# Complete one-loop electroweak corrections to $ZZZ$ production at the ILC

Su Ji-Juan, Ma Wen-Gan, Zhang Ren-You, Wang Shao-Ming, and Guo Lei
Department of Modern Physics, University of Science and Technology
of China (USTC), Hefei, Anhui 230027, P.R.China

###### Abstract

We study the complete ${\cal O}(\alpha_{ew})$ electroweak (EW) corrections to
the production of three $Z^{0}$-bosons in the the framework of the standard
model(SM) at the ILC. The leading order and the EW next-to-leading order
corrected cross sections are presented, and their dependence on the colliding
energy $\sqrt{s}$ and Higgs-boson mass $m_{H}$ is analyzed. We investigate
also the LO and one-loop EW corrected distributions of the transverse momentum
of final $Z^{0}$ boson, and the invariant mass of $Z^{0}Z^{0}$-pair. Our
numerical results show that the EW one-loop correction generally suppresses
the tree-level cross section, and the relative correction with
$m_{H}=120~{}GeV(150~{}GeV)$ varies between $-15.8\%(-13.9\%)$ and
$-7.5\%(-6.2\%)$ when $\sqrt{s}$ goes up from $350~{}GeV$ to $1~{}TeV$.

PACS: 11.15.Ex ,12.15.Lk ,13.66.Jn ,14.70.Hp

##  I. Introduction

To discover the signature of new physics beyond the standard model(SM)[1, 2]
is one of the main goals for the forthcoming collider experiments. The
precision measurements of the trilinear gauge-boson couplings are helpful for
verification of non-abelian gauge structure, and the investigation of the
quartic gauge-boson couplings can either confirm the symmetry breaking
mechanism or present the direct test on the new physics beyond the SM[3]. The
direct study of quartic gauge-boson couplings requires the investigations of
the processes involving at least three external gauge-bosons. In Refs.[4, 5,
6, 7, 8], the precise predictions for the $VVV$ productions at hadron
colliders were provided. It shows that the QCD corrections increase the $WWZ$
cross section at the LHC by more than $70\%$, and the QCD corrections to $ZZZ$
production at the LHC increase the LO cross section by about $50\%$ [6]. Thus,
any quantitative measurement of the concerned gauge couplings will have to
take QCD corrections into account,

Due to heavy backgrounds, the precise measurement at a hadron collider is more
difficult than at linear collider. The proposed International Linear Collider
(ILC) by the particle physics community will be built with the entire
colliding energy in the range of $200~{}GeV<\sqrt{s}<500~{}GeV$ and an
integrated luminosity of around $500~{}(fb)^{-1}$ in four years. The machine
should be upgradeable to $\sqrt{s}\sim 1~{}TeV$ with an integrated luminosity
of $1~{}(ab)^{-1}$ in three years[9]. Among all the ILC physics goals, the
verification of gauge theory in the SM and finding the evidence of new physics
via experimental measurement of the electroweak gauge boson couplings are
crucial tasks too. The measurement will be able to be improved considerably at
ILC compared with at Fermilab Tevatron and CERN LHC, and therefore a precise
understand of the SM phenomenology at ILC to at least one-loop order is
necessary[10]. Without the accurate theoretical predictions and reliable error
estimates for important observables at ILC, it is impossible to interpret
experimental data properly.

The process of $ZZZ$ production with the subsequential leptonic decays of
vector bosons at ILC is not only an important process as a background for
various new physics processes, but also possible to provide further tests for
the quadrilinear gauge boson couplings, including the four gauge boson
coupling, such as between $ZZZZ$, which does not exist at tree-level in the
SM, because this kind of quadrilinear couplings would induce deviations from
the SM predicted observables[11].

In this paper we present the calculations of the cross sections for the
process $e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}$ at the leading order(LO) and involving
complete electroweak (EW) one-loop (${\cal O}(\alpha_{ew})$) corrections. The
paper is organized as follows: In the next section we present the calculation
descriptions for the tree-level process $e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}$ . The
calculation of the electroweak corrections at one-loop level is reported in
section III. The numerical results and discussions are given in section IV. In
the last section we give a short summary.

