# Beyond the Standard Model Effective Field Theory:
The Singlet Extended Standard Model

Shekhar Adhikari<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of Kansas, Lawrence, Kansas, 66045 U.S.A.  Ian M. Lewis
<EMAIL_ADDRESS>Department of Physics and Astronomy, University of Kansas,
Lawrence, Kansas, 66045 U.S.A.  Matthew Sullivan<EMAIL_ADDRESS>Department of Physics and Astronomy, University of Kansas, Lawrence, Kansas,
66045 U.S.A.  High Energy Theory Group, Physics Department, Brookhaven
National Laboratory, Upton, New York, 11973 U.S.A.

###### Abstract

One of the assumptions of simplified models is that there are a few new
particles and interactions accessible at the LHC and all other new particles
are heavy and decoupled. The effective field theory (EFT) method provides a
consistent method to test this assumption. Simplified models can be augmented
with higher order operators involving the new particles accessible at the LHC.
Any UV completion of the simplified model will be able to match onto these
beyond the Standard Model EFTs (BSM-EFT). In this paper we study the simplest
simplified model: the Standard Model extended by a real gauge singlet scalar.
In addition to the usual renormalizable interactions, we include dimension-5
interactions of the singlet scalar with Standard Model particles. As we will
show, even when the cutoff scale is $3$ TeV, these new effective interactions
can drastically change the interpretation of Higgs precision measurements and
scalar searches. In addition, we discuss how power counting in a BSM-EFT
depends strongly on the processes and parameter space under consideration.
Finally, we propose a $\chi^{2}$ method to consistently combine the limits
from new particle searches with measurements of the Standard Model. Unlike
imposing a hard cutoff on heavy resonance rates, our method allows
fluctuations in individual channels that are consistent with global fits.

##  I Introduction

The Large Hadron Collider (LHC) has had two very successful runs. While no new
physics beyond the Standard Model (BSM) has been discovered, we may yet expect
it to show up in currently unanalyzed data or in future runs at the LHC. In
the absence of discoveries of more complete models such as Supersymmetry,
extra dimensions, or composite Higgs models, it is useful to study simplified
models Alves:2011wf . A frequent assumption of simplified models is that there
are at most a handful of new particles accessible at LHC energies, while all
additional new particles are too heavy to be produced. However, this raises
the question: can the effects of the inaccessible new particles be truly
neglected? For example, consider a simplified model with a new up-type vector
like quark (VLQ). If there is a new scalar in the theory, even if the scalar
cannot be directly produced, it can mediate new loop level decays of the VLQ
into photons and gluons Kim:2018mks . Indeed, in certain regions of parameter
space, these decay modes can be dominant Kim:2018mks ; Alhazmi:2018whk ;
Criado:2019mvu , fundamentally changing the phenomenology of the simplified
VLQ model.

The most “model independent” method to determine the effects of new, heavy
particles is an effective field theory (EFT). An EFT is a power expansion in
inverse powers of some new physics scale $\Lambda$:

$\displaystyle\mathcal{L}=\mathcal{L}_{\rm
ren}+\sum_{n=5}^{\infty}\sum_{k}\frac{f_{k,n}}{\Lambda^{n-4}}\mathcal{O}_{k,n},$
(1)

where $f_{k,n}$ are Wilson coefficients, $\mathcal{L}_{\rm ren}$ is the
renormalizable Lagrangian, and $\mathcal{O}_{k,n}$ are dimension-$n$
operators. In the Standard Model EFT (SMEFT) Buchmuller:1985jz ;
Grzadkowski:2010es ; Brivio:2017vri , $\mathcal{L}_{\rm ren}$ and
$\mathcal{O}_{k,n}$ consist of SM fields and are invariant under SM
symmetries. To test the stability of simplified models against heavy new
physics, this framework needs to be extended to the Beyond the Standard Model
EFT (BSM-EFT) Kim:2018mks ; Alhazmi:2018whk ; Criado:2019mvu ; Dawson:2016ugw
; Bauer:2016hcu ; Carmona:2016qgo ; Anisha:2019nzx ; Karmakar:2019vnq ;
Crivellin:2016ihg ; DiazCruz:2001tn ; Bar-Shalom:2018ure ; Chala:2017sjk ;
Alanne:2017oqj ; Goertz:2019vht ; Nagai:2014cua ; Nagai:2019tgi . In the BSM-
EFT, $\mathcal{L}_{\rm ren}$ and $\mathcal{O}_{k,n}$ consist of SM and
simplified model fields and are invariant under the symmetries of the
simplified model. This approach is agnostic about the high scale new physics
since any UV completion of a simplified model will match onto the BSM-EFT.

In this paper we study the BSM-EFT of the simplest possible extension of the
SM, the addition of a real scalar singlet $S$ OConnell:2006rsp ; Barger:2007im
; Bowen:2007ia . Beyond being the simplest extension of the SM, the singlet
model can help provide a strong first order electroweak phase transition
necessary of electroweak baryogenesis Choi:1993cv ; Profumo:2007wc ;
Espinosa:2011ax ; Curtin:2014jma ; Chen:2017qcz . At the renormalizable level,
the new singlet only enters the scalar potential, and its interactions with
fermions and gauge bosons are inherited by its mixing with the SM Higgs boson.
However, it is highly unlikely that a singlet scalar would appear without any
new physics. For example, even if it can give rise to a strong first order
electroweak phase transition, in order to successfully have electroweak
baryogenesis, new sources of CP violation are needed Xiao:2015tja ;
Cline:2017jvp ; Chao:2017oux ; Bell:2019mbn . In fact, it has been shown
Espinosa:2011eu ; Cline:2012hg ; Huang:2018aja  that the BSM-EFT for the real
scalar singlet can provide the CP violation necessary for electroweak
baryogenesis.

Our analysis will consist of two major portions: reinterpreting Higgs
precision measurements in the singlet extended SM and reinterpreting searches
for new heavy scalars. After electroweak symmetry breaking (EWSB), the new
singlet scalar and Higgs boson will mix. Without the new EFT interactions,
this mixing results in a universal suppression of Higgs boson production
rates. Hence, Higgs precision measurements have a very simple interpretation
Robens:2015gla ; Buttazzo:2015bka ; Robens:2016xkb ; Lewis:2017dme ;
Ilnicka:2018def ; Martin-Lozano:2015dja . However, the BSM-EFT will introduce
new interactions between the Higgs boson and fermions/gauge bosons. As we will
show, these can significantly alter the interpretation of Higgs measurements.
A similar argument can be made for constraints coming from heavy scalar
searches. At the renormalizable level, the new scalar inherits all of its
interactions with fermions and gauge bosons from the SM Higgs boson. Hence,
its production rates are the same as a heavy Higgs boson but suppressed by a
mixing angle. Similarly, its decay rates are the same as a heavy Higgs boson
suppressed by a mixing angle, except when a di-Higgs resonance is
kinematically available. That is, at the renormalizable model, the
phenomenology is well defined. As we will show, with the introduction of new
interactions between the scalar and fermions/gauge bosons the phenomenology
can significantly change. Even though it is typically assumed that heavy new
physics can be neglected, we will show that even in the simplest of all
simplified models this assumption must be called into question.

This paper is an extension of work in Ref. Dawson:2016ugw , where only
effective interactions between the scalar singlet and gauge bosons were
considered. We should note that the full BSM-EFT was considered in Ref.
Bauer:2016hcu . However, they also considered dimension-6 SMEFT operators.
While these effects can be important, we are interested in the question of how
the EFT including new particles can change the phenomenology of the simplified
models. Hence, we will focus on dimension-5 operators involving SM gauge
bosons, SM fermions, and the new scalar singlet. In addition, we will include
the most up-to-date Higgs precision data and searches for scalar singlets.
Also, we give a robust discussion of power counting the BSM-EFT and propose a
new $\chi^{2}$ analysis to combine heavy resonance searches with precision
measurements.

In Section II, we develop the BSM-EFT for the real scalar singlet. The EFT
power counting in a BSM-EFT can change from the usual SMEFT power counting, as
we will discuss in Sec. III. The effects of the new operators on Higgs
production and decay are shown in Sec. IV, and results from fitting to Higgs
signal strengths are given in Sec. V. In Sec. VI.1, we propose a $\chi^{2}$
analysis for heavy resonance search limits, and in Sec. VI.2, the final
results of heavy scalar resonances and their combination with Higgs signal
strengths are given. We conclude in Sec. VII. A discussion about assumptions
in Higgs signal rate calculations is given in App. A, the Feynman rules are
given in Appendix B, the experimental results we fit to are given in App. C,
and various parameter space limits are given in App. D.

##  II Model

We consider the SM extended by a real gauge singlet scalar, $S$, and will not
impose an additional $Z_{2}$ upon $S$. In order to focus on the effects of new
physics on the scalar singlet properties, we will consider only dimension-5
EFT operators. For simplicity, we will also only focus on CP even operators.
These are the lowest order effective operators that include a scalar singlet
Dawson:2016ugw ; Bauer:2016hcu . At dimension-5, the only SMEFT operators are
those that contribute to Majorana neutrino masses Weinberg:1979sa ;
Buchmuller:1985jz ; Grzadkowski:2010es , which are not relevant for LHC
analyses. Hence, these will be neglected and the BSM-EFT will only consist of
operators including the new singlet scalar.

Adapting the notation of Refs. Chen:2014ask , to order $\Lambda^{-1}$ the
scalar potential is:

$\displaystyle V(\Phi,S)$ $\displaystyle=$
$\displaystyle-\mu^{2}\Phi^{\dagger}\Phi+\lambda(\Phi^{\dagger}\Phi)^{2}+\frac{a_{1}}{2}\Phi^{\dagger}\Phi
S+\frac{a_{2}}{2}\Phi^{\dagger}\Phi
S^{2}+\frac{a_{3}}{2\Lambda}\Phi^{\dagger}\Phi
S^{3}+\frac{a_{4}}{2\Lambda}(\Phi^{\dagger}\Phi)^{2}S$ (2)
$\displaystyle+b_{1}S+\frac{b_{2}}{2}S^{2}+\frac{b_{3}}{3}S^{3}+\frac{b_{4}}{4}S^{4}+\frac{b_{5}}{5\Lambda}S^{5}$

where $\Phi=(0,\phi_{0}/\sqrt{2})^{T}$ is the SM Higgs doublet in the unitary
gauge, $\phi_{0}=h+v$ is the neutral scalar component of $\Phi$, $h$ is the
Higgs boson, and $\langle\phi_{0}\rangle=v$ is the SM Higgs vacuum expectation
value (vev). Since $S$ is not charged under any symmetry, its vev does not
break any symmetry and results in an unphysical redefinition of parameters
Chen:2014ask ; Lewis:2017dme . Hence, without loss of generality we can impose
$\langle S\rangle=0$.

