}}\left(\frac{\epsilon}{||f||_{l}}\right)\text{
.}$

Noticing that $f$ if Lipschitz iif $-f$ is, $||f||_{l}=||-f||_{l}$, if $M$ is
a median of $f$ then $-M$ is a median of $-f$ and applying what we just proved
to $-f$ the result follows. ∎

We can now prove the Gaussian concentration theorem.

###### Proof.

Let $A$ be a Borel set such that $\gamma_{n}(A)\geq\frac{1}{2}$, there exists
a half-space $H=\mathbb{R}^{n-1}\times]-\infty,a[$ such that
$\gamma_{n}(A)=\gamma_{n}(H)=\Psi(a)\geq\frac{1}{2}$. From the Isoperimetric
Gaussian Theorem

$\displaystyle\forall\epsilon\geq 0,\quad$
$\displaystyle\gamma_{n}(A_{\epsilon})\geq\gamma_{n}(H_{\epsilon})$
$\displaystyle\gamma_{n}(A_{\epsilon})\geq\Psi(a+\epsilon)\geq\Psi(\epsilon)\text{
.}$

$\Psi$ being non decreasing and $a\geq 0$ as $\Psi(a)\geq\frac{1}{2}$. Taking
the infinum on the left side and noticing that the infinimum is reached for
$H$

$\forall\epsilon>0\text{,}\\\ \inf_{A\text{ borel set
}\subset\mathbb{R}^{n}}\\{\gamma_{n}(A_{\epsilon})\;|\;\gamma_{n}(A)\geq\frac{1}{2}\\}=\Psi(\epsilon)\text{
.}$

That is to say,

$\eta_{\frac{1}{2}}=1-\Psi=\bar{\Psi}.$

∎

##  3 Limitations through Extreme Value Theory

In this section, we prove the main result of this paper: given a Lipschitz
function and a Gaussian prior $X$, if $f(X)$ is in a domain of attraction of
an extreme value distribution of parameter $\xi$ then $\xi\leq 0$. In
particular, $f(X)$ can’t be ”heavy tailed”. In fact, we prove the theorem for
a wider range of distributions.
In the following, we use the notations of Extreme Value Theory that are
introduced in [9].

######  Theorem 3.

Let $n\in\mathbb{R}^{*}$, $f:\mathbb{R}^{d}\rightarrow\mathbb{R}$ a
$\mathcal{C}^{1}$ a.e Lipschitz function with semi norm $L=||f||_{l}$. Let
$G_{k}^{d}$ be a real random variable of probability distribution function
$g_{G_{k}^{d}}(x)\propto\lVert x\rVert_{2}^{k}e^{-\frac{\lVert
x\rVert_{2}^{2}}{2}}$, where $k\in\mathbb{N}$. If $f(G_{k}^{d})$ is in the
domain of attraction of the extreme value distribution of parameter
$\xi\in\mathbb{R}$, i.e $f(G_{k}^{d})\in\mathcal{D}(\mathcal{H}_{\xi})$, then
$\xi\leq 0$.

###### Proof.

Case $d=1$. We prove by contradiction that $\xi\leq 0$. Supposing that
$\xi>0$, by theorem 8.a [1], $\forall\,\gamma\in]0,\xi[$,
$\mathbb{E}[f(G_{k}^{1})^{\gamma}]$ is finite and

$\displaystyle c_{\gamma}$
$\displaystyle=\lim_{t\rightarrow\infty}\mathbb{E}\left[\left(\frac{f(G_{k}^{1})}{t}\right)^{\gamma}|f(G_{k}^{1})>t\right]$
$\displaystyle=\left(1-\frac{\gamma}{\xi}\right)^{-1}\text{ .}$

Let $\gamma\in]0,\xi[$. We are only interested in the behaviour of the
previous integral when $t$ goes to $+\infty$ so we can suppose that $t>f(0)+1$
and $t>\sqrt{|k-1|+\gamma}L+|f(0)|$.
$f^{-1}(]t,\infty[)$ is an open set of $\mathbb{R}$ by continuity. Open
intervals are a countable base of $\mathbb{R}$, so we can write
$f^{-1}(]t,\infty[)=\bigcup_{i\in\mathcal{I}}]a_{i},b_{i}[$ where
$\mathcal{I}$ is finite or countable. We can also suppose that any of those
intervals are disjoints. So we can suppose that :

