spacing=nonfrench M[1]¿m#1 L[1]¿m#1 N@m0pt@

# Transversal Modes and Higgs Bosons
in Electroweak Vector-Boson Scattering
at the LHC

Simon Brassa, Christian Flepera, Wolfgang Kiliana,d, Jürgen Reuterb, Marco
Sekullac simon.brass@uni-siegen.dechristian.fleper@uni-
<EMAIL_ADDRESS>siegen.dejuergen.reuter<EMAIL_ADDRESS>
###### ABSTRACT

Processes where $W$ and $Z$ bosons scatter into pairs of electroweak bosons
$W$, $Z$, and Higgs, are sensitive probes of new physics in the electroweak
sector. We study simplified models that describe typical scenarios of new
physics and parameterize the range of possible LHC results between the
Standard-Model prediction and unitarity limits. Extending the study beyond
purely longitudinal scattering, we investigate the role of transversally
polarized gauge bosons. Unitarity becomes an essential factor, and limits on
parameters matched to the naive perturbative low-energy effective theory turn
out to be necessarily model-dependent. We discuss the implications of our
approach for the interpretation of LHC data on vector-boson scattering and
Higgs-pair production.

###### Contents

  1. 1 Introduction
  2. 2 Electroweak Boson Currents and Local Operators
  3. 3 Properties of the Scattering Matrix
  4. 4 (Strongly coupled) Continuum Model
  5. 5 Simplified Resonance Models
  6. 6 Implications for LHC Analyses and Conclusions
  7. A Isospin-spin amplitudes
  8. B Additional Numerical Results

##  1 Introduction

After the discovery of the Higgs boson [1, 2], particle physics faces the
question whether the new scalar sector is minimal or non-minimal, whether it
is weakly or strongly interacting, and whether it validates or invalidates the
accepted paradigm of quantum field theory as a universal description of
particle interactions. Experimental data on electroweak boson interactions
(Higgs, $W$, $Z$, and photon) will deepen our understanding in that area.
Processes of the type $VV\to VV$, where $V=W,Z,H$ (vector-boson scattering,
VBS, and associated or Higgs pair production in vector-boson fusion, VBF), are
a most sensitive probe of electroweak physics and the Higgs sector. They will
be extensively studied in the present and future runs of the LHC.

There is obvious interest in scenarios where new degrees of freedom beyond the
Standard Model (SM) couple primarily to the Higgs-Goldstone boson field. The
presence of new physics coupled to the Higgs sector might solve some of the
long-standing puzzles of particle physics. Such new modes need not have a
significant effect on existing precision data. They might be strongly-
interacting as in composite Higgs models, or weakly interacting as in models
with extra scalars that are decoupled from SM fermions. They should manifest
themselves primarily in interactions of massive electroweak bosons, namely
$W^{\pm}$, $Z$, and the Higgs itself.

The ATLAS and CMS experiments at the LHC have measured VBS processes as a
signal, embedded in partonic processes of the type $qq\to VVqq$, where $q$ is
any light quark. Numerical results have been presented in the form of limits
on parameters within the SM effective theory (SMEFT) [3, 4, 5, 6]. The usual
application of the SMEFT truncates the power expansion of the Lagrangian at
the level of dimension-six operators. A useful parameterization of VBS
processes requires dimension-eight effective operators, the second order of
the low-energy expansion beyond the SM.

In recent work [7, 8, 9], we have studied deviations from the SM interactions
that are confined to the longitudinal scattering modes of $W$ and $Z$. In the
low-energy limit, the contributing set of new interactions is small, and if
custodial (weak isospin) symmetry is imposed, there are just two free
parameters in the matching SMEFT expansion. On the other hand, recent LHC
analyses quote results for a larger set of operator coefficients which include
interactions between longitudinal and transversal modes of $W$ and $Z$ bosons.
In the current paper, we study deviations from the SM in VBS processes that
involve transversally polarized $W$ and $Z$ bosons, and also consider Higgs
bosons in the final state.

Numerical results of non-SM interactions of longitudinal scattering have
clearly shown that for the level of deviations that can be detected by the LHC
experiments, the unitarity limits are always violated in the high-energy
range, if a naive SMEFT calculation is attempted. A model-independent
parameterization beyond the SM that covers the accessible parameter range
becomes impossible. Nevertheless, reasonable assumptions on new physics lead
to unitarity constraints that limit the level of possible excess above the SM
prediction. With this knowledge, it is possible to devise simplified models
that both satisfy unitarity over the whole energy range and smoothly match
onto the SMEFT parameterization at low energy. For the purpose of an exemplary
study, we have compared a class of “continuum” models which merely extrapolate
the SMEFT expansion into asymptotically strong interactions with models that
describe single resonances with specific quantum-number assignments.

In the present paper, we extend this program to also describe transversal
modes, and to include the Higgs boson as a possible final state on the same
footing as the $W$ and $Z$ bosons. Regarding the SM processes as reference,
there are detailed calculations [10, 11, 12, 13, 14, 15] beyond leading-order
in the SM perturbation theory. Recently, there was a concise comparison of
several different codes for the precision simulation at LO and NLO for like-
sign vector boson scattering at 13 TeV [16]. For the simplified models
considered in this paper, we confine ourselves to leading-order calculations
but we remark that adding in perturbative QCD and electroweak corrections is
possible by the same methods as for the pure SM, and should eventually be done
in order to distinguish genuine deviations from uncertainties of the
approximation, in actual data analysis.

The paper is organized as follows. In Sec. 2 we discuss the structure of new
interactions in the electroweak and Higgs sector, and state the underlying
assumptions. This defines the SMEFT ansatz, and it allows us to list the
operators that describe the low-energy limit. The symmetries of those
interactions determine the eigenmodes of quasi-elastic $2\to 2$ scattering,
which allows us to diagonalize the amplitudes for all vector-boson modes, Sec.
3. We construct unitary models exhibiting a strongly interacting continuum in
Sec. 4. These models would yield the maximally allowed number of events
consistent with quantum field theory in the VBS channel, matched to the low-
energy SMEFT with specific values for the operator coefficients. In Sec. 5 we
discuss simplified models which contain a resonance and likewise parameterize
VBS amplitudes at all energies. We present numerical results and plots for
selected parameter sets and final states. In Sec. 6, we discuss the relevance
of our study in view of future analyses at the LHC and beyond.

##  2 Electroweak Boson Currents and Local Operators

Expectations for new physics beyond the SM are constrained by available data.
They may also be guided by imposing principles such as simplicity and absence
of accidental cancellations. For the current work, we base the description of
new effects beyond the SM on the following assumptions: (i) light fermions do
not participate directly in new dynamics, (ii) the observed pattern of SM
gauge invariance retains its relevance beyond the TeV range, and (iii) new
degrees of freedom beyond the SM do not carry open color. These assumptions
are not mandatory but backed by the available precision data regarding flavor
physics, QCD, new-physics searches at the LHC, and precision electroweak
observables.

If these assumptions are accepted as guiding principles for a phenomenological
description, a parameterization of dominant new effects can qualify light-
fermion currents as classical spectator fields, and focus on the bosonic SM
multiplets acting as currents that probe the new sector. The currents can be
introduced as local operators that couple to an unknown new-physics spectrum
in a manifestly $SU(2)_{L}\times U(1)_{Y}$ invariant way. New dynamics may
involve weakly coupled (comparatively) light degrees of freedom, such as extra
Higgs singlets or doublets, it may probe a strongly coupled sector which is
resolved at high energy, or it may give rise to heavy resonances, to name a
few possibilities. In any case, new physics that is coupled to SM bosons, will
manifest itself in anomalous scattering matrix elements of those bosons, and
should become accessible in high-energy VBS. As a common feature of this class
of models, we expect the scattering matrix to be self-contained and complete
in terms of SM bosons and eventual new-physics states, to a good
approximation.111Heavy fermions ($t$, $b$, $\tau$) should be taken into
account in this context, but we do not study the corresponding final states in
this paper.

For a quantitative representation, we adopt the assumption of gauge invariance
and describe new physics as coupled to gauge-covariant monomials of SM fields.
For the building blocks, we introduce the Higgs multiplet in form of a
$2\times 2$ matrix,

$\displaystyle\mathbf{H}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}v+h-\mathrm{i}w^{3}&-\mathrm{i}\sqrt{2}w^{+}\\\
-\mathrm{i}\sqrt{2}w^{-}&v+h+\mathrm{i}w^{3}\\\ \end{pmatrix}.$ (1)

The components $h,w^{\pm},w^{3}$ are the physical Higgs and unphysical
Goldstone scalars, respectively, and $v$ denotes the numerical Higgs vev,
$v=246\;\text{GeV}$. The matrix notation allows us to manifestly represent the
larger global symmetry on the Higgs field, $O(4)\sim SU(2)_{L}\times
SU(2)_{R}$ which after electroweak symmetry breaking (EWSB) becomes the
approximate custodial $SU(2)_{c}$ symmetry. $SU(2)_{L}\times
SU(2)_{R}$-symmetric monomials are invariant under bi-unitary transformations
of the form $\mathbf{H}\to U_{L}\mathbf{H}U_{R}^{\dagger}$.

The covariant derivative of the Higgs matrix is defined as

$\displaystyle\mathbf{D}_{\mu}\mathbf{H}=\partial_{\mu}\mathbf{H}-\mathrm{i}g\mathbf{W}_{\mu}\mathbf{H}+\mathrm{i}g^{\prime}\mathbf{H}\mathbf{B}_{\mu}\,,$
(2)

where

$\displaystyle\mathbf{W}_{\mu}\equiv\mathbf{W}_{\mu}^{a}\frac{\tau^{a}}{2},\qquad\mathbf{B}_{\mu}\equiv\mathbf{B}_{\mu}^{a}\frac{\tau^{3}}{2}.$
(3)

The transformation of $\mathbf{W}_{\mu}$ is $\mathbf{W}_{\mu}\to
U_{L}^{\dagger}\mathbf{W}_{\mu}U_{L}$, while $\mathbf{B}_{\mu}$ transforms
covariantly only under a $U(1)_{R}$ subgroup of $SU(2)_{R}$. The matrix-valued
field strengths are given by

$\displaystyle\mathbf{W}_{\mu\nu}=\partial_{\mu}\mathbf{W}_{\nu}-\partial_{\nu}\mathbf{W}_{\nu}-ig\left[\mathbf{W}_{\mu},\mathbf{W}_{\nu}\right],\qquad\mathbf{B}_{\mu\nu}=\partial_{\mu}\mathbf{B}_{\nu}-\partial_{\nu}\mathbf{B}_{\nu}\,.$
(4)

From these fields, we can build local composite operators which act as
currents that probe the new, possibly non-local dynamics. For instance, the
simplest Higgs-field currents are

$\displaystyle J_{H}^{(2)}$
$\displaystyle=\operatorname{tr}\left[\mathbf{H}^{\dagger}\mathbf{H}\right],$
$\displaystyle J_{H}^{(4)}$
$\displaystyle=\operatorname{tr}\left[(\mathbf{D}_{\mu}\mathbf{H})^{\dagger}(\mathbf{D}^{\mu}\mathbf{H})\right]$
$\displaystyle J_{H\mu\nu}^{(4)}$
$\displaystyle=\operatorname{tr}\left[(\mathbf{D}_{\mu}\mathbf{H})^{\dagger}(\mathbf{D}_{\nu}\mathbf{H})\right],$
(5)

while gauge-field tensors can be combined as

$\displaystyle J_{W}^{(4)}$
$\displaystyle=g^{2}\operatorname{tr}\left[\mathbf{W}_{\mu\nu}\mathbf{W}^{\mu\nu}\right],$
$\displaystyle J_{B}^{(4)}$
$\displaystyle=g^{\prime}{}^{2}\operatorname{tr}\left[\mathbf{B}_{\mu\nu}\mathbf{B}^{\mu\nu}\right],$
(6) $\displaystyle J_{W\mu\nu}^{(4)}$
$\displaystyle=g^{2}\operatorname{tr}\left[\mathbf{W}_{\mu\rho}\mathbf{W}^{\rho}_{\;\nu}\right],$
$\displaystyle J_{B\mu\nu}^{(4)}$
$\displaystyle=g^{\prime}{}^{2}\operatorname{tr}\left[\mathbf{B}_{\mu\rho}\mathbf{B}^{\rho}_{\;\nu}\right].$
(7)

These terms are electroweak singlets. Non-singlet currents can likewise be
constructed.