##  II. Leading-order $e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}$ process

In the calculations of the tree-level and one-loop level cross sections for
the $e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}$ process, we use the ’t Hooft-Feynman
gauge. The analytically calculation of the leading order cross section for
$e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}$ process is presented in this section. We
describe the lowest order $e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}$ process adopted for
evaluating the cross section as

$e^{+}(p_{1})+e^{-}(p_{2})\to Z^{0}(p_{3})+Z^{0}(p_{4})+Z^{0}(p_{5}),$ (2.1)

where $p_{i}~{}(i=1-5)$ label the four-momenta of incoming positron, electron
and outgoing $Z^{0}$-bosons, respectively. Since the mass of electron/positron
is negligible comparing with the colliding energy and the Yukawa coupling
strength between Higgs/Goldstone and fermions is proportional to the fermion
mass, in our work we ignore the contributions of the Feynman diagrams
involving the couplings of $H^{0}-e^{+}-e^{-}$ and $G^{0}-e^{+}-e^{-}$. We
depict the tree-level Feynman diagrams contributing to the cross section of
the production process of three $Z^{0}$-bosons at the ILC in Fig.1. There we
have 9 generic tree-level diagrams for the process $e^{+}e^{-}\to
Z^{0}Z^{0}Z^{0}$ in the framework of the SM. All the Born-level diagrams can
be grouped in two different topologies. Figs.1(a-c) belong to s-channel,
Figs.1(d-i) are grouped in t(u)-channel.

Figure 1:  The generic tree-level Feynman diagrams for $e^{+}e^{-}\to
Z^{0}Z^{0}Z^{0}$ process.

The differential cross section for the process $e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}$
at the tree-level is then obtained as

$\displaystyle d\sigma_{LO}=\frac{1}{3!}\frac{1}{4}\sum_{spin}|{\cal
M}_{LO}|^{2}d\Phi_{3},$ (2.2)

where ${\cal M}_{LO}$ is the amplitude of all the tree-level diagrams in
Fig.1. The factors $\frac{1}{3!}$ and $\frac{1}{4}$ are due to three identical
final $Z^{0}$-bosons and spin-averaging of the initial particles,
respectively. The summation in Eq.(2.2) is taken over the spins of the initial
and final particles, and $d\Phi_{3}$ is the three-particle phase space element
defined as

$\displaystyle
d\Phi_{3}=\delta^{(4)}\left(p_{1}+p_{2}-\sum_{i=3}^{5}p_{i}\right)\prod_{j=3}^{5}\frac{d^{3}\textbf{{p}}_{j}}{(2\pi)^{3}2E_{j}}.$
(2.3)

##  III. Electroweak (${\cal O}(\alpha_{ew})$) corrections

The ${\cal O}(\alpha_{ew})$ order electroweak corrections to the Born-level
$e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}$ process consist of two parts, i.e.,

  * •

The virtual contributions to the leading order process $e^{+}e^{-}\to
Z^{0}Z^{0}Z^{0}$ from the electroweak one-loop and their corresponding
counterterm diagrams;

  * •

The contribution from the real photon emission process $e^{+}e^{-}\to
Z^{0}Z^{0}Z^{0}$ $\gamma$. The soft photon emission process $e^{+}e^{-}\to
Z^{0}Z^{0}Z^{0}$ ($\gamma$) consists IR singularities, which will be cancelled
by the IR singularities in the contributions of the one-loop diagrams. There
is no collinear IR singularity since we keep the nonzero mass of
electron(positron);

In the following subsections, we describe in detail the calculation procedure
and discuss the calculation of each contribution part.