  * -

$f^{-1}(]t,\infty[)=\bigcup_{i\in\mathcal{I}}]a_{i},b_{i}[$ where
$\mathcal{I}$ is finite or countable

  * -

$0\notin]a_{i},b_{i}[$, $a_{i}\neq 0$, $b_{i}\neq 0$ and $f$ is strictly
positive on each interval

  * -

$-\infty\leq a_{0}<b_{0}\leq a_{1}<b_{2}\leq\ldots<b_{m^{*}}\leq+\infty$,
where $m^{*}$ is equal to $m$ or $\infty$ according to $\mathcal{I}$
cardinality

  * -

$\forall i\in\mathcal{I}\;f(a_{i})=t$ if $a_{i}>-\infty$ by continuity of $f$

  * -

$\forall i\in\mathcal{I}\;f(b_{i})=t$ if $b_{i}<\infty$ by continuity of $f$

  * -

$\forall x\in]a_{i},b_{i}[\;|x|>\frac{t-f(0)}{L}$ as $f$ is $L$-Lipschitz and
$]a_{i},b_{i}[\subset f^{-1}(]t,+\infty[$. In particular, noting
$t^{*}=\min(t,t-f(0))$, $|x|>\frac{t^{*}}{L}$

Also, we are only interested in $t\to+\infty$ so we can suppose
$t^{*^{2}}>|k-1|L^{2}$. If $\mathcal{I}=\emptyset$, the case is trivial:
$\mathbb{E}\left[\left(\frac{f(G_{k}^{1})}{t}\right)^{\gamma}|f(G_{k}^{1})>t\right]$
is not defined, which is a contradiction.
Otherwise, the conditional expectation is well defined and finite and we have:

$\mathbb{E}\left[\left(\frac{f(G_{k}^{1})}{t}\right)^{\gamma}|f(G_{k}^{1})>t\right]=\\\
\frac{\sum_{i\in\mathcal{I}}\int_{a_{i}}^{b_{i}}\left(\frac{f(x)}{t}\right)^{\gamma}|x|^{n}e^{-\frac{x^{2}}{2}}dx}{\sum_{i\in\mathcal{I}}\int_{a_{i}}^{b_{i}}|x|^{n}e^{-\frac{x^{2}}{2}}dx}\text{
.}$ (1)

Let $i\in\mathcal{I}$. For the numerator, integrating by part:

$\int_{a_{i}}^{b_{i}}\left(\frac{f(x)}{t}\right)^{\gamma}|x|^{k}e^{-\frac{x^{2}}{2}}dx=\\\
\left[-\frac{|x|^{k}}{x}\left(\frac{f(x)}{t}\right)^{\gamma}e^{-\frac{x^{2}}{2}}\right]_{a_{i}}^{b_{i}}+\\\
\int_{a_{i}}^{b_{i}}\underbrace{\left(\frac{k-1}{x^{2}}+\gamma\frac{f^{\prime}(x)}{xf(x)}\right)}_{M}\left(\frac{f(x)}{t}\right)^{\gamma}|x|^{k}e^{-\frac{x^{2}}{2}}dx$

The first integrated part is equal to
$\left[-\frac{|x|^{k}}{x}e^{-\frac{x^{2}}{2}}\right]_{a_{i}}^{b_{i}}$, as we
have seen on the interval bounds either $f$ is equal to $t$ or
$f=O_{\pm\infty}(x)$. We can bound the $M$ term as $|x|\geq\frac{t^{*}}{L}$,
$f(x)>t>0$ and $|f^{\prime}|<L$,

$|M|\leq|k-1|\frac{L^{2}}{t^{*^{2}}}+\gamma\frac{L^{2}}{t^{*^{2}}}\text{ .}$

We deduce the following inequalities:

