Hard Photons in Hadroproduction of Top Quarks with Realistic Final States
G. Bevilacqua H. B. Hartanto M. Kraus T. Weber M. Worek
 
InSe: a two-dimensional material with strong interlayer coupling
    Lijun Zhang
    December 30, 2023
================================================================









§ INTRODUCTION




The top quark, discovered by the CDF and D0 experiments at Fermilab
more than 20 years after its existence was postulated to explain the
observed CP violations in kaon decays, is the heaviest elementary
particle in the Standard Model (SM) of particle physics. Due to its
large mass and its correspondingly short lifetime the top quark decays
before hadronic bound states can be formed, thus, passing its spin
information onto its decay products. With a mass of the order of the
electroweak scale, the top quark Yukawa coupling to the Higgs boson is
of the order of unity. This alone makes the top quark unique among the
fermions and its potential to provide insights into physics beyond the
SM (BSM) is anticipated. Various BSM models introduce modifications
within the top quark sector, which can be constrained by precise
measurements of the tt̅ and tt̅+X cross sections,
 where X=H,j,γ,Z,W^±, tt̅. Examples include
composite top quarks, Randall-Sundrum extra dimensions, models with
coloured scalars or universal extra dimensions.  Studies of top quark
properties provide a unique environment for testing the SM and for
hunting BSM physics.  Investigations of the dynamics of the top quark
pair production process in association with a hard photon, for
example, directly probe the top quark electric charge and the
structure of the tt̅γ coupling. Any deviation from the SM
prediction of the measured observables could be an indication of BSM
physics and might be linked to the production of an exotic (possibly
heavier) top-like quark or the top quark with an anomalous electric
dipole moment, see e.g. Refs.
<cit.>.
Good examples of such observables comprise the transverse momentum
spectrum of the photon, p_T,γ, and the azimuthal
angle-rapidity distance between the photon and the softest b-jet,
Δ R_b_2,γ, <cit.>.

First evidence for tt̅γ production has been reported by
the CDF collaboration in pp̅ collisions at the Tevatron with
√(s)=1.96 TeV <cit.>. Observation was also
announced by the ATLAS collaboration in pp collisions at the Large
Hadron Collider (LHC) with √(s)=7 TeV
<cit.>. Meanwhile, measurements have been carried out at
the LHC by both ATLAS and CMS collaborations at √(s)=8 TeV
<cit.>. For now, due to small
available statistics, these measurements only comprise cross
sections. However, with the second run of the LHC at √(s)=13 TeV
and with increased luminosity more exclusive observables and various
properties of the top quark can be scrutinised.

On the theory side, first next-to-leading order (NLO) QCD calculations
for tt̅γ have been performed for on-shell top quarks
<cit.>. Recently, even
NLO electroweak corrections have been completed
<cit.>. Computations in the approximation of stable tops
give a general idea about the size of higher order effects. However,
they can not provide a reliable description of top quark decay
products or the magnitude of NLO corrections when specific cuts are
imposed on the final states. For a realistic analysis, not only higher
order effects to tt̅γ production need to be included but
also radiative decays of top quarks have to be incorporated. This has
been (partially) achieved by means of parton showers through matching
fixed order NLO QCD predictions for tt̅γ with parton
shower programs via the Powheg method <cit.>, albeit omitting photon emissions in the parton
shower evolution and tt̅ spin correlations
<cit.>. A more sophisticated approach has been employed
in <cit.>, where NLO QCD corrections to production and
decays in the so called narrow width approximation (NWA) have been
calculated.  Non-factorisable QCD contributions, however, that imply a
cross talk between production and decays of top quarks and which
require going beyond the NWA, have been so far neglected.  Such
contributions are formally suppressed, i.e.  O(Γ_t/m_t)
≈ 0.8%, where Γ_t ,m_t are the top quark width and
mass respectively. They proved to be small in the inclusive cross
section. Nonetheless, they can be strongly enhanced in case of
exclusive observables that are crucial for new physics searches. The
lack of any evidence of BSM at the LHC has put known new physics
scenarios under significant strain.  Our attention is focused now on
precision physics and indirect searches aiming at deviations from SM
predictions in precision observables. To probe more subtle BSM effects
also in tt̅γ production, state of the art theoretical
predictions for this process are of vital importance.

In this paper, we calculate for the first time the NLO QCD corrections
to the fully realistic final state pp → e^+ ν_e μ^-
ν̅_μ bb̅γ. We consistently take into account
resonant and non-resonant top quark and W gauge boson contributions
and interference effects among them. Our theoretical predictions are
presented in the form of the fully flexible Monte Carlo program. Thus,
various observables and cuts can be explored and their usefulness can
be demonstrated in realistic Monte Carlo simulations.  The
final results are provided as the Ntuple files
<cit.>. Specifically,  they are stored in the form of the
modified Les Houches event files <cit.> and ROOT files
<cit.> that might be directly employed in experimental studies at
the LHC.

As a final comment, we note that
NLO QCD corrections with complete top quark off-shell effects are also
known for tt̅, tt̅H and tt̅j productions
<cit.>. In case of tt̅ and
tt̅H NLO electroweak corrections have been added as well
<cit.>.

The article is organised as follows. In Section <ref> we
describe the details of our calculation. Input parameters and cuts to
simulate detector response are summarised in Section
<ref>. Numerical results for the integrated and differential
cross sections for the LHC Run II energy of 13 TeV for two
renormalisation (μ_R) and factorisation (μ_F) scale choices
are presented in Sections <ref> and <ref>, respectively.  
The theoretical uncertainties of the total cross sections and various
differential cross sections, that are associated with neglected higher
order terms in the perturbative expansion and with  different
parametrisations of the parton distribution functions (PDFs), are also
given there.  Finally, in Section <ref> our conclusions are
given.






