aainstitutetext: LMA, Département de Physique, Faculté des Sciences Exactes et
Informatique
Université Hassiba Benbouali de Chlef - Chlef 02000, Algeriabbinstitutetext:
LEPM, Faculty of Physics, Université des Sciences et de la Technologie d’Oran
Mohamed-Boudiaf USTO-MB, BP 1505, EL M’naouer, Oran, Algeriaccinstitutetext:
LMA, Département de Physique, Faculté des Sciences et de la Technologie
Université Relizane - Relizane 48000, Algéria

#  The effects of the widths on the one-loop electroweak corrections to the
$pp\to WW$ process

N. Bekheddouma Abdi b , R. Bouamrane c and K. Khelifa-Kerfa
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, we study the effects of the widths of unstable particles on the
one-loop electroweak corrections for the $pp\to WW$ process at the TeV scale
within the framework of the complex mass scheme. We also investigate, for this
same process, the unitarity of the theory at high energies.

###### Keywords:

CMS, Electroweak, Renormalisation, Cross-section, One-loop Corrections,
Unitarity, High-energy

##  1 Introduction

After the discovery of the Higgs boson in 2012 at CERN, the particle
responsible, via the Brout-Englert-Higgs mechanism, for generating the masses
of the particles of the Standard Model (SM), we entered the era of Higgs-
precision tests of SM. Since most known fundamental particles are unstable, it
becomes important to consider the width, $\Gamma$, of these particles when
evaluating physical observables CMS-Bauer-2012 ; CMS-Denner-2015 ; CMS-
Kuksa-2015 . Since The electroweak sector being the most sensitive to unstable
particles, such as the weak gauge bosons, the Higgs boson and the top quark,
we study, within the framework of the SM, the effects of the widths of the
latter on the $pp\to WW$ process.

Knowing that the inclusion of the finite width is not trivial, since it can
invoke, following a mixture of perturbative orders, a break in the gauge
invariance during calculations of radiative corrections to one-loop
Berends:1969nt ; ARGYRES1995339 . Many approaches have been proposed in the
past to incorporate the said width into perturbative computations, such as the
fixed-width scheme ARGYRES1995339 , the narrow-width approximation Baur-1995 ,
the pole scheme Beenakker-97 ; Stuart-91  and the effective field theory
Beneke-2003 ; Beneke-2004 . But the complex mass scheme CMS-Denner-2005 ; CMS-
Denner-2006 ; DENNER2012504 ; Frederix:2018nkq  is the only one, by
construction, which preserves all of the algebraic relations which satisfy
gauge invariance in the resonant and non-resonant regions, where the main idea
is based on an analytic continuation of the parameters of the Lagrangian SM,
which are related to the masses of unstable particles, in the complex plane.
Therefore, renormalizability and unitarity are also preserved. This scheme has
been implemented in various high energy software such as MadGraph5 aMC@NLO
Alwall:2014hca ; Frederix:2018nkq , OpenLoops 2 Buccioni:2019sur  and Recola
Actis_2017 . The aim of this paper is to study the effect of the width of the
weak gauge bosons, the Higgs and the top quark on the process $pp\to WW$ at
the one-loop level within the complex mass scheme.

The paper is organised as follows. In section 2 we summarise the main ideas of
the CMS. Then, in section 3, we compute the cross section of the process
considered herein, i.e., $pp\to WW$. up to one-loop, within both the usual on-
shell scheme (which we shall be referring to as “real scheme”) and the complex
mass scheme. We analyse the results obtained in the said two schemes and
discuss the effects of the width on the unitarity. Finally, we draw our
conclusions in section 4.

##  2 Complex mass scheme in a nutshell

The complex mass scheme is a renormalisable scheme which deals properly with
unstable particles in all phase space. It was first used in tree-level
calculations with W/Z resonances, then generalised to $\mathcal{O}(\alpha)$
CMS-Denner-2005 ; CMS-Denner-2006 . In this scheme and at the tree level, the
real masses of the weak gauge bosons $W/Z$ and the Higgs boson are changed to
complex quantities, defined as the position of the poles of the corresponding
propagators which have complex momenta $k$. To preserve the gauge invariance
we have to introduce the complex masses everywhere in the Feynman rules, in
such a way that the bare Lagrangian remains invariant, particularly in the
weak mixing angle;

$\hat{\mu}_{\scriptscriptstyle{W}}^{2}=M_{\scriptscriptstyle{W}}^{2}-\imath
M_{\scriptscriptstyle{W}}\Gamma_{\scriptscriptstyle{W}},\quad\hat{\mu}_{\scriptscriptstyle{Z}}^{2}=M_{\scriptscriptstyle{Z}}^{2}-\imath
M_{\scriptscriptstyle{Z}}\Gamma_{\scriptscriptstyle{Z}},\quad\hat{\mu}_{\scriptscriptstyle{H}}^{2}=M_{\scriptscriptstyle{H}}^{2}-\imath
M_{\scriptscriptstyle{H}}\Gamma_{\scriptscriptstyle{H}},$ (1)

and

$\cos^{2}\theta_{\scriptscriptstyle{W}}\equiv\hat{c}_{\scriptscriptstyle{W}}^{2}=1-\hat{s}_{\scriptscriptstyle{W}}^{2}=\frac{\hat{\mu}_{\scriptscriptstyle{W}}^{2}}{\hat{\mu}_{\scriptscriptstyle{Z}}^{2}}\simeq\frac{M_{W}^{2}}{M_{Z}^{2}}\left[1-\imath\left(\frac{\Gamma_{W}}{M_{W}}-\frac{\Gamma_{Z}}{M_{Z}}\right)\right].$
(2)

The hat over the masses and the mixing angles denotes the fact that they are
complex-valued. Since the gauge invariance is not affected by this analytical
continuation of the mass in the complex $k^{2}$ plane:
$m\longrightarrow\hat{\mu}=m-\imath\Gamma/2\longrightarrow\hat{\mu}(k)=m(k)-\imath\Gamma(k)/2$,
thus the Ward identity and that of Slavnov-Taylor are preserved. This
signifies that the elements of the S-matrix are independent of the gauge
parameters.

Although the introduction of the complex mass in the resonant propagators is
trivial,

$\displaystyle\frac{1}{k^{2}-M_{B}^{2}}$
$\displaystyle\longrightarrow\frac{1}{k^{2}-\hat{\mu}_{B}^{2}}\simeq\frac{1}{k^{2}-M_{B}^{2}\left(1-\imath\frac{\Gamma_{B}}{M_{B}}\right)},$
(3) $\displaystyle\frac{1}{\mbox{$\not{\\!p}$}-m_{F}}$
$\displaystyle\longrightarrow\frac{1}{\mbox{$\not{\\!p}$}-\hat{\mu}_{F}}=\frac{1}{\mbox{$\not{\\!p}$}-m_{F}\left(1-\imath\frac{\Gamma_{F}}{2m_{F}}\right)},$
(4)

where the subscripts $B$ and $F$ stand for bosons and fermions, respectively,
it induces in other regions (such as the weak mixing angle) when passing to
the complex mass scheme spurious terms of order $\mathcal{O}(\alpha)$ in the
tree-level amplitude, thus only affecting loop-level calculations.

To generalise the CMS to the one-loop level while keeping the bare Lagrangian
invariant we split the real masses in the latter for the unstable particles
into renormalised complex masses and complex counter-terms. The resultant
Feynman rules enable to perform perturbative calculations exactly as in the
usual renormalisation on-shell scheme. To this end, we add and subtract the
same imaginary part of each mass of the unstable particle. One of these
imaginary parts is incorporated into the free propagator to define the complex
mass of the corresponding unstable particle. The other part is introduced into
the vertex counter-term. The first term is thus resummed but the second one is
not. This prescription does not affect the gauge invariance but may invoke a
violation of unitarity of order $\mathcal{O}(\alpha^{2})$ in calculations at
order $\mathcal{O}(\alpha)$. This is due to the fact that the modified
renormalised Lagrangian is not hermitian Belanger-2006 . Apart from this
problem, the complex mass scheme is coherent and gauge invariant in the next-
to-leading order (NLO) calculations. Its implementation in a numerical code at
one-loop is feasible, since it suffices to redefine the counter-terms by
including imaginary parts for two-point functions. The complex masses are not
only introduced for the gauge bosons but for all unstable particles relevant
to the electroweak sector such as the top quark.

###  2.1 Complex renormalisation

In this section we summarise the procedure of the generalised renormalisation
which takes into account the complex masses in the ’t Hooft-Feynman gauge in a
straight-forward way CMS-Denner-2005 .

