Off-shell single-top-quark production in the Standard Model Effective Field Theory
Tobias Neumann Zack Sullivan


Optimising the Input Image to Improve Visual Relationship Detection
    
		Noel Mizzi

		Department of Computer Science 

		University of Malta, Malta 

		Adrian Muscat

		Department of Computer Science 

		University of Malta, Malta
	
    
=======================================================================================================================================================================

firstpage


We present a fully differential and spin-dependent t-channel
single-top-quark calculation at next-to-leading order () in  including off-shell effects
by using the complex mass scheme in the Standard Model () and in
the Standard Model Effective Field Theory ().  We include all relevant
 operators at 1/Λ^2 that contribute at 
in  for a fully consistent comparison to the  at .
In addition, we include chirality flipping operators that do not
interfere with the  amplitude and contribute only at
1/Λ^4 with a massless b-quark. 
Such higher order effects are usually captured by considering anomalous
right-handed Wtb and left-handed Wtb tensor couplings. 
Despite their formal suppression in
the , they describe an important class of models for new
physics.  Our calculation and analysis framework is publicly available
in .







§ INTRODUCTION


Large statistics data samples from experiments at the  Large
Hadron Collider () provide the opportunity to extract precision
information about the Standard Model (), and to look for small
deviations due to new physics that enters at energy scales beyond
direct experimental reach.  t-channel single-top-quark production
provides a unique window onto this physics, and has been well-measured
by both the  
<cit.> and  
<cit.> 
Collaborations at the √(S)=7,8 and 13 runs.  While
the Collaborations refine these measurements to extract limits on
phenomenologically motivated physics beyond the Standard Model ()
<cit.>, the theoretical models they use for
comparison are not generally as precise as the data they are fitting.
This paper describes next-to-leading order ()  calculations
of t-channel single-top-quark production including off-shell effects
that improve both the  and Standard Model Effective Field Theory
() predictions of fully differential and spin-dependent
observables.

t-channel single-top-quark production probes many aspects of the .
Measurement of the cross section provides direct access to the square
of the  matrix element
V_tb <cit.>.  The V-A nature of the
production and decay vertices are probed by spin
correlations <cit.>.  Kinematic
distributions, such as the lineshape of the b–lepton invariant mass
m_bl from the top quark decay products, allow for extraction of
the top quark mass at the  <cit.>.

This process is also a stringent test on the consistency of parton
distribution function () fits at different orders, and it
directly tests the analytic framework of improved perturbation theory.
Resummation of large logarithms of the top quark mass to the bottom
quark mass leads directly to the introduction a b-quark
s <cit.>.  Through  in  the
process becomes one of double deep inelastic scattering ()
with two independent scales, where the leading order () process
is qb → q^' t scattering <cit.>.  Because DIS
data is used to extract the s, when  scale choices are
made, the inclusive t-channel cross sections computed at different
perturbative orders should be approximately the same.  A primary
motivation for improving the  calculation is that this analytic
constraint is strongly violated by recent 
sets <cit.>.  Once this issue is resolved, this
process could provide insight into  transverse momentum
dependence <cit.>.

While improving our understanding of  physics, precision
calculations of single-top-quark production and decay establish a
baseline for controlling the backgrounds to  physics.  The final
state of the t-channel process is W + b + light jets, where the W
can decay to a lepton plus missing energy, and hence is a background
to most new physics models. Deviations in inclusive cross sections or
kinematic distributions are expected in a large class of 
physics <cit.>.  The spin correlations are especially
sensitive to new physics <cit.>, and a number of
observables have been developed in
refs. <cit.>
that are separately sensitive to new physics contributions in the
production and decay vertices of the top-quark, respectively.

In the rest of this section we briefly review the state of 
t-channel single-top-quark calculations, explain our focus on the
 as an extension of the , and summarize what we add to the
 calculations.  In <ref> we describe our setup and the
calculation. We specify the list of  operators that we use, as
well as our conventions and normalizations. Furthermore, we describe
the steps that we perform to compute and simplify the amplitudes and
provide a list of checks that we have performed. A primary goal of this
study is to allow for a direct improvement of experimental analyses,
and we describe our publicly available implementation in  and
how to use it. In <ref> we study off-shell and
W-boson/neutrino reconstruction effects in the . We define
angular observables in the top-quark rest frame that are sensitive
to  contributions. We then study the impact of off-shell
effects to these distributions, and the effect of higher order
contributions from  and stability of the .




  
Towards precise t-channel single-top-quark predictions.

The precision of single-top-quark calculations has largely coincided
with the attempts to discover, and later precisely measure, the
t-channel cross section.  Early results focused on the
inclusive  cross section with stable on-shell top quarks
<cit.>.  Once experimental
backgrounds were better understood, differential calculations were
performed <cit.> with a
stable top quark, and the results were used to improve 
kinematics in showering Monte Carlo programs.

The next step consisted of including the leptonic decay for an
on-shell top quark, preserving full spin correlations, and including
separate  corrections in production and
decay <cit.>.  The
on-shell approximation relies on the assumption that off-shell effects
are expected to be small of the order Γ_t / m_t inclusively,
where Γ_t is the top-quark decay width, and m_t is its
mass. This allows for a significant simplification of the analytical
expressions at the cost of little error for the inclusive observables
used in discovery.

Since the production of the top-quark proceeds through a b-quark,
one can distinguish between calculations that either assume an
intrinsic proton b-quark content (five-flavor scheme), or not
(four-flavor scheme).  In the latter case, and with a non-zero
b-quark mass, predictions were first calculated in the stable
top-quark approximation
at  <cit.>, and then with a decaying
on-shell top quark, retaining spin correlations
<cit.>. This is implemented in  <cit.>.
To this point, all calculations were performed in the on-shell
approximation.