We expect only weak constraints from existing data, so new dynamics, whether
parameterized by form factors, spectral functions, or inelastic scattering
into new particles, is rather arbitrary. For the purpose of this work, we
focus on two extreme scenarios: (i) a spectrum that interpolates the low-
energy description with unitarity saturation in the high-energy range, and
(ii) a spectrum that consists of separate narrow to medium-width resonances,
which we may reduce to the lowest-lying state for simplicity. For reference,
we also include (iii) the unmodified SM where any new spectral functions are
zero and all amplitudes remain weakly interacting. In terms of quasi-elastic
$2\to 2$ scattering, scenarios (i) and (iii) correspond to maximal and minimal
event yields in the asymptotic region, while (ii) exhibits unitarity
saturation at finite energy. Another extreme scenario, saturation by inelastic
scattering into new final states, asymptotically implies quasi-elastic event
rates between (i) and (iii) and should furthermore be accessible via direct
observation of new particles.

As a first step, we may confine the analysis to pure Higgs- and Goldstone-
boson scattering. This was done in our earlier paper [7]. Such a restriction
implies further assumptions on the underlying complete theory. In this work,
we remove this restriction. We investigate the bosonic $2\to 2$ scattering
matrix with Higgs, longitudinal, and transversal vector bosons included.

Unless there are undetected light particles hiding in this scattering matrix,
it allows for a local operator-product expansion. Contracting the singlet
currents listed above and ignoring terms which merely renormalize SM
parameters, the leading terms are dimension-six operators:
$(J_{H}^{(2)})^{3}$, $J_{H}^{(2)}J_{H}^{(4)}$, and $J_{H}^{(2)}J_{W}^{(4)}$.
Only the latter term is easily accessible at the LHC,222There are two
directions in the dimension-six operator space which in our context correspond
to $(J_{H}^{(2)})^{3}$ and $J_{H}^{(2)}J_{H}^{(4)}$. An unambiguous
determination of the coefficients requires measurements of Higgs-pair
production and the Higgs total width, respectively. so a phenomenological
parameterization should consider the next order of the expansion. These are
dimension-eight local interactions.

Including all singlet and non-singlet operator products, and omitting CP-odd
interactions, we can identify three distinct categories of dimension-eight
bosonic operators in the low-energy expansion that we list below.

There are two terms which couple only Higgs-field currents,

$\displaystyle\mathcal{L}_{S,0}$
$\displaystyle=F_{S,0}\text{tr}\left[(\textbf{D}_{\mu}\textbf{H})^{\dagger}(\textbf{D}_{\nu}\textbf{H})\right]\text{tr}\left[(\textbf{D}^{\mu}\textbf{H})^{\dagger}(\textbf{D}^{\nu}\textbf{H})\right],$
(8a) $\displaystyle\mathcal{L}_{S,1}$
$\displaystyle=F_{S,1}\text{tr}\left[(\textbf{D}_{\mu}\textbf{H})^{\dagger}(\textbf{D}^{\mu}\textbf{H})\right]\text{tr}\left[(\textbf{D}_{\nu}\textbf{H})^{\dagger}(\textbf{D}^{\nu}\textbf{H})\right]\;;$
(8b)

seven terms which couple Higgs- and gauge field currents,

$\displaystyle\mathcal{L}_{M,0}$
$\displaystyle=-g^{2}F_{M_{0}}\text{tr}\left[(\textbf{D}_{\mu}\textbf{H})^{\dagger}(\textbf{D}^{\mu}\textbf{H})\right]\text{tr}\left[\textbf{W}_{\nu\rho}\textbf{W}^{\nu\rho}\right],$
(9a) $\displaystyle\mathcal{L}_{M,1}$
$\displaystyle=-g^{2}F_{M_{1}}\text{tr}\left[(\textbf{D}_{\mu}\textbf{H})^{\dagger}(\textbf{D}^{\rho}\textbf{H})\right]\text{tr}\left[\textbf{W}_{\nu\rho}\textbf{W}^{\nu\mu}\right],$
(9b) $\displaystyle\mathcal{L}_{M,2}$ $\displaystyle=-g^{\prime
2}F_{M_{2}}\text{tr}\left[(\textbf{D}_{\mu}\textbf{H})^{\dagger}(\textbf{D}^{\mu}\textbf{H})\right]\text{tr}\left[\textbf{B}_{\nu\rho}\textbf{B}^{\nu\rho}\right],$
(9c) $\displaystyle\mathcal{L}_{M,3}$ $\displaystyle=-g^{\prime
2}F_{M_{3}}\text{tr}\left[(\textbf{D}_{\mu}\textbf{H})^{\dagger}(\textbf{D}^{\rho}\textbf{H})\right]\text{tr}\left[\textbf{B}_{\nu\rho}\textbf{B}^{\nu\mu}\right],$
(9d) $\displaystyle\mathcal{L}_{M,4}$
$\displaystyle=-gg^{\prime}F_{M_{4}}\text{tr}\left[(\textbf{D}_{\mu}\textbf{H})^{\dagger}\textbf{W}_{\nu\rho}(\textbf{D}^{\mu}\textbf{H})\textbf{B}^{\nu\rho}\right],$
(9e) $\displaystyle\mathcal{L}_{M,5}$
$\displaystyle=-gg^{\prime}F_{M_{5}}\text{tr}\left[(\textbf{D}_{\mu}\textbf{H})^{\dagger}\textbf{W}_{\nu\rho}(\textbf{D}^{\rho}\textbf{H})\textbf{B}^{\nu\mu}\right],$
(9f) $\displaystyle\mathcal{L}_{M,7}$
$\displaystyle=-g^{2}F_{M_{7}}\text{tr}\left[(\textbf{D}_{\mu}\textbf{H})^{\dagger}\textbf{W}_{\nu\rho}\textbf{W}^{\nu\mu}(\textbf{D}^{\rho}\textbf{H})\right]\;;$
(9g)

and eight terms which couple gauge-field currents to themselves:

$\displaystyle\mathcal{L}_{T_{0}}$
$\displaystyle=g^{4}F_{T_{0}}\text{tr}\left[\textbf{W}_{\mu\nu}\textbf{W}^{\mu\nu}\right]\text{tr}\left[\textbf{W}_{\alpha\beta}\textbf{W}^{\alpha\beta}\right],$
(10a) $\displaystyle\mathcal{L}_{T_{1}}$
$\displaystyle=g^{4}F_{T_{1}}\text{tr}\left[\textbf{W}_{\alpha\nu}\textbf{W}^{\mu\beta}\right]\text{tr}\left[\textbf{W}_{\mu\beta}\textbf{W}^{\alpha\nu}\right],$
(10b) $\displaystyle\mathcal{L}_{T_{2}}$
$\displaystyle=g^{4}F_{T_{2}}\text{tr}\left[\textbf{W}_{\alpha\mu}\textbf{W}^{\mu\beta}\right]\text{tr}\left[\textbf{W}_{\beta\nu}\textbf{W}^{\nu\alpha}\right],$
(10c) $\displaystyle\mathcal{L}_{T_{5}}$ $\displaystyle=g^{2}g^{\prime
2}F_{T_{5}}\text{tr}\left[\textbf{W}_{\mu\nu}\textbf{W}^{\mu\nu}\right]\text{tr}\left[\textbf{B}_{\alpha\beta}\textbf{B}^{\alpha\beta}\right],$
(10d) $\displaystyle\mathcal{L}_{T_{6}}$ $\displaystyle=g^{2}g^{\prime
2}F_{T_{6}}\text{tr}\left[\textbf{W}_{\alpha\nu}\textbf{W}^{\mu\beta}\right]\text{tr}\left[\textbf{B}_{\mu\beta}\textbf{B}^{\alpha\nu}\right],$
(10e) $\displaystyle\mathcal{L}_{T_{7}}$ $\displaystyle=g^{2}g^{\prime
2}F_{T_{7}}\text{tr}\left[\textbf{W}_{\alpha\mu}\textbf{W}^{\mu\beta}\right]\text{tr}\left[\textbf{B}_{\beta\nu}\textbf{B}^{\nu\alpha}\right],$
(10f) $\displaystyle\mathcal{L}_{T_{8}}$ $\displaystyle=g^{\prime
4}F_{T_{8}}\text{tr}\left[\textbf{B}_{\mu\nu}\textbf{B}^{\mu\nu}\right]\text{tr}\left[\textbf{B}_{\alpha\beta}\textbf{B}^{\alpha\beta}\right],$
(10g) $\displaystyle\mathcal{L}_{T_{9}}$ $\displaystyle=g^{\prime
4}F_{T_{9}}\text{tr}\left[\textbf{B}_{\alpha\mu}\textbf{B}^{\mu\beta}\right]\text{tr}\left[\textbf{B}_{\beta\nu}\textbf{B}^{\nu\alpha}\right].$
(10h)

Note that the enumeration of operators is not consecutive. We have adopted the
naming convention from the literature [17, 18, 19] but eliminated redundant
interactions to arrive at a linearly independent set.

We emphasize that this list only describes the model-independent low-energy
limit of the true amplitude. The actual measurement of VBS processes is not
restricted to the low-energy range and thus cannot be accurately accounted for
by the low-energy limit only. The true quasi-elastic scattering amplitudes
will resolve the local operators into non-local interactions and thus keep the
result in accordance with the applicable unitarity relations at all energies.
This is the rationale for introducing simplified models such as (i) and (ii)
above.

For the purpose of this study, we adopt a simplification that applies to all
considered models: we impose global custodial symmetry on the beyond the SM
(BSM) interactions and thus omit all terms that involve the hypercharge boson.
In the local operator basis, we are left with two parameters $F_{S_{0/1}}$,
three parameters $F_{M_{0/1/7}}$, and three parameters $F_{T_{0/1/2}}$. This
choice implies that $W$ and $Z$ amplitudes are mutually related, and that
photon interactions do not carry independent information.

##  3 Properties of the Scattering Matrix

We apply the phenomenological description of the preceding section to the set
of VBS scattering amplitudes, which we then embed in complete LHC processes.
The basic processes are all of the $2\to 2$ quasi-elastic scattering type. In
this situation, standard scattering theory applies. We may evaluate partial-
wave amplitudes and thus convert the scattering amplitudes to a finite-
dimensional matrix. This allows us to diagonalize the scattering matrix and
find a unitary projection of each eigenamplitude individually, if the
calculated model amplitude does not respect partial-wave unitarity.

This simplification is based on approximations. We ignore external and
internal photons. This implies the custodial-$SU(2)$ limit, as already
discussed in the previous section. It also implies that we ignore the Coulomb
pole in charged-$W$ scattering. The omission is justified since the forward
region is cut out in an experimental analysis, while the Coulomb singularity
is not reached for the complete process, due to the spacelike nature of the
incoming virtual vector bosons. We also treat the external particles as on-
shell, while in the real process at the LHC, the initial vector bosons are
actually space-like. In effect, these omissions amount to subleading
corrections of relative order $m_{W}^{2}/\hat{s}$ and ${q_{i}^{2}}/{\hat{s}}$
for the $2\to 2$ quasi-elastic scattering processes, where $q_{i}$ is the
space-like momentum of an incoming vector boson. We note that for VBS
kinematics, values $|q_{i}^{2}|\sim m_{W}^{2}$ dominate the cross section, but
there is a phase-space region where terms proportional to ${q^{2}}/{\hat{s}}$
become leading. In the current paper, we focus on observables inclusive in
$q^{2}$ where these terms are mostly subleading. We refer to Ref. [20] for a
more exhaustive discussion.

