11institutetext: Radboud University Nijmegen, 6500 GL Nijmegen, The
Netherlands 22institutetext: University College London, WC1E 6BT London,
United Kingdom

# Collinear Electroweak Radiation in Antenna Parton Showers

Ronald Kleissaddr1  Rob Verheyenaddr2,addr1

###### Abstract

We present a first implementation of collinear electroweak radiation in the
Vincia parton shower. Due to the chiral nature of the electroweak theory,
explicit spin dependence in the shower algorithm is required. We thus use the
spinor-helicity formalism to compute helicity-dependent branching kernels,
taking special care to deal with the gauge relics that may appear in
computation that involve longitudinal polarizations of the massive electroweak
vector bosons. These kernels are used to construct a shower algorithm that
includes all possible collinear final-state electroweak branchings, including
those induced by the Yang-Mills triple vector boson coupling and all Higgs
couplings, as well as vector boson emissions from the initial state. We
incorporate a treatment of features particular to the electroweak theory, such
as the effects of bosonic interference and recoiler effects, as well as a
preliminary description of the overlap between electroweak branchings and
resonance decays. Some qualifying results on electroweak branching spectra at
high energies, as well as effects on LHC physics are presented. Possible
future improvements are discussed, including treatment of soft and spin
effects, as well as issues unique to the electroweak sector.

##  1 Introduction

Beyond the discovery of the Higgs boson ATLASHiggs ; CMSHiggs , signs of new
physics have yet to appear at the LHC and the Standard Model has so far
survived all forms of scrutiny. It has therefore become more likely that the
Standard Model continues to describe nature accurately up to very high energy
scales. At these very high energies heavy particles like electroweak gauge
bosons, Higgs bosons and top quarks can start to appear as constituents of
jets topInJets1 ; topInJets2  or otherwise contribute to radiative
corrections. Virtual corrections have been shown to become sizeable even at
LHC energies in exclusive observables EWLarge1 ; EWLarge2 ; EWLarge3 ;
EWLarge4 ; EWLarge5 ; EWLarge6 ; EWLarge7 ; EWLarge8 ; EWLarge9 ; diJetLarge1
; diJetLarge2 ; VjLarge1 ; VjLarge2 ; VjLarge3 . For instance, corrections to
transverse momentum at LHC energies can already reach about 10% for exclusive
dijet production diJetLarge1 ; diJetLarge2 , and about 20% for single vector
boson production VjLarge1 ; VjLarge2 ; VjLarge3 , and they can be expected to
grow even larger at future collider energies 100TeV1 ; 100TeV2 .

Work such as SMFFs ; EWInclusive  has instead focussed on the incorporation of
electroweak logarithms in the resummation of inclusive observables at high
energies. However, many practical observables are not fully exclusive or fully
inclusive. In most cases, the only practical solution is to instead include EW
effects in a systematic way in parton shower algorithms as part of a general-
purpose event generator such as Pythia Pythia8.2 , Sherpa Sherpa  or Herwig
Herwig++ . These produce fully differential final states, and can be set up to
include specific phenomena exclusive to the EW sector such the effects of
symmetry breaking and gauge boson decay. Furthermore, the inclusion of EW
effects in parton shower algorithms enables automatic interleaving with the
QCD shower, as well as the interfacing with the models of nonperturbative
aspects of the event generation process, such as multi-parton interactions and
hadronization. As an example, ATLAS has reported on measurements that are
sensitive to the collinear enhancements associated with $W$ radiation in jets
ATLASEW , and compared them to event generators that incorporate these
effects. There, it is also pointed out that these types of effects will play a
significant role for several measurements at high energy scales, which will
become more abundant as the LHC gathers more data.

Electroweak corrections have been incorporated in parton showers in the past.
An implementation PythiaEW ; PythiaEW2  is available in Pythia event generator
Pythia8.2  which only includes the radiation of electroweak gauge bosons and
does not retain any spin information. The radiation of electroweak gauge
bosons was similarly included in the Sherpa event generator Sherpa  to study
$W$ emissions in jets SherpaEW . Another approach ALPGENEW  was employed in
ALPGEN ALPGEN  where fixed-order matrix element calculations are combined with
analytic Sudakov factors to achieve results similar to those of an electroweak
parton shower. A more recent work Tweedie  has implemented a final-state
electroweak shower in the Pythia $1\rightarrow 2$ transverse momentum ordered
shower formalism that retains spin information and includes all branchings
present in the electroweak sector.

In this paper, extend the antenna-based Vincia parton shower VinciaSimple ;
VinciaTimelike ; VinciaHadron1  to include all possible final-state collinear
EW branchings, including those associated with Yang-Mills triple vector boson
and Higgs couplings. Furthermore, collinear vector boson emissions off the
initial state are also included. Vincia is a plugin to the Pythia event
generator and already allows for QCD evolution with partons of definite
helicity states VinciaHelicity1 ; VinciaHelicity2 . This feature is especially
important in the electroweak theory due to its chiral nature. The electroweak
shower described in this paper will thus be responsible for the electroweak
component of the shower evolution, and is interleaved with the default Vincia
QCD shower.

The formalism descibed in this paper makes use of the spinor-helicity
formalism to compute the large number of spin-dependent branching kernels
associated with the electroweak shower. Helicity-dependent QCD antenna
functions have previously been computed with comparable methods Peskin1 ;
Peskin2 . We start with a brief overview of the spinor-helicity formalism and
the conventions used in the calculation of the branching kernels. In section
3, the spinor-helicity formalism is used to compute branching kernels for all
branching processes in the electroweak sector. Section 4 discusses the
collinear limits of those branching kernels given in terms of Altarelli-Parisi
splitting functions DGLAP3 . The results in that section are found to be in
agreement with Tweedie . Section 5 details the implementation of an
electroweak shower in the Vincia framework and treats a number of
peculiarities exclusive to the electroweak sector such as the presence of
bosonic interference and the matching to resonance decays. To show the
significance of an electroweak shower, its effects are investigated in section
6 for highly energetic particles, but also at LHC energies. We finally
conclude in section 7, discussing a number of missing features that are
required to bring the modelling of EW radiation further in line with current
QCD showers.

##  2 The Spinor-Helicity Formalism

Due to the chiral nature of the electroweak theory, it is important to
calculate electroweak branching kernels for individual spin states. We choose
to perform these calculations using the spinor-helicity formalism using
conventions similar to those described in spinorHelicity . The branching
kernels computed here describe the correct soft and collinear factorization
properties of branchings in the electroweak sector, while remaining Lorentz-
invariant and independent of a particular representation of the Dirac algebra
or explicit forms for fermionic spinors. Furthermore, their analytic nature
will be useful in dealing with issues that appear due to gauge dependence for
longitudinal polarizations of gauge bosons. We first briefly summarize our
conventions and techniques.

###  2.1 Spinors

Helicity spinors for massive fermions may be defined as

$\displaystyle u_{\lambda}(p)$
$\displaystyle=\frac{1}{\sqrt{2p{\cdot}k}}(\not{p}+m)u_{-\lambda}(k)\mbox{ and
}$ $\displaystyle v_{\lambda}(p)$
$\displaystyle=\frac{1}{\sqrt{2p{\cdot}k}}(\not{p}-m)u_{\lambda}(k),$ (1)

where $\lambda$ is the fermion helicity and $k$ is a lightlike reference
vector that defines the meaning of the helicity of the fermion. Due to its
massive nature, helicity is not a Lorentz-invariant quantity and does not
coincide with the chirality of the fermion. The spin vector associated with
the spinors defined in eq. (2.1) is

$s_{\lambda}^{\mu}=\frac{\lambda}{m}\left(p^{\mu}-\frac{m^{2}}{p{\cdot}k}k^{\mu}\right).$
(2)

We therefore choose the reference vector

$k=(1,-\vv{e}),$ (3)

where $\vv{e}$ is a unit vector pointing in the direction of $\vv{p}$. With
this choice, the massive helicity spinors retain the usual meaning of helicity
as the projection of spin along the direction of motion.

###  2.2 Polarization Vectors

The polarization vectors for a massive vector boson with momentum $p$ are
defined as

$\displaystyle\epsilon_{\pm}^{\mu}(p)$
$\displaystyle=\pm\frac{1}{\sqrt{2}}\frac{1}{2p{\cdot}k}\bar{u}_{\mp}(k)\not{p}\gamma^{\mu}u_{\pm}(k)\mbox{
and }$ $\displaystyle\epsilon_{0}^{\mu}(p)$
$\displaystyle=\frac{1}{m}\left(p^{\mu}-2\frac{m^{2}}{2p{\cdot}k}k^{\mu}\right),$
(4)

where $k$ is again a massless reference vector. Here,
$\epsilon_{\pm}^{\mu}(p)$ are the transverse polarizations and
$\epsilon_{0}^{\mu}(p)$ is the purely longitudinal polarization which only
exists for massive vector bosons. By again choosing eq. (3), it is immediately
clear that $\epsilon_{\pm}^{\mu}(p)$ are purely transverse and
$\epsilon_{0}^{\mu}(p)$ is purely longitudinal.

###  2.3 Spinor Products and Amplitude Evaluation

Having expressed all massive spinors and polarization vectors in terms of
massless spinors, amplitudes for particles with definite helicities can now be
calculated very efficiently. We first define the spinor product

$S_{\lambda}(k_{a},k_{b})\equiv\bar{u}_{\lambda}(k_{a})u_{-\lambda}(k_{b})$
(5)

for lightlike (reference) vectors $k_{a}$ and $k_{b}$, which obey

$|S_{\lambda}(k_{a},k_{b})|^{2}=2k_{a}{\cdot}k_{b}.$ (6)

Spinor products are not uniquely defined, but one possible representation is
given by

$\displaystyle S_{\lambda}(k_{a},k_{b})$ $\displaystyle=(\lambda
k_{a}^{2}+ik_{a}^{3})\sqrt{\frac{k_{b}^{0}-k_{b}^{1}}{k_{a}^{0}-k_{a}^{1}}}$
$\displaystyle-(\lambda
k_{b}^{2}+ik_{b}^{3})\sqrt{\frac{k_{a}^{0}-k_{a}^{1}}{k_{b}^{0}-k_{b}^{1}}}.$
(7)

Using the spinors and polarization vectors of the previous section, all
amplitudes can be expressed in terms of these spinor products. Structures like

$S_{\lambda}(k_{a},p_{i},p_{j},...,k_{b})\equiv\bar{u}_{\lambda}(k_{a})\not{p}_{i}\not{p}_{j}...u_{\pm\lambda}(k_{b}),$
(8)

may still appear, where $p_{i},p_{j},...$ may be massive. Such structures may
be expressed in terms of the spinor products eq. (5) by defining

$\hat{p}_{i}=p_{i}-\frac{p_{i}^{2}}{2p_{i}{\cdot}k_{i}}k_{i},$ (9)

which is explicitly massless. Eq. (8) may then be written as

$S_{\lambda}(k_{a},p_{i},p_{j},...,k_{b})=S_{\lambda}(k_{a},\hat{p}_{i})\,S_{-\lambda}(\hat{p}_{i},p_{j},...,k_{b}).$
(10)

This procedure is then repeated, the next time making $p_{j}$ massless by
subtracting $(p_{j}^{2}/2\hat{p}_{i}{\cdot}p_{j})\hat{p}_{i}$, until only
spinor products remain.

##  3 Electroweak Branching Amplitudes

We now use the spinor-helicity formalism to compute branching amplitudes for
the electroweak sector. We first recount the phase space regions where
radiative amplitudes factorize into a non-radiative amplitude and a radiative
correction. For the moment, we restrict ourselves to splittings in the final
state, where we denote the splitting momenta as $p_{ij}\rightarrow
p_{i}+p_{j}$. The parent momentum $p_{ij}$ may be considered to be off-shell
by an amount that vanishes in the singular limits.

We first consider the quasi-collinear limit QuasiCollinear1 ; QuasiCollinear2
; QuasiCollinear3 , which may be defined using the Sudakov decomposition

$\displaystyle p_{i}$ $\displaystyle=zp_{ij}+\alpha_{i}n+q_{\perp}$
$\displaystyle p_{j}$ $\displaystyle=(1-z)p_{ij}+\alpha_{j}n-q_{\perp}$ (11)

where $n$ is a lightlike reference vector usually taken to be the anti-
collinear direction, and $q_{\perp}$ is the spacelike transverse momentum with
respect to $p_{ij}$ and $n$. These momenta satisfy $q_{\perp}{\cdot}p_{ij}=0$
and $q_{\perp}{\cdot}n=0$. The variable $z$ is the collinear momentum
fraction, and the parameters $\alpha_{i}$ and $\alpha_{j}$ are fixed by the
conditions $p_{i}^{2}=m_{i}^{2}$ and $p_{j}^{2}=m_{j}^{2}$. The off-shellness
$Q^{2}$ is then given by

$\displaystyle Q^{2}$ $\displaystyle=(p_{i}+p_{j})^{2}-m^{2}_{ij}$
$\displaystyle=\frac{p_{\perp}^{2}}{z(1-z)}+\frac{m_{i}^{2}}{z}+\frac{m_{j}^{2}}{1-z}-m_{ij}^{2},$
(12)

where $p_{\perp}^{2}=-q_{\perp}^{2}$. The singular factorization of the
radiative amplitude occurs when this off-shellness becomes small with respect
to the energy scale of the process. In the massless case, the collinear region
is thus defined by $p_{\perp}\rightarrow 0$. This limit then generalizes to
the quasi-collinear limit in the massive case, defined by

$p_{\perp},m_{i},m_{j}\rightarrow 0\text{ with fixed ratios
}\frac{m_{i}}{p_{\perp}},\frac{m_{j}}{p_{\perp}}.$ (13)

In this limit, matrix elements symbolically factorize as

$|M_{n+1}|^{2}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
col.}}}{=}}\frac{1}{Q^{2}}P(\lambda_{ij},\lambda_{i},\lambda_{j},z)|M_{n}|^{2},$
(14)

where $P(\lambda_{ij},\lambda_{i},\lambda_{j},z)$ is the helicity-dependent
Altarelli-Parisi splitting kernel DGLAP1 ; DGLAP2 ; DGLAP3 .