Off-shell effects generally play a role when one considers
differential distributions.  A prime example is the top-quark
invariant mass distribution, where the region above the resonance is
severely underpopulated in the on-shell approximation. It only
receives a tiny contribution when  radiation before the
top-quark's decay is clustered with the final state b-jet. The
inclusion of off-shell effects was handled for example in an effective field theory
approach <cit.>,
which is valid only close to the resonance. The first gauge invariant
calculation valid also in the far off-shell region was performed
<cit.> in the complex mass scheme 
<cit.>, and this
is the approach we follow.

Other calculations include an attempt to improve on fixed order
matching by adding parton shower effects through implementations
in  for on-shell and off-shell single top
production <cit.> and in  for
on-shell production <cit.>. Analytical resummation has
also been performed on top of the on-shell approximated fixed order
result <cit.>.
Finally, in recent years results at  in  have been
published for stable on-shell and for decaying top-quarks
<cit.>, but these numerical results currently differ by the size of their 
correction terms.

The primary goal of this study is to provide a public implementation
in  of a fully differential spin-dependent prediction for
t-channel single-top-quark production and semi-leptonic decay at 
including the off-shell effects of initial-final state 
interference and of non-resonant interferences.  We demonstrate
in <ref> that, after cuts, off-shell effects produce
significant shifts in some key experimental observables.
Additionally, we identify kinematic regions and spin observables that
are highly sensitive to the cancellation of soft radiation in
production and decay.  We show that this sensitivity can be hidden by
the on-shell approximation, and thus such regions should be avoided in
analyses relying on fixed order predictions.




  
New physics in single-top-quark production and decay. 

Deviations from the  are frequently modeled using anomalous
couplings (see for example refs. 
<cit.>) because
they often map directly to experimental observables.  Most recent
studies of  physics in the single-top-quark sector
by  <cit.> and 
<cit.> use
this approach.  For a recent single-top-quark overview focusing on
measurements of anomalous contributions we refer the reader to
ref. <cit.>.  Without a  completion, however, it
can be difficult to systematically incorporate and renormalize higher
order perturbative corrections.  It can also be challenging to compare
limits obtained in one experimental data set with limits obtained from
other experiments or data sets since there is neither a systematic
power counting scheme, nor a definite basis of the modifying
structures <cit.> (see also the discussion in
ref. <cit.>).

In this paper we take a more systematic approach and parameterize
potentially small deviations in terms of an effective field theory
() that obeys well-established  symmetries.  A
classification of all relevant dimension six operators has been
developed in refs. <cit.>, and
goes under the name Standard Model Effective Field Theory (),
see also <cit.>.  Calculations in the 
and  have been performed in abundance at , and we can
only cite an excerpt of results, see for example
refs. <cit.>. Within
the  framework, limits obtained on operators can be compared
directly with limits obtained from B-meson decays
<cit.>, for example.

To consistently include  effects one needs to determine the
renormalization and anomalous dimension matrix of the 
operators. This framework has been fully developed in a series of
publications <cit.>. See
also ref. <cit.> for a working group report
regarding the importance of  corrections of
the . Studies including   effects for an on-shell
top quark have been performed over the years
<cit.>, and recently also for an off-shell top quark 
<cit.>, but with a limited set of operators.

The most complete treatment of the  in single-top-quark
calculations so far has been in ref. <cit.>.  The
authors consider a limited set of three operators contributing as
interference to the  at  in , but neglect the
operators that begin contributing at  in  and operators
that do not interfere with the .  Their calculation is also
performed in the complex mass scheme, and includes the effect of
parton showers.  They study different approximations for top-quark
decay in the MadGraph5 framework using MadSpin.  To estimate
uncertainties they consider additional effects at
𝒪(1/Λ^4) that come from squared 1/Λ^2
contributions and from double operator insertions.


We consider six operators that contribute at  in  at
1/Λ^2 at the amplitude level that are relevant for t-channel
production and decay.  In addition, 
we consider a limited set of two color singlet four-fermion operators
with the helicity structure (L̅L)(L̅L) and (R̅R)(R̅R)
that are relevant for matching to W^' analyses.
We ignore the corresponding color-octet operators, and
(L̅L)(R̅R), (L̅R)(L̅R), and
(L̅R)(R̅L) operators which each have color singlet and
octet contributions. We also ignore four-fermion operators involving the
neutrino and positron. In practice these later operators would only be
relevant for s-channel experimental measurements. 


Depending on whether the 
amplitudes interfere with the , one obtains effects of order
1/Λ^2 or 1/Λ^4 at the cross-section level,
respectively.  We do not include effects of 1/Λ^4 at the
amplitude level since these would require a classification of
dimension eight operators for a consistent renormalization
at . Our 1/Λ^4 effects at the cross-section level thus
come from squared amplitudes of order 1/Λ^2 and
constitute only partial effects in view of missing double insertions
and dimension eight operators.


The partial higher order effects we include firstly allow estimating
higher order effects for those operators that already enter at
1/Λ^2 as an interference with the . Secondly, they allow
for a quantification of those operators' effects that only enter
as squared contributions. These are usually neglected
in  studies, but would be present in the anomalous couplings
picture. They can be relevant as higher order effects to the
1/Λ^2 contributions, or under certain model assumptions on the
operator content.

In our study we consider a set of eight operators and all of them are
treated fully consistently at  in .  Four of them
contribute at 1/Λ^2 at the cross-section level, and four of
them only enter at 1/Λ^4. Two of the newly considered
operators here only begin to enter at  through gluon radiation
and are required for a consistent  evaluation since they mix
under renormalization. Note that while the 1/Λ^4 contributions
are partial in the  expansion, they are still computed
consistently at  in . All contributions are implemented
including off-shell top quark effects in the complex mass scheme with
a massless b-quark in the five-flavor scheme.  We do not include
all-order effects of parton shower or resummation.  