In fact, the symmetry structure of our simplified models allows for a choice
of basis that renders the scattering matrix diagonal at all energies, up to
subleading corrections. Asymptotically, the longitudinal vector boson modes
combine with the Higgs mode, while the transverse modes decouple. The external
states combine to multiplets of the custodial $SU(2)$ symmetry. This property
is well known for the SM. If we assume custodial symmetry also for the new
interactions, we can use it for expressing all quasi-elastic scattering
amplitudes of Higgs and longitudinal vector boson modes in terms of a single
scalar master amplitude, which can be used to find partial-wave
eigenamplitudes and their unitary projection [21, 22, 7]. Here, we apply the
same principle to transverse and mixed scattering amplitudes.

The key observation is that the contact interactions of the SMEFT (8, 9, 10),
although they do not provide a satisfactory phenomenological description,
already encode the most general dependence of the scattering matrix on
external quantum numbers, if we restrict the analysis to quasi-elastic $2\to
2$ scattering. To describe an arbitrary new-physics spectrum, not just the
low-energy limit, we merely have to promote the coefficients $F_{i}$ to scalar
form-factors which can depend on $s,t,u$. Turning this around, we can formally
diagonalize the scattering matrix in terms of those coefficients. Unlike the
scattering matrix for longitudinal modes only, this procedure involves the
helicities of the external vector bosons $\lambda_{i}$. Since the procedure is
required only for the high-energy range, we neglect the masses of $W$, $Z$,
and $H$ where applicable.

For the calculation below, we can thus treat the non-SM part of the amplitudes
as if they were given by the local dimension-eight operator approximation,
keeping in mind that the method works as well for non-constant coefficients.
The unitary projection that we obtain assumes the same form, with specific
functions for the coefficients, and applying the same projection a second time
will not change the asymptotic form of the result anymore.

For the transverse interactions with structures $\mathcal{L}_{T_{0/1/2}}$ and
for the mixed interactions with the operators $\mathcal{L}_{M_{0/1/7}}$ we
define the master amplitude

$\displaystyle A(s,t,u;\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})=$
$\displaystyle\mathcal{A}(W^{+}_{\lambda_{1}}W^{-}_{\lambda_{2}}\to
Z_{\lambda_{3}}Z_{\lambda_{4}})$ $\displaystyle=$
$\displaystyle-2g^{4}(F_{T_{0}}+\frac{1}{4}F_{T_{2}})\delta_{\lambda_{1},\lambda_{2}}\delta_{\lambda_{3},\lambda_{4}}s^{2}$
$\displaystyle-g^{4}(F_{T_{1}}+\frac{1}{2}F_{T_{2}})\left(\delta_{\lambda_{1},-\lambda_{3}}\delta_{\lambda_{2},-\lambda_{4}}t^{2}+\delta_{\lambda_{1},-\lambda_{4}}\delta_{\lambda_{2},-\lambda_{3}}u^{2}\right)$
$\displaystyle+\frac{1}{2}g^{4}F_{T_{2}}\delta_{\lambda_{1},\lambda_{2}}\delta_{\lambda_{3},\lambda_{4}}\delta_{\lambda_{1},-\lambda_{3}}(t^{2}+u^{2})$
$\displaystyle+\frac{1}{16}g^{2}(8F_{M_{0}}-2F_{M_{1}}+F_{M_{7}})\left(\delta_{\lambda_{1},\lambda_{2}}\delta_{\lambda_{3},0}\delta_{\lambda_{4},0}-\delta_{\lambda_{3},\lambda_{4}}16\delta_{\lambda_{1},0}\delta_{\lambda_{2},0}\right)s^{2}$
$\displaystyle+\frac{1}{16}g^{2}(2F_{M_{1}}+F_{M_{7}})\left(\delta_{\lambda_{1},-\lambda_{2}}\delta_{\lambda_{3},0}\delta_{\lambda_{4},0}-\delta_{\lambda_{3},-\lambda_{4}}\delta_{\lambda_{1},0}\delta_{\lambda_{2},0}\right)\left(s^{2}-t^{2}-u^{2}\right)$
$\displaystyle+\frac{1}{16}F_{M_{7}}\biggl{[}\left(\delta_{\lambda_{1},-\lambda_{3}}\delta_{\lambda_{2},0}\delta_{\lambda_{4},0}-\delta_{\lambda_{2},-\lambda_{4}}\delta_{\lambda_{1},0}\delta_{\lambda_{3},0}\right)\left(s^{2}-u^{2}\right)$
$\displaystyle\qquad\qquad+\left(\delta_{\lambda_{2},-\lambda_{3}}\delta_{\lambda_{1},0}\delta_{\lambda_{4},0}-\delta_{\lambda_{1},-\lambda_{4}}\delta_{\lambda_{2},0}\delta_{\lambda_{3},0}\right)\left(s^{2}-t^{2}\right)\biggr{]}.$
(11)

The decomposition of the scattering amplitudes into isospin eigenamplitudes is
identical for mixed and transverse operators and given by

$\displaystyle\mathcal{A}(W^{+}_{\lambda_{1}}W^{+}_{\lambda_{2}}\to
W^{+}_{\lambda_{3}}W^{+}_{\lambda_{4}})$
$\displaystyle=\phantom{\frac{1}{3}}\mathcal{A}_{2}(s,t,u;\boldsymbol{\lambda})$
(12a) $\displaystyle\mathcal{A}(W^{+}_{\lambda_{1}}W^{-}_{\lambda_{2}}\to
W^{+}_{\lambda_{3}}W^{-}_{\lambda_{4}})$
$\displaystyle=\frac{1}{3}\mathcal{A}_{0}(s,t,u;\boldsymbol{\lambda})+\frac{1}{2}\mathcal{A}_{1}(s,t,u;\boldsymbol{\lambda})+\frac{1}{6}\mathcal{A}_{2}(s,t,u;\boldsymbol{\lambda})$
(12b) $\displaystyle\mathcal{A}(W^{+}_{\lambda_{1}}W^{-}_{\lambda_{2}}\to
Z_{\lambda_{3}}Z_{\lambda_{4}})$
$\displaystyle=\frac{1}{3}\mathcal{A}_{0}(s,t,u;\boldsymbol{\lambda})-\frac{1}{3}\mathcal{A}_{2}(s,t,u;\boldsymbol{\lambda})$
(12c) $\displaystyle\mathcal{A}(W^{+}_{\lambda_{1}}Z_{\lambda_{2}}\to
W^{+}_{\lambda_{3}}Z_{\lambda_{4}})$
$\displaystyle=\frac{1}{2}\mathcal{A}_{1}(s,t,u;\boldsymbol{\lambda})+\frac{1}{2}\mathcal{A}_{2}(s,t,u;\boldsymbol{\lambda})$
(12d) $\displaystyle\mathcal{A}(Z_{\lambda_{1}}Z_{\lambda_{2}}\to
Z_{\lambda_{3}}Z_{\lambda_{4}})$
$\displaystyle=\frac{1}{3}\mathcal{A}_{0}(s,t,u;\boldsymbol{\lambda})+\frac{2}{3}\mathcal{A}_{2}(s,t,u;\boldsymbol{\lambda})\;.$
(12e)

Here, $\boldsymbol{\lambda}=(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})$
is a multi-index for the four different helicities of the weak vector bosons.
Using this, the isospin eigenamplitudes are given by

$\displaystyle\mathcal{A}_{0}(s,t,u;\boldsymbol{\lambda})=$
$\displaystyle\quad\;3A(s,t,u;\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})$
$\displaystyle+A(t,s,u;-\lambda_{4},\lambda_{2},\lambda_{3},-\lambda_{1})$
$\displaystyle+A(u,t,s;\lambda_{1},-\lambda_{4},\lambda_{3},-\lambda_{2})$
(13a) $\displaystyle\mathcal{A}_{1}(s,t,u;\boldsymbol{\lambda})=$
$\displaystyle\quad\;A(t,s,u;-\lambda_{4},\lambda_{2},\lambda_{3},-\lambda_{1})$
$\displaystyle-A(u,t,s;\lambda_{1},-\lambda_{4},\lambda_{3},-\lambda_{2})$
(13b) $\displaystyle\mathcal{A}_{2}(s,t,u;\boldsymbol{\lambda})=$
$\displaystyle\quad\;A(t,s,u;-\lambda_{4},\lambda_{2},\lambda_{3},-\lambda_{1})$
$\displaystyle+A(u,t,s;\lambda_{1},-\lambda_{4},\lambda_{3},-\lambda_{2}).$
(13c)

The next step is the decomposition into isospin-spin eigenamplitudes which is
done by the expansion of the isospin eigenamplitudes (12) into the Wigner
D-functions [23] $d^{J}_{\lambda,\lambda^{\prime}}(\theta)$ with
$\lambda=\lambda_{1}-\lambda_{2}$ and
$\lambda^{\prime}=\lambda_{3}-\lambda_{4}$.

$\displaystyle\mathcal{A}_{IJ}(s;\boldsymbol{\lambda})=\int_{-s}^{0}\frac{dt}{s}A_{I}(s,t,u;\boldsymbol{\lambda})\cdot
d^{J}_{\lambda,\lambda^{\prime}}\left[\arccos\left(1+2\frac{t}{s}\right)\right]$
(14)

Tables 2 (transverse operators) and 3 (mixed operators) list the complete set
of master amplitudes with their dependence on the operator coefficients, i.e.,
the asymptotically leading behavior. These helicity-dependent eigenamplitudes
can in principle be used for determining unitary projections as form factors
that multiply modified Feynman rules for the boson fields.

For the purpose of constructing a minimal unitary projection, it is sufficient
to determine a set of master amplitudes which capture the leading term
proportional to $s^{2}$ for each spin-isospin channel, uniformly for all
individual helicity combinations. The implied over-compensation of some
helicity channels that are subleading at high energy, is within the scheme
dependence that is inherent in the unitary projection. We find the following
simplified, helicity-independent expressions:

$\displaystyle\mathcal{A}_{00}(s)=$
$\displaystyle-\frac{3}{2}g^{4}\left[4F_{T_{0}}-2F_{T_{1}}+F_{T_{2}}\right]s^{2}$
$\displaystyle+\frac{3}{16}g^{2}\left[8F_{M_{0}}+2F_{M_{1}}+F_{M_{7}}\right]s^{2}$
(15a) $\displaystyle\mathcal{A}_{01}(s)=$
$\displaystyle-\frac{1}{32}g^{2}\left[4F_{M_{0}}+F_{M_{1}}-3F_{M_{7}}\right]s^{2}$
(15b) $\displaystyle\mathcal{A}_{02}(s)=$
$\displaystyle-\frac{1}{10}g^{4}\left[4F_{T_{0}}-2F_{T_{1}}+F_{T_{2}}\right]s^{2}$
$\displaystyle+\frac{1}{160}g^{2}\left[4F_{M_{0}}+F_{M_{1}}+F_{M_{7}}\right]s^{2}$
(15c) $\displaystyle\mathcal{A}_{10}(s)=$ $\displaystyle 0$ (15d)
$\displaystyle\mathcal{A}_{11}(s)=$
$\displaystyle-\frac{1}{6}g^{4}F_{T_{2}}s^{2}$
$\displaystyle-\frac{1}{32}g^{2}\left[4F_{M_{0}}+F_{M_{1}}-3F_{M_{7}}\right]s^{2}$
(15e) $\displaystyle\mathcal{A}_{12}(s)=$
$\displaystyle\frac{1}{5}g^{4}\left[-2F_{T_{0}}+F_{T_{1}}\right]s^{2}$
$\displaystyle+\frac{1}{160}g^{2}\left[4F_{M_{0}}+F_{M_{1}}+F_{M_{7}}\right]s^{2}$
(15f) $\displaystyle\mathcal{A}_{20}(s)=$ $\displaystyle 0$ (15g)
$\displaystyle\mathcal{A}_{21}(s)=$
$\displaystyle-\frac{1}{32}g^{2}\left[4F_{M_{0}}+F_{M_{1}}-3F_{M_{7}}\right]s^{2}$
(15h) $\displaystyle\mathcal{A}_{22}(s)=$
$\displaystyle-\frac{1}{10}g^{4}\left[4F_{T_{0}}-2F_{T_{1}}+F_{T_{2}}\right]s^{2}$
$\displaystyle+\frac{1}{160}g^{2}\left[4F_{M_{0}}+F_{M_{1}}+F_{M_{7}}\right]s^{2}\,.$
(15i)

In fact, comparing polarized $2\to 2$ on-shell processes we have verified that
the numerical discrepancy between a helicity-dependent treatment and the
simplified version is less than a percent for the parameter ranges considered,
negligible in comparison to the scheme dependence of the unitary projection
itself.

