tWH associated production at the LHC
Eur. Phys. J. C
tWH associated production at the LHC
Federico Demartin 2
Benedikt Maier 0 1
Fabio Maltoni 2
Kentarou Mawatari 4 5 6
Marco Zaro 3 7
0 Laboratory for Nuclear Science, Massachusetts Institute of Technology , Cambridge, MA 02139 , USA
1 Institut für Experimentelle Kernphysik, Karlsruher Institut für Technologie (KIT) , 76131 Karlsruhe , Germany
2 Centre for Cosmology , Particle Physics and Phenomenology (CP3) , Université catholique de Louvain , 1348 LouvainlaNeuve , Belgium
3 Sorbonne Universités, UPMC Univ. Paris 06, UMR 7589, LPTHE, 75005 Paris , France
4 International Solvay Institutes , Pleinlaan 2, 1050 Brussels , Belgium
5 Theoretische Natuurkunde and IIHE/ELEM, Vrije Universiteit Brussel , Pleinlaan 2, 1050 Brussels , Belgium
6 Laboratoire de Physique Subatomique et de Cosmologie, Université GrenobleAlpes , CNRS/IN2P3, Avenue des Martyrs 53, 38026 Grenoble , France
7 CNRS, UMR 7589, LPTHE , 75005 Paris , France
We study Higgs boson production in association with a top quark and a W boson at the LHC. At NLO in QCD, t W H interferes with t t¯H and a procedure to meaningfully separate the two processes needs to be employed. In order to define t W H production for both total rates and differential distributions, we consider the diagram removal and diagram subtraction techniques that have been previously proposed for treating intermediate resonances at NLO, in particular in the context of t W production. These techniques feature approximations that need to be carefully taken into account when theoretical predictions are compared to experimental measurements. To this aim, we first critically revisit the t W process, for which an extensive literature exists and where an analogous interference with t t¯ production takes place. We then provide robust results for total and differential cross sections for t W and t W H at 13 TeV, also matching shortdistance events to a parton shower. We formulate a reliable prescription to estimate the theoretical uncertainties, including those associated to the very definition of the process at NLO. Finally, we study the sensitivity to a nonStandardModel relative phase between the Higgs couplings to the top quark and to the W boson in t W H production. The study of the Higgs boson is one of the main pillars of the physics programme of the current and future LHC runs. Accurate measurements of the Higgs boson properties are crucial both to validate the standard model (SM) as well as to possibly discover new physics through the detection of

deviations from the SM predictions. Another main pillar of
the LHC research programme of the coming years is the study
of the top quark. Being the heaviest quark, the top quark
also plays a main role in Higgs boson phenomenology. In
particular, the main production channel for the Higgs boson
at the LHC entails a topquark loop, while very soon Run II
will be sensitive to onshell top–antitop pair production in
association with the Higgs boson, a process that will bring
key information on the strength of the topquark Yukawa
interaction.
Exactly as when no Higgs is present in the final state, top
quark and Higgs boson associated production can proceed
either via a top pair production mediated by QCD
interactions, or as a singletop (anti)quark process mediated by
electroweak interactions. The latter case, despite being
characterised by much smaller cross sections with respect to
the QCD production, displays a richness and peculiarities
that make it phenomenologically very interesting. For
example, it is sensitive to the relative phase between the Higgs
coupling to the top quark and to the W boson. Singletop
production (in association with a Higgs boson) can be
conveniently classified in three main channels: t channel,
schannel (depending on the virtuality of the intermediate W
boson) and t W (H ) associated production. For the first two
channels, this classification is unambiguous only up to
nexttoleading order (NLO) accuracy if a fiveflavour scheme
(5FS) is used. Beyond NLO, the two processes interfere and
cannot be uniquely separated. The associated t W (H )
production, on the other hand, can easily be defined only at
leadingorder (LO) accuracy and in the 5FS, i.e. through the
partonic process gb → t W (H ). At NLO, real corrections
of the type gg → t W b(H ) arise that can feature a
resonant t¯ in the intermediate state and therefore overlap with
gg → t t¯(H ), i.e. with t t¯(H ) production at LO. This fact
would not be necessarily a problem per se, were it not for the
fact that the cross section of t t¯(H ) is one order of magnitude
larger than t W (H ), and its subtraction – which can only be
achieved within some approximation – leads to ambiguities
that have to be carefully estimated and entails both
conceptual issues and practical complications.
A fully consistent and theoretically satisfying treatment
of resonant contributions can be achieved by starting from
the complete final state W bW b(H ) in the fourflavour
scheme (4FS), including all contributions, i.e. doubly, singly
and nonresonant diagrams. Employing the complexmass
scheme [1,2] to deal with the finite width of the top quark
guarantees the gauge invariance of the amplitude and the
possibility of consistently going to NLO accuracy in QCD.
This approach has been followed already for W bW b and
other processes calculations at NLO [3–8]. Recent advances
have also proven that these calculations can be consistently
matched to parton showers (PS) [9–11]. However, from the
practical point of view, such calculations are
computationally very expensive and would entail the generation of large
samples including resonant and nonresonant contributions
as well as their interference. This approach does not allow one
to distinguish between toppair and singletop production in
the event generation. One would then need to generate signal
and background together in the same sample (a procedure
that would entail complications from the experimental point
of view, for example in datadriven analyses) and
communicate experimental results and their comparison with theory
only via fiducial cross sections measurements. In any case,
results for W bW b H are currently available at NLO
accuracy only with massless b quarks [12], and therefore cannot
be used for studying t W H .
A more pragmatic solution is to adopt a 5FS, define final
states in terms of onshell top quarks, and remove overlapping
contributions by controlling the ambiguities to a level such
that the NLO accuracy of the computation is not spoiled,
and total cross section as well as differential distributions
can be meaningfully defined. To this aim, several techniques
have been developed with a different degree of flexibility,
some being suitable only to evaluate total cross sections,
others being employable in event generators. They have been
applied to t W production and to the production of particles
in SUSY or in other extensions of the SM, where the problem
of resonances appearing in higherorder corrections is
recurrent. Two main classes of such techniques exist for event
generation, and they are generally dubbed diagram removal
(DR) and diagram subtraction (DS). Unavoidably, all these
approaches have their own shortcomings, some of them of
more theoretical nature, such as possible violation of gauge
invariance (which, however, turns out not to be worrisome),
or ambiguities in the far offshell regions which need to be
kept into account and studied on a processbyprocess basis.
As will be recalled in the following, DR and DS actually
feature complementary virtues and vices. An important point
of the 5FS approach is that the combination of the separate
t t¯(H ) and t W (H ) results ought not to depend on the
technical details used to define the t W (H ) contribution, in the limit
where overlapping is correctly removed and possible
theoretical ambiguities are under control. In practice, the most
common approach is to organise the perturbative expansion
in poles of the top propagator, where t t¯(H ) production is
computed with onshell top quarks (this approach can also
be used in the 4FS [3–5,7]). In this case, the
complementary t W (H ) contribution should encompass all the remaining
effects, e.g. including the missing interference with t t¯(H ) if
that is not negligible. We are interested in finding a practical
and reliable procedure to generate t W (H ) events under this
scenario.
As already mentioned above, Higgs and topquark
associated processes can provide further information on the top–
Higgs interaction. While at the Run I the LHC experiments
have not claimed observation yet for these processes, setting
only limits on the signal strength [13–19], t t¯H is expected
to be soon observed at the Run II, allowing a first direct
measurement of the topquark Yukawa coupling yt. Indeed,
unlike the dominant Higgs production mode via gluon fusion,
where the extraction of yt is indirect, in the case of t t¯H
such an extraction is (rather) modelindependent. In addition,
t t¯H production is well known to be sensitive to the Higgs
CP properties [20–31]. On the other hand, Higgs
production in association with a single top quark (t H and t W H ),
though rare, is very sensitive to departures from the SM,
since the total rate can increase by more than an order of
magnitude [32,33] due to constructive interference effects,
becoming comparable to or even larger than t t¯H . In
particular, Higgs plus single top allows one to access the phase of
yt, which remains unconstrained in gluon fusion and t t¯H ;
a preliminary, yet not enough sensitive exploration has been
carried out already at Run I [19]. At variance with t channel
and schannel processes, predictions for t W H cross sections
are only available at LO. Accurate predictions for t W H are
not only important for the measurement of t W H itself, but
also as a possible background to t H production, and in view
of the observation of t t¯H and of the consequent extraction
of Higgs couplings.
The main aim of this paper is to present the first
predictions at NLO accuracy for t W H cross sections at the LHC.
In order to do that, we first review the different techniques
that can be used to remove resonant contributions from NLO
corrections and also make a proposal for an improved DS
scheme. We then study the t W process in detail, and
compare our findings with the results already available in the
literature. Finally, we apply these techniques to get novel
results for t W H production.
At this point, we stress that even though it is not really the
original motivation of this work, a critical analysis of t W is
certainly welcome. The relevance of which approach ought
to be used to describe t W production is far from being only of
academic interest: already during the Run I, singletop
production has been measured by both ATLAS and CMS in the t
channel [34–37], schannel [38,39] and t W [40–42] modes.