###  III..1  Virtual corrections

There are totally 2313 electroweak one-loop and corresponding counterterm
Feynman diagrams being taken into account in our calculation, and they can be
classified into self-energy, triangle, box, pentagon and counterterm diagrams.
We depict some representative samples among 66 pentagon diagrams in Fig.2. In
our calculation of the electroweak (${\cal O}(\alpha_{ew})$) corrections to
$e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}$ process, all the one-loop Feynman diagrams and
their relevant amplitudes are created by using $FeynArts~{}3.3$[12], and the
Feynman amplitudes are subsequently implemented by applying FormCalc5.3
programs[13] and our in-house routines. The electroweak one-loop amplitude
involves five point tensor integrals up to rank 4\. The numerical calculation
of the integral functions($n\leq 4$) are implemented by using the expressions
presented in Refs.[14, 15]. We use our independent Fortran subroutines
following the expressions for the scalar and tensor five-point integrals in
Ref.[16], and find agreement with LoopTools2.2[13]. The Grace2.2.1 program[17]
is used to accomplish five-body phase-space integration for hard photon
radiation process $e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}\gamma$ .

Figure 2:  Some representative pentagon Feynman diagrams for the process
$e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}$ .

The total unrenormalized amplitude corresponding to all the one-loop Feynman
diagrams contains both ultraviolet (UV) and infrared (IR) singularities. We
regulate all singularities adopting dimensional regularization(DR) scheme [21]
where the dimensions of spinor and space-time manifolds are extended to
$D=4-2\epsilon$. The relevant fields are renormalized by taking the on-mass-
shell (OMS) scheme [22, 23]. The IR singularity is regularized by introducing
a infinitesimal fictitious mass $m_{\gamma}$. All the tensor coefficients of
the one-loop integrals can be calculated by using the reduction formulae
presented in Refs.[16, 24]. As we expect, the UV divergence contributed by
virtual one-loop diagrams can be cancelled by that contributed from the
counterterms exactly both analytically and numerically.

###  III..2  Real photon emission process $e^{+}e^{-}\to
Z^{0}Z^{0}Z^{0}\gamma$

In our calculation for one-loop diagrams, there exists soft IR divergence. In
order to get an IR finite cross section for $e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}$ up
to the order of ${\cal O}(\alpha_{ew}^{4})$, we should consider the ${\cal
O}(\alpha_{ew})$ corrections to $e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}$ process due to
real photon emission. The soft IR divergence in virtual photonic corrections
for the process $e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}$ can be exactly cancelled by
adding the real photonic bremsstrahlung corrections to this process in the
soft photon limit. In the real photon emission process

$\displaystyle e^{+}(p_{1})+e^{-}(p_{2})\to
Z^{0}(p_{3})+Z^{0}(p_{4})+Z^{0}(p_{5})+\gamma(p_{6}),$ (3.1)

a real photon radiates from the electron/positron, and can be soft or hard.
The general phase-space-slicing (PSS) method [25] is adopted to isolate the
soft photon emission singularity part in the real photon emission process
$e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}\gamma$ , and the cross section of the real
photon emission process (3.1) is decomposed into soft and hard terms

$\Delta\sigma_{{real}}=\Delta\sigma_{{S}}+\Delta\sigma_{{H}}=\sigma_{LO}(\delta_{{S}}+\delta_{{H}}).$
(3.2)

where the ’soft’ and ’hard’ describe the energy nature of the radiated photon.
The energy $E_{6}$ of the radiated photon in the center of mass system(c.m.s.)
frame is considered soft if $E_{6}\leq\Delta E$, and hard if $E_{6}>\Delta E$,
respectively. Then both $\sigma_{{S}}$ and $\sigma_{{H}}$ should depend on the
arbitrary soft cutoff $\delta_{s}\equiv\Delta E/E_{b}$, where $E_{b}$ is the
electron beam energy in the c.m.s. frame and equals to $\sqrt{s}/2$, but the
total cross section of the real photon emission process $\sigma_{{real}}$ is
cutoff $\Delta E/E_{b}$ independent. Since the soft cutoff $\Delta E/E_{b}$ is
taken to be a small value in our calculations, the terms of order $\Delta
E/E_{b}$ can be neglected and the soft contribution can be evaluated by using
the soft photon approximation analytically [22, 23, 26]

