# Non-linear top-Higgs CP violation

Akanksha Bhardwaj<EMAIL_ADDRESS>School of Physics &
Astronomy, University of Glasgow, Glasgow G12 8QQ, UK
Department of Physics, Oklahoma State University, Stillwater, OK, 74078, USA
Christoph Englert<EMAIL_ADDRESS>School of Physics &
Astronomy, University of Glasgow, Glasgow G12 8QQ, UK
Dorival Gonçalves<EMAIL_ADDRESS>Department of Physics, Oklahoma State
University, Stillwater, OK, 74078, USA
Alberto Navarro<EMAIL_ADDRESS>Department of Physics,
Oklahoma State University, Stillwater, OK, 74078, USA

###### Abstract

Searches for additional sources of CP violation at the Large Hadron Collider
are a central part of the Higgs physics programme beyond the Standard Model.
Studies employing so-called signed observables that track CP violation through
purpose-built asymmetries bolster efforts based on Higgs boson rate analyses
under clear assumptions. A possibility, which is so far unexplored at the LHC,
is a significant non-linear realisation of CP-violation, which is naturally
described in non-linear Higgs Effective Field Theory (HEFT). We perform an
analysis of the HL-LHC potential to constrain such interactions considering a
large range of single and double Higgs production processes, including
differential information where this is statistically and theoretically
possible. A particular emphasis of our work is distinguishing expected
correlations in the Standard Model Effective Field Theory from those
attainable in HEFT.

###### pacs:

##  I Introduction

The interactions of the Higgs boson are generally considered as harbingers of
new interactions beyond the Standard Model (BSM). The precision study of the
Higgs boson at the Large Hadron Collider (LHC) has therefore opened a new
territory in our understanding of the electroweak scale. While the precise
nature of the latter is still unclear, it is reasonable to expect that
whatever the mechanism responsible for electroweak symmetry breaking, it might
have wider ramifications for the as yet unresolved questions of the SM.

In BSM scenarios, such as multi-Higgs extensions, the Higgs boson interactions
can introduce additional sources of CP violation which can address one of the
Sakharov criteria that the SM falls short of Basler et al. (2023); Anisha et
al. (2022); Gonçalves et al. (2023). From a theoretical standpoint, certain
Higgs couplings are more susceptible to pronounced new physics effects. For
instance, the extensively studied CP-odd Higgs-vector boson interactions can
appear only through operators of dimension-six or higher Buchmuller and Wyler
(1986); Grzadkowski et al. (2010), being naturally suppressed by the new
physics scale. In contrast, CP-odd Higgs-fermion couplings can already appear
at tree level leading to naturally larger CP violation effects. The top quark
Yukawa coupling, owing to its magnitude, plays a crucial role in this
discussion and emerges as a particularly sensitive probe for physics beyond
the SM.

Model-agnostic approaches employing effective field theory highlight a range
of effective interactions in a coarse-grained dimension-six approach that have
been scrutinized in a range of experimental analyses at the LHC so far. In
particular, additional (C)P violation in the top-Higgs sector

$\sim i\,\bar{t}\gamma^{5}t\,h$ (1)

can be constrained in gluon fusion Plehn et al. (2002); Hankele et al. (2006);
Klamke and Zeppenfeld (2007) and top-Higgs production Buckley and Gonçalves
(2016a); Boudjema et al. (2015); Demartin et al. (2014); Casolino et al.
(2015).111Approaches to disentangle these top-Yukawa interaction modifications
from $\sim G_{\mu\nu}\tilde{G}^{\mu\nu}h$ contact interactions have been
discussed in Englert et al. (2019). The relevance of CP-violating Yukawa
interactions for low-energy precision dipole measurements have been revisited
recently in Ref. Brod et al. (2023).

One way of pinning down such interactions phenomenologically at hadron
colliders is the construction of asymmetric observables, which then serve as
strong tests of CP-violation without relying on CP-even rate information such
as cross sections or transverse momentum spectra. However, for some processes,
the expected rate even at 3/ab of the high-luminosity (HL) LHC phase is too
limited to construct statistically sensible asymmetries. In addition, some
processes, e.g. involving scalar final states, do not show interference-
related asymmetries. Either case then warrants their inclusion under the
hypothesis that no additional sources of new physics are present, relying on
simple hypothesis testing.

In this work, we ask the question of how sensitive the LHC can be to sources
of _non-linear CP violation_. While Ref. Englert et al. (2019) discusses
approaches to disentangle gluonic from top-philic sources, the question of how
correlated CP violation across different Higgs multiplicities remains open.
Such freedom becomes apparent within the context of Higgs Effective Field
Theory (HEFT) when contrasted with correlations expected within the Standard
Model Effective Field Theory (SMEFT) Grzadkowski et al. (2010); Dedes et al.
(2017). This possibility also opens up a novel avenue to decouple dipole
moment constraints from TeV scale investigations. As shown in Brod et al.
(2023), dipole constraints are highly constraining when considering
exclusively the interaction of Eq. (1), but can be significantly relaxed when
considering analogous CP violation for light flavour Higgs interactions. This
comes at the price of a loss of phenomenological sensitivity, as such Higgs
interactions are phenomenologically not always accessible at the LHC. CP
violation measured in low energy dipole measurements dominantly sourced in di-
Higgs interactions would be further loop and light-flavour Yukawa suppressed
and will therefore be less constrained.

