#### CP-violating top quark couplings at future linear $$e^+e^-$$ colliders

Eur. Phys. J. C
CP-violating top quark couplings at future linear e+ e− colliders
W. Bernreuther 2
L. Chen 2 4
I. García 1 3
M. Perelló 1
R. Poeschl 0
F. Richard 0
E. Ros 1
M. Vos 1
0 Laboratoire de l'Accélérateur Linéaire (LAL), Centre Scientifique d'Orsay , 91898 Orsay Cedex , France
1 Instituto de Física Corpuscular (IFIC, UVEG/CSIC) , Apartado de Correos 22085, 46071 Valencia , Spain
2 Institut für Theoretische Teilchenphysik und Kosmologie, RWTH Aachen University , 52056 Aachen , Germany
3 Present address: CERN , 1211, Geneva 23 , Switzerland
4 Present address: Max-Planck-Institut f. Physik , 80805 Munich , Germany
We study the potential of future lepton colliders to probe violation of the CP symmetry in the top quark sector. In certain extensions of the Standard Model, such as the two-Higgs-doublet model (2HDM), sizeable anomalous top quark dipole moments can arise, which may be revealed by a precise measurement of top quark pair production. We present results from detailed Monte Carlo studies for the ILC at 500 GeV and CLIC at 380 GeV and use parton-level simulations to explore the potential of high-energy operation. We find that precise measurements in e+e− → t t¯ production with subsequent decay to lepton plus jets final states can provide sufficient sensitivity to detect Higgs-boson-induced CP violation in a viable two-Higgs-doublet model. The potential of a linear e+e− collider to detect CP-violating electric and weak dipole form factors of the top quark exceeds the prospects of the HL-LHC by over an order of magnitude.
1 Introduction
The top quark is by far the heaviest fundamental particle
known to date. Its large mass implies that it is the
Standard Model particle that is most strongly coupled to the
electroweak symmetry breaking sector. The top quark is also set
apart from the other quarks in that it does not form hadronic
bound-states – that is, it offers the possibility to study the
interactions of a bare quark. The experimental investigation
of single-top and top-quark pair production at the Tevatron
and at the large hadron collider (LHC) has led to a precise
knowledge of the top-quark strong and weak charged-current
interactions. These results are in good agreement with the
Standard Model (SM) predictions.
The electroweak neutral current interactions of the top
quark are much less precisely investigated. At the LHC an
accurate measurement of the top-quark neutral current
couplings to the photon (γ ) and Z boson is challenging, because
t t¯ pairs are dominantly produced by the strong interactions,
and the associated production of t t¯ and a hard photon or
Z boson is relatively rare compared to the production of
t t¯ + jets. Future lepton colliders will offer the opportunity to
precisely explore these top-quark interactions. The studies in
Refs. [
1–7
] have shown that linear collider (LC) experiments
can measure the top-quark electroweak couplings with very
competitive precision.
Several projects exist for e+e− colliders with sufficient
energy to produce top-quark pairs (i.e., with centre-of-mass
energy √s > 2mt ). A mature design exists for a linear e+e−
collider that can ultimately reach centre-of-mass energies
up to approximately 1 TeV, the international linear collider
(ILC) [
8
], to be hosted in Japan. We focus on the planned
operation at a centre-of-mass energy of 500 GeV, consider
the initial integrated luminosity scenario (500 fb−1) and the
nominal H20 scenario [
9
] (4 ab−1).
Extensive R&D into high-gradient acceleration has,
moreover, opened up the possibility of a relatively compact
multiTeV collider, the compact linear collider (CLIC) [
10
]. The
CLIC program envisages an initial stage that collects
approximately 500 fb−1 at √s = 380 GeV, followed by operation
at a centre-of-mass energy of up to 3 TeV [
11
].
Both linear collider projects offer the possibility of
polarized beams. Operation of the collider with two polarization
configurations allows one to disentangle the photon and Z
boson form factors. ILC and CLIC both envisage polarizing
the electron beam (80% longitudinal polarization). The ILC
baseline design envisages 30% positron polarization. In the
CLIC design this is left as an upgrade option.
ILC and CLIC have developed detailed detector designs
and sophisticated simulation and reconstruction software,
which allows a careful study of experimental effects in
realistic conditions. Here, we perform a full simulation of the
reaction e+e− → t t¯ → lepton plus jets in the context of
these projects.
In this paper we extend the studies of Refs. [
1, 12
] to
couplings that violate the combination of charge conjugation and
parity (CP, in the following). New physics that affects
topquark production and/or decay is parametrized in terms of
form factors that depend on kinematic invariants. The
Standard Model predicts CP violation in top-quark pair
production and decay to be very small, well beyond the experimental
sensitivity of existing and planned facilities (see the
discussion in Sect. 9). Some extensions of the SM, such as, for
instance, two-Higgs-doublet models, can give rise to sizeable
effects [13]. Any observation of CP violation in the top-quark
sector would be clear evidence of physics beyond the SM.
Early studies of non-standard CP violation (i.e. CP violation
that is not induced by the Kobayashi–Maskawa CP phase)
in e+e− → t t¯ include those in Refs. [
14–16
]. Our study is
based on the observables proposed in Ref. [
17
] for the
lepton plus jets final states that have a direct sensitivity to the
electric and weak dipole form factors of the top quark. We
present the first result of a detailed Monte Carlo simulation
of these observables in a realistic experimental environment.
This paper is organized as follows. Section 2 describes
our conventions for the form factors with which one can
parametrize top-quark pair production at lepton colliders.
We analyze in Sect. 3 the potential magnitude of the
CPviolating top-quark electric and weak dipole form factors in
two SM extensions taking into account present
experimental constraints. Moreover, we briefly discuss the potential
magnitude of CP-violating form factors in top-quark decay,
t → W b. Observables and associated asymmetries
sensitive to CP violation in t t¯ production which apply to lepton
plus jets final states are introduced in Sect. 4. We study the
effect of polarized beams in Sect. 5 and determine the
relations between the CP asymmetries and CP-violating form
factors. Full-simulation results at two centre-of-mass
energies are presented in Sect. 6 for the ILC at 500 GeV and in
Sect. 7 for CLIC at 380 GeV. In Sect. 8 we study the prospect
of 1 − 3 TeV operation in a parton-level study. Systematic
uncertainties are discussed in Sect. 9. The prospects of linear
e+e− colliders for the extraction of the CP-violating form
factors derived in this study are presented in Sect. 10 and
compared with other studies in the literature of the potential
of lepton and hadron colliders. We conclude in Sect. 11.
2 CP violation in e+ e− → t t¯
Our present knowledge of physics at the TeV scale implies
that in e+e− collisions at centre-of-mass (c.m.) energies
1 TeV top-quark pairs will be dominantly produced by
SM interactions, to wit, by s-channel photon and Z -boson
exchange. New physics interactions that involve the top quark
may modify the t t¯X ( X = γ , Z ) vertices and the overall
e+e− → t t¯ production amplitude. In order to pursue a
relatively model-independent analysis, we assume that new
CPviolating interactions, which can lead to sizeable effects in t t¯
production, have only a small effect on top-quark decay. We
will discuss in Sect. 3 the validity of this assumption within
two SM extensions.
Lorentz covariance determines the structure of the t t¯X
vertex. In the case that both top quarks are on their mass
shell and the photon and Z -boson are off-shell we can write
the t t¯X vertex as
Γμtt X (k2) = −i e γμ
F1XV (k2) + γ5 F1XA(k2)
σμν kν
+ 2mt
i F2XV (k2) + γ5 F2XA(k2)
(1)
where e = √4π α, with α the electromagnetic fine
structure constant, mt denotes the mass of the top quark, and
kμ = qμ + q¯ μ is the sum of the four-momenta qμ and
qμ of the t and t¯ quark. We use γ5 = i γ 0γ 1γ 2γ 3 and
σ¯μν = 2i γμγν − γν γμ . The Fi denote form factors which
are to be probed in the time-like domain k2 > 4mt2 by the
reaction at hand.1
For off-shell γ , Z bosons, there are in general two more
contributions, one of which could violate CP invariance.
However, if the mass of the electron is neglected, an excellent
approximation in our case, these two terms will not contribute
to the t t¯ production amplitude. We therefore omit them in the
following.
Within the Standard Model, and at tree level, the F1 are
related to the electric charge of the top quark Qt and its weak
isospin:
F1γV,SM
F1ZV,SM
= Qt = − 23 , F1γA,SM
= 0,
1
= − 4sW cW
1 − 83 sW2
, F1ZA,SM
1
= 4sW cW
where sW and cW are the sine and the cosine of the weak
mixing angle θW . The chirality-flipping form factors F2 are zero
at tree level. As in any renormalisable theory they must be
loop-induced. At zero momentum transfer F2γV (0) is related
via F2γV (0) = Qt (gt − 2)/2 to the anomalous magnetic
moment of the top quark gt , where Qt denotes its
electrical charge in units of e.
In this paper we focus on the form factors F2XA that
violate the combined charge and parity symmetry CP. The
electric dipole moment of the top quark is determined by
1 The form factors Fi are related to the F˜i of Ref. [
18
] through the
following relations:
F1XV = − F1XV + F2XV , F2XV = F2XV ,
F1XA = −F1XA, F2XA = −i F2XA. (2)
γ
F2A for an on-shell photon at zero momentum transfer,
dtγ = −(e/2mt )F2γA(0). In analogy with this relation one
may define an electric dipole form factor (EDF) and a weak
dipole form factor (WDF) for on-shell t, t¯ but off-shell γ , Z :
e
dtX (k2) = − 2mt F2XA(k2),
X = γ , Z .
(4)
For off-shell gauge bosons these form factors are in
general gauge-dependent. However, within the two SM
extensions that will be discussed in the next section, the dtX (s) are
gauge-invariant to one-loop approximation. This may justify
their use in parametrizing possible CP-violating effects in t t¯
production.