In particular, in t W analyses the difference between the two
aforementioned methods, DR and DS (without including the
t t¯–t W interference), has been added to the theoretical
uncertainties. In view of the more precise measurements at the Run
II, a better understanding of the t t¯–t W overlap is desirable, in
order to avoid any mismodelling of the process and incorrect
estimates of the associated theoretical uncertainties, both in
the total cross section and in the shape of distributions.
Furthermore, given the large amount of data expected at Run II
and beyond, a measurement aimed at studying the details of
the t t¯–t W interference may become feasible, and this gives
a further motivation to study the best modelling strategy.
Finally, a sound understanding of t W production will also be
beneficial for the numerous analyses which involve t t¯
production as a signal or as background. This is particularly true
in analyses looking for a large number of jets in the final state,
which typically employ Monte Carlo samples based on NLO
merged [43–45] events, where stable top quarks are produced
together with extra jets (t t¯ + n j ). In this case, all kinds of
nontoppair contributions, like t W , need to be generated
separately. While these effects are expected to be subdominant,
their importance has still to be assessed and may become
relevant after specific cuts, given also the plethora of analyses;
an example can be the background modelling in t t¯H or t H
searches. Note that results for W bW b plus one jet have been
recently published [46,47], but the inclusion of extra
radiation in merged samples is much more demanding if one starts
from the W bW b final state and thus may be impractical. Last
but not least, a reliable 5FS description of t W is desirable in
order to assess residual flavourscheme dependence between
the 4FS (W bW b) and the 5FS (t t¯+t W ) modelling of this
process. Such a comparison can offer insights on the relevance of
initialstate logarithms resummed in the bottomquark PDF,
which are an important source of theoretical uncertainty.
The paper is organised as follows: in Sect. 2 we review
the definitions of the DR and DS techniques, and we also
include a proposal for an improved DS scheme. In Sect. 3
we describe our setup for NLO computations, also matched
to parton shower. In Sect. 4 we review the results from these
techniques in the wellstudied case of t W production,
performing a thorough study of their possible shortcomings,
considering the impact of interference effects between
toppair and singletop processes, and investigating what happens
after typical cuts are imposed to define a fiducial region for
the t W process. In Sect. 5 we repeat a similar study for the
SM t W H process at NLO. We also include the study of the
t W H process going beyond the SM Higgs boson,
investigating results from a generic CPmixed Yukawa interaction
between the Higgs and the top quark. Our study is
complemented in the appendix by a quantitative assessment of the
t W b and t W b H channels, studied as standalone processes
in the 4FS and at the partonic level. In Sect. 6 we summarise
our findings and propose an updated method to estimate the
impact of theoretical systematics in the definition of t W and
t W H at NLO in the 5FS.
2 Subtraction of the top quark pair contribution
As discussed in the introduction, the computation of
higherorder corrections to t W (H ) requires the isolation of the
t t¯(H ) process, and its consequent subtraction. In this
section we review the techniques to remove such a resonant
contribution which appears in the NLO real emissions of the
t W (H ) process.
In the case of fixedorder calculations, and in particular
when only the total cross section is computed, a global
subtraction (GS) of the onshell top quark can be employed,
which just amounts to the subtraction of the total cross
section for t t¯(H ) production times the t → bW branching
ratio [48,49]:
where (t → W b) is the physical width, while t is
introduced in the resonant topquark propagator as a regulator,
and gauge invariance is ensured in the t → 0 limit. A
conceptually equivalent version, which can be applied locally in
the virtuality of the resonant particle and in an analytic form,1
has been employed in the NLO computations for pair
production of supersymmetric particles [50,51] and for charged
Higgs boson production [52,53].
On the other hand, NLO+PS simulations require a
subtraction which is fully local in the phase space. In order to
achieve such a local subtraction, two main schemes have
been developed, known as diagram removal (DR) and
diagram subtraction (DS) [54]. These subtraction schemes have
been studied in detail for t W production matched to parton
shower in MC@NLO [54,55] and in Powheg [56], as well
as in the case of t H − [57] and for supersymmetric particle
pair production [58–61].
To keep the discussion as compact as possible, we focus
on t W production (see Fig. 1 for the LO diagrams) and
consider the specific case of the t W −b¯ real emission and of
1 It differs only by tiny boundary effects, see [50].
At W bDR2 = A1t 2 + 2Re(A1t A2∗t ).
2
Fig. 2 Examples of doubly resonant (left), singly resonant (centre) and
nonresonant (right) diagrams contributing to W bW b production. The
first two diagrams on the left (with the t line cut) describe the NLO
realemission contribution to the t W − process
its overlap with t t¯ production. The extension to the process
with an extra Higgs boson is straightforward. Strictly
speaking, one should consider t t¯ and t W −b¯ (t¯W +b) processes as
doubly resonant and singly resonant contributions to W bW b
production, which also contains the set of nonresonant
diagrams as shown in Fig. 2. However, as discussed in detail
in the appendix, the contribution from nonresonant W bW b
production and offshell effects for the finalstate top quark
are tiny, as well as possible gaugedependent effects due to
the introduction of a finite top width. Therefore, we will treat
one top quark as a finalstate particle with zero width, so that
the only intermediate resonance appears in toppair
amplitudes. The squared matrix element for producing a t W −b¯
final state can be written as
At W b2 = A1t + A2t 2
where A1t denotes the singletop amplitudes, considered as
the realemission corrections to the t W process, while A2t
represents the resonant toppair amplitudes describing t t¯
production, where the intermediate t¯ can go onshell. The
corresponding representative Feynman diagrams are shown in
Fig. 2. In the following, we will discuss the DR and DS
techniques in detail.
DR (diagram removal): Two different version of DR have
been proposed in the literature:
– DR1 (without interference): This was firstly proposed
in [54] for t W production and its implementation in
MC@NLO. One simply sets A2t = 0, removing not only
A2t 2, which can be identified with t t¯ production, but
also the interference term 2Re(A1t A2∗t ), so that the only
contribution left is
This technique is the simplest from the implementation
point of view and, since diagrams with intermediate top
quarks are completely removed from the calculation, it
does not need the introduction of any regulator.
– DR2 (with interference): This second version of DR was
firstly proposed in [50] for squarkpair production. In this
case, one removes only A2t 2, keeping the contribution
of the interference between singly and doubly resonant
diagrams
Note that the DR2 matrix element is not positivedefinite,
at variance with DR1. In this case, while the integral is
finite even with t → 0, in practice one has to introduce
a finite t in the amplitude A2t in order to improve the
numerical stability of the phasespace integration.
An important remark concerning the DR schemes is that,
as they are based on removing contributions all over the phase
space, they are not gauge invariant. However, for t W the
issue was investigated in detail in [54], and effects due to
gauge dependence have been found to be negligible. We have
confirmed this finding for both t W and t W H in a different
way, and we discuss the details in the appendix, where we
show that gauge dependence is not an issue if one uses a
covariant gauge, such as the Feynman gauge implemented in
MadGraph5_aMC@NLO.
DS (diagram subtraction): DS methods, firstly proposed
for the MC@NLO t W implementation, have been developed
explicitly to avoid the problem of gauge dependence, which,
at least in principle, affects the DR techniques. The DS matrix
element is written as
At W bDS = A1t + A2t 2 − C2t ,
2
where the local subtraction term C2t , by definition, must [54,
56]:
1. cancel exactly the resonant matrix element A2t 2 when
the kinematics is exactly on top of the resonant pole;
2. be gauge invariant;
3. decrease quickly away from the resonant region.
Given the above conditions, a subtraction term can be
written as
2
C2t ({ pi}) = f ( pW2 b) A2t ({qi}) ,
where pW b = ( pW + pb), and { pi} is the set of momenta
of the external particles (i.e. the phasespace point), while
200 250 300
m(W–, –b) [GeV]
200 250 300
m(W–, –b) [GeV]
{qi} are the external momenta after a reshuffling that puts the
internal antitop quark on massshell, i.e.
Such a reshuffling is needed in order to satisfy gauge
invariance of C2t , which in turn implies gauge invariance of the
DS matrix element of Eq. (5) in the t → 0 limit. There
is freedom to choose the prefactor f ( pW2 b), and the Breit–
Wigner profile is a natural option to satisfy the third
condition. Here, we consider two slightly different Breit–Wigner
distributions:
– DS1:
which is just the ratio between the two Breit–Wigner
functions for the top quark computed before and after
the momenta reshuffling, as implemented in MC@NLO
and POWHEG for t W [54, 56].
– DS2:
This offshell profile of the resonance differs from DS1
by the replacement mt t → √s t [62, 63]. The exact
shape of a resonance may be processdependent, and in
the specific case of t W ( H ) we find that this profile is in
better agreement than DS1 with the offshell line shape
of the amplitudes A2t 2 (away from W b threshold), as
can be seen in Fig. 3. In particular, we have checked
that the agreement between the A2t 2 profile and the C2t
subtraction term in DS2 holds for the separate qq¯ and gg
channels; at least in the qq¯ channel there is no
gaugerelated issue, offshell effects in toppair production are
correctly described by A2t 2, and DS2 captures these
effects better. As it will be shown later, this modification
in the resonance profile leads to appreciable differences
between the two DS methods at the level of total cross
sections as well as differential distributions.