This note is organized as follows: In Sec. II, we introduce the interactions
studied in this work. Particular emphasis is given to the distinctive patterns
of CP violation predicted in SMEFT as opposed to the more general HEFT
parametrisation. The accurate discrimination of non-linear CP violation
requires a robust statistical handle on single Higgs production processes,
serving as a prerequisite for the subsequent utilization of di-Higgs
production to effectively constrain non-linearity. The processes and the
assumptions under which they are included in this work are detailed in Sec.
III. Sec. IV is devoted to the discussion of our fit to non-linear CP
violation. We summarize in Sec. V.

##  II HEFTy CP violation

As alluded to above, within the SMEFT approach, we consider the operator

${\cal{O}}_{t\Phi}=|\Phi|^{2}\bar{Q}_{L}\Phi^{c}t_{R}\,.$ (2)

$\Phi$ denotes the Higgs doublet, $\Phi^{c}=i\sigma^{2}\Phi^{\ast}$, and
$Q_{L},t_{R}$ are the left and right-chiral fermion doublet and singlet
relevant for the top interactions, respectively. This operator leads in the
broken phase to P-violating interactions for complex Wilson coefficients.
Therefore, signs of CP-violation across different Higgs multiplicities are
correlated as a consequence of the $SU(2)$ doublet structure of the Higgs
boson. For instance, the CP-violating tree-level three and four-point
irreducible vertex functions obey

${\Gamma_{\bar{t}th}\over\Gamma_{\bar{t}th^{2}}}\bigg{|}_{\gamma^{5},\text{SMEFT}}={v\over
3}\,,$ (3)

with $v\simeq 246~{}\text{GeV}$ as the vacuum expectation value of the Higgs
field. In this context, although additional sensitivity from CP-sensitive
observables in $t\bar{t}hh$ production are welcome, CP-violation in the top
Higgs sector under the assumptions of SMEFT should manifest themselves
predominantly in single Higgs physics, which provides the most significant
statistical pull in a global analysis.

A phenomenologically identical parametrisation of Eq. (2), which we will use
in the following, is given by

${\cal{L}}^{\text{SMEFT}}_{\alpha,1}=-\frac{m_{t}}{v}\,\kappa_{t}\,\bar{t}(\cos\alpha+i\gamma^{5}\sin\alpha)\,t\,h\,.$
(4)

Here, $\alpha$ represents the CP-phase and $\kappa_{t}$ is a real number that
determines the strength of the interaction. In this parametrization, the SM is
characterized by $\kappa_{t}=1$ and $\alpha=0$. Conversely, for a purely CP-
odd interaction, $\alpha$ would be equal to $\pi/2$. This parametrization can
be identified with Eq. (2) (after renormalisation of the SM Yukawa couplings
and assuming a purely CP-even SM coupling of the top quark)

${1\over\Lambda^{2}}\left(\begin{matrix}{\text{Re}\,{C_{t\Phi}}}\\\
{\text{Im}\,{C_{t\Phi}}}\end{matrix}\right)=-{\sqrt{2}\,m_{t}\over
v^{3}}\left(\begin{matrix}\kappa_{t}\cos\alpha-1\\\
\kappa_{t}\sin\alpha\end{matrix}\right)\,.$ (5)

This directly leads to

${\cal{L}}_{\alpha,2}^{\text{SMEFT}}\supset-{3m_{t}\over
2v^{2}}\,\bar{t}(\\{\kappa_{t}\cos\alpha-1\\}+i\kappa_{t}\gamma^{5}\sin\alpha)\,t\,h^{2}\,,$
(6)

which also shows that the $t\bar{t}hh$ interactions vanish for the SM point,
$(\kappa_{t},\alpha)_{\text{SM}}=(1,0)$.

Turning to HEFT, which highlights the Higgs boson as a custodial singlet
Appelquist and Bernard (1980); Longhitano (1980, 1981); Feruglio (1993);
Appelquist and Wu (1993); Brivio et al. (2014, 2016), the top quark mass
arises from the non-linear sigma model of $SU(2)_{L}\times SU(2)_{R}\to
SU(2)_{V}$ that can be parametrized as

$U(\pi^{a})=\exp\left({i\pi^{a}\tau^{a}/v}\right)\,,$ (7)

with $SU(2)$ generators $\tau^{a}$ and Goldstone fields $\pi^{a}$. This field
transforms under general $SU(2)_{L}\times SU(2)_{R}$ transformations as $U\to
L\,UR^{\dagger}$ so that the top quark mass arises from

${\cal{O}}_{\bar{t}t}=-m_{t}\,\bar{Q}_{L}Ut_{R}\,.$ (8)

Owing to the singlet character of the Higgs boson in HEFT, this operator can
be dressed with a “flare” function

$Y_{t}(h)=1+c^{(1)}{h\over v}+c^{(2)}{h^{2}\over 2v^{2}}+\dots\,,$ (9)

suppressing higher monomials of the singlet Higgs field, which are
phenomenologically not relevant. This leads to CP-violating effects analogous
to ${\cal{L}}_{\alpha}$

${\cal{L}}_{\text{HEFT}}\supset-\frac{m_{t}}{v}\,\kappa_{t}\,\bar{t}(\cos\alpha+i\gamma^{5}\sin\alpha)\,t\,h\\\
-{m_{t}\over
2v^{2}}\,\kappa_{tt}\,\bar{t}(\cos\beta+i\gamma^{5}\sin\beta)\,t\,h^{2}\,.$
(10)