Since the bare Lagrangian is unaffected the complex masses of the gauge
bosons, $W$ and $Z$, are introduced in the latter after decomposing the bare
real masses into renormalised complex masses and complex counter-terms:

$M_{W,0}^{2}=\hat{\mu}_{W}^{2}+\delta\hat{\mu}_{W}^{2},\quad\quad\quad
M_{Z,0}^{2}=\hat{\mu}_{Z}^{2}+\delta\hat{\mu}_{Z}^{2},$ (5)

where we note that the following consistency condition should be respected:

$\operatorname{Im}(\hat{\mu}_{V}^{2})=-\operatorname{Im}(\delta\hat{\mu}_{V}^{2}).$
(6)

In the above equation the subscript $0$ labels the bare quantities, and $V$
stands for $W$ or $Z$ bosons. In a similar fashion, we observe that the
renormalised gauge fields are related to the bare ones via the following
relations:

$\displaystyle
W_{0}^{\pm}=\left(1+\frac{1}{2}\delta\mathcal{Z}_{W}\right)W^{\pm},$ (7)
$\displaystyle\left(\begin{array}[]{c}Z_{0}\\\
A_{0}\end{array}\right)=\left(\begin{array}[]{cc}1+\frac{1}{2}\delta\mathcal{Z}_{ZZ}&\frac{1}{2}\delta\mathcal{Z}_{ZA}\\\
\frac{1}{2}\delta\mathcal{Z}_{AZ}&1+\frac{1}{2}\delta\mathcal{Z}_{AA}\end{array}\right)\left(\begin{array}[]{c}Z\\\
A\end{array}\right).$ (14)

Since the renormalisation conditions are the same for stable and unstable
particles, that is, the position of the poles of the propagator equals the
square of the physical mass and the residue of the propagator equals 1, they
assume similar forms in CMS as those in the usual scheme but without taking
the real part of the renormalised transverse self energy (T):

$\displaystyle\bar{\Sigma}_{T}^{W}(\hat{\mu}_{W}^{2})=0,$
$\displaystyle\bar{\Sigma}_{T}^{ZZ}(\hat{\mu}_{Z}^{2})=0,$
$\displaystyle\bar{\Sigma}_{T}^{AZ}(0)=0,$
$\displaystyle\bar{\Sigma}_{T}^{AZ}(\hat{\mu}_{Z}^{2})=0,$
$\displaystyle\bar{\Sigma}_{T}^{\prime W}(\hat{\mu}_{W}^{2})=0,$
$\displaystyle\bar{\Sigma}_{T}^{\prime
ZZ}(\hat{\mu}_{Z}^{2})=0,\quad\quad\bar{\Sigma}_{T}^{\prime AA}(0)=0,$ (15)

where the prime means differentiation with respect to the argument and the bar
on the $\Sigma$’s indicates that they are renormalised. The first two terms in
Eq. (2.1) fix the counter-terms of the masses of the $W$ and $Z$ bosons, while
the last five terms fix the counter-terms of their fields. Knowing that the
generalised renormalised transverse self energies are the same as the usual
on-shell scheme with the replacement of the real masses and counter-terms by
their complex counterparts. The solutions of the said conditions (2.1) are as
follows:

$\displaystyle\delta\hat{\mu}_{W}^{2}=\Sigma_{T}^{W}(\hat{\mu}_{W}^{2}),\quad\quad\delta\mathcal{Z}_{W}=-\Sigma_{T}^{\prime
W}(\hat{\mu}_{W}^{2}),$
$\displaystyle\delta\mathcal{Z}_{ZA}=\frac{2}{\hat{\mu}_{Z}^{2}}\Sigma_{T}^{AZ}(0),\quad\delta\mathcal{Z}_{AZ}=-\frac{2}{\hat{\mu}_{Z}^{2}}\Sigma_{T}^{AZ}(\hat{\mu}_{Z}^{2}),\qquad$
$\displaystyle\delta\mathcal{Z}_{W}=-\Sigma_{T}^{{}^{\prime}W}(\hat{\mu}_{W}^{2}),\quad\delta\mathcal{Z}_{ZZ}=-\Sigma_{T}^{{}^{\prime}ZZ}(\hat{\mu}_{Z}^{2}),\quad\delta\mathcal{Z}_{AA}=-\Sigma_{T}^{{}^{\prime}AA}(0).$
(16)

It requires analytical continuation to compute the above two-point functions
with complex arguments. To avoid this complication we expand the self energies
around real arguments. To see how to transform the renormalised self energies
and the solutions of the renormalisation conditions, we concentrate on the
case of the W gauge boson. We have

$\bar{\Sigma}_{T}^{W}(k^{2})=\Sigma_{T}^{W}(k^{2})-\delta\hat{\mu}_{W}^{2}+(k^{2}-\hat{\mu}_{W}^{2})\delta\mathcal{Z}_{W},$
(17)

with

$\displaystyle\Sigma_{T}^{W}(\hat{\mu}_{W}^{2})=\Sigma_{T}^{W}(M_{W}^{2})+\left(\hat{\mu}_{W}^{2}-M_{W}^{2}\right)\Sigma^{{}^{\prime}W}_{T}(M_{W}^{2})+\mathcal{O}(\alpha^{3}).$
(18)

Neglecting the terms at $\mathcal{O}(\alpha^{3})$ and beyond we obtain the
modified solutions, which when inserted into (17) results in a form of the
renormalised transverse self energy that resembles that of the usual on-shell
scheme but without taking the real part of the solutions:

$\displaystyle\bar{\Sigma}_{T}^{W}(k^{2})=\Sigma_{T}^{W}(k^{2})-\delta
M_{W}^{2}+(k^{2}-M_{W}^{2})\delta\mathcal{Z}_{W}$ $\displaystyle\delta
M_{W}^{2}=\Sigma_{T}^{W}(M_{W}^{2}),\quad\quad\delta\mathcal{Z}_{W}=-\Sigma_{T}^{\prime
W}(M_{W}^{2}).$ (19)

While in the on-shell scheme the self-energies are calculated with real
renormalised masses, in the CMS self-energies, Eq. (2.1), ought to be
calculated with complex masses, but with real squared momenta. This enables us
to avoid analytic continuation in the momentum space.

In order to correctly address resonances at order $\mathcal{O}(\alpha)$ one
ought to take into account the $W$ boson width, $\Gamma_{W}$, including
$\mathcal{O}(\alpha)$ corrections. This may be obtained in an iterative way
from the following equation:

$M_{W}\Gamma_{W}=\operatorname{Im}\left\\{\Sigma_{T}^{W}(M_{W}^{2})\right\\}-M_{W}\Gamma_{W}\operatorname{Re}\left\\{\Sigma_{T}^{\prime
W}(M_{W}^{2})\right\\}+\mathcal{O}(\alpha^{3}).$ (20)

This latter equation can be easily deduced from the imaginary part of (18).
Furthermore, the complex weak mixing angle is renormalised as follows:

$\frac{\delta\hat{c}_{\scriptscriptstyle{W}}}{\hat{c}_{\scriptscriptstyle{W}}}=\frac{1}{2}\left(\frac{\delta\hat{\mu}_{W}^{2}}{\hat{\mu}_{W}^{2}}-\frac{\delta\hat{\mu}_{Z}^{2}}{\hat{\mu}_{Z}^{2}}\right)=\frac{1}{2}\left[\frac{\Sigma_{T}^{W}(\hat{\mu}_{W}^{2})}{\hat{\mu}_{W}^{2}}-\frac{\Sigma_{T}^{Z}(\hat{\mu}_{Z}^{2})}{\hat{\mu}_{Z}^{2}}\right].$
(21)

For the Higgs boson, the renormalisation constants can be approached in the
same manner as before;

$M_{H,0}^{2}=\hat{\mu}_{H}^{2}+\delta\hat{\mu}_{H}^{2},\qquad\qquad\qquad\qquad\qquad\qquad\quad$
(22)

with

$\displaystyle\delta\hat{\mu}_{H}^{2}$
$\displaystyle=\Sigma^{H}(\hat{\mu}_{H}^{2}),$
$\displaystyle=\Sigma^{H}(M_{H}^{2})+(\hat{\mu}_{H}^{2}-M_{H}^{2})\Sigma^{\prime
H}(M_{H}^{2})+\mathcal{O}(\alpha^{3}).$ $\displaystyle\delta\mathcal{Z}_{H}$
$\displaystyle=-\Sigma^{{}^{\prime}H}(\hat{\mu}_{H}^{2}),$
$\displaystyle=-\Sigma^{\prime H}(M_{H}^{2})+\mathcal{O}(\alpha^{2}).$ (23)

Hence the renormalised self energy for the Higgs boson up to the
$\mathcal{O}(\alpha^{2})$ can be written as:

$\bar{\Sigma}^{H}(k^{2})=\Sigma^{H}(k^{2})-\delta
M_{H}^{2}+(k^{2}-M_{H}^{2})\,\delta\mathcal{Z}_{H}\qquad\qquad$ (24)

where

$\delta
M_{H}^{2}=\Sigma^{H}(M_{H}^{2}),\qquad\delta\mathcal{Z}_{H}=-\Sigma^{\prime
H}(M_{H}^{2}),\qquad\quad$ (25)

Since in the CMS, the complex masses are not introduced solely for gauge
bosons and the Higgs boson but for all unstable particles such as the top
quark. The renormalisation of this latter may be treated in a similar manner
as before with the introduction of its complex mass and counter-term:

$\displaystyle\hat{\mu}_{t}^{2}$ $\displaystyle=m_{t}^{2}-\imath
m_{t}\,\Gamma_{t},\quad\quad$ $\displaystyle m_{t,0}$
$\displaystyle=\hat{\mu}_{t}+\delta\hat{\mu}_{t}.$ (26)

The generalised renormalisation constants are determined by:

$\displaystyle\delta\hat{\mu}_{t}$
$\displaystyle=\frac{\hat{\mu}_{t}}{2}\left[\Sigma^{t,R}(\hat{\mu}_{t}^{2})+\Sigma^{t,L}(\hat{\mu}_{t}^{2})+2\Sigma^{t,s}(\hat{\mu}_{t}^{2})\right],$
$\displaystyle\delta\mathcal{Z}_{t,\sigma}$
$\displaystyle=-\Sigma^{t,\sigma}(\hat{\mu}_{t}^{2})-\hat{\mu}_{t}^{2}\left[\Sigma^{{}^{\prime}t,R}(\hat{\mu}_{t}^{2})+\Sigma^{{}^{\prime}t,L}(\hat{\mu}_{t}^{2})+2\Sigma^{{}^{\prime}t,s}(\hat{\mu}_{t}^{2})\right],$
(27)

where $\sigma=R,L$ indicates the left- and right-handed components of the top
self-energy, $\Sigma^{t}(p)$, following the convention of Ref.
denner1993techniques . The generalised renormalised self-energy of the top
quark is determined by:

$\bar{\Sigma}^{t}(p)=\left(\Sigma^{t,R}(p^{2})+\delta\mathcal{Z}_{t,R}\right)\not{p}\,P_{R}+\left(\Sigma^{t,L}(p^{2})+\delta\mathcal{Z}_{t,L}\right)\not{p}\,P_{L}+\\\
+\hat{\mu}_{t}\left[\Sigma^{t,s}-\frac{1}{2}\left(\delta\mathcal{Z}_{t,R}+\delta\mathcal{Z}_{t,L}\right)-\frac{\delta\hat{\mu}_{t}}{\hat{\mu}_{t}}\right],$
(28)

where the factors $P_{R,L}$ are defined below. This becomes, after the
expansion of the self-energies of (2.1) around the real mass $m_{t}^{2}$ and
negligence of higher order terms:

$\bar{\Sigma}^{t}(p)=\left(\Sigma^{t,R}(p^{2})+\delta\mathcal{Z}_{t,R}\right)\not{p}\,P_{R}+\left(\Sigma^{t,L}(p^{2})+\delta\mathcal{Z}_{t,L}\right)\not{p}\,P_{L}+\\\
+\hat{\mu}_{t}\left[\Sigma^{t,s}-\frac{1}{2}\left(\delta\mathcal{Z}_{t,R}+\delta\mathcal{Z}_{t,L}\right)-\frac{\delta
m_{t}}{m_{t}}\right],$ (29)

with

$\displaystyle\delta m_{t}$
$\displaystyle=\frac{m_{t}}{2}\left[\Sigma^{t,R}(m_{t}^{2})+\Sigma^{t,L}(m_{t}^{2})+2\Sigma^{t,s}(m_{t}^{2})\right],$
$\displaystyle\delta\mathcal{Z}_{t,\sigma}$
$\displaystyle=-\Sigma^{t,\sigma}(m_{t}^{2})-m_{t}^{2}\left[\Sigma^{{}^{\prime}t,R}(m_{t}^{2})+\Sigma^{{}^{\prime}t,L}(m_{t}^{2})+2\Sigma^{{}^{\prime}t,s}(m_{t}^{2})\right].$
(30)

Before ending this section it is important to recall that the masses of the
external particles for a given process must be real, i.e., they are considered
as stable particles as had been shown by Veltman VELTMAN1963186 . Moreover,
the same particles should not be taken in the same process as internally
unstable (resonances) and externally stable, because they cannot be treated
simultaneously by two different schemes (usual on-shell and CMS)
Denner:2019vbn .

##  3 Width effects on $pp\to WW$ at one loop

###  3.1 Cross-section at lowest order

We shall be investigating the process

$\displaystyle P(p_{1})+P(p_{2})\to
W^{+}(k_{1},\lambda_{1})+W^{-}(k_{2},\lambda_{2}),$ (31)

where $P_{i},k_{i}$ are the momenta of the protons and the W bosons
respectively, and $\lambda_{i}$ are the polarisations ($\lambda_{i}=0$ for
longitudinal polarisations, referred to as $L$, $\lambda_{i}=\pm 1$ for
transverse polarisations, referred to as $T$, and non-polarised referred to by
$U$). The tree-level Born Feynman diagrams corresponding to our process are
shown in Fig. 1.

Figure 1: Feynman diagrams for the Born process $q\bar{q}\to W^{+}W^{-}$.

The four-momenta of the protons and bosons are given by:

$\displaystyle p_{1}=\frac{\sqrt{s}}{2}(1,0,0,\beta_{q}),$
$\displaystyle\qquad
k_{1}=\frac{\sqrt{s}}{2}(1,\beta\sin\theta,0,\beta\cos\theta),$ $\displaystyle
p_{2}=\frac{\sqrt{s}}{2}(1,0,0,-\beta_{q}),$ $\displaystyle\qquad
k_{2}=\frac{\sqrt{s}}{2}(1,-\beta\sin\theta,0,-\beta\cos\theta),$ (32)

where $\beta=\sqrt{1-4M_{W}^{2}/s}$, $\beta_{q}=\sqrt{1-4m_{q}^{2}/s}$,
$M_{W}$ is the $W$ boson’s mass, $m_{q}$ is the quark’s mass and $\theta$ is
the scattering angle in the centre-of-mass of the system, with
$p_{1}^{2}=p_{2}^{2}=m_{q}^{2}$ and $k_{1}^{2}=k_{2}^{2}=M_{W}^{2}$. The
longitudinal and transverse polarisation vectors of the final bosons read:

$\displaystyle\epsilon^{1}_{L}(0)=\frac{\sqrt{s}}{2M_{W}}(\beta,\sin\theta,0,\cos\theta),$
$\displaystyle\qquad\epsilon^{1}_{T}(\pm)=(0,\cos\theta,\mp\imath,-\sin\theta)/\sqrt{2}.$
$\displaystyle\epsilon^{2}_{L}(0)=\frac{\sqrt{s}}{2M_{W}}(\beta,-\sin\theta,0,-\cos\theta).$
$\displaystyle\qquad\epsilon^{2}_{T}(\pm)=(0,-\cos\theta,\mp\imath,\sin\theta)/\sqrt{2}.$
(33)

It is worth mentioning that only the longitudinal polarisation is massive. In
CMS, the widths of the unstable particles are introduced, at tree-level,
through propagators of $Z,W,H,t$ and the weak mixing angle. We shall only be
considering, in this section, the mode $pp\to W^{+}_{L}W^{-}_{L}$ to
analytically study the effect of the widths on the tree-level amplitude in the
CMS framework, since this mode is affected by the width $\Gamma_{W}$ unlike
the transverse mode. We then compute, using MadGraph5 aMC@NLO, the Born cross-
sections for different polarisations of the $W$ gauge boson.

To see that the unitarity is preserved in the case where the masses of the
unstable particles are taken real at high energies, that is in the usual on-
shell (OS) scheme, we introduce in the calculations of the amplitudes of the
Feynman diagrams (Fig. 1) the variable $x=s/4M_{W}^{2}$ with $x\gg 1$. We
find, with the aid of FeynCalc Romao-2016, the following expression for the
total amplitude at high energy:

$\displaystyle\mathcal{M}^{\scriptscriptstyle{\text{OS}}}_{\text{tot}}$
$\displaystyle=\mathcal{M}_{\gamma}^{s}+\mathcal{M}_{Z}^{s}+\mathcal{M}_{H}^{s}+\mathcal{M}_{q}^{t}$
$\displaystyle=\frac{e^{2}(1+2T_{3}^{f})M_{Z}^{2}}{4\left(M_{Z}^{2}-M_{W}^{2}\right)M_{W}^{2}}\,\bar{v}(p_{2})\left[m_{q}(P_{R}-P_{L})-2\not{q}_{1}P_{L}\right]u(p_{1})+\mathcal{O}(1/x),$
(34)

where $M_{W}=s_{W}\,M_{Z}$, $P_{R,L}=(1\pm\gamma_{5})/2$ and
$T_{3}^{f}=+1/2(-1/2)$ for $u,c,t(d,s,b)$ quarks.

To study the effect of the decay width of the unstable massive particles
($W,Z,H$ and $t$) of figure 1 on the unitarity of the amplitudes at tree-level
and analyse the obtained results at high energies with respect to real mass
scheme (OS), we implement the CMS in the process of longitudinal $W$ boson
pair production $pp\to W_{L}W_{L}$ (since its polarisation vectors depend on
the mass). The expressions of the various amplitudes of the Feynman diagrams 1
following the prescriptions of Denner Denner:2019vbn  are reported in appendix
A (Eqs. (A)–(A)). The resultant total amplitude is as follows:

$\displaystyle\mathcal{M}_{\text{tot}}^{\scriptscriptstyle{\text{CMS}}}$
$\displaystyle=\operatorname{Re}\\{\mathcal{M}_{\text{tot}}^{\scriptscriptstyle{\text{CMS}}}\\}+\imath\operatorname{Im}\\{\mathcal{M}_{\text{tot}}^{\scriptscriptstyle{\text{CMS}}}\\},$
(35a) where
$\displaystyle\operatorname{Re}\\{\mathcal{M}_{\text{tot}}^{\scriptscriptstyle{\text{CMS}}}\\}$
$\displaystyle=\frac{e^{2}(1+2T_{3}^{f})M_{Z}^{2}}{4(M_{Z}^{2}-M_{W}^{2})M_{W}^{2}}\,\bar{v}(p_{2})\Big{[}m_{q}(P_{R}-P_{L})-2\not{q}_{1}P_{L}\Big{]}u(p_{1})+\mathcal{O}(1/x),$
$\displaystyle\operatorname{Im}\\{\mathcal{M}_{\text{tot}}^{\scriptscriptstyle{\text{CMS}}}\\}$
$\displaystyle=\frac{e^{2}(1+2T_{3}^{f})\,\Gamma_{Z}M_{Z}}{4(M_{Z}^{2}-M_{W}^{2})^{2}}\,\bar{v}(p_{2})\left[m_{q}(P_{R}-P_{L})-2\not{q}_{1}P_{L}\right]u(p_{1})+\mathcal{O}(1/x).$
(35b)

The following points are to be noticed:

  * •

If we set all widths of all internal particles introduced in our process,
$\Gamma_{W}=\Gamma_{Z}=\Gamma_{H}=\Gamma_{t}=0$, then the real parts of the
amplitudes (A)–(A) reduce to those of the real scheme (OS), while the
imaginary parts vanish.

  * •

The real part of the total amplitude
$\mathcal{M}_{\text{tot}}^{\scriptscriptstyle{\text{CMS}}}$ is not affected by
the widths of the internal particles, $Z,H$ and $t$, which cancel out in the
final expression. The said real part equals the total amplitude in the real
scheme;
$\operatorname{Re}\\{\mathcal{M}_{\text{tot}}^{\scriptscriptstyle{\text{CMS}}}\\}=\mathcal{M}_{\text{tot}}^{\scriptscriptstyle{\text{OS}}}$,
whilst, the imaginary part of
$\mathcal{M}_{\text{tot}}^{\scriptscriptstyle{\text{CMS}}}$ is affected by the
width $\Gamma_{Z}$ of the internal boson $Z$. It is proportional to
$\mathcal{M}_{\text{tot}}^{\scriptscriptstyle{\text{OS}}}$:

$\displaystyle\operatorname{Im}\\{\mathcal{M}_{\text{tot}}^{\scriptscriptstyle{\text{CMS}}}\\}=\frac{M_{W}^{2}\,\Gamma_{Z}\,M_{Z}}{M_{Z}^{2}-M_{W}^{2}}\,\mathcal{M}_{\text{tot}}^{\scriptscriptstyle{\text{OS}}}.$
(36)

  * •

The total amplitude depends on the mass and width of the internal boson $Z$
and the masses of the external particles $M_{W}$ and $m_{q}$. The other
masses, $M_{H}$ and $m_{t}$, and their respective widths, $\Gamma_{H}$ and
$\Gamma_{t}$, cancel out.

