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###### Abstract

CP violation is one of Sakharov’s conditions for the matter-antimatter
asymmetry of the Universe. The experimentally observed size of CP violation is
insufficient to account for this. Is CP violated in the Higgs sector? Could
the SM-like Higgs boson be a mixture of CP even and CP odd states of an
extended Higgs sector? With what precision could such effects be measured at
future electron-positron colliders? These questions will be discussed in the
light of the latest studies at ILC.

##  1 CP Violation

Experimentally observed size of CP violation (CPV) is not supported by the
formalism of the Standard Model (SM). New sources of CPV beyond the SM are
necessary in order to explain the still open issue of baryon asymmetry of the
Universe. CPV in the Higgs sector can be probed in the interaction of Higgs
boson with vector bosons or fermions. The SM Lagrangian can be modified by
adding CP violating terms at a loop level in case of _HVV_ vertices or at the
Born level in case of _Hff_ vertices.

So far, there are only few results on measurement of the CPV mixing angle
($\Psi_{\mathrm{CP}}$) between the Higgs scalar and pseudoscalar states. Among
the future projects (HL-LHC, HE-LHC, CEPC, FCC-ee240), ILC has the most
promising projection of precision of the $\Psi_{\mathrm{CP}}$ measurement,
($\Delta\Psi_{\mathrm{CP}}$ = 4∘), in the fermionic H
$\xrightarrow{}{\tau^{+}\tau^{-}}$ decay [1].

##  2 ILC Project

The International Linear Collider (ILC) is a high-luminosity linear e-e+
collider with centre-of-mass-energy range of 250-500 GeV (extendable to 1 TeV)
aimed for precision studies in the Higgs sector, operating as a Higgs factory.
The project is also optimised for top-physics and EW studies, as well as to
probe new physics phenomena in a direct or indirect way. The electron beam
will be polarized to 80%, and the baseline plan includes an undulator-based
positron source which will deliver 30% positron polarization. Realization of
the ILC as a Higgs factory goes inline with the 2020 Update of the European
Particle Physics Strategy [2].

Two detector concepts have been developed, ILD and SiD [3], as general-purpose
detectors designed to optimally address the ILC physics goals. The Particle-
Flow technique [4] that will play a central role in event reconstruction,
requiring highly granular calorimeters and excellent low material budget
tracking and vertexing systems in the solenoid magnetic field up to 5 T.

##  3 Probing CP Violation at ILC

Figure 1: Definition of the CPV sensitive angle $\Phi$ in the Higgs boson
production in ZZ-fusion.

There are numerous Higgs production processes available at the ILC (like HZ,
WW-fusion, ZZ-fusion) at various centre-of-mass-energies, that offer a
plethora of possibilities for individual measurements and combinations of
Higgs boson production and decay channels. The correlation between spin
orientations of vector bosons from a Higgs production (or decay) can be
extracted from the angle $\Phi$ between the production (decay) planes.
Illustration of $\Phi$ definition in the Higgs production in ZZ-fusion is
given in Fig. 1. For the same orientated unit vectors orthogonal to the
production planes (as in Fig. 1), CP-sensitive angle between the planes can be
defined as:

$\Phi=a\arccos(\hat{n}_{1}\cdot\hat{n}_{2})$ (1)

where the unit vectors are:

$\hat{n}_{1}=\frac{q_{e^{-}_{i}}\times q_{e^{-}_{f}}}{|q_{e^{-}_{i}}\times
q_{e^{-}_{f}}|}\hskip 28.45274pt\mathrm{and}\hskip
28.45274pt\hat{n}_{2}=\frac{q_{e^{+}_{i}}\times
q_{e^{+}_{f}}}{|q_{e^{+}_{i}}\times q_{e^{+}_{f}}|},$ (2)

and $q_{e^{-(+)}_{i}},q_{e^{-(+)}_{f}}$ stands for momenta of initial and
final e∓ states. The coefficient $a$ defines how the second (anti-fermion or
an off-shell boson in case of a Higgs decay to VV∗) plane is rotated w.r.t.
the first (fermion or an on-shell boson) plane and is defined as:

$a=\frac{q_{Z_{e^{-}}}\cdot(\hat{n}_{1}\times\hat{n}_{2})}{|q_{Z_{e^{-}}}\cdot(\hat{n}_{1}\times\hat{n}_{2})|}$
(3)

where $q_{Z_{e^{-}}}$ is momentum of the Z boson in the first plane. The
direction of the Z boson momentum in the first plane (or an on-shell boson in
case of a Higgs decay to VV∗) regulates the notion of forward and backward,
where if the second plane falls backwards (as illustrated in Fig. 1) $a$ =
$-1$, otherwise $a$ = 1.

###  3.1 CP violation in $H\xrightarrow{}{\tau^{+}\tau^{-}}$ at 250 GeV

Higgs fermionic decay to $\tau^{+}\tau^{-}$ at 250 GeV centre-of-mass-energy
provides excellent sensitivity to $\Psi_{\mathrm{CP}}$ to be extracted from
the fit of CP-sensitive angle $\Phi$. Dependence of the $\Phi$ distribution on
various assumptions on the CP-mixing strength ($\Psi_{\mathrm{CP}}$ value) is
illustrated in Fig. 2 (left) [5] . Figure 2 (right) is showing reconstructed
$\Phi$ of signal and background, where background $\Phi$ distribution is CP
insensitive (i.e. flat). In the full simulation of ILD detector response, with
0.9 ab-1 of data, absolute statistical uncertainty of the $\Psi_{\mathrm{CP}}$
measurement of 4∘ is found. There is an ongoing analysis in the same channel,
at 250 GeV centre-of-mass energy with the SiD detector model [6].

Figure 2: Left: Dependence of the generated angle $\Phi$ on various
assumptions on the $\Psi_{\mathrm{CP}}$ value. Right: Reconstructed $\Phi$
distribution of signal (red, fitted with a solid line) and background (blue,
fitted with a flat line) events. Distributions are normalised to 0.9 ab-1 of
data in the (80, 30)% (e${}_{\mathrm{L}}^{-}$e${}_{\mathrm{R}}^{+}$) beam
polarisation, at 250 GeV ILC.

###  3.2 CP violation in HZZ production at 1 TeV

Figure 3: Left: Distribution of the angle $\Phi$ at the generator level in the
full physical range. Right: Reconstructed distributions of $\Phi$ in the
tracker acceptance region for signal (purple) and dominant background (blue
and green) processes.

There is an ongoing analysis aiming to estimate achievable precision on
$\Psi_{\mathrm{CP}}$ (for $\Psi_{\mathrm{CP}}$ = 0) by reconstructing $\Phi$
between the Higgs production planes (as illustrated in Fig. 1), in events in
which the Higgs boson is produced by ZZ-fusion at 1 TeV ILC. Full simulation
of ILD detector for ILC is assumed. Figure 3 (left) illustrates generated
information on angle $\Phi$, in the full physical range of polar angles,
expected with 1 ab-1 of integrated luminosity and ($-80,+30$)%
(e${}_{\mathrm{L}}^{-}$e${}_{\mathrm{R}}^{+}$) beam polarization; On Fig. 3
(right), the reconstructed CP violating observable $\Phi$ is shown, in the
tracker acceptance region111Tracker acceptance region assumes that both
electron and positron from $e^{-}e^{+}\rightarrow He^{-}e^{+}$ final state are
reconstructed within the tracking detector, in the range of polar angles
between 8∘ and 172∘ (barrel)., for the exclusive $H\xrightarrow{}b\bar{b}$
decay channel, against dominant backgrounds. Events are preselected with
$\sim$ 80% efficiency, to suppress high-cross section background processes
like $e^{-}e^{+}\rightarrow qqe\nu$. As expected, background distributions are
flat exhibiting no CPV sensitive structure.

## References

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