††institutetext:  1 Department of Physics, Florida State University, 77
Chieftan Way, Tallahassee, FL 32306, USA
2Theory Center, High Energy Accelerator Research Organization (KEK), 1-1 Oho,
Tsukuba, Ibaraki 305-0801, Japan

# Energy Growth in $V_{L}V_{L}\to V_{L}V_{L},\ V_{L}V_{L}h$ Scattering to
Probe Higgs Cubic and HEFT Interactions

Shameran Mahmud1 and Kohsaku Tobioka1,2<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We compute the energy scales of perturbative unitarity violation in
$V_{L}V_{L}\to V_{L}V_{L}h$ processes and compare them to $V_{L}V_{L}\to hhh$
process. Using these energy scales, we determine which process is more
sensitive to potential modifications in the Higgs sector at high-energy
colliders. Within the Higgs Effective Field Theory (HEFT), we consider the
Higgs cubic coupling and other interactions with and without derivatives. Any
HEFT interactions predict the perturbative unitarity violation at a finite
scale, and in a generic case, the minimalistic process is $2\to 3$ scattering.
Our analysis reveals that the energy scales for unitarity violation in
$V_{L}V_{L}\to V_{L}V_{L}h$ and $V_{L}V_{L}\to hhh$ processes are similar
across all scenarios considered. If the backgrounds are similar, $V_{L}V_{L}h$
final states are more feasible because $V_{L}V_{L}h$ has higher branching
ratios in cleaner decay modes than $hhh$. We also investigate HEFT derivative
interactions derived from various UV models. In these cases, both
$V_{L}V_{L}\to V_{L}V_{L}$ and $V_{L}V_{L}\to hh$ processes exhibit unitarity
violating behavior. We demonstrate that the energy scales for unitarity
violation in $V_{L}V_{L}$ final states are comparable to or even lower than
those in the $hh$ final state.

##  1 Introduction

Since its discovery in 2012, the Higgs boson has been intensively studied and
is considered a particularly intriguing and important particle due to its
potential sensitivity to new physics beyond the Standard Model (BSM). A
significant recent advancement in understanding the Higgs boson stems from
Higgs coupling measurements, which involve assessing its couplings to fermions
and gauge bosons through production and decay processes. We observe that the
SM is consistent with these measurements at an order of 10% precision
ATLAS:2022vkf ; ATLAS:2024fkg ; CMS:2022dwd ; Mondal:2024ijr , and the
precision could be as low as a few percent at the High-Luminosity LHC (HL-
LHC). On the other hand, these measurements do not probe the pure Higgs sector
unless the Higgs boson has an unconventional kinetic term that modifies the
Higgs couplings to fermions and gauge bosons.

For the Higgs potential, the Higgs mass and the vacuum expectation value (VEV)
were already well measured, but the other terms of the Higgs potential are
poorly constrained. Therefore, probing the Higgs potential is essential. The
current major effort along this line is to measure the Higgs cubic coupling,
$h^{3}$, through the di-Higgs boson production at the LHC. The theory
predictions for di-Higgs boson production include Baglio:2012np ;
Davies:2019dfy ; Chen:2019lzz ; Grazzini:2018bsd ; Dreyer:2018qbw . Since the
cross section of this process is small, the current bound on the cubic
coupling is at $\mathcal{O}(5)$ precision compared to the SM prediction
ATLAS:2021tyg ; ATLAS:2023gzn ; CMS:2022gjd ; CMS:2022hgz , and it will remain
challenging at the HL-LHC, where the projected precision is
$\mathcal{O}(50\%)$ Cepeda:2019klc .

While measuring di-Higgs production is compelling, this observable alone is
insufficient to pinpoint the underlying new physics. If a deviation is
observed in the di-Higgs process, various modifications to the Higgs sector,
beyond a simple shift in the $h^{3}$ coupling, could account for the data.
Therefore, additional information from different processes is essential to
narrow down the possibilities.

To conduct a general analysis, we employ the Higgs Effective Field Theory
(HEFT) framework Feruglio:1992wf ; Bagger:1993zf ; Koulovassilopoulos:1993pw .
The more common framework, the Standard Model Effective Field Theory (SMEFT)
Buchmuller:1985jz ; Leung:1984ni ; PhysRevLett.43.1566  with finite-
dimensional truncation can be expressed by a finite expansion of the HEFT, but
not vice versa. Another intriguing aspect of HEFTs which are not SMEFTs, is
that the mass scale cannot be arbitrarily large, meaning that the high-energy
colliders in the near future could probe the HEFT parameter space well. See
Sec. 2 for more discussion.

Figure 1: We study the $V_{L}V_{L}\to V_{L}V_{L}h$ process (left) and the
$V_{L}V_{L}\to hhh$ process (right) within the HEFT, and show that the
$V_{L}V_{L}\to V_{L}V_{L}h$ process has a similar order in the energy-growing
behavior to the the $V_{L}V_{L}\to hhh$ process.

Our main focus of this paper is to investigate the scattering processes of
electroweak gauge bosons and Higgs boson based on the HEFT, in particular, the
ones that have perturbative unitarity violation (from now on, called unitarity
violation for brevity) at high energy. It is well-known that the $2\to 2$
scattering of electroweak gauge bosons diverges at the high energy without the
diagrams involving the Higgs boson because, in the SM, the scattering process
in the electroweak sector has subtle cancellations involving the Higgs boson.
Similarly, almost any modification of the Higgs couplings can cause an
incomplete cancellation, leading to unitarity violation at high energy.
Therefore, the exploration of Higgs/gauge boson scattering processes is
complementary to the di-Higgs measurement. Earlier works Falkowski:2019tft ;
Chang:2019vez  explicitly showed that the shift of the Higgs cubic coupling
makes the $V_{L}V_{L}\to h^{n}$ scattering processes violate unitarity with
$n\geq 3$. Another important feature is that higher collision energy will
significantly enhance the cross section, which can be realized at the proposed
FCC and muon colliders.

Regarding the unitary violating features of the HEFT, most of the previous
works are presented using $V_{L}V_{L}\to~{}h^{n}$ Falkowski:2019tft ;
Chang:2019vez ; Cohen:2021ucp ; Gomez-Ambrosio:2022qsi . The equivalence
theorem is applied in most of them, replacing the longitudinal gauge bosons
($V_{L}$) with the Nambu-Goldstone (NG) bosons ($G$) because the energy-
growing behaviors are explicit. Exploiting the geometric features of HEFT,
which are invariant under Higgs field redefinitions Alonso:2015fsp ;
Alonso:2016oah ; Cohen:2020xca , a calculation technique using sectional
curvatures has also been established by Ref. Cohen:2021ucp . However, from the
perspective of hadron collider searches, detecting multiple Higgs bosons is
significantly more challenging than detecting multiple gauge bosons due to the
large branching ratio of $h\to b\bar{b}$. This difficulty partly explains why
di-Higgs boson searches have been arduous. Conversely, final states with
multiple gauge bosons remain feasible. In fact, the ATLAS and CMS experiments
have recently started observing triple gauge boson production ATLAS:2019dny ;
CMS:2020hjs .

This paper argues that the final states with multiple gauge bosons often
exhibit a similar strength of unitarity violation as in the fully Higgs boson
final states. Roughly speaking, replacing Higgs boson pairs from the
scattering process with longitudinal gauge boson pairs would maintain the
unitarity violation, although there are exceptions for specific modifications,
which are discussed in App. A. 111For example, if only the Higgs quartic
coupling is shifted, the replacement does not work: $V_{L}V_{L}\to hhh$ has
unitary violation while $V_{L}V_{L}\to V_{L}V_{L}h$ does not.

For the demonstration, we mainly focus on the minimalistic processes, that is,
the $2\to 3$ scattering as in Fig. 1, and compare $V_{L}V_{L}\to V_{L}V_{L}h$
to $V_{L}V_{L}\to hhh$ in the various HEFT scenarios including the shift of
Higgs cubic coupling. The authors of Chang:2019vez  focused on the cubic
coupling shift and showed that the unitarity violation appears in
$Z_{L}Z_{L}\to Z_{L}Z_{L}h$, $W_{L}Z_{L}\to W_{L}Z_{L}h$, and even in
$V_{L}V_{L}\to V_{L}V_{L}V_{L}V_{L}$. The importance of $2\to 3(4)$ processes
with multiple gauge bosons in the final state was also addressed in Refs.
Belyaev:2012bm ; Abu-Ajamieh:2020yqi . In our work, we consider various
scenarios and comprehensively show which process will more effectively probe
modifications of the Higgs potential than others. If the HEFT interactions
involve derivatives, the $2\to 2$ scattering can violate unitarity, and we
show that, considering several sets of HEFT interactions, $V_{L}V_{L}\to
V_{L}V_{L}$ is at least comparable to $V_{L}V_{L}\to hh$. The past works
involving $2\to 2$ processes or derivative interactions in the HEFT include
Refs. Belyaev:2012bm ; Abu-Ajamieh:2020yqi ; Cohen:2021ucp ; Kanemura:2021fvp
; Eboli:2023mny ; Davila:2023fkk . The experimental search for the
$V_{L}V_{L}$ final state has been performed, and it was observed at
2.3$\sigma$ at the CMS experiment CMS:2020etf . See also an ATLAS study for
$q\bar{q}\to V_{L}V_{L}$ ATLAS:2023zrv .

An intriguing feature of the HEFT is that the scale of unitarity violation is
definite. In the previous works, the energy scale of unitarity violation,
$E_{*}$, for the processes at very high multiplicities can be as low as (few)
$\times 4\pi v$ (i.e., more than 3-body final states). At low multiplicities,
such as $2\to 3$ processes, the unitarity violation scales are still high, but
our focus is not to demonstrate whether the $E_{*}$ values can be reached at
the colliders. Instead, we use the $E_{*}$ values to rank the relevant
processes.

Related works within the SMEFT Henning:2018kys ; Chen:2021rid  also point out
the unitary violation of $V_{L}V_{L}\to V_{L}V_{L}h/V_{L}V_{L}V_{L}$
processes. Note that the correspondence between the Higgs cubic coupling shift
and the unitary violation rate would be slightly different from the HEFT case.
For example, a SMEFT dimension-six operator, such as $(H^{\dagger}H)^{3}$,
will have different coefficients for the relevant contact terms compared to
those from the HEFT operator shifting the Higgs cubic coupling,
$(\sqrt{2H^{\dagger}H}-v)^{3}$. Also, the SMEFT operators generically do not
predict unitary violation in arbitrarily high multiplicities, whereas the HEFT
does.

The paper is organized in the following way. In Sec. 2, we briefly review the
difference between the HEFT and the SMEFT and highlight the relevant features
of HEFT. Then, in Sec. 3, we discuss the perturbative unitarity violation of
the gauge/Higgs boson scattering processes, in particular $V_{L}V_{L}\to
V_{L}V_{L}h$ and $V_{L}V_{L}\to hhh$, and briefly mention the experimental
advantage of having multiple gauge bosons instead of Higgs bosons. In Sec. 4,
we present the results of the impact of HEFT operators, which only affect the
Higgs potential, and in Sec. 5, we do the same for HEFT derivative operators
motivated by several UV model EFTs.

##  2 HEFT vs. SMEFT

In this paper, the main focus will be on interactions in the Higgs Effective
Field Theory (HEFT) - in particular, the HEFT that do not admit a SMEFT. The
reason for this choice is explained in this section.