$\displaystyle{d}\Delta\sigma_{S}=-d\sigma_{{LO}}\frac{\alpha_{ew}}{2\pi^{2}}\int_{|\vec{p}_{6}|\leq\Delta
E}\frac{d^{3}p_{6}}{2E_{6}}\left(\frac{p_{1}}{p_{1}\cdot
p_{6}}-\frac{p_{2}}{p_{2}\cdot p_{6}}\right)^{2}.$ (3.3)

Our calculation demonstrates that the IR singularity in the soft contribution
from Eq.(3.3) is cancelled exactly with that from the virtual photonic
corrections. Therefore, $\Delta\sigma_{v}+\Delta\sigma_{real}$, the sum of the
${\cal O}(\alpha_{ew}^{4})$ cross section corrections from virtual, soft and
hard photon emission contribution parts, is independent of the cutoff value
$\delta_{s}$. The hard contribution, which is UV and IR finite, is computed by
using the Monte Carlo technique. Finally, the electroweak corrected cross
section for the $e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}$ process up to the order of
${\cal O}(\alpha^{4}_{ew})$ can be obtained by

$\displaystyle\sigma_{{tot}}=\sigma_{LO}+\Delta\sigma_{v}+\Delta\sigma_{real}=\sigma_{LO}\left(1+\delta_{tot}\right).$
(3.4)

###  III..3  QED and total ${\cal O}(\alpha_{ew})$ order corrections

In analyzing the originations of the electroweak corrections, we split the
full ${\cal O}(\alpha_{ew})$ corrections to the process $e^{+}e^{-}\to
Z^{0}Z^{0}Z^{0}$ into two parts, the QED correction part and the weak
correction part. Correspondingly we define the total relative correction as
$\delta_{tot}=\delta_{QED}+\delta_{weak}$. The QED correction part is
contributed by the diagrams with virtual photon in loop, and real photon
emission process $e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}\gamma$ . For the counterterms
involved in the QED contribution, the electron/positron wave function
renormalization constants include only photonic contribution. The remainders
of the total virtual electroweak corrections belong to weak correction part.
With above definitions we can express the full one-loop electroweak corrected
total cross section as a summation of several parts.

$\displaystyle\sigma_{tot}$ $\displaystyle=$
$\displaystyle\sigma_{LO}+\Delta\sigma_{v,QED}+\Delta\sigma_{s}+\Delta\sigma_{h}+\Delta\sigma_{v,weak}$
(3.5) $\displaystyle=$
$\displaystyle\sigma_{LO}\left(1+\delta_{QED}+\delta_{weak}\right)=\sigma_{LO}\left(1+\delta_{tot}\right),$

where $\Delta\sigma_{v}$ and $\Delta\sigma_{s}$ are the cross section
corrections contributed by the virtual electroweak one-loop diagrams and the
soft photon emission process respectively, $\Delta\sigma_{v,QED}$,
$\Delta\sigma_{s}$, $\Delta\sigma_{h}$ and $\Delta\sigma_{v,weak}$ are the
corrections from the virtual QED contribution, the soft photon emission
process, the hard photon emission process and the virtual weak contribution,
separately. $\delta_{QED}$, $\delta_{weak}$ and $\delta_{tot}$ are the
relative corrections contributed by the QED correction part, the weak
correction part and the total electroweak correction, respectively.

As we mentioned above, there exist both ultraviolet(UV) divergency and
infrared(IR) soft singularity in the contributions of the electroweak one-loop
diagrams for $e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}$ process, but no collinear IR
singularity since we keep the nonzero mass of electron/positron in our
calculation of one-loop order corrections. After doing the renormalization
procedure, we verified that the UV singularity is vanished, and the IR soft
divergency appeared in the virtual correction is cancelled by the the soft
photon emission process $e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}\gamma$ .

