compat=1.0.0

UMN-TH-4325/24

FTPI-MINN-24-16

Momentum shift and on-shell recursion relation
for electroweak theory

Yohei Ema,1,2,a<EMAIL_ADDRESS>Ting<EMAIL_ADDRESS>Wenqi
Ke,1,2,c<EMAIL_ADDRESS>Zhen Liu,1,d<EMAIL_ADDRESS>
Kun-Feng<EMAIL_ADDRESS>Ishmam Mahbub1,f<EMAIL_ADDRESS>
1 _School of Physics and Astronomy, University of Minnesota, Minneapolis, MN
55455, USA_ 2 _William I. Fine Theoretical Physics Institute, School of
Physics and Astronomy,_ _University of Minnesota, Minneapolis, MN 55455, USA_

We study the All-Line Transverse (ALT) shift which we developed for on-shell
recursion of amplitudes for particles of any mass. We discuss the validity of
the shift for general theories of spin $\leq$ 1, and illustrate the connection
between Ward identity and constructibility for massive spin-1 amplitude under
the ALT shift. We apply the shift to the electroweak theory, and various four-
point scattering amplitudes among electroweak gauge bosons and fermions are
constructed. We show explicitly that the four-point gauge boson contact terms
in massive electroweak theory automatically arise after recursive
construction, independent of UV completion, and they automatically cancel the
terms growing as (energy)4 at high energy. We explore UV completion of the
electroweak theory that cancels the remaining (energy)2 terms and impose
unitarity requirements to constrain additional couplings. The ALT shift
framework allows consistent treatment in dealing with contact term ambiguities
for renormalizable massive and massless theories, which we show can be useful
in studying real-world amplitudes with massive spinors.

###### Contents

  1. 1 Introduction
  2. 2 Momentum shift and massive constructive method
    1. 2.1 Recursion relation and factorization
    2. 2.2 All-Line Transverse (ALT) shift
  3. 3 On-shell construction of electroweak theory
    1. 3.1 Three-point amplitude
    2. 3.2 $WWWW$ amplitude
    3. 3.3 $WWZZ$ amplitude
    4. 3.4 $WWtt$ amplitude
  4. 4 UV completion of electroweak theory
    1. 4.1 A neutral scalar: Higgs doublet
    2. 4.2 Neutral and doubly charged scalar: Higgs triplet
    3. 4.3 Other UV completion
  5. 5 Conclusion and discussion
  6. A Massive spinor formalism
  7. B On-shell construction of electroweak theory in vector representation
    1. B.1 Three-point amplitude
    2. B.2 $WWWW$ amplitude
    3. B.3 $WWZZ$ amplitude
    4. B.4 $WWtt$ amplitude
  8. C UV completion of electroweak theory in vector representation
    1. C.1 A neutral scalar: Higgs doublet
    2. C.2 Neutral and doubly charged scalar: Higgs triplet

##  1 Introduction

Figure 1: Summary of recursion relation for renormalizable massive vector
theory.

Modern methods of scattering amplitude exploit unitarity, locality and
causality to construct tree-level amplitudes and explores quantum field
theory. The on-shell recursion method developed in [1, 2] has allowed the
construction of higher-point amplitudes from on-shell lower-point amplitudes.
These methods were first applied to pure Yang-Mills theory and gravity [3, 4],
and subsequently extended to various theories (see e.g. [5, 6, 7, 8, 9, 10,
11] for reviews). Although on-shell recursion of massless external particles
as well as its extension to a mixture of massive and massless particles has
been studied [12, 13, 14, 15, 16], recursion techniques for general theories
with all massive particles are still part of ongoing research [6, 17, 18, 19,
20].

The on-shell method is based on analytical continuation of an amplitude by
shifting external momenta with a complex variable $z$. Using Cauchy’s theorem,
the amplitude is written as a sum of a contribution from poles at finite $z$
where intermediate particles go on-shell and the amplitude factorizes into
products of lower-point amplitudes, and a non-factorizable contribution at
$z\rightarrow\infty$. An amplitude is “on-shell constructible” if there exists
a momentum shift under which the latter contribution is absent. The contact
term ambiguity arises here as one might find the constructed amplitudes differ
from the correct prediction of the underlying theory (e.g., from Feynman
diagram approach for a given theory) by contact terms, which can come from
either new theory inputs or inconsistent momentum shift. A useful momentum
shift for massive amplitudes would allow us to resolve these contact term
ambiguities, for instance, as recently demonstrated in Ref. [20]. In massless
theories, conditions to satisfy the on-shell constructibility are well
understood, starting with the famous example of the BCFW-shift [1, 2]. On the
contrary, momentum shifts are less explored in massive theories, and this is
the direction we pursue here.

In this paper, we apply the All-Line Transverse (ALT) shift we proposed in
[20], which deforms external momenta using transverse polarization vectors, to
amplitudes involving massive $W$\- and $Z$-bosons in the electroweak (EW)
theory, as schematically shown in Figure 1. With the dimensional analysis and
under the assumption that the theory satisfies the Ward identity in the
massless limit, we show that four- and higher-point amplitudes are
constructible under the ALT shift in a general renormalizable
theory.\natural1\natural1$\natural$1 with the exception of four scalar
amplitudes, which contain a non-factorizable term given by $A_{4}=\lambda$
with a coupling constant $\lambda$. Unless imposing additional symmetries such
as supersymmetry, this amplitude is not constructible by any momentum shift.
In particular, we explicitly construct $W$\- and $Z$-boson four-point
amplitudes, including the contact terms, solely from the three-point
amplitudes. This requires an introduction of only two dimensionless couplings,
$g_{WWZ}$ and $g_{WW\gamma}$, for three-point amplitudes among $W$ and
$Z$/photon. The contact terms arise from the ALT shift-dependent parts of the
three-point amplitudes, and therefore this would be achieved only with an
explicit introduction of the momentum shift. This is in contrast to the
approaches in [21, 22, 23, 24] where the amplitudes are glued without momentum
shift, which in general raises the contact term ambiguity (see e.g. [25, 26,
20] in the context of QED with massive fermions). The contact term derived by
the ALT shift automatically guarantees the cancellation of the term growing as
(energy)4 at high energy in all-longitudinal $W$-/$Z$-scatterings, and this is
independent of UV completion of the EW theory.

We then move our focus to different UV completions of the EW theory. Even
though the (energy)4 terms are canceled, the amplitudes still grow as
(energy)2 without additional input, and we require the cancellation of these
terms by adding weakly coupled particles. With the assumption of adding only
one neutral scalar, we show that it naturally fits into a Higgs-doublet model.
We then consider adding one neutral and one doubly charged scalar, which
naturally leads to a Higgs-triplet model, and briefly discuss UV completion by
adding vector particles.

This paper is organized as follows. In Sec. 2, the general formalism of the
on-shell recursion relation is reviewed. We then introduce the ALT shift and
discuss its large-$z$ behavior that determines the on-shell constructibility.
We highlight an important feature of ALT shift: that the large-$z$ limits of
the shifted amplitudes take a form that Ward Identity can be directly imposed.
In Sec. 3, we apply the shift for on-shell construction of $W$\- and $Z$-boson
amplitudes. Different UV completions are discussed in Sec. 4. Finally, Sec. 5
is devoted to a summary of our results and possible outlooks. The appendix
consists of three parts. In App. A, we summarize the massive spinor formalism
used throughout this paper and define our notations and conventions. In the
main text, we express our calculations in the little group covariant massive
spinor representation [17]. For readers who are more used to the vector
notation, we express our results of the EW theory and its UV completions in
the vector representation in Apps. B and C, respectively.

##  2 Momentum shift and massive constructive method

In this section, we begin with a brief overview of the on-shell methods for
tree-level amplitudes (see e.g. [7, 8, 10, 11] for reviews), and introduce the
All-Line Transverse (ALT) shift. This method will be used in Secs. 3 and 4 to
construct tree-level amplitudes in the EW theory.

###  2.1 Recursion relation and factorization

To construct higher-point amplitudes from lower-point on-shell amplitudes, we
shift the momenta of external particles as

$\displaystyle p_{i}\to\hat{p}_{i}=p_{i}+zq_{i},\quad\sum_{i}q_{i}=0,$ (2.1)

where $i$ runs over $1$ to $n$ with $n$ the total number of the external
particles, $q_{i}$ is an arbitrary momentum which we later choose to be
proportional to the transverse polarization vector, and the latter condition
guarantees the total momentum conservation. We require

$\displaystyle p_{i}\cdot q_{i}=q_{i}^{2}=0,$ (2.2)

so that the on-shell condition for external particles is not modified,
$\hat{p}_{i}^{2}=p_{i}^{2}=m_{i}^{2}$, where $m_{i}$ is the mass of the
particle $i$. The momentum shift promotes the amplitude to a complex function,
$\hat{A}_{n}(z)$, and the original amplitude is given by
$A_{n}=\hat{A}_{n}(z=0)$. Focusing on tree-level amplitudes, $\hat{A}_{n}(z)$
has only poles (and no branch cuts) in the complex $z$-plane. Each pole
corresponds to an intermediate particle being on-shell, where the amplitude
factorizes into a product of lower-point on-shell amplitudes. Therefore, by
using the complex analysis, we can express the amplitude as

$\displaystyle{A}_{n}$ $\displaystyle=\frac{1}{2\pi
i}\oint_{z=0}\frac{dz}{z}\hat{A}_{n}(z)=-\sum_{z=z_{I}}\sum_{\lambda}\mathrm{Res}_{z=z_{I}}\left[\hat{A}^{(\lambda)}_{n-m+2}\frac{1}{z}\frac{1}{\hat{p}_{I}^{2}-m_{I}^{2}}\hat{A}^{(\bar{\lambda})}_{m}\right]+B_{\infty},$
(2.3)

where $z_{I}$ satisfies $\hat{p}_{I}^{2}-m_{I}^{2}=0$ with $m_{I}$ the mass of
the intermediate particle and $\hat{p}_{I}$ the partial sum of the external
momenta. $\lambda$ denotes the helicity of the intermediate particle with
$\bar{\lambda}$ its little-group conjugate (we will explain it later), and
$B_{\infty}$ is the boundary contribution at $|z|\to\infty$. In our case, the
pole condition is solved as

$\displaystyle\hat{p}_{I}^{2}-m_{I}^{2}=0~{}~{}\Longrightarrow~{}~{}z^{\pm}_{I}=\dfrac{1}{q_{I}^{2}}\left[-p_{I}\cdot
q_{I}\pm\sqrt{(p_{I}\cdot
q_{I})^{2}-(p_{I}^{2}-m_{I}^{2})q_{I}^{2}}\,\right].$ (2.4)

The ALT shift (which we specify below) satisfies $q_{i}\cdot q_{j}\neq 0$ for
$i\neq j$, resulting in $q_{I}^{2}\neq 0$, and therefore we have a pair of the
poles, $z_{I}^{\pm}$, for each intermediate particle. By using

$\displaystyle\frac{1}{\hat{p}_{I}^{2}-m_{I}^{2}}=\frac{1}{p_{I}^{2}-m_{I}^{2}}\frac{z_{I}^{+}z_{I}^{-}}{(z-z_{I}^{+})(z-z_{I}^{-})},$
(2.5)

the amplitude is then written as

$\displaystyle{A}_{n}$
$\displaystyle=\sum_{I}\frac{1}{p_{I}^{2}-m_{I}^{2}}\frac{1}{z_{I}^{+}-z_{I}^{-}}\sum_{\lambda}\left[z_{I}^{+}\hat{A}^{(\lambda)}_{n-m+2}(z_{I}^{-})\times\hat{A}^{(\bar{\lambda})}_{m}(z_{I}^{-})-(z_{I}^{+}\leftrightarrow
z_{I}^{-})\right]+B_{\infty}.$ (2.6)

This tells us that if $B_{\infty}=0$, or if
$\lim_{z\to\infty}\hat{A}_{n}(z)=0$, the higher-point amplitudes are expressed
by lower-point on-shell amplitudes, implying the on-shell constructibility of
the amplitudes. Note that such on-shell constructibility does not mean there
exist no contact term in the theory, but rather these contact terms are
dependent on the lower point vertices through the consistent amplitude
construction. Later in this section, we investigate the large-$z$ behavior of
the shifted amplitude $\hat{A}_{n}(z)$ that controls $B_{\infty}$ under our
proposed ALT shift.

We also remark that for a given massive theory, it is possible that amplitudes
with particular combinations of spin projections are on-shell constructible,
and not all possible spin projections are on-shell constructible for a given
shift. However, since amplitudes are little group covariant, we can act the
spin-raising and spin-lowering operators on the amplitude with one particular
spin-projection to construct the rest of the amplitudes. These operators are
generators of the little group SU(2), and they are defined in the helicity
basis as

$\displaystyle J_{i}^{-}=-\lvert
i\rangle_{\alpha}\dfrac{\partial}{\partial\lvert\eta_{i}\rangle_{\alpha}}-[\eta_{i}\lvert_{\dot{\alpha}}\dfrac{\partial}{\partial[i\lvert_{\dot{\alpha}}},\qquad
J_{i}^{+}=\lvert\eta_{i}\rangle_{\alpha}\dfrac{\partial}{\partial\lvert
i\rangle_{\alpha}}+[i\lvert_{\dot{\alpha}}\dfrac{\partial}{\partial[\eta_{i}\lvert_{\dot{\alpha}}}.$
(2.7)

In calculations below, we will show this operation leads to consistent and
correct amplitudes for all longitudinal scatterings, which are not directly
constructible under ALT shift.

###  2.2 All-Line Transverse (ALT) shift

In the rest of the paper, we focus on a momentum shift scheme which is
universally applicable for both massive and massless amplitudes, dubbed ALT
shift, introduced in Ref. [20]. The ALT shift is based on the helicity
decomposition of the spinors, given by

$\displaystyle(p_{i})_{a\dot{a}}=|i\rangle_{a}[i|_{\dot{a}}-|\eta_{i}\rangle_{a}[\eta_{i}|_{\dot{a}}.$
(2.8)

For spinor notation and convention, see App. A. We construct the ALT shift
after declaring the helicity of all external particles, and the spinor shifts
are defined such that external Dirac spinors and polarization vectors are
unshifted. For spin-1/2 particles and the transverse modes of spin-1
particles, we shift the spinors such that

$\displaystyle\begin{cases}|i\rangle\to|\hat{i}\rangle=|i\rangle+zc_{i}|\eta_{i}\rangle&\mathrm{for}~{}~{}\lambda_{i}=+,\vspace{1mm}\\\
|i]\to|\hat{i}]=|i]+zc_{i}|\eta_{i}]&\mathrm{for}~{}~{}\lambda_{i}=-,\end{cases}\quad\mathrm{or}~{}~{}~{}\begin{cases}|\eta_{i}]\to|\hat{\eta}_{i}]=|\eta_{i}]+zc_{i}|i]&\mathrm{for}~{}~{}\lambda_{i}=+,\vspace{1mm}\\\
|\eta_{i}\rangle\to|\hat{\eta}_{i}\rangle=|\eta_{i}\rangle+zc_{i}|i\rangle&\mathrm{for}~{}~{}\lambda_{i}=-,\end{cases}$
(2.9)

and the other spinors are unshifted. Here, $\lambda_{i}$ is the helicity index
of the particle $i$ and $c_{i}$ is a numerical constant required to satisfy
the total momentum conservation. We can extend the shift to incorporate
massless particles with the understanding that the spinors
$|\eta_{i}\rangle,|\eta_{i}]$ are now replaced by arbitrary reference spinors,
usually denoted as $|\xi_{i}\rangle,|\xi_{i}]$, for these particles. For the
longitudinal ($L$) modes of spin-1 particles, there are two choices:

$\displaystyle\begin{cases}|i\rangle\to|\hat{i}\rangle=|i\rangle+z\dfrac{c_{i}}{2}|\eta_{i}\rangle,\vspace{1mm}\\\
[\eta_{i}|\to[\hat{\eta}_{i}|=[\eta_{i}|-z\dfrac{c_{i}}{2}[i|,\end{cases}\mathrm{or}~{}~{}~{}\begin{cases}[i|\to[\hat{i}|=[i|+z\dfrac{c_{i}}{2}[\eta_{i}|,\vspace{1mm}\\\
|\eta_{i}\rangle\to|\hat{\eta}_{i}\rangle=|\eta_{i}\rangle-z\dfrac{c_{i}}{2}|i\rangle,\end{cases}\mathrm{for}~{}~{}\lambda_{i}=L,$
(2.10)

and both choices equally work for our purpose. The Dirac spinors are given by

$\displaystyle u^{(+)}=\begin{pmatrix}|\eta_{i}\rangle_{a}\\\
|i]^{\dot{a}}\end{pmatrix},\quad u^{(-)}=\begin{pmatrix}|i\rangle_{a}\\\
|\eta_{i}]^{\dot{a}}\end{pmatrix},\quad
v^{(+)}=\begin{pmatrix}|\eta_{i}\rangle_{a}\\\
-|i]^{\dot{a}}\end{pmatrix},\quad v^{(-)}=\begin{pmatrix}|i\rangle_{a}\\\
-|\eta_{i}]^{\dot{a}}\end{pmatrix},$ (2.11)

while the polarization vectors for the gauge boson are expressed as

$\displaystyle\epsilon^{(+)}=\sqrt{2}\frac{|\eta_{i}\rangle[i|}{m_{i}},\quad\epsilon^{(-)}=-\sqrt{2}\frac{|i\rangle[\eta_{i}|}{m_{i}},\quad\epsilon^{(L)}=\frac{|i\rangle[i|+|\eta_{i}\rangle[\eta_{i}|}{m_{i}}.$
(2.12)

Hence the external polarization vectors and Dirac spinors are invariant under
the ALT shift. With the above shift of the spinors, we see that the momenta
are shifted by the transverse polarization vector

$\displaystyle p_{i}\to
p_{i}+zq_{i},~{}~{}q_{i}=c_{i}m_{i}\epsilon^{(\pm)}(p_{i}).$ (2.13)

For the external gauge boson with transverse helicity, the shifted momentum is
equal to the same helicity transverse polarization vector, and similarly for
the fermion. For the longitudinal gauge boson, we can shift the momentum by
either positive or negative transverse polarization vectors, corresponding to
the two choices in Eq. (2.10).\natural2\natural2$\natural$2 We do not shift
the momentum by the longitudinal polarization vector since this choice shifts
the external longitudinal polarization vector and modifies the on-shell
condition. The transverse polarization vector does not have the temporal
component in this basis, and hence we can satisfy the total momentum
conservation, i.e. the second equation in Eq. (2.1), with non-zero $c_{i}$ for
four and higher-point amplitudes.

#### Large-$z$ behavior

As discussed in [20], a simple dimensional analysis allows us to estimate the
large-$z$ behavior of the amplitude under the ALT shift. To see this, it is
convenient to decompose the amplitude as [27, 20]

$\displaystyle A_{n}=\left[\sum_{\mathrm{diagrams}}g\times
F\right]\times\prod\epsilon\times\prod u,$ (2.14)

where $g$ is the collection of the couplings, and $F$ is the amplitude with
the external polarization vectors and fermion wavefunctions, which we denote
collectively by $\prod\epsilon$ and $\prod u$, stripped off. ALT shift
satisfies $q_{i}\cdot q_{j}\neq 0$ for $i\neq j$, the denominators of the
propagators all scale as $z^{2}$. Combined with the numerator, we expect that
$F$ scales at most as $\hat{F}\sim z^{\gamma}$ with $\gamma\leq[F]$ at large
$z$, where $[F]$ is the mass dimension of $F$.\natural3\natural3$\natural$3 We
note here a subtle difference to our traditional mass/energy dimensional
analysis. In our explicit calculation below, we encounter the structure
$1/m_{W}^{2}$ arising from the completeness relation of the intermediate
polarization vectors. This could possibly increase the $z$ scaling counting by
two each time. We assume that this factor is compensated by massive parameters
in the numerator without increasing $z$ scaling, either due to the Ward
identity or due to the couplings with positive mass dimensions. In other
words, we focus on the theory that has a smooth massless limit, such as the
gauge theory that is only spontaneously broken. Under this assumption, this
factor does not modify our argument based on the dimensional analysis.  The
mass dimension of an $n$-point amplitude is $4-n$ which fixes $[F]$ as
$[F]=4-n-[g]-N_{F}/2$, where $N_{F}$ is the number of the external fermions
and $[g]$ is the mass dimension of $g$. Since the ALT shift does not deform
the external polarization vectors and fermion wavefunctions, we obtain the
large-$z$ behavior of the amplitude as

$\displaystyle\hat{A}_{n}\sim z^{\gamma},\quad\gamma\leq
4-n-[g]-\frac{N_{F}}{2}.$ (2.15)

If $\gamma<0$, $B_{\infty}=0$ and the theory is on-shell constructible. The
dimensional analysis suffices to prove the constructibility of four-point
amplitudes with at least one pair of fermions as well as five- and higher-
point amplitudes for renormalizable theories (which have $[g]\geq 0$).

#### Ward identity and on-shell constructibility

The estimate (2.15) based solely on the dimensional analysis does not indicate
the on-shell constructibility of four vector boson scatterings in a
renormalizable theory. But if we have (at least) one transverse gauge boson in
the external state, large-$z$ behavior under ALT shift is improved by the Ward
identity for massless gauge bosons. This is a feature of the ALT shift. Since
we shift our momentum by the transverse polarization vector, the external
momentum and polarization vector for the transverse mode become proportional
to each other at the leading order in the large $z$ limit,

$\displaystyle\lim_{z\to\infty}\hat{p}_{i}\propto\epsilon_{i}^{(\lambda_{i})},$
(2.16)

and this allows us to use the Ward identity. Saturating the inequality (2.15)
means that we consume all the dimensionful quantities by the shifted momentum
$q_{i}~{}(\propto\epsilon_{i}^{(\pm)})$. To be concrete, let us take the
particle $1$ as the transverse gauge boson, and extract the polarization
vector of the particle $1$. The leading order term saturating Eq. (2.15) at
large $z$ is then schematically expressed as

$\displaystyle\lim_{z\rightarrow\infty}\hat{A}_{n}$ $\displaystyle\sim
g\,z^{4-n-[g]-N_{F}/2}\big{[}\epsilon_{1}^{(\lambda_{1})}\cdot
F(q_{1},q_{2},...,q_{n})\big{]}\propto g\,z^{4-n-[g]-N_{F}/2}\big{[}q_{1}\cdot
F(q_{1},q_{2},...,q_{n})\big{]},$ (2.17)

where $F$ depends only on the shifted momentum $q_{i}$ and it does not contain
any other mass dimensionful parameters (such as $m_{W}$) except possibly for
their ratios. Notice that we can think of the polarization vector of the
particle $i$ as defined with respect to $q_{i}$, instead of $p_{i}$, since it
satisfies $\epsilon_{i}\cdot q_{i}=0$ in the ALT shift. This is a consequence
of our choice of the momentum shift that does not shift the external
polarization vectors. We can then _impose_ the Ward identity of the massless
theory, resulting in

$\displaystyle q_{1}\cdot F(q_{1},q_{2},...,q_{n})=0,$ (2.18)

meaning that the leading order term in large $z$ vanishes. For the amplitude
with only the longitudinal modes, the above argument does not apply since
$\epsilon_{1}^{(L)}$ is not proportional to the shifted momentum. Therefore,
we have

$\displaystyle\gamma\leq 4-n-[g]-\frac{N_{F}}{2}-c_{T},$ (2.19)

where $c_{T}=1$ if there is at least one external transverse mode, and
$c_{T}=0$ if all the external gauge bosons are longitudinal. This holds as
long as we require the Ward identity of the theory in the massless limit. This
is the on-shell equivalent of declaring that we start from the gauge theory,
which is broken only spontaneously.

The relation (2.19) also indicates that we cannot construct the scalar four-
point amplitude, but this is expected. For the scalar four-point amplitude,
there exists no momentum shift that allows the on-shell construction since we
can always add the scalar four-point interaction, $\lambda\phi^{4}$, in a
renormalizable theory (in the Lagrangian formulation). This interaction is not
related to any lower-point amplitudes, and hence they contribute to the
boundary term (unless we assume e.g. supersymmetry).

We have so far proved that renormalizable massive spin-1 theories satisfying
Ward identity are on-shell constructible for $n\geq 4$ using ALT shift. In the
previous subsection, we also showed the constructibility of a generic
renormalizable massive spin-1 theory starts at $n\geq 5$, which indicates that
the four-point contact term between vectors and scalars can be independent in
general. Now one can ask for the ALT shift, what space of massive spin-1
theories are on-shell constructible using only three-point amplitudes, and
whether theories satisfying Ward identity are the only ones fulfilling such
constructibility constraints. In this regard, for instance, we observed a
connection between the two in the construction of $WWhh$ amplitude in Sec.
4.2. The amplitude produces boundary contributions due to the presence of the
contact term. Demanding on-shell constructibility using ALT shift, we can
determine the relation among three-point couplings. We can derive the same
relation if we instead require the amplitude to obey Ward identity in the
massless limit, which illuminates the connection between on-shell
constructibility for $n\geq 4$ and Ward identity.

In the rest of this paper, we focus on the EW theory without additional higher
dimensional operators and therefore assume $[g]\geq
0$.\natural4\natural4$\natural$4We see ALT shift allows for a systematic
discussion on the contract terms, extracting which part of the coefficient is
determined from the consistent conditions from lower point amplitudes and
which part is new input. The relation (2.19) is then sufficient for our
purpose. Higher dimensional operators, in general, tend to spoil on-shell
constructibility since it is not necessarily related to lower-point
amplitudes. Indeed, higher dimensional operators correspond to $[g]<0$, and
this makes the large-$z$ behavior under the ALT shift worse. In this regard,
we may note that in the above discussion, we have focused on a transverse
polarization of a single particle, but one may ask if having more than one
transverse modes further improve the boundary behavior with ALT shift. If such
an improvement exists, one may apply it for recursion relations in the
presence of higher dimensional operators to overcome the worse large-$z$
behavior.\natural5\natural5$\natural$5 For recent developments in applying the
massive spinor formalism to SM effective field theories, see [28, 29, 30, 31,
32, 33].

##  3 On-shell construction of electroweak theory

We now construct the four-point amplitudes of the $W$\- and $Z$-bosons in the
EW theory using on-shell recursion relation. In this section, we do not
include the Standard Model (SM) Higgs, assuming that only the EW gauge bosons
and SM fermions exist in the spectrum. We will see that the ALT shift
correctly constructs the amplitudes, including the contact
terms,\natural6\natural6$\natural$6 This corresponds to the gauge four-point
interactions in the Feynman diagrammatic language.  solely from the three-
point amplitudes, independently of the presence/absence of the
Higgs.\natural7\natural7$\natural$7 Indeed, the constructibility is still
valid because the Ward identity for massless gauge boson is independent of the
details of Higgsing. The momentum-shifted parts of the three-point amplitudes
play an essential role in constructing the contact term as we see below, and
this would not be achieved without introducing an explicit momentum shift.

Our result, with the contact term derived by the ALT shift, automatically
guarantees the cancellation of the $\mathcal{O}(E^{4})$ term of all-
longitudinal gauge boson scatterings at high energy, where $E$ denotes the
center-of-mass energy. We still have finite $\mathcal{O}(E^{2})$ terms in our
amplitudes, and we have different possibilities of canceling this divergence,
e.g. by introducing the SM Higgs, as we discuss in Sec. 4. Thus, our
calculation based on the ALT shift clarifies the different natures of the
$\mathcal{O}(E^{4})$ and $\mathcal{O}(E^{2})$ terms in the four longitudinal
gauge boson scatterings; the $\mathcal{O}(E^{4})$ term is controlled by the
gauge symmetry and the Ward identity, while the $\mathcal{O}(E^{2})$ term
requires additional input for the cancellation. This is, of course, well-known
in the context of the Feynman diagrammatic calculations [34, 35, 36], but we
believe that we have derived this fact, for the first time, based solely on
the on-shell method. This is in contrast to, e.g., the discussions in [21, 22]
where the authors do not introduce a momentum shift, and both the four-point
contact terms and the Higgs contributions are added by hand, on equal footing,
to improve the high-energy behavior.

###  3.1 Three-point amplitude

For the on-shell construction of the four-point EW amplitudes, we need three-
point amplitudes involving massive spin-1 particles $W^{\pm},Z$ with masses
$m_{W},m_{Z}$, and a massless spin-1 particle $\gamma$. We denote $W^{+}$ and
$W^{-}$ as “$1$” and “$2$”, and $Z$ or $\gamma$ as “$3$”. As discussed in [17,
37, 30], the minimal three-point amplitudes
are\natural8\natural8$\natural$8Here we comment on one observation of the
minimal form of the three-point amplitude and its relation to
constructibility. For amplitudes involving the exchange of massive bosons, the
propagator has $pp/m^{2}$ in the numerator, and the amplitude takes the form
$J\Pi J$ where $\Pi$ is the propagator and $J$ is the current associated to
the gauge boson. If we require gauge invariance in the massless limit
(equivalently, Ward identity for massless gauge bosons), i.e.
$p^{\mu}J_{\mu}=\mathcal{O}(m)$, the $pp/m^{2}$ term in the propagator
contracts with $J$ on both sides to give $\mathcal{O}(m)$. Without the minimal
coupling, the $pp/m^{2}$ term will stay which in general spoils large-$z$
behavior, hence spoils constructibility.

$\displaystyle A_{WWZ}^{(\lambda_{1}\lambda_{2}\lambda_{3})}$
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}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\hbox to0.0pt{ \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), };
\hss}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[inline = (base.a),
horizontal=p1 to v] \vertex(p1) [label=${\scriptstyle W^{+}}$]; \vertex[below
right = of p1] (v); \vertex[below left = of v,label=270:${\scriptstyle
W^{-}}$] (p2); \vertex[right = of v, label=${\scriptstyle Z}$] (p3);
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\pgfsys@rect{0.24718pt}{-3.33301pt}{6.66602pt}{6.66602pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.58018pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }{bg}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\frac{g_{WWZ}}{\sqrt{2}m_{W}^{2}m_{Z}}\left[\langle\textbf{1}\textbf{2}\rangle[\textbf{2}\textbf{1}]\langle\textbf{3}\lvert
p_{1}-p_{2}\lvert\textbf{3}]+(\mathrm{cycl.})\right],$ (3.1) $\displaystyle
A_{WW\gamma}^{(\lambda_{1}\lambda_{2}\lambda_{3})}$
$\displaystyle=\leavevmode\hbox to6.67pt{\vbox
to6.67pt{\pgfpicture\makeatletter\hbox{\hskip-0.24718pt\lower-3.33301pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\hbox to0.0pt{\hbox to0.0pt{
\diagram*{ (p1) -- [photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), };
\hss}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[inline = (base.a),
horizontal=p1 to v] \vertex(p1) [label=${\scriptstyle W^{+}}$]; \vertex[below
right = of p1] (v); \vertex[below left = of v,label=270:${\scriptstyle
W^{-}}$] (p2); \vertex[right = of v, label=${\scriptstyle\gamma}$] (p3);
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\pgfsys@rect{0.24718pt}{-3.33301pt}{6.66602pt}{6.66602pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.58018pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }{bg}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\begin{cases}\displaystyle\frac{g_{WW\gamma}}{\sqrt{2}m_{W}^{2}\langle
3\xi_{3}\rangle}\left[\langle\mathbf{12}\rangle[\mathbf{21}]\langle{\xi_{3}}|(p_{1}-p_{2})|3]+(\mathrm{cycl.})\right],&\mathrm{for}~{}~{}\lambda_{3}=+,\vspace{1mm}\\\
\displaystyle-\frac{g_{WW\gamma}}{\sqrt{2}m_{W}^{2}[3\xi_{3}]}\left[\langle\mathbf{12}\rangle[\mathbf{21}]\langle{3}|(p_{1}-p_{2})|\xi_{3}]+(\mathrm{cycl.})\right],&\mathrm{for}~{}~{}\lambda_{3}=-,\end{cases}$
(3.2)

where we used the bold notation for the massive spinors,
$|\xi_{3}\rangle,|\xi_{3}]$ represent the reference spinor of the photon, and
“(cycl.)” indicates the cyclic permutation of $1,2$ and $3$. Note that the
three-point amplitudes satisfy