The second singular limit is the soft limit, where

$E_{j}\rightarrow m_{j}\mbox{ and }E_{i}\gg E_{j}$ (15)

or equivalent with $p_{i}$ and $p_{j}$ switched, with the soft particle being
a gauge boson. The amplitude exhibits the usual eikonal factorization

$M_{n+1}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny soft}}}{=}}M_{n}\times
c\,\frac{2p_{i}{\cdot}\epsilon_{\lambda_{j}}}{Q^{2}}\delta_{\lambda_{ij}\lambda_{i}},$
(16)

where $c$ is some spin-dependent coupling.

Using the spinor-helicity formalism, we compute $1\rightarrow 2$ electroweak
branchings kernels that capture these soft and quasi-collinear factorization
properties. To that end, we compute a branching amplitude

$M^{\\{ij\\}\rightarrow
ij}(\lambda_{ij},\lambda_{i},\lambda_{j})\equiv\raisebox{-18.49411pt}{\includegraphics[width=96.73918pt]{figures/EW/newSplittingDiagram/split-
eps-converted-to.pdf}}$ (17)

for every possible $1\rightarrow 2$ branching in the electroweak sector. The
branching kernel is then given by

$\displaystyle B$
${}_{\lambda_{ij},\lambda_{i},\lambda_{j}}(p_{ij},p_{i},p_{j})=\frac{1}{Q^{4}}|M^{\\{ij\\}\rightarrow
ij}(\lambda_{ij},\lambda_{i},\lambda_{j})|^{2}.$ (18)

A comprehensive list of the branching amplitudes is given in B, which also
includes vector boson emission from initial state fermions. Here, we highlight
one of these calculations to discuss the treatment of artifacts that may
appear for longitudinal polarizations.

We consider vector boson emission from a fermion. The branching amplitudes are
straightforwardly found to be

$\displaystyle M$ ${}^{f\rightarrow
f^{\prime}V}(\lambda_{ij},\lambda_{i},\lambda_{j})$
$\displaystyle=\bar{u}_{\lambda_{i}}(p_{i})(v+a\gamma^{5})\not{\epsilon}_{\lambda_{j}}(p_{j})(\not{p}_{ij}+m_{ij})u_{-\lambda_{ij}}(k_{ij}),$
(19)

where $v$ and $a$ are the vector and axial couplings of the fermion and vector
boson at hand as defined in A. Note that this amplitude is applicable to all
types of electroweak vector boson emission off fermion branching processes,
including the cases where the fermion changes flavour due to a $W$-emission,
or where the vector boson is massless and the axial coupling drops out in case
of a photon. For massive vector bosons, inserting the corresponding
polarization vector leads to the appearance of terms proportional with

$\displaystyle\frac{1}{m_{j}}\not{p}_{i}\not{p}_{j}\not{p}_{ij}=\frac{1}{m_{j}}\left((Q^{2}+m_{ij}^{2})\not{p}_{i}-m_{i}^{2}\not{p}_{ij}\right).$
(20)

The term proportional to $Q^{2}$ cancels the propagator, leaving a
contribution that is not singular in the quasi-collinear or soft limit. We
note that similar situations may appear in equivalent calculations of QCD
branchings kernels. Nonsingular gauge-dependent terms may appear that do not
contribute to the logarithmic accuracy of a parton shower NLL1 ; NLL2  and may
be used as parameters for uncertainty estimation VinciaTimelike . In the case
of emission of electroweak gauge bosons, the situation is however not the
same. Due to the $\mathcal{O}(E/m)$ scaling of the longitudinal polarization,
the leftover contributions lead to unphysical unitarity-violation at large
energies.

Such problematic terms originate from the scalar component of the longitudinal
polarization of the electroweak gauge bosons, which originate from their
corresponding Goldstone bosons. In the above example, this Goldstone boson
couples to the fermion through its Yukawa, which in this case appears as its
kinematic off-shell mass. Due to the analytic nature of the spinor-helicity
formalism, the offending terms are straightforwardly identified and corrected.
Equivalently, the unitarity-violating term of eq. (20) cannot survive if all
other Feynman diagrams of a complete branching process are included, while all
other terms are associated with a propagator factor $1/Q^{2}$ which cannot be
cancelled elsewhere. In eq. (3) the unitarity-violating terms may thus be
removed by the replacement

$\not{p}_{i}\not{p}_{j}\not{p}_{ij}\rightarrow
m_{ij}^{2}\not{p}_{i}-m_{i}^{2}\not{p}_{ij}.$ (21)

The replacements required for all other branching amplitudes are described in
B.

In leading-colour QCD showers, soft gluon interference may be approximately
incorporated by angular ordering QCDcolliderPhysics , or by $2\rightarrow 3$
dipole-like functionsHerwigDipole ; SherpaShower ; DireShower  or antennae
functions Ariadne ; VinciaTimelike ; VinciaHadron1 . In the EW sector, no
leading-colour approximation exists and many eikonal terms may contribute to
the soft limit of an EW gauge boson emission. An algorithm that includes the
full multipole structure of QED radiation in the Vincia shower was described
in QEDantenna ; Multipole . The inclusion of the all soft interference effects
is accomplished by defining a $n\rightarrow n+1$ branching kernel and
sectorizing the phase space into multiple regions to regulate the intricate
soft and collinear singular structure. An extension of this algorithm requires
the computation of an equivalent spin-dependent $n\rightarrow n+1$ branching
kernel for soft-enhanced EW gauge boson radiation, as well as an
implementation that involves a similar sectorization of phase space. This is
beyond the scope of this paper and is left for future work.

##  4 Collinear Limits of the Branchings Amplitudes

In this section we discuss the behaviour of the branching amplitudes in the
quasi-collinear limit described by eq. (13). In this limit, the reference
vectors simplify to

$k_{i}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
col.}}}{=}}k_{j}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
col.}}}{=}}k_{ij}\equiv k.$ (22)

The branching amplitudes can be expressed in terms of the energy sharing
variable $z$ by replacing

$p_{i}\rightarrow z\,p_{ij}\mbox{ and }p_{j}\rightarrow(1-z)\,p_{ij}.$ (23)

The only two remaining spinor products in the branching amplitudes are related
by

$S_{-\lambda}(k,p_{j},p_{i},k)=-S_{-\lambda}(k,p_{i},p_{j},k).$ (24)

Up to a phase factor, they are

$\displaystyle S_{-\lambda}$ $\displaystyle(k,p_{i},p_{j},k)\propto$
$\displaystyle
2\sqrt{p_{i}{\cdot}k_{i}\,p_{j}{\cdot}k_{j}\vphantom{Q^{2}+m_{ij}^{2}-m_{i}^{2}\frac{p_{ij}{\cdot}k_{ij}}{p_{i}{\cdot}k_{i}}-m_{j}^{2}\frac{p_{ij}{\cdot}k_{ij}}{2p_{j}{\cdot}k_{j}}}}\sqrt{Q^{2}+m_{ij}^{2}-m_{i}^{2}\frac{p_{ij}{\cdot}k_{ij}}{p_{i}{\cdot}k_{i}}-m_{j}^{2}\frac{p_{ij}{\cdot}k_{ij}}{p_{j}{\cdot}k_{j}}}$
$\displaystyle\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
col.}}}{=}}2p_{ij}{\cdot}k\sqrt{z(1-z)\vphantom{\tilde{Q}^{2}}}\sqrt{\tilde{Q}^{2}}$
(25)

where

$\tilde{Q}^{2}=Q^{2}+m_{ij}^{2}-\frac{m_{j}^{2}}{1-z}-\frac{m_{i}^{2}}{z}.$
(26)

Tables 1-7 contain the collinear limits of all electroweak branching
amplitudes. These limits are related to the Altarelli-Parisi splitting kernels
by

$P(\lambda_{ij},\lambda_{i},\lambda_{j},z)=|M(\lambda_{ij},\lambda_{i},\lambda_{j})|^{2}$
(27)

where $M$ is the branching amplitude. Note that we used the replacement of eq.
(4) for the sake of notation, leading to a missing phase factor that is
irrelevant for the computation of the splitting kernels. The splitting
functions found here agree with the results of Tweedie .

$\lambda_{ij}$ | $\lambda_{i}$ | $\lambda_{j}$ | $f\rightarrow f^{\prime}V\mbox{ and }\bar{f}\rightarrow\bar{f}^{\prime}V$
---|---|---|---
$\lambda$ | $\lambda$ | $\lambda$ | $\sqrt{2}\lambda(v-\lambda a)\sqrt{\tilde{Q}^{2}}\frac{1}{\sqrt{1-z}}$
$\lambda$ | $\lambda$ | $-\lambda$ | $\sqrt{2}\lambda(v-\lambda a)\sqrt{\tilde{Q}^{2}}\frac{z}{\sqrt{1-z}}$
$\lambda$ | $-\lambda$ | $\lambda$ | $\sqrt{2}\lambda\bigg{[}m_{ij}(v-\lambda a)\sqrt{z}-m_{i}(v+\lambda a)\frac{1}{\sqrt{z}}\bigg{]}$
$\lambda$ | $-\lambda$ | $-\lambda$ | 0
$\lambda$ | $\lambda$ | $0$ | $\begin{aligned} &(v-\lambda a)\bigg{[}\frac{m_{ij}^{2}}{m_{j}}\sqrt{z}-\frac{m_{i}^{2}}{m_{j}}\frac{1}{\sqrt{z}}-2m_{j}\frac{\sqrt{z}}{1-z}\bigg{]}\\\ &+(v+\lambda a)\frac{m_{i}m_{ij}}{m_{j}}\frac{1-z}{\sqrt{z}}\end{aligned}$
$\lambda$ | $-\lambda$ | $0$ | $\sqrt{\tilde{Q}^{2}}\sqrt{1-z\vphantom{\tilde{Q^{2}}}}\bigg{[}\frac{m_{i}}{m_{j}}(v-\lambda a)-\frac{m_{ij}}{m_{j}}(v+\lambda a)\bigg{]}$
Table 1: Branching amplitudes for vector boson emission off a fermion. For the antifermion, the interchange $(v-\lambda a)\leftrightarrow(v+\lambda a)$ is applied. $\lambda_{I}$ | $\lambda_{i}$ | $\lambda_{j}$ | $V\rightarrow f\bar{f}^{\prime}$
---|---|---|---
$\lambda$ | $\lambda$ | $-\lambda$ | $\sqrt{2}\lambda(v-\lambda a)\sqrt{\tilde{Q}^{2}}\,z$
$\lambda$ | $-\lambda$ | $\lambda$ | $\sqrt{2}\lambda(v+\lambda a)\sqrt{\tilde{Q}^{2}}\,(1-z)$
$\lambda$ | $\lambda$ | $\lambda$ | $\sqrt{2}\lambda\bigg{[}m_{i}(v+\lambda a)\sqrt{\frac{1-z}{z}}+m_{j}(v-\lambda a)\sqrt{\frac{z}{1-z}}\bigg{]}$
$\lambda$ | $-\lambda$ | $-\lambda$ | 0
$0$ | $\lambda$ | $\lambda$ | $\sqrt{\tilde{Q}^{2}}\bigg{[}\frac{m_{i}}{m_{ij}}(v+\lambda a)+\frac{m_{j}}{m_{ij}}(v-\lambda a)\bigg{]}$
$0$ | $\lambda$ | $-\lambda$ | $\begin{aligned} &(v-\lambda a)\bigg{[}2m_{ij}\sqrt{z(1-z)}-\frac{m_{i}^{2}}{m_{ij}}\sqrt{\frac{1-z}{z}}\\\ &-\frac{m_{j}^{2}}{m_{ij}}\sqrt{\frac{z}{1-z}}\bigg{]}+(v+\lambda a)\frac{m_{i}m_{j}}{m_{ij}}\frac{1}{\sqrt{z(1-z)}}\end{aligned}$
Table 2: Branching amplitudes for vector boson splitting to fermions. $\lambda_{I}$ | $\lambda_{i}$ | $\lambda_{j}$ | $V\rightarrow V^{\prime}V^{\prime\prime}\times g_{V}$
---|---|---|---
$\lambda$ | $\lambda$ | $\lambda$ | $\sqrt{2}\lambda\sqrt{\tilde{Q}^{2}}\sqrt{\frac{1}{z(1-z)}\vphantom{\tilde{Q^{2}}}}$
$\lambda$ | $\lambda$ | $-\lambda$ | $\sqrt{2}\lambda\sqrt{\tilde{Q}^{2}}z\sqrt{\frac{z}{1-z}\vphantom{\tilde{Q^{2}}}}$
$\lambda$ | $-\lambda$ | $\lambda$ | $\sqrt{2}\lambda\sqrt{\tilde{Q}^{2}}(1-z)\sqrt{\frac{1-z}{z}\vphantom{\tilde{Q^{2}}}}$
$\lambda$ | $-\lambda$ | $-\lambda$ | $0$
$0$ | $\lambda$ | $\lambda$ | 0
$0$ | $\lambda$ | $-\lambda$ | $m_{ij}(2z-1)+\frac{m_{j}^{2}}{m_{ij}}-\frac{m_{i}^{2}}{m_{ij}}$
$\lambda$ | $0$ | $\lambda$ | $m_{i}\left(1+2\frac{1-z}{z}\right)+\frac{m_{j}^{2}}{m_{i}}-\frac{m_{ij}^{2}}{m_{i}}$
$\lambda$ | $0$ | $-\lambda$ | 0
$\lambda$ | $\lambda$ | $0$ | $m_{j}\left(1+2\frac{z}{1-z}\right)+\frac{m_{i}^{2}}{m_{j}}-\frac{m_{ij}^{2}}{m_{j}}$
$\lambda$ | $-\lambda$ | $0$ | 0
$\lambda$ | $0$ | $0$ | $\frac{\lambda}{\sqrt{2}}\frac{m_{i}^{2}+m_{j}^{2}-m_{ij}^{2}}{m_{i}m_{j}}\sqrt{\tilde{Q}^{2}}\sqrt{z(1-z)\vphantom{\tilde{Q^{2}}}}$
$0$ | $\lambda$ | $0$ | $\frac{\lambda}{\sqrt{2}}\frac{m_{ij}^{2}+m_{j}^{2}-m_{i}^{2}}{m_{ij}m_{j}}\sqrt{\tilde{Q}^{2}}\sqrt{\frac{1-z}{z}\vphantom{\tilde{Q^{2}}}}$
$0$ | $0$ | $\lambda$ | $\frac{\lambda}{\sqrt{2}}\frac{m_{ij}^{2}+m_{i}^{2}-m_{j}^{2}}{m_{ij}m_{i}}\sqrt{\tilde{Q}^{2}}\sqrt{\frac{z}{1-z}\vphantom{\tilde{Q^{2}}}}$
$0$ | $0$ | $0$ | $\begin{aligned} &\frac{1}{2}\frac{m_{ij}^{3}}{m_{i}m_{j}}(2z-1)-\frac{m_{i}^{3}}{m_{ij}m_{j}}\left(\frac{1}{2}+\frac{1-z}{z}\right)\\\ &+\frac{m_{j}^{3}}{m_{ij}m_{i}}\left(\frac{1}{2}+\frac{z}{1-z}\right)+\frac{m_{i}m_{j}}{m_{ij}}\left(\frac{1-z}{z}-\frac{z}{1-z}\right)\\\ &+\frac{m_{ij}m_{i}}{m_{j}}(1-z)\left(2+\frac{1-z}{z}\right)-\frac{m_{ij}m_{j}}{m_{i}}z\left(2+\frac{z}{1-z}\right)\end{aligned}$
Table 3: Branching amplitudes for vector boson emission off a vector boson. $\lambda_{I}$ | $\lambda_{i}$ | $(f\rightarrow fh\mbox{ and }\bar{f}\rightarrow\bar{f}h)\times\frac{e}{2s_{w}}\frac{m_{f}}{m_{w}}$
---|---|---
$\lambda$ | $\lambda$ | $m_{f}\Big{[}\sqrt{z}+\frac{1}{\sqrt{z}}\Big{]}$
$\lambda$ | $-\lambda$ | $\sqrt{1-z\vphantom{\tilde{Q^{2}}}}\sqrt{\tilde{Q}^{2}}$
Table 4: Branching amplitudes for Higgs emission off (anti)fermions. $\lambda_{i}$ | $\lambda_{j}$ | $h\rightarrow f\bar{f}\times\frac{e}{2s_{w}}\frac{m_{f}}{m_{w}}$
---|---|---
$\lambda$ | $\lambda$ | $\sqrt{\tilde{Q}^{2}}$
$\lambda$ | $-\lambda$ | $m_{f}\Big{[}\sqrt{\frac{1-z}{z}\vphantom{\tilde{Q^{2}}}}-\sqrt{\frac{z}{1-z}\vphantom{\tilde{Q^{2}}}}\Big{]}$
Table 5: Branching amplitudes for Higgs splitting to fermions. $\lambda_{I}$ | $\lambda_{i}$ | $V\rightarrow Vh\times g_{h}$
---|---|---
$\lambda$ | $\lambda$ | $-1$
$\lambda$ | $-\lambda$ | $0$
$0$ | $\lambda$ | $\frac{1}{m_{ij}}\frac{\lambda}{\sqrt{2}}\sqrt{\tilde{Q}^{2}}\sqrt{z(1-z)\vphantom{\tilde{Q^{2}}}}$
$\lambda$ | $0$ | $\frac{1}{m_{i}}\frac{\lambda}{\sqrt{2}}\sqrt{\tilde{Q}^{2}}\sqrt{\frac{1-z}{z}\vphantom{\tilde{Q^{2}}}}$
$0$ | $0$ | $\frac{1}{2}\frac{m_{j}^{2}}{m_{i}^{2}}+\frac{1-z}{z}+z$
Table 6: Branching amplitudes for Higgs emission off a vector boson. $\lambda_{i}$ | $\lambda_{i}$ | $h\rightarrow VV\times g_{V}$
---|---|---
$\lambda$ | $\lambda$ | $0$
$\lambda$ | $-\lambda$ | $-1$
$0$ | $\lambda$ | $\frac{1}{m_{i}}\frac{\lambda}{\sqrt{2}}\sqrt{\tilde{Q}^{2}}\sqrt{\frac{1-z}{z}\vphantom{\tilde{Q^{2}}}}$
$\lambda$ | $0$ | $\frac{1}{m_{j}}\frac{\lambda}{\sqrt{2}}\sqrt{\tilde{Q}^{2}}\sqrt{\frac{z}{1-z}\vphantom{\tilde{Q^{2}}}}$
$0$ | $0$ | $\frac{1}{2}\frac{m_{ij}^{2}}{m_{i}^{2}}-1-\frac{1-z}{z}-\frac{z}{1-z}$
Table 7: Branching amplitudes for Higgs splitting to vector bosons.