However, it is important to note a significant exception: the Higgs
multiplicities remain uncorrelated in this context. The expressions for
$c^{(1)}$ and $c^{(2)}$ become

$c^{(1)}=\kappa_{t}\,e^{i\alpha}\,,~{}c^{(2)}=\kappa_{tt}\,e^{i\beta}\,.$ (11)

The relative strength of CP-violation for the three and four-point
interactions is now characterized by

${\Gamma_{\bar{t}th}\over\Gamma_{\bar{t}th^{2}}}\bigg{|}_{\gamma^{5},\text{HEFT}}={\kappa_{t}\over\kappa_{tt}}\,{\sin\alpha\over\sin\beta}\,v\,,$
(12)

where the SMEFT trajectory can be recovered by the HEFT choices

$\begin{split}\kappa_{tt}^{2}&=9(1-2\kappa_{t}\cos\alpha+\kappa_{t}^{2})\,,\\\
\tan\beta&={\kappa_{t}\sin\alpha\over\kappa_{t}\cos\alpha-1}\,.\end{split}$
(13)

CP measurements at ATLAS and CMS are typically carried by constructing
asymmetries or “signed” observables which isolate interference effects between
new physics and SM contributions. Writing the amplitude of the scattering
process ${\cal{M}}={\cal{M}}_{\text{SM}}+{\cal{M}}_{\cal{O}}$, with
${\cal{M}}_{\cal{O}}$ denoting the BSM part, the partonic cross sections scale
as

${\text{d}\sigma\over\text{dLIPS}}\sim|{\cal{M}}_{\text{SM}}|+2\text{Re}({\cal{M}}_{\text{SM}}{\cal{M}}_{\cal{O}}^{\ast})+|{\cal{M}}_{\cal{O}}|^{2}\,.$
(14)

Squared CP-odd contributions manifest in CP-even distributions, such as cross
sections, transverse momentum distributions, etc. The interference effects
between SM and new physics cancel in these CP-even distributions and are
resolved through purpose-built observables. However, for processes with
limited statistics, achieving a binned distribution might not always be
attainable, even during the high-luminosity phase of the Large Hadron Collider
(HL-LHC). We detail the processes we include in our study in the next Sec.
III.

##  III Sensitive Processes and details of the analysis

###  III.1 Inclusive $gg\to h$ production

Gluon fusion Higgs production has become one of the standard candles to study
electroweak symmetry breaking at the LHC ever since the Higgs boson’s
discovery. The phenomenological precision programme is well underway and the
experiments have laid out a detailed roadmap towards the HL-LHC phase. When
rate information is considered, the cross section and decay widths are known
to provide important handles on potential CP violation (see, e.g., Refs.
Djouadi (2008a, b)). To reflect the sensitivity of this process to phases of
Yukawa interactions as discussed above, we employ the ECFA extrapolation by
CMS outlined in Ref. Sirunyan et al. (2017). Specifically, we consider the
$h\to\gamma\gamma$ and $h\to ZZ$ signal strength extrapolations, which
forecast a sensitivity at 95% CL of

$\displaystyle{\delta\mu\over\mu}(gg\to h\to\gamma\gamma)$
$\displaystyle=3.3\%\,,$ (15) $\displaystyle{\delta\mu\over\mu}(gg\to h\to
ZZ)$ $\displaystyle=4.6\%\,.$ (16)

We also include $h\to\tau\tau$ based on an extrapolation of Ref. Sirunyan et
al. (2018a) which sets

$\displaystyle{\delta\mu\over\mu}(gg\to h\to\tau\tau)$
$\displaystyle=9.7\%\,.$ (17)

This aligns with the ECFA projection presented in Ref. Sirunyan et al. (2017).
To achieve this, we use MadGraph5_aMC@NLO to interpolate the cross section,
using a model generated with FeynRules Alloul et al. (2014), NLOCT Degrande
(2015), and UFO Degrande et al. (2012) in the finite top mass limit. This
interpolation accounts for various coupling choices and is then reweighed
based on the SM result to reflect higher-order QCD corrections Djouadi et al.
(1991); Anastasiou et al. (2015); Dittmaier et al. (2011). Throughout this
work, we take into account the modifications of the Higgs branching ratios due
to the modified top-Yukawa couplings.

###  III.2 Gluon fusion $h+2j$ production

Figure 1: Distribution for the azimuthal angle difference between the two
tagging jets $\Delta\phi_{jj}$, as defined in Eq. (18), specifically for the
$h+2j$ sample.

The production of a single Higgs boson in association with two jets is a
sensitive process due to the introduction of the ‘signed’ angular separation
between the tagging jets Plehn et al. (2002); Klamke and Zeppenfeld (2007);
Buschmann et al. (2014). Ordering the jets in rapidity $\eta_{j1}>\eta_{j2}$,
the azimuthal angular difference

$\Delta\phi_{jj}=\phi_{j1}-\phi_{j2}\,$ (18)

leads to a characteristic angular modulation, which can be exploited to set
constraints on the involved CP-odd interactions. This renders $h+2j$ as a
prime candidate for constraining the single Higgs property as compared to the
non-linear deviations.222Gluon fusion of Higgs pairs in association with two
jets has been studied in Ref. Dolan et al. (2014) and faces significant
phenomenological challenges at the LHC. Therefore, we will not discuss this
process further. Therefore, this process has been used relatively early in the
LHC Higgs programme to set constraints on sources of CP violation.