$\displaystyle\left.A_{WWZ}^{(\lambda_{1}\lambda_{2}\lambda_{3})}\right|_{\epsilon_{k}\to
p_{k}}=-g_{WWZ}\frac{m_{i}^{2}-m_{j}^{2}}{m_{i}m_{j}}\langle\mathbf{ij}\rangle[\mathbf{ji}],\quad\left.A_{WW\gamma}^{(\lambda_{1}\lambda_{2}\lambda_{3})}\right|_{\epsilon_{3}\to
p_{3}}=0,$ (3.3)

where $i,j,k$ are understood to be cyclic, and the subscript means that we
replace the polarization vector $\epsilon_{k}$ of the particle $k$ to its
momentum $p_{k}$. For notational compactness, it is useful to define

$\displaystyle V_{a\dot{a};b\dot{b};c\dot{c}}(p_{1},p_{2},p_{3})$
$\displaystyle=\frac{1}{4}\left[\epsilon_{ab}\epsilon_{\dot{a}\dot{b}}(p_{1}-p_{2})_{c\dot{c}}+\epsilon_{bc}\epsilon_{\dot{b}\dot{c}}(p_{2}-p_{3})_{a\dot{a}}+\epsilon_{ca}\epsilon_{\dot{c}\dot{a}}(p_{3}-p_{1})_{b\dot{b}}\right].$
(3.4)

With this function, the three-point amplitudes are written as

$\displaystyle A_{WWZ}^{(\lambda_{1}\lambda_{2}\lambda_{3})}$
$\displaystyle=g_{WWZ}\epsilon_{1}^{\dot{a}{a}}\epsilon_{2}^{\dot{b}b}\epsilon_{3}^{\dot{c}c}V_{a\dot{a};b\dot{b};c\dot{c}}(p_{1},p_{2},p_{3}),$
(3.5) $\displaystyle A_{WW\gamma}^{(\lambda_{1}\lambda_{2}\lambda_{3})}$
$\displaystyle=g_{WW\gamma}\epsilon_{1}^{\dot{a}{a}}\epsilon_{2}^{\dot{b}b}\epsilon_{3}^{\dot{c}c}V_{a\dot{a};b\dot{b};c\dot{c}}(p_{1},p_{2},p_{3}),$
(3.6)

where we use a short-hand notation
$\epsilon_{i}=\epsilon^{(\lambda_{i})}(p_{i})$.

###  3.2 $WWWW$ amplitude

We begin with the $WWWW$ amplitude. We use the ALT shift to construct this
amplitude and show that the contact term naturally emerges from recursion
relations. Since this is one of our main results, we explain the calculation
in some detail in the following.

We label $W^{+}W^{-}W^{+}W^{-}$ respectively as 1,2,3,4, and take all the
particles as incoming hereafter. In this process, we have the poles at

$\displaystyle\hat{p}_{12}^{2}=m_{Z}^{2},\,\mathrm{or}~{}\,0,\quad\hat{p}_{14}^{2}=m_{Z}^{2},\,\mathrm{or}~{}\,0,$
(3.7)

where we use the short-hand notation $\hat{p}_{ij}=\hat{p}_{i}+\hat{p}_{j}$.
We assume that the particle $1$ is transverse, $\lambda_{1}=\pm$. The
amplitude is constructible, and we take
$B_{\infty}=0$.\natural9\natural9$\natural$9One could take different points of
view here. A common one is that one has to prove that the amplitude is
constructible from other inputs, e.g., as shown in Sec. 2.2 with Ward
identity. Another possibility is to assume constructibility and study the
behavior of the constructed from pole terms within a given shift and determine
if it is compatible with the constructibility assumption. We defer the
detailed discussion on these for future works. The amplitude can then be
expressed solely in terms of the pole contributions (see Eq. (2.6))

$\displaystyle A_{WWWW}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}$
$\displaystyle=\sum_{I=Z,\gamma}\sum_{i=2,4}\frac{1}{p_{1i}^{2}-m_{I}^{2}}\frac{1}{{z^{+}_{1i}-z^{-}_{1i}}}$
$\displaystyle\times\sum_{\lambda}\left[{z^{+}_{1i}}\hat{A}_{WWI}^{(\lambda_{1}\lambda_{i}\lambda)}(z^{-}_{1i})\times\hat{A}_{WWI}^{(\lambda_{3}\lambda_{j}\bar{\lambda})}(z^{-}_{1i})-{z^{-}_{1i}}\hat{A}_{WWI}^{(\lambda_{1}\lambda_{i}\lambda)}(z^{+}_{1i})\times\hat{A}_{WWI}^{(\lambda_{3}\lambda_{j}\bar{\lambda})}(z^{+}_{1i})\right],$
(3.8)

where $j\neq i,1,3$ and we denote the solution of
$\hat{p}_{ij}^{2}=m_{Z}^{2},0$ as
$z=z_{ij}^{\pm}$.\natural10\natural10$\natural$10 The solution
$z=z_{ij}^{\pm}$ also depends on the mass of the intermediate particle
$m_{I}^{2}$, but we do not put an explicit “$I$”-index to $z_{ij}^{\pm}$ for
notational simplicity.

To simplify this, we use the completeness relation of the polarization vectors
of the intermediate particle. The intermediate particles have polarization
vectors whose little group indices are contracted, which we indicate by
$\lambda$ and $\bar{\lambda}$, and the contraction is expressed as

$\displaystyle[\epsilon^{IJ}(\hat{p}_{1i})]_{a\dot{a}}[\epsilon_{IJ}(-\hat{p}_{1i})]_{b\dot{b}}$
$\displaystyle=-[\epsilon^{IJ}(\hat{p}_{1i})]_{a\dot{a}}[\epsilon_{IJ}(\hat{p}_{1i})]_{b\dot{b}}$
$\displaystyle=\sum_{\lambda=\pm,L}[\epsilon^{(\lambda)}(\hat{p}_{1i})]_{a\dot{a}}[\epsilon^{(-\lambda)}(\hat{p}_{1i})]_{b\dot{b}}=-2\epsilon_{ab}\epsilon_{\dot{a}\dot{b}}+\frac{[\hat{p}_{1i}]_{a\dot{a}}[\hat{p}_{1i}]_{b\dot{b}}}{m_{Z}^{2}},$
(3.9)

for the intermediate $Z$-boson, where we used Eq. (A.18) in the first line and
Eqs. (A.36) and (A.39) in the second line. Similarly, we have

$\displaystyle[\epsilon^{IJ}(\hat{p}_{1i})]_{a\dot{a}}[\epsilon_{IJ}(-\hat{p}_{1i})]_{b\dot{b}}=-2\epsilon_{ab}\epsilon_{\dot{a}\dot{b}}+\frac{[\hat{p}_{1i}]_{a\dot{a}}[\hat{\bar{p}}_{1i}]_{b\dot{b}}+[\hat{\bar{p}}_{1i}]_{a\dot{a}}[\hat{p}_{1i}]_{b\dot{b}}}{\hat{p}_{1i}\cdot\hat{\bar{p}}_{1i}},$
(3.10)

for the intermediate photon where $\bar{p}_{i}=|\xi_{i}\rangle[\xi_{i}|$. It
then follows from Eq. (3.3) that we can write the product of the three-point
amplitudes as

$\displaystyle\sum_{\lambda}\hat{A}_{WWI}^{(\lambda_{1}\lambda_{i}\lambda)}\times\hat{A}_{WWI}^{(\lambda_{3}\lambda_{j}\bar{\lambda})}=-2g_{WWI}^{2}\epsilon_{1}^{\dot{a}a}\epsilon_{i}^{\dot{b}b}V_{a\dot{a};b\dot{b};e\dot{e}}(\hat{p}_{1},\hat{p}_{i},-\hat{p}_{1i}){V_{c\dot{c};d\dot{d};}}^{\dot{e}e}(\hat{p}_{3},\hat{p}_{j},-\hat{p}_{3j})\epsilon_{3}^{\dot{c}c}\epsilon_{j}^{\dot{d}d},$
(3.11)

where we do not have hats on the external polarization vectors since the ALT
shift does not shift them. Since $V$ is linear in momentum, the product
contains up to quadratic terms in $z_{1i}^{\pm}$

$\displaystyle
V_{a\dot{a};b\dot{b};e\dot{e}}(\hat{p}_{1},\hat{p}_{i},-\hat{p}_{1i}){V_{c\dot{c};d\dot{d};}}^{\dot{e}e}(\hat{p}_{3},\hat{p}_{j},-\hat{p}_{3j})$
$\displaystyle=V_{a\dot{a};b\dot{b};c\dot{c};d\dot{d}}^{(0)}+V_{a\dot{a};b\dot{b};c\dot{c};d\dot{d}}^{(1)}z_{1i}^{\pm}+V_{a\dot{a};b\dot{b};c\dot{c};d\dot{d}}^{(2)}\left(z_{1i}^{\pm}\right)^{2}.$
(3.12)

Among these terms, $V^{(0)}$ is simply the product of the three-point
amplitude without the momentum shift, while $V^{(1)}$ does not contribute to
the final result since the contributions from $z_{1i}^{\pm}$ cancel with each
other in Eq. (3.8). In general, any linear dependence in $z$ will cancel after
summing over all factorization channels. The most interesting contribution is
the quadratic term $V^{(2)}$. The overall $z_{1i}^{\pm}$ dependence of
$V^{(2)}$ in Eq. (3.8) is given by

$\displaystyle\frac{z_{1i}^{+}(z_{1i}^{-})^{2}-z_{1i}^{-}(z_{1i}^{+})^{2}}{z_{1i}^{+}-z_{1i}^{-}}=-\frac{p_{1i}^{2}-m_{I}^{2}}{2q_{1}\cdot
q_{i}}.$ (3.13)

The numerator of this expression cancels the propagator
$1/(p_{1i}^{2}-m_{I}^{2})$, and hence $V^{(2)}$ provides the contact term of
the $WWWW$ scattering. We thus obtain

$\displaystyle A_{WWWW}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}=$
$\displaystyle-\sum_{I=Z,\gamma}\sum_{i=2,4}\frac{2g_{WWI}^{2}}{p_{1i}^{2}-m_{I}^{2}}\epsilon_{1}^{\dot{a}a}\epsilon_{i}^{\dot{b}b}V_{a\dot{a};b\dot{b};e\dot{e}}({p}_{1},{p}_{i},-{p}_{1i}){V_{c\dot{c};d\dot{d};}}^{\dot{e}e}({p}_{3},{p}_{j},-{p}_{3j})\epsilon_{3}^{\dot{c}c}\epsilon_{j}^{\dot{d}d}$
$\displaystyle+\sum_{i=2,4}\frac{g^{2}}{q_{1}\cdot
q_{i}}\epsilon_{1}^{\dot{a}a}\epsilon_{i}^{\dot{b}b}V_{a\dot{a};b\dot{b};e\dot{e}}({q}_{1},{q}_{i},-{q}_{1i}){V_{c\dot{c};d\dot{d};}}^{\dot{e}e}({q}_{3},{q}_{j},-{q}_{3j})\epsilon_{3}^{\dot{c}c}\epsilon_{j}^{\dot{d}d},$
(3.14)

where the first term comes from $V^{(0)}$ while the second term does from
$V^{(2)}$, and we define

$\displaystyle g^{2}=g_{WWZ}^{2}+g_{WW\gamma}^{2}.$ (3.15)

To simplify the second line, we recall that we assume $\lambda_{1}=\pm$, and
hence the ALT shift takes the form

$\displaystyle\epsilon_{1}$ $\displaystyle=\frac{q_{1}}{c_{1}m_{W}}.$ (3.16)

We note that

$\displaystyle
q_{1}^{\dot{a}a}V_{a\dot{a};b\dot{b};e\dot{e}}(q_{1},q_{i},-q_{1i})$
$\displaystyle=(q_{1}\cdot
q_{i})\epsilon_{be}\epsilon_{\dot{b}\dot{e}}-\frac{1}{4}[q_{i}]_{b\dot{b}}[q_{1}]_{e\dot{e}}-\frac{1}{4}[q_{1}]_{b\dot{b}}(q_{1}+q_{i})_{e\dot{e}}.$
(3.17)

In this expression, the second term vanishes after contracting with the
polarization vector $\epsilon_{i}$, while the third term vanishes after
contracting with the other vertex function. This leaves us the first term, and
we obtain

$\displaystyle\sum_{i=2,4}\frac{\epsilon_{1}^{\dot{a}a}\epsilon_{i}^{\dot{b}b}}{q_{1}\cdot
q_{i}}V_{a\dot{a};b\dot{b};e\dot{e}}({q}_{1},{q}_{i},-{q}_{1i}){V_{c\dot{c};d\dot{d};}}^{\dot{e}e}({q}_{3},{q}_{j},-{q}_{3j})\epsilon_{3}^{\dot{c}c}\epsilon_{j}^{\dot{d}d}$
$\displaystyle=\sum_{i=2,4}\frac{\epsilon_{i}^{\dot{b}b}}{c_{1}m_{W}}V_{c\dot{c};d\dot{d};b\dot{b}}(q_{3},q_{j},-q_{3j})\epsilon_{3}^{\dot{c}c}\epsilon_{j}^{\dot{d}d}.$
(3.18)

With $\sum_{i}q_{i}=0$ and $\epsilon_{i}\cdot q_{i}=0$, we can simplify the
last expression as

$\displaystyle\sum_{i=2,4}\frac{\epsilon_{i}^{\dot{b}b}}{c_{1}m_{W}}V_{c\dot{c};d\dot{d};b\dot{b}}(q_{3},q_{j},-q_{3j})\epsilon_{3}^{\dot{c}c}\epsilon_{j}^{\dot{d}d}$
$\displaystyle=-\frac{1}{4}\epsilon_{1}^{\dot{a}a}\epsilon_{3}^{\dot{b}{b}}\left[2\epsilon_{ab}\epsilon_{\dot{a}\dot{b}}\epsilon_{cd}\epsilon_{\dot{c}\dot{d}}-\epsilon_{ac}\epsilon_{\dot{a}\dot{c}}\epsilon_{bd}\epsilon_{\dot{b}\dot{d}}-\epsilon_{ad}\epsilon_{\dot{a}\dot{d}}\epsilon_{bc}\epsilon_{\dot{b}\dot{c}}\right]\epsilon_{2}^{\dot{c}c}\epsilon_{4}^{\dot{d}d}.$
(3.19)

We thus obtain

$\displaystyle A_{WWWW}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}=$
$\displaystyle-\sum_{I=Z,\gamma}\sum_{i=2,4}\frac{2g_{WWI}^{2}}{p_{1i}^{2}-m_{I}^{2}}\epsilon_{1}^{\dot{a}a}\epsilon_{i}^{\dot{b}b}V_{a\dot{a};b\dot{b};e\dot{e}}({p}_{1},{p}_{i},-{p}_{1i}){V_{c\dot{c};d\dot{d};}}^{\dot{e}e}({p}_{3},{p}_{j},-{p}_{3j})\epsilon_{3}^{\dot{c}c}\epsilon_{j}^{\dot{d}d}$
$\displaystyle-\frac{g^{2}}{4}\epsilon_{1}^{\dot{a}a}\epsilon_{3}^{\dot{b}{b}}\left[2\epsilon_{ab}\epsilon_{\dot{a}\dot{b}}\epsilon_{cd}\epsilon_{\dot{c}\dot{d}}-\epsilon_{ac}\epsilon_{\dot{a}\dot{c}}\epsilon_{bd}\epsilon_{\dot{b}\dot{d}}-\epsilon_{ad}\epsilon_{\dot{a}\dot{d}}\epsilon_{bc}\epsilon_{\dot{b}\dot{c}}\right]\epsilon_{2}^{\dot{c}c}\epsilon_{4}^{\dot{d}d}.$
(3.20)