In <ref> we describe the full set of operators, their
relationship to anomalous couplings studies, and technical details of
our calculation.  We show in <ref> that some commonly
recommended spin-correlation observables
<cit.>
that are sensitive to these operators are relatively stable to
off-shell effects and QCD radiation, while others are highly sensitive
to soft radiation effects that are not apparent in the on-shell
calculations.









§ SETUP AND CALCULATION
 

The calculations in this study are performed in the  and in the  framework.
The  is constructed by building higher dimensional operators that respect  symmetries
out of  fields. It systematically extends
the  Lagrangian as a power series in 1/Λ, where
Λ is the scale of new physics where the  description
breaks down.

    ℒ_ = ∑_i C_i/Λ^2𝒬_i + H.c. ,

where 𝒬_i denote dimension six operators, we add Hermitian
conjugates (H.c.) for non-Hermitian operators with complex Wilson
coefficients C_i, and we add Hermitian operators (without H.c.) with
real Wilson coefficients C_i.


The  picture we consider in this paper contrasts with the
phenomenological approach of anomalous couplings, which modify the
Wtb vertex as follows:

    -g_W/√(2)b̅γ^μ ( V_L P_L + V_R P_R) t W^-_μ - g_W/√(2)b̅iσ^μνq_ν/m_W (g_L P_L + g_R P_R) t W^-_μ + H.c. ,

where in the  V_L = V_tb^*, V_R=g_L=g_R=0, and the
W-boson momentum q is chosen to be incoming.  The pictures are connected at
tree level, where the anomalous couplings vertices are generated
by  operators and can be directly mapped to
them <cit.>.  The relations between
the anomalous couplings and our operator Wilson coefficients (defined
further below) are

    δ V_L    = m_t^2/Λ^2 , where  V_L = 1 + δ V_L ,
     V_R    = ^* m_t^2/Λ^2 , 
     g_L    = -4m_W m_t/Λ^2· , 
     g_R    = -4 m_W m_t/Λ^2·^* ,

where m_W is the W-boson mass, and m_W = 1/2 g_W v has
been used to derive this equivalence.  Note that the minus sign for
g_L and g_R is different from the literature. It depends on the choice of
treating the momentum q_ν in <ref> as incoming or outgoing and
the sign convention in the covariant derivative. We treat all momenta as
incoming in this study and adopt the sign convention from the 
literature (see further below).

When considering the leading 1/Λ^2 contributions from
dimension six operators to single top observables, only V_L and
g_R contribute as interference to the  amplitude. The
contributions generated by V_R and g_L flip a b-quark chirality
and do not interfere with the  amplitude for a massless
b-quark. As such, in the  they only enter at order
1/Λ^4 as squared contributions from non-
helicity amplitudes. Operator double insertions would contribute to
the amplitudes at the same order, where renormalization requires the
inclusion of dimension eight operators and a determination of their
anomalous dimension matrix. Dimension eight operators are beyond the
scope of this study, and so we do not include them or any double
insertions.

While formally suppressed in , the couplings V_R and g_L are
relevant for the study of new charged vector currents (W^'
bosons) or scalars (H^±
bosons) <cit.> and are
strongly constrained experimentally despite their suppression.  This
is possible due to the strong spin correlations in single-top-quark
production, and a large set of observables highly sensitive to 
deviations.

To maintain a direct coupling to experiment, we follow a
hybrid approach. In the first part we include all dimension
six  operators that are relevant at  in , and
enter at order 1/Λ^2 at the cross-section level. This allows
for a fully consistent evaluation of  effects and comparison
of Wilson coefficients extracted from different experiments when
higher order effects can be neglected.

In an enhanced mode we include all contributions of order
1/Λ^2 in the amplitudes from dimension six operators of
the . The additional operator contributions do not interfere
with the  and contribute only as the operator insertions squared,
or in interference with other  contributions. This leads to
1/Λ^4 effects at the cross-section level.  This enhancement
serves two purposes.  It allows for a   mapping to
anomalous coupling studies under the assumption that dimension eight
operators can be ignored.  And it allows for a systematic
determination of whether a given observable is sensitive to
higher-order corrections in the EFT.  If one wishes to obtain
consistent limits on Wilson coefficients and compare them with other
sources, one should not be sensitive to 1/Λ^2 and
1/Λ^4 contributions at the same time. For that purpose one can
run the analysis using only the 1/Λ^2 contributions and then
compare with results obtained when including the partial 1/Λ^4
contributions.




  
 operators.

As shown in <ref>, all operators 𝒬 come with a
Wilson coefficient C and a power of 1/Λ^2.  The operators
that contribute at 1/Λ^2 in   as interference to
the  amplitude are

    𝒬_φ q^(3,33)   =  1/2 y_t^2 (φ^† i ↔D_μ^Iφ) 
    	(Q̅_L 
    	γ^μτ^I Q_L) , 
    𝒬_uW^33   = y_t g_W (Q̅_L σ^μντ^I t) φ̃ W^I_μν , 
    𝒬_uG^33   = y_t g_s (Q̅_L σ^μν T^A t) φ̃ G^A_μν , 
       = 𝒬_qq^(3,1133) = ( q̅_L γ_μτ^I q_L)  (Q̅_L γ^μτ^I Q_L)  ,

where Q_L is the third generation left handed (2) doublet
(t_L,b_L) and q_L the first generation doublet (u_L,d_L).  Here
y_t=m_t√(2)/v is the real-valued top-quark Yukawa coupling,
g_W is the electroweak coupling, and g_s is the strong
coupling. The operators  and  are Hermitian, have real
Wilson coefficients, and no Hermitian conjugate is added to the sum
in <ref>. We also add the second generation operator
𝒬_qq^(3,2233) with the same real Wilson
coefficient.The operator  modifies the t̅tg Feynman
rule vertex and enters only at  in .