The quasi-collinear limits of the branching amplitudes listed in Tables 1-7
show a rich landscape of splitting modes that are the result of several
physical effects. For instance, in the case of vector boson emission off a
fermion, listed in Table 1, we find that the splitting functions for a
transverse emission without a spin flip correspond with the familliar spin-
summed form

$P_{f\rightarrow f^{\prime}V}\propto\frac{1+z^{2}}{1-z},$ (28)

in correspondence with the QCD equivalent of gluon emission off a quark. The
presence of fermion and vector boson masses induces a shift of
$1/Q^{2}\rightarrow\tilde{Q}^{2}/Q^{4}$ in the propagator structure. The
fermionic mass corrections in $\tilde{Q}^{2}$ also appear for gluon emission
and reproduce the mass contributions to the eikonal factor in the soft limit.
For $W$ and $Z$ emission, a vector boson mass correction is also present. In
case of a transverse emission with a spin flip, the splitting functions are
suppressed with a relative factor $m^{2}/Q^{2}$, where $m$ may the the vector
boson mass or either of the fermion masses. Again, similar mass corrections
appear in QCD HerwigSpin , and cause such modes to be suppressed at off-
shellness $Q^{2}$ much larger than the electroweak scale.

The splitting function for longitudinal vector boson emission with a spin flip
behaves like the scalar splitting function

$P_{f\rightarrow f\varphi}\propto(1-z).$ (29)

In the unbroken Standard Model, this splitting mode indeed corresponds with
the emission of a Goldstone boson, and the proportionality constant is given
by the Yukawa coupling to the Goldstone. The spin flip mode is in this case
not mass suppressed, reflecting the behaviour of scalar emissions in the
unbroken phase. On the other hand, the mode without spin flip is mass
suppressed, and terms that scale as $1/m_{j}$ appear, originating from the
scalar piece of the longitudinal polarization, as well as terms that scale as
$m_{j}$, corresponding with the vector piece.

##  5 The Electroweak Shower Implementation

We use the branching kernels computed in the previous section to implement an
electroweak shower in the Vincia framework, which was set out in VinciaSimple
; VinciaTimelike ; VinciaMassive ; VinciaHelicity1 ; VinciaHelicity2 ;
VinciaHadron1 ; VinciaHadron2 ; VinciaResonance . Here we first provide a
brief summary before continuing with a description of some details specific to
the electroweak shower.

Parton showers are constructed as a Markov chain of emissions that are
distributed according to an approximation to the radiative matrix element and
an associated Sudakov factor Review1 . In most modern showers, these
branchings are kinematically modelled as $2\rightarrow 3$ processes while the
dynamics vary according to the shower model. If both branching partons are in
the final state, momenta are labelled as $I,K\rightarrow i,j,k$, and the
Vincia shower approximation to the matrix element may be written as

$|M_{n+1}|^{2}d\Phi_{n+1}=|M_{n}|^{2}d\Phi_{n}\times
a(s_{ij},s_{jk})\,d\Phi_{\mbox{\tiny{ant}}},$ (30)

where

$d\Phi_{\mbox{\tiny{ant}}}^{\mbox{\tiny{FF}}}=\frac{1}{16\pi^{2}}m_{IK}^{2}\lambda^{-\frac{1}{2}}(m_{IK}^{2},m_{I}^{2},m_{K}^{2})ds_{ij}ds_{jk}\frac{d\varphi}{2\pi},$
(31)

where $s_{ij}=2p_{i}{\cdot}p_{j}$ and $\lambda$ is the Källén function. Eq.
(31) represents an exact factorization of the radiative phase space. An
associated kinematic map is defined between the pre-branching and post-
branching momenta that conserves total momentum and is soft- and collinear-
safe VinciaMassive . The branching kernels are so-called antenna functions
$a(s_{ij},s_{jk})$ that capture the leading collinear and soft singularities
associated with QCD emissions. The equivalent expressions to eq. (30) for
radiation from the initial state, as well as the definition of the kinematic
maps, may be found in VinciaHadron2 . In the electroweak sector, we instead
use the branching kernels computed in Section 3. These kernels contain the
correct singular behaviour in the soft-collinear and collinear limits, and are
simultaneously usable for the treatment of bosonic interference effects and
recoiler selection to be discussed below.

Vincia supports QCD evolution of partons with definite helicity
VinciaHelicity1 ; VinciaHelicity2 , making for a natural framework for the
inclusion of an electroweak shower. Interference effects between intermediate
spin configurations have previously been incorporated in parton showers, such
as in the Herwig Herwig++ ; Herwig7  parton shower HerwigSpin  and in Deductor
NagyInterference2 ; NagyInterference3 . The Vincia parton shower currently
makes no attempt to incorporate such interferences, and correspondingly the
same is true for the electroweak sector.

###  5.1 Ordering and Resonance Showers

The electroweak shower includes a number of branchings that would normally be
associated with the decay of resonances by Pythia Pythia6.4 , for which Vincia
is a plugin. In particular, the Standard Model particles that have such decay-
like branchings are $Z$, $W^{\pm}$, Higgs and top quark. With the inclusion of
an electroweak shower, the decay modes of the resonances are now also all
present as shower branchings. The shower enables highly-energetic resonances
to branch and disappear at scales much higher than their width, where they
should indeed be treated as any other non-resonant particle. At scales close
to the resonance width, the Breit-Wigner character of the resonance decay
should instead dominate the distribution.

Here, we set up a method of matching the parton shower distribution to a
Breit-Wigner smoothly. However, a full-fledged incorporation of resonance
decays at high scales in the Pythia event generator framework is a much more
involved issue. In the absence of EW shower branchings, Pythia always
associates the scale of a resonance decay with the width of the resonance. As
the widths of all SM resonances are close to $\Lambda_{\mbox{\tiny{QCD}}}$,
the approach of Pythia is to factorize the QCD shower of the hard scattering
from showers in resonance decay systems. When resonances are instead able to
decay at much larger scales, this approximation is no longer valid and a more
general solution is required.

Most generally, resonances should be allowed to decay at any scale during the
shower evolution of the hard scattering. Once a decay occurs, a factorized
shower in the resonance system may be performed using Vincia’s resonance
shower described in VinciaResonance . Afterwards, the showered decay products
may be joined with the rest of the hard system and the evolution can be
continued. As these modifications require substantal restructuring of both the
Pythia and Vincia framework, they will be treated separately in
UpcomingResonance .

Here, we do however proceed to set out the matching of the EW shower to the
Breit-Wigner shape through a simple sampling procedure. To that end, we must
first choose a suitable ordering scale. An issue specific to resonance
branchings is that there are regions of phase space where the off-shellness
$Q^{2}$ is negative. As more negative values of off-shellness should
correspond with shorter-lived resonances, we are led to define an ordering
scale

$|Q^{2}|=|s_{ij}+m_{i}^{2}+m_{j}^{2}-m_{ij}^{2}|,$ (32)

and equivalent for initial state branchings. For branchings that are not of
the resonant decay type, $Q^{2}$ is strictly positive and the ordering scale
corresponds to off-shellness ordering, which indeed regulates the soft-
collinear and collinear singularities.

Note that, even without the presence of EW masses, the choice of ordering
scale is by no means unique and many shower algorithm employ different
functional forms. The inclusion of EW scale masses leads to further
ambiguities, since the virtuality or transverse momentum of the branching is
not the only scale considered to be small in the quasi-collinear limit. In
PythiaEW  the effects of shifting the ordering scale to account for the
presence of gauge boson masses were found to lead to essentially identical
results, motivating eq. (32) as it leads to simpler matching to the Breit-
Wigner shape.

For most types of non-resonant electroweak branchings the phase space is
naturally cut off due to the masses of the post-branching momenta. Beyond
resonance branchings, photon emission is the only remaining branching that is
not cut off naturally. They are instead cut off at the same scale as the QCD
shower approximately equal to $\Lambda_{\mbox{\tiny{QCD}}}$, and QED radiation
at lower scales is included as is described in Multipole .

We set up the Breit-Wigner distribution by making the replacement

$\frac{1}{Q^{4}}\rightarrow\frac{1}{Q^{4}+m^{2}\Gamma^{2}},$ (33)

in the relevant branching kernels computed in Section 3, where $\Gamma$ is the
width of the resonance. These kernels are then normalized to represent a
probability distribution from which off-shellness scales, kinematics and post-
branching spin states may be sampled. Because the ordering scale of the shower
is given by eq. (32), it is straightforward to define a matching scale
$|Q^{2}_{\mbox{\tiny{Match}}}|$ where the shower is matched to the Breit-
Wigner. However, while it may be possible to pick the matching scale to ensure
the distribution is continuous, it will in general not be smooth. Furthermore,
the shape of the shower distribution at scales close to the resonance width
depends on the starting scale and the antenna mass, so such a choice would in
any case not always be continuous. We thus ensure the distribution is smooth
by sampling the value of the matching scale for every resonance from the
distribution

$\mathcal{P}(|Q^{2}_{\mbox{\tiny{Match}}}|)\propto\frac{m^{2}\Gamma^{2}|Q_{\mbox{\tiny{Match}}}^{2}|}{(Q_{\mbox{\tiny{Match}}}^{4}+m^{2}\Gamma^{2})^{2}}.$
(34)

The shower contribution to the off-shellness spectrum is then effectively
multiplied by a factor

$\bigintssss_{0}^{|Q^{2}|}\\!\\!\\!\\!\\!\\!\\!\\!d|Q^{2}_{\mbox{\tiny{Match}}}|\,\mathcal{P}(|Q^{2}_{\mbox{\tiny{Match}}}|)=\frac{Q^{4}}{Q^{4}+m^{2}\Gamma^{2}},$
(35)

which ensures a smooth suppression of the shower kernel. In this treatment,
the total distribution is dominated by the shower at hight scales, while the
Breit-Wigner dominates at low scales.