Its explicit form for all helicity combinations is given by

$\displaystyle A_{WWWW}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}=$
$\displaystyle-\sum_{I=Z,\gamma}\sum_{i=2,4}\frac{g^{2}_{WWI}}{m_{W}^{4}(p_{1i}^{2}-m_{I}^{2})}\bigg{[}2{p}_{1}\cdot({p}_{3}-{p}_{j})\langle{\textbf{1}}{\textbf{i}}\rangle[{\textbf{i}}{\textbf{1}}]\langle{\textbf{3}}{\textbf{j}}\rangle[{\textbf{j}}{\textbf{3}}]$
$\displaystyle-\langle{\textbf{1}}{\textbf{i}}\rangle[{\textbf{i}}{\textbf{1}}]\big{(}\langle{\textbf{3}}\lvert({p}_{1}-{p}_{i})\lvert{\textbf{3}}]\langle{\textbf{j}}\lvert{p}_{3}\lvert{\textbf{j}}]-\langle{\textbf{j}}\lvert({p}_{1}-{p}_{i})\lvert{\textbf{j}}]\langle{\textbf{3}}\lvert{p}_{j}\lvert{\textbf{3}}]\big{)}+(({1},{i})\leftrightarrow({3},{j}))$
$\displaystyle+2\langle{\textbf{1}}{\textbf{3}}\rangle[{\textbf{3}}{\textbf{1}}]\langle{\textbf{i}}\lvert{p}_{1}\lvert{\textbf{i}}]\langle{\textbf{j}}\lvert{p}_{3}\lvert{\textbf{j}}]-2\langle{\textbf{1}}{\textbf{j}}\rangle[{\textbf{j}}{\textbf{1}}]\langle{\textbf{i}}\lvert{p}_{1}\lvert{\textbf{i}}]\langle{\textbf{3}}\lvert{p}_{j}\lvert{\textbf{3}}]-({1}\leftrightarrow{i})\bigg{]}$
$\displaystyle-\frac{g^{2}}{m_{W}^{4}}\bigg{[}2\langle\textbf{24}\rangle[\textbf{42}]\langle\textbf{13}\rangle[\textbf{31}]-\langle\textbf{23}\rangle[\textbf{32}]\langle\textbf{14}\rangle[\textbf{41}]-\langle\textbf{12}\rangle[\textbf{21}]\langle\textbf{34}\rangle[\textbf{43}]\bigg{]},$
(3.21)

where we use the bold notation for the polarization spinors. This is our
result of the $WWWW$ scattering based on the on-shell recursion relation.
Although the amplitude was constructed assuming $\lambda_{1}=\pm$, the little-
group covariance and spin-raising/lowering operators allow us to write the
amplitude for all polarizations. We check and find that this agrees with the
Feynman diagrammatic result.

We summarize several properties of our computation:

  * •

The final result does not depend on $q_{i}$. This is expected since the
original amplitude does not depend on how we shift the momentum. Therefore,
the $q_{i}$-independence serves as a consistency check of our argument that
$B_{\infty}=0$. Indeed, our simplification of the $z^{2}$ term is analogous to
the Ward identity in the massless theory.

  * •

We checked explicitly that the $q_{i}$-dependence does not cancel for all
longitudinal scattering when directly summing the pole contributions with ALT
shift. \natural11\natural11$\natural$11While the additional degrees of freedom
in $c_{i}$ could be further utilized to remove the explicit $q_{i}$ dependence
for this amplitude, this general $q_{i}$ dependence, which is outside the
allowed components (polarization vectors and momentum vectors) of the physical
amplitudes for all-longitudinal scattering, indicates the constructibility
issue with the vanilla ALT shift under study for this particular amplitude.
This is consistent with our discussion in Sec. 2.2, that we cannot guarantee
the constructibility of this amplitude with our shift through dimensional
analysis, even with additional external input. Instead, one can obtain the
correct longitudinal amplitude through spin raising and lowering operators.

  * •

We do not need an explicit form of $c_{i}$. The square root of the momentum
products in $z_{1i}^{\pm}$ also cancels since the poles always appear in
pairs.

  * •

The contact term arises from the $z^{2}$ term in the product of the three-
point amplitudes, which in turn arises from the parts dependent quadratically
on the external momenta. Therefore, we can reconstruct this term _only if_ we
introduce the explicit momentum shift such as the ALT
shift.\natural12\natural12$\natural$12Note that Ref. [23] also got the contact
term by dropping terms depending on $\hat{s}$, which is a process allowed by
consistent shift but not necessarily uniquely yield the correct amplitude. It
would be interesting to see how far along this method can go for higher point
amplitudes or general theories, while the consistent ALT shift introduced here
is guaranteed to apply to general theories.

In particular, we believe that we, for the first time, have derived the
contact term of the $WWWW$ scattering explicitly in the on-shell recursion
relation.

#### High-energy limit

As we have noted, even though the ALT shift by itself does not allow the
construction of the all-longitudinal $WWWW$ scattering, we can exploit the
little-group covariance to construct it from the amplitude with (at least) one
transverse mode. We can then investigate the high-energy limit of the all-
longitudinal $WWWW$ amplitude.

We take the high-energy limit with a fixed scattering angle and let
$s,t\to\infty$, where we define the Mandelstam variables
$(s,t,u)=(p_{12}^{2},p_{13}^{2},p_{14}^{2})$. In this limit, the amplitude
with all longitudinal modes behaves as

$\displaystyle\left.A_{WWWW}^{(LLLL)}\right|_{Z,\gamma}=$
$\displaystyle-\left(g^{2}-g_{WWZ}^{2}-g_{WW\gamma}^{2}\right)\frac{s^{2}+4su+u^{2}}{4m_{W}^{4}}$
$\displaystyle-\frac{s+u}{4m_{W}^{4}}\left[4(2g_{WWZ}^{2}+2g_{WW\gamma}^{2}-g^{2})m_{W}^{2}-3g_{WWZ}^{2}m_{Z}^{2}+\frac{8m_{W}^{2}u}{s}(g^{2}-g_{WWZ}^{2}-g_{WW\gamma}^{2})\right]$
$\displaystyle+\mathcal{O}(E^{0}),$ (3.22)

where we make it explicit that Eq. (3.15) guarantees the cancellation of the
$\mathcal{O}(E^{4})$ term, and the subscript indicates that this amplitude is
from the intermediate $Z$ and $\gamma$ only (and no Higgs). After using Eq.
(3.15), which we derived using the ALT shift, it is simplified as

$\displaystyle\left.A_{WWWW}^{(LLLL)}\right|_{Z,\gamma}$
$\displaystyle=-\frac{s+u}{4m_{W}^{4}}(4g^{2}m_{W}^{2}-3g_{WWZ}^{2}m_{Z}^{2})+\mathcal{O}(E^{0}).$
(3.23)

Therefore the $\mathcal{O}(E^{4})$ term cancels automatically in our recursive
construction of the amplitudes, while the $\mathcal{O}(E^{2})$ cancellation
requires additional input, such as an introduction of the Higgs, which is the
main topic of Sec. 4.

###  3.3 $WWZZ$ amplitude

We next consider the $WWZZ$ scattering. The computation is analogous to the
$WWWW$ scattering, and therefore, we omit details. Labelling $W^{+},W^{-},Z,Z$
as $1,2,3,4$, we have the poles at

$\displaystyle\hat{p}_{13}^{2}=m_{W}^{2},\quad\hat{p}_{14}^{2}=m_{W}^{2}.$
(3.24)

We take the particle $1$ as transverse, $\lambda_{1}=\pm$, which guarantees
$B_{\infty}=0$ with ALT shift, and write

$\displaystyle A_{WWZZ}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}$
$\displaystyle=\sum_{i=3,4}\frac{1}{p_{1i}^{2}-m_{W}^{2}}\frac{1}{{z^{+}_{1i}-z^{-}_{1i}}}$
$\displaystyle\times\sum_{\lambda}\left[{z^{+}_{1i}}\hat{A}_{WWZ}^{(\lambda_{1}\lambda\lambda_{i})}(z^{-}_{1i})\times\hat{A}_{WWZ}^{(\bar{\lambda}\lambda_{2}\lambda_{j})}(z^{-}_{1i})-{z^{-}_{1i}}\hat{A}_{WWZ}^{(\lambda_{1}\lambda\lambda_{i})}(z^{+}_{1i})\times\hat{A}_{WWZ}^{(\bar{\lambda}\lambda_{2}\lambda_{j})}(z^{+}_{1i})\right].$
(3.25)

We use the completeness relation of the intermediate polarization vector,

$\displaystyle[\epsilon^{IJ}(\hat{p}_{1i})]_{a\dot{a}}[\epsilon_{IJ}(-\hat{p}_{1i})]_{b\dot{b}}$
$\displaystyle=-2\epsilon_{ab}\epsilon_{\dot{a}\dot{b}}+\frac{[\hat{p}_{1i}]_{a\dot{a}}[\hat{p}_{1i}]_{b\dot{b}}}{m_{W}^{2}}.$
(3.26)

In the present case, the second term gives a finite contribution as opposed to
the $WWWW$ scattering. With Eq. (3.3), this contribution does not change under
the momentum shift, and hence we can drop the hats from the momentum for this
term. Therefore, the $z^{2}$-term arises solely from the first term, and with
a similar computation as the $WWWW$ scattering, we obtain

$\displaystyle A_{WWZZ}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}=$
$\displaystyle-\sum_{i=3,4}\frac{2g_{WWZ}^{2}}{p_{1i}^{2}-m_{W}^{2}}\epsilon_{1}^{\dot{a}a}\epsilon_{i}^{\dot{b}b}V_{a\dot{a};e\dot{e};b\dot{b}}({p}_{1},-{p}_{1i},p_{i})\left[\epsilon^{ef}\epsilon^{\dot{e}\dot{f}}-\frac{[p_{1i}]^{\dot{e}e}[p_{1i}]^{\dot{f}f}}{2m_{W}^{2}}\right]V_{f\dot{f};c\dot{c};d\dot{d}}(-{p}_{2j},{p}_{2},{p}_{j})\epsilon_{2}^{\dot{c}c}\epsilon_{j}^{\dot{d}d}$
$\displaystyle+\sum_{i=3,4}\frac{g_{WWZ}^{2}}{q_{1}\cdot
q_{i}}\epsilon_{1}^{\dot{a}a}\epsilon_{i}^{\dot{b}b}V_{a\dot{a};e\dot{e};b\dot{b}}({q}_{1},-{q}_{1i},{q}_{i}){V^{\dot{e}e;}}_{c\dot{c};d\dot{d}}(-q_{2j},{q}_{2},{q}_{j})\epsilon_{2}^{\dot{c}c}\epsilon_{j}^{\dot{d}d},$
(3.27)

where $j\neq 1,2,i$. Again, the term in the second line arising from $z^{2}$
provides the contact term. As before, we can simplify it with the relation
$\epsilon_{1}=q_{1}/c_{1}m_{W}$, and we obtain

$\displaystyle A_{WWZZ}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}=$
$\displaystyle-\sum_{i=3,4}\frac{2g_{WWZ}^{2}}{p_{1i}^{2}-m_{W}^{2}}\epsilon_{1}^{\dot{a}a}\epsilon_{i}^{\dot{b}b}V_{a\dot{a};e\dot{e};b\dot{b}}({p}_{1},-{p}_{1i},p_{i})\left[\epsilon^{ef}\epsilon^{\dot{e}\dot{f}}-\frac{[p_{1i}]^{\dot{e}e}[p_{1i}]^{\dot{f}f}}{2m_{W}^{2}}\right]V_{f\dot{f};c\dot{c};d\dot{d}}(-{p}_{2j},{p}_{2},{p}_{j})\epsilon_{2}^{\dot{c}c}\epsilon_{j}^{\dot{d}d}$
$\displaystyle+\frac{g_{WWZ}^{2}}{4}\epsilon_{1}^{\dot{a}a}\epsilon_{2}^{\dot{b}{b}}\left[2\epsilon_{ab}\epsilon_{\dot{a}\dot{b}}\epsilon_{cd}\epsilon_{\dot{c}\dot{d}}-\epsilon_{ac}\epsilon_{\dot{a}\dot{c}}\epsilon_{bd}\epsilon_{\dot{b}\dot{d}}-\epsilon_{ad}\epsilon_{\dot{a}\dot{d}}\epsilon_{bc}\epsilon_{\dot{b}\dot{c}}\right]\epsilon_{3}^{\dot{c}c}\epsilon_{4}^{\dot{d}d}.$
(3.28)

Its explicit form is given by

$\displaystyle A_{WWZZ}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}=$
$\displaystyle-\sum_{i=3,4}\frac{g^{2}_{WWZ}}{m_{W}^{2}m_{Z}^{2}(p_{1i}^{2}-m_{W}^{2})}\bigg{[}\left(2({p}_{1}\cdot{p}_{2}-{p}_{1}\cdot{p}_{j})-\frac{(m_{W}^{2}-m_{Z}^{2})^{2}}{m_{W}^{2}}\right)\langle{\textbf{1i}}\rangle[{\textbf{i1}}]\langle{\textbf{2j}}\rangle[{\textbf{j2}}]$
$\displaystyle-\langle{\textbf{1i}}\rangle[{\textbf{i1}}]\langle{\textbf{2}}\lvert({p}_{1}-{p}_{i})\lvert{\textbf{2}}]\langle{\textbf{j}}\lvert{p}_{2}\lvert{\textbf{j}}]+\langle{\textbf{1i}}\rangle[{\textbf{i1}}]\langle{\textbf{j}}\lvert({p}_{1}-{p}_{i})\lvert{\textbf{j}}]\langle{\textbf{2}}\lvert{p}_{j}\lvert{\textbf{2}}]+({1},{i})\leftrightarrow({2},{j})$
$\displaystyle+2\langle{\textbf{12}}\rangle[{\textbf{21}}]\langle{\textbf{i}}\lvert{p}_{1}\lvert{\textbf{i}}]\langle{\textbf{j}}\lvert{p}_{2}\lvert{\textbf{j}}]-2\langle{\textbf{1j}}\rangle[{\textbf{j1}}]\langle{\textbf{i}}\lvert{p}_{1}\lvert{\textbf{i}}]\langle{\textbf{2}}\lvert{p}_{j}\lvert{\textbf{2}}]-({1}\leftrightarrow{i})\bigg{]},$
$\displaystyle+\frac{g_{WWZ}^{2}}{m_{W}^{2}m_{Z}^{2}}\bigg{[}2\langle\textbf{12}\rangle[\textbf{21}]\langle\textbf{34}\rangle[\textbf{43}]-\langle\textbf{14}\rangle[\textbf{41}]\langle\textbf{23}\rangle[\textbf{32}]-\langle\textbf{13}\rangle[\textbf{31}]\langle\textbf{24}\rangle[\textbf{42}]\bigg{]}.$
(3.29)

This is our result of the $WWZZ$ scattering.

#### High-energy limit

The high-energy limit of the $WWZZ$ amplitude is given by

$\displaystyle\left.A_{WWZZ}^{(LLLL)}\right|_{W}=\frac{g_{WWZ}^{2}m_{Z}^{2}(t+u)}{4m_{W}^{4}}+\mathcal{O}(E^{0}).$
(3.30)

In particular, the $\mathcal{O}(E^{4})$ term again automatically vanishes in
our on-shell computation, while the $\mathcal{O}(E^{2})$ term needs additional
input.