§ COMPUTATIONAL FRAMEWORK





At leading order (LO) in the perturbative
expansion, the pp→ e^+ ν_e μ^- ν̅_μ bb̅γ
final state is produced via the scattering of either two gluons or a quark and
the corresponding anti-quark


    gg   → e^+ ν_e μ^- ν̅_μ bb̅γ ,
    
    qq̅   → e^+ ν_e μ^- ν̅_μ bb̅γ ,


where q stands for u,d,c,s. In total, the gg→ e^+ ν_e μ^-
ν̅_μ bb̅γ subprocess comprises 628 Feynman
diagrams and the qq̅→ e^+ ν_e μ^- ν̅_μ bb̅γ subprocess has only 346 tree level Feynman diagrams. Even
though we do not use Feynman diagrams to obtain matrix elements for
each subprocess, we provide such information here 
in order to shed some lights on the
complexity of the process at hand.  A few examples of Feynman
diagrams contributing at O(α_s^2α^5) for the gg
initial state are presented in Fig. <ref>.

The calculation of scattering amplitudes is performed by means of an
automatic off-shell iterative algorithm <cit.>,
which is implemented within the Helac-Dipoles package
<cit.> and the Helac-Phegas Monte Carlo (MC)
program <cit.>. The latter framework has been used to
cross check our LO results. For the phase-space integration depending
on the MC framework Phegas <cit.>,
Parni <cit.> and Kaleu
<cit.> have been employed.






At NLO, virtual corrections are obtained from the interference of the
one-loop dia- grams with the tree level amplitude. They might be
classified into self-energy, vertex-, box-, pentagon-, hexagon- and
heptagon-type corrections. For the gg dominant production channel we
have 36032 Feynman diagrams at one-loop, among these 90 are
heptagons and 958 are hexagons.  The latter numbers have been
obtained with the help of Qgraf <cit.>.
Virtual corrections are evaluated in d = 4 - 2ϵ dimensions in
the 't Hooft-Veltman version of the dimensional regularisation and
using the Feynman gauge for gauge bosons. The singularities coming
from infrared divergent pieces are canceled by the corresponding ones
arising from the counter-terms of the adopted subtraction scheme
integrated over the phase space of the unresolved parton. The finite
contributions of the loop diagrams are evaluated numerically in d =
4 dimensions. To ensure numerical stability of our calculations we
have used the Ward identity test. On-shell transversality of gluon
amplitudes has been checked up to the one loop level for every phase
space point. About 6% of events, that fail the gauge-invariance
check, have been recomputed in quadruple precision. For
  qq̅→ e^+ ν_e μ^- ν̅_μ bb̅γ partonic
  subprocess at O(α_s^3α^5) there are no gluons as external
particles. Since unstable electroweak bosons are treated in the
fixed-width scheme, the photon Ward identity test could not be applied
straightforwardly.  Instead the scale test, as introduced in
Ref. <cit.>, has been performed. It is based on the
simple observation that the momenta can be rescaled and the amplitude
can be recalculated and compared to the original one.  In this case
higher precision has also been used to recompute 0.15% of events
that did not pass the test. Another cross check that we have performed
comprises a verification of the cancelation of infrared poles. We
compute the virtual corrections using Helac-1Loop
<cit.> and CutTools <cit.>,
which are both parts of the Helac-NLO Monte Carlo framework
<cit.>.  The CutTools code contains an
implementation of the OPP method for the reduction of one-loop
amplitudes at the integrand level <cit.>. For unstable top quarks the complex mass scheme,
as described in Refs.  <cit.>, is used.
At the one loop level the appearance of Γ_t 0 in the
propagator requires the evaluation of scalar integrals with complex
masses, which is supported by the OneLOop program
<cit.> employed in our studies. For consistency,
mass renormalisation for the top quark is also done by applying the
complex mass scheme in the well known on-shell scheme.  Re-weighting
techniques, helicity and colour sampling methods are employed in order
to optimise the performance of our system.

The real emission corrections to the LO process arise from tree-level
amplitudes with one additional parton, i.e. an additional gluon, or a
quark anti-quark pair replacing a gluon.  All possible subprocesses
contributing to the real emission part are shown in Table
<ref>. The number of Feynman diagrams corresponding to the
subprocesses under scrutiny is also given to underline the complexity
of the calculations. The following three subprocesses qg→ e^+
ν_e μ^- ν̅_μ bb̅γ q, q̅g→ e^+ ν_e
μ^- ν̅_μ bb̅γq̅ and qq̅→ e^+
ν_e μ^- ν̅_μ bb̅γ g are related by crossing
symmetry.