The implementation within the parton shower framework is relatively
straightforward. When a shower of a resonance is initiated, a matching scale
$|Q^{2}_{\mbox{\tiny{Match}}}|$ is sampled from eq. (34), serving as a local
cutoff scale. If during showering the branching scale drops below the matching
scale, a new off-shellness scale is instead drawn from the Breit-Wigner
distribution. We emphasize that this solution serves as an approximate means
of matching the shower to a Breit-Wigner, and a more sophisticated method that
closely matches that of Pythia will be developed in UpcomingResonance .

###  5.2 Recoiler Selection

While in the QCD portion of the Vincia shower the colour structure dictates
the pairings of branching partons $I$ and $K$, no such guidance exists in the
electroweak sector. Furthermore, the branching kernels only describe the soft-
collinear and collinear singularities associated with the branching of
particle $I$. The choice of recoiler $K$ does not contribute to the accuracy
of the shower in these limits. It may however be chosen probabilistically to
minimize the physical consequences of the recoil on previously generated
branchings. For the branching of particle $i$, the probability to select a
spectator $j$ from a pool of $N$ available ones is

$\mathcal{P}_{ij}=\frac{\left|M^{x\rightarrow
ij}\right|^{2}}{\sum_{j^{\prime}=1}^{N}\left|M^{x\rightarrow
ij^{\prime}}\right|^{2}}.$ (36)

That is, all candidate spectators $j$ have a probability to be assigned as a
recoiler for particle $i$ if the pair $i,j$ may have been produced by an
electroweak branching. All of the contributions in the denominator of eq. (36)
thus correspond with possible shower histories that contribute to the current
state. The selection of a spectator is then more likely if the shower history
where the current brancher and that spectator were created by a previous
branching.

We clarify the choice of eq. (36) through an example, which may be expressed
in terms of diagrams as

$\displaystyle\Bigg{\rvert}\raisebox{-14.22636pt}{\includegraphics[width=42.67912pt]{figures/EW/ariFacEquation/lhs-
eps-converted-to.pdf}}\Bigg{\rvert}^{2}$
$\displaystyle=\frac{\Big{\rvert}\raisebox{-8.5359pt}{\includegraphics[width=34.14322pt]{figures/EW/ariFacEquation/frac1-eps-
converted-
to.pdf}}\Big{\rvert}^{2}}{\Big{\rvert}\raisebox{-8.5359pt}{\includegraphics[width=34.14322pt]{figures/EW/ariFacEquation/frac1-eps-
converted-
to.pdf}}\Big{\rvert}^{2}+\Big{\rvert}\raisebox{-9.95863pt}{\includegraphics[width=39.83368pt]{figures/EW/ariFacEquation/frac2-eps-
converted-
to.pdf}}\Big{\rvert}^{2}}\Bigg{\rvert}\raisebox{-13.08846pt}{\includegraphics[width=45.52458pt]{figures/EW/ariFacEquation/recoil1-eps-
converted-to.pdf}}\Bigg{\rvert}^{2}$
$\displaystyle+\frac{\Big{\rvert}\raisebox{-9.95863pt}{\includegraphics[width=34.14322pt]{figures/EW/ariFacEquation/frac2-eps-
converted-
to.pdf}}\Big{\rvert}^{2}}{\Big{\rvert}\raisebox{-8.5359pt}{\includegraphics[width=34.14322pt]{figures/EW/ariFacEquation/frac1-eps-
converted-
to.pdf}}\Big{\rvert}^{2}+\Big{\rvert}\raisebox{-9.95863pt}{\includegraphics[width=39.83368pt]{figures/EW/ariFacEquation/frac2-eps-
converted-
to.pdf}}\Big{\rvert}^{2}}\Bigg{\rvert}\raisebox{-17.64056pt}{\includegraphics[width=45.52458pt]{figures/EW/ariFacEquation/recoil2-eps-
converted-to.pdf}}\Bigg{\rvert}^{2}.$ (37)

The spectator for the splitting of the vector boson is chosen to be either of
the other external legs based on the probabilities that the vector boson was
emitted by either of those legs. When the vector boson splits, it is brought
off its mass shell by transferring some momentum of the spectator to the
vector boson. Because the vector boson momentum is modified, the emission
kernel that it was produced with is no longer correct. In the strong-ordering
phase space region where $Q^{2}_{\mbox{\tiny{emit}}}\gg
Q^{2}_{\mbox{\tiny{split}}}$ this type of mismodelling is absent. However, the
shower covers all of phase space, including regions where the scales
$Q^{2}_{\mbox{\tiny{emit}}}>Q^{2}_{\mbox{\tiny{split}}}$ are of comparable
size. Recoil effects of previous brancings may be especially relevant for
branchings that involve masses of the order of the electroweak scale.

The Vincia $2\rightarrow 3$ kinematic map conserves the invariant mass of the
original two-particle system. This means that the probability of eq. (36)
ensures the propagator structure of the emitter and vector boson pair that was
most important in the emission process is most often conserved. Note that a
similar procedure has in the past been used to select a recoiler for gluon
splitting Ariadne ; VinciaMassive , which also features only collinear
singularities. In Tweedie  this effect is referred to as ‘kinematic back-
reactions‘ and is accounted for as a multiplicative factor of the branching
kernels.

###  5.3 Bosonic Interference

A unique type of interference effect appears in the electroweak sector due to
the existence of multiple neutrally charged bosons bosonicInterference . It is
possible to treat such interference effects comprehensively by evolving
density matrices that contain mixed states of neutral bosons Tweedie , but
such procedures quickly become computationally prohibitive. We instead opt for
a simpler approach that incorporates the most imporant physical effects while
preventing the shower to become a bottleneck in the Monte Carlo event
generation chain Computation .

Interference between neutral bosons occurs when the electroweak shower
produces a neutral boson, and it subsequently disappears by splitting. The
heavy neutral bosons will always split due to the matching to a Breit-Wigner
distribution described in Section 5.1, but photons may survive the showering
procedure. As such, interference effects are corrected for by applying an
event weight

$w_{\mbox{\tiny{BI}}}=\sum_{x}\frac{|M^{x\rightarrow
x^{\prime}b_{1}}M^{b_{1}\rightarrow ij}+M^{x\rightarrow
x^{\prime}b_{2}}M^{b_{2}\rightarrow ij}|^{2}}{|M^{x\rightarrow
x^{\prime}b_{1}}M^{b_{1}\rightarrow ij}|^{2}+|M^{x\rightarrow
x^{\prime}b_{2}}M^{b_{2}\rightarrow ij}|^{2}}$ (38)

after the splitting, where we have dropped the helicity indices for
readability. Eq. (38) corrects the branching kernels for the interference
between two bosons $b_{1}$ and $b_{2}$. These bosons may be either a photon
and a transversely polarized $Z$ boson, or a Higgs boson and a longitudinally
polarized $Z$ boson. Interference between spin states is not taken into
account in correspondence with the rest of the shower algorithm. The weight of
eq. (38) then sums over all possible emitters $x$ of the bosons $b_{1}$ and
$b_{2}$, accounting for all possible shower histories. This includes the
different spin states of the same particle, but note that these contributions
are summed over incoherently. The weight has the property $0\leq
w_{\mbox{\tiny{BI}}}\leq 2$, and since the overal rate of electroweak boson
emissions is moderate even at high energies, there is little danger of wildly
fluctuating weights leading to inefficiencies.

###  5.4 Overestimate Determination

The implementation of electroweak radiation in the shower formalism through
the Sudakov veto algorithm Fooling ; Reloaded ; Competing  requires finding
overestimates of the associated branching kernels. Due to the large number of
types of branchings in the electroweak sector, it is desirable to automate
this procedure.

For final-state particles with an electroweak charge, a recoiler is selected
through the procedure outlined in Section 5.2. The branching phase space is
then given by eq. (31). All final-state electroweak branching kernels are
overestimated by a parameterized function

$\displaystyle\mathcal{O}^{\mbox{\tiny{FF}}}$
$\displaystyle=c_{1}^{\mbox{\tiny{FF}}}\frac{1}{|Q^{2}|}+c_{2}^{\mbox{\tiny{FF}}}\frac{1}{|Q^{2}|}\frac{E_{IK}(E_{IK}+|\vv{p}_{IK}|)}{s_{ij}+s_{ik}+m_{i}^{2}}$
$\displaystyle+c_{3}^{\mbox{\tiny{FF}}}\frac{1}{|Q^{2}|}\frac{E_{IK}(E_{IK}+|\vv{p}_{IK}|)}{s_{ij}+s_{jk}+m_{j}^{2}}+c_{4}^{\mbox{\tiny{FF}}}\frac{m_{I}^{2}}{Q^{4}},$
(39)

where $|Q^{2}|$ is given by eq. (32). The term multiplying
$c_{1}^{\mbox{\tiny{FF}}}$ reflects the $1/|Q^{2}|$ behaviour of most
branching kernels, while the second and third terms incorporate the soft
behaviour associated with vector boson emission, containing ratios
$E_{IK}/E_{i}\sim 1/z$ and $E_{IK}/E_{j}\sim 1/(1-z)$ in terms of shower
variables. The term multiplying $c_{4}^{\mbox{\tiny{FF}}}$ represents the mass
corrections that may be present for massive branchers. The contribution of
post-branching masses are typically negative, and therefore do not improve the
overestimate much.

Initial-state branchings are only allowed to recoil against other initial
states. In this case, the antenna phase space is

$d\Phi_{\mbox{\tiny{ant}}}^{\mbox{\tiny{II}}}=\frac{1}{16\pi^{2}}\frac{x_{A}^{2}}{x_{a}^{2}}\frac{x_{B}^{2}}{x_{b}^{2}}\frac{1}{s_{AB}}ds_{aj}ds_{bj}\frac{d\varphi}{2\pi},$
(40)

where $A$ branches to $a$ and $j$, and $B$ is the recoiler. The electroweak
shower currently only implements vector boson emission from fermions in the
initial state, which are treated as massless by Vincia. The ordering scale is
crossed into the initial state to give

$Q^{2}=s_{aj}-m_{j}^{2}.$ (41)

An absolute value qualification is not required here since resonance type
branchings do not occur in the initial state. A sufficient overestimate is

$\displaystyle\mathcal{O}^{\mbox{\tiny{II}}}$
$\displaystyle=c_{1}^{\mbox{\tiny{II}}}\frac{1}{Q^{2}}\frac{s_{ab}}{s_{AB}}$
$\displaystyle+c_{2}^{\mbox{\tiny{II}}}\frac{1}{Q^{2}}\frac{x_{A}^{2}s_{ab}^{2}}{x_{A}s_{bj}(s_{ab}-s_{bj})+x_{B}s_{aj}(s_{ab}-s_{aj})}.$
(42)

The factor $s_{ab}/s_{AB}$ accounts for the additional factor of $1/z$ that
shows up in the Altarelli-Parisi splitting kernels when crossed to the initial
state. The second term represents the soft enhancement $1/(1-z)$ that may
appear for vector boson emissions.

The parameters $c_{1}^{\mbox{\tiny{FF}}}$ through $c_{4}^{\mbox{\tiny{FF}}}$,
$c_{1}^{\mbox{\tiny{II}}}$ and $c_{2}^{\mbox{\tiny{II}}}$ are automatically
determined for all possible branchings in the electroweak shower. To do that,
brancher-recoiler pairs are generated from antennae with randomly chosen
invariant masses. Branchings are then generated with a distribution
$1/|Q^{2}|$ for the final state or $s_{ab}/s_{AB}\,1/Q^{2}$ for the initial
state to roughly model the branching kernel behaviour. For every event $i$,
the value of the branching kernel $B_{i}$ as well as the terms $A_{ij}$
multiplying the parameters $c_{j}$ are stored. The problem of finding suitable
values for the overestimate parameters can then be formulated as

$\displaystyle\mbox{Minimize }\sum_{i=1}^{n}(A\mathbf{c})_{i}-B_{i}$
$\displaystyle\mbox{subject to }(A\mathbf{c})_{i}\geq B_{i}\mbox{ and
}\mathbf{c}\geq 0.$ (43)

The minimization condition minimizes the average difference between the
branching kernel and its overestimate. The constraints ensure the overestimate
is larger than the branching kernel for all samples and the parameters are
positive definite. Eq. (5.4) is an instance of a linear programming problem,
for which many libraries are available. We make use of the Python Python
package PuLP PuLP .

###  5.5 Overview of the Shower Algorithm

We conclude this section with a short description of the complete shower
algorithm. Branching kernels are constructed using the formalism described in
section 3 for all possible electroweak branchings and all helicity
configurations. Overestimates are found using the optimization algorithm of
subsection 5.4, where the post-branching helicities are summed over. This
leaves a total of $277$ types of final-state branchings, of which $74$ are
resonance decays, and $90$ types of initial-state branchings.

As the shower initializes, a recoiler is selected for all final-state
particles that have electroweak charge, making use of the selection
probability described in subsection 5.2. Initial-state branchers are always
paired with the other initial-state particle, recoiling against the entire
event.

While the shower runs, electroweak branchings compete against the QCD
branchings generated by Vincia. The overestimates are used to generate trial
branchings which are accepted or rejected through the usual Sudakov veto
algorithm. For resonance decay branchings, the procedure outlined in
subsection 5.1 is used to match the shower to a Breit-Wigner distribution. We
make use of the same kinematic maps as Vincia, and first-order running of the
electroweak coupling constant is incorporated as part of the veto procedure.
Wherever applicable, the bosonic interference weight eq. (38) is included.
After accepting a branching, a helicity state is selected with probability

$\mathcal{P}_{\lambda_{I},\lambda_{i},\lambda_{j}}=\frac{B_{\lambda_{I},\lambda_{i},\lambda_{j}}(p_{I},p_{i},p_{j})}{\sum_{\lambda_{i},\lambda_{j}}B_{\lambda_{I},\lambda_{i},\lambda_{j}}(p_{I},p_{i},p_{j})},$
(44)

and equivalent for resonance decay branchings. The QCD shower and the
electroweak shower run interleaved until the QCD cutoff scale is reached,
after which only QED radiation is simulated.