For our analysis, we use these ATLAS results as a baseline for extrapolation
Aaboud et al. (2018a). We employ the Vbfnlo Arnold et al. (2009); Baglio et
al. (2011) Monte Carlo to include the finite top-mass effects that shape the
phenomenology of the $h+2j$ final state, including the phase of the top Yukawa
interaction. For illustrational purposes, we present the SM and new physics
$\phi_{jj}$ distributions in Fig. 1. We extract efficiencies for a SM sample
mapped onto the results of Aaboud et al. (2018a) and generalize these to the
BSM parameter choices involving CP-odd contributions, following the procedure
detailed in Englert et al. (2019).

###  III.3 Top-associated Higgs production $t\bar{t}h$

Figure 2: Collins-Soper angle $\theta^{*}$ (left) and azimuthal angle
distribution $\Delta\phi_{\ell\ell}^{t\bar{t}}$ (right) for $t\bar{t}h$ (top)
and $t\bar{t}hh$ (bottom) processes with dileptonic top pair final state. We
consider the SMEFT framework for demonstration purposes.

The $pp\to t\bar{t}h$ channel plays a crucial role in probing the Higgs-top
CP-structure at the tree-level, disentangling possible new physics effects
Ellis et al. (2014); Boudjema et al. (2015); Buckley and Gonçalves (2016a, b);
Gritsan et al. (2016); Gonçalves and Lopez-Val (2016); Mileo et al. (2016);
Amor Dos Santos et al. (2017); Azevedo et al. (2018); Gonçalves et al. (2018);
Aaboud et al. (2018b); Sirunyan et al. (2018b); Bortolato et al. (2021); Cao
et al. (2021); Martini et al. (2021); Gonçalves et al. (2022); Bahl and Brass
(2022); Azevedo et al. (2022); Ackerschott et al. (2023). Several kinematic
observables have been proposed in the literature to investigate the CP
structure of the Higgs-top interaction in this channel. Among those, the
Collins-Soper angle $\theta^{*}$, which is the angle between the top quark and
the beam direction in the $t\bar{t}$ CM frame, features as one of the most
sensitive observables to CP at the non-linear level Gonçalves et al. (2018,
2022) (in the sense of Eq. (14)). Genuine CP-odd observables can also be
defined exploiting the top-quark polarization that is carried over to its
decay products. It is possible to form tensor products involving the top quark
pair and their decay products, represented as
$\epsilon(p_{t},p_{t},p_{i},p_{k})\equiv\epsilon_{\mu\nu\rho\sigma}p_{t}^{\mu}p_{\bar{t}}^{\nu}p_{i}^{\rho}p_{k}^{\sigma}$
Gonçalves et al. (2018); Barman et al. (2022). This tensor product can be
simplified as ${\bf{p}}_{t}\cdot({\bf{p}}_{i}\times{\bf{p}}_{k})$ in the
$t\bar{t}$ CM frame and provides a basis for defining azimuthal angle
differences that exhibit an odd behavior under CP transformations

$\displaystyle\\!\\!\\!\Delta\phi_{ik}^{t\bar{t}}\\!=\\!\text{sgn}\left[{\bf{p}}_{t}\\!\cdot\\!({\bf{p}}_{i}\\!\times\\!{\bf{p}}_{k})\right]\arccos\\!\left(\frac{{\bf{p}}_{t}\\!\times\\!{\bf{p}}_{i}}{|{\bf{p}}_{t}\\!\times\\!{\bf{p}}_{i}|}\\!\cdot\\!\frac{{\bf{p}}_{t}\\!\times\\!{\bf{p}}_{k}}{|{\bf{p}}_{t}\\!\times\\!{\bf{p}}_{k}|}\right)\\!.$
(19)

We present both the Collins-Soper $\theta^{*}$ and the azimuthal angle
distribution $\Delta\phi_{\ell\ell}^{t\bar{t}}$ for dileptonic top pair final
states in the top panel of Fig. 2. The $t\bar{t}hh$ channel, which we will
discuss further below, may provide another complementary avenue to probe the
Higgs-top coupling at the tree-level Englert et al. (2015). Observables that
mirror those defined for the $t\bar{t}h$ process can also be established for
this additional channel as presented in the bottom panel of Fig. 2 (see also
Liu et al. (2015)).

We extract the direct Higgs-top CP sensitivity at the HL-LHC from our previous
analysis in Ref. Barman et al. (2022). In this study, we employ a synergy of
machine learning techniques and streamlined kinematic reconstruction methods
to enhance the new physics sensitivity, exploring the complex $t\bar{t}h$
multi-particle phase space. The analysis encompasses a range of final states,
including hadronic, semi-leptonic, and di-leptonic top pair decays, all in
conjunction with the Higgs decay $h\to\gamma\gamma$. It is noteworthy that the
experimental projections from ATLAS and CMS indicate that the
$h\to\gamma\gamma$ final state will display the dominant sensitivities to the
$t\bar{t}h$ channel at the HL-LHC Cepeda et al. (2019).