Finally, we note that new physics effects are often
described in the framework of effective field theory (EFT) by
anomalous couplings, i.e., constants. The ‘couplings’ dtγ and
dtZ can be related to the coefficients of certain dimension-six
operators; cf., for instance, Ref. [
19
]. However, by using EFT
for describing new physics one assumes that there is a gap
between the typical energy scale of the process under
consid√
eration ( s in our case) and the scale of new physics. This
is not the case for the models that we consider in the next
section. In particular, in the kinematic domain that we are
interested in, dtγ and dtZ show a non-negligible dependence
on √s and can develop absorptive parts, therefore becoming
complex.
3 CP-violation in SM extensions
In the SM, where CP violation is induced by the Kobayashi–
Maskawa (KM) phase in the charged weak current
interactions, resulting CP effects in flavour-diagonal amplitudes are
too small to be measurable in e+e− → t t¯ [
13
]. Sizeable
CPviolating effects involving top quarks may arise in SM
extensions with additional, non-KM CP-violating interactions. In
this section we consider two extensions of this type, namely
two-Higgs-doublet models and the mimimal
supersymmetric extension of the Standard Model (MSSM), and assess the
potential magnitude of the top-quark EDF and WDF in these
models, taking into account present experimental constraints.
At the end of this section we briefly discuss the potential size
of CP-violating form factors in top-quark decay, t → W b.
3.1 Two-Higgs-doublet models
In view of its large mass the top quark is an excellent
probe of non-standard CP violation generated by an extended
Higgs sector. We consider here two-Higgs-doublet
models (2HDMs) where the SM is extended by an additional
Higgs-doublet field and where the Yukawa couplings of the
Higgs doublets Φ1, Φ2 are such that no tree-level
flavourchanging neutral currents (FCNCs) are present. The physical
Higgs-boson spectrum of these models consists of a charged
Higgs boson and its antiparticle, H ±, and three neutral Higgs
bosons, one of which is to be identified with the 125 GeV
Higgs resonance. The Higgs potential V (Φ1, Φ2) can
violate CP, either explicitly or spontaneously by Higgs fields
developing a vacuum expectation value with non-trivial
CPviolating phase. If this is the case, then the physical CP-even
and CP-odd neutral Higgs fields mix. In the unitary gauge the
resulting three neutral Higgs mass eigenstates h j are related
to the two neutral CP-even states h, H , and the CP-odd state
A by an orthogonal transformation:
(h1, h2, h3)T = R (h, H, A)T .
(5)
The orthogonal matrix R is parametrized by three Euler
angles2 that are related to the parameters of the Higgs
potential.
For phenomenological studies it is useful to choose as
independent parameters of the 2HDM a set that includes
the masses m j and m+ of the three neutral and the charged
Higgs boson, respectively, the three Euler angles αi of R,
the parameter tan β = v2/v1 which is the ratio of the
vacuum expectation values of the two Higgs-doublet fields, and
v = v12 + v22 which is fixed by experiment to the value
1/v = (√2G F )1/2 = 246 GeV. In case of CP violation in
the Higgs sector, the Higgs mass eigenstates h j couple to
both scalar and pseudoscalar fermion currents. The Yukawa
Lagrangian is
m f
LY = − v
a j f f¯ f − b j f f¯i γ5 f h j .
(6)
Here f denotes a quark or charged lepton and the reduced
scalar and pseudoscalar Yukawa couplings a j f , b j f depend
on the type of 2HDM [
21
], on the matrix elements of R, and
on tan β.
Within the 2HDM, the CP-violating part of the
scattering amplitude of e+e− → t t¯ is determined (to one-loop
approximation and in the limit of vanishing electron mass me)
entirely by the top-quark EDF and WDF (4) that are induced
at one-loop by CP-violating neutral Higgs-boson exchange
[
22
]. There are no CP-violating box contributions. (A
CPviolating scalar form factor FSZ (s) is also generated, but it
does not contribute for me = 0.) Thus the one-loop top-quark
EDF and WDF generated in 2HDM are gauge-invariant.
The real and imaginary parts of the top-quark EDF dtγ (s)
and WDF dtZ (s) were computed for several types of 2HDM
in [
22
]. The EDF dtγ (s) is generated at one loop by the
CPviolating exchange of the three Higgs bosons h j between the
outgoing t and t¯. A C P-violating Higgs potential implies that
2 We use the conventions of [20].
the h j are not mass-degenerate. The form factor becomes
complex, i.e., it has an absorptive part for s > 4mt2. We
remark that dtγ (s) ∝ mt3: two powers of mt result from the
Yukawa interactions (6) and one power from the necessary
chirality flip. The one-loop WDF dtZ (s) receives two different
contributions: the first one is topologically identical to dtγ (s),
but with the tree-level t t¯ coupling to the photon replaced by
the vectorial t t¯ coupling to the Z boson. The second one
involves the Z Z h j coupling (where only the CP-even
component of h j is coupled) and the pseudoscalar coupling of
h j to the top quark. The second contribution, which is
proportional to m2Z mt , becomes complex for s > (m Z + m j )2,
where m j is the mass of h j .
Before evaluating the formulae for the top-quark EDF and
WDF given in [
22
] we discuss present experimental
constraints on the parameters of the type-II 2HDM. The 125 GeV
Higgs resonance must be identified with one of the neutral
Higgs bosons h j . For definiteness, we identify it with h1 and
assume the other two neutral Higgs bosons to be heavier.
The ATLAS and CMS results on the production and decay
of the 125 GeV Higgs resonance h1(125 GeV) imply that
this boson is Standard-Model like; its couplings to weak
gauge bosons, to the t and b quark, and to τ leptons have
been determined with a precision of 10–25% [
23–25
] and
these results are in reasonable agreement with the SM
predictions. Moreover, the investigation of angular correlations
in h1(125 GeV) → Z Z ∗ → 4 exclude that this Higgs
boson is a pure pseudoscalar ( J P = 0−) [
26,27
]. However,
this does not imply that h1 is purely CP-even ( J P = 0+) – it
can be a CP mixture. Because the pseudoscalar component
of such a state does not couple to Z Z and W W at tree level,
a potential pseudoscalar component is difficult to detect in
the decays of h1 to weak bosons.3
In the following we assume that h1 is a CP-mixture with
couplings to fermions and gauge bosons that are in accord
with the LHC constraints [
23,24
]. We are interested in 2HDM
parameter scenarios where the couplings of the h j to top
quarks are not suppressed as compared to the corresponding
SM Yukawa coupling. This is the case for tan β ∼ 1 or
somewhat lower than one. Moreover, we assume that the two
other neutral Higgs bosons h2, h3 are significantly heavier
than h1. In the type-II 2HDM and for tan β < 1 the Yukawa
couplings of the h j to b quarks and τ leptons are suppressed
as compared to the corresponding SM Yukawa couplings,
cf. Table 1. Moreover, the constraint that h1 has SM-like
couplings to weak gauge bosons implies that the couplings of
h2, h3 to W W and Z Z are small, irrespective of the CP nature
3 The decays h1 → τ τ , where a CP-violating effect occurs at tree level
if h1 is a CP mixture, may be used to check whether or not φ1 has
a pseudoscalar component. See, for instance [28]. Other possibilities
include the associated production of tt¯h1, once a sufficiently large event
sample will have been collected.
of these Higgs bosons. This follows from a sum rule; cf., for
instance, [
21
]. 2HDM parameter scenarios with tan β 1
and h2, h3 masses equal or larger than about 500 GeV are
compatible with the non-observation of heavy neutral Higgs
bosons at the LHC in final states with electroweak gauge
bosons [
29–33
], b quarks [
34,35
], charged leptons [
36,37
],
and top quarks [
38
]. The charged Higgs boson H ± of 2HDM
is of no concern to us here. Constraints from B-physics data,
in particular from the rare decays B → Xs + γ and B0 − B¯ 0
mixing imply that the mass of H ± must be larger than ∼ 700
GeV for low values of tan β [
39
].
In order to assess the potential size of the form factors
F2XA(s) (X = γ , Z ) in type-II 2HDM with Higgs sector CP
violation we make a scan over the independent parameters
that are of relevance for this analysis. In the kinematic range
√s 500 GeV the most important contribution to the
topquark EDF and WDF arise from h1 exchange if this Higgs
boson has top-quark Yukawa couplings such that the modulus
of the product a1t b1t is about one. We take into account recent
constraints on the couplings of h1 to W, Z , t, b, τ [
25
] and
the experimental constraints on the masses and couplings of
h2, h3 from Refs. [
23,24,29–38
]. We vary tan β in the range
0.35 ≤ tan β ≤ 1 and the three Higgs mixing angles in the
range −π/2 ≤ αi ≤ π/2, determine the resulting reduced
Yukawa couplings a j f , b j f and the couplings f j V V of the
h j to Z Z . A benchmark set of resulting couplings is given
in Table 1. Somewhat tighter constraints on the CP-violating
top-Higgs couplings were derived in Ref. [40].
For calculating the form factors F2XA(s) we assume that
the Higgs bosons h2 and h3 are heavier than 500 GeV. For
definiteness we set their masses to be 1200 GeV and 600 GeV,
respectively. Using the formulae of Ref. [
22
], mt = 173 GeV,
and the values of the Higgs couplings of Table 1, the real and
imaginary parts of F2XA(s) are shown as functions of the c.m.
energy in Fig. 1. In the kinematic range displayed in these
plots the imaginary part of the EDF is about three times larger
than that of the WDF. This holds true also for the real parts of
the form factors close to the t t¯ threshold, while they become
significantly smaller in magnitude around √s = 500 GeV
due to the strong fall-off of the dominant contribution from
h1. The values of the real and imaginary parts of these form
0.025
0.02
X 0.015
A
2
F
e 0.01
R
factors are listed in Table 2 for two c.m. energies that are
chosen for the simulations in Sects. 5–7.