##  6 Results

The inclusion of electroweak radiation in parton showers opens up a rich field
of showering phenomena, in particular at energies well above the electroweak
scale. Rather than considering specific sets of observables for specific
processes, in this section we first consider radiation spectra of several
particles at energies compatible with future colliders, showing overal
branching rates of the electroweak shower, the relative importance of the
large number of emission modes and the effects of bosonic interference. We
conclude by showing results for hadron collision processes at LHC energies and
comparing with the Pythia electroweak shower PythiaEW .

###  6.1 Branching Spectra

In this section, we show electroweak branching spectra of several highly
energetic particles as a function of the invariant mass, which closely
corresponds with the ordering scale. Figures 1, 2, 3 and 4 show this for the
first branching of a left-handed $\tau$ and top, a transverse $W^{+}$ boson
and a Higgs. Note that multiple emission rates are related to these single
branching spectra through the usual generalized Poisson statistics associated
with shower algorithms due to the multiplicative property of the Sudakov
factor 111Although the equivalence is not exact due to recoiler effects and
the favour-changing nature of many electroweak branchings.. All particles are
produced at an energy of $1$ TeV together with a recoiler that is uncharged
under electroweak interactions. For photon emissions, a cutoff around
$\Lambda_{\mbox{\tiny{QCD}}}$ is imposed. All other branchings are
automatically regulated by the particle masses or by matching to the Breit-
Wigner distribution.

Figure 1: Branching spectra of a $1$ TeV $\tau_{-}$ to $\tau\gamma$ (left) and
$\tau Z$ and $\nu_{\tau}W^{-}$ (right).

Figure 1 shows the branching spectrum of a negative-helicity $\tau$. The two
dominant photon production channels are those where the $\tau$ helicity is
conserved. The mass-suppressed spin-flip mode only contributes at very small
invariant masses, as is to be expected from the branching kernel behaviour of
$m_{\tau}^{2}/Q^{4}$. The other spin-flip mode is highly suppressed in the
collinear limit as is indicated in Table 1. For the emission of other vector
bosons, the spin-flip contributions do not become sufficiently enhanced to
show up before the kinematic limit is reached. The longitudinal vector boson
emission channels have a characteristic form which looks very similar for the
$W^{-}_{0}$ and the $Z_{0}$ channels, and which becomes comparable to the
transverse channels at scales close to the kinematic limit.

Figure 2: Branching spectra of a $1$ TeV top quark to $bW^{+}$ (left), $tZ$
(right) and $t\gamma/h$ (bottom).

Figure 2 shows the branching spectrum of a negative-helicity top. The left
graph displays the resonance branchings as generated by the sampled matching
procedure outlined in subsection 5.1. The right and bottom graphs show all
other branchings that are not of the resonance decay type. Spin-flip modes now
show up for $t\rightarrow bW^{+}$, $t\rightarrow tZ$ and $t\rightarrow
t\gamma$ due to the large top mass, and they show the expected
$m_{t}^{2}/Q^{4}$ scaling with the emission scale. The ‘natural‘ mode of spin-
flip Higgs emission is relatively flat compared with the fermion mass scaling
mode of Higgs emission without spin flip.

Figure 3: Branching spectra of a $1$ TeV $W^{+}_{+}$ to fermions (left),
$W^{+}Z$ (right) and $W^{+}\gamma/h$ (bottom).

Figure 3 shows the branching spectrum of a transverse $W^{+}$. Resonance peaks
only appear for decays to negative-helicity states due to their small masses.
The branchings $W^{+}_{+}\rightarrow t\bar{b}$ with a spin-flipped top do
occur on the other hand. The $W^{+}_{+}\rightarrow W^{+}Z$ and
$W^{+}_{-}\rightarrow W^{+}\gamma$ channels are dominated by the all-positive
helicity configuration because of its $1/z(1-z)$ scaling in the collinear
limit as can be seen in Table 3. The modes to opposite transverse helicities
are almost identical for the $W^{+}Z$ channels due to symmetry in the
collinear limit and almost identical mass, but they are widely different for
low scales in the $W^{+}\gamma$ channels. This is caused by the $z^{3}/(1-z)$
and $(1-z)^{3}/z$ scaling of the collinear limits, where the photon can attain
a very small collinear momentum fraction while that of the $W^{+}$ is
constrained by its mass. The single-longitudinal channels in $W^{+}Z$ are also
almost identical for very similar reasons. The $W^{+}_{0}Z_{0}$ is a mode that
is related to the Goldstone bosonic part of the $W^{+}$ and $Z$, and it can be
seen to be very similar to the $W_{0}^{+}h$ channel. On the other hand, the
$W^{+}_{+}h$ mode differs significantly from the $W^{+}_{+}Z_{0}$ channel
because it is dominated by the vectorial part of the longitudinal
polarization.

Figure 4: Branching spectra of a $1$ TeV Higgs boson to fermions (left) and
$VV/hh$ (right).

Figure 4 shows the branching spectrum of a Higgs. The only significant
resonance decay channels are $b_{\pm}\bar{b}_{\pm}$ and
$\tau_{\pm}\bar{\tau}_{\pm}$ as may be expected due to the coupling to the
fermion mass and the Higgs spin zero nature. On the other hand, the mass-
suppressed $t_{\pm}\bar{t}_{\mp}$ channel is comparable with the natural
$t_{\pm}\bar{t}_{\pm}$ channel. All channels to $W^{+}W^{-}$ and $ZZ$ are
almost identical since their branching kernels only differ in the gauge boson
mass and a factor of $1/c_{w}$ in the coupling. Also included is the
$h\rightarrow hh$ cubic Higgs coupling which is proportional to the Higgs mass
$m_{h}$, or equivalently the Higgs self-coupling $\lambda$. This is the only
branching where it makes an appearance, and it can be seen to provide a
significant contribution to the total branching rate.

###  6.2 Bosonic Interference

Figure 5: Differential rates of the shower histories $e_{-}\rightarrow
e_{-}\gamma/Z_{T}\rightarrow e_{-}X$ (top) and $e_{+}\rightarrow
e_{+}\gamma/Z_{T}\rightarrow e_{+}X$ (bottom) without (left) and with (right)
the bosonic interference correction. The showers are initiated from a $10$ TeV
electron with a neutral recoiler.

We now consider the effect of the application of the bosonic interference
factor described in section 5.3. Figure 5 shows rates for the shower histories
$e_{-}\rightarrow e_{-}\gamma/Z_{T}\rightarrow e_{-}X$ and $e_{+}\rightarrow
e_{+}\gamma/Z_{T}\rightarrow e_{+}X$ using a similar setup as in the previous
subsection, but starting from a $10$ TeV source electron. Multiple interesting
features appear when the bosonic interference weight eq. (38) is included. The
most striking difference occurs for the $W^{+}W^{-}$ channel, where the
bosonic interference causes an increase in case of the $e_{-}$, but a major
decrease in case of the $e_{+}$. This may be understood by considering the
structure of the interfering branching amplitudes. Factoring out coupling
constants and other kinematic components, the interference is proportional to

$\frac{1}{M_{WW}^{2}}+\frac{c_{w}}{s_{w}}\frac{1}{4s_{w}c_{w}}(1-4s_{w}^{2}-\lambda_{e})\frac{1}{M_{WW}^{2}-m_{z}^{2}+im_{z}\Gamma_{z}},$
(45)

where the factor $c_{w}/s_{w}$ comes from the $ZWW$-coupling and $\lambda_{e}$
is the electron helicity. The second term in brackets interferes destructively
with the photon contribution for sufficiently large values of $M_{WW}$, and
the remaining terms in the $Z$ contribution cancel for $\lambda=1$.

The effects of the bosonic interference factor on charged fermion rates close
to the $Z$ peak may be understood through a similar argument. The rates close
to the $Z$ peak are significantly affected by the simplified and preliminary
method of matching to resonance decays as described in section 5.1, and will
be improved upon in UpcomingResonance .

###  6.3 Electroweak Corrections to Proton Collision Processes

We finally consider the parton shower predictions of electroweak corrections
to some common proton collision processes at LHC energies and compare with the
Pythia electroweak shower PythiaEW . Since the weak vector bosons produced by
the electroweak shower at high energies are massive and thus observable, they
may provide a rich environment for phenomenological studies including
kinematic effects on the hard scattering, jet substructure due to vector boson
decay inside jet cones and external high-energy jet and lepton production.

Figure 6: Electroweak shower approximation of electroweak virtual corrections
to exclusive dijet production and exclusive $W^{+}$ \+ jet production at
center-of-mass energy $\sqrt{s}=14$ TeV as a function of the transverse
momentum of the hard scattering process $p_{\perp}^{\mbox{\tiny{Hard}}}$. The
solid line corresponds to the parton shower prediction without the QCD shower,
while the dashed line shows the effect of interleaving with QCD radiation.

With the goal of examining the general significance of electroweak Sudakov
effects in common LHC processes, we generate dijet and $W^{+}$ plus jet events
at $\sqrt{s}=14$ TeV using the default tune of Pythia 8.2 Pythia8.2  and the
NNPDF2.3 sets NNPDF2.3 . Figure 6 shows the approximate electroweak virtual
corrections as predicted by the Pythia electroweak shower, which only
incorporates vector boson emission from fermions, and the Vincia electroweak
shower as a function of the transverse momentum of the hard scattering. The
virtual corrections may be estimated by counting the events that contain at
least one weak vector boson emission. The probability for the shower to
produce no additional weak bosons is given by the Sudakov factor

$\Delta_{\mbox{\tiny{EW}}}=1-\mathcal{O}(\alpha),$ (46)

and thus the $\mathcal{O}(\alpha)$ corrections are given by the probability
for at least one weak boson emission. Virtual corrections to these processes
were calculated in for example diJetLarge1 ; diJetLarge2  for exclusive dijet
production and in VjLarge2 ; VjLarge3  for vector boson production.

For dijet production, the results of the showers are very similar. In the case
of $W^{+}$ plus jet production the substantial difference between the showers
is caused by the absence of the Yang-Mills vector boson coupling in the Pythia
shower. Furthermore, we find that the contribution to the weak boson emission
rates of the initial-state quarks is significantly smaller than that of the
final state. Because at large $x$ and high scales the PDFs are predominantly
quark-like and the hard scattering is dominated by
$q\bar{q}^{\prime}\rightarrow W^{+}g$, and the phase space for initial state
radiation is small at large $x$, virtual corrections decrease at large values
of transverse momentum.

Also shown in Figure 6 are the results of interleaving the electroweak shower
with the QCD showers of Pythia and Vincia. In the strongly ordered limit
shower branchings are unaffected by subsequent branchings, but subleading
effects due to the kinematics and the creation of weakly charged quarks still
lead to minor differences.

Figure 7: Average number of weak boson emissions in exclusive dijet production
and exclusive $W^{+}$ \+ jet production at center-of-mass energy $\sqrt{s}=14$
TeV as a function of the transverse momentum of the hard scattering process
$p_{\perp}^{\mbox{\tiny{Hard}}}$.

Figure 7 shows the average number of weak boson emissions of the showers. In
$W^{+}$ plus jet production the first purely vector boson branching is always
$W^{+}\rightarrow W^{+}Z$ explaining the large increase in the $Z$ boson
emission rates. Similarly, Pythia’s $W^{+}$ rate is small since the final-
state quark is always down-type. The increase in the Vincia shower is thus
caused entirely by secondary emissions from prior weak vector boson emissions.

##  7 Conclusion and Discussion

The effects of weak corrections in parton shower algorithms are known to
become significant already at LHC energies, in particular with the upcoming
luminosity upgrade, and will be even more relevant at future colliders. One of
the major challenges of the construction of such a shower is the calculation
of the relevant branching kernels, which in this paper was done using the
spinor-helicity formalism. Compared with QCD, the electroweak theory involves
many theoretical subtleties that have to be handled carefully. One major issue
is the chiral nature of the electroweak theory, which forces the shower to be
helicity-dependent and leads to a large number of possible types of
branchings. In particular, the scalar components of longitudinal polarizations
lead to unphysical, unitarity-violating contributions that have to be treated
carefully. The collinear limits of the computed branching kernels are found to
be in agreement with the results of Tweedie . The electroweak shower also
includes many branchings that would usually be considered to be decays of
resonances, in which case the distribution follows a Breit-Wigner peak. A
strategy to match the parton shower to a resonance decay was proposed, but
this may likely be improved upon by a better understanding of the interplay
between the virtual corrections contained in the Sudakov factor and the decay
width. A more sophisticated treatment of this matching is beyond the scope of
this paper, and will be the topic of future study UpcomingResonance . Further
electroweak effects added to the shower include a recoiler selection procedure
that compensated for recoiler effects of previous branchings and treatment of
bosonic interference effects. Results were shown that quantify the general
size of electroweak shower corrections at future collider energies and at LHC
energies.

Several features that are currently lacking from the electroweak shower were
already pointed out. They include topics such as soft and spin interference
effects, although such issues have similarly not been fully solved in the QCD
sector of commonly used shower codes. However, further issues particular to
the electroweak sector still remain. One is the inclusion of the CKM quark-
mixing matrix CKM1 ; CKM2 , which would lead to an even larger number of
possible electroweak branchings. The impact of the CKM matrix is however not
expected to be large since the off-diagonal terms of the third-generation row
and column are close to zero. The other off-diagonal terms are not as small,
but they mix quarks that are treated as massless in Vincia anyway.

One other peculiar property of the electroweak theory is the appearance of
Bloch-Nordsieck violation Bloch-Nordsieck1 ; EWLarge6 . The parton shower
formalism is fundamentally based on the principle of unitarity and the
cancellation of infrared divergences between real and virtual corrections.
Since the electroweak vector bosons are massive, divergences associated with
their emission are mass-regulated. The flavour-changing nature of $W$-boson
emission from the initial state spoils the exact cancellation of the infrared
divergences and some mass-regulated logarithms may be left-over.