###  2.1 Parametrizing the EFTs

In simplest terms, the difference between the two EFTs comes down to
parametrization. In the Higgs sector, the SMEFT would be parametrized in terms
of the $SU(2)$ Higgs doublet. One choice would be

$H=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\sqrt{2}G^{+}\\\
(v+h)+iG^{0}\end{array}\right),$ (1)

where $G^{+},G^{0}$ are the NG bosons associated with spontaneous symmetry
breaking, $v=246$ GeV is the Higgs VEV, and $h$ is the singlet Higgs field.
Using the Higgs doublet, a generic Higgs potential in the SMEFT
parametrization is usually written in integer powers of $H^{\dagger}H$,

$V_{\rm
SMEFT}=\sum_{n}\frac{c_{n}}{\Lambda^{2(n-2)}}\left(H^{\dagger}H\right)^{n}.$
(2)

where the mass scale is $\Lambda$ and $c_{n}$ is a dimensionless coefficient.
In the unitary gauge, the Higgs potential can also be given by the HEFT
parametrization by using the singlet, $h$, in Eq. (1),

$V_{\rm HEFT}=\sum_{\ell}\tilde{c}_{\ell}h^{\ell},$ (3)

where $\tilde{c}_{\ell}$ is a dimensionful coefficient. The HEFT
parametrization is more general, as different $\tilde{c}_{\ell}$’s are
correlated to express the SMEFT potential of Eq. (2) using $h$. Due to this,
HEFT allows non-analytic potentials at $H=0$, which cannot be given by the
SMEFT, for example,

$V=g\left(H^{\dagger}H\right)^{\frac{2}{3}}\supset
gv^{\frac{4}{3}}\sum_{n=3}\left(\begin{array}[]{c}\frac{4}{3}\\\
n\end{array}\right)\left(\frac{h}{v}\right)^{n}\ .$ (4)

This illustrates the fact that the HEFT is a more general EFT than the SMEFT,
as shown in Fig. 2. It can encapsulate non-analyticities in the potential,
which is described in more detail in Refs. Alonso:2016oah ; Falkowski:2019tft
; Chang:2019vez ; Cohen:2020xca . Hereafter, when the HEFT is mentioned, it
will be specifically about the space HEFT \SMEFT in Fig. 2, i.e., the green
region, while the SMEFT will refer to the blue region.

One caveat is that, with field redefinitions, some Lagrangian modifications
may look non-analytic but are actually analytic. This is an issue that
obfuscates whether or not some EFT is a SMEFT or a HEFT. This question has
been discussed in depth in Refs. Alonso:2015fsp ; Alonso:2016oah ;
Cohen:2020xca ; Gomez-Ambrosio:2022qsi , and we recommend anyone with interest
in this question to see the details in those references.

Figure 2: Schematic diagram of all possible HEFTs. Every SMEFT, where the
particles can be written as powers, is in the blue region; expanding
$H^{\dagger}H$ in terms of $h$ gives a HEFT parametrization. The green region
(along with the boundary) contains theories that are purely HEFT. Here,
writing $h$ in terms of $H^{\dagger}H$ gives non-analytic terms in the
Lagrangian. Those (here, referred to as “HEFT”) will be the focus of this
paper.

.

###  2.2 Why HEFT?

Unitary violation exists in both SMEFT and HEFT. However, the HEFT has
multiple reasons to be tested instead, from the unitary violation perspective.

First of all, the SMEFT has an unknown mass scale, $\Lambda$, which can be as
high as the Planck scale. The energy scale of unitarity violation, $E_{*}$, is
correlated to $\Lambda$, so there is no a priori reason that the unitarity
violation can be seen even at future high-energy colliders. In the HEFT case,
the energy scale of unitarity violation has to be definite, potentially as low
as a few TeV. 222As shown in Ref. Cohen:2020xca , the HEFT is induced by non-
decoupling effects due to a new particle getting the majority of its mass from
the Higgs mechanism. This leads to definite unitarity violation scales.
Otherwise, the new particle can decouple by having a very high bare mass,
leading to a SMEFT. This means HEFT is a class of testable new physics. A
well-known result shown in Falkowski:2019tft ; Cohen:2021ucp  is that the
$E_{*}$ value in the HEFT will be lower as the number of final states
increases.

Another reason is that shifting a single Higgs potential term is covered by
the HEFT rather than the SMEFT. The generic Higgs potential modification
terms, as in Eq. (3), can be written in a gauge-invariant way as

$\displaystyle h^{\ell}\to$
$\displaystyle\left(\sqrt{2H^{\dagger}H}-v\right)^{\ell}.$ (5)

A remark is that in di-Higgs measurements, the shift applied is only the cubic
coupling, which would go under the category of HEFT. However, this difference
is not important currently unless one includes NLO electroweak corrections of
the di-Higgs production or looks at relevant higher multiplicity scattering
processes which violate unitarity. Given that a sizable coefficient for the
cubic coupling is allowed by the current LHC constraints, let us show the
expansion of the cubic coupling,

$\displaystyle h^{3}\to$
$\displaystyle\left(\sqrt{2H^{\dagger}H}-v\right)^{3}\supset
h^{3}+\frac{3G^{2}}{2v}h^{2}\sum_{n=0}^{\infty}\left(-\frac{h}{v}\right)^{n}+\frac{G^{4}}{8v^{3}}\sum_{n=0}^{\infty}(n+1)\left(-\frac{h}{v}\right)^{n}\left(6vh-\frac{n+2}{2}h^{2}\right)$
(6)
$\displaystyle+\frac{G^{6}}{16v^{5}}\sum_{n=0}^{\infty}\frac{(n+1)(n+2)}{2}\left(-\frac{h}{v}\right)^{n}\left\\{2v^{2}-(n+3)(2vh-\frac{n+4}{12}h^{2})\right\\}+\cdots$

This shift of cubic coupling leads to infinite NG boson-Higgs interactions.
While the SMEFT also admits a cubic coupling shift, for example, by the
dimension-six operator,

$\left(H^{\dagger}H\right)^{3}=\frac{1}{8}((v+h)^{2}+G^{2})^{3}\supset\frac{5}{2}v^{3}h^{3},$
(7)

there are only finite contact interactions between the NG bosons and Higgs.

Similarly, other HEFTs will also have infinite contact interactions due to
expanding around $v$, meaning HEFTs, in general, have a lower $E_{*}$ value
than SMEFTs. On top of this, any $V_{L}V_{L}\to h^{n}$ or
$V_{L}V_{L}\to(V_{L}V_{L})^{n}(h)$ process with a corresponding contact term
will have unitarity violation in HEFT. Since collider searches prefer a lower
multiplicity of final states, energy-growing processes that have the least
amount of final states will be studied. In the next section, we show that in
the cases with a potential modification, 3-body final states, particularly,
the $V_{L}V_{L}\to hhh$ and the $V_{L}V_{L}\to V_{L}V_{L}h$ processes, will
always have unitarity violation in HEFT. We also show that $2\to 2$ processes,
that is, the $V_{L}V_{L}\to hh$ and the $V_{L}V_{L}\to V_{L}V_{L}$ processes,
can also have energy growth based on specific HEFTs which admit derivative
terms in the effective Lagrangian.

##  3 Unitarity violation and $2\to 3$ scattering

The modification of the Higgs sector leads to various consequences, one of
which is the energy-growing behavior in scattering processes. The most well-
known example is that the electroweak theory without the Higgs boson predicts
the $V_{L}V_{L}\to V_{L}V_{L}$ scattering cross section grows with energy, and
the (perturbative) unitarity seems to break down as the tree-level and higher-
order contributions are comparable.

We analyze tree level scattering processes of gauge/Higgs bosons as we look
for perturbative unitarity violation. The equivalence theorem for NG bosons
and gauge bosons is also used for simplicity, as gauge boson scattering would
contain unitarity violation due to incomplete cancellations between multiple
diagrams, while in the equivalence theorem picture, unitarity violation can
occur due to a single contact term diagram.

###  3.1 Averaged matrix element and $E_{*}$

Here, we examine the impact of the HEFT interactions, including the Higgs
cubic coupling, on the electroweak boson scattering processes. In order to
quantify the unitarity violation, following Refs. Falkowski:2019tft ;
Chang:2019vez ; Cohen:2021ucp , we extract the $s$-wave contribution by
evaluating the phase-space averaged scattering matrix element,

$\hat{\mathcal{M}}=\left(\frac{1}{\prod_{i}m_{i}!\prod_{i}n_{i}!\int\mathrm{dLIPS}_{m}\int\mathrm{dLIPS}_{n}}\right)^{\frac{1}{2}}\int\mathrm{dLIPS}_{m}\int\mathrm{dLIPS}_{n}\mathcal{M},$
(8)

where $m$ is the number of ingoing particles, $n$ is the number of outgoing
particles, and $m_{i}$ and $n_{i}$ are sets of indistinguishable particles in
the initial and final state, respectively, so that $\sum_{i}m_{i}=m$ and
$\sum_{i}n_{i}=n$, and
$\mathrm{dLIPS}_{n}=\prod_{i=1}^{n}\frac{\mathrm{d}^{3}p_{i}}{(2\pi)^{3}2E_{i}}(2\pi)^{4}\delta^{4}\left(\sum_{i}p_{i}\right)$.

To measure the strength of unitarity violation in each process, we find an
energy scale, $E_{*}$, at which the unitarity violates at the tree level, by
checking the condition

$|\hat{\mathcal{M}}|^{2}\simeq 1.$ (9)

The connection of unitarity violation with $\hat{\mathcal{M}}$ is discussed in
App. B. The averaged matrix element is dimensionless, and for a constant
$\mathcal{M}$, the corresponding cross section is obtained by roughly
$|\hat{\mathcal{M}}|^{2}/E_{\rm cm}^{2}$ where $E_{\rm cm}$ is the center-of-
mass energy, which is discussed in App. C.

Let us address two basic features of $E_{*}$: different scattering processes
will have different $E_{*}$ values, and the process with lower $E_{*}$ is
generally more sensitive to the underlying effects of new physics. The energy
scale of unitarity violation indicates some new physics scale with a given
modification of the Higgs sector, but new physics could cure the unitarity at
a much lower scale, although we cannot a priori know this scale. This means
$E_{*}$ is the highest possible scale for the new particle mass.

The main purpose of this paper is to sort out relevant processes in $2\to 3$
as well as $2\to 2$ scattering within various HEFT scenarios. Therefore, we
use the $E_{*}$ values to rank some processes over others in terms of how
sensitive they will be to new physics. This kind of information would be
useful for experimentalists to set up programs for high energy gauge/Higgs
boson scattering probes at the HL-LHC or other future colliders.

###  3.2 $V_{L}V_{L}\to hhh$ and $V_{L}V_{L}\to V_{L}V_{L}h$

In this section, we elaborate on why vector-boson scattering to three bosons,
in particular $V_{L}V_{L}h$ and $hhh$, will generally have energy-growing
behavior. We also compare $V_{L}V_{L}h$ and $hhh$ final states from an
experimental point of view. To be conservative, suppose only the Higgs
potential is modified, namely, no new operators involving derivatives exist.

Unitarity violation is easily understood using the equivalence theorem and the
linear parametrization of the NG boson as in Eq. (1). Expanding the HEFT
interaction by $h$ and $G$ gives almost arbitrary contact interactions (note
that $G$ always appears as a pair). Each contact term leads to the constant
contribution to the matrix element $\cal M$ of the $2\to n$ process,
independent of $E_{\rm cm}=\sqrt{s}$. Non-contact contributions have negative
powers of energy due to the propagators and, therefore, will not cancel the
contact contribution. The averaged matrix element defined in Eq. (8) has a
phase space integral in the $n$-body final state, which gives energy
dependence as

$\int\mathrm{dLIPS}_{n}=\int\prod_{i=1}^{n}\frac{\mathrm{d}^{3}p_{i}}{(2\pi)^{3}2E_{i}}(2\pi)^{4}\delta^{4}\left(\sum_{i}p_{i}\right)\simeq\frac{1}{8\pi\left(n-1\right)!\left(n-2\right)!}\left(\frac{E_{\rm
cm}}{4\pi}\right)^{2(n-2)}$ (10)

where the last expression is in the massless limit. It is clear that for the
$2\to 2$ processes, the constant matrix element does not lead to any energy-
growing behavior in $\hat{\mathcal{M}}$; thus, no unitarity violation occurs.
In contrast, $2\to 3$ or higher multiplicity processes predict unitarity
violation as long as the constant matrix element exists. As expanding HEFT
potential terms always gives contact interactions of high multiplicity, this
behavior is guaranteed in HEFT potentials.