After EWSB, the Higgs boson $h$ and scalar singlet have the same quantum
numbers and can mix:

$\displaystyle\begin{pmatrix}h_{1}\\\
h_{2}\end{pmatrix}=\begin{pmatrix}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{pmatrix}\begin{pmatrix}h\\\ S\end{pmatrix},$ (3)

where $h_{1,2}$ are mass eigenstates with masses $m_{1,2}$. We will assume
$m_{1}=125~{}{\rm GeV}<m_{2}$, since the other mass hierarchy is strongly
constrained by LEP Robens:2015gla . With the masses, mixing, and vevs, we can
now solve for five parameters in the potential

$\displaystyle\mu^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\cos^{2}\theta\,m^{2}_{1}+\sin^{2}\theta\,m_{2}^{2}\right)\,,$
(4) $\displaystyle\lambda$ $\displaystyle=$
$\displaystyle\frac{\cos^{2}\theta\,m_{1}^{2}+\sin^{2}\theta\,m_{2}^{2}}{2\,v^{2}}\,,$
$\displaystyle a_{1}$ $\displaystyle=$ $\displaystyle\sin
2\theta\,\frac{m_{1}^{2}-m_{2}^{2}}{v}-a_{4}\frac{v^{2}}{\Lambda}\,,$
$\displaystyle b_{1}$ $\displaystyle=$ $\displaystyle\frac{1}{4}\sin
2\theta\,v\,\left(m_{2}^{2}-m_{1}^{2}\right)+a_{4}\frac{v^{4}}{8\,\Lambda}\,,$
$\displaystyle b_{2}$ $\displaystyle=$
$\displaystyle\cos^{2}\theta\,m_{2}^{2}+\sin^{2}\theta\,m_{1}^{2}-\frac{1}{2}a_{2}\,v^{2}.$

These are $\mathcal{O}(v/\Lambda)$ corrections on the relationships founds in
Refs. Chen:2014ask ; Lewis:2017dme . The free parameters of the scalar
potential are then

$\displaystyle m_{1}=125~{}{\rm GeV},\,m_{2},\,v=246~{}{\rm GeV},\,\langle
S\rangle=0,\,\theta,\,a_{2},\,a_{3},\,a_{4},\,b_{3},\,b_{4},\,b_{5}.$ (5)

The scalar potential gives rise to important trilinear scalar couplings after
EWSB:

$\displaystyle
V(h_{1},h_{2})\supset\frac{1}{3!}\lambda_{111}h_{1}^{3}+\frac{1}{2}\lambda_{211}h_{1}^{2}h_{2},$
(6)

where

$\displaystyle\lambda_{111}$ $\displaystyle=$
$\displaystyle\frac{3\,m_{1}^{2}}{v}\cos^{3}\theta+2\,b_{3}\sin^{3}\theta+3\,a_{2}v\cos\theta\sin^{2}\theta+\frac{3\,a_{3}v^{2}}{2\,\Lambda}\sin^{3}\theta+\frac{3\,a_{4}v^{2}}{\Lambda}\cos^{2}\theta\sin\theta,$
(7) $\displaystyle\lambda_{211}$ $\displaystyle=$
$\displaystyle-\frac{m_{2}^{2}+2\,m_{1}^{2}}{v}\cos^{2}\theta\sin\theta+2\,b_{3}\cos\theta\sin^{2}\theta+a_{2}v\sin\theta\left(2\cos^{2}\theta-\sin^{2}\theta\right)$
(8)
$\displaystyle+\frac{3\,a_{3}v^{2}}{2\,\Lambda}\cos\theta\sin^{2}\theta+\frac{a_{4}v^{2}}{\Lambda}\cos\theta\left(\cos^{2}\theta-2\sin^{2}\theta\right).$

When kinematically allowed, the coupling $\lambda_{211}$ gives rise to
resonant double Higgs production via the decay $h_{2}\rightarrow h_{1}h_{1}$.
The Higgs trilinear coupling $\lambda_{111}$ can alter the nonresonant di-
Higgs rate away from SM predictions.

There are important theoretical constraints on the scalar potential, Eq. (2),
to consider. Limits from the potential affect the allowed values of
$\lambda_{211}$ and can have a significant impact on the $h_{2}\rightarrow
h_{1}h_{1}$ branching ratio Chen:2014ask . First, there are quintic terms
$S(\Phi^{\dagger}\Phi)^{2}$, $S^{3}\Phi^{\dagger}\Phi$ and $S^{5}$ that
dominate at large field values and can be negative, indicating an unstable
potential. We only consider parameter space where the global minimum is inside
the field value region $|S|<\Lambda$ and $|\phi_{0}|<\Lambda$ and not along
the boundaries. Above the cutoff scale, it assumed new physics comes in and
stabilizes the potential. Second, the potential is much more complicated than
the SM and has many different minimum even inside the allowed field value
regions. The singlet vev cannot contribute to the $W$ and $Z$ masses. Hence,
the Higgs vev must give the correct masses and we only consider parameter
space where the global minimum is $\langle\phi_{0}\rangle=v=246$ GeV and
$\langle S\rangle=0$. Finally, in the scalar potential we require all
dimensionless parameters to be bounded by $4\pi$ and all dimensionful
parameters to be bounded by $\Lambda$.

In addition to the scalar potential, the scalar singlet obtains new
interactions with SM fermions and gauge bosons Dawson:2016ugw ; Bauer:2016hcu
; Carmona:2016qgo . Current measurements of the observed Higgs boson are only
sensitive to third generation quarks, and second and third generation leptons.
Hence, we will only consider those interactions in addition to the gauge
bosons. The relevant effective operators in the fermion mass eigenbasis are
then:

$\displaystyle\mathcal{L}_{EFT}$ $\displaystyle\supset$
$\displaystyle\frac{g_{s}^{2}}{16\pi^{2}}\frac{f_{GG}}{\Lambda}S\,G_{\mu\nu}^{A}G^{A,\mu\nu}+\frac{g^{2}}{16\pi^{2}}\frac{f_{WW}}{\Lambda}S\,W_{\mu\nu}^{a}W^{a,\mu\nu}+\frac{{g^{\prime}}^{2}}{16\pi^{2}}\frac{f_{BB}}{\Lambda}S\,B_{\mu\nu}B^{\mu\nu}$
$\displaystyle-\sqrt{2}\left(\frac{f_{\mu}}{\Lambda}\frac{m_{\mu}}{v}S\,\overline{L}_{2}\Phi\,\mu_{R}+\frac{f_{\tau}}{\Lambda}\frac{m_{\tau}}{v}S\,\overline{L}_{3}\Phi\,\tau_{R}+\frac{f_{b}}{\Lambda}\frac{m_{b}}{v}S\,\overline{Q}_{3}\Phi\,b_{R}+\frac{f_{t}}{\Lambda}\frac{m_{t}}{v}S\,\overline{Q}_{3}\widetilde{\Phi}\,t_{R}+{\rm
h.c.}\right),$

where $L_{2,3}$ are second and third generation lepton $SU(2)_{L}$ doublets,
$Q_{3}$ is the third generation quark $SU(2)_{L}$ doublet,
$\mu_{R},\,\tau_{R},\,b_{R},\,t_{R}$ are $SU(2)_{L}$ singlets, and
$m_{\mu},m_{\tau},m_{b},m_{t}$ are the masses of the relevant fermions. All
Wilson coefficients are assumed to be real. The Feynman rules from Eqs. (2,II)
can be found in Appendix B.

##  III Power Counting

In traditional SMEFT counting, the amplitude squared terms should be truncated
to the same order as the Lagrangian. As an example, consider a baryon and
lepton number conserving SMEFT amplitude to dimension-8111In SMEFT,
dimension-5 and dimension-7 operators violate lepton and/or baryon number
Weinberg:1979sa ; Degrande:2012wf ; Lehman:2014jma ; Henning:2015alf ;
Kobach:2016ami .

$\displaystyle\mathcal{A}_{\rm SMEFT}\sim\mathcal{A}_{\rm
ren}+\frac{1}{\Lambda^{2}}\mathcal{A}_{6,{\rm
SMEFT}}+\frac{1}{\Lambda^{4}}\mathcal{A}_{8,{\rm
SMEFT}}+\mathcal{O}(\Lambda^{-6}),$ (10)

where $\mathcal{A}_{\rm ren}$ is the dimension-4 renormalizable amplitude, and
$\mathcal{A}_{n,SMEFT}$ are SMEFT amplitudes originating from operators at
dimension-$n$. The amplitude squared is then

$\displaystyle|\mathcal{A}_{\rm SMEFT}|^{2}\sim|\mathcal{A}_{\rm
ren}|^{2}+\frac{1}{\Lambda^{2}}\mathcal{A}_{\rm ren}\mathcal{A}_{6,{\rm
SMEFT}}+\frac{1}{\Lambda^{4}}|\mathcal{A}_{6,{\rm
SMEFT}}|^{2}+\frac{1}{\Lambda^{4}}\mathcal{A}_{\rm ren}\mathcal{A}_{8,{\rm
SMEFT}}+\mathcal{O}(\Lambda^{-6}).$ (11)

As can be clearly seen, at the amplitude squared level, the dimension-8 term
is of the same order as the dimension-6 squared term. Hence, for self-
consistency, if only the dimension-6 term is included in the amplitude, then
the amplitude squared should also be truncated at $\mathcal{O}(\Lambda^{-2})$.

According to this argument, since the interactions in Eqs. (2,II) are
truncated at dimension-5 the squared amplitudes should be truncated at
$\mathcal{O}(\Lambda^{-1})$. Here we note that while this is the SMEFT
procedure, in the model presented the counting is more complicated due to the
unknown scalar mixing angle. First, consider $h_{1}$ single production and
decay. The relevant singlet scalar interactions are all dimension-5 or higher.
Hence, to order $\Lambda^{-2}$, amplitudes for $h_{1}$ production and decay
are schematically

$\displaystyle\mathcal{A}_{h_{1}}\sim\cos\theta\mathcal{A}_{\rm
ren}+\cos\theta\,\frac{\mathcal{A}_{6,{\rm
SMEFT}}}{\Lambda^{2}}+\sin\theta\left(\frac{\mathcal{A}_{5,S}}{\Lambda}+\frac{\mathcal{A}_{6,S}}{\Lambda^{2}}\right)+\mathcal{O}(\Lambda^{-3}),$
(12)

where $\mathcal{A}_{5,S}$ and $\mathcal{A}_{6,S}$ are, respectively,
dimension-5 and dimension-6 operators involving the scalar singlet $S$. Note
that due to mixing among the scalars after EWSB, in the production and decay
of the mass eigenstate $h_{1}$, the SMEFT and renormalizable terms are
proportional to $\cos\theta$ and the singlet scalar EFT terms are proportional
to $\sin\theta$. The amplitude squared is then

$\displaystyle|\mathcal{A}_{h_{1}}|^{2}$ $\displaystyle\sim$
$\displaystyle\cos^{2}\theta|\mathcal{A}_{\rm
ren}|^{2}+\sin\theta\,\cos\theta\,\frac{\mathcal{A}_{\rm
ren}\mathcal{A}_{5,S}}{\Lambda}$
$\displaystyle+\frac{1}{\Lambda^{2}}\left(\sin^{2}\theta|\mathcal{A}_{5,S}|^{2}+\sin\theta\cos\theta\mathcal{A}_{\rm
ren}\mathcal{A}_{6,S}+\cos^{2}\theta\mathcal{A}_{\rm ren}\mathcal{A}_{6,{\rm
SMEFT}}\right)+\mathcal{O}(\Lambda^{-3}).$

In the small mixing angle limit, the SM and SMEFT contributions dominate, and
the usual power counting is valid. In the large mixing angle limit,
$\sin\theta\rightarrow\pm 1$, the $\cos\theta$ terms go to zero and the
amplitude squared is

$\displaystyle|\mathcal{A}_{h_{1}}|^{2}\xrightarrow[|\sin\theta|\rightarrow
1]{}\frac{|\mathcal{A}_{5,S}|^{2}}{\Lambda^{2}}+\frac{\mathcal{A}_{6,S}\mathcal{A}_{5,S}}{\Lambda^{3}}+\mathcal{O}(\Lambda^{-4}).$
(14)

Hence, the dimension-5 squared piece dominates the dimension-6 terms. That is,
in the large mixing angle limit, we can take the full dimension-5 amplitude
squared and not violate power counting rules.