Our notation, sign conventions, and operator basis follows that of the 
literature in
refs. <cit.>.
This sign convention is different from the one used in other t-channel single-top-quark
 studies <cit.>.  Both conventions exist in FeynRules model files <cit.>.
We use a plus sign for minimal coupling in the covariant derivative
D_μ = ∂_μ + i g X_μ for gauge fields X and a
corresponding minus sign in the gauge field strength tensor
X_μν = ∂_μ X_μ + ∂_ν X_μ - g X_μ X_ν
consistent with SMEFTsim package <cit.>, whereas the
dim6top package <cit.> uses the opposite
convention.


At 1/Λ^4 there are additional dimension six operators

    𝒬_φ u d ^33   = y_t^2 (φ̃^† i D_μφ) ( t̅γ^μ b) , 
    𝒬_dW^33   = y_t g_W (Q̅_L σ^μντ^I b ) Φ W^I_μν , 
    𝒬_dG^33   = y_T g_s(Q̅_L σ^μν T^A b) Φ G^A_μν , 
       = 𝒬_ud^(1,1331) + 𝒬_ud^(1,3113) = (d̅γ_μ u) ( t̅γ^μ 
    	b) + (u̅γ_μ d) ( b̅γ^μ t)  ,

where the third operator  only contributes at 
in  and modifies the b̅bg vertex. For the Hermitian
operator  no Hermitian conjugate is added to the sum in
<ref>, and we also add the corresponding second generation operator 
with the same Wilson coefficient.




  
Operator mixing and running.

The operator pair , has nonzero anomalous dimension and
mixes according to

    μ_X d/dμ_X[ ;  ] = α_s/4π C_F [ 1 0; 2 2 ][ ;  ] ,

where μ_X is a renormalization scale that is independent of
the  renormalization scale. The pair , mixes
analogously under renormalization <cit.>.  

We renormalize the Wilson coefficients in the MS
scheme following
C_i^bare = Z_ij C_j(μ), where

    Z_ij = 1
    + α_s/4πγ_ij/2ϵ .
 
A factor of (4π)^ϵ / Γ(1-ϵ) is absorbed into
the definition of α_s and from here on we define
a_s ≡α_s/(4π). We set the renormalization point of the
Wilson coefficients to the same value as the  renormalization
point, as both contributions are probed at the same scale. The effect
of the running of C_i has been studied in
refs. <cit.> and can be used to evolve the
Wilson coefficients to the scale Λ or to some lower scale for
comparisons.

We renormalize the top-quark mass and wavefunction in the complex mass
scheme with complex mass on-shell conditions.  The 
renormalization constants receive additional contributions from
. 
We confirm the  mass and wavefunction
renormalization constants in ref. <cit.> by computing
the top-quark 1PI self energy with complex mass on-shell
renormalization conditions and find

    μ_0    = (1 + a_s δ_m) μ_t , 
    δ_m    = (μ^2/μ_t^2)^ϵ C_F ( -3/ϵ -4 + 
    	 m_t μ_t/Λ^2 C_F ( 12/ϵ + 4 )
    	 ) , 
    
    	Z_Ψ   = (1 + a_s δ Z_Ψ) , 
    δ Z_Ψ   = (μ^2/μ_t^2)^ϵ C_F ( -3/ϵ - 4 +
    	 m_t μ_t/Λ^2 C_F ( 6/ϵ + 2 ) +
    	 m_t μ_t/Λ^2 C_F  iγ_5 ( 6/ϵ + 2	)
    	  ) .

Here μ_t^2 = m_t^2 - i Γ_t m_t is the squared complex
top-quark mass. One power of m_t is part of the operator
normalization and is kept real. Note that in ref. <cit.>
the covariant derivative is defined with a different sign convention which
flips the sign of the Wilson coefficient  relative to our results.

Special care has to be taken for , which receives a
wavefunction renormalization contribution proportional to
γ_5. We obtain it by adding an additional counterterm to the
top-quark 1PI self energy proportional to γ_5 and demanding
that the propagator keeps its tree-level form throughout higher
orders. For the γ_5 contribution this is analogous to the 
on-shell condition of having residue i for the renormalized
propagator.

Note: Because we are examining an off-shell top quark, the
non-resonant diagrams include contributions from b̅ b A and
b̅ b Z vertices, where A is the photon field. The
corresponding  Feynman rules receive contributions from 
in the  and renormalize the gluon contributions from 
at . In this case also the operator 𝒬_dB^33 = y_t
g_B ( Q̅σ^μν b) Φ B_μν must be included to
renormalize . The anomalous dimension matrix follows the
operator renormalization group running

    μ_X d/dμ_X[ 𝒪_dG^33; 𝒪_dW^33; 𝒪_dB^33 ] = α_s/4π C_F [    1    0    0;    2    2    0; -2/3    0    2 ][ 𝒪_dG^33; 𝒪_dW^33; 𝒪_dB^33 ] .

We include this operator  only for the renormalization of
 and set its Wilson coefficient to zero afterwards, since it
only contributes to the non-resonant amplitudes.




 §.§ Technical implementation and checks



We consider the process u(p_1) + b(p_2) →ν(p_3) + e^+(p_4) +
b(p_5) + d(p_6) at  in  in the complex mass scheme
including off-shell interference effects and non-resonant
contributions required by gauge invariance. The complex mass scheme
introduces a squared complex top-quark mass μ_t^2 = m_t^2 - i
m_tΓ_t to the otherwise real valued top-quark mass in the
Lagrangian.[Note that the  operators above have been
normalized with the real-valued on-shell top quark Yukawa coupling
y_t=√(2) m_t /v.]  We also work in the five-flavor scheme and
set m_b=0. We compute all results at the amplitude level in the
spinor helicity formalism in the 't Hooft-Veltman scheme and avoid any
ambiguities related to γ_5 in dimensional regularization and
thus treat it in the naive dimensional regularization approach.