Apart from the different resonance line shapes, another
important remark on DS is about the reshuffling of the
momenta. Such a reshuffling is not a Lorentz
transformation, since it changes the mass of the W b system, therefore
different momenta transformations could result in different
subtraction terms. Actually, there is an intrinsic arbitrariness
in defining the onshell reshuffling, potentially leading to
different counterterms and effects. Thus, on the one hand DS
ensures that gauge invariance is preserved in the t → 0
limit, at variance with DR. On the other hand, it introduces a
possible dependence on how the onshell reshuffling is
implemented, which is not present in the DR approach and needs
to be carefully assessed. To our knowledge, this problem has
not been discussed in depth in the literature; a more detailed
study is under way and will be reported elsewhere. In this
work, we adopt the reshuffling employed by MC@NLO and
POWHEG [54, 56], where the recoil is shared democratically
among the initialstate particles, also rescaling by the
difference in parton luminosities due to the change of the partonic
centreofmass energy.
Finally, we comment on the introduction of a nonzero
topquark width in the DR2 and DS methods. In order to
regularise the singularity of A2t , we have to modify the
denominator of the resonant topquark propagators as
At variance with the case of a physical resonance, here t is
just a mathematical regulator that does not necessarily need to
be equal to the physical topquark width.2 In fact, one can set
it to any number that satisfies t/mt 1 without affecting
the numerical result in a significant way [58,60]. We have
checked that the NLO DR2 and DS codes provide stable
results with t in the interval between 1.48 and 0.001 GeV.3
After all the technical details exposed in this section, we
summarise the key points in order to clearly illustrate our
rationale in assessing the results in the next sections:
– Our starting point is to assume the (common) case where
results for t t¯(H ) production are generated with onshell
top quarks. Resonance profile and correlation among
production and decay are partially recovered from the
offshell LO amplitudes with decayed top quarks, following
the procedure illustrated in [64]. In particular, after this
procedure the onshell production cross section is not
changed.
– The GS procedure is gauge invariant and ensures that all
and just the onshell t t¯(H ) contribution is subtracted.
Thus, under the working assumptions in the previous
point, GS provides a consistent definition of the missing
t W (H ) cross section, which can be combined with t t¯(H )
without double countings and including all the remaining
effects, such as interference. A local subtraction scheme
should return a cross section close to the GS result if
offshell and gaugedependent effects are small.
– DS is gauge invariant by construction. The difference
between the GS and DS cross sections can thus quantify
offshell effects in the decayed t t¯(H ) amplitudes. From
Fig. 3 and the related discussion, we already find DS2 to
provide a better treatment than DS1 in the subtraction of
the offshell t t¯(H ) contribution; the difference between
DS1 and DS2 quantifies the impact of different offshell
profiles.
2 A modified version of DS (DS∗), which requires one to know the
analytic structure of the poles over each integration channel, was
proposed in [60] to guarantee gauge invariance already with a finite width.
In practice, there is no difference between DS and DS∗ if t is small
enough.
3 However, the computational time does depend on this regulator,
because the smaller is t the larger are the numerical instabilities,
resulting in a slower convergence of the integration. For this reason, the results
presented in the paper have been generated setting this regulator close
to the physical value of the top width at LO, t 1.48 GeV.
– DR is in general gauge dependent. The difference
between GS and DR2 amounts to the impact of possible
gaugedependent contributions and offshell effects. As
it will be shown, for the t W and t W H processes this
difference is tiny. Finally, the difference between DR2 and
DR1 amounts to the interference effects between t t¯(H )
and t W (H ); the singletop process is well defined per se
only if the impact of interference is small.
As a last comment, we argue that in practice gauge
dependence in DR should not be an issue in our case. When
using a covariant gauge and only transverse external gluons,
any gaugedependent term decouples from the gg → t W b
amplitudes [54], and this remains valid also after adding
a Higgs. An independent constraint on gaugedependent
effects comes also from the offshell profiles in Fig. 3. In
the qq¯ channel, A2t 2 is free from gauge dependence and
validates the C2t DS2 offshell profile for t W (H ); the
gaugeinvariant DS2 counterterm continues to agree with A2t 2
also in the gg channel, which in turn limits the size of alleged
gaugedependent effects in DR2. Moreover, even in the case
of a significant gauge dependence, its effects should cancel
out in a consistent combination of t t¯(H ) and t W (H ) events,
if the offshell amplitudes used to decay t t¯(H ) have been
computed in the same gauge as t W (H ).
3 Setup for NLO+PS simulation
The code and events for t W production at hadron
colliders at NLOQCD accuracy can be generated in the
MadGraph5_aMC@NLO framework by issuing the following
commands:
> import model loop_smno_b_mass
> generate p p > t w [QCD]
> add process p p > t˜ w+ [QCD]
> output
> launch
and similarly for t W H production:
> import model loop_smno_b_mass
> generate p p > t w h [QCD]
> add process p p > t˜ w+ h [QCD]
> output
> launch
The output of these commands contains, among the NLO
real emissions, the t W b amplitudes that have to be treated
with DR or DS. The technical implementation of DR1 (no
interference) in the NLO code simply amounts to edit the
relevant matrix_*.f files, setting to zero the toppair
amplitudes. To implement DR2, on the other hand, one subtracts
the square of the toppair amplitudes from the full matrix
element. A subtlety is that the toppair amplitudes (and only
those) need to be regularised by introducing a nonzero width
in the topquark propagator. Note that, as we have already
remarked in Sect. 2, this width is just a mathematical
regulator. The DS is more complicated, since it also requires
the implementation of the momenta reshuffling to put the top
quark onshell before computing the subtraction term C2t .
The automation of such onshell subtraction in the
MadGraph5_aMC@NLO framework is under way and will be
become publicly available in the near future.
In our numerical simulations we set the mass of the Higgs
boson to mH = 125.0 GeV and the mass of the top quark to
mt = 172.5 GeV, which are the reference values used by the
ATLAS and CMS collaborations at the present time in Monte
Carlo generations. We renormalise the top Yukawa coupling
onshell by setting it to yt/√2 = mt/v, where v 246 GeV
is the electroweak vacuum expectation value, computed from
the Fermi constant GF = 1.16639 × 10−5 GeV−2; the
electromagnetic coupling is also fixed to α = 1/132.507. The
W and Z boson masses are set to mW = 80.419 GeV and
mZ = 91.188 GeV. In the 5FS the bottomquark mass is set to
zero in the matrix element, while mb = 4.75 GeV determines
the threshold of the bottomquark parton distribution
function (PDF), which affects the parton luminosities.4 We have
found the contributions proportional to the bottom Yukawa
coupling to be negligible, therefore we have set yb = 0 as
well.
The proton PDFs and their uncertainties are
evaluated employing reference sets and error replicas from the
NNPDF3.0 global fit [65], at LO or NLO as well as in the
5FS or 4FS (4FS numbers are shown in the appendix). The
value of the strong coupling constant at LO and NLO is set to
αs(5F,LO)(mZ) = 0.130 and, respectively, αs(5F,NLO)(mZ) =
0.118.
The factorisation and renormalisation scales (μF and μR)
are computed dynamically on an eventbyevent basis, by
setting them equal to the reference scale μd0 = HT/4, where
HT is the sum of the transverse masses of all outgoing
particles in the matrix element. The scale uncertainty in the
results is estimated varying μF and μR independently by
a factor two around μ0. Additionally, we also show total
cross sections computed with a static scale, which we fix
to μs0 = (mt + mW)/2 for t W production and to μs0 =
(mt + mW + mH)/2 for t W H .
We use a diagonal CKM matrix with Vtb = 1, ignoring
any mixing between the third generation and the first two. In
particular, this means that the top quark always decays to a
bottom quark and a W boson, Br(t → bW ) = 1, with a width
4 In the 4FS simulations presented in Appendix mb enters the calcula
tion of the hardscattering matrix elements and the phase space.
computed at LO in the 5FS equal to t = 1.4803 GeV.5 Spin
correlations can be preserved by decaying the events with
MadSpin [21], following the procedure presented in [64].
We choose to leave the W bosons stable, because we focus
on the behaviour of the b jets stemming either from the top
decay or from the initialstate gluon splitting.
Shortdistance events are matched to the Pythia8
parton shower [66] by using the MC@NLO method [67]. Jets
are defined using the antikT algorithm [68] implemented in
FastJet [69], with radius R = 0.4, and required to have
A jet is btagged if a b hadron is found among its constituents
(we ideally assume 100% btagging efficiency in our studies).
The same kinematic cuts are applied for b jets as for light
flavour jets in the inclusive study. In the fiducial phase space,
on the other hand, a requirement on the pseudorapidity of
4 t W production
is imposed, resembling acceptances of btagging methods
employed by the experiments.