In order to discuss the origin of the large correction when the colliding
energy is close to the threshold of the production of three $Z^{0}$-bosons, we
discuss the photonic (QED) corrections and the genuine total electroweak
corrections separately. The QED corrections comprise two parts: the QED
virtual corrections $\Delta\sigma_{v,QED}$ which contributed by the loop
diagrams with virtual photon exchange in loop and the corresponding QED parts
of the counterterms, and the real photon emission corrections
$\Delta\sigma_{real}$. Therefore, the QED relative correction $\delta_{QED}$
can be expressed as

$\displaystyle\delta_{QED}=\delta_{v,QED}+\delta_{real},$ (3.6)

where $\delta_{v,QED}=\Delta\sigma_{v,QED}/\sigma_{LO}$, and the genuine weak
relative correction $\delta_{w}$ can be got from

$\displaystyle\delta_{w}=\delta_{tot}-\delta_{QED}.$ (3.7)

##  IV. Numerical results and discussion

For the numerical calculation, we take the input parameters as follows[27]:

$\displaystyle m_{e}$ $\displaystyle=$ $\displaystyle 0.51099892~{}{\rm
MeV},~{}m_{\mu}~{}=~{}105.658369~{}{\rm MeV},~{}m_{\tau}~{}=~{}1776.99~{}{\rm
MeV},$ $\displaystyle m_{u}$ $\displaystyle=$ $\displaystyle 66~{}{\rm
MeV},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}m_{c}~{}=~{}1.25~{}{\rm
GeV},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}m_{t}~{}=~{}174.2~{}{\rm GeV},$
$\displaystyle m_{d}$ $\displaystyle=$ $\displaystyle 66~{}{\rm
MeV},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}m_{s}~{}=~{}95~{}{\rm
MeV},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}m_{b}~{}=~{}4.7~{}{\rm GeV},$
$\displaystyle m_{W}$ $\displaystyle=$ $\displaystyle 80.403~{}{\rm
GeV},~{}~{}~{}~{}~{}~{}~{}~{}~{}m_{Z}~{}=~{}91.1876~{}{\rm GeV}.$ (4.1)

There we use the experimental value of W-boson mass as input parameter, but
not the $m_{W}$ evaluated from $G_{\mu}$ as in $\alpha_{ew}$ scheme[28]. We
take the electric charge defined in the Thomson limit
$\alpha_{ew}(0)=1/137.036$ and the effective values of the light quark masses
($m_{u}$ and $m_{d}$) which can reproduce the hadronic contribution to the
shift in the fine structure constant $\alpha_{ew}(m_{Z}^{2})$ [29]. As we know
that the LEP II experiments provide the lower limit on the SM Higgs mass as
$114.4~{}GeV$ at the $95\%$ confidence level from the results of direct
searches for $e^{+}e^{-}\to Z^{0}H^{0}$ production[30, 31], and the
electroweak precision measurements indicate indirectly the upper bound as
$m_{H}\lesssim 182~{}GeV$ at the $95\%$ C.L., when the lower limit on $m_{H}$
is used in determination of this upper limit[31]. Therefore, in our numerical
evaluation it is reasonable to take the mass of Higgs-boson being in the range
of $115~{}GeV<m_{H}<170~{}GeV$. Then we shall not encounter the resonance
problem of Higgs-boson during our calculation.

We checked the correctness of the numerical results of the LO cross section
for process $e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}$ , by using Grace2.2.1[17] and
FeynArts3.3/FormCalc5.3 [12, 13] packages separately. In adopting Grace2.2.1
and FeynArts3.3/FormCalc5.3 programs, we used both ’t Hooft-Feynman and
unitary gauges separately in the calculations of the LO cross section to check
the gauge invariance, and we got coincident numerical results. The numerical
results of the LO cross section for the process $e^{+}e^{-}\to
Z^{0}Z^{0}Z^{0}$ , by using ’t Hooft-Feynman gauge and taking
$\sqrt{s}=500~{}GeV$ and the other input parameters shown in Eqs.(IV.), are
listed in Table 1. There it is shown that there is a good agreement between
the numerical results by adopting different packages.