$\int_{a_{i}}^{b_{i}}\left(\frac{f(x)}{t}\right)^{\gamma}|x|^{k}e^{-\frac{x^{2}}{2}}dx\leq\\\
\left[-|x|^{k-1}e^{-\frac{x^{2}}{2}}\right]_{a_{i}}^{b_{i}}+\\\
\frac{(|k-1|+\gamma)L^{2}}{t^{*^{2}}}\int_{a_{i}}^{b_{i}}\left(\frac{f(x)}{t}\right)^{\gamma}|x|^{k}e^{-\frac{x^{2}}{2}}dx\text{
,}$ (2)
$\int_{a_{i}}^{b_{i}}\left(\frac{f(x)}{t}\right)^{\gamma}|x|^{k}e^{-\frac{x^{2}}{2}}dx\geq\\\
\left[-|x|^{k-1}e^{-\frac{x^{2}}{2}}\right]_{a_{i}}^{b_{i}}-\\\
\frac{(|k-1|+\gamma)L^{2}}{t^{*^{2}}}\int_{a_{i}}^{b_{i}}\left(\frac{f(x)}{t}\right)^{\gamma}|x|^{k}e^{-\frac{x^{2}}{2}}dx\text{
.}$ (3)

The denominator has still a dependance on $f$ through the domain of
integration, so $|x|>\frac{t^{*}}{L}$ is still valid and similarly:

$\int_{a_{i}}^{b_{i}}|x|^{k}e^{-\frac{x^{2}}{2}}dx\leq\left[-|x|^{k-1}e^{-\frac{x^{2}}{2}}\right]_{a_{i}}^{b_{i}}+\\\
\frac{|k-1|L^{2}}{t^{*^{2}}}\int_{a_{i}}^{b_{i}}|x|^{k}e^{-\frac{x^{2}}{2}}dx\text{
,}$ (4)
$\int_{a_{i}}^{b_{i}}|x|^{k}e^{-\frac{x^{2}}{2}}dx\geq\left[-|x|^{k-1}e^{-\frac{x^{2}}{2}}\right]_{a_{i}}^{b_{i}}-\\\
\frac{|k-1|L^{2}}{t^{*^{2}}}\int_{a_{i}}^{b_{i}}|x|^{k}e^{-\frac{x^{2}}{2}}dx\text{
.}$ (5)

Combining equations (3) and (4) in (1), we have:

$\mathbb{E}\left[\left(\frac{f(G_{k}^{1})}{t}\right)^{\gamma}|f(G_{k}^{1})>t\right]\geq\frac{1-\frac{|k-1|L^{2}}{t^{*^{2}}}}{1+\frac{(|k-1|+\gamma)L^{2}}{t^{*^{2}}}}\text{
.}$

And combining (2) and (5) in (1), as we chose $t>\sqrt{|k-1|+\gamma}L+|f(0)|$,
we have:

$\mathbb{E}\left[\left(\frac{f(G_{k}^{1})}{t}\right)^{\gamma}|f(G_{k}^{1})>t\right]\leq\frac{1+\frac{|k-1|L^{2}}{t^{*^{2}}}}{1-\frac{(|k-1|+\gamma)L^{2}}{t^{*^{2}}}}$

So
$c_{\gamma}=\lim_{t\rightarrow\infty}\mathbb{E}\left[\left(\frac{f(G_{k}^{1})}{t}\right)^{\gamma}|f(G_{k}^{1})>t\right]=1$.
Assuming $f(G_{k}^{1})\in\mathcal{D}(\mathcal{H}_{\xi}),\xi>0$, entails
$c_{\gamma}=(1-\frac{\gamma}{\xi})^{-1}$ and $\gamma=0$. We conclude that
$\xi\leq 0$.

Case $d\in\mathbb{N}^{*}$. We prove that $\xi\leq 0$ by contradiction. If
$\xi>0$ using theorem 8.a [1], $\forall 0<\gamma<\xi$,
$\mathbb{E}[f(G_{k}^{d})^{\gamma}]$ is finite and

$\displaystyle c_{\gamma}$
$\displaystyle=\lim_{t\rightarrow\infty}\mathbb{E}\left[\left(\frac{f(G_{k}^{d})}{t}\right)^{\gamma}|f(G_{k}^{d})>t\right]$
$\displaystyle=\left(1-\frac{\gamma}{\xi}\right)^{-1}\text{ .}$

Let $\gamma\in]0,\xi[$ and $t\in\mathbb{R}_{+}^{*}$ such that $t>f(0_{d})$ and
$t>L\sqrt{|k+d-2|+\gamma}L+|f(0)|$. Using the hyperspherical coordinates, we
introduce the operator
$H:\mathcal{L}([0,\pi]^{d-2}\times[0,2\pi])\mapsto\mathbb{R}$:

$H(f)=\alpha\int_{0}^{\pi}...\int_{0}^{\pi}\int_{0}^{2\pi}\prod_{i=1}^{d-2}\sin(\theta_{i})^{d-i-1}f(\theta)d\theta\text{,}$

with $\alpha$ the normalising term for the $G_{k}^{d}$ distribution and
$d\theta=d\theta_{1}...d\theta_{d-1}$.
Then, we have:

$\mathbb{E}\left[\left(\frac{f(G_{k}^{d})}{t}\right)^{\gamma}1_{f(G_{k}^{d})>t}\right]=\\\
H\Bigl{(}\theta\mapsto\underbrace{\int_{0}^{+\infty}1_{f_{\theta}(r)>t}\left(\frac{f_{\theta}(r)}{t}\right)^{\gamma}r^{k+d-1}e^{-\frac{r^{2}}{2}}dr}_{\mathbb{E}\left[\left(\frac{f_{\theta}(G_{k+d-1}^{1})}{t}\right)^{\gamma}1_{f_{\theta}(G_{k+d-1}^{1})>t}\right]}\Bigr{)}$

with $x=r(x_{1},...,x_{d})$ and

$\displaystyle x_{1}$ $\displaystyle=\sin(\theta_{1})$ $\displaystyle x_{2}$
$\displaystyle=\sin(\theta_{1})\cos(\theta_{2})$ $\displaystyle\vdots$
$\displaystyle x_{d-1}$
$\displaystyle=\sin(\theta_{1})\sin(\theta_{2})\ldots\cos(\theta_{d-1})$
$\displaystyle x_{d}$
$\displaystyle=\sin(\theta_{1})\sin(\theta_{2})\ldots\sin(\theta_{d-1})\text{
.}$

For $\theta=(\theta_{1},...\theta_{d-1})$,
$f_{\theta}:r\in\mathbb{R}_{+}\mapsto f(rx_{1},...,rx_{d})$ is $L$-Lipschitz
as $(x_{1},\ldots,x_{d})$ is on the unit sphere. Also, $f(0)=f_{\theta}(0)$.
We can use the bounds from the 1-dimensional proof. We note:

$M_{+}=\frac{1+\frac{|k+d-2|L^{2}}{t^{*^{2}}}}{1-\frac{(|k+d-2|+\gamma)L^{2}}{t^{*^{2}}}}$
$M_{-}=\frac{1-\frac{|k+d-2|L^{2}}{t^{*^{2}}}}{1+\frac{(|k+d-2|+\gamma)L^{2}}{t^{*^{2}}}}\text{
.}$

We obtain:

$\mathbb{E}\left[\left(\frac{f(G_{k}^{d})}{t}\right)^{\gamma}1_{f(G_{k}^{d})>t}\right]\leq\\\
M_{+}H\left(\theta\mapsto\mathbb{P}(1_{f_{\theta}(G_{k+d-1}^{1})>t})\right)\text{
.}$

That is to say,

$\mathbb{E}\left[\left(\frac{f(G_{k}^{d})}{t}\right)^{\gamma}1_{f(G_{k}^{d})>t}\right]\leq
M_{+}\mathbb{P}(f(G_{k}^{d})>t)\text{ .}$

Similarly, we have:

$\mathbb{E}\left[\left(\frac{f(G_{k}^{d})}{t}\right)^{\gamma}1_{f(G_{k}^{d})>t}\right]\geq
M_{-}\mathbb{P}(f(G_{k}^{d})>t)\text{ .}$

Thus, we conclude similarly that $c_{\gamma}$ is well-defined, finite, and
$c_{\gamma}=1$ that is $\gamma=0$, which is absurd as $\gamma>0$. ∎

##  4 Conclusion and future work

Because of the intrinsic Lipschitz characteristics of Neural Networks, GANs
expressivity is limited. In particular, a Gaussian prior cannot be used to
simulate heavy tailed distributions. In the EVT framework, the question of the
existence of a tail index for the generated distribution, or the conditions
for its existence, remains. A theoretical partial answer is given in [9] for
GANs with ReLU or Leaky-ReLU activation functions and a finite number of
neurons. The general case is still an open question. Likewise, determining the
thinnest tail prior being able to simulate samples exhibiting heavy tails is
an important question needing further investigations.
Moreover, experimentally, the problem of training GANs with a heavy-tailed
prior remains too. Indeed, with such priors GANs are hard to train and exhibit
numerical instabilities.

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