The singularities from soft and/or collinear parton emissions are isolated
via subtraction methods for NLO QCD calculations: the commonly used
Catani-Seymour dipole subtraction
<cit.>, and a fairly new
Nagy-Soper subtraction scheme <cit.>, both
implemented in the Helac-Dipoles software. Specifically,
Helac-Dipoles implements the massless dipole formalism of
Catani and Seymour, as well as its massive version for arbitrary
helicity eigenstates and colour configurations of the external
partons. The Nagy-Soper subtraction scheme makes use of random
polarisation and colour sampling of the external partons. An overall
performance of this scheme has been assessed in
Ref. <cit.> where a detailed comparison with
results based on the Catani-Seymour dipole subtraction scheme has been
carried out for a collection of  processes.  
Thus, in Table <ref> we only compare the total number of
subtraction terms that need to be evaluated in both schemes.  In each
case, three times fewer terms are needed in the Nagy-Soper subtraction
scheme compared to the Catani-Seymour scheme. The difference
corresponds to the total number of possible spectators, which are
relevant in the Catani-Seymour case, but not in the Nagy-Soper case. A
phase space restriction (α_max) on the contribution of the
subtraction terms, as proposed e.g. in Refs. <cit.>, is
included for both subtraction cases.  We consider two choices, namely
α_max=1, that corresponds to the original formulation of the
Catani-Seymour and Nagy-Soper subtraction scheme, as well as
α_max=0.01. In case of the Nagy-Soper subtraction scheme,
which was our default scheme used for the calculations, we have
checked that the final results for the sum of real radiation and
integrated dipoles were independent of the α_max choice. We
have further cross checked that results for the real emission part are
in agreement with results obtained with the Catani-Seymour dipole
subtraction scheme.  For the real correction part, we also adopt the
Kaleu phase-space generator that is equipped with additional,
special channels that proved to be important for phase-space
optimisation.





§ INPUT PARAMETERS AND CUTS





In the following we present predictions for pp → e^+ ν_e  μ^-
ν̅_μ  b b̅ γ +X production at O(α_s^3 α^5) for the LHC Run II energy of √(s) =
13 TeV. We consider decays of weak bosons to different lepton
generations only, thus, neglecting the interference effects. However,
the difference between the LO cross sections for pp → e^+ ν_e  μ^-
ν̅_μ  b b̅ γ +X and pp → e^+ ν_e   e^-
ν̅_e   b b̅ γ +X is at the per mille level,
thus, the simplification is very well motivated. The complete cross
section for pp→ℓ^+ ν_ℓℓ^-ν̅_ℓ
bb̅γ, with ℓ^± = e^±,μ^±, can be obtained by
multiplying results presented in the following by a factor of 4. The
Cabibbo-Kobayashi-Maskawa mixing of the quark generations is neglected
and the following SM parameters are used



    [              m_W=80.385  GeV,                                          Γ_W = 2.0988  GeV,;           m_Z=91.1876   GeV ,                                         Γ_Z = 2.50782  GeV,;      Γ_t^ LO = 1.47848  GeV ,                                  Γ_t^ NLO = 1.35159   GeV ,;              m_t=173.2  GeV ,                               G_μ=1.166378 × 10^-5 GeV^-2 . ]



The top quark width is calculated according to <cit.>
 at the scale m_t. All other quarks, including b
quarks as well as leptons, are assumed to be massless.  Leptonic W
gauge boson decays do not receive NLO QCD corrections. To include some
effects of higher order corrections for the gauge boson widths, the
NLO QCD values of the corresponding W width are used for LO and NLO
matrix elements.  The electroweak coupling is derived from the Fermi
constant G_μ in the G_μ-scheme, where 
α_G_μ=√(2) G_μ m_W^2 sin_W^2/π and
sin_W^2=1-m_W^2/m_Z^2.  For our setup we have 
α_G_μ≈ 1/132.  In the G_μ-scheme electroweak
corrections related to the running of α_G_μ and to the ρ
parameter (proportional to m_t^2/m^2_W), are incorporated. By
parametrising the lowest order in terms of G_μ a large part of
these universal electroweak corrections is absorbed. To describe the
emission of the hard (real) photon, however, we use the
α(0)-scheme with α≡α(0) =
1/137. Consequently, the prediction for the tt̅γ cross
section is decreased by more than 3%.  Based on the fact that
relative NLO EW corrections to the on-shell tt̅γ
production at the 13 TeV LHC are negative and of the order of 2%
<cit.> we believe that this is a more consistent approach
compared to employing α_G_μ.  In the first
step we use kinematic-independent scales μ_R=μ_F=μ_0 with the
central value μ_0=m_t/2 rather than μ_0=m_t. Even though the
mass of the heaviest particle appearing in the process seems to be a
more natural option, this scale choice is motivated by the fact that
tt̅ production at the LHC is dominated by t-channel gluon
fusion, which favours smaller values of the scale. Additionally, the
contributions beyond NLO that include the resummation of
next-to-leading logarithmic soft gluon effects (NLO+NLL) are smaller
for μ_0=m_t/2 than for μ_0=m_t, as we have explicitly checked
with the help of the Top++ program
<cit.>. Taking into account that photon emission is not
a QCD effect this picture should not change for pp→ tt̅γ production. With the goal of stabilising shapes in the high
p_T regions, that are relevant for the new physics searches, we have
explored a dynamical choice for μ_R and
μ_F. Kinematic-dependent scales should help to achieve flatter
differential K-factors, thus, to describe more appropriately
regions of the phase-space far away from the tt̅ threshold. For
the process at hand, we explored several possibilities and decided in
the end to consider the following dynamical scale
μ_R=μ_F=μ_0=H_T/4 where H_T is the total transverse momentum
of the system, which we have defined as


    H_T= p_T,  e^++p_T, μ^- +p_T,  b_1 + p_T,  b_2  +
    p_T^miss + p_T, γ ,


where b_1 and b_2 are bottom-jets (not bottom quarks) and
p_T^miss is the total missing transverse momentum from escaping
neutrinos. The theoretical uncertainty is estimated with independent
scale variation μ_Rμ_F, subject to the additional restriction
0.5 < μ_R/μ_F < 2.  In practise such a  restriction amounts to
consider the following pairs


    (μ_R/μ_0,μ_F/μ_0) = {
    (2,1),(0.5,1),(1,2),(1,1),(1,0.5),(2,2),(0.5,0.5) 
    } .