For $h_{2}$ single production and decay the relevant singlet scalar
renormalizable interaction comes from the potential and induces
$h_{2}\rightarrow h_{1}h_{1}$ when kinematically allowed. This process depends
on $\lambda_{211}$. From Eq. (8), it is clear that the renormalizable piece of
$\lambda_{211}$ is proportional to $\sin\theta$. Hence, all renormalizable
contributions to $h_{2}$ single production and decay are proportional to
$\sin\theta$ and the amplitude is schematically

$\displaystyle\mathcal{A}_{h_{2}}\sim\sin\theta\mathcal{A}_{\rm
ren}+\sin\theta\,\frac{\mathcal{A}_{6,{\rm
SMEFT}}}{\Lambda^{2}}+\cos\theta\left(\frac{\mathcal{A}_{5,S}}{\Lambda}+\frac{\mathcal{A}_{6,S}}{\Lambda^{2}}\right)+\mathcal{O}(\Lambda^{-3}).$
(15)

Now, in the large mixing angle limit $\sin\theta\rightarrow\pm 1$, the
amplitude becomes SM-like and the SMEFT power counting is correct. While in
the small mixing angle limit the dimension-5 term is the leading term, similar
to Eq. (12). Hence, the leading term in the amplitude squared is the
dimension-5 squared piece and the full dimension-5 amplitude squared does not
violate power counting. Note that $h_{2}h_{2}$ production depends on
$\lambda_{221}$ which is not $\sin\theta$ mixing angle suppressed as shown in
Eq. (43) and studied in Ref. Chen:2017qcz . That is $h_{2}h_{2}$ production
the power counting changes again.

As this discussion makes clear, the power counting in BSM-EFT depends
intimately on the what parameter space is being considered and exactly what
processes are under consideration. We expect that LHC limits will force this
model into the small mixing angle limit. Hence, to test the validity of the
EFT, for Higgs precision measurements we will compare
$\mathcal{O}(\Lambda^{-1})$ rates to $\mathcal{O}(\Lambda^{-2})$ rates. For
scalar singlet searches we will always keep rates at
$\mathcal{O}(\Lambda^{-2})$.

##  IV $h_{1}$ Production and Decay

After mixing with the singlet scalar, the observed Higgs boson $h_{1}$ obtains
additional, BSM-EFT couplings to gauge bosons and fermions via Eq. (II). These
additional couplings will change the partial widths of $h_{1}$. In this
section, we show the numerical dependence of the relevant branching ratios on
the various Wilson coefficients.

The total width of $h_{1}$ is

$\displaystyle\Gamma_{1}$ $\displaystyle=$
$\displaystyle\Gamma(h_{1}\rightarrow b\overline{b})+\Gamma(h_{1}\rightarrow
c\overline{c})+\Gamma(h_{1}\rightarrow gg)$
$\displaystyle+\Gamma(h_{1}\rightarrow\gamma\gamma)+\Gamma(h_{1}\rightarrow
W^{\pm}W^{\mp,*})+\Gamma(h_{1}\rightarrow ZZ^{*})$
$\displaystyle+\Gamma(h_{1}\rightarrow\tau^{+}\tau^{-})+\Gamma(h_{1}\rightarrow\mu^{+}\mu^{-}).$

Higher order QCD corrections are included in the numerical studies. The
partial widths $\Gamma(h_{1}\rightarrow b\overline{b})$ and
$\Gamma(h_{1}\rightarrow c\overline{c})$ are calculated to next-to-next-to-
leading order (NNLO) in QCD Gorishnii:1990zu ; Chetyrkin:1996sr ;
Chetyrkin:1996ke ; Chetyrkin:1997vj ; Djouadi:2005gi ;
$\Gamma(h_{1}\rightarrow\gamma\gamma)$ Zheng:1990qa ; Djouadi:1990aj ;
Dawson:1992cy ; Djouadi:1993ji ; Melnikov:1993tj ; Inoue:1994jq ; Spira:1995rr
; Fleischer:2004vb ; Harlander:2005rq ; Anastasiou:2006hc ; Aglietti:2006tp
is calculated at NLO with the exact top mass effectsSpira:1997dg ; and
$\Gamma(h_{1}\rightarrow gg)$ Djouadi:1991tka ; Dawson:1990zj ; Spira:1995rr
and $\Gamma(h_{1}\rightarrow Z\gamma)$ Spira:1991tj ; Bonciani:2015eua ;
Kara:2015oes ; Gehrmann:2015dua  are calculated to NLO in QCD Djouadi:2005gi
by reweighting the exact LO quark loop amplitudes, including all quark mass
effects, by the NLO top loop amplitudes calculated in the infinite top quark
mass limit:

$\displaystyle\mathcal{A}_{q,NLO}=\mathcal{A}_{q,LO}\frac{\mathcal{A}_{NLO,m_{t}\rightarrow\infty}}{\mathcal{A}_{LO,m_{t}\rightarrow\infty}},$
(17)

where $\mathcal{A}_{q,NLO}$ are the quark loop amplitudes we use in our fits,
$\mathcal{A}_{q,LO}$ are the exact LO quark loop amplitudes including all
quark mass effects, and
$\mathcal{A}_{LO,m_{t}\rightarrow\infty},\mathcal{A}_{NLO,m_{t}\rightarrow\infty}$
are the LO and NLO top loop amplitudes, respectively, including the top quark
contributions calculated in the infinite top quark mass limit. For
$h_{1}\rightarrow gg$ there is also a three point contribution, $g-g-h_{1}$,
from the BSM-EFT. The NLO correction differs from the $m_{t}\rightarrow\infty$
limit of the SM by $(1+11\alpha_{s}/4\pi)$ Djouadi:1991tka ; Dawson:1990zj ;
Spira:1995rr ; deBlas:2018tjm , which we take into account. Finally, for loop
level decays $\Gamma(h_{1}\rightarrow Z\gamma)$ and
$\Gamma(h_{1}\rightarrow\gamma\gamma)$ we include contributions from
$t,b,c,\tau,\mu$ and $W$, while for $\Gamma(h_{1}\rightarrow gg)$ we include
$t,b$ and $c$.

In addition to the Higgs boson mass, our input parameters are the same as the
LHC Higgs Cross Section Working Group deFlorian:2016spz :

$\displaystyle G_{F}=1.16637\times 10^{-5}~{}{\rm
GeV}^{-2},\,M_{W}=80.35797~{}{\rm GeV},\,M_{Z}=91.15348~{}{\rm GeV},$
$\displaystyle m_{t}^{OS}=173~{}{\rm
GeV},\,\overline{m}_{b}(\overline{m}_{b})=4.18~{}{\rm
GeV},\,\overline{m}_{c}(3~{}{\rm GeV})=0.986~{}{\rm GeV},$ $\displaystyle
m_{\tau}=1.77682~{}{\rm GeV},\,m_{\mu}=0.1056583715~{}{\rm GeV},$
$\displaystyle\alpha_{s}(M_{Z})=0.118,$

where bars indicate $\overline{\rm MS}$ parameters, the superscript $OS$
indicates parameters evaluated in the on-shell scheme, and masses inside
parentheses indicate the renormalization scale at which the parameters are
set.

For calculating $h_{1}$ decay rates, we use one renormalization scale for all
parameters, including running quark masses. We set the renormalization scale
to the Higgs mass for tree level processes, while for loop level processes we
set the renormalization scale to half of the Higgs mass. For partial widths to
quarks, light quark masses are evaluated in the $\overline{\rm MS}$ scheme.
For the partial widths $\Gamma(h_{1}\rightarrow Z\gamma)$ and
$\Gamma(h_{1}\rightarrow gg)$, pole masses for all quarks are used, while for
the loop level $\Gamma(h_{1}\rightarrow\gamma\gamma)$, the running
$\overline{\rm MS}$ masses normalized to the pole masses are usedSpira:1997dg
. For this purpose, we use the relationship between the the on-shell and
$\overline{\rm MS}$ masses in Ref. Djouadi:2005gi .

Figure 1:  Dependence of various branching ratios of $h_{1}$ and $h_{1}$ total
width normalized to their SM values as a function of the gauge boson Wilson
coefficients (a) $f_{BB}$, (b) $f_{WW}$, and (c) $f_{GG}$. All other Wilson
coefficients are set to zero. The partial widths are calculated at both
(solid) $\mathcal{O}(\Lambda^{-1})$ and (dotted) $\mathcal{O}(\Lambda^{-2})$.
Different colors indicate different final states.

Figure 2:  Same as Fig. 1 for the fermion Wilson coefficients (a) $f_{b}$, (b)
$f_{\tau}$, and (c) $f_{t}$.

In Figs. 1 and 2, we show the dependence of various Higgs branching ratios and
the total Higgs width on the gauge boson and fermion Wilson coefficients with
a scalar mixing angle of $\sin\theta=0.1$. We consider one Wilson coefficient
at a time and show branching ratios for which the Wilson coefficients make a
direct contribution to the partial widths. Additionally, deviations in the
total width are important in fits to Higgs precision data since $\Gamma_{1}$
enters all Higgs branching ratios. Hence, we also show the dependence of the
total width on the Wilson coefficients.

From Fig. 1, it is clear that the $WW$ and $ZZ$ partial widths have very
little dependence on the Wilson coefficients $f_{BB}$ and $f_{WW}$. This can
be understood by noting that these decays are tree level in the SM, while the
EFT contributions are suppressed by a loop factor, a small mixing angle, and a
heavy scale. The situation changes for SM loop level decays. Both
$h_{1}\rightarrow\gamma\gamma$ and $h_{1}\rightarrow Z\gamma$ depend strongly
on $f_{BB}$ and $f_{WW}$, with deviations from SM predictions up to $15\%$.
Similarly, $h_{1}\rightarrow gg$ strongly depends on $f_{GG}$, with order one
deviations from the SM. Finally, the total width has little dependence on
$f_{BB}$ and $f_{WW}$ since $\gamma\gamma$ and $Z\gamma$ have negligible
contributions to $\Gamma_{1}$. However, $\Gamma_{1}$ depends more strongly on
$f_{GG}$ due to the larger $h_{1}\rightarrow gg$ partial width. These results
are consistent with Ref. Dawson:2016ugw .

The branching ratios into fermionic final states depend strongly on the
fermion Wilson coefficients, as evidenced in Figs. 2(a,b). The decay to
$\tau\tau$ varies as much as $\sim 30\%$ from SM predictions, while the bottom
quark final state varies by around $\sim 20\%$. It is striking that
$h_{1}\rightarrow bb$ depends less on $f_{b}$ than $h_{1}\rightarrow\tau\tau$
depends on $f_{\tau}$. This can be understood by noting that the total width
of $h_{1}$ depends strongly on $f_{b}$, but very little on $f_{\tau}$. Hence,
the variation in the $h_{1}\rightarrow bb$ partial width is somewhat
compensated by the variation in $\Gamma_{1}$. Similarly, while the partial
width of $h_{1}\rightarrow gg$ has little dependence on $f_{b}$, the ${\rm
BR}(h_{1}\rightarrow bb)$ varies up to $\sim 20-30\%$ due to the variation in
the total width. Finally, all the loop level processes depend relatively
strongly on the top quark Wilson coefficient, as seen in Fig. 2(c). In
particular, ${\rm BR}(h_{1}\rightarrow\gamma\gamma)$ and ${\rm
BR}(h_{1}\rightarrow gg)$ vary upward of $\sim 15\%$ and $\sim 30$%,
respectively.

Finally, the branching ratios and widths calculated to (solid)
$\mathcal{O}(\Lambda^{-1})$ and (dotted) $\mathcal{O}(\Lambda^{-2})$ are shown
in Figs. 1 and 2. For most final states and the total width, both the
$\mathcal{O}(\Lambda^{-1})$ and $\mathcal{O}(\Lambda^{-2})$ results agree
well. This indicates that the BSM-EFT is valid in these regions of parameter
space. The only exception is the dependence of $h_{1}\rightarrow gg$ on
$f_{GG}$. However, as we will show in the next section, the fits to the Higgs
precision data also indicate the BSM-EFT is valid in the allowed parameter
regions.