In the kinematic range that we are interested in (√s 500
GeV) the imaginary parts of the EDF and WDF are rather
insensitive to the values of the heavy Higgs-boson masses,
as long as m2,3 > 500 GeV. This is also the case for the
real parts of the form factors close to the t t¯ threshold that are
dominated by the contribution from h1 exchange. This term
falls off strongly with increasing c.m. energy. Moreover, at
c.m. energies √s 500 GeV the contributions from h2, h3 to
the real parts of the form factors may no longer be negligible.
We find that the real parts of the EDF and WDF at √s = 500
GeV depend, for fixed Higgs-boson couplings, sensitively on
the masses of h2, h3, but do not exceed 10−3 in magnitude
for the couplings of Table 1.
As mentioned above, the formulae of [
22
] apply to any
type of 2HDM where tree-level FCNC are absent. In fact,
the results shown in Fig. 1 and given in Table 2 apply also to
other types of 2HDM in the low tan β region; for instance,
to the type-I model where all right-chiral quarks and charged
leptons are coupled to the Higgs doublet Φ2 only, or to
the so-called lepton specific model where the right-chiral
quarks (right-chiral charged leptons) are coupled to Φ2 (Φ1)
only.
In summary, within the 2HDM the real (imaginary) part of
the top-quark electric dipole form factor F2γA can be as large
as ∼ 0.02 (∼ 0.01) in magnitude near the t t¯ production
threshold, taking into account the present constraints from
LHC data.
3.2 The minimal supersymmetric SM extension
The Higgs sector of the MSSM corresponds to a type-II
2HDM. Supersymmetry (SUSY) forces the tree-level Higgs
potential V (Φ1, Φ2) of the MSSM to conserve CP.
Nevertheless, the MSSM contains in its general form many
CP-violating phases besides the KM phase, especially in
the supersymmetry-breaking terms of the model, including
phases of the complex Majorana mass terms of the
neutral gauginos and of the complex chargino and sfermion
mass matrices. Motivated by assumptions as regards SUSY
breaking at very high energies, one often puts constraints on
the SUSY-breaking terms, in particular on the CP-violating
phases, in order to restrict the number of unknown
parameters of the model. Nevertheless, generic features of SUSY
CP violation remain. Unlike the case of Higgs-boson induced
one-loop EDMs, fermion EDMs generated at one-loop can
be large, also for u, d quarks and the electron. The
experimental upper bounds on the EDM of the neutron and of
atoms/molecules strongly constrain in particular the
CPphases associated with the sfermion mass matrices of the
first and second generation, barring fine-tuned cancellations.
See, for instance, Ref. [
41
] for a review. However, the phases
of the sfermion mass matrices need not be flavour-universal.
For the top flavour the associated phase ϕt˜ can still be of
order one. Often a common phase of the gaugino masses
is assumed. Using phase redefinitions of the fields in the
MSSM Lagrangian, one can choose for the
parametrization of MSSM CP violation in the top-quark sector [
42,43
]
the phase ϕt˜, the corresponding b-flavour phase ϕb˜, and the
phase ϕμ = arg(μ) of the so-called μ term in the MSSM
Lagrangian that generates a Dirac mass of the higgsinos. For
a rather recent analysis of constraints on the CP-violating
phases in the MSSM, see Ref. [44].
The one-loop top-quark EDF and WDF induced by the
CP-violating interactions of the MSSM are gauge-invariant.
They are generated by one-loop γ t t¯ and Z t t¯ vertex diagrams
involving t˜ and b˜ squarks, gluinos g˜, neutralinos χ˜ 0, and
charginos χ˜ ± in the loop. The t˜t˜∗g˜ contributions to the EDF
and WDF were determined in [
45–47
]. The complete set of
1-loop contributions were computed in [
42,43,48,49
]. They
consist, apart from the gluino contribution, of the chargino
contribution (with χ˜ +χ˜ −b˜ and b˜b˜∗χ˜ + in the loop), and of the
neutralino contribution (with t˜t˜∗χ˜ 0 and χ˜ 0χ˜ 0t˜ in the loop).
If light neutralinos and charginos and/or light t˜, b˜ squarks
with masses mi , m j of order 100–200 GeV would exist there
would be strong enhancements of the top-quark EDF and
WDF F2γA,Z (s) in the range 2mt √s 500 GeV near the
two-particle production threshold √sth = mi + m j .
References [
42,43
] computed these form factors for light gauginos
and t˜, b˜ squarks. Reference [43] found maximal values of the
form factors at √s = 500 GeV for some favourable set of
SUSY parameters of the order of 10−3. However, the input
parameters of these computations have since been excluded.
Searches for supersymmetry were negative so far, and the
LHC searches put strong lower bounds on the masses of
SUSY particles that are, in most cases, model-dependent, to
wit: mg˜ > 1.8 TeV, mb˜1,2 > 840 GeV, mχ˜ ± > 715 GeV, and
mt˜1,2 > 800 GeV for mχ˜10 < 200 GeV. For a recent review,
we refer to Ref. [
50
]. However, a light stop with mass ∼ 200
GeV is not yet excluded if the decay t˜1 → t χ˜10 exists. The
limit mχ˜ ± > 715 GeV results from an analysis in Ref. [
51
]
using a simplified SUSY model.
In order to estimate the potential size of the top-quark
EDF and WDF, we evaluated the chargino, gluino, and
neutralino contributions using the formulae of [
42,43
] with
SUSY masses that are in accord with these experimental
constraints. The phases ϕμ, ϕt˜, ϕb˜ were chosen such that they
maximize the EDF and WDF for given masses. Using the
lower bounds on the masses of SUSY particles cited in the
previous paragraph we find
|Re F2γA|, |Re F2ZA| < 10−3,
|Im F2γA|, |Im F2ZA| < 10−4 for √s
As mentioned above, a light top squark t˜1 with mass ∼ 200
GeV and a light neutralino χ˜10 are not yet excluded. In this
case non-zero but small imaginary parts are generated by the
gluino and neutralino contribution to the EDF and WDF in
the considered range of c.m. energies.
In the case of the MSSM there are also CP-violating box
contributions to e+e− → t t¯ that involve neutralino (e˜χ˜ 0χ˜ 0t˜)
and chargino (ν˜eχ˜ ±χ˜ ∓b˜) exchanges in the one-loop
amplitudes. They are, as shown in [
43,49
], in general not
negligible compared to the top-quark EDF and WDF contributions.
We shall, however, refrain from evaluating these box
contributions, which goes beyond the scope of this paper. In the
simulations performed in Sects. 5–7 we shall stick to the
parametrization (1) of CP-violating effects in t t¯ production
in terms of the EDF and WDF.
3.3 CP-violating form factors in t → W b
So far, the only top-quark decay mode that has been observed
is t → W b with subsequent decay of the W boson into
leptons or quarks. In the SM the branching ratio of this decay
is almost 100%. The decay amplitude for t → W +b with
all particles on-shell can be parametrized in terms of two
chirality-conserving and two chirality-flipping form factors
fL , f R and gL , gR , respectively; cf., for instance, [
16
]. The
measurements of these form factors [
52,53
] are in agreement
with the SM predictions.
Let us denote the corresponding form factors in the
chargeconjugate decay t¯ → W −b¯ by fi , gi (i = L , R). CPT
invariance implies that fi∗ = fi and gi∗ = gi . CP
invariance requires that the corresponding form factors are equal.
These relations imply the following: if final-state
interactions can be neglected in top-quark decay, then CP violation
induces non-zero imaginary parts that are equal in magnitude
but differ in sign [
16,54
]: Im fi = −Im fi , Imgi = −Imgi ,
i = L , R.
In Ref. [
54
] the potential size of CP-violating (and
CPconserving) contributions to the form factors in t → W b was
investigated for several SM extensions. Within the 2HDM
it was found that |Im fi |, |Imgi | 3 × 10−4 for tan β
0.6. In the MSSM the CP-violating effects were found to be
smaller by at least one order of magnitude. The observables
and CP-violating asymmetries that we introduce in the next
section and in Sect. 5 are insensitive to CP violation in
topquark decay. Therefore we can neglect CP violation in
topquark decay in the following and parametrize CP violation
in t t¯ production with subsequent decay into lepton plus jets
final states solely by the top-quark EDF and WDF defined in
Eq. (1). One may probe CP violation in semi-leptonic t and
t¯ decay with a CP-odd asymmetry constructed from suitable
triple product correlations [
16,46
].
Let us summarize the discussion of the previous subsections.
We analyzed the potential size of CP-violating effects in t t¯
production in e+e− collisions and subsequent t and t¯ decay
within two popular and motivated SM extensions, taking into
account present experimental constraints. As to the BSM
scenarios investigated above, an extended Higgs sector with
CP-violating neutral Higgs-boson exchange has the largest
potential to generate observable effects in this reaction. If
the observed h1(125GeV) Higgs resonance has both scalar
and pseudoscalar couplings to top quarks whose strengths
are of order one compared to the SM top Yukawa coupling
γ
then the magnitude of Im F2 A can be ∼ 1% in the energy
range √s 500 GeV that we consider in the following. The
real part of this form factor can become of the same order
of magnitude near the t t¯ threshold. The real and imaginary
parts of the top-quark WDF are in general smaller by a
factor of about 0.3; cf. Table 2. Within the MSSM the top-quark
EDF and WDF are smaller, with maximum values
compatible with current experimental constraints below 10−3. The
CP-violating form factors in the t → W b decay amplitude
that can be generated within the 2HDM or the MSSM are
very small and of no further interest to us here. Moreover,
we recall that within the 2HDM there are no CP-violating
box contributions to the e+e− → t t¯ amplitude to one-loop
approximation if the electron mass is neglected. These results
motivate the use of the parametrization of Eq. (1) in the
simulations of the following sections.