Consequently, the minimum and maximum of the resulting cross section
is chosen. Let us mention here that while calculating the scale
dependence for the NLO cross section we keep Γ_t^ NLO
fixed independently of the scale choice. For two scales μ =
μ_0/2 and μ= 2μ_0 with μ_0 = m_t/2 the change in the
value of Γ_t^ NLO is smaller than ± 1.2%. The error
introduced by this treatment is, however, of higher orders. We have
checked that for the simpler case, i.e. for the pp → e^+ν_e μ^-
ν̅_μ bb̅ +X process, the variation in Γ_t^
NLO has introduced deviations in the total cross section up to ±
1.5% only <cit.>. Let us further note
that in Ref. <cit.> a similar procedure has been
discussed. In this paper the mismatch between the scale used in
partial and total top quark decay widths has been compensated by the
so-called partial width correction. The latter has been studied for
total cross section for pp/pp̅→ e^+ ν_e μ^- ν̅
bb̅+X within cuts for Tevatron and LHC at different center of
mass system energies both with fixed and dynamical scales. In the case
of the LHC at √(s)=14 TeV, for example, these partial width
corrections amounted to 1%-3% depending on the scale choice
employed. At NLO (LO) in QCD we employ CT14nlo (CT14llo)
<cit.> PDFs and describe the running of the strong
coupling constant α_s with two-loop (one-loop) accuracy. The
calculations are performed in the so-called 5 flavour scheme, however,
the b-initiated contributions are not taken into account due to
their numerical insignificance. To be more precise already at LO they
are below 0.1%. All final-state partons with pseudorapidity |η|
<5 are recombined into infrared-safe jets via the anti-k_T
jet algorithm <cit.>. The cone size and jet resolution
parameter R is set to R=0.4. We require exactly two b-jets, one
photon, two charged leptons and missing transverse momentum,
p_T^miss.  The hard photon is defined with p_T, γ>25
  GeV and |y_γ|<2.5. To avoid QED collinear singularities
in photon emission, caused by q → q γ splittings, a
separation between quark and photon is required. Since distinguishing
between quark and gluon jets is impossible on the experimental side,
at the same time a separation between photons and gluons is induced as
well. As a consequence, at a given photon p_T an angular restriction
on the soft gluon emission phase-space is introduced. Thus, soft
divergences in the real emission part are different from those in the
virtual correction impairing the cancelation of infrared divergences.
To ensure soft and collinear safety we use a modified cone approach as
described in Ref. <cit.>, which implements a (smooth)
isolation condition treating quarks and gluons the same way. With the
isolation cone of R_γ j=0.4 for each parton i we evaluate
Δ R_γ i between this parton and the photon. We reject
the event unless the following condition is fulfilled



    ∑_i E_T, i Θ(R - R_γ i)  ≤ E_T, γ(
    1-cos(R)/1-cos(R_γ j))
     ,



for all R≤ R_γ j, where E_T, i is the transverse
energy of the parton i and E_T, γ is the transverse energy
of the photon. Jets reconstructed inside the cone size R_γ j
are not subject to additional selection criteria.  We apply the
following inclusive cuts to simulate detector response



    [    p_T, ℓ>30   GeV ,          |y_ℓ|<2.5 ,       Δ R_ℓℓ > 0.4 ,;    p_T, b>40   GeV ,         |y_b|<2.5  ,         Δ R_bb>0.4 ,; p^miss_T >20   GeV ,         Δ R_ℓγ>0.4 ,      Δ R_ℓ b > 0.4 ,;                      ]



where ℓ stands for μ^-,e^+.  We set no restriction on the
kinematics of the extra jet. 





§ RESULTS FOR THE LHC RUN II ENERGY OF 13 TEV 
FOR THE FIXED   SCALE CHOICE










 §.§ Integrated cross section and its scale 
dependence
 










With the input parameters and cuts specified above, we arrive at
the following  predictions for μ_R=μ_F=μ_0=m_t/2



    σ^ LO_pp→ e^+ν_eμ^-ν̅_μ bb̅γ
    ( CT14, μ_0=m_t/2)    =
    8.27 ^+2.92   (35%)_-2.01  (24%) fb , 
    σ^ NLO_pp→ e^+ν_eμ^-ν̅_μ bb̅γ 
    ( CT14, μ_0=m_t/2)   =
    7.44 ^+0.07  (   1%)_-1.03  (14%) fb .



At the central scale μ_0=m_t/2, the gg channel dominates the
total LO pp cross section by 79%, followed by the
qq̅+q̅q channel with 21%. Photons are, therefore,
predominantly radiated off the top quark and top quark decay
products. The full pp cross section receives negative and moderate
NLO corrections of 10%. The theoretical uncertainties resulting
from scale variations, where μ_R and μ_F have been varied
independently, and taken in a very conservative way as a
maximum of the lower and upper bounds are 35% at LO and 14% at
NLO. Thus, a reduction of the theoretical error by a factor of 2.5
is observed. Should we instead vary μ_R and μ_F
simultaneously, up and down by a factor of 2 around μ_0, the
uncertainties would remain unchanged. This is due to the fact that the
scale variation is driven solely by the changes in μ_R. In the
case of truly asymmetric uncertainties, however, it is always more
appropriate to symmetrise the errors. After symmetrisation the scale
uncertainty at LO is assessed to be instead of the order of
30%. After inclusion of the NLO QCD corrections, the scale
uncertainty is reduced down to 7%. The graphical presentation of
the behaviour of LO and NLO cross sections upon varying the scale by a
factor ξ∈{0.125,…,8} is shown in
Fig. <ref>.  At LO the individual
contributions of the partonic subprocesses are additionally presented.
The final scale dependence of the NLO cross section as emerged
out of the two contributions (the virtual plus the LO part and the
real emission part) is also
depicted in Fig.  <ref>.  Of course, the
separation is entirely unphysical, but well defined once we state that
we use the 't Hooft-Veltman version of the dimensional regularisation,
with the integrals as defined in the OneLOop library.