We now turn to the unitary projection of the scattering amplitudes, applicable
to the high-energy range where the leading behavior in the presence of nonzero
operator coefficients is given by (15). We follow the T-matrix projection
scheme introduced in Ref. [7] and apply it to the simplified helicity-
independent eigenamplitudes, for the case where only one type of new
interactions (mixed or transversal) is active at a time. The simplifications
combined allow us to evaluate the projection and thus the compensating terms
in closed form. (If coefficients are non-zero for both classes simultaneously
or a detailed separation of helicities is intended, we have to resort to
numerical evaluation of the T-matrix projection. This is beyond the scope of
the present work.)

The unitary projection of a spin-isospin eigenamplitude ${\mathcal{A}}_{IJ}$
is given by the expression

$\displaystyle\hat{\mathcal{A}}_{IJ}=\frac{1}{\mathrm{Re}{\frac{1}{\mathcal{A}_{IJ}}}-\frac{\mathrm{i}}{32\pi}}.$
(16)

This projection may be recast as an $s$-dependent correction counterterm for
each eigenamplitude,

$\displaystyle\Delta\mathcal{A}_{IJ}=\hat{\mathcal{A}}_{IJ}-\mathcal{A}_{IJ}.$
(17)

The limit $A_{IJ}\to\infty$ lets us recover the universal unitarity bound for
each eigenamplitude,

$\displaystyle|\hat{\mathcal{A}}_{IJ}|\leq 32\pi\,.$ (18)

In particular, the truncated SMEFT expansion, i.e. constant coefficients in
the eigenamplitude above, yields $\lim_{s\to\infty}\mathcal{A}_{IJ}\to\infty$
for all partial waves with nonzero coefficients. The T-matrix projection then
asymptotically saturates unitarity. A model with a pole in the amplitude at
some value $s=M^{2}$, projected according to this prescription, saturates the
unitarity limit at this point and follows a Breit-Wigner shape for the energy
dependence in the vicinity of the pole. The actual pole of the amplitude gets
shifted away from the real axis.

##  4 (Strongly coupled) Continuum Model

As mentioned in Sec. 2, the simplified models that we actually consider are
(i) continuum models which smoothly interpolate between high-energy unitarity
saturation and the low-energy SMEFT, and (ii) resonance models where distinct
features arise in the spectrum. Including also the unmodified SM in the
discussion, these models cover the whole range of possible interaction
strengths that future VBS measurements may observe, and thus yield a fairly
robust projection for the sensitivity of a collider experiment. With the
exception of the unmodified SM, neither of these models is UV complete, and
the actual results should behave differently in the asymptotic regime. For
instance, new inelastic channels may appear as final states. However, as long
as the initial assumptions about unitarity, gauge invariance and minimal
flavor violation hold true, we should not expect event rates in this sector
which exceed the strongly-interacting continuum scenario that we consider
here.

For our numerical studies of the continuum scenario, we have adopted the
amplitudes with the local operators of Sec. 2 added to the SM Feynman rules
and converted this to a unitary model according to (16) after diagonalization.
This has been re-expressed in terms of form-factor modified Feynman rules
along the lines of Refs. [21, 7] and implemented in the Monte-Carlo event
generator WHIZARD [24]333WHIZARD is a multi-purpose event generator which
ships with its a tree-level matrix element generator [25, 26]. It uses the
color-flow formalism for QCD [27], and comes with its own parton shower
implementations [28]. While it allows to simulate almost arbitrary BSM
interactions (cf. e.g. the SUSY implementation [29]) via its FeynRules
interface [30], WHIZARD has been also successfully extended towards next-to-
leading order and matched to gluon and photon resummation [31]. Using Feynman
rules and a straightforward on-shell projection of the boson momenta, the
interactions can be evaluated off-shell in the context of an automatic
amplitude evaluation, and thus enter the standard WHIZARD framework that
ultimately yields simulated event samples. There are some subtleties hidden in
the on-shell projection; this is discussed in detail, along with some
refinement of the method, in Ref. [20].

Figure 1: Cross sections differential in the diboson invariant mass for the
process $pp\to W^{+}W^{+}jj$. The solid black line shows the Standard Model
differential cross section, the green, blue and red lines the cross sections
with anomalous couplings $F_{M_{i}}=50\,\
\text{TeV}^{-4},F_{M_{i}}=10\,\text{TeV}^{-4}$ and
$F_{M_{i}}=2\,\text{TeV}^{-4}$ for i = 0 (upper left panel), i = 1 (upper
right panel), and i = 7 (lower panel), respectively. Solid: unitarized;
dashed: naive result. Cuts: $M_{jj}>500\,\ \text{GeV}$, $\Delta\eta_{jj}>2.4$,
$|\eta_{j}|<4.5$, $p_{T}^{j}>20\,\ \text{GeV}$.

Figure 2: Cross sections differential in the diboson invariant mass for the
process $pp\to HHjj$. The solid black line shows the Standard Model
differential cross section, the green, blue and red lines the cross sections
with anomalous couplings $F_{M_{i}}=50\,\
\text{TeV}^{-4},F_{M_{i}}=10\,\text{TeV}^{-4}$ and
$F_{M_{i}}=2\,\text{TeV}^{-4}$ for i = 0 (upper left panel), i = 1 (upper
right panel), and i = 7 (lower panel), respectively. Solid: unitarized;
dashed: naive result. Cuts: $M_{jj}>500\,\ \text{GeV}$, $\Delta\eta_{jj}>2.4$,
$|\eta_{j}|<4.5$, $p_{T}^{j}>20\,\ \text{GeV}$.

The processes $pp\to jjW^{+}W^{+}$ and $pp\to jjHH$ have received special
attention. The former process exhibits a characteristic signature of like-sign
dileptons and has the largest signal-to-branching ratio of all VBS processes
at the LHC, while the latter is difficult to isolate but carries a dependence
on the triple-Higgs coupling which is among the most elusive SM parameters. In
fact, an anomalous triple-Higgs coupling can be attributed to a gauge-
invariant dimension-six operator, while in this work we are considering
dimension-eight contributions. Clearly, an unambiguous determination of a
dimension-six parameter in a systematic low-energy expansion is only possible
if the next higher order is under control.

In Fig. 1 and in Fig. 2 we show results for the process $pp\to W^{+}W^{+}jj$
and for $pp\to HHjj$ within the continuum simplified model with nonzero
coefficients for the longitudinal-transverse mixed operators with parameters
$F_{M_{0/1/7}}$, respectively. We choose three distinct values,
$F=2,10,50\;\text{TeV}^{-4}$, with one nonzero coefficient at a time. The
solid lines show the distribution in the invariant mass of the $W^{+}W^{+}/HH$
pair, which coincides with the effective energy $\sqrt{\hat{s}}$ for the basic
VBS process. We note that in the presence of background and finite jet-energy
resolution, this distribution is not actually measurable for $W^{+}W^{+}$.
Rather, $W$-boson decay leptons are detected. However, the plots describe most
clearly the expected physics, which will only be diluted in actual
observables.

For reference, we also display the results which would be obtained if the
naive dimension-eight SMEFT amplitude, without T-matrix correction, were used
for the calculation (dashed). Clearly, such a calculation overestimates the
achievable event yield by a huge amount and suggests a sensitivity to the
model parameters which is unphysical.

We observe that regardless of parameters, the solid curves approach an
asymptotic differential cross section which for the $W^{+}W^{+}$ final state
is enhanced by about an order of magnitude over the SM prediction. In the case
of the $HH$ final state, the enhancement amounts to more than two orders of
magnitude. These asymptotic limits correspond to a maximally strong
interaction, saturation of the unitarity limit within the quasi-elastic
channel. The residual parameter dependence is confined to a certain transition
region. Beyond this region, from the saturated quasi-elastic amplitudes we can
read off the maximally allowed event number for the given spin-isospin
channel.

Figure 3: Cross sections differential in the diboson invariant mass for the
process $pp\to W^{+}W^{+}jj$. The solid black line shows the SM differential
cross section, the green, blue and red lines the cross sections with anomalous
couplings $F_{T_{i}}=50\,\ \text{TeV}^{-4},F_{T_{i}}=10\,\text{TeV}^{-4}$ and
$F_{T_{i}}=2\,\text{TeV}^{-4}$ for i = 0 (upper left panel), i = 1 (upper
right panel), and i = 2 (lower panel), respectively. Solid: unitarized;
dashed: naive result. Cuts: $M_{jj}>500\,\ \text{GeV}$, $\Delta\eta_{jj}>2.4$,
$|\eta_{j}|<4.5$, $p_{T}^{j}>20\,\ \text{GeV}$.

In Fig. 3 we plot results for the purely transverse interactions with
parameters $F_{T_{0/1/2}}$. Again, the studied process is $pp\to
W^{+}W^{+}jj$. The $HH$ channel is not affected by these interactions, because
the purely transverse operators do not contribute to any anomalous coupling
involving a Higgs. We choose three distinct values,
$F_{T}=2,10,50\;\text{TeV}^{-4}$, with one nonzero coefficient at a time.

In these scenarios, the asymptotic enhancement of the continuum model over the
SM approaches two orders of magnitude. We may read this observation as an
indication for much larger freedom for new-physics effects in purely
transverse vector-boson interactions, compared to mixed and purely
longitudinal interactions. This fact should be accounted for in data analysis.
Nevertheless, also in this class of models, the naive SMEFT result represented
by the dashed lines overestimates the possible event rates by a large factor.

We emphasize that the above plots, which only indicate the variation with
respect to one of the model parameters, should not be taken individually as
realistic predictions, even if accepting the basic assumptions regarding a
strongly interacting continuum in the electroweak sector. They sweep a range
of predictions, within the given model class. In reality, we expect more than
one coefficient to be present, so a global fit would be required to determine
the correct parameter dependence and the sensitivity of a collider experiment.
On the other hand, we can already conclude that due to the failure of the
naive SMEFT, there is no meaningful description of these processes that can be
viewed as model-independent. An analysis that compares actual data to a
prediction, apart from the SM result, must choose among the conceivable
(simplified) models for a comparison, of which we can only show a set of
examples.

If several anomalous couplings are present in a model, it is essential to
increase the number of independent observables that enter a global fit of all
parameters. At the LHC, there is a number of di-boson final state that can be
produced in VBS. In Appendix B we present results for the additional VBS final
states $W^{+}W^{-}$, $W^{+}Z$, and $ZZ$ which are not as easily accessible or
have smaller leptonic rates compared to $W^{+}W^{+}$ but should be considered
in this context, particularly as there are already results from the LHC
experiments for the latter two. For those results, we choose a value of
$2\;\text{TeV}^{-4}$ for each of the parameters $F_{i}$, with only one
parameter nonzero at a time.