###  III.4 $Z$ boson-associated Higgs production

Although the leading contribution for the Higgstrahlung channel $Zh$ arises at
tree level with $q\bar{q}\to Zh$, this channel displays relevant
$\mathcal{O}(\alpha_{s}^{2})$ corrections through the loop-induced gluon
fusion $gg\to Zh$ Altenkamp et al. (2013); Harlander et al. (2014), which are
particularly important in the boosted regime, $p_{Th}\sim m_{t}$ Englert et
al. (2014); Gonçalves et al. (2015); Hespel et al. (2015); Rossia et al.
(2023). Setting limits in these exclusive phase-space regions is an
experimental challenge and to obtain a qualitative sensitivity estimate, we
perform a more detailed signal vs. background investigation.

We denote the $q\bar{q}$ and $gg$ subprocesses as $Zh_{\rm DY}$ and $Zh_{\rm
GF}$, respectively. The $Zh_{\rm GF}$ process exhibits sensitivity to the
linear and quadratic terms of the top-Higgs Yukawa coupling. Owing to the
large destructive interference for the top Yukawa terms, the $Zh_{\rm GF}$
contribution can be sensitive to the magnitude and sign of a possible non-
standard top-Higgs coupling $(\kappa_{t},\alpha)$.333A comprehensive study of
the angular moments for the $Z$ boson in the $Zh_{\rm GF}$ channel is
presented in Appendix A. These probes work as additional analyzers for the
Higgs-top CP violation effects.

We now investigate the sensitivity to new physics in the ${gg\to
Z(\ell\ell)h(b\bar{b})}$ channel. Our signal comprises two charged leptons,
$\ell=e$ or $\mu$, reconstructing a boosted $Z$ boson recoiling against two
$b$-jets. The main background processes are $Zb\bar{b}$, $t\bar{t}$+jets, and
$ZZ$. For our analysis, we generate the signal sample $Zh_{GF}$ using
MadGraph5_aMC@NLO Alwall et al. (2014); Hirschi and Mattelaer (2015), while
the background samples are simulated with Sherpa+OpenLoops Gleisberg et al.
(2009); Cascioli et al. (2012); Denner et al. (2017), following the study
presented in Ref. Gonçalves and Nakamura (2018). The $Zh_{\mathrm{DY}}$,
$Zb\bar{b}$, and $ZZ$ background samples are merged at LO with up to one
additional jet emission using the CKKW algorithm Catani et al. (2001); Hoeche
et al. (2009). We normalize their cross sections to the NLO rates obtained
from Ref. Gonçalves et al. (2015). Additionally, we generate the $t\bar{t}$
background at NLO using the MC@NLO algorithm Frixione and Webber (2002);
Hoeche et al. (2012), considering hadronization and underlying event effects
in our simulation.

To reconstruct the signal events, we require two same-flavor leptons with
opposite-sign charges satisfying ${p_{T\ell}>30}$ GeV and $|\eta_{\ell}|<2.5$,
within the invariant mass range ${75~{}\text{GeV}<m_{\ell\ell}<105}$ GeV. The
$Z$ boson is required to have a large boost, $p_{\mathrm{T}\ell\ell}>200$GeV.
We adopt the BDRS analysis for the $h\to b\bar{b}$ tagging Butterworth et al.
(2008), which involves re-clustering the hadronic activity using the
Cambridge-Aachen jet algorithm Cacciari et al. (2012) with ${R=1.2}$. We
impose at least one boosted fat-jet with $p_{TJ}>200$ GeV and
$|\eta_{J}|<2.5$, Higgs-tagged using the BDRS algorithm, which demands three
sub-jets with the two leading sub-jets being $b$-tagged. We assume a flat 70%
$b$-tagging efficiency and a 1% mis-tag rate. To further improve the signal-
to-background ratio, we impose a constraint on the filtered Higgs mass within
the range ${|m_{h}^{\text{BDRS}}-m_{h}|<10}$ GeV, where $m_{h}=125$ GeV. The
resulting event rate is presented in Tab. 1.

| $Zh_{\mathrm{GF}}$ | $Zh_{\mathrm{DY}}$ | $Zb\bar{b}$ | $t\bar{t}$ | $ZZ$
---|---|---|---|---|---
BDRS reconstruction | 0.03 | 0.16 | 0.35 | 0.02 | 0.02
$|m_{h}^{\rm BDRS}-m_{h}|<10$ GeV
Table 1: Signal rate for $Zh_{\mathrm{GF}}$ and background rates for
$Zh_{\mathrm{DY}}$, $Zb\bar{b}$, $t\bar{t}$, and $ZZ$. The signal is generated
for the SM hypothesis $(\kappa_{t},\alpha)_{\text{SM}}=(1,0)$ and rates are
given in fb after the BDRS analysis.

###  III.5 Beyond linearity: $t\bar{t}hh$ inclusive $hh$ production

We now turn to the discussion of processes that provide genuine sensitivity to
non-linearity via the production of final states containing a pair of Higgs
bosons. Such processes are statistically limited at the LHC, yet in the case
of gluon fusion production $gg\to hh$ relatively well understood, both
theoretically and experimentally. In particular, Higgs pair production has
been subject to considerable experimental scrutiny already, and detailed
experimental forecasts for the HL-LHC frontier have been made available,
similar to the case of $gg\to h$ production. To this end, we consider the
$b\bar{b}\gamma\gamma+b\bar{b}\tau\tau$ extrapolation of Aad et al. (2022)

${\sigma(hh)\over\sigma(hh)_{\text{SM}}}<2\,,$ (20)

at 95% confidence level (CL), which could lower to $1.1$ if systematics become
sufficiently well-controlled. Both $b\bar{b}\tau\tau$ and
$b\bar{b}\gamma\gamma$ have comparable statistical sensitivity and we include
them on an equal footing to our statistical analysis, again taking into
account the effect of modified Higgs branching ratios as a function of
$(\kappa_{t},\alpha)$. Similar to the $gg\to h$ process, we interpolate $gg\to
hh$ production using MadGraph5_aMC@NLO in the finite top mass limit to reflect
the constraint from Eq. (20) within our combined analysis in Sec. IV.