To obtain a gauge invariant result with an off-shell top quark,
requires the inclusion of both resonant and non-resonant
contributions. We include all such contributions and show a partial
sample of the diagrams
in <ref>. In addition to 
corrections, we allow for exactly one  operator insertion in
each diagram (at positions denoted by the crossed circles in <ref>).
We do not include the W+2 jets contributions that have a gluon
exchange at tree level. These diagrams are separately gauge invariant,
do not interfere with our contributions through , and are
considered a background that can be computed fully independently.








Apart from gluon radiation, in the  and throughout  only
two helicity amplitudes contribute. The first amplitude only
encompasses left-handed particles and a right-handed positron. Its
predominant contribution is from the resonant diagrams, which are
purely left-handed charged current mediated. It also receives
non-resonant contributions from Z-boson and photon exchanges.  The
second amplitude with flipped b-quark helicities comes purely from
non-resonant pieces. When  operators are added, one has
additional helicity amplitudes where either one of the b-quark
helicities is flipped (,,), or the helicities of
the light quark line are flipped together with the initial-state
b-quark helicity ().

The one-loop amplitudes we want to compute and simplify have large
tensor ranks that are further increased with the 
contributions.  Their evaluation with a standard framework like
FeynCalc <cit.> and
Passarino-Veltman reduction, for example, would be prohibitively
difficult due to the size of resulting and intermediate
expressions. We instead develop our own setup in
Mathematica <cit.> which performs the tensor reduction
with dimensional shift relations <cit.>. We implement
the dimensionally shifted integrals with increased propagator powers
in terms of standard one-loop master integrals by means of integration
by parts reduction performed with Kira <cit.>. The
scalar one-loop integrals are evaluated with 
2.0 <cit.>.  The few necessary Feynman
diagrams are generated with  <cit.> and
translated into initial  <cit.> code
with  <cit.> to output Mathematica
code. Feynman rules are generated using 
<cit.>
and checked by-hand, as well as compared with
refs. <cit.>. We make use of the
Mathematica packages  <cit.> and FeynCalc
<cit.> for debugging purposes. The 
simplification of large expressions is accelerated enormously with the
multivariate polynomial greatest common divisor implementation in
Fermat <cit.>, and it is used through an interface to
Mathematica <cit.>.

A major part of our calculation involves the reduction of spinor
(helicity) chains to a set of basis structures, as outlined in
ref. <cit.>, where the elements of the minimal set are
referred to as standard matrix elements. We extended these
reduction prescriptions to spinor chains of type
left-right ⟨⋯⟩ and
right-left [ ⋯ ], which appear in our 
contributions.  In principle, a reduction to one spinor master
structure is possible for the  helicity
amplitudes <cit.>; and a reduction to two master
structures can be performed for the amplitudes with one flipped
b-quark helicity (left-right and right-left types).  In practice, we
balance the number of structures we use against the number of terms
produced when we express the coefficients in terms of kinematic
invariants.

We begin by reducing all spinor combinations to a set of 59
structures. To achieve a full reduction to two master structures one
can directly write down a set of 58 linearly independent equations in terms
of nine Lorentz-invariants <cit.>. The use of Lorentz-invariants to
parameterize the kinematics enforces one additional Gram determinant
constraint <cit.>, which leaves 57 independent
equations.  This system of equations is highly complicated, but can be solved with the aid
of Kira <cit.> for example. The
resulting expressions are huge and do not directly to lead to
simplifications when inserted in the amplitudes.  Instead we follow
the suggestion of ref. <cit.>, and perform the reduction
using only equations that do not introduce additional denominator
structures.  For example, we express the  amplitude with
left-handed b-quarks and contributions from
, , , to the same helicity configuration in
terms of five spinor chains.  We reduce to a larger basis set for the
amplitudes with flipped quark helicities, since the equations would
either introduce additional complicated denominator structures, or do
not lead to simpler final results.
 
Both resonant and non-resonant contributions contain box diagrams that
naively lead to huge expressions. While expressing the results in
terms of scalar box integrals in six dimensions removes some of the
cancellations between the box and triangle diagrams, we do not use a
basis that lends itself to simple expressions for the loop amplitudes.
In order to deal with leftover spurious cancellations that eventually will
impair numerical stability we implement a simple rescaling scheme
stability control mechanism.

This stability control works as follows: For each phase space point we
evaluate our matrix elements twice — in double precision, and again
in double precision where all dimensionful quantities are rescaled by
a constant. Taking into account the dimensionality of the matrix
element, we then divide out the constant and check how many digits
agree to get an estimate for the numerical precision of the result. We
find that for precision runs the integration eventually focuses its
sampling on numerically unstable points and the integration becomes
unstable. If the stability check fails with less than four digits
precision left, we reevaluate them using the 
library <cit.>, which implements twice the precision of 
doubles (approximately 32 decimal digits) using two double precision
variables. This is faster than an evaluation with full  quad
precision and allows for Fortran compilers without such quad precision
support.

We directly compute all amplitudes for single-top-quark production. The
matrix elements for single-top-antiquark production are obtained by
crossing the single-top-quark matrix elements after applying a CP
transformation. After the CP transformation one crosses the light
quarks and reidentifies electron and neutrino particle labels to
obtain the single-top-antiquark result.  The CP transformation itself
introduces sign flips for the imaginary parts of the Wilson
coefficients, which we take into account, but leaves the other parts
unaffected since we assume V_tb is real.

In order to maintain a connection to  fits, we implement the use
of double deep inelastic scattering ()
scales <cit.> in .  We label the DIS momentum
transfer between the light-quark line and the b-quark line as
Q^2=-q^2. We then set the renormalization and factorization scales
for the light-quark line to μ^2=Q^2, and for the b-quark line
μ^2 = Q^2 + m_t^2. The implementation of  scales at 
is a non-trivial effort, since light line and heavy line corrections
have to be handled separately, and Catani-Seymour dipole contributions
have also to be accounted for with the right scales.