While the earlier works tend to focus on the $V_{L}V_{L}\to h^{n}$ processes
with $n\geq 3$ Falkowski:2019tft ; Chang:2019vez ; Cohen:2021ucp ; Gomez-
Ambrosio:2022qsi ; Davila:2023fkk , we would like to emphasize the
$V_{L}V_{L}\to V_{L}V_{L}h$ process because the scale of the unitarity
violation is often lower than $V_{L}V_{L}\to hhh$ process or at least
comparable, as we will show in Secs. 4 and 5. Additionally, we point out that
the experimental analysis would be easier than the multi-Higgs boson final
states. In the multi-Higgs final states, extracting clean signatures would be
more difficult than $V_{L}V_{L}\to V_{L}V_{L}h$ because the Higgs boson decays
dominantly to a bottom-quark pair, which suffers from the QCD background, and
the clean decays of $h\to\gamma\gamma,\tau^{+}\tau^{-},4\ell$ have small
branching ratios. On the other hand, $V_{L}V_{L}h$ has more signal yield in
the clean channels because the branching ratios of electroweak gauge bosons
decaying to leptons is $\cal O$(10%).

final state $hAB$ | possible measured decays | branching ratio products
---|---|---
$hhh$ | ($b\bar{b}$)($b\bar{b}$)($\gamma\gamma$) | $2.08\times 10^{-3}$
($b\bar{b}$)($b\bar{b}$)($\tau^{+}\tau^{-}$) | $5.06\times 10^{-2}$
($b\bar{b}$) ($\gamma\gamma$) ($\gamma\gamma$) | $8.27\times 10^{-6}$
($b\bar{b}$)($\tau^{+}\tau^{-}$)($\tau^{+}\tau^{-}$) | $5.72\times 10^{-3}$
($b\bar{b}$)($\gamma\gamma$)($\tau^{+}\tau^{-}$) | $4.77\times 10^{-4}$
($\gamma\gamma$)($\tau^{+}\tau^{-}$)($\tau^{+}\tau^{-}$) | $2.70\times 10^{-5}$
($\gamma\gamma$)($\gamma\gamma$)($\tau^{+}\tau^{-}$) | $1.13\times 10^{-6}$
$ZZh$ | $(b\bar{b})(\ell^{+}\ell^{-})(\ell^{+}\ell^{-})$ | $5.41\times 10^{-3}$
$(b\bar{b})(q\bar{q})(\ell^{+}\ell^{-})$ | $7.47\times 10^{-2}$
$WWh$ | $(b\bar{b})(\ell\nu)(\ell\nu)$ | $5.62\times 10^{-2}$
$(b\bar{b})(q\bar{q})(\ell\nu)$ | $0.233$
Table 1: Examples of some branching ratios to compare a 3 Higgs boson final
state to one with one Higgs boson and two gauge bosons. Higgs boson to
$\gamma\gamma$ was chosen in this example as it is the cleanest channel
available. Looking for multiple Higgs boson to $b\bar{b}$ decays is
statistically difficult, as the $b$ pairs come in jets, which are much harder
to analyze due to the difficulties of QCD. Also, processes involving $V_{L}$
also have a transverse background, which makes the signal gain more non-
trivial. All the branching ratio values have been obtained from the PDG or the
LHC Higgs working group.

Figure 3: Most reasonable decay processes for both the $V_{L}V_{L}h$ and the
$hhh$ final states at hadron colliders. The $V_{L}V_{L}h$ final state has the
$V_{L}V_{L}$ decaying to $\bar{q}q$+$\ell\nu/\ell\ell$ and the $h$ decaying to
a $\bar{b}b$ pair, compared to the $hhh$ final state where $hh$ decays to a
$\bar{b}b\bar{b}b$ and the third $h$ decays to $\tau^{+}\tau^{-}$. As shown in
Table. 1, the $V_{L}V_{L}h$ final state has a higher branching ratio than the
$hhh$ final state with this choice of decay products.

In Table 1, we show the products of branching ratios of Higgs and gauge
bosons, assuming at least one Higgs boson decays to $b\bar{b}$; otherwise, a
significant fraction of the signal would be lost. The representative processes
are as in Fig. 3. As we will show that the production cross sections would be
similar, it is more efficient to search for $W_{L}W_{L}h$ as opposed to $hhh$
due to the branching ratios. For example, in decay modes with four jets,
$WWh\to b\bar{b}q\bar{q}\ell\nu$ is larger than $hhh\to
b\bar{b}b\bar{b}\tau^{+}\tau^{-}(b\bar{b}b\bar{b}\gamma\gamma)$ by a factor of
40 (100), and in modes with two jets, $WWh\to b\bar{b}\ell\nu\ell\nu$ is
larger than $hhh\to
b\bar{b}\tau^{+}\tau^{-}\tau^{+}\tau^{-}(b\bar{b}\gamma\gamma\tau^{+}\tau^{-})$
by a factor of 10 (100).

Since the sensitivities depend on the background, we briefly discuss the
possible backgrounds, although the detailed examination of the background is
beyond the scope of this paper. For the di-Higgs boson search, the $hh\to
b\bar{b}\gamma\gamma$ process is difficult at the LHC, because the continuum
background ($\gamma\gamma$+jets) and single-Higgs background are significant
ATLAS:2021ifb , and the $hh\to b\bar{b}\tau^{+}\tau^{-}$ process is suffered
from $t\bar{t}$, jets+$\tau_{\rm fake}$ and $Zb\bar{b}$ backgrounds
ATLAS:2022xzm . If we naively extend this knowledge to the $pp\to hhhjj$
process, the continuum background would still be large for $hhh\to
b\bar{b}b\bar{b}\gamma\gamma$. The di-Higgs boson processes could be a
background for the triple Higgs boson signal. In the decay of $hhh\to
b\bar{b}b\bar{b}\tau^{+}\tau^{-}$, we expect the background of
$Z+b\bar{b}b\bar{b}/b\bar{b}jj$, $t\bar{t}b\bar{b}$, and jets+$\tau_{\rm
fake}$. On the other hand, for the $pp\to V_{L}V_{L}hjj$, the main background
would be processes with two gauge bosons (without scattering each other) such
as $hV_{T}V_{T}jj$ and $V_{T}V_{T}b\bar{b}jj$, which is inferred from the
measurement of $W_{L}W_{L}\to W_{L}W_{L}$ at the CMS experiment CMS:2020etf .
On top of this, Given that calculating the background processes at high
multiplicities cannot be easily done, we focus on the signal yields in this
paper.

Let us comment on the other $2\to 3$ scattering processes, such as
$W_{L}W_{L}\to W_{L}Z_{L}W_{L}$ and $W_{L}W_{L}\to Z_{L}hh$. Their averaged-
matrix element may grow with energy as well, despite having a propagator,
which would bring $(E_{\rm cm})^{-2}$ dependence. However, their unitarity
violations are not necessarily as strong as $V_{L}V_{L}h$ and $hhh$ since
there are no diagrams from the contact term. Therefore, for simplicity and
direct comparison to other references, the main focus in this paper will be
$V_{L}V_{L}\to V_{L}V_{L}h$. Note, recently, the experimental search for
triple gauge bosons has been performed ATLAS:2019dny ; CMS:2020hjs . The
unitarity violation in the $V_{L}V_{L}V_{L}$ final state is potentially
interesting to constrain the Higgs couplings to the light quarks, top-quark,
and electroweak gauge bosons Abu-Ajamieh:2020yqi ; Falkowski:2020znk .

Figure 4: $V_{L}V_{L}\to V_{L}hh$ and $V_{L}V_{L}\to V_{L}V_{L}V_{L}$
processes may have energy-growing behavior due to new derivative interactions
shifting the vertices. Because of the addition of propagators potentially
reducing the energy growth, these are not studied in the paper.

###  3.3 $V_{L}V_{L}\to hh$ and $V_{L}V_{L}\to V_{L}V_{L}$

Until now, we have considered HEFT interactions without derivatives. If one
considers UV models that can give a HEFT at low energy, the resulting set of
HEFT interactions generally includes derivative terms that are as sizable as
the potential terms. The derivative interactions introduce energy dependence
into the contact interaction, adding energy-growing behavior to the matrix
element $\cal M$. Therefore, the average matrix element of $2\to 2$ processes
can violate the unitarity even if the phase space does not grow with energy.
We investigate these processes in Sec. 5 and see that the strength of energy
growth in the $V_{L}V_{L}\to V_{L}V_{L}$ process is at least comparable to the
one in the $V_{L}V_{L}\to hh$ process. Again, the experimental search for the
two electroweak gauge bosons would be easier than the di-Higgs channel, as
longitudinal gauge bosons have been produced at the LHC CMS:2020etf ;
ATLAS:2023zrv . Such $2\to 2$ processes were also studied in Refs.
Cohen:2021ucp ; Kanemura:2021fvp ; Eboli:2023mny , which focuses on energy-
growing behavior from derivative interactions in their Lagrangian.

###  3.4 Connection to the SMEFT

Note that when it comes to energy-growing behavior/unitary violation, the
formalism in this paper can be applied to both SMEFT and HEFT. The HEFT was
chosen for the reasons given in Sec. 2. The two major differences are that in
the case of SMEFT operators, the number of processes growing with energy is
finite and that the $E_{*}$ values can be arbitrarily high because the new
physics effect can decouple. Previous works on the SMEFT include Refs.
Corbett:2014ora ; Henning:2018kys ; Chen:2021rid .

##  4 Unitarity violation for Higgs potential modifications

We consider various interactions that affect only the Higgs potentials and
investigate the relative importance of different scattering processes. One can
view this as a more bottom-up approach because the EFTs from some UV models
generically predict the HEFT interactions with derivatives, which is
considered in Sec. 5. As discussed in Sec. 4, the HEFT interactions merely
modifying the Higgs potential do not lead to the unitarity violation at the
$2\to 2$ processes. Therefore, we focus on the $2\to 3$ scattering processes.

Firstly, a generic Higgs cubic coupling shift is looked at, i.e.,
$V(h)\supset\frac{m_{h}^{2}}{2v}\delta_{3}h^{3}$. This was also discussed in
Refs. Falkowski:2019tft  and Chang:2019vez , in the context of unitary
violation in $V_{L}V_{L}\to h^{n}$ Falkowski:2019tft , and more generally
including a few NG boson processes, such as $Z_{L}Z_{L}\to Z_{L}Z_{L}h$ and
$W_{L}Z_{L}\to W_{L}Z_{L}h$ Chang:2019vez . In this paper, we also study other
HEFT interactions as defined in Refs. Cohen:2020xca ; Cohen:2021ucp . As
mentioned in Sec. 2, many HEFTs manifestly involve an infinite series of
$\frac{h}{v}$ in all orders of the EFT expansion parameter. The Higgs cubic
coupling shift, if it exists, would belong in the category of HEFT since
writing $h^{3}$ in a gauge invariant manner gives

$h^{3}=\left(\sqrt{2H^{\dagger}H}-v\right)^{3}\supset
3v^{2}\sqrt{2H^{\dagger}H}+\left(2H^{\dagger}H\right)^{\frac{3}{2}}.$ (11)

Also, for all models, Custodial symmetry is assumed for simplicity; in
general, HEFT can also violate custodial symmetry.

###  4.1 Other Higgs potential modifications in HEFT

Aside from a direct cubic coupling shift, we check a few other potential
modifications in a framework of HEFT,

$\displaystyle
V\supset\frac{1}{4\pi^{2}}\left(H^{\dagger}H\right)^{2}\ln\left(H^{\dagger}H\right),$
(12) $\displaystyle V\supset\left(H^{\dagger}H\right)^{\frac{2}{3}},$ (13)
$\displaystyle V\supset\sqrt{H^{\dagger}H}.$ (14)

The first potential commonly arises when integrating a heavy particle out at
one loop, explaining the $4\pi^{2}$ factor. The other two potentials are
obtained at different parameter spaces after integrating out a second $SU(2)$
scalar doublet at the tree level. For example, $\sqrt{H^{\dagger}H}$ is
obtained in Ref. Galloway:2013dma . Usually, integrating out the new particle
is also associated with non-analytic terms involving derivatives, which makes
the theory grow faster. The $E_{*}$ values obtained from these terms will be
checked in Sec. 5.