While we do not explicitly show the variation of the Higgs production cross
section, it should be noted that gluon fusion is the main production mode. For
on-shell $h_{1}$ decay, the LO gluon fusion production rate is

$\displaystyle\sigma_{ggF}(pp\rightarrow
h_{1})=\frac{\pi^{2}}{8\,m_{1}\,S}\,L\,\Gamma(h_{1}\rightarrow gg),$ (18)

where the parton luminosity is

$\displaystyle
L=\int_{\ln(\sqrt{\tau_{0}})}^{-\ln(\sqrt{\tau_{0}})}dy\,g(\sqrt{\tau_{0}}e^{y})\,g(\sqrt{\tau_{0}}e^{-y}),$
(19)

where $\sqrt{S}$ is the hadronic center-of-momentum energy and
$\tau_{0}=m_{1}^{2}/S$. As shown in Figs. 1(c) and 2(c), this production rate
will have a strong dependence on $f_{GG}$ and $f_{t}$. Other subdominant but
important production modes are Higgs production in association with $W/Z$
($Wh_{1}/Zh_{1}$) and vector boson fusion (VBF). The relevant Wilson
coefficients for these production modes are $f_{BB}$ and $f_{WW}$. However, as
evidenced in Figs. 1(a,b) $Wh_{1}$, $Zh_{1}$, and VBF production will have
little dependence on $f_{BB}$ and $f_{WW}$.

##  V Higgs Signal Strengths

Now we perform a fit to the Higgs precision data. The effects of the
additional interactions on the Higgs measurements are parameterized using
Higgs signal strengths:

$\displaystyle\mu_{i}^{f}=\frac{\sigma_{i}(pp\rightarrow
h_{1})}{\sigma_{i,SM}(pp\rightarrow h_{1})}\frac{{\rm BR}(h_{1}\rightarrow
f)}{{\rm BR}_{SM}(h_{1}\rightarrow f)},$ (20)

where $i$ is the initial state, $f$ is the final state, and the subscript $SM$
indicates SM values. We combine the signal strengths into a chi-square:

$\displaystyle\chi^{2}_{h_{1}}=\sum_{i,j}\frac{(\mu_{i}^{f}-\hat{\mu}_{i}^{f})^{2}}{(\delta_{i}^{f})^{2}},$
(21)

where $\mu_{i}^{f}$ is a calculated signal strength, $\hat{\mu}_{i}^{f}$ is a
signal strength measured at the LHC, and $\delta_{i}^{f}$ is the one standard
deviation uncertainty on $\hat{\mu}_{i}^{f}$. We combine measurements from
both ATLAS and CMS at the 13 TeV LHC. The set of signal strengths we use can
be found in Tables 1 and 2 in Appendix C.

Figure 3:  95% CL regions from Higgs precision data for $\sin\theta$ vs (a)
$f_{GG}$, (b) $f_{t}$, (c) $f_{b}$, (d) $f_{\tau}$, (e) $f_{+}$, and (f)
$f_{\mu}$. Parameters not shown in a plot are profiled over. For $\mu^{i}_{f}$
expanded to $\mathcal{O}(\Lambda^{-1})$: (blue dashed) $tth_{1}+th_{1}+ggF$
initial states only, and (black solid) all initial and final states. For
widths and cross sections kept to $\mathcal{O}(\Lambda^{-2})$: (red solid) all
initial and final states and (red dotted) keeping $|f_{i}|<4\pi$ at a new
physics scale of $\Lambda=3$ TeV. Regions inside contours are allowed.

For the gluon fusion (ggF) production rate, the NLO cross section results are
only known for the effective gluon and top quark couplings Deutschmann:2017qum
. To be consistent across our fits, we also include the bottom quark EFT
couplings. Since these cross sections are not available, we use the
approximation

$\displaystyle\frac{\sigma_{ggF}(pp\rightarrow
h_{1})}{\sigma_{ggF,SM}(pp\rightarrow
h_{1})}\approx\frac{\Gamma(h_{1}\rightarrow gg)}{\Gamma_{SM}(h_{1}\rightarrow
gg)},$ (22)

to calculate the contributions from $f_{b}$. The validity of this
approximation for top quark-Higgs and gluon-Higgs effective operators is
discussed in Appendix A. We note that this is a standard approximation
Dawson:2016ugw ; deBlas:2018tjm , and is indeed good to $~{}\sim 3-5\%$ for
most top quark-Higgs and gluon-Higgs EFT contributions. In our fits we use the
NLO cross section results of Ref. Deutschmann:2017qum  to calculate the
$f_{t}$ and $f_{GG}$ contributions to the signal strength, and the
approximation of Eq. (22) for the $f_{b}$ contributions:

$\displaystyle\frac{\sigma_{ggF}(pp\rightarrow
h_{1})}{\sigma_{ggF,SM}(pp\rightarrow h_{1})}$ $\displaystyle=$
$\displaystyle\cos^{2}\theta+\frac{\cos\theta\,\sin\theta}{(\Lambda/{\rm
TeV})}\left(0.517\,f_{t}+1.45\,f_{GG}-0.0281\,f_{b}\right)$ (23)
$\displaystyle+\frac{\sin^{2}\theta}{(\Lambda/{\rm
TeV})^{2}}\left(0.0626\,f_{t}^{2}+0.492\,f_{GG}^{2}+8.53\times
10^{-4}\,f_{b}^{2}\right.$
$\displaystyle\left.+0.351\,f_{t}\,f_{GG}-8.62\times
10^{-3}\,f_{b}\,f_{t}-0.0226\,f_{b}\,f_{GG}\right).$

Corrections up to N3LO in QCD are known for gluon fusion Harlander:2016hcx ;
Anastasiou:2016hlm ; Brooijmans:2016vro  and Higgs decays to gluons
Contino:2014aaa ; Djouadi:2018xqq . We also include $Wh_{1}$, $Zh_{1}$, Higgs
production in association with a $t\overline{t}$ pair ($t\overline{t}h_{1}$),
Higgs production in association with a top plus jet or top plus W
(collectively $th_{1}$), and VBF. For these production modes the model is
implemented in MadGraph5_aMC@NLO Alwall:2014hca  via FeynRules Alloul:2013bka
. The default NNPDF2.3LO pdf sets Ball:2012cx  are used and for $h_{1}$
production modes the renormalization and factorization scales are set to the
sum of the final state particle masses. For the VBF mode, we apply the cuts
Azzi:2019yne

$\displaystyle p_{T}^{j}>20~{}{\rm
GeV},\,|\eta^{j}|<5,\,|\Delta\eta^{jj}|>3,\,{\rm and}\,m_{jj}>130~{}{\rm
GeV},$ (24)

where $p_{T}^{j}$ are jet transverse momenta, $\eta^{j}$ are jet
pseudorapidity, $\Delta\eta^{jj}$ is the difference in the jet pseudorapidity,
and $m_{jj}$ is the di-jet invariant mass. These production and decay modes
are calculated at LO in QCD, and it is hoped that most of the QCD corrections
cancel in the ratio of the cross sections used for the signal strengths.
However, it should be pointed out that in the SMEFT, for some observables the
QCD corrections can be strongly dependent upon the EFT operators
Baglio:2017bfe ; Baglio:2018bkm ; Baglio:2019uty ; Baglio:2020oqu .

Once all the signal strengths are known, we perform a fit to the Wilson
coefficients and scalar mixing angle. As shown in Appendix B, the
$h_{1}\rightarrow\gamma\gamma$ decay depends on the combination
$f_{BB}+f_{WW}$ but not $f_{BB}-f_{WW}$. Also, Fig. 1 shows that processes
with external $W$ and $Z$ bosons do not depend strongly on $f_{WW}$ and
$f_{BB}$. Hence, we define

$\displaystyle f_{\pm}=\frac{1}{\sqrt{2}}\left(f_{BB}\pm f_{WW}\right).$ (25)

Now $h_{1}\rightarrow\gamma\gamma$ will constrain only $f_{+}$, and
$Wh_{1},\,Zh_{1},\,$VBF$,\,h_{1}\rightarrow WW^{*},$ and $h_{1}\rightarrow
ZZ^{*}$ have negligible dependence on both $f_{\pm}$. Hence, we set $f_{-}=0$.
Hence, the following parameters are fit using just Higgs signal strengths:

$\displaystyle\sin\theta,\,f_{GG},\,f_{+},\,f_{b},\,f_{t},\,f_{\mu},\,{\rm
and}\,f_{\tau}.$ (26)

In Fig. 3 we show the results of the $\chi^{2}$ fits to Higgs data at 95% CL.
As can be seen from Eq. (III), the squared amplitudes are invariant under the
simultaneous parity transformations: $\sin\theta\rightarrow-\sin\theta$ and
all Wilson coefficients $f_{i}\rightarrow-f_{i}$. Hence, only $f_{i}>0$
results are shown and contain all the information. The results are shown for
(blue-dashed) just $tth_{1}+th_{1}$+ggF production modes and (black/red)
including all signal strengths. As can be seen, if only gluon fusion and Higgs
production in association with top quarks are used, all values of $\sin\theta$
are allowed. This is because ggF,$\,tth_{1},$ and $th_{1}$ depend relatively
strongly on $f_{t}$ and $f_{GG}$. Hence, deviations in $\sin\theta$ can be
compensated for by changes in $f_{t}$ and $f_{GG}$. The major effect of vector
boson fusion and Higgs production in association with $W^{\pm}$ or $Z$ is to
eliminate the largest $\sin\theta$ regions. As discussed above in Sec. IV,
VBF,$\,Wh_{1},$ and $Zh_{1}$ do not depend strongly on the Wilson
coefficients222This is also true in SMEFT Corbett:2015ksa .. Hence, the
production rates for these modes are approximately the SM rate suppressed by
the mixing angle $\cos^{2}\theta$:

$\displaystyle\sigma_{VBF/Vh_{1}}(pp\rightarrow
h_{1})\approx\cos^{2}\theta\,\sigma_{VBF/Vh_{1},SM}(pp\rightarrow h_{1}),$
(27)

where $Vh_{1}=Zh_{1},Wh_{1}$. Limits on these production rates then
essentially place limits on the scalar mixing angle. Additionally, at large
Wilson coefficient values, fits including only $ggF,tth_{1},$ and $th_{1}$
agree well with the full fit, except for $f_{b}$, $f_{\mu}$, and $f\tau$. This
is can be understood by noting that the strong constraints on
$h_{1}\rightarrow\tau^{+}\tau^{-}$, $h_{1}\rightarrow\mu^{+}\mu^{-}$, and
$h_{1}\rightarrow bb$ come from VBF,$\,Wh_{1},$ and $Zh_{1}$.

Figure 3 also compares various calculations of the full $\chi^{2}$ to
determine the validity of the limits on the BSM-EFT. As discussed in Sec. III,
for small mixing angles, the power counting of $h_{1}$ production and decay is
expected to follow the usual power counting of the SMEFT. To check the
validity of the power counting in our fits, Fig. 3 shows the (black dash-dot-
dot) signal strengths expanded to $\mathcal{O}(\Lambda^{-1})$ and (red solid)
signal strengths calculated by keeping cross sections and widths to
$\mathcal{O}(\Lambda^{-2})$. For the bulk of the distributions, these two
scenarios largely agree with each other showing that the the limits are
compatible with the BSM-EFT power counting. However, if the full
$\Lambda^{-2}$ dependence is kept, new 95% CL regions open up at large Wilson
coefficients.

From general perturbativity arguments, it is expected that the Wilson
coefficients are bounded $|f_{i}|<4\pi$. The red dotted contours in Fig. 3
show the results of requiring $|f_{i}|<4\pi$ for a new physics scale of
$\Lambda=3$ TeV. Comparing to the red solid lines, it can be seen that in the
relevant regions, the fits are consistent with perturbative Wilson
coefficients. Comparing the power counting and perturbativity constraints, it
is clear that the new allowed parameter regions that appear at
$\mathcal{O}(\Lambda^{-2})$ but not $\mathcal{O}(\Lambda^{-1})$ are not
consistent with perturbativity. Hence, imposing the perturbativity constraints
automatically guarantees Higgs precision measurements are fully compatible
with BSM-EFT power counting.