4 Optimal CP-odd observables
As demonstrated in Ref. [
1
], at a future linear e+e−
collider precise measurements of the t t¯ cross-section and the
top-quark forward–backward asymmetry for two different
beam polarizations allow the extraction of the top-quark
CPconserving electroweak form factors with a precision that
exceeds that of the HL-LHC. In this section the prospects
for the measurement of CP-violating form factors F2γA,Z are
investigated, as an extension of the previous study. The
CPviolating effects in e+e− → t t¯ manifest themselves in
specific top-spin effects, namely CP-odd top spin–momentum
correlations and t t¯ spin correlations. If one considers the
dileptonic decay channels, t t¯ → + − + · · · , then it is
appropriate to consider CP-odd dileptonic angular
correlations [
16
], which efficiently trace CP-odd t t¯ spin correlations.
We recall the well-known fact that the charged lepton in
semileptonic t or t¯ decay is by far the best analyzer of the top spin.
Here we consider t t¯ decay to lepton plus jets final states which
yield more events than the dileptonic channels and, moreover,
allow for a straightforward experimental reconstruction of
the t and t¯ rest frames. For these final states the most efficient
way to probe for CP-violating effects in t t¯ production is to
construct observables that result from t and t¯ single-spin–
momentum correlations, that is, from correlations which
involve only the spin of the semi-leptonically decaying t or t¯.
Here, we adopt the observables proposed in [
17
] for detecting
these correlations in lepton plus jets final states.
We consider in the following the production of a top-quark
pair via the collision of longitudinally polarized electron and
positron beams:
e+(p+, Pe+ ) + e−(p−, Pe− ) → t (kt ) + t¯(kt¯).
(8)
Here, p± and kt , kt¯ denote the e±, t , and t¯ three-momenta
in the e+e− c.m. frame. The spin degrees of freedom of the
t and t¯ are not exhibited. Moreover, Pe− ( Pe+ ) is the
longitudinal polarization degree of the electron (positron) beam.
In our notation, Pe− = −1 ( Pe+ = −1) refers to left-handed
electrons (positrons). For our purpose the most useful final
states are, as mentioned, the lepton plus jets final states from
semi-leptonic t decay and hadronic t¯ decay and vice versa:
(9)
(10)
t t¯ →
t t¯ →
+(q+) + ν
Xhad(qX ) +
+ b + X had(qX¯ ),
−(q−) + ν¯ + b¯,
where the three-momenta in (9) and (10) also refer to the
e+e− c.m. frame.
We compute the reactions (8)–(10) at tree level, both in
the SM and with non-zero CP-odd form factors F2γA,Z , taking
the polarizations and spin correlations of the intermediate t
and t¯ into account. As discussed in the previous section these
form factors can have imaginary parts. Non-zero real parts
Re F2γA,Z (s) induce a difference in the t and t¯ polarizations
orthogonal to the scattering plane of the reaction. Non-zero
absorptive parts, Im F2γA,Z (s), lead to a difference in the t
and t¯ polarizations along the top-quark direction of flight
and along the direction of the electron or positron beam. At
the level of the intermediate t and t¯ these effects manifest
themselves in non-zero expectation values of the following
CP-odd observables:
p
ˆ + × kˆ t · (st −st¯),
kˆ t · (st −st¯),
p
ˆ + · (st − st¯),
(11)
where st and st¯ denote the spin operators of t and t¯,
respectively, and hats denote unit vectors. In (11) two-body
kinematics is used, i.e., kt¯ = −kt . The expectation value of the
first observable of the list (11) depends on Re F2γA,Z , while the
expectation values of the other two observables depend on
Im F2γA,Z . Each observable listed in (11) is the difference of
two terms that involve the t and t¯ spin, respectively. The term
that contains the t (t¯) spin can be translated, in the case of the
lepton plus jets final states, into a correlation that involves
the + ( −) direction of flight. This is the most efficient way
to analyze the t (t¯) spin. These correlations can be measured
with the + + jets and − + jets events (9) and (10),
respectively.
Based on these considerations, so-called optimal
observables [
15
], i.e., observables with a maximal signal-to-noise
ratio to a certain parameter appearing in the squared matrix
element, were constructed in Ref. [
17
] for tracing CP
violation in the lepton plus jets final states (9) and (10). These
optimal observables are, in essence, given by those parts of
the squared matrix element that are linear in the CP-violating
form factors Re F2γA,Z or Im F2γA,Z . One may simplify these
(13)
I m
O+
= −
1 +
expressions and use for the final states (9) the following two
observables [
17
] that are nearly optimal:
5 Polarized beams
√s
+ 2mt ˆ X¯ · pˆ +qˆ +∗ · pˆ+.
q
2mt − 1 (qˆ X¯ · pˆ +)
2
qˆ +∗ · qˆ X¯
The corresponding observables O− for the final states (10)
are defined to be the CP image of O+ and are obtained from
O+ by the substitutions qˆ X¯ → −qˆ X , qˆ +∗ → −qˆ −∗, pˆ+ →
pˆ +. The unit vectors qˆ ±∗ refer to the ± directions of flight
defined in the t and t¯ rest frame, respectively. The differences
of the expectation values of O+ and O− that we consider in
the next section probe for CP-violating effects.
The observables (12) and (13) are approximations to the
rather unwieldy optimal observables listed in the appendix of
Ref. [
17
]. Using the optimal observables at low energy leads
to a minor increase in sensitivity. Between the t t¯ production
threshold and √s ∼ 500 GeV the sensitivity to the CP-odd
form factors increases by a few percent. At very high energy
the difference is somewhat more pronounced: at 3 TeV the
sensitivity is expected to increase by approximately 30%.
As discussed in Sect. 3.3, non-standard CP-violating
interactions can induce, besides CP violation in t t¯ production, also
anomalous couplings in the t → W +b and t¯ → W −b¯ decay
amplitudes. However, observables such as (12) and (13) and
their CP images, where the t and t¯ spins are analyzed by
charged lepton angular correlations, are insensitive to these
anomalous couplings, as long as one uses the linear
approximation [
46, 55, 56
] which is legitimate here. This justifies the
parametrization of the CP asymmetries O+ − O− solely
in terms of F2γA,Z .
We study the distributions of O−Re and O −Im at
leadingorder (LO) in the SM couplings, putting F2γA,Z = 0, with
the WHIZARD 1.95 event generator [
57
]. Distributions of
both observables are shown in Fig. 2 for a centre-of-mass
energy of 500 GeV. The three histograms in each panel
correspond to unpolarized beams (dashed line), to a left-handed
electron beam and a right-handed positron beam (e−Le+,
R
Pe− , Pe+ = −80%, +30%, red continuous histogram) and
for a right-handed electron beam and a left-handed positron
beam (e−Re+L, Pe− , Pe+ = +80%, −30%, black
continuous histogram). The degree of longitudinal polarization that
is used follows the design values of the ILC: Pe− , Pe+ =
±80%, ∓30%. As the top-quark EDF and WDF are
negligible in the SM (and set to zero in the simulation), the
distributions for unpolarized beams are symmetric around the
origin.
Initial-state polarization affects the normalization, but
leaves the shape of the O−Re distribution unaffected. The
total cross-section increases strongly for e+e− beams in the
e−e+R configuration as compared to unpolarized beams, and
L
somewhat less strongly for the polarization configuration
e−e+.
R L
I m as
Beam polarization has a more profound impact on O−
shown in Fig. 2 (right panel). With unpolarized beams the
distribution is symmetric around zero, but the distributions
corresponding to polarized beams show significant distortions.
This is expected because the initial state with different beam
polarization for electrons and positrons is not CP-symmetric.
Asymmetries A can be defined [
17
] as the difference of
the expectation values and
O− :
Fig. 2 The WHIZARD LO Standard Model prediction for the
normalized distribution of the CP observables O−Re (left panel) and O−Im (right
panel) defined in (12) and (13). The results correspond to e+e−
collisions at a centre-of-mass energy of 500 GeV and three different beam
polarizations: the dashed line corresponds to unpolarized beams, the red
(black) solid lines to − 80% (+ 80%) polarization of the electron beam
and + 30% (− 30%) polarization of the positron beam. The histogram
for LR polarized beams is normalized to unit area. The area of the other
histograms is scaled so as to maintain the cross-section ratios. The O±Re
distribution is confined to [
− 1, 1
] by construction, the O±Im distribution
is truncated to the same interval
cγ (s)
In the asymmetry, many experimental effects are expected
I m
disto cancel. This applies also to the distortion of the O±
tributions by beam polarization. The O+ I m
distriI m and O−
butions are shifted by approximately equal amounts, but in
I m observable
opposite directions. The mean value of the O−
is − 0.08 ± 0.01 for Pe− , Pe+ = −80%, +30% and + 0.09
± 0.01 for Pe− , Pe+ = +80%, −30%. The distributions of
O+ I m .
There
I m are distorted in the same way as those of O−
fore, the effect of initial-state polarization cancels in the
difference of the two observables.
The asymmetries ARe, AI m are sensitive to CP violation
effects in the t t¯ production amplitude through the
contributions of ReF2γA,Z and Im F2γA,Z , respectively:
A
A
Re
I m
Re
= O+
Re
− O−
I m
= O+
I m
− O−
= cγ (s)ReF2γA + cZ (s)ReF2ZA,
γ
= c˜γ (s)Im F2A + c˜Z (s)Im F2ZA.
(16)
(15)
The values of these coefficients depend on the
polarizations Pe− and Pe+ . In our approach, where we normalize
the expectation values O by the SM cross section (that
is, neglecting the contributions bilinear in the CP-violation
form factors), the asymmetries ARe, AI m are strictly linear in
the form factors. Analytical expressions for the coefficients
cγ (s), cZ (s), c˜γ (s) and c˜Z (s) of relations (15) and (16) for
arbitrary beam polarization are given in the appendix.