For the potential modifications, there are two ways to compute the $E_{*}$
value. Firstly, a linear parametrization is used, with the non-analytic
potential expanded up to the needed number of NG bosons, for example,

$\displaystyle\left(H^{\dagger}H\right)^{2}\ln\left(H^{\dagger}H\right)\supset$
$\displaystyle\frac{v^{4}}{2}\ln\left(1+\frac{h}{v}\right)+\frac{v^{4}}{4}\ln\left(1+\frac{G^{2}}{\left(v+h\right)^{2}}\right)\supset
v^{2}\frac{G^{2}}{4}\sum_{n=0}^{\infty}\left(n+1\right)\left(-\frac{h}{v}\right)^{n}+$
(15)
$\displaystyle\frac{G^{4}}{8}\sum_{n=0}^{\infty}\frac{\left(n+1\right)\left(n+2\right)\left(n+3\right)}{6}\left(-\frac{h}{v}\right)^{n}+\cdots$

For this paper, we truncate the expansion at $\mathcal{O}(G^{4}h)$, as the
$V_{L}V_{L}\to V_{L}V_{L}$ $(h)$ scattering is the main concern. If a process
with $V_{L}^{6}$ was wanted, the truncation should be done a term higher, and
so on.

Another approach, following Cohen:2021ucp , involves a covariant and geometric
approach, which uses the non-linear parametrization, so that
$H^{\dagger}H=\frac{(v+h)^{2}}{2}$. We check that both linear and non-linear
parametrizations gave consistent results. One thing of note is that we
exclusively used non-linear parametrization for the calculations in the Sec. 5
following Ref. Cohen:2021ucp .

After obtaining a matrix element, $\mathcal{M}$, the momentum was averaged
over using Eq. (8). The massless limit was taken for the potential case since
such low multiplicities in the final states will have a much higher $E_{*}$
value than the mass of the $Z$, $W$, or $h$.

Figure 5: Plot of $E_{*}$ values of multiple $2\to 3$ processes. As discussed
in Sec. 4, four different interactions are considered. The first three
processes for $(1+\delta_{3})h^{3}$ were also studied in Ref. Chang:2019vez .
The factor in front of the logarithm is due to the loop-level potential
modification. The $E_{*}$ values in the $V_{L}V_{L}h$ final states are usually
the same as, or better than, those in the $hhh$ final state, which does not
depend on the choices of parameters. This is most apparent for the logarithmic
term (blue circle).

###  4.2 Finding $|\hat{\mathcal{M}}|^{2}$ for the potential modifications

For a potential modification only, the matrix element $\mathcal{M}$ can be
found either by the standard QFT methods after expanding the potential or by
taking the covariant derivative of the potential in the non-linear
parametrization, as done in Cohen:2021ucp . 333Note that here, the convention
of Cohen:2021ucp  is followed. This gives an extra minus sign in
$\mathcal{M}$. Both methods will give the same result, but using the covariant
derivative is more straightforward and can be generalized more easily for a
given potential. For $V_{L}V_{L}\to h^{3}$,

$\mathcal{M}_{V_{L}V_{L}\to
h^{3}}=\partial_{h}^{3}\left(f(h)\partial_{h}V(h)\right)|_{h=0},$ (16)

where $f(h)=\frac{1}{v+h}$. And for $V_{L}V_{L}\to V_{L}V_{L}h$, in the
$(G^{+},G^{-},G^{0})$ basis,

$\displaystyle\mathcal{M}_{V_{L}V_{L}\to
V_{L}V_{L}h}=\frac{1}{5}T_{ijkl}\partial_{h}\left\\{3f(h)\partial_{h}\left(f(h)\partial_{h}V(h)\right)-\partial_{h}^{2}f(h)\partial_{h}V(h)-2\partial_{h}f(h)\partial_{h}^{2}V(h)\right\\}|_{h=0}.$
(17)

where
$T_{ijkl}=\left(\sigma_{ij}\sigma_{kl}+\sigma_{ik}\sigma_{jl}\sigma_{il}\sigma_{jk}\right)$,
and

$\sigma=\left(\begin{array}[]{c c c}0&1&0\\\ 1&0&0\\\
0&0&1\end{array}\right).$ (18)

The main energy-growing contribution for both processes comes from the contact
interaction terms in Eqs. (16, 17), so for the purposes of finding the $E_{*}$
value, the contact terms are the only ones that need to be accounted for, as
other terms just add corrections to it.

Given that the contact interactions are constant, the only non-trivial part is
evaluating the phase space integral. Eq. (10) can be used as a good
approximation of the integral, making obtaining $\hat{\mathcal{M}}$
straightforward. The details for the phase space integral are in App. D.

###  4.3 Results for the Higgs potential modifications

After doing the (tree level) analysis of the potential-only modifications, we
find that the $V_{L}V_{L}\to V_{L}V_{L}h$ processes have similar $E_{*}$
values to the $V_{L}V_{L}\to hhh$ processes. The results are shown in Fig. 5,
and the tabulated values are given in Table 2 of App. E. In the case of the
logarithmic potential modification, all the $V_{L}V_{L}h$ final states have
lower $E_{*}$ values. In other cases, some of the $V_{L}V_{L}h$ final states
have lower $E_{*}$ values than the $hhh$ final state. For example, in the case
of the fractional potential modifications, $(H^{\dagger}H)^{\frac{1}{2}}$ and
$(H^{\dagger}H)^{\frac{2}{3}}$, the $W_{L}W_{L}\to W_{L}W_{L}h$ and the
$Z_{L}Z_{L}\to Z_{L}Z_{L}h$ processes have lower $E_{*}$ values than the
$W_{L}W_{L}\to hhh$ process.

While the $E_{*}$ values are too high to be accessed at the colliders, we
focus on the minimalistic processes and demonstrate that the $V_{L}V_{L}\to
V_{L}V_{L}h$ processes and the $V_{L}V_{L}\to hhh$ processes have similar
energy scales of unitarity violation in the various HEFT scenarios. The
results for the Higgs cubic modification match what was shown in Ref.
Chang:2019vez . In order to reach even lower $E_{*}$ values, we have to
explore higher multiplicity processes beyond 3-body final states. One
intriguing process is $V_{L}V_{L}\to V_{L}V_{L}V_{L}V_{L}$, and it is shown in
Ref. Chang:2019vez  that this process has significantly lower $E_{*}$ value
and suitable to constraint the Higgs cubic coupling.

##  5 Unitarity violation for the HEFT with derivatives

Another case to evaluate the importance of multi-gauge boson final states is
HEFT interactions involving derivatives. Here, instead of picking arbitrary
operators, we investigate sets of HEFT operators motivated by viable UV
models, which were studied in Ref. Banta:2021dek . In this approach, to obtain
derivative terms, three UV models were chosen. A heavy singlet scalar and a
heavy vector-like fermion are chosen based on the viable loryon searches using
Ref. (Banta:2021dek, , Fig. 8, Fig. 10). These models admit a potential term
similar to
$\frac{1}{4\pi^{2}}\left(H^{\dagger}H\right)^{2}\ln\left(H^{\dagger}H\right)$.
On top of this, a model based on the two-Higgs doublet model, briefly
mentioned in Ref. Falkowski:2019tft , was chosen, with parameters that will
admit a $(H^{\dagger}H)^{\frac{2}{3}}$ term in the potential. For this
analysis, because the $E_{*}$ values obtained from the derivative terms are
much lower than those from the potential terms due to the very strong energy-
growing behavior, the main focus is on the derivative terms. These terms have
energy dependence as well, even at the $2\to 2$ level, so both the $2\to 2$
and $2\to 3$ processes are looked at.

The two models chosen from Ref. Banta:2021dek  induce HEFTs via a loop. The
first one is

$\mathcal{L}_{\mathrm{UV},S}=\mathcal{L}_{\mathrm{Higgs}}-\frac{1}{2}S\left(\partial^{2}+m^{2}+\lambda
H^{\dagger}H\right)S,$ (19)

where $S$ is a singlet under $SU(2)$, and $\mathcal{L}_{\mathrm{Higgs}}$ is
the standard model Higgs sector Lagrangian. The other EFT is a set of vector-
like fermions where a non-zero bare mass is allowed,

$\mathcal{L}_{\mathrm{UV},\Psi}=\mathcal{L}_{\mathrm{Higgs}}+\bar{\Psi}\left(i\not{D}-m-\Xi\right)\Psi,$
(20)

where

$\Psi=\left(\begin{array}[]{c}\psi\\\ \chi\end{array}\right),$ (21)

and $\psi$ and $\chi$ are Dirac fermion doublets for which the handed
components transform under the SM $SU(2)_{L}$ and $SU(2)_{R}$ respectively,
and

$\Xi=\left(\begin{array}[]{c c}0&y\xi\\\
y^{*}\xi^{\dagger}&0\end{array}\right),$ (22)

where $\xi=\left(\begin{array}[]{c c}i\sigma_{2}H^{*}&H\end{array}\right)$.
These choices were made to enforce Custodial symmetry in the product
$y\bar{\psi}\xi\chi+(\rm h.c.)$ and to have a non-zero gauge invariant bare
mass. In this paper, the bare mass is turned off. One thing to note is that
the fermion model, by itself, actually breaks $\kappa_{\gamma}$ constraints.
However, this could be fixed by adding other BSM particles, which may bring
$\kappa_{\gamma}$ to the allowed range.

Another (more constrained) model, briefly discussed in Falkowski:2019tft , is
the addition of a scalar $SU(2)$ doublet, $\Sigma$, with no bare mass,

$\mathcal{L}_{\mathrm{UV},\Sigma}=\mathcal{L}_{\mathrm{Higgs}}+|D\Sigma|^{2}+\kappa^{2}(\Sigma^{\dagger}H+\mbox{h.c.})-\lambda_{\Sigma}(\Sigma^{\dagger}\Sigma)^{2}.$
(23)

We choose one viable benchmark point per model. Additionally, we include one
more viable benchmark point for the fermion to demonstrate that the $hhh$
final state has different coupling scaling on $E_{*}$ compared to the
$V_{L}V_{L}h$ final state. For the SU(2) doublet addition, one benchmark point
based on the allowed ranges of the Higgs cubic coupling shift was chosen. From
these benchmark points, we obtain sets of HEFT interactions and calculate the
$E_{*}$ values for each one.

To make sure to be in the parameter space where the HEFT-like behaviors are
the strongest, the bare mass of all of the loop theories are set to $0$ as
well. These theories still receive a mass from the Higgs.

In this case, since the unitarity violation scales for the $V_{L}V_{L}h$ final
states had momentum and angular dependences, the full phase space integral was
done without the approximation in Eq. (64). The integral was not simply
$\int\mathrm{dLIPS}_{3}$, but $\int\mathrm{dLIPS}_{3}\mathcal{M}$ . Details
are given in Sec. 5.2 and App. D.

###  5.1 Obtaining the EFT

As stated in Cohen:2020xca , for HEFT, the EFT must be all orders in the $h$
and two orders in the derivative. Suppose there exists a heavy new particle,
$\phi$. The simplest way to find the lowest order EFT for $\phi$ is by using
tree-level matching. This is done by minimizing the potential for $\phi$, and
hence obtaining $\phi[H]$, where $H$ is the Higgs field. Due to the derivative
terms for $\phi$, the effective Lagrangian will naturally contain the 2nd
order derivative terms. For example, in the case of the scalar doublet
addition,

$\frac{\partial
V_{\Sigma}}{\partial\Sigma^{\dagger}}=\kappa^{2}H-\lambda_{\Sigma}(\Sigma^{\dagger}\Sigma)\Sigma=0\to\Sigma[H]=\left(\frac{\kappa^{2}}{2\lambda_{\Sigma}}\right)^{\frac{1}{3}}\frac{H}{\left(H^{\dagger}H\right)^{\frac{1}{3}}}.$
(24)

Then, the effective Lagrangian reads

$\mathcal{L}_{\mathrm{EFT},\Sigma}=\mathcal{L}_{\mathrm{Higgs}}+\left(\frac{\kappa^{2}}{2\lambda_{\Sigma}}\right)^{\frac{2}{3}}\frac{\left|DH\right|^{2}-\frac{8}{9}\left(\partial\sqrt{H^{\dagger}H}\right)^{2}}{\left(H^{\dagger}H\right)^{\frac{2}{3}}}+\frac{3}{4}\left(\frac{\kappa^{8}}{\lambda_{\Sigma}}\right)^{\frac{1}{3}}\left(2H^{\dagger}H\right)^{\frac{2}{3}}.$
(25)

Using $H=\frac{v+h}{\sqrt{2}}U$ gives the HEFT interactions.