  * •

The effect of the width $\Gamma_{Z}$ of the internal gauge boson $Z$ on the
amplitude at tree-level at high energy is around $2\%$. In effect, the ratio
of the amplitudes is defined by

$\displaystyle\frac{\delta\mathcal{M}_{\text{tot}}}{\mathcal{M}_{\text{tot}}^{\scriptscriptstyle{\text{OS}}}}=\frac{\mathcal{M}_{\text{tot}}^{\scriptscriptstyle{\text{OS}}}-\mathcal{M}_{\text{tot}}^{\scriptscriptstyle{\text{CMS}}}}{\mathcal{M}_{\text{tot}}^{\scriptscriptstyle{\text{OS}}}}=-\imath\,\frac{M_{W}^{2}}{M_{Z}^{2}-M_{W}^{2}}\,\frac{\Gamma_{Z}}{M_{Z}}.$
(37)

  * •

Following the instructions of Denner Denner:2019vbn  we have taken, in all
amplitudes of Feynman diagrams, the width of the external gauge boson
$\Gamma_{W}=0$. We may, however, see the effect of $\Gamma_{W}$ on the tree-
level amplitude at high energy by introducing it at the level of longitudinal
polarisation vectors and the weak mixing angle. We find, as before, that only
the imaginary part of
$\mathcal{M}_{\text{tot}}^{\scriptscriptstyle{\text{CMS}}}$ is affected by the
widths $\Gamma_{Z}$ and $\Gamma_{W}$ (Eqs. (A) – (A)). It is not affected,
however, by the widths of the Higgs boson and the top quark.

$\displaystyle\operatorname{Re}\\{\mathcal{M}_{\text{tot}}^{\scriptscriptstyle{\text{CMS}}}\\}$
$\displaystyle=\mathcal{M}_{\text{tot}}^{\scriptscriptstyle{\text{OS}}},$
$\displaystyle\operatorname{Im}\\{\mathcal{M}_{\text{tot}}^{\scriptscriptstyle{\text{CMS}}}\\}$
$\displaystyle=\frac{e^{2}(1+2T_{3}^{f})\,M_{Z}}{2(M_{Z}^{2}-M_{W}^{2})^{2}}\,\,\bar{v}(p_{2})\,\Biggl{\\{}\,\frac{\Gamma_{Z}}{M_{Z}}\,\Biggl{[}m_{q}\,\left(1-2P_{L}\right)-2\not{q_{1}}P_{L}\Biggr{]}+$
$\displaystyle+\frac{\Gamma_{W}}{M_{W}}\,\frac{1}{M_{Z}}\Bigg{[}m_{q}\left(\frac{T_{3}^{f}}{1+2T_{3}^{f}}-P_{L}\right)-\not{q_{1}}\Bigg{(}2\frac{Q_{f}}{M_{Z}}\left(M_{Z}^{2}-M_{W}^{2}\right)+$
$\displaystyle+P_{L}\Bigg{)}\Bigg{]}\Biggr{\\}}\times
u(p_{1})+\mathcal{O}(1/x).$ (38)

The above equations (• ‣ 3.1) reduce to those in Eq. (35) if $\Gamma_{W}=0$.

  * •

The total tree-level amplitude at high energy is finite in the case where the
external bosons are considered either as stable or unstable, thus respecting
the unitarity condition.

###  3.2 Numerical results

To perform the numerical calculations of the cross sections at tree-level both
in CMS and OS schemes, we use the fixed-order Monte Carlo program
MadGraph5_aMC@NLO. The following input parameters have been implemented:

$\displaystyle M_{H}$ $\displaystyle=125\,\text{GeV},\qquad
M_{Z}=91.188\,\text{GeV},\qquad M_{W}=80.401\,\text{GeV},$
$\displaystyle\Gamma_{H}$
$\displaystyle=0.008\,\text{GeV},\quad\,\,\,\Gamma_{Z}=2.4952\,\text{GeV},\qquad\,\,\Gamma_{W}=2.092698\,\text{GeV},$
$\displaystyle\alpha^{-1}$
$\displaystyle=137.0359895,\,\,\,\,m_{t}=173.2\text{GeV},\qquad\quad\,\,\,\Gamma_{t}=1.3\text{GeV}.$
(39)

In Figure 2, we present, for different polarisations of the external gauge
bosons, the effect of the widths $\Gamma_{Z},\Gamma_{W},\Gamma_{H}$ and
$\Gamma_{t}$ on the Born cross sections in the CMS framework in comparison
with OS.

Figure 2: (Left) plots for real (OS) and CMS cross sections as a function of
the scattering energy $\sqrt{s}$ for various polarisations of the $W^{+}W^{-}$
bosons. (Right) ratio of the real (OS) and CMS cross sections.

We have added here the effect of the width of the $W$ boson since it does not
appear at the internal of Feynman diagrams at tree-level (Fig. 1). Hence it
does not pose the problem highlighted by Denner Denner:2019vbn  at one-loop,
which states that a given particle cannot be treated by two different schemes
when it is taken to be stable externally and unstable internally.

For different combinations of polarisations of external bosons, LL, TT and LT
+ TL, the cross sections behave as $1/s$ at high energy in the real scheme and
preserves such behaviour in CMS. These results confirm that the instability of
internal and external particles does not affect unitarity at tree-level. We
note the following points for the cross sections in both schemes: they are of
the same order for all energies used; the maximum is around $1000\,$GeV; in
the range $4000-6000\,$GeV, the contributions of the LL and LT+TL modes are
almost null, whilst the TT and UU modes have non-vanishing values. The TT mode
is dominant and makes the principal contribution for the UU mode.

In figures 3 and 4, we see that relative to Fig. 2 the widths $\Gamma_{H}$ and
$\Gamma_{t}$ have almost no effect. This is due to the weak ratio
$\Gamma_{H}/M_{H}\sim 6\times 10^{-3}\%$ and $\Gamma_{t}/m_{t}\sim 0.7\%$,
although the latter is relatively higher.

Figure 3: Cross sections in OS and CMS at tree-level for $pp\to W_{U}W_{U}$ as
a function of $\sqrt{s}$ for the case $\Gamma_{H}=0$ (left) and their ratio
(right).

Figure 4: Cross sections in OS and CMS at tree-level for $pp\to W_{U}W_{U}$ as
a function of $\sqrt{s}$ for the case $\Gamma_{t}=0$ (left) and their ratio
(right).

Fig. 5 shows that the introduction of $\Gamma_{Z}$ affects, both at low and
high energies, the Born cross section at around $2\%$. This effect has its
origin in the relatively high value of the ratio $\Gamma_{Z}/M_{Z}\sim 2.7\%$.

Figure 5: Cross sections in OS and CMS at tree-level for $pp\to W_{U}W_{U}$ as
a function of $\sqrt{s}$ for the case $\Gamma_{W}=0$ (left) and their ratio
(right).

Moreover, if we take into consideration the gauge boson $W$ as an unstable
particle with a ratio $\Gamma_{W}/M_{w}\sim 2.6\%$, Fig. 6 shows that
$\Gamma_{W}$ has a comparable effect to that of $\Gamma_{Z}$ if taken
separately but opposite if combined together (see Fig. 2). In the next section
we consider the CMS effect on the one-loop correction of the $pp\to
W^{+}W^{-}$ process.

Figure 6: Cross sections in OS and CMS at tree-level for $pp\to W_{U}W_{U}$ as
a function of $\sqrt{s}$ for the case $\Gamma_{Z}=0$ (left) and their ratio
(right).

###  3.3 One-loop corrections (NLO)

In Fig. 7, we show a selected sample of self-energy, vertex and box diagrams,
which are most sensitive to the widths of the unstable particles, among about
1000 diagrams contributing to the $pp\to WW$ process that we have generated
using the fixed-order Monte Carlo program MadGraph5_aMC@NLO. In the following,
we present the results of NLO cross sections (Born + corrections) in both CMS
and OS schemes for different widths of unstable particles. We consider in what
follows below only the effect of the internal particle widths $\Gamma_{H}$,
$\Gamma_{Z}$ and $\Gamma_{t}$ on the one-loop corrections in the case where
the external bosons are unpolarized and stable. This is for two reasons; one
is linked to the fact that the Monte Carlo program MadGraph5_aMC@NLO has not
yet implemented one-loop polarizations for this type of process; the other
reason is linked to Denner’s remark that the external particles must be
considered as stable in accordance with the work of Veltman VELTMAN1963186 ;
Bouamerane-90 . We therefore take the width of the $W$ boson, $\Gamma_{W}$, to
be zero everywhere in the one-loop Feynman diagrams, since the same particle
cannot be treated by two different renormalisation schemes simultaneously for
the same process.

It is worthwhile mentioning that our study does not involve the one-loop non-
abelian QCD corrections. We only deal with the one-loop renormalisation of the
electroweak contributions in both CMS and the usual OS schemes.

Figure 7: A sample of one loop self-energy, vertex and box Feynman diagrams
contributing to the process $pp\to WW$.