In this section we (re)compute NLO+PS calculations for
t W production at the LHC, running with a centreofmass
energy √s = 13 TeV. With the shorthand t W we mean the
sum of the two processes pp → t W − and pp → t¯W +,
which have the same rates and distributions at the LHC. We
carefully quantify the impact of theoretical systematics in
the event generation. Our discussion is split in two parts,
focusing first on the inclusive event generation and the related
theoretical issues, and then on what happens when fiducial
cuts are applied.
4.1 Inclusive results
We start by showing in Fig. 4 the renormalisation and
factorisation scale dependence of the pp → t W cross section,
computed at LO and NLO accuracy, keeping the t stable.
Results are obtained by employing the static and dynamic
scales μs0 and μd0 (defined in Sect. 3) in the left and right
plot, respectively. We show results where we simultaneously
vary the renormalisation and factorisation scales on the
diagonal μR = μF; on top of this, for LO and NLO DR results,
we also present two offdiagonal profiles where μR = √2μF
and μR = μF/√2. In the two plots we present predictions
5 In the 4FS, due to a nonzero bottom mass, the LO width is slightly
reduced to t = 1.4763 GeV.
Fig. 4 Scale dependence of the total cross section for pp → t W − and
t¯W + at the 13TeV LHC, computed in the 5FS at LO and NLO accuracy,
presented for μF = μR ≡ μ using a static scale (left) and a dynamic
scale (right). The NLO t W b channels are treated using DR and DS; see
Sect. 2 for more details. Furthermore, we show NLO results from GS
t W (13 TeV)
NLO DS2
NLO GS
tW at the LHC13
5FS inclusive cross section
tW at the LHC13
5FS inclusive cross section
obtained employing both DR, neglecting (DR1, red) or taking
into account (DR2, orange) the interference with t t¯, and DS,
with the two Breit–Wigner forms in Eq. (8) (DS1, blue) or in
Eq. (9) (DS2, green). We also report results using global
subtraction (GS, squares) for the static scale choice. The details
for the various NLO schemes can be found in Sect. 2. We
remark that we have validated our NLO DR1 and DS1 codes
against the MC@NLO code, finding very good agreement.
The values of the total rate computed at the central scale μ0
are also quoted in Table 1. Unlike in Fig. 4, in this case scale
variations are computed by varying μF and μR independently
by a factor two around μ0.
As expected, NLO corrections visibly reduce the scale
dependence with respect to LO predictions. Comparing DR1
and DR2, we see that interference effects are negative at
this centreofmass energy, and reduce significantly the NLO
cross section, by about 13%. Also, the cross section scale
dependence is different, in particular for very small scales.
This effect is driven by the LO scale dependence in t t¯
amplitudes, which is larger at low scales. Moving to DS, we find
that DS1 and DS2 predictions show a 8% difference.
Therefore, the dependence on the subtraction scheme is large, being
comparable to the scale uncertainty or even larger.
We note that the total rate predictions obtained with DR2
and DS2 agree rather well within uncertainties, especially at
the reference scale choice, and also agree with the predictions
from the GS scheme. This result is quite satisfactory because
it supports some important observations. First, that the
offshell effects of the topquark resonant diagrams are small, and
indeed well described by the (gaugeinvariant)
parametrisation of Eq. (9). Second, that possible gauge dependence in
DR2 is in practice not an issue if one uses a covariant gauge,
(only for a static scale), and two offdiagonal profiles of the scale
dependence, (μR = √2μ , μF = μ/√2) and (μR = μ/√2 , μF = √2μ),
for LO and NLO DR. Finally, the scale dependence of pp → t t¯ at LO
is also reported for comparison
μd0 = HT/4. We also report the scale and PDF uncertainties and the
NLOQCD K factors; the numerical uncertainty affecting the last digit
is quoted in parentheses
where the subtraction of A2t 2 turns out to be very close to
an onshell gaugeinvariant subtraction. On the other hand,
DR1, which does not include the interference in the
definition of the signal, and DS1, which has a different profile over
the virtuality of the intermediate top quark, do not describe
well the NLO effects and extrapolate to a biased total cross
section, even in the t → 0 limit. Thus, a third observation is
that interference terms are not negligible, and it is mandatory
to keep them in the definition of the t W process in order to
have a complete simulation. Finally, a fourth point is that to
include interference effects is not enough, but one also needs
to subtract the toppair process with an adequate profile over
the phase space. This picture is confirmed at the level of
differential distributions in the following discussion, and also
at the total cross section level in the 4FS; see the appendix.
We now turn to differential distributions, and we show
some relevant observables in Figs. 5 and 6. Here, we employ
a dynamical scale choice, μ0 = HT/4 and we do not impose
any cut on the finalstate particles. Note that, for simplicity
and after the shorthand t W , we label as t both the undecayed
Fig. 6 Same as Fig. 5, but for the btagged jets. Note that the secondhardest b jet is described by the parton shower at LO, while by the matrix
element at NLO
top quark in t W − production and the antitop in t¯W +;
similarly, W indicates the W − in the first process and W + in the
second one, i.e. the boson produced in association with t , and
not the one coming from the t decay. Particles (not) coming
from the top decay are identified by using the eventrecord
information. We see that the DR1 and DS1 simulations tend
to produce harder and more central distributions, while the
DR2 and DS2 results, very similar one another, tend to be
softer and more forward. In any case, NLO corrections
cannot be taken into account by the LO scale uncertainty, nor
be described by a K factor, especially for the physics of b
jets. The hardest b jet ( jb,1) dominantly comes from the top
decay, while the secondhardest b jet is significantly softer
due to the initialstate g → bb¯ splitting. As seen for DR2,
the high pT W boson and b jets are highly suppressed due
to the negative interference with the t t¯ process. In fact, due
to this interference the cross section can become negative in
some corners of the phase space, for example in the high pT
tail of the second b jet. We interpret this fact as a sign that t W
cannot be separated from t t¯ in this region, and the two
conTable 2 Total cross sections in pb at the LHC 13 TeV for the processes
pp → tt¯ and pp → t W , in the 5FS at NLO+PS accuracy. Results
are presented before any cut (left), after fiducial cuts (centre), and also
adding top reconstruction on the event sample (right). We also report the
scale and PDF uncertainties, as well as the cut efficiency with respect
to the case with no cuts. All numbers are computed with the reference
dynamic scale μ0 = HT/4, and the numerical uncertainty affecting the
last digit is reported in parentheses
t W DR1
t W DR2
t W DS1
t W DS2
% %
σNLO ± δμ ± δPDF
744.1 (9) +−48..78 ±1.7
73.22 (9) +−56..17 ±2.0
65.12 (9) +−26..88 ±2.0
70.93 (9) +−46..70 ±2.0
66.09 (9) +−26..88 ±1.9
Fiducial cuts
% %
σNLO ± δμ ± δPDF
44.9 (3) +−69..05 ±1.9
44.70 (7) +−46..07 ±1.9
43.88 (8) +−37..20 ±1.9
44.65 (8) +−36..88 ±1.9
44.05 (8) +−36..39 ±1.9
tributions must be combined in order to obtain a physically
observable (positive) cross section.
In summary, the t W –t t¯ interference significantly affects
the inclusive total rate as well as the shapes of various
distributions at NLO. In particular, different schemes give rise
to different NLO results, with ambiguities which in
principle can be larger than the scale uncertainty. Such differences
arise from two sources: the interference between resonant
(toppair) and nonresonant (singletop) diagrams, which is
relevant and ought to be taken into account, and (in the case
of DS) the treatment of the offshell tails of the toppair
contribution. These ambiguities are intrinsically connected to the
attempt of separating two processes that cannot be physically
separated in the whole phase space. On the other hand, we
have also found that two of such schemes, DR2 and DS2,
give compatible results among themselves and integrate up
to the total cross section defined in a gaugeinvariant way
in the GS scheme. We are now ready to explore whether a
region of phase space (possibly accessible from the
experiments) exists where the two processes can be separated in a
meaningful way.
4.2 Results with fiducial cuts
In this section we would like to investigate whether t W can be
defined separately from t t¯ at least in some fiducial region of
the phase space, in the sense that in such a region interference
terms between the two processes and thus theoretical
ambiguities are suppressed. In practice, this goal can be achieved
by comparing results among different NLO schemes, since
the difference among them provides a measure of
interference effects and related theoretical systematics (gauge
dependence in DR, subtraction term in DS). We remark that the
following toy analysis is mainly for illustrative purposes, since
the same procedure can be applied to any set of fiducial cuts
defined in a real experimental analysis, also imposing a
selection on specific decay products of the W bosons.
Fiducial cuts + top reco.
% %
σNLO ± δμ ± δPDF
44.9 (3) +−69..05 ±1.9
41.70 (7) +−36..88 ±1.9
41.85 (8) +−37..70 ±1.9
41.90 (8) +−36..88 ±1.9
41.91 (8) +−36..89 ±1.9
Motivated by the bjet spectra in Fig. 5 and by
experimental t W searches, a popular strategy to suppress the t t¯
background as well as t W –t t¯ interference is to select events
with exactly one central b jet [40–42,48,55,70]. We define
our set of “fiducial cuts” for t W by selecting only events with
1. exactly one b jet with pT( jb) > 20 GeV and η( jb) <
2.5,
2. exactly two central W bosons with rapidity y(W ) <
2.5.