There is no straightforward method to incorporate these violations in the
shower formalism, since they explicitly break unitarity. We note that, while
Bloch-Nordsieck violations are not particularly significant at the LHC
EWLarge6 ; diJetLarge2 , they will be important at future collider energies
and a comprehensive treatment will be necessary.

Finally, hard processes initiated by vector bosons have been considered for a
long time EffectiveVBA1 ; EffectiveVBA2 . PDF sets with QED corrections have
been available for some time photonPdfs1 ; photonPdfs2 ; photonPdfs3 ;
photonPdfs4 , and recent progress was made towards PDFs with complete
electroweak corrections EWPdfs1 ; EWPdfs2 ; EWPdfs3 . The current shower
implementation only allows for the emission of vector bosons from the initial
state. The calculation of the other required initial-state branching kernels
is in principle as straightforward as the calculation of those available
already, but an implementation in the Pythia framework is likely not simple.

###### Acknowledgements.

We are grateful to Helen Brooks and Peter Skands for many useful discussions
and help with the implementation in the Vincia parton shower. R. V.
acknowledges support by the Foundation for Fundamental Research of Matter
(FOM) via program 156 ’Higgs as Probe and Portal’ and by the Science and
Technology Facilities Council (STFC) via grant award ST/P000274/1.

##  Appendix A Relevant Feynman Rules of the Electroweak Theory

$\displaystyle\raisebox{-31.2982pt}{\includegraphics[width=85.35826pt]{figures/EW/feynRules/ffV-
eps-converted-to.pdf}}=i(v+a\gamma^{5})\gamma^{\mu}$
$\displaystyle\raisebox{-31.2982pt}{\includegraphics[width=85.35826pt]{figures/EW/feynRules/ffh-
eps-converted-to.pdf}}=i\frac{e}{2s_{w}}\frac{m_{f}}{m_{W}}$
$\displaystyle\raisebox{-31.2982pt}{\includegraphics[width=85.35826pt]{figures/EW/feynRules/hVV-
eps-converted-to.pdf}}=ig_{h}g^{\mu\nu}$
$\displaystyle\raisebox{-31.2982pt}{\includegraphics[width=85.35826pt]{figures/EW/feynRules/hhh-
eps-converted-to.pdf}}=-i\frac{3}{2}\frac{m_{h}^{2}}{m_{w}s_{w}}.$
$\displaystyle\raisebox{-31.2982pt}{\includegraphics[width=91.04872pt]{figures/EW/feynRules/VVV-
eps-converted-to.pdf}}=ig_{V}Y(p_{1},\mu,p_{2},\nu,p_{3},\alpha)$ Figure 8:
The vertex interactions of the electroweak theory.

This appendix lists the Feynman rules of the electroweak theory that are
relevant for the calculation of branching amplitudes. We elect to make use of
a practical notation for the electroweak Feynman rules which makes for simpler
results, but obfuscates some of the underlying group structure. The relevant
vertex interactions are given in Figure 8, where

$\displaystyle Y(p_{1},\mu,p_{2},\nu,p_{3},\alpha)$
$\displaystyle=(p_{1}-p_{2})^{\alpha}g^{\mu\nu}+(p_{2}-p_{3})^{\mu}g^{\nu\alpha}$
$\displaystyle+(p_{3}-p_{1})^{\nu}g^{\mu\alpha}$ (47)

is the Yang-Mills vertex. As usual, the weak mixing angle is defined as

$c_{w}\equiv\cos\theta_{w}=\frac{m_{W}}{m_{Z}}\quad
s_{w}\equiv\sin\theta_{w}.$ (48)

The coupling constants are defined in Table 8.

| $v$ | $a$ |  | $g_{V}$
---|---|---|---|---
| $\gamma$ | $W$ | $Z$ | $\gamma$ | $W$ | $Z$ |  | $WW\gamma$ | $-e$
$d$ | $-\frac{1}{3}e$ | $\frac{-e}{\sqrt{8}s_{w}}$ | $\frac{-e}{4s_{w}c_{w}}\left(1-\frac{4}{3}s_{w}^{2}\right)$ | $0$ | $\frac{-e}{\sqrt{8}s_{w}}$ | $\frac{-e}{4s_{w}c_{w}}$ |  | $WWZ$ | $-e\frac{c_{w}}{s_{w}}$
$u$ | $\frac{2}{3}e$ | $\frac{-e}{\sqrt{8}s_{w}}$ | $\frac{e}{4s_{w}c_{w}}\left(1-\frac{8}{3}s_{w}^{2}\right)$ | $0$ | $\frac{-e}{\sqrt{8}s_{w}}$ | $\frac{e}{4s_{w}c_{w}}$ |  |  | $g_{h}$
$e$ | $-e$ | $\frac{-e}{\sqrt{8}s_{w}}$ | $\frac{-e}{4s_{w}c_{w}}\left(1-4s_{w}^{2}\right)$ | $0$ | $\frac{-e}{\sqrt{8}s_{w}}$ | $\frac{-e}{4s_{w}c_{w}}$ |  | $hWW$ | $-e\frac{m_{w}}{s_{w}}$
$\nu$ | $0$ | $\frac{-e}{\sqrt{8}s_{w}}$ | $\frac{e}{4s_{w}c_{w}}$ | $0$ | $\frac{-e}{\sqrt{8}s_{w}}$ | $\frac{e}{4s_{w}c_{w}}$ |  | $hZZ$ | $-e\frac{m_{w}}{s_{w}}$

Table 8: Values of the coupling constants

##  Appendix B Branching amplitudes

All branching amplitudes are multiplied by a propagator factor $1/Q^{2}$ where

$\displaystyle
Q^{2}=\begin{cases}2p_{i}{\cdot}p_{j}+m_{i}^{2}+m_{j}^{2}-m_{ij}^{2}&\text{(Final
State)}\\\ 2p_{a}{\cdot}p_{j}-m_{j}^{2}&\text{(Initial State)}.\end{cases}$
(49)

###  B.1 Vector Boson Emission

We define the prefactors

$\displaystyle A^{\mbox{\tiny{emit}}}_{\perp}$
$\displaystyle=\frac{1}{2\sqrt{2}}\frac{\lambda}{\sqrt{p_{i}{\cdot}k_{i}}\sqrt{p_{ij}{\cdot}k_{ij}}\,p_{j}{\cdot}k_{j}}$
$\displaystyle A^{\mbox{\tiny{emit}}}_{L}$
$\displaystyle=\frac{1}{2}\frac{1}{m_{j}}\frac{1}{\sqrt{p_{i}{\cdot}k_{i}}\sqrt{p_{ij}{\cdot}k_{ij}}}.$
(50)

####  B.1.1 Vector Boson Emission from Fermion

$\displaystyle M$ ${}^{f\rightarrow
f^{\prime}V}(\lambda,\lambda,\lambda)=A^{\mbox{\tiny{emit}}}_{\perp}$
$\displaystyle\times$ $\displaystyle\begin{aligned} &\bigg{[}(v-\lambda
a)S_{-\lambda}(k_{i},p_{i},p_{j},k_{j})S_{-\lambda}(k_{j},p_{ij},k_{ij})\\\
&+(v+\lambda
a)m_{i}m_{ij}S_{-\lambda}(k_{i},k_{j})S_{-\lambda}(k_{j},p_{j},k_{ij})\bigg{]}\end{aligned}$
$\displaystyle M$ ${}^{f\rightarrow
f^{\prime}V}(\lambda,\lambda,-\lambda)=A^{\mbox{\tiny{emit}}}_{\perp}$
$\displaystyle\times$ $\displaystyle\begin{aligned} &\bigg{[}(v-\lambda
a)S_{-\lambda}(k_{i},p_{i},k_{j})S_{-\lambda}(k_{j},p_{j},p_{ij},k_{ij})\\\
&+(v+\lambda
a)m_{i}m_{ij}S_{-\lambda}(k_{i},p_{j},k_{j})S_{-\lambda}(k_{j},k_{ij})\bigg{]}\\\
\end{aligned}$ $\displaystyle M$ ${}^{f\rightarrow
f^{\prime}V}(\lambda,-\lambda,\lambda)=A^{\mbox{\tiny{emit}}}_{\perp}$
$\displaystyle\times$ $\displaystyle\begin{aligned} &\bigg{[}m_{ij}(v+\lambda
a)S_{\lambda}(k_{i},p_{i},k_{j})S_{-\lambda}(k_{j},p_{j},k_{ij})\\\
&-m_{i}(v-\lambda
a)S_{\lambda}(k_{i},p_{j},k_{j})S_{-\lambda}(k_{j},p_{ij},k_{ij})\bigg{]}\end{aligned}$
$\displaystyle M$ ${}^{f\rightarrow
f^{\prime}V}(\lambda,-\lambda,-\lambda)=A^{\mbox{\tiny{emit}}}_{\perp}$
$\displaystyle\times$ $\displaystyle\begin{aligned} &\bigg{[}m_{ij}(v+\lambda
a)S_{\lambda}(k_{i},p_{i},p_{j},k_{j})S_{\lambda}(k_{j},k_{ij})\\\
&-m_{i}(v-\lambda
a)S_{\lambda}(k_{i},k_{j})S_{\lambda}(k_{j},p_{j},p_{ij},k_{ij})\bigg{]}\end{aligned}$
$\displaystyle M$ ${}^{f\rightarrow
f^{\prime}V}(\lambda,\lambda,0)=A^{\mbox{\tiny{emit}}}_{L}$
$\displaystyle\times$ $\displaystyle\begin{aligned}
&\bigg{[}S_{-\lambda}(k_{i},(v-\lambda a)(m_{ij}^{2}p_{i}-m_{i}^{2}p_{ij})\\\
&+(v+\lambda a)m_{i}m_{ij}p_{j},k_{ij})\\\
&-\frac{m_{j}^{2}}{p_{j}{\cdot}k_{j}}\bigg{(}(v-\lambda
a)S_{-\lambda}(k_{i},p_{i},k_{j},p_{ij},k_{ij})\\\ &+(v+\lambda
a)m_{ij}m_{i}S_{-\lambda}(k_{i},k_{j},k_{ij})\bigg{)}\bigg{]}\end{aligned}$
$\displaystyle M$ ${}^{f\rightarrow
f^{\prime}V}(\lambda,-\lambda,0)=A^{\mbox{\tiny{emit}}}_{L}$
$\displaystyle\times$ $\displaystyle\begin{aligned} &\bigg{[}m_{i}(v-\lambda
a)S_{-\lambda}(k_{i},p_{j}-\frac{m_{j}^{2}}{p_{j}{\cdot}k_{j}}k_{j},p_{ij},k_{ij})\\\
&+m_{ij}(v+\lambda
a)S_{-\lambda}(k_{i},p_{i},p_{j}-\frac{m_{j}^{2}}{p_{j}{\cdot}k_{j}}k_{j},k_{ij})\bigg{]}\end{aligned}$

####  B.1.2 Vector Boson Emission from Antifermion

$\displaystyle M$
${}^{\bar{f}\rightarrow\bar{f}^{\prime}V}(\lambda,\lambda,\lambda)=A^{\mbox{\tiny{emit}}}_{\perp}$
$\displaystyle\times$ $\displaystyle\begin{aligned} &\bigg{[}(v+\lambda
a)S_{\lambda}(k_{ij},p_{ij},k_{j})S_{-\lambda}(k_{j},p_{j},p_{i},k_{i})\\\
&+(v-\lambda
a)m_{i}m_{ij}S_{\lambda}(k_{ij},p_{j},k_{j})S_{-\lambda}(k_{j},k_{i})\bigg{]}\end{aligned}$
$\displaystyle M$
${}^{\bar{f}\rightarrow\bar{f}^{\prime}V}(\lambda,\lambda,-\lambda)=A^{\mbox{\tiny{emit}}}_{\perp}$
$\displaystyle\times$ $\displaystyle\begin{aligned} &\bigg{[}(v+\lambda
a)S_{\lambda}(k_{ij},p_{j},k_{j})S_{-\lambda}(k_{j},p_{i},k_{i})\\\
&+(v-\lambda
a)m_{i}m_{ij}S_{\lambda}(k_{ij},k_{j})S_{-\lambda}(k_{j},p_{i},k_{i})\bigg{]}\end{aligned}$
$\displaystyle M$
${}^{\bar{f}\rightarrow\bar{f}^{\prime}V}(\lambda,-\lambda,\lambda)=A^{\mbox{\tiny{emit}}}_{\perp}$
$\displaystyle\times$ $\displaystyle\begin{aligned} &\bigg{[}m_{ij}(v-\lambda
a)S_{\lambda}(k_{ij},p_{j},k_{j})S_{-\lambda}(k_{j},p_{i},k_{i})\\\
&-m_{i}(v+\lambda
a)S_{\lambda}(k_{ij},p_{ij},k_{j})S_{-\lambda}(k_{j},p_{j},k_{i})\bigg{]}\end{aligned}$
$\displaystyle M$
${}^{\bar{f}\rightarrow\bar{f}^{\prime}V}(\lambda,-\lambda,-\lambda)=A^{\mbox{\tiny{emit}}}_{\perp}$
$\displaystyle\times$ $\displaystyle\begin{aligned} &\bigg{[}m_{ij}(v-\lambda
a)S_{\lambda}(k_{ij},k_{j})S_{-\lambda}(k_{j},p_{j},p_{i},k_{i})\\\
&-m_{i}(v+\lambda
a)S_{\lambda}(k_{ij},p_{ij},p_{j},k_{j})S_{-\lambda}(k_{j},k_{i})\bigg{]}\end{aligned}$
$\displaystyle M$
${}^{\bar{f}\rightarrow\bar{f}^{\prime}V}(\lambda,\lambda,0)=A^{\mbox{\tiny{emit}}}_{L}$
$\displaystyle\times$ $\displaystyle\begin{aligned}
&\bigg{[}S_{\lambda}(k_{ij},(v+\lambda a)(m_{ij}^{2}p_{i}-m_{i}^{2}p_{ij})\\\
&+(v-\lambda a)m_{ij}m_{i}p_{j},k_{i})\\\
&-\frac{m_{j}^{2}}{p_{j}{\cdot}k_{j}}\bigg{(}(v+\lambda
a)S_{\lambda}(k_{ij},p_{ij},k_{j},p_{i},k_{i})\\\ &+(v-\lambda
a)m_{ij}m_{i}S_{\lambda}(k_{ij},k_{j},k_{i})\bigg{)}\bigg{]}\end{aligned}$
$\displaystyle M$
${}^{\bar{f}\rightarrow\bar{f}^{\prime}V}(\lambda,-\lambda,0)=A^{\mbox{\tiny{emit}}}_{L}$
$\displaystyle\times$ $\displaystyle\begin{aligned} &\bigg{[}m_{i}(v+\lambda
a)S_{\lambda}(k_{ij},p_{ij},p_{j}-\frac{m_{j}^{2}}{p_{j}{\cdot}k_{j}}k_{j},k_{i})\\\
&+m_{ij}(v-\lambda
a)S_{\lambda}(k_{ij},p_{j}-\frac{m_{j}^{2}}{p_{j}{\cdot}k_{j}}k_{j},p_{i},k_{i})\bigg{]}\end{aligned}$