Using the effective potential formalism expanded to 2 derivative order gives
us the loop EFT, in the same way as App. D in Cohen:2020xca  or App. A in
Banta:2021dek . The results are restated here in the $m=0$ limit for
convenience:

$\mathcal{L}_{\mathrm{EFT},S}=\mathcal{L}_{\mathrm{SM}}+\frac{\lambda}{96\pi^{2}}\left(\partial\sqrt{H^{\dagger}H}\right)^{2}+\frac{\lambda^{2}}{64\pi^{2}}\left(H^{\dagger}H\right)^{2}\left(\ln\frac{\mu^{2}}{\lambda
H^{\dagger}H}+\frac{3}{2}\right),$ (26)

for the scalar and

$\mathcal{L}_{\mathrm{EFT},\Psi}=\mathcal{L}_{\mathrm{SM}}+\frac{y^{2}}{4\pi^{2}}\left(|DH|^{2}\ln\frac{\mu^{2}}{y^{2}H^{\dagger}H}-\frac{2}{3}\left(\partial\sqrt{H^{\dagger}H}\right)^{2}\right)-\frac{y^{4}}{4\pi^{2}}(H^{\dagger}H)^{2}\left(\ln\frac{\mu^{2}}{y^{2}H^{\dagger}H}+\frac{3}{2}\right),$
(27)

for the fermion, where $y$ was chosen to be real for simplicity, and we choose
$\mu=m_{f}=\frac{yv}{\sqrt{2}}$ to minimize the logarithm. Note that the
scalar case in Eq. (26) does not have any $\mu$ dependence in the derivative
terms. 444In fact, even with a logarithmic potential modification in the
contact terms, $\mu$ is only important in the $2\to 2$ case. As stated in the
beginning of this section, the potential terms will be ignored for this
analysis.

While Cohen:2021ucp  covered the scalar singlet model as well, different
benchmarks were chosen, and it was in the context of multiple Higgs final
states. Here, the main addition will be multi $(V_{L}V_{L})^{n}$ $(h)$ final
states (for now, $V_{L}V_{L}$).

Figure 6: Plots of $E_{*}$ values of multiple $2\to 2$ (left) and $2\to 3$
(right) processes. The HEFT interections involve derivatives, as discussed in
Sec. 5. The $V_{L}V_{L}\to h^{n}$ processes in the scalar case are studied in
Cohen:2021ucp , but we choose different parameters. Here,
$M\equiv\left(\frac{\kappa^{2}}{\lambda_{\Sigma}}\right)^{\frac{1}{2}}$, and
$\mu=m_{f}$, where $m_{f}$ is the mass of the fermion being integrated out.
The $W_{L}W_{L}\to h^{n}$ processes had no $E_{*}$ values of
$\mathcal{O}$(TeV) in the scalar case, as a bare mass is required for
$\mathcal{K}_{h}|_{h=0}$ to be non-zero. Also, we adopt two benchmark points
of the fermion case because the scaling in the couplings is different between
the $hh(h)$ final states and the $V_{L}V_{L}(h)$ final states due to the
differences in $\mathcal{K}_{h,\Psi}\propto y^{4}$ and
$\mathcal{K}_{\pi,\Psi}\propto y^{2}$.

###  5.2 Finding $|\hat{\mathcal{M}}|^{2}$ for the derivative terms

In this case, due to the derivative interactions, the derivative terms of the
Higgs and NG bosons will no longer be canonically normalized and, in the non-
linear parametrization, will look like

$\mathcal{L}_{\mathrm{kin}}=\frac{1}{2}K^{2}(h)\left(\partial
h\right)^{2}+\frac{1}{2}F^{2}(h)\left|\partial U\right|^{2}.$ (28)

Here, the Higgs doublet, $H$, is written as $\frac{v+h}{\sqrt{2}}U$, where
$U=\exp(i\frac{T^{a}G^{a}}{v})\left(\begin{array}[]{c}0\\\
1\end{array}\right)$, with $T^{a}$ being the $SU(2)$ generators. $U$ encodes
all the information on the NG bosons, and $h$ is the Higgs field. For example,
in the case of the additional $SU(2)$ doublet,

$\mathcal{L}_{\mathrm{kin},\Sigma}=\frac{1}{2}\left[1+\frac{1}{9}\left(\frac{M}{v+h}\right)^{\frac{4}{3}}\right]\left(\partial
h\right)^{2}+\frac{1}{2}\left[1+\left(\frac{M}{v+h}\right)^{\frac{4}{3}}\right]\left|\partial
U\right|^{2},$ (29)

where $M\equiv\left(\frac{\kappa^{2}}{\lambda_{\Sigma}}\right)^{\frac{1}{2}}$.

From $K$ and $F$, one may get sectional curvatures $\mathcal{K}_{h}$ and
$\mathcal{K}_{\pi}$, as defined in Cohen:2021ucp :

$\mathcal{K}_{h}=-\frac{1}{K^{2}}\left(\frac{F^{\prime\prime}}{F}-\frac{K^{\prime}F^{\prime}}{KF}\right),$
(30)

$\mathcal{K}_{\pi}=-\frac{1}{v^{2}F^{2}}\left(1-\frac{v^{2}F^{\prime}}{K^{2}}\right),$
(31)

where the prime denotes a derivative with respect to $h$. Using these forms,
the matrix element, $\mathcal{M}$, is written down like in Ref. Cohen:2021ucp
as

$\mathcal{M}_{V_{L}V_{L}\to
h^{n}}=-\left(s_{12}-\frac{2m_{h}^{2}}{n+1}\right)\left(\frac{\partial_{h}}{K(h)}\right)^{n-2}\mathcal{K}_{h}(h)\bigg{|}_{h=0},$
(32)

where $n=2$ or $3$ for both the $2\to 2$ case and the $2\to 3$ case. The gauge
boson scattering process has the matrix element

$\displaystyle\mathcal{M}_{V_{L}V_{L}\to V_{L}V_{L}}$
$\displaystyle=\frac{1}{6}T_{ijkl}s_{1234}$ (33)
$\displaystyle-\frac{1}{2}\left[\sigma_{ij}\sigma_{kl}(s_{12}+s_{34})+\sigma_{ik}\sigma_{jl}(s_{13}+s_{24})+\sigma_{il}\sigma_{jk}(s_{14}+s_{23})\right]\mathcal{K}_{\pi}(h)\bigg{|}_{h=0},$

where $\sigma_{ij}$ and $T_{ijkl}$ are defined as in Eq. (18), and
$s_{ij...k}=(p_{i}+p_{j}+\cdots p_{k})^{2}$, with all the momenta pointing
inwards. For the $V_{L}V_{L}\to V_{L}V_{L}h$ process, one just replaces
$\mathcal{K}_{\pi}(h)$ with
$\left(\frac{\partial_{h}}{K(h)}\right)\mathcal{K}_{\pi}(h)$ in Eq. (33). The
potential terms from Eqs. (16, 17) still contribute, but as these are not
energy-growing, they can safely be neglected, as they do not violate
perturbative unitarity in $\mathcal{O}$(TeV).

As can be seen, the matrix elements have energy dependence, both for the $2\to
2$ and the $2\to 3$ processes. This means that there will be unitary violation
for both sets of processes. However, one thing of note is that when all
$V_{L}=Z_{L}$ in Eq. 33,

$\displaystyle s_{12}+s_{34}+s_{13}+s_{24}+s_{14}+s_{23}=8m_{Z}^{2},$
$\displaystyle s_{1234}=0,$ $\displaystyle\mbox{($2\to 2$ case)},$ (34a)
$\displaystyle
s_{12}+s_{34}+s_{13}+s_{24}+s_{14}+s_{23}=8m_{Z}^{2}+m_{h}^{2},$
$\displaystyle s_{1234}=m_{h}^{2},$ $\displaystyle\mbox{($2\to 3$ case)}.$
(34b)

This shows that the $Z_{L}Z_{L}\to Z_{L}Z_{L}h$ only has a constant
modification to $\mathcal{M}$, even for the derivative terms, similar to the
potential modification cases. For this reason, the $Z_{L}Z_{L}\to Z_{L}Z_{L}h$
process is omitted in this analysis.

As before, only the contact term is kept. However, complications arise due to
the fact that now, the matrix elements are momentum-dependent, so the phase
space integral, $\int\mathrm{dLIPS}_{3}\mathcal{M}$, has non-trivial angular
and momentum parts, which makes computing $\hat{\mathcal{M}}$ more difficult.
Due to this, a numerical approach was taken, where a Monte Carlo integration
was done to obtain $\mathrm{dLIPS}_{3}$. For more details, see App. D.

###  5.3 Results for the derivative terms

The results are seen in Fig. 6, and the tabulated values in Tables 3 and 4 of
App. E. There are a few things to note. First of all, we show that the
$V_{L}V_{L}h$ processes are a viable alternative to the $hhh$ processes. The
same tendency is seen in the $2\to 2$ processes, where the $V_{L}V_{L}$ final
states have similar or lower $E_{*}$ values compared to the $hh$ final state.
As mentioned in Sec. 3, the mass scale of the new particles are lower than any
of the calculated $E_{*}$ values. For example, in the benchmark points, we
have $m_{f}=452$ GeV ($y=2.6$) and $m_{S}=501$ GeV, much lower than the lowest
possible $E_{*}$ value in both cases (3.1 TeV in the $W_{L}W_{L}\to
Z_{L}Z_{L}$ and $Z_{L}Z_{L}\to W_{L}W_{L}$ processes in the fermion case with
$y=2.6$ and 8.9 TeV in the $W_{L}W_{L}\to Z_{L}Z_{L}h$ and $Z_{L}Z_{L}\to
W_{L}W_{L}h$ processes in the scalar singlet case). So if these theories are
true in nature, the particles will be seen before reaching the given $E_{*}$
value. 555All the loryons searched in Ref. Banta:2021dek  have masses of $800$
GeV or less. As mentioned in Sec. 5.2, the potential terms are neglected for
the $E_{*}$ computation. This is because, as seen by comparing Figs. 5 and 6,
the unitarity violation scales for the derivative terms are much lower than
those of the potential terms. The potential contribution barely changes the
$E_{*}$ value.

Next, two benchmarks were chosen in the fermion portion to show that the
scaling in the $hhh$ case is different from the scaling in the $V_{L}V_{L}(h)$
case. This is due to the fact that $\mathcal{K}_{h}$ and $\mathcal{K}_{\pi}$
have different dependences on $y$ in the regime of the parameter space chosen.
In general, for $\mu=m_{f}$ and $m=0$, using Eqs. (27, 30, 31),

$\mathcal{K}_{h,\Psi}=\frac{3y^{4}}{8\pi^{2}v^{2}}\frac{8\pi^{2}-y^{2}}{\left(6\pi^{2}-y^{2}\right)^{2}},$
(35)
$\mathcal{K}_{\pi,\Psi}=\frac{y^{2}}{8\pi^{2}v^{2}}\frac{16\pi^{2}-3y^{2}}{6\pi^{2}-y^{2}}.$
(36)

In the part which is rational in $y$, given the choices in parameters in this
paper, the constant terms dominate over the coupling terms, so roughly
speaking, $\mathcal{K}_{h}\propto y^{4}$ and $\mathcal{K}_{\pi}\propto y^{2}$.
Then, from Eqs. (32, 33, 8), it can be shown that for the $2\to 2$ processes,
the energy growing behavior is proportional to $|\mathcal{K}_{a}|^{2}E^{4}$,
$a=h,\pi$. This would mean for this case if the phase space factor were to be
neglected,

$\frac{E_{i}}{E_{j}}\simeq\sqrt{\frac{|\mathcal{K}_{a}^{j}|}{|\mathcal{K}_{a}^{i}|}},$
(37)

where $i$ and $j$ label two different parameter choices. This means that for
the $2\to 2$ case,

$\displaystyle\frac{E_{i}}{E_{j}}\simeq\frac{y_{j}^{2}}{y_{i}^{2}},\quad\quad\mbox{($hh$
final state),}$ (38a)
$\displaystyle\frac{E_{i}}{E_{j}}\simeq\frac{y_{j}}{y_{i}},\quad\quad\mbox{($V_{L}V_{L}$
final state).}$ (38b)