In this regard we stress that the first selection is the key to
suppress the contributions from t t¯ amplitudes, hence both
the pure t t¯ “background” as well as the t W –t t¯ interference
(i.e. theoretical ambiguities). Note that we would like to draw
general conclusions about the generation of t W events,
therefore we have chosen to define a pseudo event category that
does not depend on the particular decay channel of the W
bosons. The second selection is added to mimic a good
reconstructability of these bosons inside the detector regardless of
their finalstate daughters; it affects less than 7% of the events
surviving selection 1.
Looking at Table 2 we can see that, before any cut is
applied, the event category is largely dominated by the t t¯
contribution. Once the above fiducial cuts are applied, the t t¯
contribution is reduced by more than a factor 16, while the
t W rate shrinks by about just one third (for DR2 and DS2),
bringing the signaltobackground ratio σ (t W )/σ (t t¯) close
to unity, which is exactly the aim of t W searches. The impact
of interference has been clearly reduced by the cuts; The
fiducial cross sections computed with the different NLO schemes
agree much better with each other, than before selections are
applied. Still, there is a minor residual difference in the rates,
which amounts to about 2%.
From the distributions in Figs. 7 and 8 we can see once
more an improved agreement among the different NLO
schemes in the fiducial region. The lower panels show flatter
and positive K factors and a lower scale dependence in the
high pT tail than before the cuts, since we have suppressed
Fig. 7 pT and η distributions the top quark and the W boson as in Fig. 5, but after applying the fiducial cuts to suppress interference between t W b
and t t¯
the interference with LO t t¯ amplitudes. Although
considerably mitigated, some differences are still visible among
the four schemes in the high pT region of the btagged jet
( jb,1). Monte Carlo information shows that the central b
jet coincides with the one stemming from the top decay
( jb,t ) for the vast majority of events. In the high pT region,
however, the b jet can also originate from a hard
initialstate g → bb¯ splitting, similar to the case of t channel t H
production [33].
This suggests that, if on top of the fiducial cuts we also
demand the central b jet to unambiguously originate from the
top quark, then we may be able to suppress even further the
t W –t t¯ interference and the related theoretical systematics.
In fact, we can see from Table 2 and from the right plot
in Fig. 8 that, after such a requirement is included in the
event selection, the total rates as well as the distributions
end up in almost perfect agreement, and one can effectively
talk about t W and t t¯ as separate processes in this region:
interference effects have been suppressed at or below the
level of numerical uncertainty in the predictions. A possible
remark is that the topreconstruction requirement shaves off
another ∼ 2 pb of the cross section, i.e. more than the residual
discrepancy between the different NLO schemes before this
last selection is applied.
To summarise, a naturally identified region of phase space
exists where t W is well defined, i.e. gauge invariant and
basically independent of the scheme used (either DR1, DR2,
DS1, DS2) to subtract the t t¯ contribution. Given the fact that
DS2 and DR2 also give consistent results outside the fiducial
region and integrate to the same total cross section, equal
to the GS one, they can both be used in MC simulations.
In practice, given the fact that the gaugedependent effects
are practically small when employing a covariant gauge, and
that the implementation in the code is rather easy, DR2 is
certainly a very convenient scheme to use in simulations of
t W production in the 5FS, including the effects of
interference with the t t¯ contribution. In addition, one can use the
difference between DR1 and DR2 (i.e. the amount of t W –
t t¯ interference) to assess whether the fiducial region where
the measurements are performed is such that the
processdefinition uncertainties are under control (smaller than the
missing higherorder uncertainties), and to estimate the
residual processdefinition systematics. We have seen that
requiring the presence of exactly one central b jet is a rather
effective way to identify such a fiducial region. We have also found
that, especially in DR2 and DS2 schemes, the perturbative
series for the t W process is well behaved, NLOQCD
corrections mildly affect the shape of distributions but reduce the
scale dependence considerably with respect to LO. A
further handle to suppress processdefinition systematics can
be given by a reconstruction of the top quark, identifying the
central b jet as coming from its decay. Toptagging techniques
are being developed (theoretical and experimental reviews
can be found at [71] and [72,73]), and may help to define
a sharper fiducial region, although this may depend on the
tradeoff between the toptagging efficiency and the amount
of residual processdefinition ambiguities to be suppressed.
5 t W H production
In this section we present novel NLO+PS results for t W H
production in the 5FS at the 13TeV LHC (diagrams are
shown in Figs. 9, 10). Similar to what we have done for
t W in the previous section, we address the theoretical
systematics both at the inclusive level and with fiducial cuts. We
anticipate that our findings for t W H are qualitatively similar
to the ones for t W , but the larger numerical ratio between the
toppair and singletop contributions enhances the impact of
interference effects and exacerbates theoretical systematics
in the simulation, which are clearly visible in the t , W , H and
bjet observables. We will see that this can be alleviated after
applying suitable cuts. Finally, we investigate the impact of
nonSM couplings of the Higgs boson on this process.
5.1 Inclusive results
As for t W , we start by showing the renormalisation and
factorisation scale dependence of the t W H cross section
in Fig. 11, both at LO and NLO accuracy, using
different schemes to treat the t W b H realemission channels (the
details for the various NLO schemes can be found in Sect. 2).
The values of the total rate computed at the central scale μ0
are also quoted in Table 3. Unlike in Fig. 11, in this case scale
variations are computed by varying μF and μR independently
by a factor two around μ0.
The same pattern we have found for t W is repeated.
Comparing DR results obtained by neglecting (DR1, red) or
taking into account (DR2, orange) interference with t t¯H , we
observe again that these interference effects are negative, but
their relative impact on the cross section is even more
sizeable. The interference reduces the NLO rate by about 5 fb,
which amounts to a hefty −25%, leading to a K factor close
to 1. Since interference effects are driven by the LO t t¯H
contribution, they grow larger for lower scale choices. The
cross sections obtained employing the two DS techniques,
DS1 (blue) and DS2 (green), show large differences which
go beyond the missing higher orders estimated by scale
variations, and can be traced back to the different Breit–Wigner
prefactor in the subtraction term C2t . As it has been the case
for t W production, we find that DR2 and DS2 are in good
agreement with GS.
In complete analogy with the case of the t W b channel in
t W production at NLO, we perform a study of the theoretical
systematics in the modelling of the t W b H channel
(employing the 4FS to isolate this contribution), which can be found
in the appendix.
In Figs. 12 and 13 we collect some differential
distributions. Observables related to the Higgs boson can essentially
be described by a constant K factor for each subtraction
scheme. On the other hand, similar to the t W case, the NLO
distributions for the top quark and the W boson are quite
different among the four NLO techniques. As we know, these
differences are driven essentially by whether the
interference with t t¯H is included or not (in DR), and by the profile
Fig. 11 Scale dependence of the total cross section for pp → t W − H
and t¯W + H at the 13TeV LHC, computed in the 5FS at LO and NLO
accuracy, presented for μF = μR ≡ μ using a static scale (left)
and a dynamic scale (right). The NLO t W b H channels are treated
using DR and DS; see Sect. 2 for more details. Furthermore, we show
NLO results from GS (only for a static scale), and two offdiagonal
(pμroRfil=es μo/f√th2e, sμcaFle=de√p2enμd)e,nfcoer,L(OμRand=N√LO2μD,Rμ.FFi=nallμy,/√the2)scaanlde
dependence of pp → t t¯H at LO is also reported as a reference
scale μd
0 = HT/4. We also report the scale and PDF uncertainties and
the NLOQCD K factors; the numerical uncertainty affecting the last
digit is quoted in parentheses
tWH at the LHC13
5FS inclusive cross section
tWH at the LHC13
5FS inclusive cross section
t W H (13 TeV)
of the subtraction term (in DS). These NLO effects are quite
remarkable for the b jets, since the negative interference with
t t¯H drastically suppresses central hard b jets.
Summarising, in analogy with the t W process, effects due
to the interference between t t¯H and t W H which appear in
NLO corrections of the latter process are significant, and
hence the details of how the t t¯H contribution is subtracted
enormously affect the predictions for both the total rate and
the shape of distributions. On the one hand, a LO description
of t W H in the 5FS is apparently not sufficient. On the other
hand, the NLO prediction strongly depends on the subtraction
scheme employed. This last point is only a relative issue, if
we take into account the fact that DR2 and DS2 results are
quite consistent with each other and integrate to the same total
cross section as GS, which suggests that they provide a better
description of the physics not included in t t¯H than DR1 and
DS1. Nevertheless, as in the case of t W production, it is clear
that fiducial cuts are crucial to obtain a meaningful separation
of t W H from t t¯H , and their effects will be discussed in the
next subsection.
5.2 Results with fiducial cuts
We now move to investigate whether the separation between
t W H and t t¯H can become meaningful in a fiducial region,
where interference between the two processes and
theoretical systematics are suppressed. The problem is exactly
analogous to the t W –t t¯ separation. In practice, for any selection
defined by suitable cuts, one needs to quantify the residual
difference among different subtraction schemes and see if it
is small enough.