Next, we have checked the dependence on the parameters introduced 
in the photon isolation procedure. Specifically,  the general photon
isolation formula is given  by 



    ∑_i E_T, i Θ(R - R_γ i)  ≤ϵ_γ    E_T, γ(
    1-cos(R)/1-cos(R_γ j))^n
     ,



with two additional parameters ϵ_γ and n.  The default
choice, which should guarantee moderate corrections, is
ϵ_γ=1 and n=1, see
Ref. <cit.>. Nevertheless, both ϵ_γ and 
n a priori can have arbitrary values. We have recalculated the
subtracted real emission part of the  NLO results with a different
choice, namely ϵ_γ=1/2 and n=1/2. Within the
integration errors our new results have agreed with the old ones. Thus, 
NLO QCD results for the pp→ e^+ν_e μ^-ν̅_μ bb̅γ +X production process are not sensitive to moderate changes in
values of  ϵ_γ and  n. 

In the subsequent step, the size of the top quark non-factorisable
corrections has been estimated for the total cross section. To achieve
this the full result has been compared with the result in the NWA.
The latter has been obtained by rescaling the coupling of the top
quark to the W boson and the b quark by several large factors, 
as described in Ref. <cit.>, to mimic
the limit Γ_t → 0 when the scattering cross section
factorizes into on-shell production and decay. The top quark
non-factorisable corrections for the pp→ e^+ν_e
μ^-ν̅_μ bb̅γ +X production process amount to
1.5% (2.5%) for LO (NLO). They are consistent with the expected
uncertainty of the NWA, which is of the order of O(Γ_t/m_t).

Coming back to the theoretical uncertainties, we note that, another
source of theoretical uncertainties comes from the PDF
parametrisation. To that end, we have recomputed NLO QCD corrections
to the pp→ e^+ ν_e μ^-ν̅_μ bb̅γ +X
production process with different PDF sets. Following recommendations
of PDF4LHC for the usage of PDFs suitable for applications at the LHC
Run II <cit.> we employ additionally to the CT14
PDF set the MMHT14 PDF set <cit.> and NNPDF3.0
<cit.>. Let us say here, that differences coming from
NLO results for various PDF sets are comparable (usually even higher)
to the individual estimates of PDF systematics.  We have checked that
this is the case for the similar process, namely for pp → e^+
ν_eμ^- ν̅_μ bb̅ j +X production
<cit.>. In this paper, we take the PDF uncertainties
to be the difference between our default PDF set (CT14) and the other
two PDF sets considered (MMHT14 and NNPDF3.0).




Our findings for  
MMHT14 and NNPDF3.0 PDF  can be summarised as follows



    σ^ NLO_pp→ e^+ν_eμ^-ν̅_μ bb̅γ
    ( MMHT14, μ_0=m_t/2)    = 7.49   fb ,
    
    σ^ NLO_pp→ e^+ν_eμ^-ν̅_μ bb̅γ 
    ( NNPDF3.0, μ_0=m_t/2)   =  7.72   fb .



The PDF uncertainties for the process under scrutiny are, therefore, given by
+0.05   fb (1%) for the MMHT14 PDF set and +0.28
  fb (4%) for NNPDF3.0. Our result for the integrated
cross section at NLO in QCD with the CT14 PDF set and for
μ_0=m_t/2 is given by


    σ^ NLO_pp→ e^+ν_eμ^-ν̅_μ bb̅γ 
    ( CT14, μ_0=m_t/2)=
    7.44 ^+0.07  (   1%)_-1.03  (14%)  [ scales]  ^+0.05   (1%)_+0.28   (4%)  [ PDF] fb .


Taken in a very conservative way, the PDF uncertainties are of the
order of 4% (to be compared to the theoretical uncertainties of
14% from the scale dependance). After symmetrisation they are
reduced down to 2% (to be compared to 7%). Overall, the PDF
uncertainties are well below the theoretical uncertainties due to the
scale dependence. The latter remain the dominant source of the
theoretical systematics.