This shows why the parametric dependence changes from the two processes in the
regime where $y^{2}<6\pi^{2}$. In the case of the 3-body final state, the
energy growing behavior is $|\mathcal{K}_{a}|^{2}E^{6}$, as
$\mbox{dLIPS}_{3}\propto E^{2}$, and the derivative of $\mathcal{K}_{a}$ still
adds terms where the constants dominate. This means for the $2\to 3$ case,
when $y^{2}<6\pi^{2}$,

$\displaystyle\frac{E_{i}}{E_{j}}\simeq\left(\frac{y_{j}}{y_{i}}\right)^{\frac{4}{3}},\quad\quad\mbox{($hhh$
final state)},$ (39a)
$\displaystyle\frac{E_{i}}{E_{j}}\simeq\left(\frac{y_{j}}{y_{i}}\right)^{\frac{2}{3}},\quad\quad\mbox{($V_{L}V_{L}h$
final state)}.$ (39b)

This same analysis may be done for the singlet and doublet scalar cases using
Eqs. (30, 31, 26, 29). However, for the singlet scalar case, one thing to note
is that for $m=0$, $K(h)$ is actually a constant; this leads to a vanishing
$\mathcal{K}_{h}$. This is why there are no low $E_{*}$ values for the Higgs
final state processes. For the $V_{L}V_{L}$ and $V_{L}V_{L}h$ processes, a
similar derivation as before may be done to get that, for $\lambda<96\pi^{2}$,

$\displaystyle\frac{E_{i}}{E_{j}}\simeq\sqrt{\frac{\lambda_{j}}{\lambda_{i}}},\quad\quad\mbox{($V_{L}V_{L}$
final state),}$ (40a)
$\displaystyle\frac{E_{i}}{E_{j}}\simeq\left(\frac{\lambda_{j}}{\lambda_{i}}\right)^{\frac{1}{3}},\quad\quad\mbox{($V_{L}V_{L}h$
final state).}$ (40b)

Then, for the SU(2) doublet, if $\kappa^{2}<\lambda_{\Sigma}v^{2}$, the two
cases have similar behavior, and

$\displaystyle\frac{E_{i}}{E_{j}}\simeq\left(\frac{M_{j}}{M_{i}}\right)^{\frac{2}{3}},\quad\quad\mbox{(2-body
final state),}$ (41a)
$\displaystyle\frac{E_{i}}{E_{j}}\simeq\left(\frac{M_{j}}{M_{i}}\right)^{\frac{4}{9}},\quad\quad\mbox{(3-body
final state).}$ (41b)

The point $\lambda=8.3$ for the scalar and $y=2.6$ for the fermion are the
maximum possible coupling values for no bare mass given in the results of
Banta:2021dek . This was done to compare the lowest unitarity violation scales
for the given parameter space, to show that the $V_{L}V_{L}(h)$ processes are
compared to the $h^{n}$ processes. Using these two values and their given
energies in Fig. 6, any other $E_{*}$ value in the allowed regimes with no
bare mass can be found using the above equations.

##  6 Discussions and Conclusions

Finding the shape of the Higgs potential is one of the most important current
goals in physics. While the di-Higgs boson search is a very promising probe of
the Higgs cubic coupling, looking at gauge boson/Higgs scattering will allow
us to probe the Higgs potential independently as well. Hence, high energy
electroweak boson scattering processes are essential and complementary to the
di-Higgs boson searches. However, experimental programs at high multiplicities
are not yet established.

From this point of view, we ask, “Is the process with multiple gauge bosons in
the final state just as good as the one with multiple Higgs bosons to examine
modifications of the Higgs sector at high-energy colliders?” Answering this
question was the main goal of this work, and it is done in two ways. Firstly,
we evaluate the energy scales of unitarity violation for the Higgs cubic
coupling shift as well as for multiple different Higgs potential modifications
that fall under the HEFT. We use the equivalence theorem and compute
scattering processes at the tree level. Our results in Fig. 5 suggest that the
$V_{L}V_{L}\to V_{L}V_{L}h$ processes have similar unitarity violation scales
to the $V_{L}V_{L}\to hhh$ process. Secondly, we also examine the $2\to 2$
scattering because the energy growth would appear in this case if HEFT
derivative interactions exist. Such derivative terms are common for many
HEFTs, which are obtained from known UV models. We show, in Fig. 6, that the
unitarity violation scales of the $V_{L}V_{L}\to V_{L}V_{L}$ processes are
similar or even lower than that of the $V_{L}V_{L}\to hh$ process. We find the
same tendency between the $V_{L}V_{L}\to V_{L}V_{L}h$ processes and the
$V_{L}V_{L}\to hhh$ processes. Furthermore, we address the experimental
advantage in the final states with multiple gauge bosons in Table 1.

Let us clarify our method regarding $E_{*}$, the scale of perturbative
unitarity violation in the scattering. We use the value of $E_{*}$ to compare
different processes in each modification of the Higgs sector, mainly between
$hh(h)$ and $V_{L}V_{L}(h)$ final states. We checked in several cases our
$E_{*}$ values are consistent with Ref. Abu-Ajamieh:2020yqi ;
Falkowski:2019tft ; Chang:2019vez ; Cohen:2021ucp . Note that reaching the
given $E_{*}$ value was not the main focus of this paper. The mass of new
particles can be much lower than the $E_{*}$ value, and the values of $E_{*}$
depend on the size of couplings. However, the relative importance of different
processes is robust.

Given that prospective modification of the Higgs sector is unknown,
experiments need to set up observables that cover a wide class of models and
EFTs and are simultaneously more feasible. We emphasize that this paper’s
results will help pick up good observables, such as $V_{L}V_{L}\to
V_{L}V_{L}h$, that can be sensitive to not only the cubic coupling but also
other possible modifications. For the $2\to 2$ processes, we even show that
all the considered HEFT interactions are probed better in the $V_{L}V_{L}\to
V_{L}V_{L}$ processes. This result enhances the impact of the current
experimental program at the LHC.

For realistic sensitivities, one needs to perform the background studies to
see whether or not the $V_{L}V_{L}h$ final state has an advantage over the
$hhh$ in the various collider setups, including the HL-LHC, FCC, and muon
collider. However, they are beyond the scope of this paper. If one confirms
that the background level for both processes is similar, the promising
benchmark $2\to 3$ process for exploring generic modifications of the Higgs
potential is $V_{L}V_{L}\to V_{L}V_{L}h$ because the $V_{L}V_{L}h$ final state
has higher branching ratios in the cleaner decay modes. For the $V_{L}V_{L}\to
V_{L}V_{L}$ processes, it is necessary to look at the high energy part of the
di-Higgs process from the vector-boson fusion, while the CMS experiment tested
the $V_{L}V_{L}\to V_{L}V_{L}$ process CMS:2023rcv . Beyond 3-body final
states, an interesting process is $V_{L}V_{L}\to V_{L}V_{L}V_{L}V_{L}$ as
discussed in Ref. Chang:2019vez , since this process has much smaller $E_{*}$
values in the case of cubic coupling modification, which could be true for
other modifications. However, this has the complications of being a 4-body
final state, leading to many diagrams (of $\mathcal{O}(10^{5})$) and tiny
cross sections. The practical sensitivity studies need more computational
efforts.

##  7 Acknowledgements

We thank Maria Mazza, Takemichi Okui, Xiaochuan Lu, and Yoshihiro Shigekami
for the useful discussion. This work was supported by, in part, the US
Department of Energy grant DE-SC0010102. KT is also supported by JSPS KAKENHI
21H01086 and FSU Summer Research Support award Program.

##  Appendix A How many Higgs bosons can be replaced by longitudinal gauge
bosons?

In general, the number of Higgs bosons that can be replaced by the
longitudinal gauge bosons in some $V_{L}V_{L}\to h^{n}$ process depends on the
type of new interactions. Firstly, all the NG bosons must be added in pairs as

$\displaystyle H^{\dagger}H\supset 2vh+h^{2}+G^{2}\quad\mbox{(Potential
terms),}$ (42a) $\displaystyle|\partial H|^{2}=\frac{1}{2}(\partial
h)^{2}+\frac{1}{2}(\partial G)^{2}\quad\mbox{(Derivative terms),}$ (42b)

so if there is an odd number of Higgs bosons, one of them must remain.

Let us consider a shift in some power of the Higgs potential, $h^{\ell}$.
This, to the lowest order in the NG bosons, can be written as

$\displaystyle
h^{\ell}\to\left(\sqrt{2H^{\dagger}H}-v\right)^{\ell}=\left(h+\frac{G^{2}}{2(v+h)}+\cdots\right)^{\ell}.$
(43)

Here, by expanding, we get $G^{2\ell}$ as the lowest contact interaction of
only $G$, and every term with a lower power of $G$ has to be multiplied by
some number of $h$’s. This shows that if an $h^{\ell}$ term is added to the
potential, the $G^{2(\ell-k)}h^{k}$, $k<\ell$ contact interaction can not have
any Higgs bosons replaced. This is why, for example, the shift of Higgs cubic
coupling gives no $G^{4}$ contact interaction, while $G^{2}h^{2}$ interaction
exists but does not cause the unitarity violation. Another example is that the
shift of Higgs quartic coupling brings a $G^{2}h^{3}$ interaction leading to
the $V_{L}V_{L}\to hhh$ unitarity violation, but the $G^{4}h$ interaction is
absent thus no unitarity violation in $V_{L}V_{L}\to V_{L}V_{L}h$.

From Eq. (43), one can find a $G^{2k}h^{2(\ell-k)}$ interaction by expanding
the $\frac{1}{v+h}$ part, leading to the $V_{L}V_{L}\to
V_{L}^{2(k-1)}h^{2(\ell-k)}$ process for $k<\ell$. Here, all the $\ell-k$
Higgs boson pairs of this process can be replaced by the gauge bosons because
the $G^{2\ell}$ contact interaction exists. More specifically, for $k=1$, in
the $V_{L}V_{L}\to h^{2(\ell-1)}$ process, it is possible to replace all Higgs
boson pairs.

If there is an odd number of Higgs bosons, a $2\to(2\ell-3)$ process can also
have all Higgs boson pairs replaced as well, leaving just one Higgs boson.
This is why in the $G^{2}h^{3}$ case for a cubic coupling shift, two Higgs
bosons can still be replaced. In general, for a $G^{2k}h^{2(\ell-k)-1}$
contact interaction for $k<\ell$, $\ell-k-1$ Higgs boson pairs may be replaced
by NG boson pairs due to the presence of $G^{2\ell-2}h$ contact interaction.

In the case of some other potentials, such as those derived from UV models,
the situation changes because they are typically a function of $H^{\dagger}H$,
not $\sqrt{2H^{\dagger}H}-v$. This means that in a generic case, there is
always a constant term, such as $v^{2}$, that may be multiplied by the $G^{2}$
terms after expanding the potential. For example, consider Eq. (15),

$\displaystyle\left(H^{\dagger}H\right)^{2}\ln\left(H^{\dagger}H\right)\supset$
$\displaystyle\frac{v^{4}}{2}\ln\left(1+\frac{h}{v}\right)+\frac{v^{4}}{4}\ln\left(1+\frac{G^{2}}{\left(v+h\right)^{2}}\right)\supset
v^{2}\frac{G^{2}}{4}\sum_{n=0}^{\infty}\left(n+1\right)\left(-\frac{h}{v}\right)^{n}+$
(44)
$\displaystyle\frac{G^{4}}{8}\sum_{n=0}^{\infty}\frac{\left(n+1\right)\left(n+2\right)\left(n+3\right)}{6}\left(-\frac{h}{v}\right)^{n}+\cdots$

Here, the second term gives an infinite number of $G^{2}h^{n}$ terms, but
because every higher order $G^{2}$ term has a series expansion in $h$ starting
with a constant, any number of (even) Higgs bosons may be replaced by NG
bosons. This is also true for $\left(H^{\dagger}H\right)^{\frac{2}{3}}$ or
$\sqrt{H^{\dagger}H}$ as well, as most UV models do not give an EFT with a
function like $\sqrt{2H^{\dagger}H}-v$.