As part of our calculation we compute the decay width t→ W b
at  including the  operators, with an on-shell W-boson
and a massless b-quark. This is consistent with the complex mass
scheme at . We follow the steps of ref. <cit.>
to perform the necessary real emission phase space integrals. In
addition to the integrals listed in ref. <cit.> table
I, we find that three additional finite phase space integrals ⟨
y ⟩, ⟨ z ⟩ and ⟨ y z ⟩ are
necessary. We compute these using the Mathematica package
HypExp <cit.>.



  
Crosschecks.
We compute Feynman rules in the  and the  with
both  and by hand, ensuring proper relative signs with the
help of ref. <cit.>. Our  Feynman rules agree
with those in ref. <cit.>. The
relative signs between  and , and  and
 are fixed by the anomalous dimension matrix in
ref. <cit.>, and we agree with this through our
operator renormalization.

We compared our results analytically to the  tree level matrix
elements printed in ref. <cit.>, eq. (27). We also compared
our results with the   decay width results in eq. (120) of
ref. <cit.> as well as with the mass and wavefunction renormalization
constants. We fully agree with all those results when taking into account
the different sign convention used in the covariant derivative for
the  operator definition.

Using our anomalous couplings parametrization in <ref> with the
W-boson momentum defined to be incoming, we find
a global sign difference for  and  when comparing our results against the Protos
code <cit.>. This difference
is explained by the different momentum convention in Protos, where the momentum is outgoing.
We do agree on the relative signs between real and imaginary parts, so this difference
is indeed purely a global sign of  and .

Another check on our signs that eliminates a consistency problem with defining
momenta as incoming or outgoing in the Feynman rules is as follows.
The needed Feynman rules in off-shell single top production at 
from  and , all have a linear momentum
dependence, and one might argue that a sign difference can appear
be because we define particle momenta as incoming. However, because
we are considering off-shell effects, we must introduce the operator
, which adds an additional contribution from the momentum
independent W^+ W^- b̅ b vertex. The relative sign between this
contribution and the contributions from  and 
at  is set by the cancellation between the poles for 
renormalization and  subtraction. As we are overall consistent
with the signs in ref. <cit.>, this fixes the signs of
both  and . Since  and  have the same
structure and mixing, it also fixes their signs. This is the
first calculation to include the operators  and 
at  (in the off-shell process), so this check is new.

We explicitly checked  gauge invariance for our amplitudes —
analytically for pole terms and numerically for finite pieces. We note
that all amplitudes contributing to  and  are
separately gauge invariant. As part of our setup we reproduce
the   decay width, which is calculated in detail in
refs. <cit.>. We extensively compare
our off-shell  calculation to on-shell results for
compatibility. We check the proper cancellation of poles
between real and virtual corrections by checking independence of the
α parameter in the  Catani-Seymour dipole
implementation <cit.> to the
per-mille level. We also check that all infrared singular limits of
the real emission amplitude are approached correctly as predicted by
the Catani-Seymour dipole terms.





 §.§ Implementation in MCFM-8.3


Our results are implemented in the release version 8.3
of . Here we describe the user-visible modifications
of .  The code allows one to directly and easily reproduce the
plots in the following phenomenology section.  We implement b
tagging, top-quark and W reconstruction, as well as preconfigured
histograms for the most common observables in the  and spin
correlation studies. Our implementation provides an easy analysis
framework to perform further studies.

Dynamical double deep inelastic scattering scales can be
consistently used at  by setting  to `DDIS'
and = to 1d0. In this way the
momentum transfer along the W-boson Q^2 is used as the scale for
the light-quark-line corrections μ^2=Q^2, and μ^2=Q^2+m_t^2 for
the heavy-quark-line corrections. These scales are also consistently
used for the non-resonant contributions, with  corrections on the
ud-quark line, and separate  corrections on the bottom-quark
line.

The new block Single top SMEFT, nproc=164,169 in the input
file governs the inclusion of  operators and corresponding
orders.  The scale of new physics Λ can be separately set, and
has a default value of 1000.  The flag  enables the 1/Λ^4 contributions, where operators
, , and  can contribute for the first
time.  For the non-Hermitian operators we allow complex Wilson
coefficients.  We also have a flag to disable the pure 
contribution, leaving only contributions from  operators
either interfered with the  amplitudes or as squared
contributions at 1/Λ^4.  This can be used to directly and
quickly extract kinematical distributions and the magnitudes of
pure  contributions.

To allow for easier comparisons with previous anomalous couplings
results, and possibly estimate further higher order effects, we allow
for an anomalous couplings mode at  by enabling the corresponding
flag.  The relations between our operators and the anomalous couplings
are the same as in <ref>.

The analysis/plotting routine is contained in the file
`', where all observables
presented in this study are implemented, and the W-boson/neutrino
reconstruction is implemented and can be switched on or off. With this
one can directly reproduce all the phenomenological results in this
study.







§ PHENOMENOLOGY
 

In this section we examine kinematic distributions in off-shell single-top-quark
production and decay in both the  and the .
We begin by examining the effects of a
W-boson / neutrino reconstruction on the top-quark reconstruction. We
then study a set of angular observables in the top-quark rest frame
for the  before we focus our attention on 
contributions. We address the importance of unique  perturbative
corrections to the  contributions compared to using 
predictions with  K-factors. We also show the behavior of the
operators  and  that only enter at  and are
shown here for the first time for the full process.

Our set of cuts is given in <ref>. We require at least
one b-jet and one non-b-jet, but also allow for a third jet of
either kind. We refer to the leading non-b-jet as the spectator
jet. On top of these cuts, experimental anomalous couplings studies in
t-channel production select exactly two jets and have further cuts
on the rapidities of the b and spectator jet
<cit.>. We find that these 
additional cuts decrease the acceptance, but do not alter any of our conclusions here.