#### Input parameters:

For the purpose of our numerical investigations, we have used the following
input values, in addition to those used for the tree level case (3.2):

$\displaystyle m_{e}$ $\displaystyle=0.51099906\,\text{MeV},\qquad
m_{u}=47.0\,\text{MeV},\qquad m_{d}=47.0\,\text{MeV},$ $\displaystyle m_{\mu}$
$\displaystyle=105.658389\,\text{MeV},\qquad m_{c}=1.55\,\text{GeV},\qquad
m_{s}=150\,\text{MeV},$ $\displaystyle m_{\tau}$
$\displaystyle=177.1\,\text{MeV},\qquad\;\;\qquad m_{b}=4.5\,\text{GeV}.$ (40)

For the sake of deriving our results we have replaced the one-loop squared
amplitude by the following formula Denner-1998 :

$|\mathcal{M}|^{2}\to|\mathcal{M}_{Born}|^{2}(1+\delta_{soft})+2\,\mathcal{R}e\left(\mathcal{M}^{*}_{Born}\delta\mathcal{M}\right),$
(41)

where $\delta_{soft}$ takes into consideration soft bremstrahlung and is
required in dealing with infrared divergences. Moreover, $\delta\mathcal{M}$
contains all one-loop Feynman diagrams as well as their corresponding counter-
terms.

#### Results and discussion:

In figures 8 and 9, we present, for stable and unpolarized external bosons,
the full cross sections at one-loop in OS and CMS schemes respectively. At
high energies, they behave like $1/s$, which confirms that the two schemes do
not break the unitarity of the one-loop results. At low energies, the
corrections are positive and of the order of $5-10\,\%$ in both schemes. Once
the full NLO cross sections reach their maxima, around $1000\,\text{GeV}$, the
corrections become negative and large. They maintain this behaviour throughout
the remaining energy range. In the range $2000-6000\,\text{GeV}$ the
corrections are around $10-30\,\%$ in OS and $10-40_{,}\%$ in CMS.
Furthermore, they reach about $55\,\%$ in OS and $65\,\%$ in CMS around
$14\,$TeV. Thus at high energies, we notice that the corrections are of the
order of the Born cross section in both schemes. Hence the
$\mathcal{O}(\alpha^{2})$ corrections should be taken into consideration in
the calculation of the corrected cross section, in order to obtain more
reliable and precise results.

Figure 8: Plots for OS cross sections as a function of the scattering energy
$\sqrt{s}$, comparing Born and NLO corrections.

v

Figure 9: Plots for CMS cross sections as a function of the scattering energy
$\sqrt{s}$, comparing Born and NLO corrections.

Figures 10 and 11 show that only $\Gamma_{Z}$ significantly affects the
corrections at one-loop. Whereas $\Gamma_{H}$ and $\Gamma_{t}$ present
insignificant effects, even if the former preserves its ability to restore
unitarity as in the case of the actual OS scheme.

Figure 10: Plots for CMS cross sections as a function of the scattering energy
$\sqrt{s}$, comparing Born and NLO corrections, for the case $\Gamma_{H}=0$.

Figure 11: Plots for CMS cross sections as a function of the scattering energy
$\sqrt{s}$, comparing Born and NLO corrections, for the case $\Gamma_{t}=0$.

Figure 12 shows a comparison between the full cross sections in the two
aforementioned schemes. At low energy, the effect of the width $\Gamma_{Z}$ on
the full cross section is around $2.5\,\%$, which vanishes around
$1000\,\text{GeV}$ and then increases with energy until it reaches around
$15\,\%$ at $14\,$ TeV. This behaviour is expected to hold for further high
energies. That is, the effect of $\Gamma_{Z}$ increases with energy. This,
however, remains to be verified, especially if this process is considered as
an internal part of another process with stable external particles. This is to
be investigated in our future work.

Figure 12: Plots for SM (real) and SMc (CMS) cross sections as a function of
the scattering energy $\sqrt{s}$ at NLO.

##  4 Conclusion

In the present work, we made an extension of the calculations of the
electroweak radiative corrections to a loop of the process $pp\to WW$ to the
complex mass scheme, where the widths of the unstable particles are introduced
respecting the identity of Ward and that of Slavnov- Taylor.

At tree level, we obtained for the longitudinal mode of the external gauge
bosons a total amplitude proportional to the width $\Gamma_{Z}$ of the
internal gauge boson Z. We have shown that upon the inclusion of the width
$\Gamma_{W}$ of the external $W$ gauge boson, the total amplitude depends on
both $\Gamma_{W}$ and $\Gamma_{Z}$. The other widths $\Gamma_{H}$ and
$\Gamma_{t}$ of the Higgs boson and the top quark have practically no effect
on neither the amplitude nor the Born cross section. The effect of
$\Gamma_{Z}$ on the Born cross section is about $2\,\%$ for unpolarized and
stable W bosons. When the width $\Gamma_{W}$ of the $W$ external gauge boson
is included, it produces an effect on the Born cross section comparable to
that of $\Gamma_{Z}$ if they are introduced separately, and an opposite effect
to $\Gamma_{Z}$ if they are introduced together.

The Born and full one-loop cross sections behave as $1/s$ at high energies in
both CMS and OS schemes, thus preserving the unitarity of the theory. The one-
loop corrections are affected by $\Gamma_{Z}$ up to $65\%$ in the CMS scheme
at energy of about $14\,$ TeV. They reach for the same case about $55\%$ in
the real OS scheme. They are therefore of the order of the Born cross section
(at high energy), and hence higher orders, $\mathcal{O}(\alpha^{2})$ and
above, have to be included for any meaningful predictions.

Finally, comparisons of the full cross sections in OS and CMS schemes,
revealed that the width $\Gamma_{Z}$ affects the NLO cross section by about
$5\%$ around 2 TeV and $15\%$ around 14 TeV. This effect therefore increases
with increasing energy and its behaviour after 14 TeV remains to be verified.
For a full study of this type of problems where the effect of the width
$\Gamma_{W}$ is not neglected, as well as the widths of the other unstable
particles, $\Gamma_{Z},\Gamma_{H}$ and $\Gamma_{t}$, on the one-loop
corrections, it suffices to relate this process, i.e, $pp\to WW$, to another
one where the external particles are stable. Another important point to
consider for future work is the effect of various widths in the case where
polarisation is not neglected in tree and one-loop calculations.

###### Acknowledgements.

We would like to thank Dr. Noureddine Bouayed for support and encouragement to
carry out this work, valuable discussions particularly on complex
renormalisation as well as collaboration on the usage of FeynCalc.

## Appendix

##  Appendix A Amplitudes in CMS

The different high-energy amplitudes at the tree level in CMS in the case
where the longitudinal external gauge bosons are stable and unstable are given
by the following expressions:

$\displaystyle\operatorname{Re}\\{\mathcal{M}_{\gamma}^{s}\\}$
$\displaystyle=-Q_{f}\,e^{2}\,\frac{\bar{v}(p_{2})\,\
\not{q_{1}}u(p_{1})}{M_{W}^{2}}+\mathcal{O}(1/x),\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\,$
$\displaystyle\operatorname{Im}\\{\mathcal{M}_{\gamma}^{s}\\}$
$\displaystyle=0+\mathcal{O}(1/x).$ (42)
$\displaystyle\operatorname{Re}\\{\mathcal{M}_{Z}^{s}\\}$
$\displaystyle=e^{2}\,\frac{\bar{v}(p_{2})}{M_{W}^{2}}\Big{[}Q_{f}\,\
\not{q_{1}}+\frac{M_{Z}^{2}\,\
T_{3}^{f}}{2(M_{Z}^{2}-M_{W}^{2})}\Big{(}m_{q}(1-2P_{L})-2\not{q_{1}}P_{L}\Big{)}\Big{]}u(p_{1})+\mathcal{O}(1/x),\quad\,$
$\displaystyle\operatorname{Im}\\{\mathcal{M}_{Z}^{s}\\}$
$\displaystyle=e^{2}\,\frac{\Gamma_{Z}\,\ M_{Z}\,\
T_{3}^{f}}{2(M_{Z}^{2}-M_{W}^{2})^{2}}\,\
\bar{v}(p_{2})\Big{(}m_{q}(1-2P_{L})-2\not{q_{1}}P_{L}\Big{)}u(p_{1})+\mathcal{O}(1/x).$
(43) $\displaystyle\operatorname{Re}\\{\mathcal{M}_{q}^{s}\\}$
$\displaystyle=e^{2}\,\frac{-M_{Z}^{2}}{2M_{W}^{2}(M_{Z}^{2}-M_{W}^{2})}\,\
\bar{v}(p_{2})\Big{(}m_{q}+\not{q_{1}}\Big{)}P_{L}u(p_{1})+\mathcal{O}(1/x),\qquad\qquad\qquad\qquad\quad\,$
$\displaystyle\operatorname{Im}\\{\mathcal{M}_{q}^{s}\\}$
$\displaystyle=e^{2}\,\frac{-\Gamma_{Z}M_{Z}}{2(M_{Z}^{2}-M_{W}^{2})^{2}}\,\
\bar{v}(p_{2})\Big{(}m_{q}+\not{q_{1}}\Big{)}P_{L}u(p_{1})+\mathcal{O}(1/x).$
(44) $\displaystyle\operatorname{Re}\\{\mathcal{M}_{H}^{s}\\}$
$\displaystyle=e^{2}\,\frac{m_{q}M_{Z}^{2}}{4M_{W}^{2}(M_{Z}^{2}-M_{W}^{2})}\,\
\bar{v}(p_{2})u(p_{1})+\mathcal{O}(1/x),\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad\,\,$
$\displaystyle\operatorname{Im}\\{\mathcal{M}_{H}^{s}\\}$
$\displaystyle=e^{2}\,\frac{\Gamma_{Z}m_{q}M_{Z}}{4(M_{Z}^{2}-M_{W}^{2})^{2}}\,\
\bar{v}(p_{2})u(p_{1})+\mathcal{O}(1/x).$ (45)
$\displaystyle\operatorname{Re}\\{\mathcal{M}_{\gamma}^{s}\\}$
$\displaystyle=-Q_{f}\,e^{2}\,\frac{\bar{v}(p_{2})\,\
\not{q_{1}}u(p_{1})}{M_{W}^{2}}+\mathcal{O}(1/x),\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\,$
$\displaystyle\operatorname{Im}\\{\mathcal{M}_{\gamma}^{s}\\}$
$\displaystyle=-Q_{f}\,e^{2}\,\frac{\Gamma_{W}}{M_{W}}\,\frac{\bar{v}(p_{2})\not{q_{1}}u(p_{1})}{M_{W}^{2}}+\mathcal{O}(1/x).$
(46) $\displaystyle\operatorname{Re}\\{\mathcal{M}_{Z}^{s}\\}$
$\displaystyle=e^{2}\,\frac{\bar{v}(p_{2})}{M_{W}^{2}}\Big{[}Q_{f}\,\
\not{q_{1}}+\frac{M_{Z}^{2}\,\
T_{3}^{f}}{2(M_{Z}^{2}-M_{W}^{2})}\Big{(}m_{q}(1-2P_{L})-2\not{q_{1}}P_{L}\Big{)}\Big{]}u(p_{1})+\mathcal{O}(1/x),\,\,\,\,\,$
$\displaystyle\operatorname{Im}\\{\mathcal{M}_{Z}^{s}\\}$
$\displaystyle=e^{2}\,\frac{M_{Z}\,T_{3}^{f}}{2(M_{Z}^{2}-M_{W}^{2})^{2}}\,\bar{v}(p_{2})\Biggl{\\{}\Gamma_{Z}+\frac{\Gamma_{W}}{M_{W}}\,\frac{M_{Z}\left(M_{Z}^{2}-2M_{W}^{2}\right)}{M_{W}^{2}}\Biggr{\\}}\Biggl{\\{}m_{q}\left(1-2P_{L}\right)$
$\displaystyle-2\not{q_{1}}P_{L}\Biggr{\\}}\,u(p_{1})+Q_{f}\frac{\Gamma_{W}}{M_{W}}\,\frac{\left(M_{Z}^{2}-2M_{W}^{2}\right)}{M_{Z}^{2}-M_{W}^{2}}\,\not{q_{1}}\,\bar{v}(p_{2})\,u(p_{1})+\mathcal{O}(1/x).$
(47) $\displaystyle\operatorname{Re}\\{\mathcal{M}_{q}^{s}\\}$
$\displaystyle=e^{2}\,\frac{-M_{Z}^{2}}{2M_{W}^{2}(M_{Z}^{2}-M_{W}^{2})}\,\
\bar{v}(p_{2})\Big{(}m_{q}+\not{q_{1}}\Big{)}P_{L}u(p_{1})+\mathcal{O}(1/x),\qquad\qquad\qquad\qquad\,\,$
$\displaystyle\operatorname{Im}\\{\mathcal{M}_{q}^{s}\\}$
$\displaystyle=e^{2}\,\frac{-M_{Z}}{2\left(M_{Z}^{2}-M_{W}^{2}\right))^{2}}\,\bar{v}(p_{2})\Biggl{\\{}\Gamma_{Z}+\frac{\Gamma_{W}}{M_{W}}\,\frac{M_{Z}\left(M_{Z}^{2}-2M_{W}^{2}\right)}{M_{W}^{2}}\Biggr{\\}}$
$\displaystyle+\left(m_{q}+\not{q_{1}}\right)P_{L}u(p_{1})+\mathcal{O}(1/x).$
(48) $\displaystyle\operatorname{Re}\\{\mathcal{M}_{H}^{s}\\}$
$\displaystyle=e^{2}\,\frac{m_{q}M_{Z}^{2}}{4M_{W}^{2}(M_{Z}^{2}-M_{W}^{2})}\,\
\bar{v}(p_{2})\,u(p_{1})+\mathcal{O}(1/x),\qquad\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle\operatorname{Im}\\{\mathcal{M}_{H}^{s}\\}$
$\displaystyle=e^{2}\,\frac{\Gamma_{Z}\,m_{q}\,M_{Z}}{4\left(M_{Z}^{2}-M_{W}^{2}\right)^{2}}\,\bar{v}(p_{2})\,u(p_{1})+\mathcal{O}(1/x).$
(49)

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# An extreme wave field in the winter Antarctic marginal ice zone during an
explosive polar cyclone

Alberto Alberello
The University of Tokyo, Japan
Luke G. Bennetts
University of Adelaide, Australia
Miguel Onorato
Università di Torino & INFN, Italy
Marcello Vichi
University of Cape Town, South Africa
Keith MacHutchon
University of Cape Town, South Africa
Clare Eayrs
New York University Abu Dhabi, UAE
Butteur Ntamba Ntamba
Cape Peninsula University of Technology, South Africa
Alvise Benetazzo
Istituto di Scienze Marine, Consiglio Nazionale delle Ricerche, Italy
Filippo Bergamasco
Università Ca’ Foscari, Italy
Filippo Nelli
Swinburne University of Technology, Australia
Rohinee Pattani
Atkins, UK
Hans Clarke
The University of Melbourne, Australia
Alessandro Toffoli
The University of Melbourne, Australia Corresponding author:
<EMAIL_ADDRESS>
###### Abstract

Propagation of energetic surface gravity waves over a $>40$ km transect of the
winter Antarctic marginal ice zone comprised of pancake floes and interstitial
frazil ice during an explosive polar cyclone are presented, obtained with a
shipborne stereo-camera system. The waves are shown to attenuate at an
exponential rate over distance, but, despite this, remain large, even at the
deepest measurement locations and in 100% ice concentration, where they are up
to 8 m in amplitude—the largest waves measured in comparable conditions. The
occurrence of large waves in the marginal ice zone is shown to be consistent
with linear theory. Using concomitant measurements of wind speeds, evidence is
given that wind-to-wave momentum transfer occurs through a 100% pancake/frazil
ice cover, which is not permitted in most contemporary models.

##  1 Introduction

The marginal ice zone (MIZ) is the region of the ocean where surface gravity
waves and sea ice interact [1, 2]. In winter, when sea ice extent reaches its
maximum, the MIZ (defined by ice concentration between 15% and 80%) occupies
20–30% of the total sea ice area [3] and consists mostly of small (1–10 m
diameter), thin (order 0.1 m thickness), roughly circular, pancake ice floes
and interstitial frazil ice [2]. Pancake-dominated MIZs are highly mobile and
rapidly evolve in response to the frequent intense storms and cyclones in the
winter Southern Ocean [4]. Moreover, they are associated with rapid ice
production, as the unconsolidated mix of thin ice keeps the ocean in closer
contact with the cold atmosphere than the thicker, consolidated inner pack ice
[5].

Pancake floes are formed in wavy conditions [6], and, once formed, waves
prevent pancakes from consolidating [7]. Therefore, it is important to
understand how the large waves that characterise the Southern Ocean propagate
through the winter Antarctic MIZ, and whether strong winds over the MIZ
transfer momentum to waves. A key unknown is the attenuation rate of large-
amplitude waves over distance through a pancake/frazil ice cover, as this
informs predictions of the width of the ice-covered region impacted by waves
and hence the MIZ extent [8]. Moreover, the evolution of the wave spectrum
through the MIZ, particularly the reduction in amplitudes, frequency
downshifting and directional spread, are required to predict the floe size
distribution [9]. Momentum transfer from winds over the ice cover is a
prominent knowledge gap, with most contemporary models simply assuming wind
energy input is identical to the open ocean but scaled by the fraction of the
ocean surface not covered by sea ice, i.e. no wave generation in 100% ice
concentration [10, 11].

Despite advances over the past decade in measuring wave propagation through
the Arctic [12] and Antarctic MIZ [13, 14], there is a lack of measurements in
the harsh winter Antarctic MIZ and for the extreme sea states in the winter
Southern Ocean, where, over the open ocean, the mean significant wave height
in many sectors reaches $5$ m and the 90th percentiles (extremes) are up to 7
m [15]. In the Arctic, an array of wave buoys was used to measure wave
propagation through an autumn MIZ consisting of 50–100% concentration
pancake/frazil ice, during a storm event where the incident significant wave
height from the open ocean was up to 5 m [16]. Large waves (significant wave
heights $>3$ m) were found to attenuate at a linear rate,and small waves ($<3$
m) at an exponential rate. In the Southern Ocean, arrays of wave buoys were
used to measures waves in the Antarctic MIZ during spring to early winter, in
which floes were 10–100 m in diameter, $\approx{}0.5$ m thick and covered a
range of concentrations [17]. Significant wave heights in excess of 6 m were
recorded, but only in ice concentration $\approx 50\%$ in the outskirts of the
MIZ (within the first few tens of kilometres from the ice edge).

Stereo-imaging techniques have emerged as a tool for in-situ wave measurements
in the MIZ. The images are used to reconstruct the sea surface elevation in
time and space, thus enabling analysis of wave dynamics in both two-
dimensional physical space and the frequency–direction spectral domain,
including statistical analysis of the wave field [18]. Remote sensing
techniques have also been developed to measure the directional wave spectrum
in the MIZ [19]. Airborne SAR configurations can provide robust directional
spectra over 60–80 km long transects of the MIZ [20, 21]. However, stereo-
imaging, being an in-situ technique, can be used to simultaneously measure sea
ice geometrical properties [22] and can be combined with co-located
meteorological measurements, e.g. wind velocities.

To date, use of stereo-imaging techniques for wave measurements in sea ice has
been limited in scope. Campbell et al. [23] and Smith and Thomson [24] both
use stereo-camera systems to derive one-dimensional wave spectra and bulk wave
properties (peak periods and significant wave heights) from images over small
domains (order 100 m2) for small waves propagating through pancake ice.
Campbell et al. [23] use a camera system on a fixed platform on the edge of a
lake, whereas Smith and Thomson [24] use a camera system on a moored vessel in
the Arctic MIZ, and hence have to correct the images for wave-induced ship
motions.