##  5 Simplified Resonance Models

In this section, we consider simplified models where an anomalous local
interaction resolves into a resonance which saturates a partial-wave spin-
isospin amplitude. A resonance saturates an elastic channel for finite energy
and exhibits a falloff of the amplitude beyond the peak, before strong
interactions may set in again at higher energy. This is observed, e.g., for
some isolated bound states that precede a strongly interacting continuum in
QCD. In Ref. [8], we described this class of models in the context of VBS and
studied couplings of the resonance to longitudinal gauge bosons via the scalar
current $J_{H}^{(4)}$ (5). In this work, we extend the allowed coupling to
transversal bosons. As an example, we take a single scalar with a coupling to
the current $J^{(4)}_{W}$ (6).

There are various models of a non-minimal Higgs sector which effectively lead
to a phenomenology of this type. In general, we expect couplings of the
resonance both to longitudinal and transverse vector-boson modes. BSM models
which allow a direct coupling of a new physics particle only to the transverse
mode of electroweak gauge bosons are often very constrained by data [32]. Only
a few extra-dimensional models [33, 34] including a directly and strongly
coupled spin-2 resonance, for example a KK-graviton, are not as hampered by
experimental data. Other BSM models introduce the coupling of transverse
vector bosons to a new physics particle not directly, but due to loop
contributions. In Randall-Sundrum [35] or ADD [36] models this could also be
achieved through a top loop [37].

Models with extra scalar resonances typically introduce additional new heavy
particles. For instance, in composite Higgs models the coupling to the
transversal gauge sector can be mediated by technipions [38] or by heavy
fermions [39, 40]. If the mass scale of such extra heavy particles is beyond
the experimental reach of LHC, the loop contributions are small and can be
parametrized within an EFT. Effective couplings of a resonance involve both
longitudinal and transversal vector bosons. In recent diphoton studies, this
EFT framework was also used to estimate the effect of a possible diphoton
resonance [32], [41, 42, 43]. Another class of models with heavy resonances
are Little Higgs models [44, 45]. For these models, the coefficients of the
SMEFT as the low-energy expansion have been calculated, e.g., in Ref. [46].

In the present paper, we do not refer to a specific scenario. We construct a
simplified model with transverse couplings of a generic heavy resonance
$\sigma$. The effective Lagrangian takes the following form,

$\displaystyle\mathcal{L}_{\sigma}$
$\displaystyle=-\frac{1}{2}\sigma(m_{\sigma}^{2}-\partial^{2})\sigma+\sigma(J_{\sigma\parallel}+J_{\sigma\perp})$
(19a) $\displaystyle J_{\sigma\parallel}$ $\displaystyle=F_{\sigma
H}\operatorname{tr}\left[(\mathbf{D}_{\mu}\mathbf{H})^{\dagger}(\mathbf{D}^{\mu}\mathbf{H})\right]$
(19b) $\displaystyle J_{\sigma\perp}$
$\displaystyle=g^{2}F_{W\sigma}\sigma\operatorname{tr}\left[\mathbf{W}_{\mu\nu}\mathbf{W}^{\mu\nu}\right]+{g^{\prime}}^{2}F_{B\sigma}\sigma\operatorname{tr}\left[\mathbf{B}_{\mu\nu}\mathbf{B}^{\mu\nu}\right]$
(19c)

with three independent coupling parameters.

In the low-energy limit, the scalar resonance can be integrated out, and we
obtain the SMEFT Lagrangian with the following nonzero coefficients of the
dimension-8 operators at leading order:

$\displaystyle F_{S_{0}}$ $\displaystyle=\phantom{-}{F_{\sigma
H}^{2}}/{2m_{\sigma}^{2}}$ (20a) $\displaystyle F_{M_{0}}$
$\displaystyle=-{F_{\sigma H}F_{\sigma W}}/{m_{\sigma}^{2}}$ (20b)
$\displaystyle F_{M_{2}}$ $\displaystyle=-{F_{\sigma H}F_{\sigma
B}}/{m_{\sigma}^{2}}$ (20c) $\displaystyle F_{T_{0}}$
$\displaystyle=\phantom{-}{F_{\sigma W}^{2}}/{2m_{\sigma}^{2}}$ (20d)
$\displaystyle F_{T_{5}}$ $\displaystyle=\phantom{-}{F_{\sigma W}F_{\sigma
B}}/{m_{\sigma}^{2}}$ (20e) $\displaystyle F_{T_{8}}$
$\displaystyle=\phantom{-}{F_{\sigma B}^{2}}/{2m_{\sigma}^{2}}.$ (20f)

To set the relation between the coupling constant to the electroweak currents
and the resonance mass, we also compute the width of the scalar resonance:

$\displaystyle\Gamma(m_{\sigma})=\int
d\Omega\frac{|\vec{p}|}{32\pi^{2}m_{\sigma}^{2}}$
$\displaystyle\left(|\mathcal{M}_{\sigma\rightarrow
W^{+}W^{-}}|^{2}+\frac{1}{2}|\mathcal{M}_{\sigma\rightarrow
ZZ}|^{2}+\frac{1}{2}|\mathcal{M}_{\sigma\rightarrow HH}|^{2}\right.$
$\displaystyle\left.\quad+|\mathcal{M}_{\sigma\rightarrow
Z\gamma}|^{2}+\frac{1}{2}|\mathcal{M}_{\sigma\rightarrow\gamma\gamma}|^{2}\right)\,,$
(21)

with $|\vec{p}|=m_{\sigma}/2$. Here, we neglect the masses of the electroweak
gauge bosons in the kinematics of the phase space vectors.

The model with only $F_{\sigma H}$ nonzero has been covered in Ref. [8]. For
this paper, we set $F_{\sigma H}=F_{\sigma B}=0$ and keep only $F_{\sigma W}$.
The resonance width becomes

$\displaystyle\Gamma_{W}(m_{\sigma})$
$\displaystyle=\;\frac{3m_{\sigma}^{3}}{16\pi}g^{4}F_{\sigma
W}^{2}\left(1+\mathcal{O}(1/m^{2}_{\sigma})\right)\,.$ (22)

The low-energy limit contains only the operator $\mathcal{L}_{T_{0}}$. We can
thus easily compare distributions with a resonance to the anologous
distributions with a continuum, where both models reduce to the same low-
energy limit. While the low-energy approximation has a single parameter, the
dimension-eight operator coefficient $F_{T_{0}}$, the resonance model has two
free parameters, the resonance mass and the resonance coupling, or
alternatively the width.

We have implemented the resonance model in the Monte-Carlo generator WHIZARD
[24], using the same unitarity projection algorithm as for the continuum
models. In Fig. 4, we show the invariant-mass distribution of the $ZZ$ final
state for a scalar resonance with mass $m_{\sigma}=1\,\mathrm{TeV}$ and
different couplings $F_{\sigma W}=10.0,4.5,2.0\,\text{TeV}^{-1}$. These values
correspond to the anomalous quartic coupling
$F_{T_{0}}=50,10,2\,\text{TeV}^{-4}$ if the scalar resonance is integrated
out. The dashed lines show the naive result of implementing the scalar
resonance as an extra particle with its width given by the formula (22). The
solid lines show the unitary projection for each coupling value, respectively.

This plot illustrates two properties of resonance models. First of all, we
observe that the unitary projection has two effects: on the resonance, the
peak becomes narrower and more pronounced. This is the result of subleading
terms in the width formula, which we did not include in the naive result but
which are accounted for by the unitary projection. Asymptotically, the
amplitude is suppressed by the projection. This is the result of saturating
partial waves by $s^{2}$ terms which originate from the derivative coupling.

Since a derivative coupling is a typical feature of strong interactions where
couplings involve form factors, and a necessary property of resonances with
higher spin, the asymptotic effect of unitarity saturation is essential for a
complete description. The T-matrix projection is a method for implementing
unitarity in the model for the whole kinematical range.

Figure 4: Cross sections differential in the diboson invariant mass for the
process $pp\to ZZjj$. The solid black line shows the Standard Model
differential cross section, the green, blue and red lines the cross sections
with a scalar resonance with mass $m_{\sigma}=1\,\text{TeV}$ and coupling of
$F_{W\sigma}=10.0\,\ \text{TeV}^{-1},F_{W\sigma}=4.5\,\text{TeV}^{-1}$ and
$F_{W\sigma}=2.0\,\text{TeV}^{-1}$, respectively. Solid: unitarized; dashed:
naive result. Cuts: $M_{jj}>500\,\ \text{GeV}$, $\Delta\eta_{jj}>2.4$,
$|\eta_{j}|<4.5$, $p_{T}^{j}>20\,\ \text{GeV}$. Figure 5: Cross sections
differential in the diboson invariant mass for the process $pp\to ZZjj$. The
solid black line shows the Standard Model, the blue lines show an anomalous
coupling $F_{T_{i}}=2\,\text{TeV}^{-4}$ and the red lines show a scalar
resonance with mass $m_{\sigma}=1\,\text{TeV}$ and coupling of $F_{\sigma
W}=2.0\,~{}\text{TeV}^{-2}$. Solid: unitarized; dashed: naive result. Cuts:
$M_{jj}>500\,\ \text{GeV}$, $\Delta\eta_{jj}>2.4$, $|\eta_{j}|<4.5$,
$p_{T}^{j}>20\,\ \text{GeV}$.

In Fig. 5, we compare the simplified model with a scalar resonance with mass
$m_{\sigma}=1\,\mathrm{TeV}$ and coupling $F_{\sigma W}=2.0\,\text{TeV}^{-1}$
(red) to the corresponding EFT result with the matching anomalous quartic
coupling $F_{T}=2\,\text{TeV}^{-4}$ (blue), with and without unitary
projection (solid vs. dashed). This is a rather small coupling, and the
resonance behaves almost like an elementary particle. The peak is not
approximated at all by the EFT operator description. We may argue that for
such a type of model, the EFT is useful only in the case of strong couplings
and broad resonances. We also see that the high-energy behavior of the EFT
approximation has no resemblance to the high-energy behavior of the resonance
model, regardless of unitarity projection.

We conclude that including resonance models in the description allows us to
smoothly interpolate between weakly and strongly interacting models. This
interpolation may leave the applicability range of perturbative expansions,
but does not require to deal with unphysical behavior as a calculational
artefact.

##  6 Implications for LHC Analyses and Conclusions

The ATLAS and CMS experiments have analyzed the early-stage LHC data with
respect to the sensitivity to VBS parameters. Table 1 summarizes published
results, expressed in terms of the unmodified SMEFT parameterization with
dimension-eight operators included.