Values for 100% polarization are given in Tables 3 and 4, using
mt = 173.34 GeV, m Z = 91.1876 GeV, mW = 80.385 GeV,
and sin2 θW = 1 − m2W /m2Z .
Table 3 The values of the coefficients in the expressions for the
asymmetries ARe and AIm . The values are calculated for several c.m.
energies used in this paper and for the e−Le+R beam-polarization configuration
(Pe− = − 1, Pe+ = + 1)
c.m. energy √s (GeV)
Table 4 Same as Table 3, but for the opposite e−Re+L beam polarization:
Pe− = + 1, Pe+ = − 1
c.m. energy √s (GeV)
The polarization of the e− and e+ beams provides a means
to disentangle the contributions of the CP-violating photon
and Z -boson vertices. The coefficients cγ (s) and c˜Z (s)
corresponding to the LR and RL configurations have opposite
signs. The measurement of the two CP asymmetries ARe
and AI m for two beam polarizations provides sufficient
constraints to solve the system of equations formed by Eqs. (15)
and (16).
For √s 2mt the coefficients cγ (s), cZ (s) that appear
in the expression for ARe grow with the c.m. energy √s. The
interactions associated with F2γA,Z involve a factor kν , which
is the sum of the t and t¯ four-momenta (cf. Eq. (1)). Therefore,
the sensitivity of the asymmetry ARe to F2A increases with
centre-of-mass energy.
I m consist of a sum of terms, two
The observables O±
of which contain the factor √s. Therefore the coefficients
c˜γ (s), c˜Z (s) that determine AI m grow with s for √s 2mt .
However, this does not imply that this asymmetry has a
significantly higher sensitivity than ARe to CP-violating effects
in t t¯ production at high energies, because the widths of the
I m grow accordingly.
distributions of O±
6 Full simulation: ILC at 500 GeV
In this section we study the 500 GeV run of the ILC, assuming
an integrated luminosity of 500 fb−1. The sample is divided
into two beam-polarization configurations: the L R sample
has − 80 and + 30% electron and positron polarization,
respectively. In the R L sample the signs of both electron and
positron polarization are inverted: the electron polarization
is + 80% and the positron polarization is − 30%.
The full-simulation study is based on samples produced
for the ILC TDR [
58
]. The event sample is generated with
WHIZARD 1.95 [
59
] by the LCC generator group. It includes
all six-fermion processes that produce a lepton plus jets final
state, e+e− → bb¯l±νl qq¯ . This includes top-quark pair
production and a number of other processes that lead to the
same final state, with the largest non-doubly-resonant
contribution coming from single-top production [
60
]. The effect
of initial-state radiation (ISR) is included in the generator.
Events are generated with the nominal ILC luminosity
spectrum described in Ref. [
58
], which includes the effects of
beam energy spread and beamstrahlung. The events
generated are restricted to the physics of the SM, hence the F2γA,Z
are set to zero. Fragmentation and hadronization is modelled
using PYTHIA 6.4 [
61
] with a parameter set tuned to e+e−
data recorded at LEP.
The generated events are processed with the ILD
detector simulation software based on GEANT4 [
62
]. The ILD
detector model is described in the Detailed Baseline Design
included in the ILC TDR [
58
]. The ILD detector consists of
cylindrical barrel detectors and two end-caps. Together these
provides nearly hermetic coverage down to a polar angle of
approximately 6 degrees. For the reconstruction of charged
particles ILD relies on a combination of a solid and gaseous
tracking system in a 3.5 Tesla magnetic field. Precise
silicon pixel and micro-strip detectors occupy the inner radii,
from r = 1.5 cm to r = 33 cm. A large Time Projection
Chamber provides measurements out to 1.8 m. The tracker
is surrounded by a highly granular calorimeter designed for
particle flow. A highly segmented tungsten electromagnetic
calorimeter provides up to 30 samples in depth with a
transverse cell size of 5 × 5 mm2. This is followed by a highly
segmented hadronic calorimeter with 48 steel absorber layers
and 3 × 3 cm2 read-out tiles.
The γ γ → hadrons background corresponding to a single
bunch crossing is overlaid. The data from the different
subdetectors are combined into particle-flow objects (PFO) using
the Pandora [
63
] particle-flow algorithm. Jets are
reconstructed using a robust algorithm [
64
] specifically designed
for high-energy lepton colliders with non-negligible
background levels. Particle-flow objects are clustered into exactly
four jets. Heavy-flavour jets are identified using the LCFI
algorithm [
65,66
].
The selection and reconstruction of the top-quark
candidates proceeds as described in Ref. [
1
]. The event selection
relies primarily on the presence of the b-jets and the lepton
from the W -boson decay.
Leptons are identified by the particle-flow algorithm. A
number of criteria is applied to the lepton and the jet
containing the lepton to ensure that the lepton is isolated: the
ratio plT /M j of the lepton pT and the invariant mass of the
jet must be greater than 0.25. The energy of the lepton must
be greater than 60% of the jet energy. Exactly one isolated
lepton must be present in the event. The isolated lepton is
removed from the collection of particle-flow objects and jet
clustering is repeated on the remaining objects.
The LCFI flavour tagger returns a likelihood for the four
reconstructed jets, that is, based on track and vertex
information. At least one jet must satisfy a stringent requirement
(btag likelihood greater than 0.9). A second jet must be found in
the event that satisfies a looser requirement (b-tag likelihood
greater than 0.6).
After these basic requirements the non-W bW b
background is reduced to a manageable level. No strong cuts on
kinematic observables are required to isolate the signal. A
number of loose cuts are applied to the invariant mass of
the hadronic final state (180 < mhad. < 420 GeV) and on
the mass of the reconstructed W -bosons and top quarks (120
< mW < 250 GeV and 120 < mt < 270 GeV. These have
virtually no effect on the signal selection efficiency, but are
helpful to reduce the background due to two-fermion and
four-fermion events.
The e+e− → bb¯l±νl qq¯ process includes a small fraction
of single top, which is considered part of the signal, and
less than 1% of W W Z events. The selection is based on
extensive studies in Refs. [
1,67
]. The contamination of the
signal sample by events due to processes other than those
included in the e+e− → bb¯l±νl qq¯ sample is less than 5%
and is neglected in the following.
The average selection efficiency for signal events is
approximately 54% for the L R sample and 56% for the R L
polarized case. The efficiency is over 70% for events with
muons and 2% lower for events with electrons or positrons.
Events with τ -leptons enter the signal selection with an
efficiency of 20%, thanks to τ -decays to electrons and muons. As
expected, no significant difference is observed between the
selection efficiencies for positively and negatively charged
leptons.
The hadronic top candidate is reconstructed by pairing the
two light-quark jets with the b-jet that minimizes a χ 2 based
on the expected W -boson and top quark energy and mass and
on the angle between the W -boson and the b-jet. For e−Le+R
polarization migrations strongly affect the distributions. A
maximum χ 2 is required to retain only well-reconstructed
events. This requirement reduces the overall selection
efficiency to approximately 30%. This quality cut is not applied
for e−Re+L polarization, where migrations have a small effect.
The reconstructed distributions for the observables O±Re
I m are shown in Fig. 3. In the same figure the true
disand O±
tribution is shown, that is, the distribution of the observable
constructed with the lepton and top quark from the Monte
Carlo record, before any detector effects or selection cuts are
applied.
The event selection has a clear impact on the distributions
of O±Re. A dip in the central part of the reconstructed
distributions is observed that is due to the limited acceptance of the
experiment in the forward region. The cuts on lepton energy
and isolation have a very small effect. The energy resolution
of the reconstructed hadronic top-quark candidate and
ambiguities in the assignment of b-jets to W-boson candidates
leads to a slight broadening of the distribution. The
distribu
I m , moreover, exhibit the expected asymmetry due
tions of O±
to the beam polarization.
The response of the experiment is the same for positively
and negatively charged leptons and for the hadronic top and
anti-top quark decay products. Therefore, any distortions in
the reconstructed distributions are expected to cancel in the
asymmetries ARe and AI m . Experimental effects generally
do not generate spurious asymmetries. The reconstructed
asymmetries in Table 5 are found to be compatible with zero
within the statistical uncertainty of 0.003–0.004.
7 Full simulation: CLIC at 380 GeV
In this section we study the potential of CLIC operation at
√s = 380 GeV. The baseline CLIC design allows for up
0.12
Fig. 3 The CP-odd observables O±Re,I m for the ILC at √s = 500 GeV.
The four distributions correspond to the reconstructed (solid) and true
(dashed) distributions for two beam polarizations. The red histogram
(e−Le+) corresponds to − 80% electron polarization and + 30% positron
R
polarization, theblack histogram (e−Re+L) to + 80% electron polarization
Table 5 Reconstructed values of the CP-odd asymmetries from a
Monte Carlo simulation of the ILD detector response to t t¯ events
produced in electron–positron collisions at √s = 500 GeV. The quoted
uncertainties are due to the limited statisitcs of the simulated samples
to ± 80% longitudinal electron polarization ( Pe− = ±0.8).
Space is reserved in the layout for positron polarization as an
upgrade option. No positron polarization is assumed in the
following. An integrated luminosity of 500 fb−1 is assumed.
Events are generated with WHIZARD 1.95 [
59
], again
including all six-fermion processes that produce the
relevant final state. The effect of ISR and the CLIC luminosity
spectrum are taken into account. The machine parameters
correspond to the settings reported in the CLIC Conceptual
Design Report [
10
].
and − 30% positron polarization. The histogram for the left-handed
electron beam is normalized to unit area. The area of the histogram for
right-handed polarization is scaled so as to maintain the cross section
ratios
The generated events are processed with a full simulation
of the CLIC_ILD detector [
10
]. The CLIC_ILD detector is
an adaptation of the ILD detector described in Sect. 6 to
the high-energy environment. To deal with machine-induced
backgrounds the vertex detector is moved out to r = 2.5 cm
and the time stamping capabilities of the detector are
reinforced. The thickness of the calorimeter is enhanced to fully
contain energetic jets: the combination of electromagnetic
and hadronic systems corresponds to 8.5 interaction lengths.