 §.§ Differential cross sections

















While the size of higher order corrections to the total cross section
is certainly interesting, it is crucial to study the corrections to
differential distributions. In Fig. <ref> we present
representative differential distributions, that are relevant for BSM
searches <cit.>. We display p_T of the
hard photon and Δ R_b_2, γ between the hard photon and
the softer b-jet. The upper panels show the distributions themselves
and their scale dependence. The lower panels reveal the differential
K-factor with its error and the relative scale uncertainties
of the LO cross section.  To be more precise we plot K^
NLO(μ)=(dσ^ NLO(μ)/dX)/(dσ^ LO(μ_0)/dX)
and K^ LO(μ)=(dσ^ LO(μ)/dX)/(dσ^
LO(μ_0)/dX) where μ_0=m_t/2 is the central value of the scale
and X denotes the observable that is scrutinised.  Higher order
corrections have strongly altered the shape of Δ R_b_2,
γ where corrections range from -29% to +122%, causing
distortions of up to 150%. Similar effects have been noticed for
other observables, most notably for other angular observables 
shown in Fig. <ref>. Among
others the most affected by higher order corrections are the
separation in the rapidity-azimuthal angle plane between the hard
photon and the hardest b-jet, Δ R_b_1, γ (NLO
corrections from -24% to +93%), as well as the separation
between the hard photon and the hardest or softer charged lepton,
Δ R_ℓ_1, γ (NLO corrections ranging from -25% to +91%)
and Δ R_ℓ_2, γ (NLO corrections ranging from -16% to
+132%).  In each case the large differential K-factor for Δ
R ≳ 4 is associated with photon emission from
initial state quark from the qg+gq partonic subprocess, where q
stands for quark and antiquark. Such a contribution appears only
starting at NLO and adds significantly at large Δ R. Let
us mention here, that the qg+gq channel contribution to σ^
NLO_pp→ e^+ν_e μ^-ν̅_μ bb̅γ is estimated
at the level of 29%.  Moreover, due to the leading order like
nature of the contribution also the scale dependence in this region is
enlarged. Let us additionally note here, that emission of the
photon from the charged lepton leads to collinear enhancement at small
values of the separation between the photon and the softer charged
lepton, Δ R_ℓ_2,γ as can be clearly observed in
Fig. <ref>. Moreover, in the case of the
separation between the photon and the softer b-jet,
R_b_2,γ, depicted in Fig. <ref>,
 events are produced over a wide range of Δ R_b_2,γ
 values rather than in the back-to-back configuration. This confirms
 the findings of Ref. <cit.> that photon radiation off
 top quark decay products yields a significant contribution to the
 cross-section.

 In case of p_T,γ the differential K-factor is rather
constant  only in the plotted range from -8% to -18%.
Thus, p_T of the photon is more stable against higher order
corrections and hence better suited for BSM searches. Nevertheless,
both observables require higher order calculations to be properly
described. In view of ongoing indirect searches for BSM physics, where
the emphasis is looking for small deviations from the most accurate
SM predictions, such state of the art results are indispensable.

Finally, in Fig. <ref> we
present dimensionful observables. Specifically, we display
the transverse momentum distributions of the hardest and the softer
b-jet and charged lepton. In all cases negative and large higher
order QCD corrections can be detected. In the high p_T regions they
amount to -38%, -53%, -43% and -76% for p_T,  b_1,
p_T,  b_2, p_T, ℓ_1 and p_T, ℓ_2
respectively. Moreover, the NLO error bands do not fit within the LO
ones as one would expect from a well-behaved perturbative expansion.
Thus, the fixed scale choice does not ensure a stable shape when going
from LO to NLO for these observables. Through the implementation of a
dynamical scale, the large discrepancies between the shapes of these
distributions at NLO and LO should disappear. Thus, in the next step
we shall examine NLO results for μ_R=μ_F=μ_0=H_T/4 with the
goal of stabilising differential K-factor, i.e. decreasing 
NLO QCD corrections in the tails, for p_T,  b_1, p_T,  b_2, 
p_T, ℓ_1 and p_T, ℓ_2 while keeping the behaviour 
of K almost unchanged for p_T, γ.





 §.§ Theoretical uncertainties for differential 
cross sections








At this point we would like to estimate theoretical uncertainties
inherent in our LO and NLO differential cross sections as obtained
with μ_0=m_t/2 and the CT14 PDF set. The scale uncertainties are
again estimated conservatively by scanning bin by bin values of the
lower and upper bounds and by choosing the maximal number. To get a
general idea about the size of theoretical errors we quote here only
this maximal value. In this way, for the transverse momentum
distribution of the hard photon we have obtained theoretical errors up
to ± 40% at LO and up to ± 22% at NLO. For dimensionful
observables these maximal values come from the high p_T regions. A
similar pattern can be seen for the angular separation between the
hard photon and b-jet or the charged lepton. Specifically, for
Δ R_b_2,γ we have ± 40% at LO to be compared with
± 33% at NLO and for Δ R_b_1,γ is ± 42% at LO
versus ± 28% at NLO. In case of the charged lepton the situation is
very similar as we have estimated the theoretical error at the level
of ± 42% (± 40%) at LO and ± 28% (± 27%) at NLO
for Δ R_ℓ_1,γ (Δ R_ℓ_2,γ). Thus, in
all above mentioned cases a reduction by a factor of 1.5-2 is
achieved by increasing the order in perturbative expansion. However, in
case of transverse momentum distributions of b-jets and leptons the
picture has changed and there is a large residual scale dependence in
these observables even at NLO. Actually, for the p_T,b_1
distribution the theoretical error is at the same level independently of
the perturbative order and amounts to ± 46%. For the p_T,ℓ_1
and p_T,b_2 at NLO the theoretical error is larger than at LO,
respectively ± 56% and ± 78%. Finally, for the last plotted
observable, i.e.  p_T,ℓ_2 huge uncertainties of the order of
± 186% can be seen. This clearly tell us that μ_0=m_t/2 is
not equipped to properly describe tails of p_T distributions even at
NLO.  Many of these features can be improved by performing NLO
computation with the kinematic-dependent choices of the scales.