Coefficient | CMS&ATLAS [47] | ATLAS[48, 49] | CMS [50, 51] | CMS reweighted
---|---|---|---|---
[TeV-4] | 8 TeV, EFT | 8 TeV, T-matrix | 13 TeV, EFT | 13 TeV, EFT
$f_{S_{0}}/\Lambda^{4}$ | [-38,40] |  | [-7.7,7.7] | [-7.7,7.7]
$f_{S_{1}}/\Lambda^{4}$ | [-118,120] |  | [-21.6,21.8] | [-21.6,21.8]
$F_{S_{0}}$ | [-70,70] | [-104,130] |  |
$F_{S_{1}}$ | [-118,120] | [-122,144] |  |
$f_{M_{0}}/\Lambda^{4}$ | [-18,18] |  | [-6.0,5.9] | [-13.8,14.1]
$f_{M_{1}}/\Lambda^{4}$ | [-44,47] |  | [-8.7,9.1] | [-21.4,20.4]
$f_{M_{6}}/\Lambda^{4}$ | [-65,63] |  | [-11.9,11.8] | [-27.7,27.9]
$f_{M_{7}}/\Lambda^{4}$ | [-70,66] |  | [-13.3,12.9] | [-30.3,31.2]
$f_{T_{0}}/\Lambda^{4}$ | [-4.2,4.6] |  | [-0.46,0.44] | [-2.53,2.42]
$f_{T_{1}}/\Lambda^{4}$ | [1.9,2.2] |  | [-0.28,0.31] | [-1.54,1.71]
$f_{T_{2}}/\Lambda^{4}$ | [-5.2,6.4] |  | [-0.89,1.02] | [-4.9,5.6]
$f_{T_{9}}/\Lambda^{4}$ | [-6.9,6.9] |  | [-1.8,1.8] | [-7.5,7.5]
Table 1:  Observed Limits of ATLAS and CMS of complete LHC data at
$\sqrt{s}=8\mathrm{TeV}$ and current observed limits of CMS at
$\sqrt{s}=13\mathrm{TeV}$ using the naive EFT model and the T-matrix model.
The last column show the limits in natural reweighting of field strength
tensors: $\mathbf{W}^{\mu\nu}\rightarrow\mathrm{i}g\mathbf{W}^{\mu\nu}$,
$\mathbf{B}^{\mu\nu}\rightarrow\mathrm{i}g^{\prime}\mathbf{B}^{\mu\nu}$.

In view of the results presented in the preceding sections, we have to discuss
the physical relevance of the published exclusion bounds. In principle, the
SMEFT approach provides a well-defined framework. However, our findings
confirm the expectation that the SMEFT expansion, applied to VBS as a LHC
process, does not provide a systematic expansion or meaningful description of
the complete data set. For nonvanishing dimension-eight coefficients, the
amplitudes rise steeply with energy, such that a problem invariably arises
within the accessible kinematic range. This happens for _any_ set of parameter
values, unless the dimension-eight coefficients are so small that the
prediction remains entirely indistinguishable from the SM.

The measurements acquire a physical interpretation only within the context of
a unitary model. For instance, we may apply a straightforward T-matrix
projection to the naive extrapolation and thus consider a unitary simplified
model that is smoothly matched to the low-energy SMEFT, depending on the same
parameters that in the low-energy act as dimension-eight operator
coefficients. We find that the sensitivity of this unitary model to the free
parameters is much weaker than the naive calculation would suggest, likely by
an order of magnitude. Since the minimal T-matrix projection interpolates the
low-energy behavior with asymptotic saturation of the elastic channel, this
particular projection provides us with the ultimate limitation to the
achievable parameter sensitivity.

We conclude that any such description or theoretical prediction of non-SM
behavior has to depart from the model-independent paradigm. Otherwise, data
analysis has to artificially remove kinematical regions from the data sample,
losing valuable information. A well-defined universal but model-dependent
parameterization is certainly possible, however, without losing contact to the
SMEFT as a systematic description of the low-energy region.

In this work, we have demonstrated the construction of unitary projections
that yield usable simplified models for otherwise unknown new physics.
Extending previous work, we have included transverse vector-boson polarization
modes together with final-state Higgs bosons in the completed framework. None
of our models is UV complete or otherwise meaningful as a prediction. However,
for the purpose of estimating the prospects for future measurements in
quantitative terms, such a set of simplified models becomes a useful tool.
Applying the direct T-matrix projection to the straightforward extrapolation
of the SMEFT amplitudes with dimension-eight operators, we obtain a natural
interpolation between the low-energy range which is well understood, and high-
energy amplitudes which saturate the unitarity limits. This sets the scale for
more refined models, such as the model of a singlet scalar coupled to
transverse gauge bosons which we also have considered in some detail. In
essence, we obtain parameter-dependent upper limits for event rates of all
processes for all energy ranges, which refined models have to respect.

The lesson to be learned from such results is twofold. Firstly, we read off
the range of event rates and distributions that we can possibly expect from
LHC experiments, for any underlying model. This range can only be exceeded if
some natural, basic assumptions are violated by Nature. More precisely,
violations of the assumptions would point to (a) fermions directly involved in
new (strong) Higgs-sector interactions, or (b) gauge symmetry being just a
low-energy accident, or (c) four-dimensional quantum field theory becoming
invalid. Either scenario appears to be unlikely given the success of the SM in
describing low-energy data, in particular in the flavor sector. For this
reason, we believe that the quantitative results obtained within the framework
of unitary simplified models reliably exhaust the range that can be expected
from real data.

Secondly, the framework of unitary projection, now extended to transverse
polarizations, enables any theoretical idea or model of the Higgs sector as a
viable model for Monte-Carlo simulations. I.e., the projection satisfies the
applicable unitarity constraints, correctly couples to fermionic currents, and
the collider environment is described in consistency with the analogous SM
calculation. In short, the model can be compared to data without further
approximations or simplifications. The downside is that for VBS processes,
there is no usable model-independent framework, and any study has to agree on
a particular model class and assumptions for interpreting the results.
Furthermore, the arbitrariness in the parameterization mandates the inclusion
of all quantum-number combinations and global fits, which would greatly
benefit from a larger set of observables such as can be obtained at high-
energy lepton colliders supplementing the LHC.

Finally, we add a remark on Higgs pair-production. This final state has
received particular attention since it is sensitive to the triple-Higgs
coupling and thus to the Higgs potential. Higgs pairs can result from gluon or
massive vector-boson fusion. The latter channel has the particular feature of
extra taggable forward jets. In a generic EFT description, the Higgs pair-
production process in VBF receives various contributions that can be
attributed to higher-dimensional operators, and the Higgs potential correction
is only one of those. Furthermore, our results show that dimension-eight
operators can drastically enhance the Higgs pair-production rate by three
orders of magnitude before unitarity limits set in. Since there is no reason
to expect the operator series expansion to stop at dimension six, we are
forced to argue that any analysis of Higgs pair-production data that confines
itself to a truncated expansion has to be taken with a grain of salt. On the
other hand, linear gauge invariance relates anomalous effects in Higgs pair-
production to anomalous effects in VBS at the same order. Future LHC Higgs
pair-production analyses thus should correlate all accessible boson-production
channels. The interpretation, however, will rely on model-dependent approaches
such as the one that we present in this paper.

### Acknowledgments

C.F. has been supported by a HYT Fellowship of the University of Siegen. W.K.
thanks the CLICdp and Theory Groups for their hospitality at CERN, where part
of this work was completed. We also acknowledge valuable discussions with C.
Grojean, M.-A. Pleier, M. Rauch, M. Szleper and A. Wulzer, and we thank D.
Zeppenfeld for carefully reading and useful comments on the manuscript. J.R.R.
acknowledges support by the EU COST Network ”VBScan” (COST Action CA16108).

##  Appendix A Isospin-spin amplitudes

In this section we collect the different isospin-spin eigenamplitudes for the
different combinations of helicities of the electroweak gauge bosons according
to decomposition into Wigner functions in Eq. 14. As explained in the last
paragraph of Sec. 2, we take here only the Wilson coefficients of those
transversal and mixed operators $\mathcal{L}_{T_{i}}$ and
$\mathcal{L}_{M_{i}}$, respectively, with indices $i=0,1,2$ into account for
which there is custodial $SU(2)_{C}$ conserved.

Already in Ref. [8] it was shown that the kinematic functions for the
unitarized amplitudes for resonances are not simple powers of $s$, but contain
logarithms and pole-like rational functions of $s$ in general. For the
isospin-spin amplitudes in the case of a isoscalar scalar resonance in Table
4, we define the following kinematic functions:

$\displaystyle X(s,m)$ $\displaystyle=\frac{3s^{2}}{s-m^{2}}$ (23)
$\displaystyle\mathcal{S}_{0}(s,m)$
$\displaystyle=2m^{2}+2\frac{m^{4}}{s}\log\left(\frac{m^{2}}{s+m^{2}}\right)-s$
(24) $\displaystyle\mathcal{S}_{1}(s,m)$
$\displaystyle=4\frac{m^{4}}{s}+6m^{4}\left(2m^{2}+s\right)\log\left(\frac{m^{2}}{s+m^{2}}\right)+\frac{s}{3}$
(25) $\displaystyle\mathcal{S}_{2}(s,m)$
$\displaystyle=6\frac{m^{4}}{s^{2}}\left(2m^{2}+s\right)+2\frac{m^{4}}{s^{3}}\left(6m^{4}+6m^{2}s+s^{2}\right)\log\left(\frac{m^{2}}{s+m^{2}}\right)$
(26) $\displaystyle\tilde{\mathcal{S}_{2}}(s,m)$
$\displaystyle=\frac{m^{2}}{3}-\frac{m^{4}}{2s}+\frac{m^{6}}{s^{2}}+\frac{m^{8}}{s^{3}}\log\left(\frac{m^{2}}{s+m^{2}}\right)-\frac{s}{4}$
(27)