The electromagnetic calorimeter and barrel hadron
calorimeter use Tungsten as absorber material. The end-cap has iron
absorber layers. The electromagnetic calorimeter is read out
by 30 sampling layers with finely segmented silicon
detectors, with a pad size of 5 × 5 mm2. The hadronic calorimeter
is read out by 75 layers (60 in the end-cap) of scintillator
material with a cell size of 30 × 30 mm2.
To deal with the background from γ γ → hadrons,
particle-flow objects are selected using a set of timing and
energy cuts, corresponding to the loose selection of Ref. [
68
].
The event selection is identical to that described in Sect. 6.
The b-tagging likelihood cut is reoptimized to achieve a
similar signal efficiency. The overall selection efficiency is
somewhat higher than for the ILC at 500 GeV: 58% for the average
over lepton flavours and beam polarizations. The efficiencies
for the two beam polarizations agree within 1%. A similar
pattern is observed for the lepton flavours: the efficiency for
events with muons, electrons and τ -leptons are ∼ 82, ∼ 74
and ∼ 20%.
Reconstruction of the W -boson and top quark candidates
proceeds as described in Sect. 6. At a centre-of-mass energy
of 380 GeV the observables are reconstructed quite
accurately. The distributions are centered at zero. A slight dip is
visible at the centre of the reconstructed O±Re distribution due
to the limited acceptance in the forward region of the
experiment. Other than that, the differences between reconstructed
and generated distributions are very small.
Again, we find that the reconstructed asymmetries given in
Table 6 are compatible with zero within the statistical
uncertainty. The entry of ARe for Pe− = +0.8, that is, 2 σ away
from 0, is taken to be a statistical fluctuation. Studies of
selection and reconstruction at parton level with much larger
samples fail to generate spurious non-zero values for the
asymmetry. Higher-order QCD corrections to continuum t t¯
production and decay are known to be moderate to small (cf.
the brief discussion in Sect. 9, where references are given).
Therefore we expect that these corrections affect the shape of
the distributions of the observables O±Re, O±Im (Figs. 3 and 4),
presented in this and in the previous section, only in a rather
moderate way. Much more important are the experimental
effects of limited acceptance, efficiency, and bin migration
on the shape of these distributions, discussed in Sect. 9.
Moreover, the QCD corrections (which are, needless to say,
CPinvariant) cancel in the CP-odd asymmetries ARe and AI m .
Only a small residual effect remains via the effect of the QCD
corrections on the normalization of the expectation values
that enter these asymmetries. We will estimate the resulting
theoretical uncertainties in Sect. 9.
8 Parton-level study for high-energy operation
In this section we study the potential of the high-energy
stages of the CLIC programme that could reach 3 TeV. The
instantaneous luminosity scales approximately proportional
to the centre-of-mass energy and one may expect an
integrated luminosity of several ab−1.
The decay of boosted top quarks produces a topology [
69
]
that is very different from that of t t¯ events close to the
production threshold. Therefore, the reconstruction of the 1–3
TeV collisions must be performed with an algorithm
specifically developed for high energy, where the collimated decay
products of the hadronic top quark are captured in a single
large-R jet (i.e. a jet reconstructed with a radius parameter
R greater than 1). In this reconstruction scheme the
combinatoric problem of pairing W -boson and b-tagged jets is
entirely avoided.
The γ γ → hadrons background in multi-TeV collisions is
more severe than at low energy. The reconstruction of boosted
top quarks at CLIC was studied in a detailed simulation,
including realistic background levels in Ref. [
70
]. With tight
pre-selection cuts on the particle-flow objects and the robust
algorithm of Ref. [
64
] the top-quark energy can be
reconstructed with a resolution of 8%. Also the jet mass and other
substructure observables can be reconstructed precisely, with
much better resolution than at the LHC. As background
processes have cross-sections that are similar to that of top-quark
production, it seems safe to assume that t t¯ events with
centreof-mass energies of 1–3 TeV can be efficiently selected and
distinguished from background processes.
An evaluation based on a detailed simulation of the
experimental response for the optimal observables is not yet
available. We identify the most important effects using a
partonlevel simulation. A representative selection is applied to
parton-level e+e− → t t¯ → bb¯qq¯ lν events generated with
MG5_aMC@NLO [
71
]. The detector resolution is
implemented by smearing of the parton four-vectors.
The limited acceptance in the forward region shapes the
distributions significantly. For partons emitted at shallow
angle, part of the jet energy flow disappears down the beam
pipe. We mimic this effect by requiring that all partons have
| cos θ | < 0.98 (the detector coverage extends to well beyond
| cos θ | = 0.99; some margin is added as jets have a finite
size). In Fig. 5 the distribution for selected events is
compared to the full distribution. The effect is more pronounced
and more localized than in the low-energy analysis.
We furthermore apply a smearing to mimic the resolution
for the hadronic top quark candidate. The reconstructed
topquark four-vector is used to boost the lepton to the top-quark
system. The finite energy resolution and angular resolution
may lead to distortions of the reconstructed distribution. The
effects of a 10% energy resolution and 0.02 radian angular
resolution, twice the size of the resolution found in the study
of Ref. [
70
], are indicated in Fig. 5. The reconstruction has
a much less severe impact than in the low-energy analysis.
As for the low-energy analysis, these experimental effects
are identical for positively and negatively charged leptons and
for quarks and anti-quarks. We therefore expect that
experFig. 4 The CP-odd observables O±Re,I m for CLIC at √s = 380 GeV.
The four distributions correspond to the reconstructed and true
distributions for two beam polarizations. The red histogram (e−Le+) corresponds
0
to − 80% electron polarization, the blackhistogram (e−Re0+) to + 80%
imental effects do not create spurious asymmetries. Rough,
but conservative, limits on systematic effects are presented
in the next section.
A more detailed study of a detailed detector simulation is
required for a quantitative study of the high-energy
performance. In the following we estimate the potential of
highenergy operation, assuming an acceptance of 40% for lepton
+ jets events.
9 Systematic uncertainties
Before we discuss the prospects of linear colliders to extract
the real and imaginary parts of the form factors F2γA,Z , a
number of potential sources of systematic uncertainties are briefly
discussed.
The polarization of the electron and positron beams is
the key machine parameter in the extraction of the form
factors. A combination of polarimeters and in-situ
measurements allows for a precise determination of Pe− and Pe+ . The
detailed study of the ILC case in Ref. [
72
] envisages a
determination to the 10−3 level. The study of (single) W -boson
electron polarization. The histogram for the left-handed electron beam
is normalized to unit area. The area of the histogram for right-handed
polarization is scaled so as to maintain the cross-section ratios
production is expected to provide per-mille level precision at
high energy. This precision is well beyond what is needed to
avoid significant uncertainties in the form factor extraction.
The uncertainties of other machine parameters, such as the
integrated luminosity or the centre-of-mass energy, have a
negligible effect on the result.
The analysis is found to be quite robust against the effects
of event selection and reconstruction of the t t¯ system. The
limited acceptance and efficiency do lead to significant
distortions of the distributions of O±Re and O ±Im . Also, the impact
of migrations is clearly visible in each of the distributions.
However, these effects cancel in the asymmetry. Therefore,
none of these effects generate a non-zero asymmetry when
the true value is 0. This type of uncertainty is referred to as
bias. The full-simulation study shows that a spurious
nonzero result due to systematic effects is expected to be smaller
than 0.005.
For arbitrary values of the true asymmetry the analysis of
the systematics is a bit more involved. We must also
consider the possibility that the selection and reconstruction of
the events lead to a non-linearity in the response to non-zero
CP asymmetries ARe and AI m . These effects are labelled as
−1
−0.5
−1
−0.5
−1
−0.5
resolution on O±Re are shown. Panels c and d represent the effect of the
I m , g and h the impact of the angular and energy
resoluselection on O±
I m distribution
tion. All histograms are normalized to unit area. The O±
is truncated to the interval [
− 5, 5
]
0.5
0.5
1
ORe
+
1
ORe
+
4
OI+m
4
OI+m
non-linearity in the following. They are evaluated in a
partonlevel study using events generated with non-zero WDF and
EDF. Distributions and asymmetries with non-zero values of
the top-quark EDF and WDF are generated using a
MadGraph [
71
] UFO model developed in Ref. [
73
]. The most
important cuts in the analysis, namely on the charged
lepton energy, its isolation, and the polar angle of final-state
quarks are applied to the six-fermion final state. The finite
resolution in the reconstruction of the hadronic top-quark
candidate is implemented by smearing the top-quark
threemomentum vector. The migrations due to ambiguities in
pairing b-jets and W -bosons at low energy are simulated
by implementing the incorrect pairing for 15% of events.
The selection tends to enhance the reconstructed asymmetry.
This effect is particularly pronounced at very high
centreof-mass energy, where it can reach up to 10% of the true
asymmetry (for √s = 3 TeV). Migrations and resolution
effects dilute the asymmetry, yielding reconstructed values
that are reduced by 5–15%. For centre-of-mass energies of
380 or 500 GeV migrations are the most important systematic
effect. At higher energy the resolution is the dominant effect.
Theory uncertainties are estimated as follows. Radiative
corrections to t t¯ production in e+e− collision are known to
high precision. The next-to-leading order (NLO) QCD
corrections have been known for a long time [
74
]. The NLO
electroweak corrections were determined in Refs. [
75–77
].
Off-shell t t¯ production and decay including non-resonant
and interference contributions at NLO QCD were
investigated in Ref. [
78
]. The NNLO QCD corrections to t t¯
production, including differential distributions, were calculated
in [
79, 80
]. Although not done in this work, the coefficients
of Re F2γA,Z and Im F2γA,Z in the asymmetries of Eqs. (15)
and (16) can be computed at NLO in the SM couplings.