Figure 4:  One-dimensional fits to $\sin\theta$ using Higgs signal strengths
with all other parameters profiled over for (black dash-dot-dot) signal
strengths expanded to $\mathcal{O}(\Lambda^{-1})$, (red dotted) cross sections
and widths kept at $\mathcal{O}(\Lambda^{-2})$, and (blue dashed) all
dimension-5 operators set to zero. The new physics scale is $\Lambda=3$ TeV
and Wilson coefficients are required to be $|f_{i}|<\pi$.

Finally, in Fig. 4, we show the one-dimensional fits to $\sin\theta$ using
Higgs data with dimension-5 operators and in the renormalizable singlet extend
SM without dimension-5 operators. As can be seen, even with a new physics
scale of $\Lambda=3$ TeV, the dimension-5 operators make a substantial impact
on the interpretation of Higgs data. Also, consistently expanding signal
strengths to $\mathcal{O}(\Lambda^{-1})$ gives the same result as keeping all
cross sections and widths to $\mathcal{O}(\Lambda^{-2})$. The conclusion is
that the BSM-EFT is valid for Higgs measurements and, even if we assume a new
physics scale beyond the current reach of the LHC, the effects of this new
physics on the singlet extended SM cannot be ignored.

##  VI Including Heavy Resonance Searches

Heavy scalars are searched for regularly at the LHC. The EFT couplings of
$h_{1},h_{2}$ are inherited by the mixing of the SM Higgs with the scalar $S$.
Hence, in production and decay of $h_{1}$, the Wilson coefficients and mixing
angle always appear in the combination $\sin\theta f_{i}$ whereas in the
production and decay of $h_{2}$, they appear in the combination
$\cos\theta\,f_{i}$. As a result, heavy resonance searches are expected to
give complementary information to Higgs signal strengths.

First, we describe how heavy resonance searches are incorporated into our
$\chi^{2}$ fits, then we give the results.

###  VI.1 $\chi^{2}$ for Heavy Resonance Searches

Similar to the Higgs signal strengths, it is assumed that scalar resonance
searches are Gaussian and a $\chi^{2}$ fit is performed:

$\displaystyle\left(\chi^{f}_{i,h_{2}}\right)^{2}=\left(\frac{\sigma_{i}^{f}-\hat{\sigma}_{i}^{f}}{\delta\sigma_{i}^{f}}\right)^{2},$
(28)

where $(\chi^{f}_{i,h_{2}})^{2}$ is a chi-square of a single $h_{2}$ process,
$\sigma_{i}^{f}$ is the calculated cross section for initial state $i$ into
final state $f$, $\hat{\sigma}_{i}^{f}$ is the measured cross section at the
LHC, and $\delta\sigma_{i}^{f}$ is the one standard deviation uncertainty on
$\hat{\sigma}_{i}^{f}$. To calculate the cross section, both the SM rate as
well as the new physics contribution must be included. Using the narrow width
approximation, we have

$\displaystyle\sigma_{i}^{f}=\sigma_{i,SM}^{f}+\sigma(i\rightarrow h_{2}){\rm
BR}(h_{2}\rightarrow f).$ (29)

While applying experimental bounds using the narrow width approximation is
standard Bechtle:2011sb ; Bechtle:2013wla ; Bechtle:2020pkv , depending on the
region of parameter space, the interference between the SM and heavy scalar
resonances can be $\mathcal{O}(10\%)$ for $gg\rightarrow ZZ$, $gg\rightarrow
W^{+}W^{-}$, and $gg\rightarrow h_{1}h_{1}$ Kauer:2015hia ; Dawson:2015haa ;
Greiner:2015ixr ; Carena:2018vpt  even in the on-shell region. The
$gg\rightarrow ZZ$ and $gg\rightarrow WW$ processes contribute $\sim 5\%$ to
the total $ZZ$ and $WW$ rates Dicus:1987dj ; Glover:1988fe ; Binoth:2005ua ;
Dawson:2013lya ; Binoth:2006mf ; Caola:2015ila ; vonManteuffel:2015msa ;
Caola:2015psa ; Caola:2015rqy . As we will show, after Higgs measurements and
resonant searches are combined, the width is indeed narrow.

Typically, the observed and expected 95% CL upper limits on resonance
production are reported. Assuming that there are no large fluctuations away
from the SM predictions, a SM cross section is measured in all new physics
searches. The allowed fluctuations away from the SM cross section at 95% CL
are then the expected 95% CL upper limits on new resonance cross sections.
That is, the uncertainty on the cross section is approximated as

$\displaystyle\delta\sigma_{i}^{f}\approx\hat{\sigma}_{i,Exp}^{f}/1.96,$ (30)

where $\hat{\sigma}_{i,Exp}^{f}$ is the expected 95% CL upper limit on the
resonance cross section. Again assuming there are no large excesses, the
measured cross section is mostly SM-like with a small deviation given by the
difference in the observed and expected bounds:

$\displaystyle\hat{\sigma}_{i}^{f}\approx\sigma_{i,SM}^{f}+\hat{\sigma}_{i,Obs}^{f}-\hat{\sigma}_{i,Exp}^{f},$
(31)

where $\hat{\sigma}_{i,Obs}^{f}$ is the observed 95% CL upper limit on the
resonance cross section. With these approximations, we finally have

$\displaystyle\left(\chi^{f}_{i,h_{2}}\right)^{2}=\left(\frac{\sigma(i\rightarrow
h_{2}){\rm BR}(h_{2}\rightarrow
f)+\hat{\sigma}_{i,Exp}^{f}-\hat{\sigma}_{i,Obs}^{f}}{\hat{\sigma}_{i,Exp}^{f}/1.96}\right)^{2}.$
(32)

One final complication is if
$\hat{\sigma}_{i,Obs}^{f}<\hat{\sigma}_{i,Exp}^{f}$ then according to Eq. (32)
the best fit signal cross section $\sigma(i\rightarrow h_{2}){\rm
BR}(h_{2}\rightarrow f)$ will be negative, which is nonsensical. We propose to
alter the definition in Eq. (32) to

$\displaystyle\left(\chi^{f}_{i,h_{2}}\right)^{2}=\begin{cases}\displaystyle\left(\frac{\sigma(i\rightarrow
h_{2}){\rm BR}(h_{2}\rightarrow
f)+\hat{\sigma}_{i,Exp}^{f}-\hat{\sigma}_{i,Obs}^{f}}{\hat{\sigma}_{i,Exp}^{f}/1.96}\right)^{2}&{\rm
if~{}}\hat{\sigma}_{i,Obs}^{f}\geq\hat{\sigma}_{i,Exp}^{f}\\\
\displaystyle\left(\frac{\sigma(i\rightarrow h_{2}){\rm BR}(h_{2}\rightarrow
f)}{\hat{\sigma}_{i,Obs}^{f}/1.96}\right)^{2}&{\rm
if~{}}\hat{\sigma}_{i,Obs}^{f}<\hat{\sigma}_{i,Exp}^{f}.\end{cases}~{}$ (33)

The second line forces the best fit value of $\sigma(i\rightarrow h_{2}){\rm
BR}(h_{2}\rightarrow f)$ to be bounded from below by zero. Also in the second
line, the uncertainty has been changed from the expected to observed signal
rate. If the best fit value of the signal cross section is at zero, then
$\hat{\sigma}_{i,Obs}^{f}$ is how far away it can fluctuate from zero at 95%
CL. Hence, this form of the $\chi^{2}$ allows for upward fluctuations with a
best fit value of the signal cross section away from zero as well as bounding
the best fit value of the cross section to be positive.

The usual use of the reported 95% CL upper bounds is to put a strict upper
bound on resonance cross sections: $\sigma(i\rightarrow h_{2}){\rm
BR}(h_{2}\rightarrow f)<\hat{\sigma}_{i,Obs}^{f}$ Bechtle:2011sb ;
Bechtle:2013wla ; Bechtle:2020pkv . To check that our proposal is consistent,
it must be checked that this interpretation can be derived from Eq. (33).
Assuming a one-parameter fit, the value of the resonance cross section at the
minimum $\chi^{2}$ is

$\displaystyle\left[\sigma(i\rightarrow h_{2}){\rm BR}(h_{2}\rightarrow
f)\right]_{\chi^{2}_{\rm
min}}=\begin{cases}\hat{\sigma}_{i,Obs}^{f}-\hat{\sigma}_{i,Exp}^{f}&{\rm
if~{}}\hat{\sigma}_{i,Obs}^{f}\geq\hat{\sigma}_{i,Exp}^{f}\\\ 0&{\rm
if~{}}\hat{\sigma}_{i,Obs}^{f}<\hat{\sigma}_{i,Exp}^{f}\end{cases}$ (34)

Then, the one-parameter fit limit is found by requiring
$\Delta\chi^{2}=\chi^{2}-\chi^{2}_{\rm min}<3.84$, where $\chi_{\rm min}^{2}$
is the minimum $\chi^{2}$. It can then be shown that Eq. (33) gives the limit

$\displaystyle\sigma(i\rightarrow h_{2}){\rm BR}(h_{2}\rightarrow
f)<\hat{\sigma}_{i,Obs}^{f},$ (35)

which is consistent with the usual interpretation of these bounds.

With these results, all heavy scalar searches can be combined into one
$\chi^{2}$:

$\displaystyle\chi^{2}_{h_{2}}=\sum_{i,f}\left(\chi^{f}_{i,h_{2}}\right)^{2}.$
(36)

Unlike imposing a hard cutoff on the heavy resonance rates, this method will
allow for fluctuations in some channels that are consistent with a global fit
at 95% CL. The combined limits from scalar searches and Higgs measurements are
found by combing the $\chi^{2}$ in Eqs. (21) and (36):

$\displaystyle\chi^{2}_{\rm Tot}=\chi^{2}_{h_{1}}+\chi^{2}_{h_{2}}.$ (37)

###  VI.2 Results for Heavy Resonance Searches and Higgs Precision

As with the Higgs boson, the main production channel of $h_{2}$ is gluon
fusion due to the large gluon parton luminosities. Hence, we will only
consider the ggF initial state. To calculate this we reweight partial widths
with the NNLO+NNLL SM-like Higgs predictions provided by the LHC Higgs Cross
Section Working Group deFlorian:2016spz :

$\displaystyle\sigma_{ggF}(pp\rightarrow
h_{2})=\sigma_{ggF,SM}^{NNLO+NNLL}(pp\rightarrow
h_{2})\frac{\Gamma(h_{2}\rightarrow gg)}{\Gamma_{SM}(h_{2}\rightarrow gg)},$
(38)

where the subscript $SM$ indicates the prediction for a SM-like Higgs boson at
a mass $m_{2}$. The same higher order corrections used for $h_{1}$ decays, as
discussed in Sec. IV, are incorporated into $h_{2}$ decays. We also correctly
account for the $(1+11\alpha_{s}/4\pi)$ difference in the NLO contributions to
$h_{2}\rightarrow gg$ from quark loops and $g-g-h_{2}$ contact interactions
Djouadi:1991tka ; Dawson:1990zj ; Spira:1995rr ; deBlas:2018tjm , as discussed
previously. For $h_{2}$ decay rates the renormalization scale is set to
$m_{2}$ for tree level decays and $m_{2}/2$ for loop level decays. The values
of $\hat{\sigma}_{ggF,Obs}^{f}$ and $\hat{\sigma}_{ggF,Exp}^{f}$ for the final
states under consideration are given in Tables 3 and 4 in App. C.

For all results presented in this section we keep cross sections and widths to
order $\Lambda^{-2}$. As discussed in Sec. III, this is valid power counting
for $h_{2}$ processes in the small mixing limit. Finally, we always require
that Wilson coefficients are bounded by $|f_{i}|<4\pi$ and fit to all relevant
Wilson coefficients and scalar trilinear couplings:

$\displaystyle\sin\theta,\,f_{GG},\,f_{+},f_{-},\,f_{b},\,f_{t},\,f_{\tau},\,f_{\mu},\,{\rm
and}\,\lambda_{211}.$ (39)

As with Higgs rates the $h_{2}$ rates are invariant under the simultaneous
parity transformation $\sin\theta\rightarrow-\sin\theta$ and
$f_{i}\rightarrow-f_{i}$. Hence, we only show results for $f_{i}>0$ and
results for $f_{i}<0$ can be found by performing the transformation.