####  B.1.3 Vector Boson Emission from Vector Boson

The branching amplitude can be written as

$\displaystyle M$
${}^{\mbox{\tiny{V$\rightarrow$V'V''}}}(\lambda_{ij},\lambda_{i},\lambda_{j})$
$\displaystyle=-2g_{V}\left(p_{j}{\cdot}\epsilon_{i}\,\epsilon_{j}{\cdot}\bar{\epsilon}_{ij}-p_{i}{\cdot}\epsilon_{j}\,\epsilon_{i}{\cdot}\bar{\epsilon}_{ij}+p_{i}{\cdot}\bar{\epsilon}_{ij}\,\epsilon_{i}{\cdot}\epsilon_{j}\right)$
(52)

To compute the branching amplitude for all helicity configurations, we write
out all possible products of momenta and polarization vectors

$\displaystyle\epsilon$ ${}_{\lambda}(p_{a}){\cdot}\epsilon_{\lambda}(p_{b})$
$\displaystyle=-\frac{1}{4}\frac{1}{p_{a}{\cdot}k_{a}\,p_{b}{\cdot
k_{b}}}S_{-\lambda}(k_{a},p_{a},p_{b},k_{b})S_{\lambda}(k_{b},k_{a})$
$\displaystyle\epsilon$ ${}_{\lambda}(p_{a}){\cdot}\epsilon_{-\lambda}(p_{b})$
$\displaystyle=-\frac{1}{4}\frac{1}{p_{a}{\cdot}k_{a}\,p_{b}{\cdot
k_{b}}}S_{\lambda}(k_{a},p_{a},k_{b})S_{-\lambda}(k_{a},p_{b},k_{b})$
$\displaystyle\epsilon$ ${}_{\lambda}(p_{a}){\cdot}\epsilon_{0}(p_{b})$
$\displaystyle=\frac{\lambda}{2\sqrt{2}}\frac{1}{m_{b}}\frac{1}{p_{a}{\cdot}k_{a}}\begin{aligned}
&\bigg{(}S_{-\lambda}(k_{a},p_{a},p_{b},k_{a})\\\
&-\frac{m_{b}}{p_{b}{\cdot}k_{b}}S_{-\lambda}(k_{a},p_{a},k_{b},k_{a})\bigg{)}\end{aligned}$
$\displaystyle\epsilon$ ${}_{0}(p_{a}){\cdot}\epsilon_{0}(p_{b})$
$\displaystyle=\frac{1}{m_{a}m_{b}}\begin{aligned}
&\bigg{(}p_{a}{\cdot}p_{b}-\frac{m_{a}}{p_{a}{\cdot}k_{a}}k_{a}\\\
&-\frac{m_{b}}{p_{b}{\cdot}k_{b}}k_{b}+\frac{m_{a}}{p_{a}{\cdot}k_{a}}\frac{m_{b}}{p_{b}{\cdot}k_{b}}k_{a}{\cdot}k_{b}\bigg{)}\\\
\end{aligned}$ $\displaystyle\epsilon$ ${}_{\lambda}(p_{a}){\cdot}p_{b}$
$\displaystyle=\frac{\lambda}{\sqrt{2}}\frac{1}{p_{a}{\cdot}k_{a}}S_{-\lambda}(k_{a},p_{a},p_{b},k_{a})$
$\displaystyle\epsilon$ ${}_{0}(p_{a}){\cdot}p_{b}$
$\displaystyle=\frac{1}{m_{a}}\left(p_{a}{\cdot}p_{b}-\frac{m_{a}^{2}}{p_{a}{\cdot}k_{a}}\right)$
(53)

The unitarity-violating terms are then removed by the substitutions

$\displaystyle 2p_{i}{\cdot}p_{j}$ $\displaystyle\rightarrow
m_{ij}^{2}-m_{i}^{2}-m_{j}^{2}$ $\displaystyle 2p_{ij}{\cdot}p_{i}$
$\displaystyle\rightarrow m_{ij}^{2}+m_{i}^{2}-m_{j}^{2}$ $\displaystyle
2p_{ij}{\cdot}p_{j}$ $\displaystyle\rightarrow m_{ij}^{2}-m_{i}^{2}+m_{j}^{2}$
(54)

The same substitutions are used in the computation of amplitudes involving a
Higgs and two vector bosons, where similar products of polarization vectors
occur.

###  B.2 Higgs Emission

####  B.2.1 Higgs Emission from Fermion

$\displaystyle M$ ${}^{f\rightarrow
fh}(\lambda,-\lambda,h)=\frac{e}{4s_{w}}\frac{m_{i}}{m_{w}}\frac{1}{\sqrt{p_{ij}{\cdot}k_{ij}}\sqrt{p_{i}{\cdot}k_{i}}}$
$\displaystyle\times\bigg{[}S_{-\lambda}(k_{i},p_{i},p_{ij},k_{ij})+m_{i}^{2}S_{-\lambda}(k_{i},k_{ij})\bigg{]}$
$\displaystyle M$ ${}^{f\rightarrow
fh}(\lambda,\lambda,h)=\frac{e}{4s_{w}}\frac{m_{i}^{2}}{m_{w}}\frac{1}{\sqrt{p_{ij}{\cdot}k_{ij}}\sqrt{p_{i}{\cdot}k_{i}}}$
$\displaystyle\times S_{-\lambda}(k_{i},p_{i}+p_{ij},k_{ij})$ (55)

####  B.2.2 Higgs Emission from Antifermion

$\displaystyle M$
${}^{\bar{f}\rightarrow\bar{f}h}(\lambda,-\lambda,h)=\frac{e}{4s_{w}}\frac{m_{i}}{m_{w}}\frac{1}{\sqrt{p_{ij}{\cdot}k_{ij}}\sqrt{p_{i}{\cdot}k_{i}}}$
$\displaystyle\times\bigg{[}S_{\lambda}(k_{ij},p_{ij},p_{i},k_{i})+m_{i}^{2}S_{\lambda}(k_{ij},k_{i})\bigg{]}$
$\displaystyle M$
${}^{\bar{f}\rightarrow\bar{f}h}(\lambda,\lambda,h)=\frac{e}{4s_{w}}\frac{m_{i}^{2}}{m_{w}}\frac{1}{\sqrt{p_{ij}{\cdot}k_{ij}}\sqrt{p_{i}{\cdot}k_{i}}}$
$\displaystyle\times S_{\lambda}(k_{ij},p_{i}+p_{ij},k_{i})$ (56)

####  B.2.3 Higgs Emission from Vector Boson

$\displaystyle M$ ${}^{V\rightarrow
Vh}(\lambda,\lambda,h)=-\frac{g_{h}}{4}\frac{1}{p_{ij}{\cdot}k_{ij}\,p_{i}{\cdot}k_{i}}$
$\displaystyle\times
S_{-\lambda}(k_{ij},p_{ij},k_{i})S_{-\lambda}(k_{ij},p_{i},k_{i})$
$\displaystyle M$ ${}^{V\rightarrow
Vh}(\lambda,-\lambda,h)=-\frac{g_{h}}{4}\frac{1}{p_{ij}{\cdot}k_{ij}\,p_{i}{\cdot}k_{i}}$
$\displaystyle\times
S_{-\lambda}(k_{i},k_{ij})S_{-\lambda}(k_{ij},p_{ij},p_{i},k_{i})$
$\displaystyle M$ ${}^{V\rightarrow
Vh}(0,\lambda,h)=-\frac{g_{h}}{2\sqrt{2}}\frac{1}{m_{ij}}\frac{\lambda}{p_{i}{\cdot}k_{i}}$
$\displaystyle\times
S_{-\lambda}(k_{i},p_{i},p_{ij}-\frac{m_{ij}^{2}}{p_{ij}{\cdot}k_{ij}}k_{ij},k_{i})$
$\displaystyle M$ ${}^{V\rightarrow
Vh}(\lambda,0,h)=-\frac{g_{h}}{2\sqrt{2}}\frac{1}{m_{i}}\frac{\lambda}{p_{ij}{\cdot}k_{ij}}$
$\displaystyle\times
S_{-\lambda}(k_{ij},p_{ij},p_{i}-\frac{m_{i}^{2}}{p_{i}{\cdot}k_{i}}k_{i},k_{ij})$
$\displaystyle M$ ${}^{V\rightarrow Vh}(0,0,h)=-\frac{g_{h}}{m_{ij}^{2}}$
$\displaystyle\times\bigg{[}\frac{1}{2}m_{j}^{2}+m_{ij}^{2}\left(\frac{p_{i}{\cdot}k_{i}}{p_{ij}{\cdot}k_{ij}}+\frac{p_{j}{\cdot}k_{j}}{p_{i}{\cdot}k_{i}}\right)\bigg{]}$
(57)

####  B.2.4 Higgs Emission from Higgs

$M^{h\rightarrow hh}(h,h,h)=\frac{3}{2}\frac{m_{ij}^{2}}{m_{w}s_{w}}$ (58)

###  B.3 Vector Boson Splitting

####  B.3.1 Vector Boson Splitting to Fermion-antifermion

Defining the prefactors

$\displaystyle A^{\mbox{\tiny{split}}}_{\perp}$
$\displaystyle=-\frac{1}{2\sqrt{2}}\frac{\lambda}{p_{ij}{\cdot}k_{ij}\sqrt{p_{i}{\cdot}k_{i}}\sqrt{p_{j}{\cdot}k_{j}}}$
$\displaystyle A^{\mbox{\tiny{split}}}_{L}$
$\displaystyle=\frac{1}{2}\frac{1}{m_{I}}\frac{1}{\sqrt{p_{i}{\cdot}k_{i}}\sqrt{p_{j}{\cdot}k_{j}}}$
(59)

the branching amplitudes are

$\displaystyle M$ ${}^{V\rightarrow
f\bar{f}}(\lambda,\lambda,-\lambda)=A^{\mbox{\tiny{split}}}_{\perp}$
$\displaystyle\times$ $\displaystyle\begin{aligned} &\bigg{[}(v-\lambda
a)S_{-\lambda}(k_{i},p_{i},k_{ij})S_{-\lambda}(k_{ij},p_{ij},p_{j},k_{j})\\\
&+(v+\lambda
a)m_{i}m_{j}S_{-\lambda}(k_{i},p_{ij},k_{ij})S_{-\lambda}(k_{ij},k_{j})\bigg{]}\end{aligned}$
$\displaystyle M$ ${}^{V\rightarrow
f\bar{f}}(\lambda,-\lambda,\lambda)=A^{\mbox{\tiny{split}}}_{\perp}$
$\displaystyle\times$ $\displaystyle\begin{aligned} &\bigg{[}(v+\lambda
a)S_{-\lambda}(k_{i},p_{i},p_{ij},k_{ij})\\\ &+(v-\lambda
a)m_{i}m_{j}S_{-\lambda}(k_{i},k_{ij})S_{-\lambda}(k_{ij},p_{ij},k_{j})\bigg{]}\end{aligned}$
$\displaystyle M$ ${}^{V\rightarrow
f\bar{f}}(\lambda,\lambda,\lambda)=A^{\mbox{\tiny{split}}}_{\perp}$
$\displaystyle\times$ $\displaystyle\begin{aligned} &\bigg{[}(v+\lambda
a)m_{i}S_{-\lambda}(k_{i},p_{ij},k_{ij})S_{-\lambda}(k_{ij},p_{j},k_{j})\\\
&+(v-\lambda
a)m_{j}S_{-\lambda}(k_{i},p_{i},k_{ij})S_{-\lambda}(k_{ij},p_{ij},k_{j})\bigg{]}\end{aligned}$
$\displaystyle M$ ${}^{V\rightarrow
f\bar{f}}(\lambda,-\lambda,-\lambda)=A^{\mbox{\tiny{split}}}_{\perp}$
$\displaystyle\times$ $\displaystyle\begin{aligned} &\bigg{[}(v-\lambda
a)m_{i}S_{-\lambda}(k_{i},k_{ij})S_{-\lambda}(k_{ij},p_{ij},p_{j},k_{j})\\\
&+(v+\lambda
a)m_{j}S_{-\lambda}(k_{i},p_{i},p_{ij},k_{ij})S_{-\lambda}(k_{ij},k_{j})\bigg{]}\end{aligned}$
$\displaystyle M$ ${}^{V\rightarrow
f\bar{f}}(0,\lambda,-\lambda)=A^{\mbox{\tiny{split}}}_{L}$
$\displaystyle\times$ $\displaystyle\begin{aligned}
&\bigg{[}S_{-\lambda}(k_{i},(v-\lambda a)(m_{i}^{2}p_{j}+m_{j}^{2}p_{i})\\\
&-(v+\lambda
a)m_{i}m_{j}(p_{ij}-\frac{m_{ij}^{2}}{p_{ij}{\cdot}k_{ij}})k_{ij},k_{j})\\\
&-\frac{m_{ij}^{2}}{p_{ij}{\cdot}k_{ij}}(v-\lambda
a)S_{-\lambda}(k_{i},p_{i},k_{ij},p_{j},k_{j})\bigg{]}\end{aligned}$
$\displaystyle M$ ${}^{V\rightarrow
f\bar{f}}(0,\lambda,\lambda)=A^{\mbox{\tiny{split}}}_{L}$
$\displaystyle\times$ $\displaystyle\begin{aligned} &\bigg{[}m_{i}(v+\lambda
a)S_{-\lambda}(k_{i},p_{ij}-\frac{m_{ij}^{2}}{p_{ij}{\cdot}k_{ij}}k_{ij},p_{j},k_{j})\\\
&-m_{j}(v-\lambda
a)S_{-\lambda}(k_{i},p_{i},p_{ij}-\frac{m_{ij}^{2}}{p_{ij}{\cdot}k_{ij}},k_{j})\bigg{]}\end{aligned}$