###  3.4 $WWtt$ amplitude

In this subsection, we illustrate the on-shell construction of four-point
massive gauge boson and fermion amplitude with dimensionless couplings. We
will explicitly work out $WWtt$ amplitude as an example, but the discussion is
general and it can be extended to other processes. For this, we introduce one
generation of quarks, namely $t$ and $b$ quarks with masses $m_{t}$ and
$m_{b}$, and assume their electric charges are $+2/3$ and $-1/3$ respectively.
With these additional particles, we can write down the following minimal
three-point interaction

$\displaystyle A_{tbW}^{(\lambda_{1}\lambda_{2}\lambda_{3})}$
$\displaystyle=\leavevmode\hbox to6.67pt{\vbox
to6.67pt{\pgfpicture\makeatletter\hbox{\hskip-0.24718pt\lower-3.33301pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\hbox to0.0pt{\hbox to0.0pt{\hbox
to0.0pt{ \diagram*{ (p1) -- [photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), };
\hss}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[inline = (base.a),
horizontal=p1 to v] \vertex(p1) [label=${\scriptstyle t}$]; \vertex[below
right = of p1] (v); \vertex[below left = of
v,label=270:${\scriptstyle\bar{b}}$] (p2); \vertex[right = of v,
label=${\scriptstyle W^{-}}$] (p3);
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\pgfsys@rect{0.24718pt}{-3.33301pt}{6.66602pt}{6.66602pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.58018pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }{bg}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\frac{\sqrt{2}g_{tbW}}{m_{W}}\langle\textbf{1}\textbf{3}\rangle[\textbf{3}\textbf{2}],$
(3.31) $\displaystyle A_{ttZ}^{(\lambda_{1}\lambda_{2}\lambda_{3})}$
$\displaystyle=\leavevmode\hbox to6.67pt{\vbox
to6.67pt{\pgfpicture\makeatletter\hbox{\hskip-0.24718pt\lower-3.33301pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\hbox to0.0pt{\hbox to0.0pt{\hbox
to0.0pt{\hbox to0.0pt{ \diagram*{ (p1) -- [photon,momentum=${\scriptstyle
p_{1}}$] (v) -- [photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) --
[photon, rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), };
\hss}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[inline = (base.a),
horizontal=p1 to v] \vertex(p1) [label=${\scriptstyle{t}}$]; \vertex[below
right = of p1] (v); \vertex[below left = of
v,label=270:${\scriptstyle\bar{t}}$] (p2); \vertex[right = of v,
label=${\scriptstyle Z}$] (p3);
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\pgfsys@rect{0.24718pt}{-3.33301pt}{6.66602pt}{6.66602pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.58018pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }{bg}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\frac{\sqrt{2}}{m_{W}}\left[g_{ttZ}^{L}\langle\textbf{1}\textbf{3}\rangle[\textbf{3}\textbf{2}]+g_{ttZ}^{R}[\textbf{1}\textbf{3}]\langle\textbf{3}\textbf{2}\rangle\right],$
(3.32) $\displaystyle A_{tt\gamma}^{(\lambda_{1}\lambda_{2}\lambda_{3})}$
$\displaystyle=\leavevmode\hbox to6.67pt{\vbox
to6.67pt{\pgfpicture\makeatletter\hbox{\hskip-0.24718pt\lower-3.33301pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\hbox to0.0pt{\hbox to0.0pt{\hbox
to0.0pt{\hbox to0.0pt{\hbox to0.0pt{ \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), };
\hss}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[inline = (base.a),
horizontal=p1 to v] \vertex(p1) [label=${\scriptstyle t}$]; \vertex[below
right = of p1] (v); \vertex[below left = of
v,label=270:${\scriptstyle\bar{t}}$] (p2); \vertex[right = of v,
label=${\scriptstyle\gamma}$] (p3);
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\pgfsys@rect{0.24718pt}{-3.33301pt}{6.66602pt}{6.66602pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.58018pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }{bg}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\begin{cases}\displaystyle\frac{\sqrt{2}g_{tt\gamma}}{\langle
3\xi_{3}\rangle}\bigg{[}\langle\mathbf{2}\xi_{3}\rangle[3\mathbf{1}]+\langle\mathbf{1}\xi_{3}\rangle[3\mathbf{2}]\bigg{]},&\mathrm{for}~{}~{}\lambda_{3}=+,\vspace{1mm}\\\
\displaystyle-\frac{\sqrt{2}g_{tt\gamma}}{[3\xi_{3}]}\bigg{[}[\mathbf{2}\xi_{3}]\langle
3\mathbf{1}\rangle+[\mathbf{1}\xi_{3}]\langle
3\mathbf{2}\rangle\bigg{]},&\mathrm{for}~{}~{}\lambda_{3}=-.\end{cases}$
(3.33)

Here we have added parity violating interactions to align with the Standard
Model, but on-shell constructibiliy is independent of parity violation of
these three-point amplitudes. Now let us denote $W^{+},W^{-},{t},\bar{t}$ as
$1,2,3,4$, then the poles of the factorization channels are

$\displaystyle\hat{p}_{12}^{2}=0,\,~{}\mathrm{or}~{}\,m_{Z}^{2},\quad\hat{p}_{14}^{2}=m_{b}^{2}.$
(3.34)

and the amplitude is given by

$\displaystyle A_{WWtt}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}$
$\displaystyle=\sum_{I=Z,\gamma}\frac{1}{p_{12}^{2}-m_{I}^{2}}\frac{1}{{z^{+}_{12}-z^{-}_{12}}}$
$\displaystyle\times\sum_{\lambda}\left[{z^{+}_{12}}\hat{A}_{WWI}^{(\lambda_{1}\lambda_{2}\lambda)}(z^{-}_{12})\times\hat{A}_{ttI}^{(\lambda_{3}\lambda_{4}\bar{\lambda})}(z^{-}_{12})-{z^{-}_{12}}\hat{A}_{WWI}^{(\lambda_{1}\lambda_{2}\lambda)}(z^{+}_{12})\times\hat{A}_{ttI}^{(\lambda_{3}\lambda_{4}\bar{\lambda})}(z^{+}_{12})\right]$
$\displaystyle+\frac{1}{p_{14}^{2}-m_{b}^{2}}\frac{1}{{z^{+}_{14}-z^{-}_{14}}}$
$\displaystyle\times\sum_{\lambda}\left[{z^{+}_{14}}{\hat{A}}^{(\lambda_{4}\lambda\lambda_{1})}_{tbW}(z^{-}_{14})\times\hat{A}_{tbW}^{(\lambda_{3}\bar{\lambda}\lambda_{2})}(z^{-}_{14})-{z^{-}_{14}}\hat{A}_{tbW}^{(\lambda_{4}\lambda\lambda_{1})}(z^{+}_{14})\times\hat{A}_{tbW}^{(\lambda_{3}\bar{\lambda}\lambda_{2})}(z^{+}_{14})\right].$
(3.35)

In contrast to the four gauge boson scatterings, here we do not assume that
$\lambda_{1}=\pm$ since the external fermions improve the large-$z$ behavior;
see Eq. (2.19). This amplitude has internal massive fermion and vector, and we
use the completeness relation for fermion Eq. (A.20) and vectors in Eq. (3.9).
Then, we can rewrite

$\displaystyle\sum_{\lambda}\hat{A}_{WWI}^{(\lambda_{1}\lambda_{2}\lambda)}\times\hat{A}_{ttI}^{(\lambda_{3}\lambda_{4}\bar{\lambda})}$
$\displaystyle=-2g_{WWI}\epsilon_{1}^{\dot{a}a}\epsilon_{2}^{\dot{b}b}{V_{a\dot{a};b\dot{b}}}^{c\dot{c}}(\hat{p}_{1},\hat{p}_{2},-\hat{p}_{12})(g_{ttI}^{L}\lvert\textbf{3}\rangle_{c}[\textbf{4}\lvert_{\dot{c}}+g_{ttI}^{R}\lvert\textbf{4}\rangle_{c}[\textbf{3}\lvert_{\dot{c}}),$
(3.36)
$\displaystyle\sum_{\lambda}\hat{A}_{tbW}^{(\lambda_{4}\lambda\lambda_{1})}\times\hat{A}_{tbW}^{(\lambda_{3}\bar{\lambda}\lambda_{2})}$
$\displaystyle=-\frac{2g_{tbW}^{2}}{m_{W}^{2}}[{\textbf{4}}{\textbf{1}}]\langle{\textbf{2}}{\textbf{3}}\rangle\langle\textbf{1}\lvert\hat{p}_{14}\lvert{\textbf{2}}],$
(3.37)

where we defined $g_{tt\gamma}^{L}=g_{tt\gamma}^{R}=g_{tt\gamma}$. Now, we
notice that all terms in Eq. (3.35) have one momentum insertion at maximum in
the numerator which originates from
${V_{a\dot{a};b\dot{b}}}^{c\dot{c}}(\hat{p}_{1},\hat{p}_{2},-\hat{p}_{12})$ or
internal fermion. They contain only up to linear terms in $z$, and the linear
terms cancel after summing over the poles of each factorization channel.
Therefore, in contrast to the $WWWW$ and $WWZZ$ amplitudes, the $WWtt$
amplitude does not have a four-point contact term. Then, we can write the full
amplitude in spinor representation as

$\displaystyle A_{WWtt}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}=$
$\displaystyle-\sum_{I=Z,\gamma}\frac{1}{m_{W}^{2}(p_{12}^{2}-m_{I}^{2})}\bigg{(}g_{WWI}g_{ttI}^{L}(\langle{\textbf{1}}{\textbf{2}}\rangle[{\textbf{2}}{\textbf{1}}]\langle{\textbf{3}}\lvert{p}_{1}-{p}_{2}\lvert{\textbf{4}}]-2\langle{\textbf{3}}{\textbf{1}}\rangle[{\textbf{1}}{\textbf{4}}]\langle{\textbf{2}}\lvert{p}_{1}\lvert{\textbf{2}}]$
$\displaystyle+2\langle{\textbf{3}}{\textbf{2}}\rangle[{\textbf{2}}{\textbf{4}}]\langle{\textbf{1}}\lvert{p}_{2}\lvert{\textbf{1}}])+g_{WWI}g_{ttI}^{R}(\langle{\textbf{1}}{\textbf{2}}\rangle[{\textbf{2}}{\textbf{1}}]\langle{\textbf{4}}\lvert{p}_{1}-{p}_{2}\lvert{\textbf{3}}]-2\langle{\textbf{4}}{\textbf{1}}\rangle[{\textbf{1}}{\textbf{3}}]\langle{\textbf{2}}\lvert{p}_{1}\lvert{\textbf{2}}]$
$\displaystyle+2\langle{\textbf{4}}{\textbf{2}}\rangle[{\textbf{2}}{\textbf{3}}]\langle{\textbf{1}}\lvert{p}_{2}\lvert{\textbf{1}}])\bigg{)}-\frac{2g_{tbW}^{2}}{m_{W}^{2}(p_{14}^{2}-m_{b}^{2})}[{\textbf{4}}{\textbf{1}}]\langle{\textbf{2}}{\textbf{3}}\rangle\langle\textbf{1}\lvert{p}_{14}\lvert{\textbf{2}}].$
(3.38)

##  4 UV completion of electroweak theory

In Sec. 3, we have seen that we can construct the EW gauge boson amplitudes,
including the contact terms, with the ALT shift even in the absence of the
Higgs boson. There we see that our calculation results in the automatic
cancellation of the $\mathcal{O}(E^{4})$ term in the all-longitudinal EW gauge
boson amplitude at high energy, which leaves us only the $\mathcal{O}(E^{2})$
terms.

Canceling the $\mathcal{O}(E^{2})$ terms requires additional input. Here, we
have different options, e.g., we can introduce new particles such as scalar
and vector particles with different interactions. The main topic of this
section is to study these UV completions in the on-shell method, again with
the ALT shift. We mainly focus on UV completion by scalar particles,
corresponding to Higgs multiplets, and only briefly mention UV completion by
vector particles at the end. We can see that studying the unitarity behavior
will further restrict relations between the coefficients of different three-
point amplitudes.

###  4.1 A neutral scalar: Higgs doublet

We begin with adding one real scalar particle, $h$, with its mass $m_{h}$,
which we call the Higgs. Indeed, after requiring the cancellations of the
$\mathcal{O}(E^{2})$ terms, it will be clear that this naturally leads to UV
completion by a Higgs doublet.

The minimal three-point amplitudes of a real scalar particle and vector
particles are given by

$\displaystyle A_{WWh}^{(\lambda_{1}\lambda_{2})}$
$\displaystyle=\leavevmode\hbox to6.67pt{\vbox
to6.67pt{\pgfpicture\makeatletter\hbox{\hskip-0.24718pt\lower-3.33301pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\hbox to0.0pt{\hbox to0.0pt{\hbox
to0.0pt{\hbox to0.0pt{\hbox to0.0pt{\hbox to0.0pt{ \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), };
\hss}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[inline = (base.a),
horizontal=p1 to v] \vertex(p1) [label=${\scriptstyle W^{\pm}}$];
\vertex[below right = of p1] (v); \vertex[below left = of
v,label=270:${\scriptstyle W^{\mp}}$] (p2); \vertex[right = of v,
label=${\scriptstyle h}$] (p3);
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\pgfsys@rect{0.24718pt}{-3.33301pt}{6.66602pt}{6.66602pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.58018pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }{bg}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\frac{g_{WWh}}{m_{W}^{2}}\langle\textbf{1}\textbf{2}\rangle[\textbf{2}\textbf{1}],\quad
A_{ZZh}^{(\lambda_{1}\lambda_{2})}=\leavevmode\hbox to6.67pt{\vbox
to6.67pt{\pgfpicture\makeatletter\hbox{\hskip-0.24718pt\lower-3.33301pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\hbox to0.0pt{\hbox to0.0pt{\hbox
to0.0pt{\hbox to0.0pt{\hbox to0.0pt{\hbox to0.0pt{\hbox to0.0pt{ \diagram*{
(p1) -- [photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), };
\hss}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[inline = (base.a),
horizontal=p1 to v] \vertex(p1) [label=${\scriptstyle Z}$]; \vertex[below
right = of p1] (v); \vertex[below left = of v,label=270:${\scriptstyle Z}$]
(p2); \vertex[right = of v, label=${\scriptstyle h}$] (p3);
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\pgfsys@rect{0.24718pt}{-3.33301pt}{6.66602pt}{6.66602pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.58018pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }{bg}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>(4.1)

In the on-shell method, $g_{WWh}$ and $g_{ZZh}$ are arbitrary at this point,
and we will fix it by requiring the cancellation of the $\mathcal{O}(E^{2})$
terms. On top of these couplings, we have three-point amplitudes of the Higgs
particles, but they do not lead to $\mathcal{O}(E^{2})$ terms. Therefore these
contributions are out of our interest, and we ignore them in the rest part. In
the following, we construct four-point amplitudes from these three-point
amplitudes using the on-shell method.

####  $WWWW$ amplitude

We start from the $WWWW$ amplitude. The Higgs contribution is given by

$\displaystyle\left.A_{WWWW}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}\right|_{h}$
$\displaystyle=\sum_{i=2,4}\frac{1}{p_{1i}^{2}-m_{h}^{2}}\frac{1}{{z^{+}_{1i}-z^{-}_{1i}}}$
$\displaystyle\times\left[{z^{+}_{1i}}\hat{A}_{WWh}^{(\lambda_{1}\lambda_{i})}(z^{-}_{1i})\times\hat{A}_{WWh}^{(\lambda_{3}\lambda_{j})}(z^{-}_{1i})-{z^{-}_{1i}}\hat{A}_{WWh}^{(\lambda_{1}\lambda_{i})}(z^{+}_{1i})\times\hat{A}_{WWh}^{(\lambda_{3}\lambda_{j}}(z^{+}_{1i})\right],$
(4.2)

where $z_{1i}^{\pm}$ are the roots for the pole condition
$\hat{p}_{1i}^{2}=m_{h}^{2}$. The three-point amplitudes do not depend on $z$
since the polarization vectors are not shifted, and therefore it is simplified
as the product of the three-point amplitudes without the momentum shift as

$\displaystyle\left.A_{WWWW}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}\right|_{h}$
$\displaystyle=\frac{g_{WWh}^{2}}{m_{W}^{4}}\left[\frac{\langle\mathbf{12}\rangle[\mathbf{21}]\langle\mathbf{34}\rangle[\mathbf{43}]}{p_{12}^{2}-m_{h}^{2}}+\frac{\langle\mathbf{14}\rangle[\mathbf{41}]\langle\mathbf{32}\rangle[\mathbf{23}]}{p_{14}^{2}-m_{h}^{2}}\right].$
(4.3)

Incorporating this contribution, the high energy limit of the all-longitudinal
amplitude is given by

$\displaystyle A_{WWWW}^{(LLLL)}$
$\displaystyle=-\frac{s+u}{4m_{W}^{4}}\left[4g^{2}m_{W}^{2}-3g_{WWZ}^{2}m_{Z}^{2}-g_{WWh}^{2}\right]+\mathcal{O}(E^{0}).$
(4.4)

Therefore good high-energy behavior of this channel requires

$\displaystyle g^{2}_{WWh}=4g^{2}m_{W}^{2}-3g_{WWZ}^{2}m_{Z}^{2}.$ (4.5)

####  $WWZZ$ amplitude

The computation of the $WWZZ$ scattering is analogous to the $WWWW$
scattering, and we obtain

$\displaystyle\left.A_{WWZZ}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}\right|_{h}$
$\displaystyle=\frac{g_{WWh}g_{ZZh}}{m_{W}^{2}m_{Z}^{2}}\frac{\langle\mathbf{12}\rangle[\mathbf{21}]\langle\mathbf{34}\rangle[\mathbf{43}]}{p_{12}^{2}-m_{h}^{2}}.$
(4.6)

Including this contribution, the high-energy limit of the all-longitudinal
amplitude is given by

$\displaystyle A_{WWZZ}^{(LLLL)}$
$\displaystyle=\frac{t+u}{4m_{W}^{2}m_{Z}^{2}}\left[\frac{g_{WWZ}^{2}m_{Z}^{4}}{m_{W}^{2}}-g_{WWh}g_{ZZh}\right]+\mathcal{O}(E^{0}).$
(4.7)

Therefore UV completion of this channel requires

$\displaystyle g_{WWh}g_{ZZh}$
$\displaystyle=\frac{g_{WWZ}^{2}m_{Z}^{4}}{m_{W}^{2}}.$ (4.8)

We also have the $ZZZZ$ scattering, in the presence of the Higgs, but it grows
only as $\mathcal{O}(E^{0})$ due to $s+t+u=4m_{Z}^{2}$, and we do not consider
it.