Figure 5:  95% CL allowed regions from (black solid) heavy scalar searches,
(black dotted) heavy scalar searches with $\Gamma_{2}<0.1\,m_{2}$, (blue dot-
dashed) Higgs measurements, (blue dotted) Higgs measurements with
$\Gamma_{2}<0.1\,m_{2}$, (red dashed) combined heavy scalar searches and Higgs
measurements, and (red dotted) combined heavy scalar searches and Higgs
measurements with $\Gamma_{2}<0.1\,m_{2}$. These are shown for $\lambda_{211}$
vs $\sin\theta$ with all other parameters profiled over. Two scalar masses are
considered: (a) $m_{2}=400$ GeV and (b) $m_{2}=600$ GeV. The new physics scale
is $\Lambda=3$ TeV. The regions within the contours are allowed.

Figure 6:  95% CL allowed regions from (black solid) heavy scalar searches,
(black dotted) heavy scalar searches with $\Gamma_{2}<0.1\,m_{2}$, (blue dot-
dashed) Higgs measurements, and (red dotted) combined heavy scalar searches
and Higgs measurements. The regions within the contours are allowed. These are
shown for (a,b,c) $f_{GG}$ and (d,e,f) $f_{t}$ vs $\sin\theta$ with all other
parameters profiled over. Three scalar masses are considered: (a,d)
$m_{2}=200$ GeV, (b,e) $m_{2}=400$ GeV, and (c,f) $m_{2}=600$ GeV. The new
physics scale is $\Lambda=3$ TeV.

Figure 5 shows the two-dimensional 95% CL allowed regions for $\lambda_{211}$
vs $\sin\theta$ with all other parameters profiled over. Only two scalar
masses are considered, $m_{2}=400$ GeV and $m_{2}=600$ GeV, since the decay
$h_{2}\rightarrow h_{1}h_{1}$ is then kinematically possible. The scalar
searches do not put meaningful limits on $\lambda_{211}$: the upper bounds on
$\lambda_{211}$ come from the theoretical requirement that the scalar
potential be bounded and that the global minimum be the correct EWSB minimum
as discussed in Sec. II. As can be seen, the Higgs measurements mainly limit
the values of $\sin\theta$.

Many of the scalar searches that are included require that $h_{2}$ be a narrow
resonance. Hence, the results from requiring that the $h_{2}$ total width,
$\Gamma_{2}$, be less than 10% of the $h_{2}$ mass are also shown in Fig. 5.
For the scalar searches, the width constraint limits both $\sin\theta$ and
$\lambda_{221}$. When $\sin\theta$ is large, the decays $h_{2}\rightarrow WW$
and $h_{2}\rightarrow ZZ$ are SM-like and large for large $m_{2}$
deFlorian:2016spz . Hence, $\Gamma_{2}/m_{2}<0.1$ places stronger constraints
on $\sin\theta$ than just blindly applying the scalar searches. Also, if
$\lambda_{211}$ is too large the partial width of the decay $h_{2}\rightarrow
h_{1}h_{1}$ becomes large. As a result, requiring a narrow resonance puts
strong constraints on $\lambda_{211}$. Higgs measurements already strongly
constrain $\sin\theta$, so the effect of requiring a narrow resonance is much
less pronounced here. When the Higgs measurements are combined with scalar
searches, the narrow width requirement does not meaningfully constrain the
parameter space.

The two-dimensional 95% CL allowed regions for $f_{GG}$ vs $\sin\theta$ and
$f_{t}$ vs $\sin\theta$ with all other parameters profiled over are shown in
Fig. 6333The results for the remaining Wilson coefficients can be found in
App. D.. As with the $\lambda_{211}$ limits, requiring a narrow width in the
scalar search results squeezes the allowed parameter region for $m_{2}=400$
and $m_{2}=600$ GeV. The narrow width requirement does not significantly
change the Higgs precision and combination constraints. For the smaller
$m_{2}$, the narrow width requirement makes no difference on any of the
limits.

As can be seen in Fig. 6, Higgs measurements and scalar searches are
complementary. That is, the allowed regions for scalar searches and Higgs
measurements do not fully overlap. Indeed, the combined allowed region is
smaller than the individual allowed regions. This is particularly striking for
$m_{2}=600$ GeV.

Figure 7:  $\Delta\chi^{2}$ for combined Higgs measurements and scalar
searches as a function of $\sin\theta$ with all other parameters profiled
over. Both (black solid) Wilson coefficients with $|f_{i}|<4\pi$ and (blue
dashed) $f_{i}=0$, i.e. dimension-5 terms set to zero. Three $h_{2}$ masses
are considered: (a) $m_{2}=200$ GeV, (b) $m_{2}=400$ GeV, and (c) $m_{2}=600$
GeV.

Finally, in Fig. 7, we show the $\Delta\chi^{2}$ distributions as a function
$\sin\theta$ for the BSM-EFT and renormalizable model with all other
parameters profiled over. In the BSM-EFT, the shape of $\Delta\chi^{2}$
changes dramatically. The 95% CL and 68% CL allowed regions also change
drastically and exactly how they change depends strongly on the $h_{2}$ mass.
It is clear that even 3 TeV new physics effects can make a significant impact
on the interpretation of current measurements.

##  VII Conclusion

A common assumption of simplified models at the LHC is that there are a few
new BSM particles that can be produced, while all other new particles are
heavy and decoupled. Under these assumptions, most studies of simplified
models are renormalizable. However, using EFT techniques, it is possible to
test the basic assumption that all other new particles are indeed decoupled.

In this paper, we studied a popular simplified model, the real singlet
extended SM, and supplemented it with all possible dimension-5 operators
involving the scalar singlet. We studied the effects of the effective
operators on the interpretation of Higgs signal strengths as well as searches
for heavy new resonances. As we showed, even if the new physics occurs at 3
TeV, the interpretation of these measurements and searches are changed
drastically. This study shows that even in the simplest of simplified model,
the heavy new physics is not “decoupled” even when the BSM-EFT expansion is
valid. That is, it cannot be neglected and the BSM-EFT should generically be
considered.

In addition to the numerical results, we also gave a comprehensive discussion
of the counting in BSM-EFT for production and decay rates. We showed that
while in the linear SMEFT power counting is relatively straightforward, power
counting in a BSM-EFT is strongly process and parameter space dependent. We
also developed a new proposal to consistently combine the limits from new
resonance searches and precision measurements via an $\chi^{2}$. This method
allows for fluctuations in individual channels, while keeping the global
$\chi^{2}$ within allowable limits. This is unlike the usual cutoff method
where all resonance cross sections are strictly cutoff at the observed limits
Bechtle:2011sb ; Bechtle:2013wla ; Bechtle:2020pkv .

## Acknowledgments

We thank Jeong Han Kim, KC Kong, Tilman Plehn, Daniel Tapia Takaki, and Yajuan
Zheng for helpful discussions. Chris Rogan is thanked for reassuring IML that
he is not crazy. IML would like to thank the Institute for Theoretical Physics
at Universität Heidelberg for their hospitality during the completion of this
work. This work was performed in part at the Aspen Center for Physics, which
is supported by National Science Foundation grant PHY-1607611. SA, IML, and MS
are supported in part by United States Department of Energy grant number DE-
SC0017988. MS is supported in part by the United States Department of Energy
under Grant Contract DE-SC0012704. SA and MS are also supported in part by the
State of Kansas EPSCoR grant program. The data to reproduce the plots have
been uploaded with the arXiv submission or is available upon request.

##  Appendix A The Gluon Fusion Cross Section and Gluon Partial Width

In this appendix, we will inspect numbers for the gluon partial width and
gluon fusion cross section and confirm the validity of the signal strength
approximation in Eq. (22) in this EFT. The cross section results of Ref.
Deutschmann:2017qum  gives

$\displaystyle\frac{\sigma_{ggF}(pp\rightarrow
h_{1})}{\sigma_{ggF,SM}(pp\rightarrow h_{1})}$ $\displaystyle=$
$\displaystyle\cos^{2}\theta+\frac{\cos\theta\,\sin\theta}{(\Lambda/{\rm
TeV})}\left(0.517\,f_{t}+1.45\,f_{GG}\right)$ (40)
$\displaystyle+\frac{\sin^{2}\theta}{(\Lambda/{\rm
TeV})^{2}}\left(0.0626\,f_{t}^{2}+0.492\,f_{GG}^{2}+0.351\,f_{t}\,f_{GG}\right).$

The corresponding gluon partial width calculation in the scalar EFT (dropping
bottom quark EFT contributions for the purposes of comparison) gives

$\displaystyle\frac{\Gamma(h_{1}\rightarrow gg)}{\Gamma_{SM}(h_{1}\rightarrow
gg)}$ $\displaystyle=$
$\displaystyle\cos^{2}\theta+\frac{\cos\theta\,\sin\theta}{(\Lambda/{\rm
TeV})}\left(0.526\,f_{t}+1.38\,f_{GG}\right)$ (41)
$\displaystyle+\frac{\sin^{2}\theta}{(\Lambda/{\rm
TeV})^{2}}\left(0.0702\,f_{t}^{2}+0.481\,f_{GG}^{2}+0.367\,f_{t}\,f_{GG}\right).$

Comparing the size of the linear terms, the $f_{t}$ terms differ by less than
$2\%$ and the $f_{GG}$ terms differ by around $5\%$. For the quadratic terms,
the $f_{GG}^{2}$ term and the $f_{GG}f_{t}$ term differ by around $2\%$ and
$5\%$, respectively, while the $f_{t}^{2}$ term differs by around $12\%$.
Naively, one might be worried that the large difference in the $f_{t}^{2}$
term might challenge the validity of the approximation in Eq. (22). However,
as seen in Fig. 3 and the discussion that followed, the Higgs signal strengths
constrained the mixing angle to be small, and so the linear terms dominated
over the quadratic terms. We thus conclude that, for the parameter space
allowed by observation, the approximation is good to within around $5\%$ or
less.