Figure 3: Collins-Soper angle $\theta^{*}$ for the $t\bar{t}hh$ process in the
HEFT framework. We use $\kappa_{t}=\kappa_{tt}=1$ and $\alpha=0$ for
illustration.

In comparison, the $t\bar{t}hh$ process is rather more complex and currently
only proof-of-principle analyses exist, e.g., Refs. Englert et al. (2015); Aad
et al. (2016) for the HL-LHC. The former predicts around 10 signal events in
the SM for a $b$-rich final state. Being statistically limited, shape analyses
of signed observables, which can be constructed similar to the $t\bar{t}h$
process, will not yield relevant exclusion constraints. Selected, relevant
observables for this channel are illustrated in Fig. 2 (bottom panel) and Fig.
3 within the SMEFT and HEFT frameworks, respectively. Given the statistical
limitation, we incorporate the 95% CL cross section exclusion limit for
$t\bar{t}hh$

${\sigma(t\bar{t}hh)\over\sigma(t\bar{t}hh)_{\text{SM}}}<1.4...6.8$ (21)

based on the analyses of Refs. Englert et al. (2015); Aad et al. (2016). This
limit does not include the impact of background systematics, which can reduce
this estimate. However, it is worth highlighting that within the experimental
context, the potential to improve this channel remains relatively unexplored.
We note that the cross section is driven by the four-point interactions
Banerjee et al. (2019), similar to $gg\to hh$, and has been a focus of studies
like for the composite Higgs framework Gröber and Mühlleitner (2011).

##  IV A fit to non-linear CP violation in the top-Higgs sector

The asymmetries and total rates are used to set CL limits on the parameter
space $(\kappa_{t},\alpha,\kappa_{tt},\beta)$, assuming the SM as the null
hypothesis. To this end, we consider a $\chi^{2}$ statistic defined as

$\chi^{2}=\sum_{i}{(N_{i}-N_{i}^{\text{SM}})^{2}\over\sigma^{2}_{i}}\,.$ (22)

Here, the index $i$ runs over a binned distribution where this is
statistically warranted, or $i=1$ for constraints from cross sections. $N_{i}$
denotes the event count in a particular bin (or the entire signal event count
for cross sections) for a given luminosity, which we set to
${\cal{L}}=3~{}\text{ab}^{-1}$. We tune the uncertainties $\sigma^{2}_{i}$ to
reproduce the quoted single channel sensitivities. Given these individual
$\chi^{2}$ contributions, we can then consider their combination, increasing
the degrees of freedom depending on the hypothesis under investigation.

Figure 4: $95\%$ CL limits on $(\kappa_{t},\alpha)$ for the 13 TeV HL-LHC with
$3~{}\text{ab}^{-1}$ in the SMEFT framework, for which $(\kappa_{tt},\beta)$
are related to the former via Eq. (13).

Before we turn to combinations and the comparison between SMEFT and HEFT, it
is instructive to highlight the sensitivity of each of these channels and how
multi-Higgs production serves as means to distinguish non-linearity. We will
focus on the HL-LHC data set in the following.

In Fig. 4, we show the sensitivity of all channels before combination,
focusing on the SMEFT parametrization that singles out the correlation of Eq.
(13) across the different Higgs multiplicities. As expected, the most
sensitive channels in SMEFT are those with highest statistical abundance.
Under the assumption of suppressed competing coupling modification in SMEFT,
this is given by the inclusive gluon fusion rate along the $\kappa_{t}$
direction. Exploitable angular correlations in the $h+2j$ mode augment the
sensitivity along the direction of the CP angle.444This information is also
accessible at the interference level in Eq. (14) and is therefore relatively
robust with regard to linearizing differential cross sections in the EFT
expansion. Directly probing the top Yukawa coupling through the $t\bar{t}h$
channel also leads to relevant complementary constraints.

Figure 5:  $95\%$ CL limits on the $(\kappa_{t},\alpha)$ plane at the 13 TeV
HL-LHC with $3~{}\text{ab}^{-1}$ of data for $hh$ and $t\bar{t}hh$ channels in
SMEFT (top) and HEFT (bottom) frameworks with $(\kappa_{tt}=0,\beta=0)$. We
also highlight the importance of the $t\bar{t}hh$ channel in collapsing the
available parameter space, a non-trivial combination is shown for the
stringent $t\bar{t}hh$ assumption.

Given the reduced sensitivity in the multi-Higgs production, the
$(\kappa_{t},\alpha)$ constraints carry over from the SMEFT parametrization to
HEFT, modulo changes in the number of degrees of freedom and the small pull
provided by the SMEFT correlation in light of the correlation of Eq. (13). The
importance of the latter correlation becomes clear when contrasting the $gg\to
hh$ and $t\bar{t}hh$ combination in SMEFT against HEFT for
$\kappa_{tt}=\beta=0$, as depicted in Fig. 5. This comparison highlights the
relevance of quartic $t\bar{t}hh$ contact interactions for these final states.
As can be seen, these particular BSM contact interactions drive the cross
section for the di-Higgs production modes.