Our default choice of renormalization and factorization scales for the
off-shell results are the  scales, where for the light-quark
line the momentum transfer Q^2 to the b-quark line is used,
μ^2=Q^2, and for the b-quark line μ^2 = Q^2 + m_t^2. It has been shown that the  scales
lead to small perturbative corrections in inclusive
observables <cit.>.  We confirm that
the difference between using a fixed scale μ^2=m_t^2 and
the  scales is tiny for most  accurate observables, even
differentially. For  observables like the subleading b or
subleading light quark jet transverse momentum, which only enter
through the real emission, the scale choices lead to significant
differences.

We note that in on-shell results, we use the fixed scale m_t which
is used throughout the literature.  While a comparison between
off-shell results with  scales and on-shell results with m_t
as a scale is not on precisely equal footing, we consider the 
scales to be an improvement over the current calculations that do not
allow for this natural scale choice.  While the  paradigm
formally breaks down at  in , interference between the
light- and heavy-quark lines is expected to be
small <cit.>.  In addition, there
remains an analytic relationship to DIS in PDF
fits <cit.> that is directly constrained by the
consistency of the calculation of .


We use  parton distribution functions
(s) <cit.> at the corresponding perturbative
orders with a value of α_s^(m_Z)=0.118 at , and at
α_s^(m_Z)=0.13 at . The  matrix is chosen to be
diagonal and all other parameters have recent PDG values as
implemented in -8.3. The top-quark width is evaluated at the
corresponding perturbative order for t→ W b at the fixed scale
m_t and takes into account the  contributions at 
and .




 §.§ Off-shell and W-reconstruction effects in the Standard Model.



It is well known that in fixed order perturbation theory colored
resonances are sensitive to soft radiation <cit.>.  At
higher orders in perturbation theory soft and collinear parton
configurations between virtual corrections and real emission
corrections cancel in the singular limit.  However, configurations
approaching the soft/collinear limits are still present. In our case,
the top-quark is reconstructed from a reconstructed b-jet and
W-boson.  Depending on whether such radiation configurations get
clustered with the b-jet, and whether the radiation is
produced before the resonance or in its decay, one can observe a mass
enhancement or diminution. 

Assuming the top-quark is in the on-shell approximation, the
cancellation between virtual corrections and real emission is pinched
to the phase space with an on-shell reconstructed top-quark.  Having
an off-shell top quark makes the approach of the cancellation
explicit, with large positive and negative contributions around
≃ m_t ±Γ_t.  To obtain a smooth invariant mass
distribution near the peak one can either choose a larger binning with
radius ≃Γ_t or include all-order effects through parton
shower or resummation.

A further complication is that experimental analyses have to use a
reconstruction scheme for the leptonically decaying W-boson.  The
neutrino's transverse component can be derived by requiring the
event's transverse momentum to be vanishing.  On the other hand, the
longitudinal component is completely unknown and needs to be
reconstructed.  This reconstruction induces a smearing, not just of
the W-boson, but also of the reconstructed top quark.  We follow the
most recent  study on anomalous coupling
contributions <cit.> and reconstruct the neutrino's
four-vector by requiring that the invariant mass of the
neutrino-electron system in the top-quark decay equals the on-shell
W-boson mass.  With this condition one has either two real solutions
or two complex solutions for the neutrino's longitudinal component.
In the former case the solution closer to zero is taken.  For the
latter case of complex solutions, the neutrino's transverse component
is rescaled by 0.9 until a real and positive solution is found.

As a result of the neutrino reconstruction the reconstructed top-quark
invariant mass distribution gets smeared, and the aforementioned problem
is somewhat ameliorated, although not fully removed. We show this
in <ref>, where we compare the reconstructed top-quark
invariant mass distribution using the full neutrino four momentum to
using the reconstructed neutrino (smeared). For comparison the
on-shell distributions are also shown. The full off-shell result
receives large negative (not shown on the logarithmic scale) and
positive contributions close to the resonance. These are smeared by
using the reconstructed neutrino, but one can still see a noticeable
dip just below m_t. These off-shell effects in comparison to the
on-shell approximation are well known
<cit.>.



We note that the previous off-shell  calculation in the complex
mass scheme in ref. <cit.> with an on-shell
W-boson seems to obtain a smooth m_t distribution
with 1 bins without applying any smearing procedures.  This
behavior could be due to the way the W-boson is handled, leading to
a smoothing effect, although no indication of how to treat the W-boson
decay is given in their study.  It is also conceivable that since the
affected region is not well-defined in fixed order perturbation
theory, their use of a different subtraction procedure leads to a
differently distributed result there.

The off-shell effects we consider are important for experimental
observables.  In <ref> we show the positron transverse
momentum distribution, which is sensitive to soft  radiation
corrections only through the recoil of the W boson.  The off-shell
distribution is up to ∼ 15% harder at 300 compared to
the on-shell result, while corrections at low p_T are at the few
percent level.  The K-factor
(σ_NLO/σ_LO) in an on-shell
calculation and the K-factor in the off-shell calculation differ by
at most a few percent.  Similar
corrections can be seen in the leading b-jet transverse momentum
distribution in <ref>. There, off-shell effects at 
are about 5-10% in the tail. The ratio of the K-factors
for the off-shell and on-shell production is not flat, and shows
deviations with up to ∼ 10%.

Deviations in the distributions of these top-quark decay products will
have a significant effect on  measurements of the top-quark mass.
To avoid neutrino reconstruction uncertainties, it is common
practice to fit the top-quark mass based on the line shape of the
b-jet/lepton invariant mass
m_bl <cit.>.  In <ref>
we observe that off-shell effects lead to a large ∼ 20% shift in
the m_bl line shape close to the kinematic endpoint.  In a template
fit this effect is similar to a few GeV shift in the top-quark mass,
though the difference might be partially ameliorated by further final
state showering.