##  Appendix B Feynman Rules

###  B.1 Trilinear Scalar Couplings

The trilinear scalar couplings are defined as

$\displaystyle
V(h_{1},h_{2})\supset\frac{1}{3!}\lambda_{111}h_{1}^{3}+\frac{1}{2}\lambda_{211}h_{1}^{2}h_{2}+\frac{1}{2}\lambda_{221}h_{1}h_{2}^{2}+\frac{1}{3!}\lambda_{222}h_{2}^{3}.$
(42)

where $\lambda_{111}$ and $\lambda_{211}$ are given in Sec. II, and

$\displaystyle\lambda_{221}$ $\displaystyle=$
$\displaystyle\frac{m_{1}^{2}+2\,m_{2}^{2}}{v}\cos\theta\sin^{2}\theta+2\,b_{3}\cos^{2}\theta\sin\theta+a_{2}v\,\cos\theta\left(\cos^{2}\theta-2\sin^{2}\theta\right)$
(43)
$\displaystyle+\frac{3\,a_{3}v^{2}}{2\Lambda}\cos^{2}\theta\sin\theta-\frac{a_{4}v^{2}}{\Lambda}\sin\theta\left(2\cos^{2}\theta-\sin^{2}\theta\right)\,,$
$\displaystyle\lambda_{222}$ $\displaystyle=$
$\displaystyle-\frac{3\,m_{2}^{2}}{v}\sin^{3}\theta+2\,b_{3}\cos^{3}\theta-3\,a_{2}v\,\cos^{2}\theta\sin\theta+\frac{3\,a_{3}v^{2}}{2\,\Lambda}\cos^{3}\theta$
$\displaystyle+\frac{3\,a_{4}v^{2}}{\Lambda}\cos\theta\sin^{2}\theta\,.$

###  B.2 $h_{i}-f-f$ and $h_{i}-V-V^{\prime}$ couplings

The vertex rules, with all momenta outgoing, are

$\displaystyle\mathcal{V}_{h_{1}ff}$ $\displaystyle=$
$\displaystyle-i\frac{m_{f}}{v}(\cos\theta+\frac{f_{f}}{\Lambda}v\sin\theta)\,,$
(45) $\displaystyle\mathcal{V}_{h_{1}g_{\mu}(p_{1})g_{\nu}(p_{2})}$
$\displaystyle=$
$\displaystyle-i\frac{g_{s}^{2}}{4\pi^{2}}\frac{f_{GG}}{\Lambda}\sin\theta\,(\eta_{\mu\nu}p_{1}\cdot
p_{2}-p_{1\nu}p_{2\mu})\,,$ (46)
$\displaystyle\mathcal{V}_{h_{1}\gamma_{\mu}(p_{1})\gamma_{\nu}(p_{2})}$
$\displaystyle=$
$\displaystyle-i\frac{e^{2}}{4\pi^{2}}\frac{f_{BB}+f_{WW}}{\Lambda}\sin\theta\,(\eta_{\mu\nu}p_{1}\cdot
p_{2}-p_{1\nu}p_{2\mu})\,,$ (47)
$\displaystyle\mathcal{V}_{h_{1}W^{+}_{\mu}(p_{1})W^{-}_{\nu}(p_{2})}$
$\displaystyle=$ $\displaystyle
2i\frac{M_{W}^{2}}{v}\eta_{\mu\nu}\cos\theta-i\frac{g^{2}}{4\pi^{2}}\frac{f_{WW}}{\Lambda}\sin\theta\,(\eta_{\mu\nu}p_{1}\cdot
p_{2}-p_{1\nu}p_{2\mu})\,,$ (48)
$\displaystyle\mathcal{V}_{h_{1}Z_{\mu}(p_{1})Z_{\nu}(p_{2})}$
$\displaystyle=$ $\displaystyle
2i\frac{M_{Z}^{2}}{v}\eta_{\mu\nu}\cos\theta-i\frac{g_{Z}^{2}}{4\pi^{2}}\frac{f_{ZZ}}{\Lambda}\sin\theta(\eta_{\mu\nu}p_{1}\cdot
p_{2}-p_{1\nu}p_{2\mu})\,,$ (49)
$\displaystyle\mathcal{V}_{h_{1}Z_{\mu}(p_{1})\gamma_{\nu}(p_{2})}$
$\displaystyle=$
$\displaystyle-i\frac{g_{Z}e}{2\pi^{2}}\frac{f_{Z\gamma}}{\Lambda}\sin\theta\,(\eta_{\mu\nu}p_{1}\cdot
p_{2}-p_{1\nu}p_{2\mu})\,,$ (50) $\displaystyle\mathcal{V}_{h_{2}ff}$
$\displaystyle=$
$\displaystyle-i\frac{m_{f}}{v}(-\sin\theta\,+\frac{f_{f}}{\Lambda}v\cos\theta)\,,$
(51) $\displaystyle\mathcal{V}_{h_{2}g_{\mu}(p_{1})g_{\nu}(p_{2})}$
$\displaystyle=$
$\displaystyle-i\frac{g_{s}^{2}}{4\pi^{2}}\frac{f_{GG}}{\Lambda}\cos\theta\,(\eta_{\mu\nu}p_{1}\cdot
p_{2}-p_{1\nu}p_{2\mu})\,,$ (52)
$\displaystyle\mathcal{V}_{h_{2}\gamma_{\mu}(p_{1})\gamma_{\nu}(p_{2})}$
$\displaystyle=$
$\displaystyle-i\frac{e^{2}}{4\pi^{2}}\frac{f_{BB}+f_{WW}}{\Lambda}\cos\theta\,(\eta_{\mu\nu}p_{1}\cdot
p_{2}-p_{1\nu}p_{2\mu})\,,$ (53)
$\displaystyle\mathcal{V}_{h_{2}W^{+}_{\mu}(p_{1})W^{-}_{\nu}(p_{2})}$
$\displaystyle=$
$\displaystyle-2i\frac{M_{W}^{2}}{v}\eta_{\mu\nu}\sin\theta-i\frac{g^{2}}{4\pi^{2}}\frac{f_{WW}}{\Lambda}\cos\theta\,(\eta_{\mu\nu}p_{1}\cdot
p_{2}-p_{1\nu}p_{2\mu})\,,$ (54)
$\displaystyle\mathcal{V}_{h_{2}Z_{\mu}(p_{1})Z_{\nu}(p_{2})}$
$\displaystyle=$
$\displaystyle-2i\frac{M_{Z}^{2}}{v}\eta_{\mu\nu}\sin\theta-i\frac{g_{Z}^{2}}{4\pi^{2}}\frac{f_{ZZ}}{\Lambda}\cos\theta\,(\eta_{\mu\nu}p_{1}\cdot
p_{2}-p_{1\nu}p_{2\mu})\,,$ (55)
$\displaystyle\mathcal{V}_{h_{2}Z_{\mu}(p_{1})\gamma_{\nu}(p_{2})}$
$\displaystyle=$
$\displaystyle-i\frac{g_{Z}e}{2\pi^{2}}\frac{f_{Z\gamma}}{\Lambda}\cos\theta\,(\eta_{\mu\nu}p_{1}\cdot
p_{2}-p_{1\nu}p_{2\mu})\,.$ (56)

with $g_{Z}=\frac{g}{\cos\theta_{W}}$,
$f_{ZZ}=f_{BB}\sin^{4}\theta_{w}+f_{WW}\cos^{4}\theta_{W}$,
$f_{Z\gamma}=f_{BB}\sin^{2}\theta_{w}+f_{WW}\cos^{4}\theta_{W}$, and
$\theta_{W}$ is the weak mixing angle.

##  Appendix C Signal Strengths/Bounds

###  C.1 Higgs Signal Strengths

We now give the signal strengths used in our fits. These are chosen to be the
measured signal strengths with the most integrated luminosity in a given
channel. All results are from Run 2, with up to 139 fb-1 of accumulated data.
The guide to nonobvious abbreviations: ggF = gluon fusion, VBF = vector boson
fusion, $Wh_{1}$ = Higgs associated production with a $W$, $Zh_{1}$ = Higgs
associated production with a $Z$, $Vh_{1}$ = combination of $Wh_{1}$ and
$Zh_{1}$, and $VV^{*}$ = combination of $ZZ^{*}$ and $WW^{*}$.

| $\gamma\gamma$ | $ZZ^{*}$ | $WW^{*}$ | $VV^{*}$ | $\tau\tau$ | $bb$ | $\mu\mu$ | $Z\gamma$
---|---|---|---|---|---|---|---|---
ggF |  $1.03^{+0.11}_{-0.11}$ ATLAS:2020qdt  | 0.94${}^{+0.11}_{-0.10}$ ATLAS:2020qdt  | 1.08${}^{+0.19}_{-0.18}$ ATLAS:2020qdt  | * |  $1.02^{+0.60}_{-0.55}$ ATLAS:2020qdt  | * |  $0.61^{+0.75}_{-0.75}$ Aad:2020xfq  | *
VBF |  $1.31^{+0.26}_{-0.23}$ ATLAS:2020qdt  |  $1.25^{+0.50}_{-0.41}$ ATLAS:2020qdt  |  $0.60^{+0.36}_{-0.34}$ ATLAS:2020qdt  | * |  $1.15^{+0.57}_{-0.53}$ ATLAS:2020qdt  |  $0.99^{+0.35}_{-0.34}$ Aad:2020ago  |  $1.8^{+1.0}_{-1.0}$ Aad:2020xfq  | *
$Vh_{1}$ |  $1.32^{+0.33}_{-0.30}$ ATLAS:2020qdt  |  $1.53^{+1.13}_{-0.92}$ ATLAS:2020qdt  | * | * | * |  $1.02^{+0.18}_{-0.17}$ ATLAS:2020qdt  | * | *
$Wh_{1}$ | * | * |  $2.3^{+1.2}_{-1.0}$ Aad:2019lpq  | * | * | * | * | *
$Zh_{1}$ | * | * |  $2.9^{+1.9}_{-1.3}$ Aad:2019lpq  | * | * | * | * | *
$t{\overline{t}}h_{1}+th_{1}$ |  $0.90^{+0.27}_{-0.24}$ ATLAS:2020qdt  | * | * |  $1.72^{+0.56}_{-0.53}$ ATLAS:2020qdt  |  $1.20^{+1.07}_{-0.93}$ ATLAS:2020qdt  | * | * | *
$t{\overline{t}}h_{1}$ | * | * | * | * | * |  $0.43^{+0.36}_{-0.33}$ ATLAS-CONF-2020-058  | * | *
$Vh_{1}+t{\overline{t}}h_{1}$ | * | * | * | * | * | * |  $5.0^{+3.5}_{-3.5}$ Aad:2020xfq  | *
ggF+VBF+$t{\overline{t}}h_{1}$ | * | * | * | * | * | * | * |  $2.0^{+1.0}_{-0.9}$ Aad:2020plj

Table 1: ATLAS signal strengths at 13 TeV. The asterisk indicates those signal
strengths are not used in our fits. The production modes are listed in the
rows and the decay modes are listed in the columns.

| $\gamma\gamma$ | $ZZ^{*}$ | $WW^{*}$ | $\tau\tau$ | $bb$ | $\mu\mu$
---|---|---|---|---|---|---
ggF |  $0.98^{+0.13}_{-0.10}$ CMS:2020omd  |  $0.98^{+0.12}_{-0.11}$ CMS:2020gsy  |  $1.28^{+0.20}_{-0.19}$ CMS:2020gsy  |  $0.39^{+0.38}_{-0.39}$ CMS:2020gsy  |  $2.45^{+2.53}_{-2.35}$ CMS:2020gsy  |  $0.63^{+0.65}_{-0.64}$ CMS:2020eni
VBF |  $1.15^{+0.36}_{-0.31}$ CMS:2020omd  |  $0.57^{+0.46}_{-0.36}$ CMS:2020gsy  |  $0.63^{+0.65}_{-0.61}$ CMS:2020gsy  |  $1.05^{+0.30}_{-0.29}$ CMS:2020gsy  | * |  $1.36^{+0.69}_{-0.61}$ CMS:2020eni
$Vh_{1}$ |  $0.71^{+0.31}_{-0.28}$ CMS:2020omd  |  $1.10^{+0.96}_{-0.74}$ CMS:2020gsy  | * | * | * |  $5.48^{+3.10}_{-2.83}$ CMS:2020eni
$Wh_{1}$ | * | * |  $2.85^{+2.11}_{-1.87}$ CMS:2020gsy  |  $3.01^{+1.65}_{-1.51}$ CMS:2020gsy  |  $1.27^{+0.42}_{-0.40}$ CMS:2020gsy  | *
$Zh_{1}$ | * | * |  $0.90^{+1.77}_{-1.43}$ CMS:2020gsy  |  $1.53^{+1.60}_{-1.37}$ CMS:2020gsy  |  $0.93^{+0.33}_{-0.31}$ CMS:2020gsy  | *
$t{\overline{t}}h_{1}+th_{1}$ |  $1.40^{+0.33}_{-0.27}$ CMS:2020omd  |  $0.13^{+0.93}_{-0.13}$ CMS-PAS-HIG-19-001  | * | * | * | *
$t{\overline{t}}h_{1}$ | * | * |  $0.93^{+0.48}_{-0.45}$ CMS:2020gsy  |  $0.92^{+0.26}_{-0.23}$ CMS:2020iwy  |  $1.13^{+0.33}_{-0.30}$ CMS:2020gsy  |  $2.32^{+2.27}_{-1.95}$ CMS:2020eni
$th_{1}$ | * | * | * |  $5.7^{+4.1}_{-4.0}$ CMS:2020iwy  | * | *

Table 2: CMS signal strengths at 13 TeV. The asterisk indicates those signal
strengths are not used in our fits. The production modes are listed in the
rows and the decay modes are listed in the columns.