###  B.4 Higgs Splitting

####  B.4.1 Higgs Splitting to Fermion-antifermion

$\displaystyle M$ ${}^{f\rightarrow
fh}(\lambda,\lambda,h)=\frac{e}{4s_{w}}\frac{m_{i}}{m_{w}}\frac{1}{\sqrt{p_{i}{\cdot}k_{i}}\sqrt{p_{j}{\cdot}k_{j}}}$
$\displaystyle\times\bigg{[}S_{-\lambda}(k_{i},p_{i},p_{j},k_{j})-m_{i}^{2}S_{-\lambda}(k_{i},k_{j})\bigg{]}$
$\displaystyle M$ ${}^{f\rightarrow
fh}(\lambda,-\lambda,h)=\frac{e}{4s_{w}}\frac{m_{i}^{2}}{m_{w}}\frac{1}{\sqrt{p_{i}{\cdot}k_{i}}\sqrt{p_{j}{\cdot}k_{j}}}$
$\displaystyle\times S_{-\lambda}(k_{i},p_{i}-p_{j},k_{j})$ (61)

####  B.4.2 Higgs Splitting to Vector Bosons

$\displaystyle M$ ${}^{h\rightarrow VV}(h,\lambda,-\lambda)=-\frac{g_{h}}{4}$
$\displaystyle\times$ $\displaystyle
S_{-\lambda}(k_{i},p_{i},k_{j})S_{-\lambda}(k_{i},p_{j},k_{j})$ $\displaystyle
M$ ${}^{h\rightarrow VV}(h,\lambda,\lambda)=-\frac{g_{h}}{4}$
$\displaystyle\times$ $\displaystyle
S_{-\lambda}(k_{j},k_{i})S_{-\lambda}(k_{i},p_{i},p_{j},k_{j})$ $\displaystyle
M$ ${}^{h\rightarrow
VV}(h,0,\lambda)=-\frac{g_{h}}{2\sqrt{2}}\frac{1}{m_{i}}\frac{\lambda}{p_{j}{\cdot}k_{j}}$
$\displaystyle\times$ $\displaystyle
S_{-\lambda}(k_{j},p_{j},p_{i}-\frac{m_{i}^{2}}{p_{i}{\cdot}k_{i}}k_{i},k_{j})$
$\displaystyle M$ ${}^{h\rightarrow
VV}(h,\lambda,0)-\frac{g_{h}}{2\sqrt{2}}\frac{1}{m_{j}}\frac{\lambda}{p_{i}{\cdot}k_{i}}$
$\displaystyle\times$
$\displaystyle=S_{-\lambda}(k_{i},p_{i},p_{j}-\frac{m_{j}^{2}}{p_{j}{\cdot}k_{k}}k_{j},k_{i})$
$\displaystyle M$ ${}^{h\rightarrow VV}(h,0,0)=\frac{g_{h}}{m_{i}m_{j}}$
$\displaystyle\times$
$\displaystyle\bigg{[}\frac{1}{2}\left(m_{ij}^{2}-m_{i}^{2}-m_{j}^{2}\right)-m_{j}^{2}\frac{p_{i}{\cdot}k_{i}}{p_{j}{\cdot}k_{j}}-m_{i}^{2}\frac{p_{j}{\cdot}k_{j}}{p_{i}{\cdot}k_{i}}\bigg{]}$
(62)

###  B.5 Vector Boson Emission (Initial State)

####  B.5.1 Vector Boson Emission from Fermion

We define the prefactors

$\displaystyle\tilde{A}^{\mbox{\tiny{emit}}}_{\perp}$
$\displaystyle=\frac{1}{2\sqrt{2}}\frac{\lambda}{\sqrt{p_{a}{\cdot}k_{j}}\sqrt{p_{aj}{\cdot}k_{aj}}\,p_{j}{\cdot}k_{j}}$
$\displaystyle\tilde{A}^{\mbox{\tiny{emit}}}_{L}$
$\displaystyle=\frac{1}{2}\frac{1}{m_{j}}\frac{1}{\sqrt{p_{a}{\cdot}k_{a}}\sqrt{p_{aj}{\cdot}k_{aj}}}.$
(63)

$\displaystyle M$
${}^{\tilde{f}\rightarrow\tilde{f}^{\prime}V}(\lambda,\lambda,\lambda)=\tilde{A}^{\mbox{\tiny{emit}}}_{\perp}$
$\displaystyle\times$ $\displaystyle\begin{aligned} &\bigg{[}(v-\lambda
a)S_{-\lambda}(k_{aj},p_{aj},p_{j},k_{j})S_{-\lambda}(k_{j},p_{a},k_{a})\\\
&-(v+\lambda
a)m_{a}m_{aj}S_{-\lambda}(k_{aj},k_{j})S_{-\lambda}(k_{j},p_{j},k_{a})\bigg{]}\end{aligned}$
$\displaystyle M$
${}^{\tilde{f}\rightarrow\tilde{f}^{\prime}V}(\lambda,\lambda,-\lambda)=\tilde{A}^{\mbox{\tiny{emit}}}_{\perp}$
$\displaystyle\times$ $\displaystyle\begin{aligned} &\bigg{[}(v-\lambda
a)S_{-\lambda}(k_{aj},p_{aj},k_{j})S_{-\lambda}(k_{j},p_{j},p_{a},k_{,}a)\\\
&-(v+\lambda
a)m_{a}m_{aj}S_{-\lambda}(k_{aj},p_{j},k_{j})S_{-\lambda}(k_{j},k_{a})\bigg{]}\end{aligned}$
$\displaystyle M$
${}^{\tilde{f}\rightarrow\tilde{f}^{\prime}V}(\lambda,-\lambda,\lambda)=\tilde{A}^{\mbox{\tiny{emit}}}_{\perp}$
$\displaystyle\times$ $\displaystyle\begin{aligned} &\bigg{[}(v+\lambda
a)m_{aj}S_{-\lambda}(k_{aj},k_{j})S_{-\lambda}(k_{j},p_{j},p_{a},k_{a})\\\
&-(v-\lambda
a)m_{a}S_{-\lambda}(k_{aj},p_{aj},p_{j},k_{j})S_{-\lambda}(k_{j},k_{a})\bigg{]}\end{aligned}$
$\displaystyle M$
${}^{\tilde{f}\rightarrow\tilde{f}^{\prime}V}(\lambda,-\lambda,-\lambda)=\tilde{A}^{\mbox{\tiny{emit}}}_{\perp}$
$\displaystyle\times$ $\displaystyle\begin{aligned} &\bigg{[}(v+\lambda
a)m_{aj}S_{-\lambda}(k_{aj},p_{j},k_{j})S_{-\lambda}(k_{j},p_{a},k_{a})\\\
&-(v-\lambda
a)m_{a}S_{-\lambda}(k_{aj},p_{aj},k_{j})S_{-\lambda}(k_{j},p_{j},k_{a})\bigg{]}\end{aligned}$
$\displaystyle M$
${}^{\tilde{f}\rightarrow\tilde{f}^{\prime}V}(\lambda,\lambda,0)=\tilde{A}^{\mbox{\tiny{emit}}}_{L}$
$\displaystyle\times$ $\displaystyle\begin{aligned}
&\bigg{[}S_{-\lambda}(k_{aj},(v-\lambda a)(m_{a}^{2}p_{aj}-m_{aj}^{2}p_{a})\\\
&+(v+\lambda a)m_{a}m_{aj}p_{j},k_{a})\\\
&-\frac{m_{j}^{2}}{p_{j}{\cdot}k_{j}}\bigg{(}(v-\lambda
a)S_{-\lambda}(k_{aj},p_{aj},k_{j},p_{a},k_{a})\\\ &-(v+\lambda
a)m_{aj}m_{a}S_{-\lambda}(k_{aj},k_{j},k_{a})\bigg{)}\bigg{]}\\\
\end{aligned}$ $\displaystyle M$
${}^{\tilde{f}\rightarrow\tilde{f}^{\prime}V}(\lambda,-\lambda,0)=\tilde{A}^{\mbox{\tiny{emit}}}_{L}$
$\displaystyle\times$ $\displaystyle\begin{aligned} &\bigg{[}m_{aj}(v-\lambda
a)S_{-\lambda}(k_{aj},p_{j}-\frac{m_{j}^{2}}{p_{j}{\cdot}k_{j}}k_{j},p_{a},k_{a})\\\
&+m_{aj}(v+\lambda
a)S_{-\lambda}(k_{aj},p_{aj},p_{j}-\frac{m_{j}^{2}}{p_{j}{\cdot}k_{j}}k_{j},k_{a})\bigg{]}\end{aligned}$

####  B.5.2 Vector Boson Emission from Antifermion

$\displaystyle M$
${}^{\tilde{\bar{f}}\rightarrow\tilde{\bar{f}}^{\prime}V}(\lambda,\lambda,\lambda)=\tilde{A}^{\mbox{\tiny{emit}}}_{\perp}$
$\displaystyle\times$ $\displaystyle\begin{aligned} &\bigg{[}(v+\lambda
a)S_{\lambda}(k_{a},p_{a},k_{j})S_{-\lambda}(k_{j},p_{j}p_{aj},k_{aj})\\\
&-(v-\lambda
a)m_{a}m_{aj}S_{\lambda}(k_{a},p_{j},k_{j})S_{-\lambda}(k_{j},k_{aj})\bigg{]}\end{aligned}$
$\displaystyle M$
${}^{\tilde{\bar{f}}\rightarrow\tilde{\bar{f}}^{\prime}V}(\lambda,\lambda,-\lambda)=\tilde{A}^{\mbox{\tiny{emit}}}_{\perp}$
$\displaystyle\times$ $\displaystyle\begin{aligned} &\bigg{[}(v+\lambda
a)S_{\lambda}(k_{a},p_{a},p_{j},k_{j})S_{-\lambda}(k_{j},p_{aj},k_{aj})\\\
&-(v-\lambda
a)m_{a}m_{aj}S_{\lambda}(k_{a},k_{j})S_{-\lambda}(k_{j},p_{j},k_{a})\bigg{]}\end{aligned}$
$\displaystyle M$
${}^{\tilde{\bar{f}}\rightarrow\tilde{\bar{f}}^{\prime}V}(\lambda,-\lambda,\lambda)=\tilde{A}^{\mbox{\tiny{emit}}}_{\perp}$
$\displaystyle\times$ $\displaystyle\begin{aligned} &\bigg{[}(v-\lambda
a)m_{a}S_{\lambda}(k_{a},p_{j},k_{j})S_{-\lambda}(k_{j},p_{aj},k_{aj})\\\
&-(v+\lambda
a)m_{aj}S_{\lambda}(k_{a},p_{a},k_{j})S_{-\lambda}(k_{j},p_{j},k_{aj})\bigg{]}\end{aligned}$
$\displaystyle M$
${}^{\tilde{\bar{f}}\rightarrow\tilde{\bar{f}}^{\prime}V}(\lambda,-\lambda,-\lambda)=\tilde{A}^{\mbox{\tiny{emit}}}_{\perp}$
$\displaystyle\times$ $\displaystyle\begin{aligned} &\bigg{[}(v-\lambda
a)m_{a}S_{\lambda}(k_{a},k_{j})S_{-\lambda}(k_{j},p_{j},p_{aj},k_{aj})\\\
&-(v+\lambda
a)m_{aj}S_{\lambda}(k_{a},p_{a},p_{j},k_{j})S_{-\lambda}(k_{j},k_{aj})\bigg{]}\end{aligned}$
$\displaystyle M$
${}^{\tilde{\bar{f}}\rightarrow\tilde{\bar{f}}^{\prime}V}(\lambda,\lambda,0)=\tilde{A}^{\mbox{\tiny{emit}}}_{L}$
$\displaystyle\times$ $\displaystyle\begin{aligned}
&\bigg{[}S_{\lambda}(k_{a},(v+\lambda a)(m_{a}^{2}p_{aj}-m_{aj}^{2}p_{a})\\\
&+(v-\lambda a)m_{a}m_{aj}p_{j},k_{aj})\\\
&-\frac{m_{j}^{2}}{p_{j}{\cdot}k_{j}}\bigg{(}(v+\lambda
a)S_{\lambda}(k_{a},p_{a},k_{j},p_{aj},k_{aj})\\\ &-(v-\lambda
a)m_{aj}m_{a}S_{\lambda}(k_{a},k_{j},k_{aj})\bigg{)}\bigg{]}\end{aligned}$
$\displaystyle M$
${}^{\tilde{\bar{f}}\rightarrow\tilde{\bar{f}}^{\prime}V}(\lambda,-\lambda,0)=\tilde{A}^{\mbox{\tiny{emit}}}_{L}$
$\displaystyle\times$ $\displaystyle\begin{aligned} &\bigg{[}m_{a}(v-\lambda
a)S_{\lambda}(k_{a},p_{j}-\frac{m_{j}^{2}}{p_{j}{\cdot}k_{j}}k_{j},p_{aj},k_{aj})\\\
&+m_{aj}(v+\lambda
a)S_{\lambda}(k_{a},p_{a},p_{j}-\frac{m_{j}^{2}}{p_{j}{\cdot}k_{j}}k_{j},k_{aj})\bigg{]}\end{aligned}$

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