In this Letter, we address the gap in measurements of large waves in the
winter Antarctic MIZ. Stereo-camera measurements are presented of extreme sea
states ($\approx{}90$th percentile for the sector at the ice edge) during an
explosive polar cyclone along an $>40$ km transect into the Antarctic MIZ
closely aligned with the mean wave direction ($\approx 30^{\circ}$
difference). Large individual wave events are recorded in the MIZ, even in
100% ice concentration at the deepest measurement locations, notwithstanding
the exponential attenuation of wave energy with distance. The occurrence of
large waves is shown to be consistent with linear wave theory. Evidence is
given of momentum transfer from winds over the first 40 km into the MIZ, which
could explain the milder attenuation rate than previously reported for high
ice concentrations. Analysis of the wave spectra reveals frequency
downshifting of the dominant wave energy, which is attributed to stronger
attenuation of high-frequency components by the ice cover and evolution of the
incident wave field from the open ocean, and broadening of the directional
spectrum.

##  2 Field Measurements

Between 07:00 and 08:00 UTC on the 4th July 2017, the South African icebreaker
S.A. Agulhas II entered the Antarctic MIZ (62∘ South and 30∘ East) during an
intense polar cyclone [4]. Ice concentration on the day, sourced from AMSR2
satellite at 3.125 km resolution [25], is shown in Fig. 1a. The icebreaker
moved South encountering the ice edge at 08:00 UTC, i.e. the northernmost
location where sea ice concentration exceeds 10% in a 1 km radius around the
vessel (ASPeCt; [26]). It reached 100% sea-ice concentration at 09:00 UTC
(Fig. 1a), approximately 10 km from the ice edge, after which it continued
South, whilst remaining in 100% ice concentration [22]. Over this time,
intense cyclonic conditions (wind speed 18–19 m s-1 from NW, turning by
$<10^{\circ}$ during the measurements) intensified ocean waves at the ice
edge, from 6.7 m at 08:00 UTC to 7.8 m at 13:00 UTC [27], therefore exceeding
the 90th percentile of waves in the Indian sector of Southern Ocean, noting
that it is the sector containing the most intense waves in the Southern Ocean
[15]. The mean wave direction at the ice edge was aligned with the wind and
directional spread remained steady [27].

Figure 1: (a) Overview of study area with ice concentration from AMSR2 on the
4th July 2017 and wind vectors from ERA5 at 11:00 UTC, with crosses denoting
average location of wave measurements during a 30 mins acquisition and the
circle location of wave measurements obtained by a buoy deployed on an ice
floe during the voyage [28]. (b) Pair of synchronised camera images, and (c)
reconstructed surface elevation, $\eta$.

A system of two GigE monochrome industrial CMOS cameras with a 2/3 inch sensor
was installed on the monkey bridge of the icebreaker $\approx 34$ m from the
waterline and with their axes 5 m apart, tilted 20∘ below the horizon. The
cameras were equipped with 5 mm lenses to provide a field of view of the ocean
surface $\approx{}90^{\circ}$ around the port side of the ship (Fig. 1b).
Images were recorded with resolution 2448$\times$2048 pixels and a sampling
rate 2 Hz during daylight on the 4th July (from 07:00 to 14:00 UTC).
Acquisition was performed over $\approx{}30$ mins intervals ($\approx{}6$
times longer than Smith and Thomson [24]).

The wave acquisition stereo-camera system (WASS; [29]) is used to reconstruct
the water surface elevation. Camera images are first calibrated to remove lens
distortion and find the pixel to meter conversion factor. The algorithm
detects and matches common features in each pair of synchronised images,
allowing three-dimensional reconstruction of the free surface (Fig. 1c). The
WASS system captures $\approx{}150$ m$\times 200$ m snapshots of the ocean
surface ($\approx{}5$ times larger than Smith and Thomson [24]). The system
operated from 07:00 to 14:00 UTC but images acquired in the early hours of the
day (07:00–08:00 UTC) lacked contrast and light to reconstruct sea surface
elevation. An inertial measurement unit (IMU) installed close to the cameras
and synchronised with the camera images is used to correct for ship motions
and reconstruct the surface elevation with respect to a common horizontal
plane defining the mean sea level.

The sea surface elevation is reconstructed in a frame of reference moving with
the ship, and, therefore, is affected by Doppler shift. The wave spectrum is
retrieved by computing the directional spectrum in the ship frame of reference
with the wavelet method [30] using timeseries extracted on a virtual array,
i.e. at multiple points within the field of view, and then correcting the
directional spectrum for Doppler shift using the dispersion relation from
linear wave theory and deep water conditions [31]. The two-dimensional
frequency–direction spectra, $E(f,\theta)$, are subsequently integrated with
respect to direction to give one-dimensional frequency spectra, $E(f)$.

WASS was also used to monitor in-situ ice properties along the transect.
Remotely sensed concentration varied from 0% at 08:00 UTC to 100% at 09:00
UTC. In-situ measurements revealed that from 09:00 UTC onwards the sea ice
cover was comprised of a mixture of $\approx 60\%$ pancake ice floes, with
median diameter 3.1 m, and the remaining $\approx 40\%$ interstitial frazil
ice [22].

##  3 Results

Surface elevation timeseries around the largest waves recorded ($H_{\max}$;
maximum crest to trough distance) at the six locations denoted by crosses in
Fig. 1a are shown in Fig. 2a, where $d$ denotes distance from the ice edge
projected along the mean wave direction. The first timeseries (top panel) was
obtained close to the ice edge ($d=5$ km) in $\approx 50\%$ ice concentration.
Subsequent timeseries are obtained deeper into the MIZ ($d\geq{}15$ km) in
100% ice concentration. The significant wave height, i.e.
$H_{S}=4\sigma_{\eta}$ where $\sigma_{\eta}$ is the surface elevation standard
deviation, is also shown for each timeseries (Fig. 2a; dashed lines).

The significant wave height close to the ice edge is 6.6 m (Fig. 2a top
panel). It is approximately the same as ERA5 reanalysis in open water just
outside the ice edge (ERA5 only provides wave data up to 15% ice
concentration). The significant wave height decreases deeper into the MIZ, to
$H_{S}=4.6$ m at $d=44$ km (bottom panel). The value at this location agrees
with co-located buoy measurements (black circle in Fig. 1a; [4]).

At each location, the largest wave is found as part of an energetic wave
group. The maximum wave height also tends to diminish with distance into the
MIZ. But large waves, $H_{\max}\approx 8$ m, are recorded in 100% ice
concentration $43$ km into the MIZ (second to bottom panel). To our knowledge,
this is the largest recorded wave in 100% ice concentration. The maximum to
significant wave height ratio ($H_{\max}/H_{S}$) slightly increases over
distance, from 1.3 close to the edge to 1.5 deeper in the MIZ, but remains
lower than the rogue wave threshold, $H_{\max}/H_{S}=2$ [32]. Fig. 2b shows
the probability density function of the normalised wave intensity
($\eta^{2}/\langle\eta^{2}\rangle$). The exponential distribution associated
to a linear (Gaussian) sea surface elevation is shown for reference [33].

Figure 2: (a) Samples of surface elevation time series at progressive
distances from the ice edge (top–to–bottom; color-coding corresponds to
location shown in Fig. 1a) shown around the highest wave in the record. Dashed
lines indicate $H_{S}$ and cyan background denotes intermediate ice
concentration. (b) Probability density function of the wave intensity, where
red line indicates linear wave theory benchmark. (c) Evolution of $H_{S}$ with
distance from the ice edge. Dots are present data (cross the buoy) and gray
line denotes best (exponential) fit. Dashed lines are attenuation derived from
Kohout et al. [17] for $i_{c}<0.8$ (green; thicker for $T_{p}>14$ s) and
$i_{c}>0.8$ (red; thicker for $T_{p}>14$ s).

The significant wave height (normalised by its open water counterpart) decays
exponentially with distance $d$ (Fig. 2c) at a rate $\alpha\approx 8\times
10^{-6}$ m-1. For comparison, significant wave heights versus distance into
the Antarctic MIZ based on buoy measurements [17] are also shown in Fig. 2b
for high ice concentration ($>80\%$; red) and low ice concentration ($<80\%$;
green), which have attenuation rates $26.5$–$32.7\times 10^{-6}$ m-1 and
$1.6$–$5.0\times 10^{-6}$ m-1, respectively. In both cases, the lower
attenuation is for long waves ($T_{p}>14$ s; thick dashed) while the higher
for short waves ($T_{p}\leq 14$ s; thin dashed).

The two- and one-dimensional spectra, i.e. $E(f,\theta)$ and $E(f)$, at the
six locations are shown in Fig. 3a–Fig. 3b. The peak period (the most
energetic period; $T_{p}$) and the mean period (the reciprocal frequency
moment of the wave spectrum; $T_{m}$) at each location are indicated on the
one-dimensional spectra (Fig. 3b), along with the corresponding open ocean
peak period ($T_{p,ERA5}$) and mean period ($T_{m,ERA5}$) at the ice edge at
the time of the measurement and accounting for advection time. The open ocean
peak and mean periods increase from $T_{p,ERA5}=12.4$ s and $T_{m,ERA5}=10.4$
s when the measurements were made at the ice edge to $T_{p,ERA5}=13.5$ s and
$T_{m,ERA5}=11.4$ s when the measurements were made at $d=43$–44 km. The
values of $T_{p}$ and $T_{m}$ at $d=43$–44 km agree with co-located buoy
measurements (Fig. 3b), and at the ice edge with open ocean values.

Attenuation of wave energy with distance into the MIZ is evident in the one-
dimensional spectra, with higher frequencies (shorter periods) experiencing
greater attenuation than lower frequencies (longer periods). The attenuation,
combined with the increasing peak and mean periods over time at the ice edge,
causes the peak and mean periods to increase over distance and time. The peak
period increases by only a small amount between $d=5$ km and $d=36$ km, from
$T_{p}=12.0$ s to $T_{p}=12.6$ s. Beyond $d=36$ km, it increases sharply to
$T_{p}=15.1$ s at 43–44 km. In contrast, $T_{m}$ increases steadily over
distance, from 10.3 s at $d=5$ km to 12.9 s at $d=43$–44 km. The difference in
the evolution of the peak and mean periods is attributed to the dependence of
the peak period on spectral resolution, whereas the mean period, being an
integral over frequencies, is a more robust parameter. Attenuation causes the
peak and mean periods at the deepest locations ($d=43$–44 km) to be
$12$–$15\%$ longer than the corresponding open ocean peak and mean periods.