We can then estimate the theory uncertainties of these
coefficients as follows. The uncertainty of the t t¯ cross-section
associated with renormalization scale variations in the range
√s/2 ≤ μ ≤ 2 s is at NLO (NNLO) QCD about 2% (1%)
√
at √s = 380 GeV and ∼ 0.9 (0.2%) at √s = 500 GeV
[80]. Assuming that the NLO SM corrections to the squared
matrix element including the EDF and WDF to t t¯ production
and decay are known, we take these NLO QCD values as
theory uncertainties. They are labelled “theory (non-linearity)”
in Table 7. We believe that these uncertainty estimates are
not unrealistic because the uncertainties of these coefficients
are, in fact, associated with the expectation values O±Re ,
I m , which are ratios that are usually expanded in powers
O±
of the SM couplings. QCD scale uncertainties of expanded
ratios are in general smaller than the scale uncertainty of
the cross-section. An example is the top-quark forward–
backward asymmetry AtFB which is known to NNLO QCD
accuracy [
79, 80
]. The scale uncertainty of the expanded AtFB
is below 0.5% at these c.m. energies [80].
Table 7 The main systematic uncertainties on the asymmetry ARe for
left-handed polarized electron beam (and right-handed positron beam in
the case of 500 GeV operation). Entries labelled bias represent estimates
of upper bounds on systematic effects that yield a spurious non-zero
result in the Standard Model. Entries labelled non-linearity represent
systematic uncertainties that affect the proportionality of the response to
non-zero values of the asymmetry (induced by physics beyond the
Standard Model). Positive signs indicate effects that enhance the observed
asymmetry. Negative signs correspond to effects that dilute the
asymmetry
Source
Machine parameters (bias)
Machine parameters
(non-linearity)
Experimental (bias)
Exp. acceptance (non-linearity)
Exp. reconstruction
(non-linearity)
Theory (bias)
Theory (non-linearity)
380 GeV
500 GeV
3 TeV
–
<< 1%
The numbers in the row “theory (bias)” in Table 7 are a
very conservative estimate of CP-violating SM contributions
induced by higher-order W -boson exchange to e+e− → t t¯.
At one loop in the electroweak couplings there are no
CPviolating SM contributions to this flavour-diagonal
reaction. Beyond one loop the CP-violating SM contributions
to the asymmetries of Eqs. (15), (16) are smaller than
[gW2 /(16π 2)]2Im J , where gW = e/ sin θW and Im J is the
imaginary part of a product of four quark mixing matrix
elements, which is invariant under phase-changes of the quark
fields. Its value is |Im J | ∼ 2 × 10−5.
The estimates of the systematic uncertainties on ARe for
several centre-of-mass energies are presented in Table 7. Our
study has not found any sources of systematic uncertainty
that yield a spurious asymmetry when the true asymmetry is
zero. Upper limits on a systematic bias in ARe are given in the
table with the label “(bias)”. Several sources can, however,
enhance or dilute a non-zero true asymmetry. These are
indicated as the expected relative modification of the asymmetry,
with the label “(non-linearity)”. Of course, these effects can
be corrected to a good extent using Monte Carlo simulation.
The selection bias can, moreover, be reduced by comparing
the measured and predicted results in an appropriate fiducial
region.
10 Prospects for CP-violating form factors
The prospects for a measurement of the top-quark form
factors F2γA,Z are presented in Table 8. Rows two and three of the
table show the result of our simulations described in the
preto predictions in the literature, from fast-simulation studies in the
context of the TESLA TDR [
81
], and from studies on the prospects at the
(high-luminosity) LHC [
5,82–84
], on the potential of a 100 TeV proton
collider [85], and of the LHeC electron–proton collider [
86
]
Lint (ab−1)
γ
Re F2A
ceding sections for a 380 GeV stage of the Compact Linear
Collider CLIC and the initial 500 GeV run at the
International Linear Collider. In both cases an integrated luminosity
of 500 fb−1 is assumed. We find that both projects have a
very similar sensitivity to these form factors, reaching
limγ
its of |F2 A| < 0.01 for the EDF. Assuming that systematic
uncertainties can be controlled to the required level a
luminosity upgrade of either of these machines may bring about
a further improvement. The fourth line of Table 8 shows the
prospects for the nominal ILC scenario, which envisages an
integrated luminosity of 4 ab−1.
The prospects for these measurements at a multi-TeV
electron–positron collider are listed in the row labelled
“CLIC3000” of Table 8. The sensitivity of the CP-odd
observables studied in this paper to F2γA,Z increases, for
√s 2mt , approximately linearly with the centre-of-mass
energy. On the other hand the cross-section for t t¯
production via s-channel Z /γ ∗-boson exchange decreases as 1/s.
At linear colliders this is partly compensated by the higher
luminosity at high energy: typically the instantaneous
luminosity increases linearly with √s. All in all, for the 3 TeV
stage of CLIC the precision is expected to be significantly
higher than for the initial stage at √s = 380 GeV.
We recall here that the two-Higgs-doublet extensions of
the three-generation standard model investigated in Sect. 3
give rise to sizeable form factors predominantly at
centre-ofmass energies close to the t t¯ production threshold. However,
CP-violating new physics models with new heavy particles
are conceivable that lead to enhancements of the CP-violating
top-quark form factors F2γA,Z in the TeV energy range.
The next row in Table 8 lists the results given in the
TESLA Technical Design Report [
81
]. The results of our
fullsimulation analysis are in agreement with the expectations
of this parton-level study, once differences in the
assumptions on polarization and integrated luminosity are taken into
account.
10.1 Prospects at hadron colliders
A complete study of measurement prospects on F2γA,Z in the
associated production of top-quark pairs and gauge bosons,
t t¯Z and t t¯γ , at hadron colliders was made in Refs. [
82, 83
].
The constraints on the four CP-violating form factors are
listed in Table 8 under the header “prospects for hadron
colliders”. These results are compared to our results for the
initial ILC and CLIC stages in Fig. 6. Clearly, the measurements
at hadron colliders are expected to be considerably less
precise than those that can be made at lepton colliders, even
after completion of the full LHC programme including the
planned luminosity upgrade.
Furthermore, Table 8 summarizes the results of more
recent studies of the potential of hadron colliders. The
chirality-flipping terms proportional to σμν in the effective
Lagrangian used in Ref. [
5
] (cf. also Refs. [
84, 85
]) differ by
a factor 2mt /m Z ∼ 4 from our convention defined in Eq. (1).
Thus the form factors F2 A used in this paper are related to the
couplings C2 A of Ref. [5] by F2 A = C2 A2mt /m Z . The 95%
C.L. limits on C2V /A given in Refs. [
5, 84, 85
] are translated
into 68% C.L. limits on F2V /A to facilitate comparison.
The ultimate prospects of the LHC and the luminosity
upgrade depend crucially on the control of systematic
uncertainties. Reference [
5
] finds a theory uncertainty of 15% on
the total cross-section calculated at NLO precision, leading to
a 20–40% improvement of the constraint on Re F2ZA obtained
at LO. Reference [
84
] shows that cross-section ratios σtt¯Z /σtt¯
10−1
10−2
10−3
HL-LHC, s = 14 TeV, L = 3000 fb-1
Phys.Rev.D71 (2005) 054013, Phys.Rev.D73 (2006) 034016
ILC initial, s = 500 GeV, L = 500 fb-1
ILC nominal, s = 500 GeV, L = 4000 fb-1
CLIC initial, s = 380 GeV, L = 500 fb-1
CLIC, s = 3 TeV, L = 3000 fb-1
Re[Fγ2A] Re[F2ZA] Im[Fγ2A] Im[F2ZA]
Fig. 6 Graphical comparison of 68% C.L. limits on CP-violating form
factors expected at the LHC [
82,83
], and at the ILC and CLIC (this
3w0o0r0k)f.bT−h1eaLt1H4CTseiVm.uFlaotriothnes aILssCumwee aanssiunmteegraanteidniltuiamliLno=sity50o0f fLb−=1
at 500 GeV and a beam polarization Pe− = ±0.8, Pe+ = ∓0.3. The
nominal scenario envisages an integrated luminosity of 4 ab−1. For
CLIC we assume L = 500 fb−1 at 380 GeV for the initial stage and
L = 3000 ab−1 at 3 TeV for the high-energy stage. The electron beam
polarization is Pe− = ±0.8 and no positron polarization is envisaged
and σtt¯γ /σtt¯ may be calculated to approximately 3%
precision. The HL-LHC and FCChh prospects from Ref. [85]
listed in Table 8 assume a systematic uncertainty of 15 and
5%, respectively.
A lepton–proton collider such as the LHeC [
87
] can
provide constraints on anomalous top-quark electroweak
couplings through measurements of the single-top production
rate (ep → νt X ) and the t t¯ photo-production rate [
86
].
These measurements constrain the combination of the
CPconserving and CP-violating form factors of the top-quark
γ γ
interaction with the photon, i.e., on F2V and F2 A in the
notation of Sect. 2. Assuming a large integrated
luminosity (100 fb−1) of energetic ep collisions (E p = 7 TeV,
Ee = 140 GeV), Ref. [
86
] derives the expected limit on
F2γA that is listed in the last row of Table 8.
10.2 Comparison to indirect constraints
Direct experimental bounds on CP-violating contributions
to the t t¯Z and t t¯γ vertices are not available. However, with
mild assumptions measurements that yield information about
the W t b vertex can be recast into limits on the form
factors of the γ t t¯ and Z t t¯ interactions. In a dimension-six
effective-operator framework based on the SM gauge
symmetry [
19, 88, 89
] the operator Ot W (with Wilson coefficient
Ct W ) generates an anomalous chirality-flipping coupling gR
of the W -boson (cf. Sect. 3.3) and non-zero values for the
real part of the F2γA,Z form factors in e+e− → t t¯ production.