2.5 No cuts
2.5 No cuts
2.5 No cuts
Motivated by the same rationale behind our t W
discussion, we define our set of “fiducial cuts” for t W H selecting
only events with
We recall that the first selection is the key to suppress the
doubletop amplitudes and hence t W H –t t¯H interference
and theoretical ambiguities. We do not assume any particular
decay channel for the heavy bosons and hence the second and
third selections are added to mimic a good
reconstructability of the W and H bosons in the detector. However, they
are not crucial since they affect just 5% of the events after
surviving selection 1. Our pseudo event category is defined
mainly for illustrating the issues behind the simulation of the
t W H signal, but the same procedure can be applied to any
realistic set of fiducial cuts in experimental analyses,
including a selection on specific decay products of the W and H
bosons.
Looking at Table 4, we can see that the situation for t W H
is very similar to the one we have already seen for t W . Before
the fiducial cuts, the category is largely dominated by t t¯H
events. Once the fiducial cuts are applied, the contribution
from t t¯H is reduced by more than a factor 20, while the
one from t W H just by about 1/4 (for DR2), enhancing the
signaltobackground ratio (t W H/t t¯H ) to about 0.5, which
is encouraging from the search point of view. The
interference with LO t t¯H amplitudes has been visibly reduced, with
fiducial cross sections among the four techniques agreeing
much better than in the inclusive case; this is also apparent
in the differential distributions of Figs. 14 and 15, and in
particular in the much smaller scale dependence in the tails of
t W H distributions at NLO.
Nevertheless, a residual difference of about 6% (0.7 fb)
is present between the DR1 and DR2 fiducial cross sections,
and this discrepancy is also visible in the shape of some
pT distributions. Once again, if we use MC information to
additionally require the central b jet to come unambiguously
from the top quark, the residual interference effects are
further reduced to less than 1% at a tiny cost on the signal
efficiency. This brings the differential predictions in excellent
agreement among the four schemes and with this selection
one can effectively consider t W H and t t¯H as separate
processes.
Fig. 13 Same as Fig. 12, but for the btagged jets. Note that the secondhardest b jet is described by the parton shower at LO, while by the matrix
element at NLO
Finally, we briefly comment on the possibility to observe
the t W H signal at the LHC. Naturally, one may wonder
whether it will be possible to observe it over the (already quite
rare) t t¯H process, in an experimental analysis that applies a
selection similar to our fiducial cuts. For example, the LHC
Run II is expected to deliver an integrated luminosity in the
100 fb−1 ballpark. In our pseudo event category (with top
reconstruction), the difference between including or
excluding the t W H contribution amounts to
Unfortunately, once branching ratios of the Higgs and W
bosons and realistic efficiencies are taken into account, these
numbers disfavour the possibility to observe t W H over t t¯H
at the Run II. On top of that, there are many more background
processes contributing to our event category than just t t¯H .
This makes the searches for the SM t W H signal extremely
Table 4 Total cross sections in fb at the LHC 13 TeV for the processes
pp → t t¯H and pp → t W H , in the 5FS at NLO+PS accuracy. Results
are presented before any cut (left), after fiducial cuts (centre), and also
adding top reconstruction on the event sample (right). We also report the
last digit is reported in parentheses
scale and PDF uncertainties, as well as the cut efficiency with respect
to the case with no cuts. All numbers are computed with the reference
t W H DR1
t W H DR2
t W H DS1
t W H DS2
tWH at the LHC13
t)( 1.5
η
d
/
dσ 1.0
% %
σNLO ± δμ ± δPDF
Fiducial cuts
% %
σNLO ± δμ ± δPDF
tWH at the LHC13
Fiducial cuts + top reco.
% %
σNLO ± δμ ± δPDF
tWH at the LHC13
3.0 tWH at the LHC13
2.5 Fiducial cuts
3.0 tWH at the LHC13
2.5 Fiducial cuts
3.0 tWH at the LHC13
2.5 Fiducial cuts
challenging, and the highluminosity upgrade of the LHC
(also called t H q by experiments) with Higgs decay into a
is definitely needed in order to have a sufficient number of
pair of bottom quarks ( H → bb¯), where semileptonic t W H
events can lurk in the signal region defined by a large (b)jet
On the other side, simulated t W H events should be taken
multiplicity. In fact, including the t W H simulation in the
into account in other searches for Higgs boson and top quark
signal definition (as opposed to considering it a background)
associated production, which are not necessarily going to
apply t W H specific fiducial cuts, in order to complete the
in the case of either t t¯H or t channel t H searches will lead to
a more comprehensive view on Higgs boson and topquark
MC modelling. In particular, this will be relevant in searches
associated production, e.g. being relevant when setting limits
for the t t¯H signal, and also for the t channel t H process
or measuring the signal strength.
tWH at the LHC13
5FS (N)LO+PYTHIA8
Fiducial cuts + top reco.
tWH at the LHC13
5FS (N)LO+PYTHIA8
Fiducial cuts + top reco.
5.3 Higgs characterisation
In this section we explore the sensitivity of t W H production
to beyond the standard model (BSM) physics in the Higgs
sector. In particular, we start by studying the total
production rate in the socalled “κ framework” [74, 75] where the
SM Higgs interactions are simply rescaled by a
dimensionless constant κ . Then we move to characterising the Yukawa
interaction between the Higgs boson and the top quark, which
in general can be a mixture of CPeven and CPodd terms,
similar to what has been done for t channel t H production
in Sect. 5 of [33]. To describe the Yukawa interaction, we
consider the following Lagrangian for a generic spin0 mass
eigenstate X0 that couples to both scalar and pseudoscalar
fermionic currents:
t
L0 = −ψ¯ t cα κHtt gHtt + i sα κAtt gAtt γ5 ψt X0,
where cα ≡ cos α and sα ≡ sin α are the cosine and sine
of the CPmixing phase α; κHtt,Att are real dimensionless
parameters that rescale the magnitude of the CPeven and
CPodd couplings, and gHtt = gAtt = mt/v (= yt/√2),
with v 246 GeV. While redundant (only two
independent real quantities are needed to parametrise the most
general CPviolating interaction between a spin0 particle and
the top quark at dimension four), this parametrisation has
the practical advantage of easily interpolating between the
purely CPeven (cα = 1, sα = 0) and purely CPodd
(cα = 0, sα = 1) cases, as well as to easily recover the
SM when cα = 1 , κHtt = 1 . In the κframework cα = 1,
and only the part proportional to κHtt is considered. On the
other hand, the SMlike interactions between the Higgs and
the EW vector bosons is described by
where gHVV = 2m2V /v (V = W, Z ). For the full Higgs
characterisation (HC) Lagrangian, including CPeven and
CPodd higherdimensional X0V V operators, we refer to [76,
77]. The Feynman rules from these Lagrangians are coded
in the publicly available HC_NLO_X0 model [78]. The code
and events for t W X0 production at NLO can be generated in
a way completely analogous to SM t W H :
> import model HC_NLO_X0no_b_mass
> generate p p > t w x0 [QCD]
> add process p p > t˜ w+ x0 [QCD]
In this section we show results obtained only with the DR
techniques. We start by showing results in the κframework
in Fig. 16. We can see that a CPeven Higgs boson is highly
sensitive to the relative sign of Higgs couplings to fermions
(t ) and EW bosons (W ). Depending on the (κHtt, κSM)
configuration, the inclusive t W H rate (DR2, including
interference with t t¯H ) can be enhanced from 15 fb to almost 800 fb.
The t W H process can thus be exploited to further constrain
the allowed regions in the twodimensional plane spanned by
κHtt and κSM together with the already sensitive t H
production.
Given the experimental constraints after the LHC Run I
[79], we can reasonably fix the Higgs interaction with the
EW bosons to be the SM one, and turn to study CPmixing
effects in the Higgs–fermion sector. It is also reasonable to
assume that gluon fusion is dominated by the topquark loop,
and consequently the X0–top interaction must reproduce the
SM gluonfusion rate at NLO accuracy to comply with
experimental results. This fixes the values of the rescaling factors
in Eq. (13) to
κHtt = 1, κAtt =  gHgg/gAgg  = 2/3,
leaving the value of the CPmixing angle α free.