$m_{H}(GeV)$ |  $\sigma_{LO}(fb)$(Grace) |  $\sigma_{LO}(fb)$ (FeynArts)
---|---|---
115 | 1.0056(4) | 1.0055(2)
120 | 1.0139(4) | 1.0138(2)
150 | 1.0975(4) | 1.0975(2)
170 | 1.2565(4) | 1.2564(2)

Table 1:  The numerical results of the LO cross sections for the process
$e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}$ by using Grace2.2.1 and
FeynArts3.3/FormCalc5.3 packages separately, and taking the input parameters
as shown in Eqs.(IV.) and $\sqrt{s}=500~{}GeV$.

In the one-loop calculation, we must set the values of the IR regulator
$m_{\gamma}$, the fictitious photon mass, and soft cutoff $\delta_{s}=\Delta
E/E_{b}$ besides the parameters mentioned in Eqs.(IV.). As we know, the total
cross section should have no relation with these two parameters if the IR
divergency does really vanish. Our numerical results show that the cross
section correction at ${\cal O}(\alpha^{4}_{ew})$ order
$\Delta\sigma_{tot}=\Delta\sigma_{real}+\Delta\sigma_{v}$ is invariable within
the calculation errors when $m_{H}=120~{}GeV$, $\sqrt{s}=500~{}GeV$,
$\delta_{s}=10^{-3}$ and the fictitious photon mass $m_{\gamma}$ varies from
$10^{-15}~{}GeV$ to $10^{-1}~{}GeV$.

Fig.3(a) presents a verification of the correctness of our calculation for
process $e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}$ including ${\cal O}(\alpha_{ew})$
order corrections. The amplified curve for $\Delta\sigma_{tot}$ including
calculation errors is depicted in Fig.3(b). Both figures are to show the
independence of the total ${\cal O}(\alpha_{{\rm ew}})$ electroweak correction
on the soft cutoff $\delta_{s}$, when we take $m_{H}=120~{}{{\rm GeV}}$ and
$\sqrt{s}=500~{}{\rm GeV}$. From Fig.3(b) we can say that the total EW
relative correction $\Delta\sigma_{tot}$ has no relation to the value of
$\delta_{s}$ within the calculation error range. In the further calculations,
we set $m_{\gamma}=10^{-2}~{}GeV$ and $\delta_{s}=10^{-3}$.

Figure 3:  (a) The dependence of the ${\cal O}(\alpha_{{\rm ew}})$ correction
to cross section of $e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}$ on the soft cutoff
$\delta_{s}$ with $m_{H}=120~{}GeV$ and $\sqrt{s}=500~{}GeV$. (b) the
amplified plot of the curve for the total correction $\Delta\sigma_{tot}$ in
Fig.3(a), where it includes calculation errors.

In Table 2, we list some representative numerical results of the LO and one-
loop EW corrected cross sections($\sigma_{LO}$, $\sigma_{tot}$), the QED and
total EW corrections to the cross sections for $e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}$
process($\Delta\sigma_{QED}$, $\Delta\sigma_{tot}$), and their corresponding
relative corrections($\delta_{QED}$, $\delta_{tot}$) when $\sqrt{s}=500~{}GeV$
and the values of Higgs-boson mass are taken to be $115~{}GeV$, $150~{}GeV$
and $170~{}GeV$ separately. From these data we can see that the one-loop EW
corrections suppress the LO cross section of the process $e^{+}e^{-}\to
Z^{0}Z^{0}Z^{0}$ , and the relative corrections are about minus few percent.
We can conclude also the LO and EW corrected cross sections increase with the
increment of $m_{H}$, while the absolute total EW correction decreases when
the value of Higgs-boson mass goes up.