We use this approach to convert constraints from
measurements of the W-helicity fractions in top-quark decay [
90–92
],
of the single-top production cross-sections, and from
studies of the polarization of W -boson in t-channel single-top
production [
53, 93
] into constraints on F2γA,Z .
Reference [
90
] presents a combined fit to W -boson
helicity fractions and single-top production cross-sections
measured at the LHC, resulting in a 95% C.L. limit of ImgR ∈
[−0.30, 0.31], where gR is one of the two chirality-flipping
form factors in the t → W b decay amplitude; see Sect. 3.3.
We translate this result into a bound on F2γA,Z . First we use the
following expression from Ref. [
19
] in order to relate gR to
the Wilson coefficient Ct W of the effective (dimension-six)
operator Ot W :
The result of Ref. [
90
] can then be converted into an allowed
band for Re F2γA and Re F2ZA using the following relations4:
Im[cW2 Ct W − sW2 Ct B ]
(17)
(18)
(19)
gR =
√
v2
2Ct W Λ2 .
Re F2ZA =
and
γ
Re F2 A =
√2
√2
4mt2
Λ2sW cW
4mt2
Λ2
Im[Ct W + Ct B ].
ATLAS has recently released two measurements of the
decay of polarized top quarks in t-channel single-top
production [
53, 93
] and presented the 95% C.L. limit: Im(gR / VL ) ∈
[−0.18, 0.06]. Setting VL = Vtb ∼ 1 this leads to a slightly
tighter limit on the CP-violating dipole operators. The bands
corresponding to both limits are drawn in Fig. 7, where the
prospects listed Table 8 are also shown for comparison.
Further indirect bounds can be extracted from data at lower
energies. Reference [
96
] used electroweak precision data to
derive constraints on top-quark electroweak couplings, but
CP-violating operators were not taken into account. Using in
addition experimental upper bounds on the electric dipole
moments of the neutron and atoms/molecules a powerful
indirect constraint was derived in Refs. [
97, 98
] on the static
γ
moment F2 A of the top quark.
4 As a cross-check the relations between form factors and Wilson
coefficients in Eqs. (18) and (19) have been verified using a MadGraph [71]
UFO model of the dimension-six operators that affect the top-quark
electroweak vertices. The basis of the model is presented in Ref. [
94
].
More recent additions, in particular the extension to the CP-violating
imaginary parts of the coefficients will be reported in a future
publication [
95
]. With this setup and conversion relations we are able to
reproduce several key results of Refs. [
19
] and [
5
].
γ 2A 0.4
F
e
R 0.2
0
−0.2
Fig. 7 The 68% C.L. limits on Re F2ZA and Re F2γA derived from
measurements of the W t b vertex performed by ATLAS and CMS during
run I of the LHC. This interpretation assumes that there is a relation
between the imaginary part of the anomalous chirality-flipping
coupling gR that affects the t W b-vertex and the real part of the form factors
F2γA,Z measured in e+e− → t t¯ production, as is generally the case in an
effective-operator interpretation. The prospects of future colliders are
indicated for comparison
11 Conclusions
CP violation in the top-quark sector is relatively
unconstrained by direct measurements. While the Standard Model
predicts very small effects, which are beyond the
sensitivity of current and future colliders, sizeable effects may
occur within well-motivated extensions of the SM. We have
updated, within the type-II two-Higgs-doublet model and the
MSSM, the potential magnitude of CP violation in the
topquark sector, taking into account constraints of LHC
measurements. The CP-violating top-quark form factors F2γA,Z
whose static limits are the electric and weak dipole moments
of the top quark can be as large as 0.01 in magnitude in a
viable 2HDM.
We have investigated the prospects of detecting CP
violation in t t¯ production at a future e+e− collider. The top spin–
momentum correlations proposed in Ref. [
17
] for t t¯ decay
to lepton plus jets final states were evaluated with a full
simulation of polarized electron and positron beams including
a detailed model of the detector response. Biases due to the
selection and migrations in the distributions of observables
O±Re and O ±Im due to ambiguities in the reconstruction of the
top-quark candidates were found to cancel in the CP
asymmetries ARe and AI m defined in Eqs. (15), (16). We expect
therefore that these asymmetries, which are sensitive to the
CP-violating top-quark form factors F2γA,Z , are robust against
such effects and can be measured with good control over
experimental and theoretical systematic uncertainties. Thus,
our results validate the findings of an earlier parton-level
study [
81
] for the TESLA collider.
Measurements of these top spin–momentum correlations
at a future lepton collider can provide a tight constraint on
CP violation in the top-quark sector. The 68% C.L. limits
on the magnitudes of the form factors Re F2γA,Z and Im F2γA,Z
derived from our analysis of assumed 500 fb−1 of data
collected at 380 or 500 GeV are expected to be better than
0.01. An improvement by a further factor of three may be
achieved in the luminosity upgrade scenario of the ILC or in
the high-energy stage of CLIC. These prospects constitute an
improvement by two orders of magnitude over the existing
indirect limits. With this precision, a linear collider can probe
the level of CP violation in the top-quark sector predicted by
a viable 2HDM model of Higgs-boson induced CP violation.
A comparison with the expectations for hadron
colliders, as derived in Refs. [
5, 82–85
], shows that the
sensitivity of a future e+e− collider to CP-violating dipole form
factors is very competitive. The constraints on form factors
represent an order of magnitude improvement of the limits
expected after the complete LHC programme, including the
planned luminosity upgrade. The potential even exceeds that
of a 100 TeV hadron collider, such as the FCChh.
Acknowledgements We would like to thank our colleagues in the ILD
and CLICdp groups. The study in Sect. 6 was carried out in the
framework of the ILD detector concept, that of Sect. 7 in the CLICdp
collaboration. We gratefully acknowledge in particular the LCC generator
group and the core software group which developed the simulation
framework and produced the Monte Carlo simulation samples used in
this study. We thank several members of both collaborations for a
careful review of the manuscript and many helpful suggestions to improve
the paper. This work benefits from services provided by the ILC Virtual
Organisation, supported by the national resource providers of the EGI
Federation. This research was done using resources provided by the
Open Science Grid. The authors at IFIC (UVEG/CSIC) are supported
under Grant MINEICO/FEDER-UE, FPA2015-65652-C4-3-R and by
the “Severo Ochoa” excellence centre program, under Grant
SEV-20140398. L. Chen is supported by a scholarship from the China Scholarship
Council (CSC).
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Appendix: coefficients for ARe and AI m
Here we give formulae for the coefficients cγ (s), cZ (s) and
c˜γ (s), c˜Z (s) that determine the CP asymmetries (15) and
(16), respectively. They can be represented as ratios,
cγ (s) =
c˜γ (s) =
Nγ (s)
D(s)
Nγ (s)
D(s)
cZ (s) =
c˜Z (s) =
NZ (s)
D(s)
NZ (s)
D(s)
(20)
We compute the matrix elements for the lepton plus jets finals
states at tree level, use the narrow width approximation for the
intermediate t and t¯, and integrate over the full phase space.
Moreover, we neglect the width in the Z -boson propagator,
since we are sufficiently far away from the Z peak and we
work to lowest order in the electroweak couplings. With the
conventions defined in Eqs. (1) and (3) we obtain
Nγ (s) = −
NZ (s) = −
4βt √s
3mt
4βt s3/2
3mt
(s − m2Z )veγ
(1 − P− P+)saeZ vtZ
− ( P− − P+)[(m2Z − s)veγ vtγ − sveZ vtZ ] ,
(21)
( P− − P+)s(aeZ )2vtZ + ( P− − P+)
×veZ [(s − m2Z )veγ vtγ + sveZ vtZ ]
− (1 − P− P+)aeZ [(m2Z − s)veγ vtγ − 2sveZ vtZ ] ,
Nγ (s) =
NZ (s) =
2βt
15mt2
2βt s(16mt2 + s)
15mt2
(16mt2 + s)(s − m2Z )veγ
( P− − P+)saeZ vtZ
− (1 − P P ) (m2Z − s)veγ vtγ − sveZ vtZ
− +
, (23)
(1 − P− P+)s(aeZ )2vtZ
+(1 − P− P+)veZ [(s − m2Z )veγ vtγ + sveZ vtZ ]
− ( P− − P+)aeZ [(m2Z − s)veγ vtγ − 2sveZ vtZ ] ,
4
D = s
(1 − P− P+)s2(s − 4mt2)(veZ atZ )2
+ (1− P− P+)(s + 2mt2)[(s −m2Z )veγ vtγ +sveZ vtZ ]2
+ (1 − P− P+)s2(aeZ )2[(atZ )2(s − 4mt2)
+(s + 2mt2)(vtZ )2]
+ 2( P− − P+)saeZ
+ vtZ (s + 2mt2)[(s − m2Z )veγ vtγ + sveZ vtZ ]
veZ (atZ )2s(s − 4mt2)
where P− ≡ Pe− and P+ ≡ Pe+ are the longitudinal
polarization degrees of the e∓ beams, mt and m Z denote the
mass of the t quark and Z boson, respectively, and βt =
1 − 4mt2/s. The electroweak couplings of f = e−, t are
1
v Z
f = 2sW cW
vγf = Q f ,
aγf = 0,
T3 f − 2sW2 Q f ,
1
a Zf = − 2sW cW T3 f ,
where T3 f is the third component of the weak isospin of f ,
Q f is the electric charge of f in units of e > 0, and sW , cW
are the sine and cosine of the Weinberg mixing angle θW . We
have neglected terms bilinear in the CP-violating form factors
(22)
(24)
,
(25)
(26)
in the computation of the denominator D, because we know a
γ ,Z
posteriori that |F2 A | must be significantly smaller than one.
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