####  $WWZh$ amplitude

In the presence of the Higgs in the spectrum, we have amplitudes with external
Higgs particles, and the cancellation of the $\mathcal{O}(E^{2})$ term in
these processes puts additional relations among the couplings. Let us begin
with the $WWZh$ amplitude and label $W^{+},W^{-},Z,h$ as $1,2,3,4$. The
amplitude has the poles at

$\displaystyle\hat{p}_{12}^{2}=m_{Z}^{2},\quad\hat{p}_{13}^{2}=m_{W}^{2},\quad\hat{p}_{14}^{2}=m_{W}^{2},$
(4.9)

and is given by

$\displaystyle A_{WWZh}^{(\lambda_{1}\lambda_{2}\lambda_{3})}$
$\displaystyle=\frac{1}{p_{12}^{2}-m_{Z}^{2}}\frac{1}{{z^{+}_{12}-z^{-}_{12}}}\sum_{\lambda}\left[{z^{+}_{12}}\hat{A}_{WWZ}^{(\lambda_{1}\lambda_{2}\lambda)}(z^{-}_{12})\times\hat{A}_{ZZh}^{(\bar{\lambda}\lambda_{3})}(z^{-}_{12})-{z^{-}_{12}}\hat{A}_{WWZ}^{(\lambda_{1}\lambda_{2}\lambda)}(z^{+}_{12})\times\hat{A}_{ZZh}^{(\bar{\lambda}\lambda_{3})}(z^{+}_{12})\right]$
$\displaystyle+\frac{1}{p_{13}^{2}-m_{W}^{2}}\frac{1}{{z^{+}_{13}-z^{-}_{13}}}\sum_{\lambda}\left[{z^{+}_{13}}\hat{A}_{WWZ}^{(\lambda_{1}\lambda\lambda_{3})}(z^{-}_{13})\times\hat{A}_{WWh}^{(\bar{\lambda}\lambda_{2})}(z^{-}_{13})-{z^{-}_{13}}\hat{A}_{WWZ}^{(\lambda_{1}\lambda\lambda_{3})}(z^{+}_{13})\times\hat{A}_{WWh}^{(\bar{\lambda}\lambda_{2})}(z^{+}_{13})\right]$
$\displaystyle+\frac{1}{p_{14}^{2}-m_{W}^{2}}\frac{1}{{z^{+}_{14}-z^{-}_{14}}}\sum_{\lambda}\left[{z^{+}_{14}}\hat{A}_{WWZ}^{(\lambda\lambda_{2}\lambda_{3})}(z^{-}_{14})\times\hat{A}_{WWh}^{(\lambda_{1}\bar{\lambda})}(z^{-}_{14})-{z^{-}_{14}}\hat{A}_{WWZ}^{(\lambda\lambda_{2}\lambda_{3})}(z^{+}_{14})\times\hat{A}_{WWh}^{(\lambda_{1}\bar{\lambda})}(z^{+}_{14})\right],$
(4.10)

As usual, we simplify this by using the completeness relation of the
intermediate polarization vectors. With the help of Eq. (3.3), the products of
the three-point amplitudes are given by

$\displaystyle\hat{A}_{WWZ}^{(\lambda_{1}\lambda_{2}\lambda)}\times\hat{A}_{ZZh}^{(\bar{\lambda}\lambda_{2})}$
$\displaystyle=-g_{WWZ}g_{ZZh}\epsilon_{1}^{\dot{a}a}\epsilon_{2}^{\dot{b}b}\epsilon_{3}^{\dot{c}c}V_{a\dot{a};b\dot{b};c\dot{c}}(\hat{p}_{1},\hat{p}_{2},-\hat{p}_{12}),$
(4.11)
$\displaystyle\hat{A}_{WWZ}^{(\lambda_{1}\lambda\lambda_{3})}\times\hat{A}_{WWh}^{(\bar{\lambda}\lambda_{2})}$
$\displaystyle=-g_{WWZ}g_{WWh}\left[\epsilon_{1}^{\dot{a}a}\epsilon_{2}^{\dot{b}b}\epsilon_{3}^{\dot{c}c}V_{a\dot{a};b\dot{b};c\dot{c}}(\hat{p}_{1},-\hat{p}_{13},\hat{p}_{3})+\frac{m_{Z}^{2}-m_{W}^{2}}{2m_{W}^{2}}(\epsilon_{1}\cdot\epsilon_{3})(\epsilon_{2}\cdot\hat{p}_{4})\right],$
(4.12)
$\displaystyle\hat{A}_{WWZ}^{(\lambda\lambda_{2}\lambda_{3})}\times\hat{A}_{WWh}^{(\lambda_{1}\bar{\lambda})}$
$\displaystyle=-g_{WWZ}g_{WWh}\left[\epsilon_{1}^{\dot{a}a}\epsilon_{2}^{\dot{b}b}\epsilon_{3}^{\dot{c}c}V_{a\dot{a};b\dot{b};c\dot{c}}(-\hat{p}_{23},\hat{p}_{2},\hat{p}_{3})-\frac{m_{Z}^{2}-m_{W}^{2}}{2m_{W}^{2}}(\epsilon_{2}\cdot\epsilon_{3})(\epsilon_{1}\cdot\hat{p}_{4})\right].$
(4.13)

All these terms contain only up to linear in $z_{1i}^{\pm}$, and the terms
linear in $z_{1i}^{\pm}$ cancel with each other in Eq. (4.10). Therefore the
amplitude reduces to the products of the three-point amplitudes without the
momentum shift as

$\displaystyle A_{WWZh}^{(\lambda_{1}\lambda_{2}\lambda_{3})}=$
$\displaystyle-\frac{g_{WWZ}g_{ZZh}}{p_{12}^{2}-m_{Z}^{2}}\epsilon_{1}^{\dot{a}a}\epsilon_{2}^{\dot{b}b}\epsilon_{3}^{\dot{c}c}V_{a\dot{a};b\dot{b};c\dot{c}}({p}_{1},{p}_{2},-{p}_{12})$
$\displaystyle-\frac{g_{WWZ}g_{WWh}}{p_{13}^{2}-m_{W}^{2}}\epsilon_{1}^{\dot{a}{a}}\epsilon_{3}^{\dot{c}c}{{V_{a\dot{a};}}^{b\dot{b};}}_{c\dot{c}}({p}_{1},-{p}_{13},{p}_{3})\left[\epsilon_{bd}\epsilon_{\dot{b}\dot{d}}-\frac{[p_{13}]_{b\dot{b}}[p_{13}]_{d\dot{d}}}{2m_{W}^{2}}\right]\epsilon_{2}^{\dot{d}d}$
$\displaystyle-\frac{g_{WWZ}g_{WWh}}{p_{14}^{2}-m_{W}^{2}}\epsilon_{2}^{\dot{b}{b}}\epsilon_{3}^{\dot{c}c}{V^{a\dot{a};}}_{b\dot{b};c\dot{c}}(-{p}_{23},{p}_{2},{p}_{3})\left[\epsilon_{ad}\epsilon_{\dot{a}\dot{d}}-\frac{[p_{14}]_{a\dot{a}}[p_{14}]_{d\dot{d}}}{2m_{W}^{2}}\right]\epsilon_{1}^{\dot{d}d}.$
(4.14)

In particular, it does not contain any contact terms. Its explicit form is
given by

$\displaystyle A_{WWZh}^{(\lambda_{1}\lambda_{2}\lambda_{3})}$
$\displaystyle=\frac{g_{ZZh}g_{WWZ}}{m_{W}^{2}m_{Z}(p_{12}^{2}-m_{Z}^{2})}\bigg{[}\langle\textbf{12}\rangle[{\textbf{21}}]\langle{\textbf{3}}\lvert({p}_{1}-{p}_{2})\lvert{\textbf{3}}]-2\langle{\textbf{13}}\rangle[{\textbf{31}}]\langle{\textbf{2}}\lvert{p}_{1}\lvert{\textbf{2}}]+2\langle{\textbf{23}}\rangle[{\textbf{32}}]\langle{\textbf{1}}\lvert{p}_{2}\lvert{\textbf{1}}]\bigg{]}$
$\displaystyle+\frac{g_{WWZ}g_{WWh}}{m_{W}^{2}m_{Z}(p_{13}^{2}-m_{W}^{2})}$
$\displaystyle\times\left[\langle{\textbf{13}}\rangle[{\textbf{31}}]\left(\langle{\textbf{2}}\lvert({p}_{1}-{p}_{3})\lvert{\textbf{2}}]-\frac{m_{W}^{2}-m_{Z}^{2}}{m_{W}^{2}}\langle{\textbf{2}}\lvert{p}_{4}\lvert{\textbf{2}}]\right)-2\langle{\textbf{12}}\rangle[{\textbf{21}}]\langle{\textbf{3}}\lvert{p}_{1}\lvert{\textbf{3}}]+2\langle{\textbf{23}}\rangle[{\textbf{32}}]\langle{\textbf{1}}\lvert{p}_{3}\lvert{\textbf{1}}]\right]$
$\displaystyle+\frac{g_{WWZ}g_{WWh}}{m_{W}^{2}m_{Z}(p_{14}^{2}-m_{W}^{2})}$
$\displaystyle\times\left[\langle{\textbf{23}}\rangle[{\textbf{32}}]\left(\langle{\textbf{1}}\lvert({p}_{2}-{p}_{3})\lvert{\textbf{1}}]-\frac{m_{Z}^{2}-m_{W}^{2}}{m_{W}^{2}}\langle{\textbf{1}}\lvert{p}_{4}\lvert{\textbf{1}}]\right)-2\langle{\textbf{12}}\rangle[{\textbf{21}}]\langle{\textbf{3}}\lvert{p}_{2}\lvert{\textbf{3}}]+2\langle{\textbf{13}}\rangle[{\textbf{31}}]\langle{\textbf{2}}\lvert{p}_{3}\lvert{\textbf{2}}]\right].$
(4.15)

The all-longitudinal amplitude behaves as

$\displaystyle A_{WWZh}^{(LLL)}$
$\displaystyle=\frac{g_{WWZ}(t-u)}{4m_{W}^{4}m_{Z}}\left(g_{ZZh}m_{W}^{2}-g_{WWh}m_{Z}^{2}\right)\
.$ (4.16)

At high energy, the cancellation of the $\mathcal{O}(E^{2})$ term gives

$\displaystyle g_{ZZh}m_{W}^{2}-g_{WWh}m_{Z}^{2}=0.$ (4.17)

The physical meaning of the above equation is straightforward; to unitarize
the $WWZh$ amplitudes with only one neutral Higgs that couples to $WW$ and
$ZZ$, this Higgs field needs to be fully in charge of the electroweak symmetry
breaking, such that the mass ratio between $W$ and $Z$ bosons is their Higgs
couplings ratio.

####  $WWhh$ amplitude

Now we come to calculate the $WWhh$ scattering. As we will see, the on-shell
method with the ALT shift generates a contact term, which guarantees the
absence of the $\mathcal{O}(E^{2})$ term in this channel. Therefore, this
process does not require any additional relations among the couplings.

Let us label $W^{+},W^{-},h,h$ as $1,2,3,4$. The amplitude has the poles at

$\displaystyle\hat{p}_{13}^{2}=m_{W}^{2},\quad\hat{p}_{14}^{2}=m_{W}^{2},$
(4.18)

and is given by

$\displaystyle A_{WWhh}^{(\lambda_{1}\lambda_{2})}$
$\displaystyle=\sum_{i=3,4}\frac{1}{p_{1i}^{2}-m_{W}^{2}}\frac{1}{{z^{+}_{1i}-z^{-}_{1i}}}$
$\displaystyle\times\sum_{\lambda}\left[{z^{+}_{1i}}\hat{A}_{WWh}^{(\lambda_{1}\lambda)}(z^{-}_{1i})\times\hat{A}_{WWh}^{(\bar{\lambda}\lambda_{2})}(z^{-}_{1i})-{z^{-}_{1i}}\hat{A}_{WWh}^{(\lambda_{1}\lambda)}(z^{+}_{1i})\times\hat{A}_{WWh}^{(\bar{\lambda}\lambda_{2})}(z^{+}_{1i})\right],$
(4.19)

where, again, by taking $\lambda_{1}=\pm$ we have $B_{\infty}=0$. By using the
completeness condition of the intermediate polarization vectors, the product
of the three-point amplitude is given by

$\displaystyle\sum_{\lambda}\hat{A}_{WWh}^{(\lambda_{1}\lambda)}\times\hat{A}_{WWh}^{(\bar{\lambda}\lambda_{2})}$
$\displaystyle=-\frac{g_{WWh}^{2}}{2}\epsilon_{1}^{\dot{a}{a}}\left[\epsilon_{ab}\epsilon_{\dot{a}\dot{b}}-\frac{[\hat{p}_{1i}]_{a\dot{a}}[\hat{p}_{1i}]_{b\dot{b}}}{2m_{W}^{2}}\right]\epsilon_{2}^{\dot{b}{b}}.$
(4.20)

This contains the $z^{2}$-term, arising from
$\hat{p}_{1i}\hat{p}_{1i}/m_{W}^{2}$ in the bracket, which results in the
contact term. After simplifying the overall $z$-dependence of the $z^{2}$-term
as before, we obtain

$\displaystyle A_{WWhh}^{(\lambda_{1}\lambda_{2})}$
$\displaystyle=-\frac{g_{WWh}^{2}}{2}\sum_{i=3,4}\frac{1}{p_{1i}^{2}-m_{W}^{2}}\epsilon_{1}^{\dot{a}{a}}\left[\epsilon_{ab}\epsilon_{\dot{a}\dot{b}}-\frac{[{p}_{1i}]_{a\dot{a}}[{p}_{1i}]_{b\dot{b}}}{2m_{W}^{2}}\right]\epsilon_{2}^{\dot{b}{b}}-\frac{g_{WWh}^{2}}{2m_{W}^{2}}\sum_{i=3,4}\frac{(\epsilon_{1}\cdot
q_{1i})(\epsilon_{2}\cdot q_{1i})}{q_{1}\cdot q_{i}},$ (4.21)

where the first term comes from the $z$-independent term, while the second
term comes from the $z^{2}$-term. Again, the terms linear in $z_{1i}^{\pm}$ do
not contribute since the $z_{1i}^{\pm}$ contributions cancel with each other.
Recalling that $\lambda_{1}=\pm$ and hence $\epsilon_{1}=q_{1}/c_{1}m_{W}$, we
can simplify the second term and obtain

$\displaystyle A_{WWhh}^{(\lambda_{1}\lambda_{2})}$
$\displaystyle=-\frac{g_{WWh}^{2}}{2}\sum_{i=3,4}\frac{1}{p_{1i}^{2}-m_{W}^{2}}\epsilon_{1}^{\dot{a}{a}}\left[\epsilon_{ab}\epsilon_{\dot{a}\dot{b}}-\frac{[{p}_{1i}]_{a\dot{a}}[{p}_{1i}]_{b\dot{b}}}{2m_{W}^{2}}\right]\epsilon_{2}^{\dot{b}{b}}-\frac{g_{WWh}^{2}}{2m_{W}^{2}}\epsilon_{1}\cdot\epsilon_{2},$
(4.22)

where we used $\sum_{i}q_{i}=0$ and $\epsilon_{i}\cdot q_{i}=0$. Its explicit
form is given by

$\displaystyle A_{WWhh}^{(\lambda_{1}\lambda_{2})}$
$\displaystyle=-\frac{g_{WWh}^{2}}{m_{W}^{2}}\sum_{i=3,4}\frac{1}{p_{1i}^{2}-m_{W}^{2}}\left[\langle\mathbf{12}\rangle[\mathbf{21}]+\frac{\langle\mathbf{1}|p_{i}|\mathbf{1}]\langle\mathbf{2}|p_{j}|\mathbf{2}]}{m_{W}^{2}}\right]-\frac{g_{WWh}^{2}}{m_{W}^{4}}\langle\mathbf{12}\rangle[\mathbf{21}],$
(4.23)

where $j\neq 1,2,i$. We can see that, with the contact term we derived,
$A_{WWhh}^{(LL)}$ behaves only as $\mathcal{O}(E^{0})$ at high energy.
Therefore, this process does not provide any relations among the couplings.