Assuming a SM value in HEFT for the single Higgs modes,
$(\kappa_{t},\alpha)=(1,0)$, the expected constraints from purely non-linear
CP violation are given in Fig. 6. As can be seen, the multi-Higgs production
modes can be used to set constraints mostly on the magnitude of the contact
interaction, whilst the expected sensitivity is not large enough to constrain
its phase. This blind direction could potentially be explored through the
multi-particle final state kinematics as illustrated in Fig. 3. However,
achieving this may necessitate a higher event rate and might realistically
only become feasible at upcoming higher-energy colliders, such as the FCC-hh
Banerjee et al. (2019).

SMEFTy extensions close to the decoupling limit select a subspace of HEFT.
Given the correlation predicted by SMEFT-like extensions of the SM, we can
therefore employ these production modes to highlight the expected sensitivity
for $\kappa_{tt},\beta$ when comparing SMEFT and HEFT in Fig. 6. The SMEFT
contour highlights the correlation of a combined fit of the most sensitive
single Higgs channels in SMEFT projected onto the $(\beta,\kappa_{tt})$ plane
given the correlations of Eq. (13). For illustration purposes, we limit the
HEFT parameter space to SM couplings in the single Higgs sector (the
corresponding couplings will be relatively well measured at 3/ab and the di-
Higgs cross sections are predominantly sensitive to the multi-Higgs
couplings). Clearly, a SM-like outcome of the single Higgs measurements
renders the available parameter space in the di-Higgs couplings relatively
limited in SMEFT. Even if the optimistic $t\bar{t}hh$ constrain is relaxed to
looser constraints, $gg\to hh$ production is still sensitive to significant
quartic $t\bar{t}hh$ vertices and associated CP violation in HEFT.

When reducing the size of $\kappa_{tt}$, the sensitivity to $\beta$ is
naturally suppressed. Higher sensitivity in the relevant channels is therefore
key to further maximise the LHC potential: the $t\bar{t}hh$ contour in Fig. 6
only slightly bends for larger angles $\beta$. Perhaps an unrealistic
improvement over the quoted constraints would extend the $\beta$ sensitivity.
The role of $t\bar{t}hh$ production remains critical, even when only the
$\kappa_{tt}$ effects are considered. Feasibility studies beyond a first
exploratory studies, e.g. Aad et al. (2016), should continue to maximise the
value of LHC data. Of course, the statistical limitations present for multi-
Higgs mode at the LHC are naturally relaxed at a future hadron machine such as
FCC-hh, envisioned to operate at 100 TeV with a target luminosity of
$30~{}\text{ab}^{-1}$. A more fine grained approach exploiting angular
correlations as demonstrated in Fig. 2 (bottom panel) and Fig. 3 will become
possible, which will lead to a qualitatively new $t\bar{t}hh$ exclusion.

Figure 6:  $95\%$ CL limits on $(\kappa_{tt},\beta)$ at the 13 TeV HL-LHC with
$3~{}\text{ab}^{-1}$ of data for $hh$ and $t\bar{t}hh$ channels in the HEFT
framework with $(\kappa_{t}=1,\alpha=0)$. The SMEFT region selected from a fit
to single Higgs data is highlighted for comparison.

##  V Conclusions

In this work, we have examined the potential of the LHC to constrain CP phases
of the top-Yukawa interactions combining the sensitivity of a range of single-
and double Higgs production processes. Single Higgs processes encompass all
the relevant correlations in dimension-six SMEFT, and multi-Higgs production
does not lead to significant sensitivity gain. However, this paradigm shifts
when considering non-linear sources of CP violation. Given the limited rates
of multi-Higgs production at the LHC, the resulting constraints are naturally
less stringent than those anticipated from single Higgs physics, especially
when incorporating rate information under appropriate assumptions.
Nonetheless, the LHC shows sensitivity, in particular when discriminating
between SMEFTy and HEFTy CP violation in the top-Higgs sector.

| $A_{0}$ | $A_{1}$ | $A_{2}$ | $A_{3}$ | $A_{4}$ | $A_{5}$ | $A_{6}$ | $A_{7}$
---|---|---|---|---|---|---|---|---
SM | 0.027 $\pm 0.003$ | 0.001 $\pm 0.005$ | -0.976 $\pm 0.005$ | -0.003 $\pm 0.004$ | -0.003 $\pm 0.004$ | 0.000 $\pm 0.004$ | 0.001 $\pm 0.005$ | -0.003 $\pm 0.005$
$\alpha=\pi/4$ | 0.016 $\pm 0.003$ | -0.006 $\pm 0.005$ | -0.968 $\pm 0.005$ | 0.002 $\pm 0.004$ | 0.003 $\pm 0.004$ | -0.009 $\pm 0.004$ | 0.000 $\pm 0.005$ | 0.011 $\pm 0.005$
$\alpha=-\pi/4$ | 0.017 $\pm 0.003$ | 0.001 $\pm 0.005$ | -0.962 $\pm 0.005$ | -0.002 $\pm 0.004$ | 0.002 $\pm 0.004$ | -0.002 $\pm 0.004$ | 0.003 $\pm 0.005$ | 0.003 $\pm 0.005$
Table 2: Angular coefficients $A_{i}$ for the loop-induced process
$gg\rightarrow\ell^{+}\ell^{-}h$ with minimum selections $p_{T,\ell\ell}>200$
GeV, $75~{}\text{GeV}<m_{\ell\ell}<105$ GeV. We derive the angular
coefficients $A_{i}$ in the Collins-Soper frame Collins and Soper (1977).
Uncertainties are derived from Monte Carlo statistics.