Lastly, we have examined PDF uncertainties for the differential cross
sections with the fixed scale choice. For all observables that we have
studied PDF uncertainties are negligible in comparison to the
theoretical uncertainties from the scale dependence. As an example we
show in Fig. <ref> NLO differential 
distributions as a function of the transverse momentum of the hard
photon and the azimuthal angle-rapidity distance between the hard
photon and the softest b-jet. The upper panels present the NLO
predictions for three different PDF sets at the central scale value
μ_R=μ_F=μ_0=m_t/2.  In addition to the CT14 PDF set, we employ
the MMHT14 and NNPDF3.0 PDF sets.  The lower panels of
Fig. <ref> give the ratio of the MMHT14 (NNPDF3.0) PDF set
to CT14.

To summarise this part, for pp → e^+ ν_e μ^- ν̅_μ
bb̅γ +X production at the LHC Run II with √(s)=13 TeV
with our selection of cuts and input parameters, the PDF uncertainties
are insignificant both at the level of total and differential cross
sections once contrasted with theoretical errors from the scale
dependence.  Let us note at this point, however, that
additional theoretical effects should be examined for the process at
hand.  These include among others NLO electroweak effects, the size
of which has to be estimated and compared to the size of NLO QCD effects.
Moreover, dedicated analyses of complete NLO QCD off-shell effects of
the top quark at the differential level have to be carried out. We
leave both aspects for  future studies. Thus, from now on we shall
concentrate only on theoretical uncertainties from the scale
dependence.





§ RESULTS FOR THE LHC RUN II ENERGY OF 13 TEV 
FOR  THE DYNAMICAL   SCALE CHOICE










 §.§ Integrated cross section and its scale 
dependence






For the kinematic-dependent scale μ_R=μ_F=μ_0=H_T/4 our
results can be summarised as follows 



    σ^ LO_pp→ e^+ν_eμ^-ν̅_μ bb̅γ
    ( CT14, μ_0=H_T/4)    =
    7.32 ^+2.44  (33%)_-1.71  (23%) fb , 
    σ^ NLO_pp→ e^+ν_eμ^-ν̅_μ bb̅γ 
    ( CT14, μ_0=H_T/4)   =
    7.50 ^+0.10  (1%)_-0.46  (6%) fb .



As expected they are in agreement with results provided at LO and at
NLO for μ_0=m_t/2. Precisely, within quoted theoretical errors
they agree at the level of 0.2σ at LO and 0.05σ at NLO.
This time, however, the full pp cross section receives positive and
small NLO corrections of 2.5%. The theoretical uncertainties
resulting from scale variations are 33% at LO and 6% at NLO. A
reduction of the theoretical error by a factor of 5.5 is observed
for μ_0=H_T/4. After symmetrisation of theoretical errors the
scale uncertainty at LO is estimated to be instead of the order of
28% and at NLO is reduced down to 4%. Therefore, by going from
LO to NLO we have reduced theoretical error by a factor of 7. The
graphical display of scale dependence is shown in
Fig. <ref>. The new scale choice indeed
captures parts of unknown higher order corrections. After all not only
the size of NLO corrections is diminished but also the theoretical
error is smaller when comparing to the results with the fixed scale
choice.  Scale dependence of the LO cross section with the individual
contributions of the partonic subprocesses and scale dependence of the
NLO order cross section decomposed into the contribution of the
virtual corrections plus LO and the real radiation part are
additionally given in Fig. <ref>.  Moreover,
the variation of μ_R (μ_F) with fixed μ_F (μ_R) is
presented in Fig. <ref> as well. Here
qualitatively our findings remain the same as for the fixed scale
choice.











 §.§ Differential cross sections

















We turn now our attention to differential cross sections
for μ_0=H_T/4. We have examined the same set of observables as in
the case of μ_0=m_t/2.  Our goal being to find flatter results for
differential K factors for dimensionful observables without
introducing major changes in the differential K factor of
p_T, γ. In the case of p_T, γ already for
μ_0=m_t/2 quite stable (negative) corrections have been
observed. Specifically, shape distortions up to only 10% have been
detected. 

We start with differential distribution for the transverse momentum of
the hard photon that is displayed in
Fig. <ref>. For the dynamical scale choice of
μ_0=H_T/4 positive corrections up to 13% are obtained. We can
also notice that the NLO error band as calculated through the scale
variation is within the LO error band as it should be for a
well behaved observable that is described by the perturbative
expansion in α_s. For the dimensionless observable Δ
R_b_2, γ, that is also shown in
Fig. <ref>, the size of NLO corrections has been
moderately reduced. The higher order corrections range now from
-12% up to +116%. Thus, the shape distortion up to 128% has
been obtained for this observable, which should be compared with
150% for the fixed scale choice. Other dimensionless observable are
presented in Fig. <ref>.  For the differential
distribution as a function of the separation in the rapidity-azimuthal
angle plane between the hard photon and the b-jet or lepton,
i.e. Δ R_b_1, γ, Δ R_ℓ_1, γ and Δ
R_ℓ_2, γ we have acquired NLO QCD corrections in the
following range {-11% , +90%},
{-10%,+90%} and {-5%,+130%}
respectively. In each case shape distortions have been decreased by
about 15%-20%.

Finally, we have reexamined dimensionful observables. Specifically,
transverse momentum distributions of the hardest and the softer lepton
as well as transverse momentum distributions of the hardest and the
softer b-jet. They are given in
Fig. <ref>. Higher order corrections in high
p_T regions have been substantially reduced. For the transverse
momentum distribution of the hardest b-jet we have attained +19%
instead of -38% and for the softer b-jet -16% to be compared
with -53% for the fixed scale choice. The same pattern can be
noticed for the p_T differential cross section of the hardest (the
softer) lepton. In details, for the hardest one we have obtained
+8% as a substitute to -43% whereas for the softer charged
lepton -30% rather than -76%.