i j |  | 0 |  |  | 1 |  |  | 2 |  | $\boldsymbol{\lambda}$
---|---|---|---|---|---|---|---|---|---|---
| $\frac{3}{2}$ | $-\frac{3}{8}$ | $\frac{3}{16}$ | 0 | 0 | 0 | 0 | 0 | 0 |  | $+$ | $+$ | $0$ | $0$
---|---|---|---
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\frac{1}{20}\sqrt{\frac{3}{2}}$ | $\frac{1}{40}\sqrt{\frac{3}{2}}$ |  | $+$ | $-$ | $0$ | $0$
---|---|---|---
0 | 0 | 0 | 0 | $-\frac{1}{8}$ | $\frac{1}{32}$ | $\frac{7}{192}$ | $\frac{1}{40}$ | $-\frac{1}{160}$ | $\frac{3}{320}$ |  | $+$ | $0$ | $-$ | $0$
---|---|---|---
| 0 | 0 | 0 | $\frac{1}{8}$ | $-\frac{1}{32}$ | $-\frac{7}{192}$ | $\frac{1}{40}$ | $-\frac{1}{160}$ | $\frac{3}{320}$ |  | $+$ | $0$ | $0$ | $-$
---|---|---|---
| 0 | 0 | 0 | 0 | $\frac{1}{12}$ | $\frac{1}{24}$ | 0 | 0 | 0 |  | $+$ | $0$ | $+$ | $0$
---|---|---|---
| 0 | 0 | 0 | 0 | $-\frac{1}{12}$ | $-\frac{1}{24}$ | 0 | 0 | 0 |  | $+$ | $0$ | $0$ | $+$
---|---|---|---
| 0 | 0 | 0 | 0 | 0 | $\frac{1}{24}$ | 0 | 0 | 0 |  | $+$ | $+$ | $0$ | $0$
---|---|---|---
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |  | $+$ | $-$ | $0$ | $0$
---|---|---|---
1 | 0 | 0 | 0 | $-\frac{1}{8}$ | $\frac{1}{32}$ | $\frac{1}{96}$ | $\frac{1}{40}$ | $-\frac{1}{160}$ | $\frac{1}{160}$ |  | $+$ | $0$ | $-$ | $0$
---|---|---|---
| 0 | 0 | 0 | $-\frac{1}{8}$ | $\frac{1}{32}$ | $\frac{1}{96}$ | $-\frac{1}{40}$ | $\frac{1}{160}$ | $-\frac{1}{160}$ |  | $+$ | $0$ | $0$ | $-$
---|---|---|---
| 0 | 0 | 0 | 0 | $\frac{1}{12}$ | $\frac{1}{24}$ | 0 | 0 | 0 |  | $+$ | $0$ | $+$ | $0$
---|---|---|---
| 0 | 0 | 0 | 0 | $\frac{1}{12}$ | $\frac{1}{24}$ | 0 | 0 | 0 |  | $+$ | $0$ | $0$ | $+$
---|---|---|---
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |  | $+$ | $+$ | $0$ | $0$
---|---|---|---
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |  | $+$ | $-$ | $0$ | $0$
---|---|---|---
2 | 0 | 0 | 0 | $-\frac{1}{8}$ | $\frac{1}{32}$ | $-\frac{1}{24}$ | $\frac{1}{40}$ | $-\frac{1}{160}$ | 0 |  | $+$ | $0$ | $-$ | $0$
---|---|---|---
| 0 | 0 | 0 | $\frac{1}{8}$ | $-\frac{1}{32}$ | $\frac{1}{24}$ | $\frac{1}{40}$ | $-\frac{1}{160}$ | 0 |  | $+$ | $0$ | $0$ | $-$
---|---|---|---
| 0 | 0 | 0 | 0 | $\frac{1}{12}$ | $\frac{1}{24}$ | 0 | 0 | 0 |  | $+$ | $0$ | $+$ | $0$
---|---|---|---
| 0 | 0 | 0 | 0 | $-\frac{1}{12}$ | $-\frac{1}{24}$ | 0 | 0 | 0 |  | $+$ | $0$ | $0$ | $+$
---|---|---|---
| $c_{0}$ | $c_{1}$ | $c_{2}$ | $c_{0}$ | $c_{1}$ | $c_{2}$ | $c_{0}$ | $c_{1}$ | $c_{2}$ |
Table 2: Coefficients of the isospin-spin amplitudes calculated with eq. (14) for the mixed operators $\mathcal{L}_{M_{i}}$ depending on the helicity of the incoming and outgoing particles. The isospin spin amplitudes are given by $A_{ij}(s;\boldsymbol{\lambda})=(c_{0}F_{M_{0}}+c_{1}F_{M_{1}}+c_{2}F_{M_{7}})g^{2}s^{2}$. i j |  | 0 |  |  | 1 |  |  | 2 |  | $\boldsymbol{\lambda}$
---|---|---|---|---|---|---|---|---|---|---
| -6 | -2 | $-\frac{5}{2}$ | 0 | 0 | 0 | 0 | 0 | 0 |  | $+$ | $+$ | $+$ | $+$
---|---|---|---
0 | 0 | 0 | 0 | 0 | 0 | 0 | $-\frac{2}{5}$ | $-\frac{4}{5}$ | $-\frac{1}{2}$ |  | $+$ | $-$ | $+$ | $-$
---|---|---|---
| 0 | 0 | 0 | 0 | 0 | 0 | $-\frac{2}{5}$ | $-\frac{4}{5}$ | $-\frac{1}{2}$ |  | $+$ | $-$ | $-$ | $+$
---|---|---|---
| $-\frac{22}{3}$ | $-\frac{14}{3}$ | $-\frac{11}{6}$ | 0 | 0 | 0 | $-\frac{2}{15}$ | $-\frac{4}{15}$ | $-\frac{1}{30}$ |  | $+$ | $+$ | $-$ | $-$
---|---|---|---
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |  | $+$ | $+$ | $+$ | $+$
---|---|---|---
1 | 0 | 0 | 0 | 0 | 0 | 0 | $\frac{2}{5}$ | $-\frac{1}{5}$ | 0 |  | $+$ | $-$ | $+$ | $-$
---|---|---|---
| 0 | 0 | 0 | 0 | 0 | 0 | $-\frac{2}{5}$ | $\frac{1}{5}$ | 0 |  | $+$ | $-$ | $-$ | $+$
---|---|---|---
| 0 | 0 | 0 | $\frac{2}{3}$ | $-\frac{1}{3}$ | $\frac{1}{6}$ | 0 | 0 | 0 |  | $+$ | $+$ | $-$ | $-$
---|---|---|---
| 0 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |  | $+$ | $+$ | $+$ | $+$
---|---|---|---
2 | 0 | 0 | 0 | 0 | 0 | 0 | $-\frac{2}{5}$ | $-\frac{1}{5}$ | $-\frac{1}{5}$ |  | $+$ | $-$ | $+$ | $-$
---|---|---|---
| 0 | 0 | 0 | 0 | 0 | 0 | $-\frac{2}{5}$ | $-\frac{1}{5}$ | $-\frac{1}{5}$ |  | $+$ | $-$ | $-$ | $+$
---|---|---|---
| $-\frac{4}{3}$ | $-\frac{8}{3}$ | $-\frac{1}{3}$ | 0 | 0 | 0 | $-\frac{2}{15}$ | $-\frac{1}{15}$ | $-\frac{1}{30}$ |  | $+$ | $+$ | $-$ | $-$
---|---|---|---
| $c_{0}$ | $c_{1}$ | $c_{2}$ | $c_{0}$ | $c_{1}$ | $c_{2}$ | $c_{0}$ | $c_{1}$ | $c_{2}$ |
Table 3: Coefficients of the isospin-spin amplitudes calculated with eq. (14) for the transversal operators $\mathcal{L}_{T_{i}}$ depending on the helicity of the incoming and outgoing particles. The isospin-spin amplitudes are given by $A_{ij}(s;\boldsymbol{\lambda})=(c_{0}F_{T_{0}}+c_{1}F_{T_{1}}+c_{2}F_{T_{2}})g^{4}s^{2}$. i j | 0 | 1 | 2 | $\boldsymbol{\lambda}$
---|---|---|---|---
| $X(s,m)$ | 0 | 0 |  | $+$ | $+$ | $+$ | $+$
---|---|---|---
0 | 0 | 0 | $\tilde{\mathcal{S}_{2}}(s,m)$ |  | $+$ | $-$ | $+$ | $-$
---|---|---|---
| 0 | 0 | $\tilde{\mathcal{S}_{2}}(s,m)$ |  | $+$ | $-$ | $-$ | $+$
---|---|---|---
| $X(s,m)+\mathcal{S}_{0}(s,m)$ | 0 | $\mathcal{S}_{2}(s,m)$ |  | $+$ | $+$ | $-$ | $-$
---|---|---|---
| 0 | 0 | 0 |  | $+$ | $+$ | $+$ | $+$
---|---|---|---
1 | 0 | 0 | $-\tilde{\mathcal{S}_{2}}(s,m)$ |  | $+$ | $-$ | $+$ | $-$
---|---|---|---
| 0 | 0 | $\tilde{\mathcal{S}_{2}}(s,m)$ |  | $+$ | $+$ | $+$ | $+$
---|---|---|---
| 0 | $\mathcal{S}_{1}(s,m)$ | 0 |  | $+$ | $+$ | $-$ | $-$
---|---|---|---
| 0 | 0 | 0 |  | $+$ | $+$ | $+$ | $+$
---|---|---|---
2 | 0 | 0 | $\tilde{\mathcal{S}_{2}}(s,m)$ |  | $+$ | $-$ | $+$ | $-$
---|---|---|---
| 0 | 0 | $\tilde{\mathcal{S}_{2}}(s,m)$ |  | $+$ | $-$ | $-$ | $+$
---|---|---|---
| $\mathcal{S}_{0}(s,m)$ | 0 | $\mathcal{S}_{2}(s,m)$ |  | $+$ | $+$ | $-$ | $-$
---|---|---|---
Table 4: Coefficients $c$ of the isospin spin amplitudes calculated with eq.
(14) for the isoscalar scalar resonance $\mathcal{L}_{\sigma W}$ depending on
the helicity of the incoming and outgoing particles. The isospin spin
amplitudes are given by $A_{ij}(s;\boldsymbol{\lambda})=cg^{4}s^{2}$.

##  Appendix B Additional Numerical Results

In this section, we display results for the invariant-distribution of the LHC
processes $pp\to W^{+}W^{-}jj$, $ZZjj$, and $W^{+}Zjj$ which supplement the
results for the $W^{+}W^{+}$ and $HH$ channels in the main text. For all
processes, we present the SM distribution together with the corresponding
distribution of the continuum simplified model, one free parameter varied at a
time, with a universal parameter value of $2\;\text{TeV}^{-4}$.

(a) $\mathcal{O}_{T_{0}}$

(b) $\mathcal{O}_{T_{1}}$

(c) $\mathcal{O}_{T_{2}}$

(d) $\mathcal{O}_{M_{0}}$

(e) $\mathcal{O}_{M_{1}}$

(f) $\mathcal{O}_{M_{7}}$

Figure 6: The plot shows the differential cross section as a function of the
invariant mass $m_{VV}$ of the two colliding vector bosons. The solid black
line is the standard model and the colored lines are the contributions of
$\mathcal{O}_{T_{0,1,2}}$ and $\mathcal{O}_{M_{{0,1,7}}}$ (dashed: naive EFT,
solid: unitarized model). Cuts: $M_{jj}>500$ GeV; $\Delta\eta_{jj}>2.4$;
$p^{j}_{T}>20$ GeV; $|\eta_{j}|>4.5$

(a) $\mathcal{O}_{T_{0}}$

(b) $\mathcal{O}_{T_{1}}$

(c) $\mathcal{O}_{T_{2}}$

(d) $\mathcal{O}_{M_{0}}$

(e) $\mathcal{O}_{M_{1}}$

(f) $\mathcal{O}_{M_{7}}$

Figure 7: The plot shows the differential cross section as a function of the
invariant mass $m_{VV}$ of the two colliding vector bosons. The solid black
line is the standard model and the colored lines are the contributions of
$\mathcal{O}_{T_{0,1,2}}$ and $\mathcal{O}_{M_{0,1,7}}$ (dashed: naive EFT,
solid: unitarized model). Cuts: $M_{jj}>500$ GeV; $\Delta\eta_{jj}>2.4$;
$p^{j}_{T}>20$ GeV; $|\eta_{j}|>4.5$

(a) $\mathcal{O}_{M_{0}}$

(b) $\mathcal{O}_{M_{1}}$

(c) $\mathcal{O}_{M_{7}}$

Figure 8: The plot shows the differential cross section as a function of the
invariant mass $m_{VV}$ of the two colliding vector bosons. The solid black
line is the standard model and the colored lines are the contributions of
$\mathcal{O}_{M_{0,1,7}}$ (dashed: naive EFT, solid: unitarized model). Cuts:
$M_{jj}>500$ GeV; $\Delta\eta_{jj}>2.4$; $p^{j}_{T}>20$ GeV; $|\eta_{j}|>4.5$