In Fig. 17 we plot the total NLO cross section for Higgs
production in association with a topquark pair t t¯X0 (red),
and for the combined contribution of t t¯X0 and t W X0
including their interference (orange), which is simply obtained by
summing the t W X0 DR2 cross section to the t t¯X0 one. We
can immediately see that the inclusion of the t W X0 process
lifts the yt → −yt degeneracy that is present in t t¯X0
production. For a flippedsign Yukawa coupling, the interference
between singletop diagrams where the Higgs couples to the
top and the ones where it couples to the W becomes
constructive, and the total cross section is augmented from roughly
Fig. 16 Left: inclusive t W H cross sections with DR2 scanned over
different values for κHtt and κSM. Note that the standard model
configuration (+1, +1) almost lies in a minimum, which means the process is
suited for constraining this place due to enhanced rates for deviations
from the SM. Right: the t W H cross section is shown for three different
intensities of the X0W W coupling κSM, as a function of κHtt, where
DR1 results are also reported, to gauge the impact of interference with
tt¯H
500 fb (SM, α = 0◦) to more than 600 fb (α = 180◦). This
enhancement can help in a combined analysis of the Higgs
interactions, though it is less striking than the one which
takes place in the t channel Higgs plus singletop process
(which is also reported in blue for comparison). For the sake
of clarity we point out that, going along the αaxis in Fig. 17,
Inclusive NLO cross section at LHC 13 TeV
Gluon fusion @ SM rate (κHtt = 1, κAtt = 2/3)
L = – y—t2 ψ−t (cακHtt + isακAtt γ5 )ψt X0
yt = −yt,SM
the t W X0 cross section includes in fact two different
interference effects. On the one hand, there is the interference
between singletop amplitudes with Higgstofermion and
Higgstogaugeboson interactions, similar to the t H
process. This is already present at LO, and it drives the growth
of the cross section from the SM case (maximally destructive
interference) to the case of a reversedsign top Yukawa
(maximally constructive). On the other hand, employing DR2 for
the computation of the t W X0 NLO cross section means that
also the interference with t t¯H is included. This is an effect
present only at NLO, and its size depends as well on the
CPmixing angle α (due to the different ratio between t t¯H and
t W H amplitudes).
In Fig. 18 we compare some differential distributions for
the SM hypothesis (blue), the purely CPodd scenario (red)
and the flippedsign CPeven case (green), before any cuts.
We can see that the interference between the doubly resonant
t t¯H and the singly resonant t W H amplitudes is largest for
the SM case. For the case of flipped Yukawa coupling the
interference gives a minor contribution, while for the CPodd
case it is very tiny because the doubly resonant contribution
is at its minimum. The W and Higgs transverse momentum
distributions become harder when the mixing angle is larger.
Once the fiducial cuts are applied (Fig. 19), the difference
between DR1 and DR2 decreases as expected.
In conclusion, we find that the t W H process can help to
lift the yt → −yt degeneracy for t t¯H and put constraint on
BSM Yukawa interactions of the Higgs boson in a combined
analysis, on top of the most sensitive t channel t H
production mode. Finally we recall that, if one also assumes a SM
interaction between the Higgs and the W bosons, one can
Fig. 17 NLO cross sections (with scale uncertainties) for pp → t t¯X0,
pp → t W X0 (with DR2) and pp → t X0 (t channel) at the 13TeV
LHC as a function of the CPmixing angle α, where κHtt and κAtt are set
to reproduce the SM gluonfusion cross section for every value of α.
The t t¯X0 and t W X0 processes have been computed using the dynamic
scale μ0 = HT/4, while t X0 results are taken from [33]
Fig. 18 pT and η distributions for the top quark, the W boson and the
Higgs boson at NLO+PS accuracy in t W H production at the 13TeV
LHC with different values of the CPmixing angles between the Higgs
boson and the top quark, where κHtt and κAtt are set to reproduce the SM
gluonfusion cross section for every value of α. The results are obtained
employing DR2 (solid) and DR1 (dashed), without any cut
Fig. 19 Same as in Fig. 18, but after applying the fiducial cuts
further include the γ γ decay channel data to put limits on
the CPmixing phase α.
6 Summary
In this work we have provided for the first time NLO accurate
predictions for the t W H process, including partonshower
effects. In order to achieve a clear understanding of the
ambiguities associated to the very definition of the process at NLO
accuracy due to its mixing with t t¯H , we have revisited the
currently available subtraction schemes in the case of t W
production. We have therefore carefully analysed t W at NLO in
the fiveflavour scheme, and then we have proceeded in an
analogous way for t W H . On the one hand, NLO corrections
to these processes are crucial for a variety of reasons,
ranging from a reliable description of the b quark kinematics to a
better modelling of backgrounds in searches for Higgs
production in association with single top quark or a top pair. On
the other hand, they introduce the issue of interference with
t t¯ or t t¯H production, which has a significant impact on the
phenomenology of these processes.
Our first aim has been to study the pro’s and the con’s
of the various techniques (which fall in the GS, DR and DS
classes) that are available to subtract the resonant
contributions appearing in the NLO corrections. At the inclusive level
these techniques can deliver rather different results, with
differences which can often exceed the theoretical uncertainties
on the NLO cross sections estimated via scale variations.
These differences have been traced back to whether a given
technique accounts for the interference between the t W (H )
and t t¯(H ) processes, and to how the offshell tails of the
resonant diagrams are treated. They become visible at the
total cross section level as well as in distributions,
particularly those involving bjet related observables. We find the
DR2 and DS2 techniques to provide a more faithful
description of the underlying physics in t W and t W H than that of
DS1 and DR1, therefore we deem them as preferable to
generate events for these two processes at NLO. We stress that
the aim of our work is to provide a practical and reliable
technique to simulate t W and t W H at NLO, when the
corresponding t t¯ and t t¯H process are generated separately in the
onshell approximation. Our results have no claim of
generality, and cannot be immediately extended to other SM or BSM
processes. A study of subtraction techniques should be
performed on a processbyprocess basis, in particular for BSM
physics, where different widthtomass ratios and different
amplitude structures (i.e. resonance profiles) can appear.
Our second aim has been to study what happens once event
selections similar to those performed in experimental
analyses are applied, and in general whether one can find a fiducial
region where the singletop processes t W and t W H can be
considered well defined per se, and they are stable under
perturbative corrections. A simple cut as requiring exactly one
btagged jet in the central detector (which becomes three b jets
in the case of t W H if the Higgs decays to bottom quarks) can
greatly reduce interference effects, and thus all the
processdefinition systematics of t W (H ) at NLO. In such a fiducial
region, we find the perturbative description of t W (H ) to be
well behaved, and the inclusion of NLO corrections
significantly decreases the scale dependence; differences between
the various DR and DS subtraction techniques are reduced
below those due to missing perturbative orders, making the
separation of the singletop and toppair processes
meaningful. Given a generic set of cuts, we have provided a simple
and robust recipe to estimate the leftover processdefinition
systematics, i.e. use the difference between the DR1 and
DR2 predictions (which amounts to the impact of
interference effects). In general, such approach provides a covenient
way to quantify the limits in the separation of t t¯(H ) and
t W (H ) and the quality of fiducial regions. In particular, this
is essential for a reliable extraction of the Higgs couplings in
t W H production.
Finally, we have investigated the phenomenological
consequences of considering a generic CPmixed Yukawa
interaction between the Higgs boson and the top quark in t W H
production. While the SM cross section is tiny, due to
maximally destructive interference between the H –t and H –W
interactions, and direct searches for this process may only be
feasible after the highluminosity upgrade of the LHC, BSM
Yukawa interaction tend to increase the production rate. For
example, in the case of a reversedsign Yukawa coupling with
respect to the SM, the t W H cross section is enhanced by an
order of magnitude, similar to what happens for the dominant
singletop associated mode, i.e. the t channel t H production.
The large event rate predicted after the combination of these
Higgs plus singletop modes will help to exclude a
reversedsign top Yukawa coupling already during the LHC Run II.
Acknowledgements We thank the LHCHXSWG and in particular the
members of the t t H/t H task force for giving us the motivation to
pursue this study. We are grateful to Simon Fink, Stefano Frixione,
Dorival GonçalvesNetto, Michael Krämer, David LopezVal, Davide
Pagani, Tilman Plehn and Francesco Tramontano for many stimulating
discussions, and to the MadGraph team (in particular Pierre Artoisenet,
Rikkert Frederix, Valentin Hirschi, Olivier Mattelaer and Paolo
Torrielli) for their valuable help. KM would like to acknowledge the Mainz
Institute for Theoretical Physics (MITP) for providing support during
the completion of this work. This work has been performed in the
framework of the ERC grant 291377 “LHCtheory: Theoretical predictions
and analyses of LHC physics: advancing the precision frontier” and
of the FP7 Marie Curie Initial Training Network MCnetITN
(PITNGA2012315877). It is also supported in part by the Belgian
Federal Science Policy Office through the Interuniversity Attraction Pole
P7/37. The work of FD and FM is supported by the IISN “MadGraph”
convention 4.4511.10 and the IISN “Fundamental interactions”
convention 4.4517.08. BM acknowledges the support by the DFGfunded
Doctoral School “Karlsruhe School of Elementary and Astroparticle
Physics: Science and Technology”. The work of KM is supported by
the TheoryLHCFrance initiative of the CNRS (INP/IN2P3). The work
of MZ is supported by the European Union’s Horizon 2020 research and
innovation programme under the Marie SklodovskaCurie grant
agreement No 660171 and in part by the ILP LABEX (ANR10LABX63),
in turn supported by French state funds managed by the ANR within
the “Investissements d’Avenir” programme under reference
ANR11IDEX000402.