$m_{H}(GeV)$ | $\sigma_{LO}(fb)$ | $\sigma_{tot}(fb)$ | $\Delta\sigma_{QED}(fb)$ | $\Delta\sigma_{tot}(fb)$ | $\delta_{QED}(\%)$ | $\delta_{tot}(\%)$
---|---|---|---|---|---|---
115 | 1.0055(2) | 0.9159(7) | -0.0451(7) | -0.0896(7) | -4.49(7) | -8.91(7)
150 | 1.0975(2) | 1.0194(8) | -0.0444(8) | -0.0780(8) | -4.04(7) | -7.11(7)
170 | 1.2564(2) | 1.1989(9) | -0.0393(8) | -0.0575(9) | -3.12(7) | -4.58(7)

Table 2:  The numerical results of $\sigma_{LO}$, $\sigma_{tot}$,
$\Delta\sigma_{QED}$, $\Delta\sigma_{tot}$(in femto bar), and their
corresponding relative EW and QED corrections($\delta_{tot}$, $\delta_{QED}$)
for the process $e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}$ , when $\sqrt{s}=500~{}GeV$
and $m_{H}=115~{}GeV,~{}150~{}GeV,~{}170~{}GeV$ respectively.

The numerical results of the LO, ${\cal O}(\alpha_{ew})$ EW, QED corrected
cross sections($\sigma_{LO}$, $\sigma_{tot}$, $\sigma_{QED}$) and the total
relative EW, QED corrections($\delta_{tot}$, $\delta_{QED}$) for the process
$e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}$ with $m_{H}=120~{}GeV,~{}150~{}GeV$ as the
functions of colliding energy $\sqrt{s}$ are plotted in Figs.4(a) and (b)
respectively. As indicated in Fig.4(a), The curves for the cross sections of
$\sigma_{LO}$, $\sigma_{tot}$ and $\sigma_{QED}$ increase quickly in the
$\sqrt{s}$ region of $[350~{}GeV,~{}550~{}GeV]$ and decrease when
$\sqrt{s}>600~{}GeV$. The two figures show the ${\cal O}(\alpha_{ew})$
corrections always suppress the corresponding LO cross sections of process
$e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}$ in both cases of $m_{H}=120~{}GeV$ and
$m_{H}=150~{}GeV$ separately, but the QED correction parts can enhance the LO
cross sections when $\sqrt{s}>700~{}GeV$. We can read out from Fig.4(b) that
the corresponding total EW relative corrections for $m_{H}=120~{}GeV$ and
$150~{}GeV$ vary in the ranges of $[-15.8\%,~{}-7.5\%]$ and
$[-13.9\%,~{}-6.2\%]$ respectively, when $\sqrt{s}$ runs from $350~{}GeV$ to
$1~{}TeV$. We can see also from these two plots that in the colliding energy
$\sqrt{s}$ region near the threshold of the production of three
$Z^{0}$-bosons, the main contribution to the total EW relative
correction($\delta_{tot}$) comes from the QED correction part. That is due to
the Coulomb singularity effect coming from the instantaneous photon exchange
in loops which has a small spatial momentum.

Figure 4:  (a) The LO($\sigma_{LO}$), ${\cal O}(\alpha_{ew})$ EW, QED
corrected cross sections($\sigma_{tot}$, $\sigma_{QED}$) for the process
$e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}$ as the functions of colliding energy
$\sqrt{s}$ with $m_{H}=120~{}GeV$, $150~{}GeV$ separately. (b) The
corresponding relative EW, QED relative corrections($\delta_{tot}$,
$\delta_{QED}$) versus $\sqrt{s}$.