Another remark is that going the “other way”, i.e., replacing NG boson pairs
with Higgs boson pairs is also not always guaranteed to have energy-growing
behavior. For example, consider the HEFT interactions involving derivatives
from the singlet scalar model, which is discussed in Sec. 5. We show that the
$V_{L}V_{L}\to hh$ does not violate unitarity in some cases, while the
$V_{L}V_{L}\to V_{L}V_{L}$ case does. This is understood by the different
curvatures $\mathcal{K}_{h,S}$ and $\mathcal{K}_{\pi,S}$,

$\displaystyle\mathcal{K}_{h,S}$
$\displaystyle=\frac{1}{48\pi^{2}}\frac{\lambda^{2}m^{2}}{(2m^{2}+\lambda(1+\frac{\lambda}{96\pi^{2}})(v+h)^{2})^{2}},$
(45a) $\displaystyle\mathcal{K}_{\pi,S}$
$\displaystyle=\frac{1}{96\pi^{2}}\frac{\lambda^{2}}{2m^{2}+\lambda(1+\frac{\lambda}{96\pi^{2}})(v+h)^{2}}.$
(45b)

which are given in Ref. Cohen:2021ucp . If the bare mass $m$ is non-zero, the
unitarity violation appears in the $V_{L}V_{L}\to h^{n}$ processes. However,
as $m$ gets smaller, $\mathcal{K}_{h,S}$ decreases making $V_{L}V_{L}\to
h^{n}$ processes smaller, while the $V_{L}V_{L}\to V_{L}V_{L}(h)$ process
shows a larger unitarity violation. In the limit of $m=0$, the unitarity
violation exists in $V_{L}V_{L}\to V_{L}V_{L}(h)$ but not in $V_{L}V_{L}\to
hh(h)$. This shows the reverse case of the $h^{\ell}$ coupling shift where the
$G^{4}(h)$ is the leading contact interaction which has $2\to 2(3)$ unitarity
violation. And, since
$\left(\frac{\partial_{h}}{K(h)}\right)^{n}\mathcal{K}_{h,S}=0$ for all $n$ in
the massless case, all $V_{L}V_{L}\to h^{n}$ processes will lack the stronger
unitarity violation from derivative terms in this parameter choice. Note that
the weaker unitarity violation from UV potential modifications still exists
for higher multiplicities of $V_{L}V_{L}\to h^{n}$, but they will give much
higher $E_{*}$ values compared to new derivative terms, as demonstrated in
Secs. 4 and 5.

##  Appendix B Average Matrix Element and Unitarity Violation

The following discussion is taken from App. A in Chang:2019vez  and App. B in
Cohen:2021ucp . To get bounds from perturbative unitarity, the $S$-matrix and
the optical theorem are used. Firstly, writing the $S$ matrix, 666This is,
again, following convention of Cohen:2021ucp , so $T$ is negative of that in
ex. Peskin and Schroeder.

$\displaystyle S=1-iT,$ (46)

where $T$ denotes the interactions of the particles. Then consider the quantum
state $\ket{p,\alpha}$, where $p$ is the momentum, and $\alpha$ is other
quantum mechanical information assumed to be discrete (for example, relative
angular momentum or particle type). The normalization condition is

$\displaystyle\braket{q,\beta}{p,\alpha}=(2\pi)^{4}\delta^{4}(p-q)\delta_{\alpha\beta}.$
(47)

and the matrix element is defined as

$\displaystyle\braket{q,\beta}{S}{p,\alpha}=(2\pi)^{4}\delta^{4}(p-q)S_{\alpha\beta},$
(48)
$\displaystyle\braket{q,\beta}{T}{p,\alpha}=(2\pi)^{4}\delta^{4}(p-q)\hat{\mathcal{M}}_{\alpha\beta},$
(49)

to factor out the delta function. We obtain

$\displaystyle
S_{\alpha\beta}=\delta_{\alpha\beta}-i\hat{\mathcal{M}}_{\alpha\beta}.$ (50)

By unitarity arguments, $|S_{\alpha\beta}|^{2}\leq 1$. Setting
$\alpha\neq\beta$ gives

$\displaystyle|\hat{\mathcal{M}}_{\alpha\beta}|^{2}\leq 1.$ (51)

Next, consider $\alpha=\beta$. We apply the optical theorem,

$\displaystyle
1=\sum_{\beta}|S_{\alpha\beta}|^{2}=1+2\Im\hat{\mathcal{M}}_{\alpha\alpha}+\sum_{\beta}|\hat{\mathcal{M}}_{\alpha\beta}|^{2}$
(52)
$\displaystyle\to-2\Im\hat{\mathcal{M}}_{\alpha\alpha}=\sum_{\beta}|\hat{\mathcal{M}}_{\alpha\beta}|^{2}\geq|\hat{\mathcal{M}}_{\alpha\alpha}|^{2}=|\Re\hat{\mathcal{M}}_{\alpha\alpha}|^{2}+|\Im\hat{\mathcal{M}}_{\alpha\alpha}|^{2}$
(53) $\displaystyle\to
1\geq|\Re\hat{\mathcal{M}}_{\alpha\alpha}|^{2}+|\Im\hat{\mathcal{M}}_{\alpha\alpha}+1|^{2}.$
(54)

From Eq. (54), we get that $|\Re\hat{\mathcal{M}}_{\alpha\alpha}|\leq 1$, and
$0\geq\Im\hat{\mathcal{M}}_{\alpha\alpha}\geq-2$. As the analysis is a tree-
level one, we may omit the imaginary part restriction and say that

$\displaystyle|\hat{\mathcal{M}}_{\alpha\beta}|\leq
1,\quad\quad\forall\alpha,\beta\quad\quad\mbox{(tree level),}$ (55)

Now, we have a unitarity condition for any particle state $\alpha$ and
$\beta$. Next, for the normalization factor to the canonically normalized
matrix element, $\mathcal{M}$, we use Eqs. (47, 49), and for simplicity, use a
set of scalar states $1,2,\cdots r$,

$\displaystyle\ket{p,k_{1},k_{2},\cdots,k_{r}}=A_{k_{1},k_{2},\cdots,k_{r}}\int\mathrm{d}^{4}xe^{ip\cdot
x}\prod_{i=1}^{r}\left[\phi_{i}(x)\right]^{k_{i}}\ket{0},$ (56)

where $\phi_{i}(x)$ is the field with the creation operator for the $i$th
particle. By braketing the $T$-matrix with $\bra{q,m_{1},m_{2},\cdots,m_{r}}$
and $\ket{p,n_{1},n_{2},\cdots,n_{r}}$ and applying Eq. (49), we can relate
the canonically normalized matrix element $\mathcal{M}$ to the unitarity
bounded matrix element as

$\displaystyle\hat{\mathcal{M}}_{\alpha\beta}=A^{*}_{n_{1},n_{2},\cdots,n_{r}}A_{m_{1},m_{2},\cdots,m_{r}}\int\mathrm{dLIPS}_{m}\int\mathrm{dLIPS}_{n}\mathcal{M}_{\alpha\beta},$
(57)

where $m_{i}$ and $n_{i}$ denote identical initial and final particles,
respectively. Note that $\hat{\mathcal{M}}$ only picks out $s$-wave states, as
they have the simplest unitarity behavior (an example of more complicated
behavior would be the $t$ and $u$ channels each giving a constant term in the
$ZZ\to hh$ cross section despite the matrix element being constant as well).
All that is left now is to find the normalization factor
$A_{k_{1},k_{2},\cdots,k_{r}}$, and this is done by braketing the states in
(56) and applying Eq. (47). It can be shown that

$\displaystyle\frac{1}{|A_{k_{1},k_{2},\cdots,k_{r}}|^{2}}=\prod_{i=1}^{r}k_{i}!\int\mathrm{dLIPS}_{k},$
(58)

where $\sum_{i=1}^{r}k_{i}=k$, and dLIPSk is defined in Eq. (10). Then
plugging this result into Eq. (57) finally gives Eq. (8), where the indices
are dropped for brevity:

$\hat{\mathcal{M}}=\left(\frac{1}{\prod_{i}m_{i}!\prod_{i}n_{i}!\int\mathrm{dLIPS}_{m}\int\mathrm{dLIPS}_{n}}\right)^{\frac{1}{2}}\int\mathrm{dLIPS}_{m}\int\mathrm{dLIPS}_{n}\mathcal{M}.$
(59)

##  Appendix C Cross section growth for higher multiplicity final states

We use the phase-space averaged matrix element, $\hat{\mathcal{M}}$, to
quantify the perturbative unitarity violation of electroweak boson scattering.
In this appendix, we relate $\hat{\mathcal{M}}$ to the cross section. We show
that a phase-space averaged matrix element growing faster in energy, which is
typically at higher multiplicities, also leads to a fast growth in the cross
section.

For simplicity, we assume the massless limits, an initial state of
$W_{L}W_{L}$, and a constant matrix element $\cal M$. From Eq. (8), the
averaged matrix element of the $2\to n$ process is given by

$\displaystyle\hat{\mathcal{M}}=$
$\displaystyle\left(\frac{1}{8\pi\Pi_{i}n_{i}!\int\mathrm{dLIPS}_{n}}\right)^{\frac{1}{2}}\int\mathrm{dLIPS}_{n}\mathcal{M}\
\to$ (60a)
$\displaystyle|\hat{\mathcal{M}}|^{2}=|{\mathcal{M}}|^{2}\left(\frac{\int\mathrm{dLIPS}_{n}}{8\pi\Pi_{i}n_{i}!}\right),$
(60b)

where we use $\mathcal{M}$ being constant in the second line. Since the cross
section is proportional to

$\sigma\propto\frac{1}{E_{\rm
cm}^{2}}\int\mathrm{dLIPS_{n}}\left|\mathcal{M}\right|^{2}\ ,$ (61)

the relation with $\hat{\mathcal{M}}$ is

$\sigma\propto\frac{8\pi\Pi_{i}n_{i}!}{E_{\rm
cm}^{2}}|\hat{\mathcal{M}}|^{2}.$ (62)

The explicit energy dependence of the cross section is seen using Eq. (10),

$\sigma\propto\frac{\left|\mathcal{M}\right|^{2}}{\left(n-1\right)!\left(n-2\right)!}\left(\frac{E_{\rm
cm}}{4\pi}\right)^{2(n-3)}.$ (63)

This shows that for a constant matrix element, the energy dependence of the
cross section gets higher powers for the larger multiplicities. For the HEFT
interactions, most powers of $E_{\rm cm}$ are compensated by the Higgs VEV,
such as $(E_{\rm cm}/v)$. Therefore, in the case of HEFT, a process with
higher multiplicities tends to have a lower energy scale of unitarity
violation. This statement is supported by Ref. Chang:2019vez , as it
recommends looking at the $V_{L}V_{L}\to V_{L}V_{L}V_{L}V_{L}$ process at the
collider. Hence, if the goal is to have an $E_{*}$ value which can be
accessible to colliders, the $V_{L}V_{L}\to V_{L}V_{L}V_{L}V_{L}$ process
would be a better candidate to search for rather than one of the 3-body final
state processes.

##  Appendix D Evaluating the phase space integral

Here, we compute

$\mathrm{dLIPS}_{n}=\prod_{i=1}^{n}\frac{\mathrm{d}^{3}p_{i}}{(2\pi)^{3}2E_{i}}(2\pi)^{4}\delta^{4}\left(\sum_{i}p_{i}\right).$
(64)

In this paper, we need to evaluate the dLIPSn integral for the $n=2$ and 3
cases. We can perform the integral with $n=2$ analytically, and the result is

$\int\mathrm{dLIPS}_{2}=\frac{F}{8\pi},$ (65)

where
$F=\sqrt{\left(1-\frac{m_{1}^{2}}{s}-\frac{m_{2}^{2}}{s}\right)^{2}-4\frac{m_{1}^{2}m_{2}^{2}}{s^{2}}}$
for the masses of $m_{1,2}$. This simplifies to $F=\sqrt{1-\frac{4m^{2}}{s}}$
for the same masses $m_{1}=m_{2}=m$.

In general, for $2\to 2$ scattering, all the kinematics can be expressed in
terms of $s$, $t$, and $u$, the Mandelstam variables. The $s$ variable is
independent of the integral, keeping the results the same. Regarding $t$ and
$u$ dependence, the contact term matrix element, $\mathcal{M}$, will have
terms proportional to $\cos\theta$, where $\theta$ is the final state
momenta’s angle. Those terms vanish due to the fact that
$\int\mathrm{dLIPS}_{2}\mathcal{M}\propto\int_{-1}^{1}\cos\theta\mathrm{d}\cos\theta$.
This leaves only constant terms, so one can use Eq. (65).