###  C.2 Scalar Search Bounds

Now we give the relevant observed and expected scalar cross section upper
bounds from ATLAS in Tab. 3 and CMS in Tab. 4. For searches for $pp\rightarrow
h_{2}\rightarrow h_{1}h_{1}$, several of the experimental papers
Aaboud:2018ftw ; Aaboud:2018zhh ; Aaboud:2018ksn ; CMS-PAS-HIG-18-013  report
bounds on the production cross section $\sigma(pp\rightarrow h_{2}\rightarrow
h_{1}h_{1})$, not including the $h_{1}$ decays. To do this, the CMS and ATLAS
collaborations assume that $h_{1}$ decays are SM-like. However, in our model
we also change the branching ratios of $h_{1}$ and we need bounds on the cross
section $\sigma(pp\rightarrow h_{2}\rightarrow h_{1}h_{1}\rightarrow 2X\,2Y)$
including $h_{1}$ decays, where $X$ and $Y$ are $h_{1}$ decay products. Hence,
when the experimental searches are reported as bounds on $\sigma(pp\rightarrow
h_{2}\rightarrow h_{1}h_{1})$ we multiply the bounds by the relevant SM
$h_{1}$ branching ratios as provided by the LHC Higgs Cross Section Working
Group deFlorian:2016spz . This step eliminates the assumption that $h_{1}$
decays are SM-like and provides the relevant cross section bounds for our
model.

$\sigma$(pb) |  $m_{2}=200$ GeV |  $m_{2}=400$ GeV |  $m_{2}=600$ GeV
---|---|---|---
| Obs. | Exp. | Obs. | Exp. | Obs. | Exp.
$pp\rightarrow h_{2}\rightarrow Z\gamma$ | * | * | 0.027 pb Aaboud:2017uhw  | 0.027 pb Aaboud:2017uhw  | 0.0081 pb Aaboud:2017uhw  | 0.0125 pb Aaboud:2017uhw
$pp\rightarrow h_{2}\rightarrow\gamma\gamma$ | 0.0053 pb ATLAS:2020tws  | 0.0041 pb ATLAS:2020tws  | 0.0013 pb ATLAS:2020tws  | 0.0010 pb ATLAS:2020tws  | 0.00057 pb ATLAS:2020tws  | 0.00046 pb ATLAS:2020tws
$pp\rightarrow h_{2}\rightarrow ZZ$ | 0.11 pb Aad:2020fpj  | 0.12 pb Aad:2020fpj  | 0.050 pb Aad:2020fpj  | 0.039 pb Aad:2020fpj  | 0.023 pb Aad:2020fpj  | 0.017 pb Aad:2020fpj
$pp\rightarrow h_{2}\rightarrow WW$ | 6.4 pb Aaboud:2017gsl  | 5.6 pb Aaboud:2017gsl  | 1.29 pb Aaboud:2017gsl  | 1.22 pb Aaboud:2017gsl  | 0.30 pb Aaboud:2017gsl  | 0.43 pb Aaboud:2017gsl
$pp\rightarrow h_{2}\rightarrow h_{1}h_{1}\rightarrow b\bar{b}b\bar{b}$ | * | * | 0.222 pb Aaboud:2018knk  | 0.210 pb Aaboud:2018knk  | 0.0273 pb Aaboud:2018knk  | 0.0391 pb Aaboud:2018knk
$pp\rightarrow h_{2}\rightarrow h_{1}h_{1}\rightarrow b\bar{b}\gamma\gamma$ | * | * | 0.00209 pb Aaboud:2018ftw  | 0.00176 pb Aaboud:2018ftw  |  $7.11\times 10^{-4}$ pb Aaboud:2018ftw  |  $8.33\times 10^{-4}$ pb Aaboud:2018ftw
$pp\rightarrow h_{2}\rightarrow h_{1}h_{1}\rightarrow b\bar{b}\tau^{+}\tau^{-}$ | * | * | 0.0397 pb Aaboud:2018sfw  | 0.0445 pb Aaboud:2018sfw  | 0.00907 pb Aaboud:2018sfw  | 0.0117 pb Aaboud:2018sfw
$pp\rightarrow h_{2}\rightarrow h_{1}h_{1}\rightarrow b\bar{b}WW$ | * | * | * | * | 0.301 pb Aaboud:2018zhh  | 0.319 pb Aaboud:2018zhh
$pp\rightarrow h_{2}\rightarrow h_{1}h_{1}\rightarrow\gamma\gamma WW$ | * | * | 0.0090 pb Aaboud:2018ewm  | 0.00630 pb Aaboud:2018ewm  | * | *
$pp\rightarrow h_{2}\rightarrow h_{1}h_{1}\rightarrow WWWW$ | * | * | 0.242 pb Aaboud:2018ksn  | 0.309 pb Aaboud:2018ksn  | * | *
$pp\rightarrow h_{2}\rightarrow\tau^{+}\tau^{-}$ | 0.23 pb Aad:2020zxo  | 0.424 pb Aad:2020zxo  | 0.0836 pb Aad:2020zxo  | 0.0408 pb Aad:2020zxo  | 0.015 pb Aad:2020zxo  | 0.0128 pb Aad:2020zxo
$pp\rightarrow h_{2}\rightarrow\mu^{+}\mu^{-}$ | 0.0442 pb Aaboud:2019sgt  | 0.0258 pb Aaboud:2019sgt  | 0.0074 pb Aaboud:2019sgt  | 0.0079 pb Aaboud:2019sgt  | 0.006 pb Aaboud:2019sgt  | 0.0041 pb Aaboud:2019sgt
$pp\rightarrow h_{2}\rightarrow jj$ | * | * | * | * | 9.95 pb Aaboud:2018fzt  | 10.5 pb Aaboud:2018fzt

Table 3: Observed (Obs.) and expected (Exp) 95 $\%$ CL ATLAS upper limit on
cross section times branching ratio for heavy resonances at center of mass
energy 13 TeV. The asterisk indicates that either there were no relevant
results or they were not included in the fit.

$\sigma$(pb) |  $m_{2}=200$ GeV |  $m_{2}=400$ GeV |  $m_{2}=600$ GeV
---|---|---|---
| Obs. | Exp. | Obs. | Exp. | Obs. | Exp.
$pp\rightarrow h_{2}\rightarrow Z\gamma$ | * | * | 0.022 pb Sirunyan:2017hsb  | 0.027 pb Sirunyan:2017hsb  | 0.016 pb Sirunyan:2017hsb  | 0.013 pb Sirunyan:2017hsb
$pp\rightarrow h_{2}\rightarrow\gamma\gamma$ | * | * | * | * | 0.0014 Sirunyan:2018wnk  pb | 0.0016 pb Sirunyan:2018wnk
$pp\rightarrow h_{2}\rightarrow ZZ$ | 0.24 pb Sirunyan:2018qlb  | 0.33 pb Sirunyan:2018qlb  | 0.067 pb Sirunyan:2018qlb  | 0.083 pb Sirunyan:2018qlb  | 0.025 pb Sirunyan:2018qlb  | 0.032 pb Sirunyan:2018qlb
$pp\rightarrow h_{2}\rightarrow WW$ | 6.8 pb CMS:2019kjn  | 5.9 pb CMS:2019kjn  | 1.0 pb CMS:2019kjn  | 1.21 pb CMS:2019kjn  | 0.43 pb CMS:2019kjn  | 0.33 pb CMS:2019kjn
$pp\rightarrow h_{2}\rightarrow h_{1}h_{1}\rightarrow b\bar{b}b\bar{b}$ | * | * | 0.163 pb Sirunyan:2018zkk  | 0.179 pb Sirunyan:2018zkk  | 0.0355 pb Sirunyan:2018zkk  | 0.0396 pb Sirunyan:2018zkk
$pp\rightarrow h_{2}\rightarrow h_{1}h_{1}\rightarrow b\bar{b}\gamma\gamma$ | * | * | 0.0012 pb Sirunyan:2018iwt  | 0.0012 pb Sirunyan:2018iwt  |  $4.4\times 10^{-4}$ pb Sirunyan:2018iwt  |  $4.4\times 10^{-4}$ pb Sirunyan:2018iwt
$pp\rightarrow h_{2}\rightarrow h_{1}h_{1}\rightarrow b\bar{b}\tau^{+}\tau^{-}$ | * | * | 0.0860 pb Sirunyan:2017djm  | 0.106 pb Sirunyan:2017djm  | 0.0314 pb Sirunyan:2017djm  | 0.0200 pb Sirunyan:2017djm
$pp\rightarrow h_{2}\rightarrow h_{1}h_{1}\rightarrow b\bar{b}ZZ$ | * | * | 0.659 pb Sirunyan:2020qcq  | 0.92 pb Sirunyan:2020qcq  | 0.16 pb Sirunyan:2020qcq  | 0.305 pb Sirunyan:2020qcq
$pp\rightarrow h_{2}\rightarrow\tau^{+}\tau^{-}$ | 0.823 pb Sirunyan:2018zut  | 0.887 pb Sirunyan:2018zut  | 0.0743 pb Sirunyan:2018zut  | 0.0851 pb Sirunyan:2018zut  | 0.0495 pb Sirunyan:2018zut  | 0.0294 pb Sirunyan:2018zut
$pp\rightarrow h_{2}\rightarrow\mu^{+}\mu^{-}$ | 0.0175 pb Sirunyan:2019tkw  | 0.0112 pb Sirunyan:2019tkw  | 0.0028 pb Sirunyan:2019tkw  | 0.003 pb Sirunyan:2019tkw  | 0.00175 pb Sirunyan:2019tkw  | 0.00175 pb Sirunyan:2019tkw
$pp\rightarrow h_{2}\rightarrow dijets(gg)$ | * | * | * | * | 71.8 pb Sirunyan:2018xlo  | 38.4 pb Sirunyan:2018xlo

Table 4: Observed (Obs.) and expected (Exp) 95 $\%$ CL CMS upper limit on
cross section times branching ratio for heavy resonances at center of mass
energy 13 TeV. The asterisk indicates that either there were no relevant
results or they were not included in the fit.

##  Appendix D 95% C.L. Limits

Here we show all other Wilson coefficient 95% CL allowed regions not shown in
the main text. Fig. 8 shows the limits on $f_{b}$ and $f_{\tau}$, Fig. 9 shows
the limits on $f_{-}$ and $f_{+}$, and Fig. 10 shows the limits on $f_{\mu}$.

Figure 8: Same as Fig. 6 for (a,b,c) $f_{b}$and (d,e,f) $f_{\tau}$ vs
$\sin\theta$ with all other parameters profiled over. Three scalar masses are
considered: (a,d) $m_{2}=200$ GeV, (b,e) $m_{2}=400$ GeV, and (c,f)
$m_{2}=600$ GeV. The new physics scale is $\Lambda=3$ TeV.

Figure 9: Same as Fig. 6 for (a,b,c) $f_{-}$and (d,e,f) $f_{+}$ vs
$\sin\theta$ with all other parameters profiled over. Three scalar masses are
considered: (a,d) $m_{2}=200$ GeV, (b,e) $m_{2}=400$ GeV, and (c,f)
$m_{2}=600$ GeV. The new physics scale is $\Lambda=3$ TeV.

Figure 10: Same as Fig. 6 for $f_{\mu}$ vs $\sin\theta$ with all other
parameters profiled over. Three scalar masses are considered: (a,d)
$m_{2}=200$ GeV, (b,e) $m_{2}=400$ GeV, and (c,f) $m_{2}=600$ GeV. The new
physics scale is $\Lambda=3$ TeV.

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