To summarise this part, the validity of the proposed dynamical scale
μ_0=H_T/4, that is blind to the underlining top quark resonance
history, is confirmed. The size of NLO QCD corrections to all
presented observables has been reduced. Moreover, this judicious
choice of the scale has allowed us to obtain nearly constant K-factors in all dimensionful distributions that we have studied.






 §.§ Theoretical uncertainties for differential 
cross sections




As a final step we have examined theoretical uncertainties for
differential cross sections for the dynamical scale choice
μ_0=H_T/4.  The μ_0=m_t/2 scale choice has proved to be
inadequate for the modelling of various differential distributions and
more importantly for the estimation of their theoretical errors in the
high p_T regions. The latter phase space regions are simply not very
sensitive to the threshold contributions for the tt̅γ
production that are well described by the fixed scale choice. For each
considered observable we have observed reduced theoretical errors as
compared to the μ_0=m_t/2 scale choice. The effect is more
pronounce in the case of dimensionful observables in the high p_T
regions. Thus, for example we can see from
Fig. <ref> where the differential cross section
as a function of the transverse momentum of the hard photon is
plotted, that theoretical error is now up to  ± 8% at
NLO (up to ± 36% at LO) as compared to ± 22% at NLO (±
40% at LO) for μ_0=m_t/2. When considering p_T distribution of
the hardest and the softer b-jet, depicted in
Fig. <ref>, the theoretical error at NLO is
reduced from ± 47% and ± 78% down  to ± 10% and ±
 18% respectively. The most dramatic effect can be seen in the case
 of p_T distribution of the hardest and the softer lepton, also
 given in Fig. <ref>. In that case instead of
 theoretical errors up to ± 56% and ± 186% we have received 
 theoretical errors up to only ± 7% and ± 31% respectively.
  
To recapitulate this part, the dynamical scale μ_0=H_T/4, which
has been considered in our analysis, has proven to be very effective
in stabilising the perturbative convergence in the phase space regions
far away from the 2m_t threshold and in providing small theoretical
uncertainties as estimated by the scale variation. For all considered
observables the latter are below 10%-30%.





§ SUMMARY AND OUTLOOK




 
We have presented the first complete higher order predictions for the
pp→ tt̅γ process in the di-lepton channel for the LHC
run II energy of √(s)=13 TeV. With our inclusive cuts and for
μ_R=μ_F=μ_0=m_t/2, NLO predictions reduced the unphysical
scale dependence by a factor of 2.5 and lowered the total rate by
about 10% compared to LO predictions. The theoretical
uncertainty of the NLO cross  section as obtained from the scale
dependence has been estimated to be 14%. By comparison the PDF
uncertainties are negligible at the level of 4% only.  On the other hand, for
μ_R=μ_F=μ_0=H_T/4 the full pp cross section has received
positive and small NLO QCD corrections of 2.5%. Additionally, the
inclusion of higher order effects has reduced the theoretical error by
a factor of 5.5. Specifically, the theoretical
uncertainties due to scale dependence are now at the 6% level only,
however,  they are still larger than the PDF uncertainties. 

Even though NLO QCD corrections to the total cross section vary from
moderate to small depending on the scale choice their impact on
differential distributions is much larger.  Independently of the scale
choice for some dimensionless observables shape distortions of more
than 100% have been observed. The prominent example comprises the
differential cross section as a function of the separation in the
rapidity-azimuthal angle plane between the hard photon and the softer
b-jet, Δ R_b_2,γ. In the case of this observable,
which is relevant for new physics searches, shape distortions up to
150% (128%) have been obtained for μ_0=m_t/2
(μ_0=H_T/4). For the dimensionful observables presented in this
paper, however, the dynamical scale choice has helped to obtain almost
flat differential K-factors as well as to stabilise the high
p_T regions, which are very poorly described by NLO results with the
fixed scale choice.  Also in the case of differential
observables the PDF uncertainties have been examined. Similarly to the
total cross section case their size is negligible when comparing to scale
uncertainties. We repeat at this point  that additional
theoretical effects should be investigated. Among others the
size of NLO electroweak effects has to be calculated for the pp →
e^+ν_e μ^- ν̅_μ bb̅γ cross section and for
various differential cross sections. We plan to include such effects
in a future publication.

In addition, the size of
the top quark off-shell effects for the total cross section has been
estimated to be ≲ 2.5%. Their influence on differential
distributions and extraction of the SM parameters, however, might be
much stronger, as has already been shown in case of tt̅ and
tt̅j production <cit.>. Again, we leave such studies for the
future.

Our theoretical predictions are stored in the form of the Ntuples
files  and are available upon request. Specifically, they
are stored in the form of modified Les Houches event files and ROOT
files, that might be directly employed in experimental studies at the
LHC. They can be used for example to change kinematical cuts or to
define new observables. The latter can be obtained without need of any
additional rerunning of the code. Moreover, any change in the
renormalisation or factorisation scale choice or in the PDF set can be
accommodated by simple reweighting of these files. Thus, they can be
employed to study broad phenomenological aspects of top quark physics
at the LHC.





§ ACKNOWLEDGEMENT




The work of M.W. and T.W.  was supported by
the DFG Research Grant "Top-Quarks under the LHCs Magnifying
Glass: From Process Modelling to Parameter Extraction”. Furthermore,
the work of H.B.H. was supported by a Rutherford Grant
ST/M004104/1. The research of G.B. was supported by grant K 125105 of
the National Research, Development and Innovation Office in
Hungary. Simulations were performed with computing resources granted
by RWTH Aachen University under project rwth0211.




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