We present the distributions of the transverse momenta of final $Z^{0}$-bosons
at leading order and up to one-loop order, $d\sigma_{LO}/dp_{T}^{Z}$,
$d\sigma_{NLO}/dp_{T}^{Z}$, when $m_{H}=120~{}GeV$ and $\sqrt{s}=500~{}GeV$ in
Fig.5. There we pick the $p_{T}^{Z}$ of each of the three $Z^{0}$-bosons as an
entry in this histograms, then the final result of the differential cross
section is obtained by multiplying factor $1/3$. In this figure we can see
that the EW one-loop correction suppresses obviously the LO differential cross
section $d\sigma_{LO}/dp_{T}^{Z}$ when $p_{T}^{Z}>50~{}GeV$, but the EW
correction is small when $p_{T}^{Z}<25~{}GeV$. It also shows that the EW
corrections do not observably change the LO distribution line-shape of
$p_{T}^{Z}$, and both the differential cross sections of
$d\sigma_{LO}/dp_{T}^{Z}$ and $d\sigma_{NLO}/dp_{T}^{Z}$ have their maximal
values at about $p_{T}^{Z}\sim 50~{}GeV$ respectively.

Figure 5:  The distributions of the transverse momenta of $Z^{0}$-bosons
($P_{T}^{Z}$) at the LO and up to EW one-loop order with $m_{H}=120~{}GeV$ and
$\sqrt{s}=500~{}GeV$.

We plot the distributions of the invariant mass of $Z^{0}Z^{0}$-pair, denoted
as $M_{ZZ}$, at the LO and up to EW one-loop order with $m_{H}=120~{}GeV$ and
$\sqrt{s}=500~{}GeV$ in Fig.6. Here we can see that the EW correction slightly
enhances the LO differential cross section when $M_{ZZ}<250~{}GeV$, but
suppresses $d\sigma_{LO}/dM_{ZZ}$ obviously when $M_{ZZ}>250~{}GeV$. The
suppression of $d\sigma_{LO}/dM_{ZZ}$ is due to the contribution from the hard
photon emission process $e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}\gamma$ at the ${\cal
O}(\alpha_{ew}^{4})$ order, in which the momentum balance between the sum of
the momenta of three $Z^{0}$-bosons and that of the radiated hard photon will
reduce the value of invariant mass $M_{ZZ}$ and change the line-shape in the
range with large $M_{ZZ}$.

Figure 6:  The distributions of the invariant mass of
$Z^{0}Z^{0}$-pair($M_{ZZ}$) at the LO and up to EW one-loop order when
$m_{H}=120~{}GeV$ and $\sqrt{s}=500~{}GeV$.

##  V. Summary

In this paper we describe the impact of the complete one-loop EW corrections
to the scattering process $e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}$ in the SM. This
channel can be used to measure the quartic vector boson coupling at ILC. We
investigate the dependence of the LO, ${\cal O}(\alpha_{ew})$ EW and QED
corrected cross sections on colliding energy $\sqrt{s}$ and Higgs-boson mass,
and present the LO and EW one-loop corrected distributions of the transverse
momenta of final $Z^{0}$-bosons and the LO and EW corrected differential cross
sections of invariant mass of $Z^{0}Z^{0}$-pair. To see the origin of some of
the large corrections clearly, we calculate the QED and genuine weak
corrections separately. We conclude that both the Born cross section and the
EW corrected cross section for $e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}$ process are
sensitive to the Higgs boson mass in the range of $115~{}GeV<m_{H}<170~{}GeV$.
We find the ${\cal O}(\alpha_{ew})$ corrections generally suppress the LO
cross section, the LO distribution of the momenta of $Z^{0}$-bosons and the LO
differential cross sections of invariant mass of $Z^{0}Z^{0}$-pair for process
$e^{+}e^{-}\to Z^{0}Z^{0}Z^{0}$ . Our numerical results show that when
$m_{H}=120~{}GeV(150~{}GeV)$ and the colliding energy goes up from $350~{}GeV$
to $1~{}TeV$, the relative EW correction varies from $-15.8\%(-13.9\%)$ to
$-7.5\%(-6.2\%)$.

Acknowledgments: This work was supported in part by the National Natural
Science Foundation of China, Specialized Research Fund for the Doctoral
Program of Higher Education(SRFDP) and a special fund sponsored by Chinese
Academy of Sciences.

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