####  $WWtt$ amplitude

After the addition of a single scalar particle, we can write down the
following minimal three-point amplitude between scalar and fermions

$\displaystyle A_{ffh}^{(\lambda_{1}\lambda_{2})}$
$\displaystyle=\leavevmode\hbox to6.67pt{\vbox
to6.67pt{\pgfpicture\makeatletter\hbox{\hskip-0.24718pt\lower-3.33301pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\hbox to0.0pt{\hbox to0.0pt{\hbox
to0.0pt{\hbox to0.0pt{\hbox to0.0pt{\hbox to0.0pt{\hbox to0.0pt{\hbox to0.0pt{
\diagram*{ (p1) -- [photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), };
\hss}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[inline = (base.a),
horizontal=p1 to v] \vertex(p1) [label=${\scriptstyle t}$]; \vertex[below
right = of p1] (v); \vertex[below left = of
v,label=270:${\scriptstyle\bar{t}}$] (p2); \vertex[right = of v,
label=${\scriptstyle h}$] (p3);
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\pgfsys@rect{0.24718pt}{-3.33301pt}{6.66602pt}{6.66602pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.58018pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }{bg}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=g_{ffh}\left(\langle\textbf{2}\textbf{1}\rangle+[\textbf{2}\textbf{1}]\right)$
(4.24)

where $g_{ffh}$ is arbitrary dimensionless coupling. Given we have only
introduced two quarks $t$ and $b$, $g_{ffh}$ could be $g_{tth}$ or $g_{bbh}$.
Then, the Higgs contribution for $WWtt$ amplitude is given by

$\displaystyle\left.A_{WWtt}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}\right|_{h}$
$\displaystyle=\frac{1}{p_{12}^{2}-m_{h}^{2}}\frac{1}{{z^{+}_{12}-z^{-}_{12}}}$
$\displaystyle\times\left[{z^{+}_{12}}\hat{A}_{WWh}^{(\lambda_{1}\lambda_{2})}(z^{-}_{12})\times\hat{A}_{tth}^{(\lambda_{3}\lambda_{4})}(z^{-}_{12})-{z^{-}_{12}}\hat{A}_{WWh}^{(\lambda_{1}\lambda_{2})}(z^{+}_{12})\times\hat{A}_{tth}^{(\lambda_{3}\lambda_{4})}(z^{+}_{12})\right].$
(4.25)

where $z_{12}^{\pm}$ is found by solving $\hat{p}_{12}^{2}=m_{h}^{2}$. After
summing over the pole contributions, we find

$\displaystyle\left.A_{WWtt}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}\right|_{h}$
$\displaystyle=\frac{g_{tth}g_{WWh}}{m_{W}^{2}}\frac{(\langle\textbf{1}\textbf{2}\rangle[\textbf{2}\textbf{1}]\langle\textbf{4}\textbf{3}\rangle+\langle\textbf{1}\textbf{2}\rangle[\textbf{2}\textbf{1}][\textbf{4}\textbf{3}])}{p_{12}-m_{h}^{2}}$
(4.26)

Now we take the high energy limit of $WWtt$ amplitude. As in the case of gauge
boson amplitude, this amplitude exhibits bad high-energy behavior.
$A_{WWtt}^{(LL\pm\mp)}$ scales as $\mathcal{O}(E^{2})$ and it vanishes after
we impose the following condition among the couplings

$\displaystyle g_{WWZ}g_{ttZ}^{R}$ $\displaystyle=-g_{WW\gamma}g_{tt\gamma},$
(4.27) $\displaystyle g_{WWZ}g_{ttZ}^{L}$
$\displaystyle=-g_{WW\gamma}g_{tt\gamma}+g_{tbW}^{2}.$ (4.28)

These relations make the leading order energy dependence to be a constant for
$A_{WWtt}^{(LL\pm\mp)}$. But, the above relation does not cure the
$\mathcal{O}(E)$ scaling of $A_{WWtt}^{(LL\pm\pm)}$ amplitude without the
Higgs contribution and unitarity requirement imposes the following constraint
among the couplings

$\displaystyle{m_{t}}g_{tbW}^{2}=-g_{tth}g_{WWh}$ (4.29)

#### Coupling relations and $\rho$ parameter

By requiring the cancellation of the $\mathcal{O}(E^{2})$ terms, we have
obtained three relations in Eqs. (4.5), (4.8) and (4.17) and they restrict
three of the six independent parameters
$m_{W},m_{Z},g_{WWZ},g_{WW\gamma},g_{WWh},$ $g_{ZZh}$. Two relations fix the
Higgs couplings to the gauge bosons as

$\displaystyle g_{WWh}=gm_{W},\quad g_{ZZh}=\frac{gm_{Z}^{2}}{m_{W}}.$ (4.30)

The remaining relation fixes the ratio of the gauge boson masses times the
couplings, known as the $\rho$ parameter

$\displaystyle\rho=\frac{g^{2}m_{W}^{2}}{g_{WWZ}^{2}m_{Z}^{2}}=1.$ (4.31)

In the Lagrangian formulation, it is well known that $\rho$ is linked to the
representation of the Higgs multiplet, and $\rho=1$ in particular indicates
the doublet. It is interesting that we can derive the same relation, $\rho=1$,
without assuming that the additional scalar particle $h$ is involved in the
multiplet of SU(2). Our result makes sense since the cancellation of the
$\mathcal{O}(E^{2})$ term effectively forces $h$ to be involved in the SU(2)
multiplet, and the only option of having one additional neutral scalar
particle is to embed $h$ into the doublet.

###  4.2 Neutral and doubly charged scalar: Higgs triplet

In Sec. 4.1, we have seen that the cancellation of $\mathcal{O}(E^{2})$ with a
single neutral scalar particle naturally leads to the UV completion by the
Higgs doublet, in particular $\rho=1$. It is then interesting to study other
combinations of the scalar particles and see if we conclude that they are also
in SU(2) multiplets. Therefore, in this subsection, we add a neutral scalar,
$h^{0}$, and a doubly charged scalar, $h^{++}$, with masses $m_{0}$ and
$m_{2}$, respectively. In the Lagrangian formulation, we know this corresponds
to the UV completion by the complex Higgs triplet, and our focus is to see if
we can recover $\rho=1/2$, corresponding to the Higgs triplet, in the bottom-
up approach. It turns out that this example reveals a new feature of the on-
shell computation; the Ward identity of four-point amplitudes, the
cancellation of the boundary terms, require relations among seemingly distinct
three-point couplings (from the amplitude construction point of view), as we
explain below. Of course, all of these are also guaranteed traditionally by
the non-broken $U(1)$ gauge symmetry.

The relevant three-point amplitudes are given by

$\displaystyle A_{WWh^{0}}^{(\lambda_{1}\lambda_{2})}$
$\displaystyle=\leavevmode\hbox to6.67pt{\vbox
to6.67pt{\pgfpicture\makeatletter\hbox{\hskip-0.24718pt\lower-3.33301pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\hbox to0.0pt{\hbox to0.0pt{\hbox
to0.0pt{\hbox to0.0pt{\hbox to0.0pt{\hbox to0.0pt{\hbox to0.0pt{\hbox
to0.0pt{\hbox to0.0pt{ \diagram*{ (p1) -- [photon,momentum=${\scriptstyle
p_{1}}$] (v) -- [photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) --
[photon, rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), };
\hss}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[inline = (base.a),
horizontal=p1 to v] \vertex(p1) [label=${\scriptstyle W^{\pm}}$];
\vertex[below right = of p1] (v); \vertex[below left = of
v,label=270:${\scriptstyle W^{\mp}}$] (p2); \vertex[right = of v,
label=${\scriptstyle h^{0}}$] (p3);
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\pgfsys@rect{0.24718pt}{-3.33301pt}{6.66602pt}{6.66602pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.58018pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }{bg}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\frac{g_{WWh}^{(0)}}{m_{W}^{2}}\langle\textbf{1}\textbf{2}\rangle[\textbf{2}\textbf{1}],\quad
A_{ZZh^{0}}^{(\lambda_{1}\lambda_{2})}=\leavevmode\hbox to6.67pt{\vbox
to6.67pt{\pgfpicture\makeatletter\hbox{\hskip-0.24718pt\lower-3.33301pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\hbox to0.0pt{\hbox to0.0pt{\hbox
to0.0pt{\hbox to0.0pt{\hbox to0.0pt{\hbox to0.0pt{\hbox to0.0pt{\hbox
to0.0pt{\hbox to0.0pt{\hbox to0.0pt{ \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), };
\hss}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[inline = (base.a),
horizontal=p1 to v] \vertex(p1) [label=${\scriptstyle Z}$]; \vertex[below
right = of p1] (v); \vertex[below left = of v,label=270:${\scriptstyle Z}$]
(p2); \vertex[right = of v, label=${\scriptstyle h^{0}}$] (p3);
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\pgfsys@rect{0.24718pt}{-3.33301pt}{6.66602pt}{6.66602pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.58018pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }{bg}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\frac{g_{ZZh}^{(0)}}{m_{Z}^{2}}\langle\textbf{1}\textbf{2}\rangle[\textbf{2}\textbf{1}],$
(4.32) $\displaystyle A_{WWh^{\mp\mp}}^{(\lambda_{1}\lambda_{2})}$
$\displaystyle=\leavevmode\hbox to6.67pt{\vbox
to6.67pt{\pgfpicture\makeatletter\hbox{\hskip-0.24718pt\lower-3.33301pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\hbox to0.0pt{\hbox to0.0pt{\hbox
to0.0pt{\hbox to0.0pt{\hbox to0.0pt{\hbox to0.0pt{\hbox to0.0pt{\hbox
to0.0pt{\hbox to0.0pt{\hbox to0.0pt{\hbox to0.0pt{ \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), };
\hss}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[inline = (base.a),
horizontal=p1 to v] \vertex(p1) [label=${\scriptstyle W^{\pm}}$];
\vertex[below right = of p1] (v); \vertex[below left = of
v,label=270:${\scriptstyle W^{\pm}}$] (p2); \vertex[right = of v,
label=${\scriptstyle h^{\mp\mp}}$] (p3);
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\pgfsys@rect{0.24718pt}{-3.33301pt}{6.66602pt}{6.66602pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.58018pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }{bg}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\frac{g_{WWh}^{(2)}}{m_{W}^{2}}\langle\textbf{1}\textbf{2}\rangle[\textbf{2}\textbf{1}],$
(4.33)

for those involving one scalar particle, and

$\displaystyle A_{h^{++}h^{--}Z}^{(\lambda_{3})}$
$\displaystyle=\leavevmode\hbox to6.67pt{\vbox
to6.67pt{\pgfpicture\makeatletter\hbox{\hskip-0.24718pt\lower-3.33301pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\hbox to0.0pt{\hbox to0.0pt{\hbox
to0.0pt{\hbox to0.0pt{\hbox to0.0pt{\hbox to0.0pt{\hbox to0.0pt{\hbox
to0.0pt{\hbox to0.0pt{\hbox to0.0pt{\hbox to0.0pt{\hbox to0.0pt{ \diagram*{
(p1) -- [photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[scalar,momentum=${\scriptstyle p_{1}}$] (v) --
[scalar,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), };
\hss}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[inline = (base.a),
horizontal=p1 to v] \vertex(p1) [label=${\scriptstyle h^{++}}$]; \vertex[below
right = of p1] (v); \vertex[below left = of v,label=270:${\scriptstyle
h^{--}}$] (p2); \vertex[right = of v, label=${\scriptstyle Z}$] (p3);
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\pgfsys@rect{0.24718pt}{-3.33301pt}{6.66602pt}{6.66602pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.58018pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }{bg}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\frac{g_{hhZ}^{(2)}}{\sqrt{2}m_{Z}}\langle\mathbf{3}|(p_{1}-p_{2})|\mathbf{3}],$
(4.34) $\displaystyle A_{h^{++}h^{--}\gamma}^{(\lambda_{3})}$
$\displaystyle=\leavevmode\hbox to6.67pt{\vbox
to6.67pt{\pgfpicture\makeatletter\hbox{\hskip-0.24718pt\lower-3.33301pt\hbox
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\diagram*{ (p1) -- [photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[fermion,momentum=${\scriptstyle p_{1}}$] (v) --
[fermion,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[photon,momentum=${\scriptstyle p_{1}}$] (v) --
[photon,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [scalar,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[scalar,momentum=${\scriptstyle p_{1}}$] (v) --
[scalar,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), }; \hss} \diagram*{ (p1) --
[scalar,momentum=${\scriptstyle p_{1}}$] (v) --
[scalar,rmomentum'=${\scriptstyle p_{2}}$] (p2), (v) -- [photon,
rmomentum'=${\scriptstyle p_{3}}$] (p3), };
\hss}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[inline = (base.a),
horizontal=p1 to v] \vertex(p1) [label=${\scriptstyle h^{++}}$]; \vertex[below
right = of p1] (v); \vertex[below left = of v,label=270:${\scriptstyle
h^{--}}$] (p2); \vertex[right = of v, label=${\scriptstyle\gamma}$] (p3);
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.58018pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }{bg}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\begin{cases}\displaystyle\frac{g_{hh\gamma}^{(2)}}{\sqrt{2}\langle
3\xi_{3}\rangle}\langle\xi_{3}|(p_{1}-p_{2})|3],&\mathrm{for}~{}~{}\lambda_{3}=+,\vspace{1mm}\\\
\displaystyle-\frac{g_{hh\gamma}^{(2)}}{\sqrt{2}[3\xi_{3}]}\langle
3|(p_{1}-p_{2})|\xi_{3}],&\mathrm{for}~{}~{}\lambda_{3}=-,\end{cases}$ (4.35)

for those involving two scalar particles. Note that two distinct scalar
particles can couple to a vector particle. In addition, we have Higgs three-
point amplitudes, but we again ignore them as they do not generate
$\mathcal{O}(E^{2})$ contributions.