## References

  * [1] G. Aad et al. [ATLAS Collaboration], Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B 716, 1 (2012) doi:10.1016/j.physletb.2012.08.020 [arXiv:1207.7214 [hep-ex]].
  * [2] S. Chatrchyan et al. [CMS Collaboration], Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC, Phys. Lett. B 716, 30 (2012) doi:10.1016/j.physletb.2012.08.021 [arXiv:1207.7235 [hep-ex]].
  * [3] W. Buchmüller and D. Wyler, Effective Lagrangian Analysis of New Interactions and Flavor Conservation, Nucl. Phys. B 268 (1986) 621.
  * [4] K. Hagiwara, S. Ishihara, R. Szalapski and D. Zeppenfeld, Low-energy constraints on electroweak three gauge boson couplings, Phys. Lett. B 283 (1992) 353.
  * [5] K. Hagiwara, S. Ishihara, R. Szalapski and D. Zeppenfeld, Low-energy effects of new interactions in the electroweak boson sector, Phys. Rev. D 48 (1993) 2182.
  * [6] B. Grzadkowski, M. Iskrzynski, M. Misiak and J. Rosiek, Dimension-Six Terms in the Standard Model Lagrangian, JHEP 1010 (2010) 085 [arXiv:1008.4884 [hep-ph]].
  * [7] W. Kilian, T. Ohl, J. Reuter and M. Sekulla, High-Energy Vector Boson Scattering after the Higgs Discovery, Phys. Rev. D 91 (2015) 096007 [arXiv:1408.6207 [hep-ph]].
  * [8] W. Kilian, T. Ohl, J. Reuter and M. Sekulla, Resonances at the LHC beyond the Higgs boson: The scalar/tensor case, Phys. Rev. D 93, no. 3, 036004 (2016) [arXiv:1511.00022 [hep-ph]].
  * [9] C. Fleper, W. Kilian, J. Reuter and M. Sekulla, Scattering of W and Z Bosons at High-Energy Lepton Colliders, Eur. Phys. J. C 77, no. 2, 120 (2017) doi:10.1140/epjc/s10052-017-4656-5 [arXiv:1607.03030 [hep-ph]].
  * [10] B. Jäger, C. Oleari and D. Zeppenfeld, Next-to-leading order QCD corrections to $W^{+}W^{+}jj$ and $W^{-}W^{-}jj$ production via weak-boson fusion, Phys. Rev. D 80, 034022 (2009) doi:10.1103/PhysRevD.80.034022 [arXiv:0907.0580 [hep-ph]].
  * [11] T. Melia, K. Melnikov, R. Röntsch and G. Zanderighi, Next-to-leading order QCD predictions for $W^{+}W^{+}jj$ production at the LHC, JHEP 1012, 053 (2010) doi:10.1007/JHEP12(2010)053 [arXiv:1007.5313 [hep-ph]].
  * [12] T. Melia, K. Melnikov, R. Röntsch and G. Zanderighi, NLO QCD corrections for $W^{+}W^{-}$ pair production in association with two jets at hadron colliders, Phys. Rev. D 83, 114043 (2011) doi:10.1103/PhysRevD.83.114043 [arXiv:1104.2327 [hep-ph]].
  * [13] B. Jäger and G. Zanderighi, NLO corrections to electroweak and QCD production of $W^{+}W^{+}$ plus two jets in the POWHEGBOX, JHEP 1111, 055 (2011) doi:10.1007/JHEP11(2011)055 [arXiv:1108.0864 [hep-ph]].
  * [14] J. Baglio et al., Release Note - VBFNLO 2.7.0, arXiv:1404.3940 [hep-ph].
  * [15] B. Biedermann, A. Denner and M. Pellen, Large electroweak corrections to vector-boson scattering at the Large Hadron Collider, Phys. Rev. Lett. 118, no. 26, 261801 (2017) doi:10.1103/PhysRevLett.118.261801 [arXiv:1611.02951 [hep-ph]].
  * [16] A. Ballestrero et al., arXiv:1803.07943 [hep-ph].
  * [17] O. J. P. Eboli, M. C. Gonzalez-Garcia and J. K. Mizukoshi, $pp\to jje^{\pm}\mu^{\pm}\nu\nu$ and $jje^{\pm}\mu^{\mp}\nu\nu$ at $\mathcal{O}(\alpha_{em}^{6})$ and $\mathcal{O}(\alpha_{em}^{4}\alpha_{s}^{2})$ for the study of the quartic electroweak gauge boson vertex at CERN LHC, Phys. Rev. D 74, 073005 (2006) [hep-ph/0606118].
  * [18] C. Degrande, A basis of dimension-eight operators for anomalous neutral triple gauge boson interactions, JHEP 1402, 101 (2014) doi:10.1007/JHEP02(2014)101 [arXiv:1308.6323 [hep-ph]].
  * [19] M. Baak et al., Working Group Report: Precision Study of Electroweak Interactions, arXiv:1310.6708 [hep-ph].
  * [20] A. Perez, M. Sekulla and D. Zeppenfeld, Anomalous quartic gauge couplings and unitarization for the vector boson scattering process $pp\rightarrow W^{+}W^{+}jjX\rightarrow\ell^{+}\nu_{\ell}\ell^{+}\nu_{\ell}jjX$, KA-TP-15-2018
  * [21] A. Alboteanu, W. Kilian and J. Reuter, Resonances and Unitarity in Weak Boson Scattering at the LHC, JHEP 0811, 010 (2008) [arXiv:0806.4145 [hep-ph]].
  * [22] M. Sekulla, Anomalous couplings, resonances and unitarity in vector boson scattering, PhD thesis, University of Siegen (2015)
  * [23] E.P. Wigner, Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren, Vieweg Vlg, Braunschweig, 1931\.
  * [24] W. Kilian, T. Ohl and J. Reuter, WHIZARD: Simulating Multi-Particle Processes at LHC and ILC, Eur. Phys. J. C 71, 1742 (2011) [arXiv:0708.4233 [hep-ph]].
  * [25] M. Moretti, T. Ohl and J. Reuter, O’Mega: An Optimizing matrix element generator, In *2nd ECFA/DESY Study 1998-2001* 1981-2009 [hep-ph/0102195].
  * [26] B. Chokoufe Nejad, T. Ohl and J. Reuter, Comput. Phys. Commun. 196, 58 (2015) doi:10.1016/j.cpc.2015.05.015 [arXiv:1411.3834 [physics.comp-ph]].
  * [27] W. Kilian, T. Ohl, J. Reuter and C. Speckner, JHEP 1210, 022 (2012) [arXiv:1206.3700 [hep-ph]].
  * [28] W. Kilian, J. Reuter, S. Schmidt and D. Wiesler, JHEP 1204, 013 (2012) [arXiv:1112.1039 [hep-ph]].
  * [29] T. Ohl and J. Reuter, Eur. Phys. J. C 30, 525 (2003) doi:10.1140/epjc/s2003-01301-7 [hep-th/0212224].
  * [30] N. D. Christensen, C. Duhr, B. Fuks, J. Reuter and C. Speckner, Eur. Phys. J. C 72, 1990 (2012) [arXiv:1010.3251 [hep-ph]].
  * [31] F. Bach, B. C. Nejad, A. Hoang, W. Kilian, J. Reuter, M. Stahlhofen, T. Teubner and C. Weiss, JHEP 1803, 184 (2018) doi:10.1007/JHEP03(2018)184 [arXiv:1712.02220 [hep-ph]]; B. Chokoufé Nejad, W. Kilian, J. M. Lindert, S. Pozzorini, J. Reuter and C. Weiss, JHEP 1612, 075 (2016) doi:10.1007/JHEP12(2016)075 [arXiv:1609.03390 [hep-ph]]; B. Chokoufe Nejad, W. Kilian, J. Reuter and C. Weiss, PoS EPS -HEP2015, 317 (2015) [arXiv:1510.02739 [hep-ph]]; C. Weiss, B. Chokoufe Nejad, W. Kilian and J. Reuter, PoS EPS -HEP2015, 466 (2015) [arXiv:1510.02666 [hep-ph]]; N. Greiner, A. Guffanti, T. Reiter and J. Reuter, Phys. Rev. Lett. 107, 102002 (2011) [arXiv:1105.3624 [hep-ph]]; T. Binoth, N. Greiner, A. Guffanti, J. Reuter, J.-P. Guillet and T. Reiter, Phys. Lett. B 685, 293 (2010) [arXiv:0910.4379 [hep-ph]]; W. Kilian, J. Reuter and T. Robens, Eur. Phys. J. C 48, 389 (2006) [hep-ph/0607127].
  * [32] D. Buttazzo, A. Greljo and D. Marzocca, Knocking on new physics’ door with a scalar resonance, Eur. Phys. J. C 76 (2016) no.3, 116 doi:10.1140/epjc/s10052-016-3970-7 [arXiv:1512.04929 [hep-ph]].
  * [33] S. B. Giddings and H. Zhang, Kaluza-Klein graviton phenomenology for warped compactifications, and the 750 GeV diphoton excess, Phys. Rev. D 93 (2016) no.11, 115002 doi:10.1103/PhysRevD.93.115002 [arXiv:1602.02793 [hep-ph]].
  * [34] C. Han, H. M. Lee, M. Park and V. Sanz, The diphoton resonance as a gravity mediator of dark matter, Phys. Lett. B 755 (2016) 371 doi:10.1016/j.physletb.2016.02.040 [arXiv:1512.06376 [hep-ph]].
  * [35] L. Randall and R. Sundrum, A Large mass hierarchy from a small extra dimension, Phys. Rev. Lett. 83, 3370 (1999) doi:10.1103/PhysRevLett.83.3370 [hep-ph/9905221].
  * [36] N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, The Hierarchy problem and new dimensions at a millimeter, Phys. Lett. B 429, 263 (1998) doi:10.1016/S0370-2693(98)00466-3 [hep-ph/9803315].
  * [37] C. Q. Geng and D. Huang, Note on spin-2 particle interpretation of the 750 GeV diphoton excess, Phys. Rev. D 93 (2016) no.11, 115032 doi:10.1103/PhysRevD.93.115032 [arXiv:1601.07385 [hep-ph]].
  * [38] R. Pasechnik, V. Beylin, V. Kuksa and G. Vereshkov, Chiral-symmetric technicolor with standard model Higgs boson, Phys. Rev. D 88 (2013) no.7, 075009 doi:10.1103/PhysRevD.88.075009 [arXiv:1304.2081 [hep-ph]].
  * [39] Y. W. Yoon, K. Cheung, S. K. Kang and J. Song, Phys. Rev. D 96, no. 5, 055041 (2017) doi:10.1103/PhysRevD.96.055041 [arXiv:1705.05486 [hep-ph]].
  * [40] M. Bauer, C. Hörner and M. Neubert, Diphoton Resonance from a Warped Extra Dimension, JHEP 1607 (2016) 094 doi:10.1007/JHEP07(2016)094 [arXiv:1603.05978 [hep-ph]].
  * [41] R. Franceschini et al., What is the $\gamma\gamma$ resonance at 750 GeV?, JHEP 1603 (2016) 144 doi:10.1007/JHEP03(2016)144 [arXiv:1512.04933 [hep-ph]].
  * [42] R. S. Gupta, S. Jäger, Y. Kats, G. Perez and E. Stamou, Interpreting a 750 GeV Diphoton Resonance, JHEP 1607 (2016) 145 doi:10.1007/JHEP07(2016)145 [arXiv:1512.05332 [hep-ph]].
  * [43] J. S. Kim, J. Reuter, K. Rolbiecki and R. Ruiz de Austri, A resonance without resonance: scrutinizing the diphoton excess at 750 GeV, Phys. Lett. B 755, 403 (2016) doi:10.1016/j.physletb.2016.02.041 [arXiv:1512.06083 [hep-ph]].
  * [44] N. Arkani-Hamed, A. G. Cohen and H. Georgi, Electroweak symmetry breaking from dimensional deconstruction, Phys. Lett. B 513, 232 (2001) doi:10.1016/S0370-2693(01)00741-9 [hep-ph/0105239].
  * [45] N. Arkani-Hamed, A. G. Cohen, E. Katz and A. E. Nelson, The Littlest Higgs, JHEP 0207, 034 (2002) doi:10.1088/1126-6708/2002/07/034 [hep-ph/0206021].
  * [46] W. Kilian and J. Reuter, The Low-energy structure of little Higgs models, Phys. Rev. D 70, 015004 (2004) doi:10.1103/PhysRevD.70.015004 [hep-ph/0311095].
  * [47] D. R. Green, P. Meade and M. A. Pleier, Multiboson interactions at the LHC, Rev. Mod. Phys. 89, no. 3, 035008 (2017) doi:10.1103/RevModPhys.89.035008 [arXiv:1610.07572 [hep-ex]].
  * [48] M. Aaboud et al. [ATLAS Collaboration], Phys. Rev. D 95, no. 3, 032001 (2017) doi:10.1103/PhysRevD.95.032001 [arXiv:1609.05122 [hep-ex]].
  * [49] M. Sekulla, W. Kilian, T. Ohl and J. Reuter, PoS LHCP 2016, 052 (2016) [arXiv:1610.04131 [hep-ph]].
  * [50] A. M. Sirunyan et al. [CMS Collaboration], arXiv:1709.05822 [hep-ex].
  * [51] A. M. Sirunyan et al. [CMS Collaboration], Phys. Lett. B 774, 682 (2017) doi:10.1016/j.physletb.2017.10.020 [arXiv:1708.02812 [hep-ex]].