Our work re-advertises the relevance of the $t\bar{t}hh$ and inclusive $hh$
sensitivity studies. For scenarios that are more closely related to the HEFT
parametrisation, the multi-Higgs rates also play central roles in honing
sensitivity to non-linear CP violation. Although these processes suffer from
limitations at the LHC and their resulting constraints are relatively weak
when compared to SMEFT correlations, they provide unique avenues for probing
such interactions, in particular because low energy precision experiments
(e.g. dipole measurements) will have reduced sensitivity compared to SMEFT.

###### Acknowledgements.

A.B. and C.E. are supported by the STFC under grant ST/T000945/1. C.E. is
supported by the Leverhulme Trust under grant RPG-2021-031 and the IPPP
Associateship Scheme. DG and AN thank the U.S. Department of Energy for the
financial support, under grant number DE-SC 0016013.

##  Appendix A CP-violation effects to the $Z$ boson angular moments in the
$gg\to Zh$ process

The angular moments for the $Z$ boson can be used as analyzers for the Higgs-
top CP violation effects in the loop-induced $gg\to Zh$ process. In general,
the differential cross-section for the described process can be represented as

$\displaystyle\frac{1}{\sigma}\frac{\text{d}\sigma}{\text{d}\cos{\theta}\text{d}\phi}=$
$\displaystyle\frac{3}{16\pi}[1+\cos^{2}{\theta}+\frac{A_{0}}{2}(1-3\cos^{2}{\theta})+A_{1}\sin{2\theta}\cos{\phi}$
$\displaystyle+\frac{A_{2}}{2}\sin^{2}{\theta}\cos{2\phi}+A_{3}\sin{\theta}\cos{\phi}+A_{4}\cos{\theta}$
$\displaystyle+A_{5}\sin^{2}{\theta}\sin{2\phi}+A_{6}\sin{2\theta}\sin{\phi}+A_{7}\sin{\theta}\sin{\phi}]\,,$
(23)

where $\theta$ and $\phi$ denote the polar and azimuthal angles of the
$\ell^{-}$ lepton in the $Z$ boson rest frame. The eight coefficients $A_{i}$,
$i=0,..,7$, correspond to the degrees of freedom for the polarization density
matrix of a spin-1 particle. Remarkably, the three coefficients $A_{5,6,7}$
are proportional to the relative complex phases of the scattering amplitudes
Gonçalves and Nakamura (2018). Hence, when associated to depleted strong phase
contributions from loop contributions, these coefficients can be sensitive to
truly CP-violation effects.

To extract the angular coefficients $A_{i}$ from our Monte Carlo simulation,
we recognize that Eq. (23) represents a spherical harmonic decomposition for
the differential cross-section, utilizing real spherical harmonics
$Y_{lm}(\theta,\phi)$ of order $l\leq 2$ Mammen Abraham and Gonçalves (2022).
Consequently, we can access the angular coefficients by exploring the
orthogonality relations of the spherical harmonics. The angular coefficients
are projected out using the following relations

$\displaystyle A_{0}=4-\left<10\cos^{2}\theta\right>,$ $\displaystyle A_{1}$
$\displaystyle=\left<5\sin 2\theta\cos\phi\right>,$ $\displaystyle
A_{2}=\left<10\sin^{2}\theta\cos 2\phi\right>,$ $\displaystyle A_{3}$
$\displaystyle=\left<4\sin\theta\cos\phi\right>,$ $\displaystyle
A_{4}=\left<4\cos\theta\right>,$ $\displaystyle A_{5}$
$\displaystyle=\left<5\sin^{2}\theta\sin 2\phi\right>,$ $\displaystyle
A_{6}=\left<5\sin 2\theta\sin\phi\right>,$ $\displaystyle A_{7}$
$\displaystyle=\left<4\sin\theta\sin\phi\right>,$ (24)

and the weighted normalization is defined as

$\displaystyle\left<f(\theta,\phi)\right>\equiv\int_{-1}^{1}\text{d}\cos\theta\int_{0}^{2\pi}\text{d}\phi\,\frac{f(\theta,\phi)}{\sigma}\frac{\text{d}\sigma}{\text{d}\cos\theta\text{d}\phi}\,.$
(25)

In Table 2, we present the angular coefficients $A_{i}$ for the
$gg\to\ell^{+}\ell^{-}h$ process, considering the SM and new physics scenarios
$\alpha=\pi/4$ and $-\pi/4$. Two comments are in order. First, we observe sub-
leading strong phase contributions from the one-loop calculation to the
coefficients $A_{5,6,7}$, as evident from the SM scenario. Second, CP-
violation effects are also depleted in the same coefficients as seen for the
$\alpha=\pi/4$ and $-\pi/4$ scenarios. Notably, the only statistically
significant CP-phase $\alpha$ for the spin density parametrization arises in
the coefficient $A_{0}$.

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