####  $WWWW$ amplitude

In the presence of $h^{++}$, the $WWWW$ amplitude has the poles at

$\displaystyle\hat{p}_{12}^{2}=m_{0}^{2},\quad\hat{p}_{14}^{2}=m_{0}^{2},\quad\hat{p}_{13}^{2}=m_{2}^{2}.$
(4.36)

The calculation is analogous to Sec. 4.1, and therefore we show only the final
result, given by

$\displaystyle\left.A_{WWWW}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}\right|_{h}$
$\displaystyle=\left(\frac{g_{WWh}^{(0)}}{m_{W}^{2}}\right)^{2}\left[\sum_{i=2,4}\frac{\langle\mathbf{1i}\rangle[\mathbf{i1}]\langle\mathbf{3j}\rangle[\mathbf{j3}]}{p_{1i}^{2}-m_{0}^{2}}\right]+\left(\frac{g_{WWh}^{(2)}}{m_{W}^{2}}\right)^{2}\frac{\langle\mathbf{13}\rangle[\mathbf{31}]\langle\mathbf{24}\rangle[\mathbf{42}]}{p_{13}^{2}-m_{2}^{2}}.$
(4.37)

Including this contribution, the high energy limit of the all-longitudinal
amplitude is given by

$\displaystyle A_{WWWW}^{(LLLL)}$
$\displaystyle=-\frac{s+u}{4m_{W}^{4}}\left[4g^{2}m_{W}^{2}-3g_{WWZ}^{2}m_{Z}^{2}-\left(g_{WWh}^{(0)}\right)^{2}+\left(g_{WWh}^{(2)}\right)^{2}\right]+\mathcal{O}(E^{0}).$
(4.38)

Therefore UV completion of this channel requires

$\displaystyle 0$
$\displaystyle=4g^{2}m_{W}^{2}-3g_{WWZ}^{2}m_{Z}^{2}-\left(g_{WWh}^{(0)}\right)^{2}+\left(g_{WWh}^{(2)}\right)^{2}.$
(4.39)

In particular, the coupling relation is affected by the presence of the doubly
charged scalar.

####  $WWZZ$ amplitude

In this process, only the neutral scalar shows up as the intermediate state,
and we obtain the same result as Sec. 4.1:

$\displaystyle\left.A_{WWZZ}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}\right|_{h}$
$\displaystyle=\frac{g_{WWh}^{(0)}g_{ZZh}^{(0)}}{m_{W}^{2}m_{Z}^{2}}\frac{\langle\mathbf{12}\rangle[\mathbf{21}]\langle\mathbf{34}\rangle[\mathbf{43}]}{p_{12}^{2}-m_{0}^{2}}.$
(4.40)

Including this contribution, the all-longitudinal amplitude behaves as

$\displaystyle A_{WWZZ}^{(LLLL)}$
$\displaystyle=\frac{t+u}{4m_{W}^{2}m_{Z}^{2}}\left[\frac{g_{WWZ}^{2}m_{Z}^{4}}{m_{W}^{2}}-g_{WWh}^{(0)}g_{ZZh}^{(0)}\right]+\mathcal{O}(E^{0}),$
(4.41)

and the cancellation of the $\mathcal{O}(E^{2})$ term requires

$\displaystyle g_{WWh}^{(0)}g_{ZZh}^{(0)}$
$\displaystyle=\frac{g_{WWZ}^{2}m_{Z}^{4}}{m_{W}^{2}}.$ (4.42)

####  $WWZh$ amplitude

The doubly charged scalar does not affect the $W^{+}W^{-}Zh^{0}$ scattering,
Therefore, in the same way as Sec. 4.1, this amplitude is given by

$\displaystyle A_{WWZh^{0}}^{(\lambda_{1}\lambda_{2}\lambda_{3})}=$
$\displaystyle-\frac{g_{WWZ}g_{ZZh}^{(0)}}{p_{12}^{2}-m_{Z}^{2}}\epsilon_{1}^{\dot{a}a}\epsilon_{2}^{\dot{b}b}\epsilon_{3}^{\dot{c}c}V_{a\dot{a};b\dot{b};c\dot{c}}(\hat{p}_{1},\hat{p}_{2},-\hat{p}_{12})$
$\displaystyle-\frac{g_{WWZ}g_{WWh}^{(0)}}{p_{13}^{2}-m_{W}^{2}}\epsilon_{1}^{\dot{a}{a}}\epsilon_{3}^{\dot{c}c}{{V_{a\dot{a};}}^{b\dot{b};}}_{c\dot{c}}(\hat{p}_{1},-\hat{p}_{13},\hat{p}_{3})\left[\epsilon_{bd}\epsilon_{\dot{b}\dot{d}}-\frac{[p_{13}]_{b\dot{b}}[p_{13}]_{d\dot{d}}}{2m_{W}^{2}}\right]\epsilon_{2}^{\dot{d}d}$
$\displaystyle-\frac{g_{WWZ}g_{WWh}^{(0)}}{p_{14}^{2}-m_{W}^{2}}\epsilon_{2}^{\dot{b}{b}}\epsilon_{3}^{\dot{c}c}{V^{a\dot{a};}}_{b\dot{b};c\dot{c}}(-\hat{p}_{23},\hat{p}_{2},\hat{p}_{3})\left[\epsilon_{ad}\epsilon_{\dot{a}\dot{d}}-\frac{[p_{14}]_{a\dot{a}}[p_{14}]_{d\dot{d}}}{2m_{W}^{2}}\right]\epsilon_{1}^{\dot{d}d},$
(4.43)

and the high-energy limit is given by

$\displaystyle A_{WWZh^{0}}^{(\lambda_{1}\lambda_{2}\lambda_{3})}$
$\displaystyle=\frac{g_{WWZ}(t-u)}{4m_{W}^{4}m_{z}}\left[g_{ZZh}^{(0)}m_{W}^{2}-g_{WWh}^{(0)}m_{Z}^{2}\right]+\mathcal{O}(E^{0}).$
(4.44)

From this result, we obtain the relation

$\displaystyle 0$
$\displaystyle=g_{ZZh}^{(0)}m_{W}^{2}-g_{WWh}^{(0)}m_{Z}^{2}.$ (4.45)

This is the same condition as the doublet case in Eq. 4.17 and the
interpretation is the same.

In the presence of the doubly charged scalar, there is also the
$W^{+}W^{+}Zh^{--}$ four-point scattering amplitude. We label
$W^{+},W^{-},Z,h^{--}$ as $1,2,3,4$, respectively. In this process, we have
the poles at

$\displaystyle\hat{p}_{12}^{2}=m_{2}^{2},\quad\hat{p}_{13}^{2}=m_{W}^{2},\quad\hat{p}_{14}^{2}=m_{W}^{2}.$
(4.46)

The products of the three-point amplitudes contain the terms up to only linear
in $z_{1i}^{\pm}$, and therefore we obtain

$\displaystyle A_{WWZh^{--}}^{(\lambda_{1}\lambda_{2}\lambda_{3})}=$
$\displaystyle-\frac{2g_{WWh}^{(2)}g_{hhZ}^{(2)}}{p_{12}^{2}-m_{2}^{2}}(\epsilon_{1}\cdot\epsilon_{2})(\epsilon_{3}\cdot
p_{4})$
$\displaystyle-\sum_{i=1,2}\left[\frac{g_{WWZ}g_{WWh}^{(2)}}{p_{i3}^{2}-m_{W}^{2}}\epsilon_{i}^{\dot{a}{a}}\epsilon_{3}^{c\dot{c}}{{V_{a\dot{a};}}^{b\dot{b};}}_{c\dot{c}}(p_{i},-p_{i3},p_{3})\left(\epsilon_{bd}\epsilon_{\dot{b}\dot{d}}-\frac{[p_{i3}]_{b\dot{b}}[p_{i3}]_{d\dot{d}}}{2m_{W}^{2}}\right)\epsilon_{j}^{d\dot{d}}\right].$
(4.47)

The high-energy limit of the all-longitudinal amplitude is given by

$\displaystyle A_{WWZh^{--}}^{(LLL)}$
$\displaystyle=\frac{t+u}{4m_{W}^{4}m_{Z}}\left[g_{WWZ}g_{WWh}^{(2)}(4m_{W}^{2}-m_{Z}^{2})+2m_{W}^{2}g_{WWh}^{(2)}g_{hhZ}^{(2)}\right]+\mathcal{O}(E^{0}).$
(4.48)

Therefore we obtain

$\displaystyle 0$
$\displaystyle=g_{WWZ}g_{WWh}^{(2)}\left(4-\frac{m_{Z}^{2}}{m_{W}^{2}}\right)+2g_{WWh}^{(2)}g_{hhZ}^{(2)}.$
(4.49)

####  $WW\gamma h$ amplitude

To fix $g_{hh\gamma}^{(2)}$, we consider the $W^{+}W^{+}\gamma h^{--}$
scattering. For the external photon, we fix the reference vector so that the
transverse polarization vector has vanishing temporal component and the photon
momentum is shifted by this transverse polarization vector. The computation is
analogous to $WWZh^{--}$. The amplitude has the poles at

$\displaystyle\hat{p}_{12}^{2}=m_{2}^{2},\quad\hat{p}_{13}^{2}=m_{W}^{2},\quad\hat{p}_{14}^{2}=m_{W}^{2},$
(4.50)

and the pole contributions contain terms only up to linear in $z$. Therefore
we obtain

$\displaystyle A_{WW\gamma h^{--}}^{(\lambda_{1}\lambda_{2}\lambda_{3})}=$
$\displaystyle-\frac{2g_{WWh}^{(2)}g_{hh\gamma}^{(2)}}{p_{12}^{2}-m_{2}^{2}}(\epsilon_{1}\cdot\epsilon_{2})(\epsilon_{3}\cdot
p_{4})$
$\displaystyle-\sum_{i=1,2}\frac{g_{WWh}^{(2)}g_{WW\gamma}}{p_{i3}^{2}-m_{W}^{2}}\epsilon_{i}^{\dot{a}a}\epsilon_{3}^{\dot{c}c}{{V_{a\dot{a};}}^{b\dot{b};}}_{c\dot{c}}(p_{i},-p_{i3},p_{3})\left[\epsilon_{bd}\epsilon_{\dot{b}\dot{d}}-\frac{[p_{i3}]_{b\dot{b}}[p_{i3}]_{d\dot{d}}}{2m_{W}^{2}}\right]\epsilon_{j}^{\dot{d}d},$
(4.51)

where $j\neq 3,4,i$. Now, if we replace the photon polarization vector by its
momentum, $\epsilon_{3}\to p_{3}$, we obtain

$\displaystyle\left.A_{WW\gamma
h^{--}}^{(\lambda_{1}\lambda_{2}\lambda_{3})}\right|_{\epsilon_{3}\to p_{3}}=$
$\displaystyle-\left(g_{hh\gamma}^{(2)}+2g_{WW\gamma}\right)(\epsilon_{1}\cdot\epsilon_{2}).$
(4.52)

This indicates that the Ward identity of the photon does not hold for
arbitrary $g_{hh\gamma}^{(2)}$, and we need

$\displaystyle g_{hh\gamma}^{(2)}=-2g_{WW\gamma}.$ (4.53)

We require this condition for the consistency of our procedure from both gauge
symmetry perspective and constructibility consistency, i.e., $B_{\infty}=0$.
This relation makes sense since we assume that $h^{++}$ is doubly charged
under the photon, but the on-shell method does not automatically tell us this
relation. Seemingly, if the gauge symmetry relates different three-point
couplings such as $g_{hh\gamma}^{(2)}$ and $g_{WW\gamma}$, we need to impose
these conditions.

####  $WWhh$ amplitude

Finally, we consider the $WWhh$ scattering. The $W^{+}W^{-}h^{0}h^{0}$
amplitude is not affected by $h^{++}$, while $W^{+}W^{+}h^{0}h^{--}$ is
analogous to the $W^{+}W^{-}h^{0}h^{0}$. Therefore, we focus on the amplitude
of $W^{+}W^{-}h^{++}h^{--}$, labeled as $1,2,3,4$, respectively. In this
process, we have the poles at

$\displaystyle\hat{p}_{12}^{2}=m_{Z}^{2},\,\mathrm{or}\,~{}0,\quad\hat{p}_{14}^{2}=m_{W}^{2},$
(4.54)

and the amplitude is given by

$\displaystyle A_{WWh^{++}h^{--}}^{(\lambda_{1}\lambda_{2})}=$
$\displaystyle\sum_{I=Z,\gamma}\frac{1}{p_{12}^{2}-m_{I}^{2}}\frac{1}{z_{12}^{+}-z_{12}^{-}}\sum_{\lambda}\left[z_{12}^{+}\hat{A}_{WWI}^{(\lambda_{1}\lambda_{2}\lambda)}(z_{12}^{-})\times\hat{A}_{h^{++}h^{--}I}^{(\bar{\lambda})}(z_{12}^{-})-(z_{12}^{+}\leftrightarrow
z_{12}^{-})\right]$
$\displaystyle+\frac{1}{p_{14}^{2}-m_{W}^{2}}\frac{1}{z_{14}^{+}-z_{14}^{-}}\sum_{\lambda}\left[z_{14}^{+}\hat{A}_{WWh^{--}}^{(\lambda_{1}\lambda)}(z_{14}^{-})\times\hat{A}_{WWh^{++}}^{(\bar{\lambda}\lambda_{2})}(z_{14}^{-})-(z_{14}^{+}\leftrightarrow
z_{14}^{-})\right],$ (4.55)

where, as usual, we assume $\lambda_{1}=\pm$. The products of the three-point
amplitudes are given by

$\displaystyle\hat{A}_{WWI}^{(\lambda_{1}\lambda_{2}\lambda)}\times\hat{A}_{h^{++}h^{--}I}^{(\bar{\lambda})}$
$\displaystyle=-g_{WWI}g_{hhI}^{(2)}\epsilon_{1}^{\dot{a}a}\epsilon_{2}^{\dot{b}b}V_{a\dot{a};b\dot{b};c\dot{c}}(\hat{p}_{1},\hat{p}_{2},-\hat{p}_{12})(\hat{p}_{3}-\hat{p}_{4})^{\dot{c}c},$
(4.56)
$\displaystyle\hat{A}_{WWh^{--}}^{(\lambda_{1}\lambda)}\times\hat{A}_{WWh^{++}}^{(\bar{\lambda}\lambda_{2})}$
$\displaystyle=-\frac{\left(g_{WWh}^{(2)}\right)^{2}}{2}\epsilon_{1}^{\dot{a}a}\left[\epsilon_{ab}\epsilon_{\dot{a}\dot{b}}-\frac{[\hat{p}_{14}]_{a\dot{a}}[\hat{p}_{14}]_{b\dot{b}}}{2m_{W}^{2}}\right]\epsilon_{2}^{b\dot{b}}$
(4.57)

Each term contains up to the $z^{2}$-terms. By using $\lambda_{1}=\pm$ and
hence $q_{1}=c_{1}m_{W}\epsilon_{1}$, the $z^{2}$-term is given by

$\displaystyle\left[A_{WWh^{++}h^{--}}^{(\lambda_{1}\lambda_{2})}\right]_{\mathrm{contact}}$
$\displaystyle=\frac{1}{\sqrt{2}c_{1}m_{W}}\left[\left(g_{WWZ}g_{hhZ}^{(2)}+g_{WW\gamma}g_{hh\gamma}^{(2)}\right)\langle\mathbf{2}|(q_{3}-q_{4})|\mathbf{2}]+\frac{\left(g_{WWh}^{(2)}\right)^{2}}{2m_{W}^{2}}\langle\mathbf{2}|q_{3}|\mathbf{2}]\right].$
(4.58)

For arbitrary choice of $g_{hhZ}^{(2)}$ and $g_{WWh}^{(2)}$, this _does not_
lead to the result independent of $q_{i}$. Since the original amplitude does
not depend on $q_{i}$ (except for $q_{1}\propto\epsilon_{1}$), we need a non-
zero $B_{\infty}$ to cancel the $q_{i}$-dependence in general, indicating that
the Ward identity does not hold for arbitrary $g_{hhZ}^{(2)}$ and
$g_{WWh}^{(2)}$. In the current case, the result is independent of these
variables only when we have

$\displaystyle 0$
$\displaystyle=4\left(g_{WWZ}g_{hhZ}^{(2)}+g_{WW\gamma}g_{hh\gamma}^{(2)}\right)m_{W}^{2}+\left(g_{WWh}^{(2)}\right)^{2},$