For $n=3$, we use the integration method similar to the one used in Ref.
Girmohanta:2023tdr . We chose to keep the mass despite using the equivalence
theorem, as it did not add much extra complexity to the Monte-Carlo
integration. Three of the integrals, for example, the Higgs boson momentum
integrals in $V_{L}V_{L}h$, can be removed by the delta functions, leading to

$\mathrm{dLIPS}_{3}=\frac{1}{2E_{3}}\frac{\mathrm{d}^{3}p_{1}}{(2\pi)^{3}2E_{1}}\frac{\mathrm{d}^{3}p_{2}}{(2\pi)^{3}2E_{2}}2\pi\delta(E_{\rm
cm}-E_{1}-E_{2}-E_{3}),$ (66)

where $E_{1,2}^{2}=p_{1,2}^{2}+m_{1,2}^{2}$,
$E_{3}=\sqrt{p_{1}^{2}+p_{2}^{2}+2p_{1}p_{2}f+m_{h}^{2}}$, $m_{h}$ is the
Higgs boson mass, and $p_{1}$ and $p_{2}$ are the magnitudes of the three-
momenta of the two $V_{L}$. And, $f$ represents

$f\equiv
f(\theta,\phi)=\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}\sin\Delta\phi=x_{1}x_{2}+\sqrt{1-x_{1}^{2}}\sqrt{1-x_{2}^{2}}\sin\Delta\phi,$
(67)

where $\theta_{1}$ and $\theta_{2}$ are the angles of $\overrightarrow{p_{1}}$
and $\overrightarrow{p_{2}}$ respectively, $x_{i}=\cos\theta_{i}$ and
$\Delta\phi=\phi_{1}-\phi_{2}$ is the difference of the azimuthal angles of
$\overrightarrow{p_{1}}$ and $\overrightarrow{p_{2}}$. The range of $f$ is
$[-1,1]$. A change of variables is made such that $\Delta\phi\to\phi_{2}$ as
well, and then one of the $\phi$ integrals is trivial giving a factor of
$2\pi$. So, the 3-body integral becomes

$\int\mathrm{dLIPS}_{3}=\int\frac{p_{1}^{2}\mathrm{d}p_{1}p_{2}^{2}\mathrm{d}p_{2}\mathrm{d}x_{1}\mathrm{d}x_{2}\mathrm{d}\phi_{2}}{8(2\pi)^{4}E_{1}E_{2}E_{3}}\delta(E_{\rm
cm}-E_{1}-E_{2}-E_{3}),$ (68)

where $E_{1,2}$ is a function of $p_{1,2}$ and $m_{1,2}$, and $E_{3}$ is a
function of those as well as $x_{1}$, $x_{2}$, and $\phi$, as defined above
Eq. (67). Now, the common delta function property is used,

$\delta(g(x))=\sum_{i}\frac{\delta(x-x_{i})}{|g^{\prime}(x_{i})|},$ (69)

where $g(x_{i})=0$. This can be used to remove one of the $p_{i}$’s (for the
case with identical $V_{L}$, this choice is irrelevant; otherwise, assume the
heavier particle, $p_{2}$, is chosen). In this case,

Figure 7: Example of the Monte-Carlo 3-body phase space integration region for
the $W_{L}W_{L}\to W_{L}W_{L}h$ case using $E_{\rm cm}=2(2m_{W}+m_{h})$. Here,
$y_{1}=\frac{E_{1}}{E_{\rm cm}}$, and $f(\theta,\phi)$ is defined in Eq. (67).
For the solution $p_{2}^{+}$, both the orange and blue regions are integrated
over, as $p_{2}^{+}$ is real and non-negative in both regions. For
$p_{2}^{-}$, only the orange region has real and non-negative values. The
$y_{1}E_{\rm cm}<m_{W}$ region is also removed due to the fact that $E_{1}\geq
m_{W}$.

.

$\displaystyle g(p_{2})=E_{\rm cm}-E_{1}-E_{2}-E_{3}$ (70) $\displaystyle
g^{\prime}(p_{2})=\frac{p_{2}}{E_{2}}+\frac{p_{2}+fp_{1}}{E_{3}}=\frac{(E_{\rm
cm}-E_{1})p_{2}+E_{2}fp_{1}}{E_{2}E_{3}},$ (71)

where $E_{\rm cm}=E_{1}+E_{2}+E_{3}$ was used to replace $E_{2}+E_{3}$ in the
last step, and $E_{2}$ and $E_{3}$ are functions of $p_{2}$ as shown earlier.
Using the first condition, one can solve for $p_{2}$. Next, using the
condition that $p_{2}$ is real and non-negative, the boundaries for $p_{1}$
and $f$ are found, and $p_{2}$ is integrated out. It turns out that there are
two viable solutions for $p_{2}$. For example, in the case of $m_{1}=m_{2}=m$,
the solutions are

$p_{2}^{\pm}=\frac{-B\pm\sqrt{B^{2}-4AC}}{2A},$ (72)

and

$\displaystyle A=E_{\rm cm}^{2}+2E_{\rm
cm}E_{1}+E_{1}^{2}(1-f^{2})+m^{2}f^{2}\ ,$ (73) $\displaystyle
B=p_{1}f^{2}\left(E_{\rm cm}^{2}-2E_{\rm cm}E_{1}-m_{h}^{2}+2m^{2}\right),$
(74) $\displaystyle C=-\left(E_{\rm cm}^{2}-2E_{\rm
cm}E_{1}-m_{h}^{2}\right)^{2}+4\left(p_{1}^{2}+m_{h}^{2}\right)m^{2}.$ (75)

Both solutions are real and non-negative in different regions of the phase
space. This can be seen in Fig. 7, where different regions are integrated
based on which $p_{2}$ is used. Both the blue and orange regions are
integrated for $p_{2}^{+}$, and only the orange region is integrated for
$p_{2}^{-}$. The other regions either have a complex $p_{2}$ or have
$p_{2}<0$.

After finding the bounds, a change of variables is done, specifically using
the energies instead of the momenta, as
$E_{1}\mathrm{d}E_{1}=p_{1}\mathrm{d}p_{1}$, and then
$y_{i}=\frac{E_{i}}{E_{\rm cm}}$, and $\tilde{p}_{i}=\frac{p_{i}}{E_{\rm
cm}}$. Also, we check the consistency of the massive phase space as a function
of $\frac{\sum_{i}m_{i}}{E_{\rm cm}}$ with Cohen:2021ucp . Here,
$\sum_{i}m_{i}$ is the sum of all the final state masses.

Finally, it is obtained that

$\mathrm{dLIPS}_{3}=\frac{E_{\rm
cm}^{2}}{128\pi^{4}}\frac{\tilde{p_{1}}\tilde{p_{2}}}{1-y_{1}+\frac{\tilde{p_{1}}}{\tilde{p_{2}}}y_{2}f}\mathrm{d}y_{1}\mathrm{d}x_{1}\mathrm{d}x_{2}\mathrm{d}\phi_{2},$
(76)

where either the $p_{2}^{+}$ or $p_{2}^{-}$ solution is taken based on the
region of the phase space.

After this, a Monte-Carlo integration code was written in Mathematica. The
size of random sampling is $10^{5}$. An example of the phase space integral
region is shown in Fig. 7. We scan multiple $E_{\rm cm}$ values and
interpolate them to obtain the final $\int\mathrm{dLIPS}_{3}$ as a function of
$E_{\rm cm}$.

If the matrix element $\mathcal{M}$ has a dependence on any of the variables
being integrated, such as in all $V_{L}V_{L}\to V_{L}V_{L}h$ cases in Sec. 5,
those parts are included in the numerical integration, i.e., the integral
being evaluated is

$\int\mathcal{M}\mathrm{dLIPS}_{3}=\frac{E_{\rm
cm}^{2}}{128\pi^{4}}\int\frac{\tilde{p_{1}}\tilde{p_{2}}}{1-y_{1}+\frac{\tilde{p_{1}}}{\tilde{p_{2}}}y_{2}f}\mathrm{d}y_{1}\mathrm{d}x_{1}\mathrm{d}x_{2}\mathrm{d}\phi_{2}\mathcal{M}.$
(77)

##  Appendix E Tabulated values of the results

In this section, we include the tabulated version of the results in Figs. (5,
6).

Process | $\frac{y^{2}}{(4\pi^{2})}(H^{\dagger}H)^{2}\ln(H^{\dagger}H)$ | $\kappa_{\frac{2}{3}}(H^{\dagger}H)^{\frac{2}{3}}$ | $\frac{m_{h}^{2}}{2v}\delta\left(\sqrt{2H^{\dagger}H}-v\right)^{3}$ | $\kappa_{\frac{1}{2}}(H^{\dagger}H)^{\frac{1}{2}}$
---|---|---|---|---
$W_{L}W_{L}\to hhh$ | 2.7 | 1.7 | 0.12 | 0.68
$Z_{L}Z_{L}\to Z_{L}Z_{L}h$ | 0.72 | 0.77 | 0.094 | 0.37
$W_{L}Z_{L}\to W_{L}Z_{L}h$ | 1.1 | 1.2 | 0.14 | 0.55
$Z_{L}Z_{L}\to W_{L}W_{L}h$ | 1.6 | 1.6 | 0.20 | 0.79
$W_{L}W_{L}\to Z_{L}Z_{L}h$ | 1.6 | 1.6 | 0.20 | 0.79
$W_{L}W_{L}\to W_{L}W_{L}h$ | 0.55 | 0.58 | 0.071 | 0.28
Table 2: Energy scales of unitarity violation for the bottom-up potential terms that have been added to the SM Lagrangian given in $10^{3}$ TeV. This corresponds to Fig. 5 Process | Fermion $(y=1.4)$ | Fermion $(y=2.6)$ | Scalar singlet | Scalar doublet
---|---|---|---|---
$W_{L}W_{L}\to hh$ | 25 | 6.9 | - | 9.7
$W_{L}W_{L}\to W_{L}W_{L}$ | 6.8 | 3.7 | 18 | 9.0
$W_{L}W_{L}\to Z_{L}Z_{L}$ | 5.7 | 3.1 | 15 | 7.6
$Z_{L}Z_{L}\to W_{L}W_{L}$ | 5.7 | 3.1 | 15 | 7.6
$W_{L}Z_{L}\to W_{L}Z_{L}$ | 6.8 | 3.7 | 19 | 9.0
Table 3: Energy scales of unitarity violation for the chosen HEFTs involving derivatives that modify the SM Lagrangian given in TeV for the $2\to 2$ processes. For the scalar singlet, $\lambda=8.3$, for the scalar doublet, $M=45$ GeV, where $M^{2}=\frac{\kappa^{2}}{\lambda_{\Sigma}}$. The $hh$ final state has no unitarity violation scale from the derivative terms when the bare mass is set to zero. This corresponds to the left plot in Fig. 5 Process | Fermion $(y=1.4)$ | Fermion $(y=2.6)$ | Scalar singlet | Scalar doublet
---|---|---|---|---
$W_{L}W_{L}\to hhh$ | 14 | 7.2 | - | 6.1
$W_{L}W_{L}\to W_{L}W_{L}h$ | 5.2 | 3.8 | 9.9 | 5.3
$W_{L}W_{L}\to Z_{L}Z_{L}h$ | 4.7 | 3.4 | 8.9 | 4.8
$Z_{L}Z_{L}\to W_{L}W_{L}h$ | 4.6 | 3.4 | 8.9 | 4.7
$W_{L}Z_{L}\to W_{L}Z_{L}h$ | 5.2 | 3.8 | 10 | 5.3
Table 4: Energy scales of unitarity violation for the chosen HEFTS involving
derivatives that modify the SM Lagrangian given in TeV for the $2\to 3$
processes. For the scalar singlet, $\lambda=8.3$, for the scalar doublet,
$M=45$ GeV, where $M^{2}=\frac{\kappa^{2}}{\lambda_{\Sigma}}$. Similarly to
the $2\to 2$ case, the $hhh$ state has no unitarity violation scale from the
derivative terms when the bare mass is set to zero. This corresponds to the
right plot in Fig. 5

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