The spectral bandwidths (the breadth of the spectrum in frequency evaluated at
half of the spectral peak energy, $\Delta f$; [32]) are indicated by
horizontal bars on the one-dimensional spectra in Fig. 3b. They show narrowing
of the spectrum with distance into the MIZ, with the bandwidth at $d=43$–44 km
$\approx{}80$% of the bandwidth at the ice edge.

The two-dimensional spectra (Fig. 3a) indicate the directional properties of
the spectrum also evolve in the MIZ. Close to the ice edge (top panel), the
wave spectrum is bimodal, with two distinct energy peaks of comparable
magnitude $\approx 25^{\circ}$ from each other with the lower-frequency peak a
swell and the higher-frequency peak a wind sea locally generated. At the next
measurement point ($d=15$ km), the spectrum is dominated by one energy peak,
i.e. it is unimodal, which is attributed to wave transformation, e.g.
refraction, caused by the local conformation of the intermediate ice
concentration area West of the transect (see Fig. 1a). The spectrum remains
unimodal until the deepest measurement locations ($d=43$–44 km), where some
small additional peaks appear. The frequency averaged directional spread, i.e.
a measure of the directional distribution of wave energy [16], is indicated by
vertical bars for each location. Directional spread tends to increase over
distance, from $\approx 40^{\circ}$ at $d=5$ km to $\approx 60^{\circ}$ at
$d=43$–44 km, but the trend is not well defined, and the maximum is $\approx
75^{\circ}$ at $d=36$ km. The directional spread agrees with typical open
water values during storms in the Southern Ocean [34].

Figure 3: (a) Two-dimensional and (b) one-dimensional wave spectra at
progressive distances from the ice edge (top–to–bottom; color-coding
corresponds to the locations shown in Fig. 1a). Two-dimensional spectra are
normalised by the peak energy to highlight directional properties, and one-
directional spectra are normalised by a common factor to highlight energy
loss. In (a), Cartesian coordinates are used as the spectra cover only a
narrow directional range, $-60^{\circ}<\theta<60^{\circ}$, and bars denote
directional spread ($\sigma_{0}$). In (b), bars denote spectral bandwidth,
crosses and dots peak and mean period ($T_{p}$; $T_{m}$), dotted and dashed
grey lines peak and mean period in open ocean ($T_{p,ERA5}$; $T_{m,ERA5}$). In
the bottom panel, the spectrum from the buoy is also shown (black dotted). (c)
Wave age at progressive distances from the ice edge. Dotted line denotes the
threshold between young and old waves. In (a–c), cyan backgrounds denote the
measurement location in intermediate ice concentration.

The ratio $c_{p}/u_{10}$, where $c_{p}$ is the phase velocity (ratio of
wavelength to wave period) and $u_{10}$ the wind speed, is known as the wave
age [35]. It is used to separate young waves, which are growing in size and
length under the action of wind ($c_{p}/u_{10}<1.25$), from old (swell) waves,
which propagate without active wind input ($c_{p}/u_{10}>1.25$). The wave age
(Fig. 3c), computed using wind speed from the onboard met-station, indicate
waves are young over the first $\approx{}40$ km of the MIZ
($c_{p}/u_{10}\approx 0.75$), but sharply switch to old ($c_{p}/u_{10}\approx
1.40$) at the deepest two measurement locations ($d=43$–44 km), caused by the
sudden drop in wind speed (from $\approx 25$ m s-1 to $\approx 17$ m s-1).

##  4 Discussion and Conclusions

The exponential attenuation of significant wave height with distance into the
Antarctic MIZ is consistent with the recent findings of Kohout et al. [17].
However, the attenuation rate found here is $\approx 4$ times less than the
attenuation rate Kohout et al. [17] report for high ice concentrations, and
slightly greater than the rate they report for low ice concentrations. The
differences indicate the attenuation rate depends on ice cover properties, as
Kohout et al. [17] used measurements when the MIZ was comprised of floes
10–100 m in diameter, whereas the measurements reported here were taken when
the MIZ was comprised of small pancake ice floes (1–10 m) and interstitial
frazil ice. For small pancake floes in the Arctic (diameters $\approx 0.3$ m)
interspersed with water and/or grease ice, significant wave height attenuation
rates have not been reported, but attenuation rates at the frequency
corresponding to peak periods reported here are of the order
$10^{-6}$–$10^{-5}$ m-1 [36, 37].

The low attenuation rate found herein for the winter MIZ corresponds to
significant wave heights halving every $\approx 75$ km, e.g. $H_{S}=8.5$ m in
open ocean remains $H_{S}>0.5$ m at 300 km in the ice-covered ocean,
preventing floe welding and maintaining an unconsolidated ice cover [6, 28].
Thus, during the passage of intense storms and cyclones, wave–ice interactions
are likely to persist hundreds of kilometres in from the ice edge, even in
100% ice concentration. This finding corroborates satellite observations by
Stopa et al. [38] spanning multiple seasons of waves propagating hundreds of
kilometers into the Antarctic MIZ.

Most models scale wind input according to the open water fraction [10, 39],
i.e. no wind input in 100% ice concentration. In contrast, Rogers et al. [11]
suggest wind energy transfer can occur in ice-covered regions, particularly in
frazil, brash and/or pancake ice fields, albeit less effectively than in the
open water. Moreover, Zhao and Zhang [40] recently presented a theoretical
model in which wind can actively transfer energy to waves through the ice
cover. The wave age values presented here (Fig. 3c) indicate strong winds
potentially feed wave growth in 100% ice concentration for tens of kilometres
into a pancake/frazil MIZ. Wind input would offset wave attenuation due to the
ice cover, providing an alternative explanation for the lower attenuation rate
than reported by Kohout et al. [17], noting Kohout et al. [17] do not report
the prevailing winds during their measurements.

Few measurements of the directional wave spectrum exist in the MIZ, with most
using buoys, which can be unreliable [16, 17], and some using airborne SAR
[20, 21]. There is no consensus on the evolution of the directional spectrum
in the MIZ. Previous measurements in MIZs consisting of floes much smaller
than wavelengths (e.g. pancake floes), range from narrowing [16, 21] to no
change [41] to broadening [1, 42, 20]. The measurements presented here show
evidence of directional broadening. The physical mechanism responsible for the
broadening is unknown, noting that theoretical models generally predict
directional narrowing due to wave dissipation in a MIZ consisting of small
floes [43].

Attenuation of wave energy over distance into the MIZ reduces the wave
steepness ($\varepsilon=k_{p}H_{S}/2$, where $k_{p}$ is the wavenumber
associated to $T_{p}$) from $0.10$ at $d=5$ km to $0.04$ at $d=43$–44 km.
Despite the large waves measured (Fig. 2a), the small steepness values in the
MIZ indicate wave probabilities should conform to linear wave predictions
[44], which is backed by the similarity between the probability density
function and the linear wave theory benchmark (Fig. 2b). The finding is in
agreement with observations by Thomson et al. [45] in the Arctic MIZ for
milder wave conditions. Our measurements strengthens the argument that
nonlinear wave mechanisms responsible for the generation of large waves (e.g.
modulational instability) are weakened in the MIZ. Note, however, there is no
theoretical framework for wave nonlinearity in the MIZ at present, as it is
defined only for waves in the open ocean [32].

In conclusion, measurements have been reported of extreme wave conditions in
the winter Antarctic MIZ during an explosive polar cyclone, which were
captured by a stereo-camera system over a $44$ km transect. The findings from
the measurements have implications for the treatment of wave–ice interactions
in wave and sea ice forecasting models. In particular, most contemporary
models tend to dissipate wave energy over short distances in 100% ice
concentration regardless of the ice type, whereas evidence has been presented
to support dependence of attenuation rates on ice type, and that waves can
propagate long distances into MIZs consisting of pancake–frazil ice even at
100% ice concentration. As a consequence, the width of the winter Antarctic
MIZ is likely to be under-predicted by an order of magnitude, especially if a
concentration-based MIZ definition is used, as in many coupled models [46] and
studies [3, 47], for which the MIZ width would have been only $\approx 10$ km
during the reported observations. Consequently, heat and momentum exchanges
promoted by wave-induced motion in 100% sea ice are likely to be poorly
represented in models at present [48, 42, 49, 9].

## Acknowledgments

The expedition was funded by the South African National Antarctic Programme
through the National Research Foundation. This work was motivated by the
Antarctic Circumnavigation Expedition (ACE) and partially funded by the ACE
Foundation and Ferring Pharmaceuticals. AA, LB and AT were supported by the
Australian Antarctic Science Program (project 4434). AA is supported by the
Japanese Society for the Promotion of Science (PE19055). LGB is supported by
the Australia Research Council (FT190100404). LGB and AT are supported by the
Australia Research Council (DP200102828). MO was supported from the Simons
Collaboration on Wave Turbulence, Award No. 617006, and from the “Departments
of Excellence 2018–2022” Grant awarded by the Italian Ministry of Education,
University and Research (MIUR, L.232/2016). MV and KM were supported by the
NRF SANAP contract UID118745. CE was supported under NYUAD Center for global
Sea Level Change project G1204. We are indebted to Captain Knowledge Bengu and
the crew of the SA Agulhas II for their invaluable contribution to data
collection. ERA5 reanalysis was obtained using Copernicus Climate Change
Service Information. MO acknowledges B GiuliNico for interesting discussions.
AA, AT and MO thank L Fascette for technical support during the cruise.

Datasets for this research (reconstructed surface elevations) are available
through the Australian Antarctic Data Centre (AADC): Alberello et al. (2021)
Wave acquisition stereo-camera system measurements (WASS) from a voyage of the
S.A. Agulhas II, July 2017, Ver. 1, Australian Antarctic Data Centre –
doi:10.26179/q9bd-5f74.

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