Open Access This article is distributed under the terms of the Creative
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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: The t W b and t W b H channels in the 4FS
diagram subtraction techniques, which are used to eliminate
the t t¯ resonant contribution. Since the issue appears just in
the matrixelement description, the study in this appendix is
simply performed at the partonic level. The t W b channel is
more easily addressed in the 4FS, where it appears as a finite
and independent LO contribution, thus it can be isolated from
the other channels contributing to t W . The only difference
from the 5FS is that bottom mass effects are included in
the 4FS description, which act as an IR cutoff; the Feynman
diagrams are the same ones describing the 5FS NLO
realemission channel, and the features and shortcomings of DR
and DS are independent of the flavour scheme employed. An
analogous study is then repeated for the t W b H channel in
the 4FS.
The problem of the LO t t¯ contribution in the t W −b¯
channel has first been addressed in [48], where it is subtracted
at the cross section level (see Eq. (4) in the reference). This
global subtraction procedure (GS) is described in Sect. 2;
an important point in the calculation is that the two pieces
(t W −b¯ and t t¯) are separately integrated before the
subtraction is performed. The GS procedure ensures that the
remainder of the subtraction converges to a welldefined
limit t → 0, where the result is fully gauge invariant,
and exactly all and just the LO onshell t t¯ contribution is
subtracted. Therefore, combining the t t¯ simulation with the
t W −b¯ obtained this way, one gets a welldefined total rate for
producing the common physical final state, without double
counting and also including interference effects; this
procedure provides a consistent way to define the t W cross section.
Actually, the only way to perform a theoretically
consistent simulation that encompasses both the toppair and
the singletop contributions, that is gauge invariant and that
includes interference and other finite t effects, is to compute
pp → W +bW −b¯ in the 4FS and using a complex topquark
mass. This W bW b simulation will also contain the
contribution from amplitudes without any resonant top propagator
A0t , and also interference between singletop and
singleantitop contributions A1t A1∗t¯, which are not present in the
t W b simulation
2 2
AW bW b = A2t + A1t + A1t¯ + A0t 
= A2t 2 + A1t 2 + 2Re(A2t A1∗t )
2 + 2Re(A2t A1∗t¯)
+ 2Re(A1t A1∗t¯)
2 + 2Re (A2t + A1t + A1t¯)A0∗t
In this appendix we perform a study of the various ways
to treat the t W b channel, in particular we will discuss the
performance and shortcomings of the diagram removal and
nonetheless, we expect the last two lines in Eq. (A.1) to be
negligible compared to the previous two lines, which
encompass toppair t t¯ and singletop t W b production.
Table 5 LO cross sections in the 4FS at the 13TeV LHC for the
processes pp → W +bW −b¯ (complexmass scheme), pp → t t¯ (t stable),
and singly resonant pp → t W −b¯ plus pp → t¯W +b computed using
the GS, DR and DS prescriptions. For these t W b results we also report
the ratio R defined in Eq. (A.2). All numbers are computed using the
static scale μs0 = (mt +mW)/2, and the numerical uncertainty affecting
the last digit is reported in parentheses
In the end, the reference result will be the difference
between the W bW b cross section (computed in the
complexmass scheme, with a physical t ) and the t t¯ cross section
(computed with onshell top’s), which in general guarantees
a correct description of t W b production. If the nonresonant
contributions A0t to W bW b, the A1t A1∗t¯ interference, and
the offshell effects related the single top kept stable in t W b
simulations are small enough, this cross section will be close
to the one obtained from GS.
The global subtraction schemes cannot be applied to event
generation, where a fully local subtraction of the toppair
contribution must be performed in the 2 → 3 phase space; this
is exactly the reason why alternative techniques such as DR
and DS have been developed and implemented in MC@NLO
and POWHEG for t W production. Nevertheless, a simple but
powerful way to test the adequacy of DR and DS can be
carried out by comparing their total cross section with the GS
one, which is the number we expect to be returned from a
consistent local subtraction scheme. We perform this
comparison in Table 5, where cross sections are computed with
s
the static scale μ0, also showing the cross section ratio R
defined as
From the results in Table 5 we first notice that the
W bW b − t t¯ cross section (computed with a physical t)
is in good agreement with the t W b one computed with the
GS prescription (which is independent on the actual value of
t), thus either can be considered as the reference value. This
also confirms that nonresonant contributions from A0t and
A1t A1∗t¯ interference are small, and justifies the 5FS treatment
where one top is always onshell.
Ratio to DR1
200 250 300
m(W–, –b) [GeV]
Fig. 20 Invariant mass m(W −, b¯) in the pp → t W −b¯ process,
computed with DR and DS
Among the two diagram removal techniques, the DR1
modelling does not capture the A2t A1∗t interference, which
amounts to more than 9 pb (this was evident already in
Table 1). On the other hand, there is excellent agreement
between the DR2 cross section and the desired one from
W bW b − t t¯, thus any possible violation of gauge invariance
in the DR2 total rate must be negligible.6 When we compute
AW bW b2 − A2t 2 (namely W bW b − A2t 2 in Table 5),
we can see that the difference with t W b DR2 is a modest
2%; this provides a further confirmation that effects related
equivalent to an onshell t t¯ subtraction (compare W bW b −t t¯
and W bW b − A2t 2).
Moving to diagram subtraction, we can see that DS2 is in
rather good agreement with GS and DR2, while DS1 clearly
overestimates the total rate, which tends to be much closer
to DR1.
The situation can be understood also at the differential
level by looking at the m W b distribution in Fig. 20. The
missing of interference in DR1 leads to an underestimate of the
rate in the lowmass region m W b < mt, and to an
overestimate in the tail m W b > mt; at the LHC energy, the latter
region dominates, leading to a net overestimate of the total
6 We recall that in our simulations we have included only transverse
polarisations of initialstate gluons, and we have employed a
covariant gauge for gluon propagators. A noncovariant gauge (axial) was
shown to lead to differences at the level of permille in the case of t W
production [54].
Table 6 LO cross sections in the 4FS at the LHC with √s = 13 TeV for
the processes pp → W +bW −b¯ H (complexmass scheme), pp → t t¯H
(t stable), and singly resonant pp → t W −b¯ H plus pp → t¯W +b H
computed using the GS, DR and DS prescriptions. For these t W b H
results we also report the ratio R, which is analogous to the one defined
in Eq. (A.2). All numbers are computed using the static scale μs0 =
(mt + mW + mH)/2, and the numerical uncertainty affecting the last
digit is reported in parentheses
rate.7 DR2 and DS2 nicely reproduce the peakdip
interference pattern, with small differences between the two curves;
since DS2 is gauge invariant, this fact can be interpreted as
that gauge effects in DR2, when employing a covariant gauge,
are small also at the level of differential shapes. Finally, while
DS1 includes interference effects as well, it also introduces
a significant distortion in the profile of the subtraction term
C2t , as already shown in Fig. 3; the net effect is an unreliable
m W b profile, with an inverted dippeak structure and a too
large tail.
We now move on to studying the t W b H channel in t W H
production at NLO, which overlaps with LO t t¯H . We follow
a procedure completely analogous to the one employed for
t W b, therefore we do not repeat all the details in the following
discussion.
Our reference total rate is the difference between the
W bW b H cross section, computed in the complex topquark
mass scheme, and the t t¯H cross section computed in the
approximation of stable finalstate top quarks. Once again
we find GS to be in very good agreement with this
reference value, so both results can be taken as a reference for
comparison with DR and DS; see Table 6.
We can see that the ratio between toppair and
singletop amplitudes is even higher than for t t¯ versus t W , and
this exacerbates the same problems we have observed in that
case. Interference effects are very large and neglecting them
results in an error of O (100%) in DR1, where the cross
section is more than twice that from GS. Once again, we find
DR2 results to be in excellent agreement within the
numer
7 We have verified that the net sum of interference effects in the total rate
is positive at collider energies below ∼2 TeV, while becomes more and
more negative at higher energies, where the phase space for mW b > mt
is larger.
Ratio to DR1
200 250 300
m(W–, –b) [GeV]
Fig. 21 Invariant mass m(W −, b¯) in the pp → t W −b¯ H process,
computed with DR and DS
ical accuracy. The impact of nonresonant amplitudes and
of interference between singletop and singleantitop
contributions is very small, less than 2% of the DR2 rate in
this channel. The rate obtained from DS1 is overestimated
by more than a factor two, while DS2 looks again in better
agreement with GS and DR2, although there is a residual
difference of about 0.7 fb (slightly larger than the 0.3 fb in
the 5FS scheme).
In Fig. 21 we show the m W b differential distribution. A
similar pattern of the one for t W b is repeated: interference
effects are large and positive in the m W b < mt region, while
negative for m W b > mt, where DR1 clearly overestimates
the event rate. The interference pattern is nicely reproduced
by the DR2 and DS2 shapes, although there are some minor
differences between the two methods; instead, DS1 fails to
return a physical shape, due to the visibly distorted profile of
the subtraction term C2t ; see Fig. 3.
We would like to stress one final remark: the fact that gauge
dependence is apparently not an issue in the DR2 procedure
should be regarded as a peculiarity of the t W b and t W b H
channels, and not as a general result. We cannot exclude that
gauge dependence could become a significant issue at higher
perturbative orders (NNLO t W ( H )), or in other processes
with a more complex colour flow, or using a different (i.e.
noncovariant) gauge.
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