 §.§ Angular observables in the top-quark rest frame


Apart from common kinematical distributions like transverse momenta,
rapidities and invariant masses, t-channel single-top-quark analyses
are characterized by their use of angular distributions.  In
particular, the angle between the leading non-b jet and the lepton
from the top-quark decay is strongly correlated in the top-quark rest
frame <cit.>, and this is used in part to
identify the t-channel state <cit.>.  Any
modification (other than scaling) of the production or decay vertices
is expected to be observed as a modification of one or more angular
observables <cit.>.
While most of the angles are well-behaved under the inclusion of
off-shell effects from fixed order perturbation theory, we find that
one angle is highly unstable to soft radiation.



The first set of angles we examine is sensitive to operators that
modify the production of the top quark.  A coordinate system is
established by using the direction of the spectator quark p⃗_j
in the top-quark rest frame to define a corresponding axis ẑ.
A second axis ŷ is defined by the direction orthogonal to the
plane made by the spectator quark and the initial-state light-quark,
while the third axis x̂ is defined by requiring the coordinate
system to be right-handed <cit.>. The
direction of the initial-state quark is taken to be that of the proton
beam that shares the same sign of rapidity as that of the spectator
jet.

    ẑ = p⃗_j/|p⃗_j|,  ŷ = p⃗_j ×p⃗_q/|p⃗_j ×p⃗_q|,  x̂ = ŷ×ẑ .

We refer to angles of the lepton in the top-quark rest frame with
respect to these axes as , ,.

The second coordinate system we consider is sensitive to the decay
vertex, and starts with the direction q⃗ of the W-boson in
the top-quark rest frame as one axis q̂. The second axis
N̂ is orthogonal to the plane defined by q̂ and the
top-quark spin direction, as implemented by the spectator quark
direction s⃗_t in the top-quark rest frame. The last axis
T̂ is again defined by the right-handedness of the coordinate
system <cit.>:

    q̂ = q⃗/|q⃗|,  N̂ = s⃗_t ×q⃗/|s⃗_t ×q⃗|,  T̂ = q̂×N̂ .

From this basis we construct the angles between the lepton in the
W-boson rest frame with respect to these three axes.  We refer to
them as  for the angle with respect to the q̂ axis,
and ,  with respect to N̂ and T̂,
respectively.  In addition to the angles described here, one can find
angles between the N̂ and T̂ axes, and projections of
the lepton in the W boson rest frame onto the N̂-T̂
plane being used in analyses.  We have examined those projection
angles, but do not find interesting results regarding the 
contributions.




  
Discussion of angular observables.

First we note that neutrino reconstruction has a noticeable impact on
most of the observables in the top-quark rest frame.  This is expected
as the top-quark rest frame has a direct dependence on the neutrino
four-momentum.  Since the reconstruction procedure we use is based on
an experimental algorithm described in <ref>, we do not
comment further on angular differences due to other reconstruction
procedures.  Both on-shell and off-shell results that follow use a
reconstructed neutrino.

It was previously observed <cit.> that, after
cuts, going from  to  had little effect on the 
angular distributions used to measure t-channel single-top-quark
production.  When comparing off-shell distributions to on-shell
results, this similarity in most angular distributions is maintained.
For example, for  the K-factor ratio is ≃ 0.97 -
0.98 and flat within 1-2% of integration noise. In the ,
off-shell effects have rather uniform impact.  With one notable
exception, the largest deviations in shape we find are ∼ 10% in
 and  which are modestly relevant when
considering backgrounds to  operators. For completeness we
include corresponding plots with K-factor ratios
in <ref>
in <ref>.

One angular distribution that is currently used in experimental
analyses <cit.> demands further discussion.  The
angular distribution of  becomes unphysical at  for an
off-shell top quark in fixed order perturbation theory,
see <ref>.  This is because the top-quark invariant mass
distribution is not well-defined close to the resonance, where soft
radiation from production and decay cancels. It turns out
that the angle  is highly sensitive to this cancellation and
the cross section prediction becomes negative for ≳ 0,
which is compensated by an according increase for ≲ 0.



When using this observable at  or with the on-shell approximation
one seemingly does not have this issue, since no negative cross
section is observed. But the intrinsic sensitivity to the cancellation
of soft radiation in this angle is merely hidden.  Effects from
resummation and parton showers can change the distribution drastically.
As such we do not recommend to use this angle for precision studies in
the  and the . One must be especially careful in the latter
case, as  operators can modify gluon radiation. A
resummation or parton shower without taking into account the 
operators might lead to incorrect conclusions or limits.



 §.§  contributions


We now turn our attention to modifications of the angular
distributions by  operators in the off-shell  
calculation.  We distinguish how  operators modify the
distributions compared to   effects.  We also discuss the
importance of higher order corrections in the  for the
consistent extraction of limits on  Wilson coefficients.
Operators that begin to enter at 1/Λ^2 receive corrections at
order 1/Λ^4.  To obtain universal results one needs to make
sure that these higher order  corrections are small.

We limit ourselves to some representative examples here, as a detailed
study of all operators and their effects on all observables used in
single-top studies is beyond the scope of this paper. Additional
observables can easily and quickly be predicted with our published
code. We only show off-shell results here and present K-factors for
them, if applicable. We do not display  in our plots since this
operator is just a rescaling of the  results with an effective
modification of V_tb. We consider the case where just one Wilson
coefficient is non-zero and choose Wilson coefficients of one or i, with a
scale Λ = 1000. This choice is not very important
here, except for the consideration of higher order effects
1/Λ^4. Otherwise, our presentation of the pure (
subtracted) and normalized  contributions divides out the
Wilson coefficient.



  
 and 1/Λ^4 effects. 

We start with a discussion of higher order effects in  and in
the . It has already been pointed out in
ref. <cit.> that inclusively the K-factors
for the  contributions are different from the  K-factor
broadly by 10-25% depending on the operator combination.  We show
below that differentially this worsens somewhat.  For the
actual distributions used to constrain  physics,  
corrections to the  operators are essential.

We begin with one of the most important operators , that
leads to the richest phenomenological structure.
In <ref> we show the pure  contribution
(with the  contribution subtracted) at  and  in 
to get an impression of the perturbative corrections. We also include
effects of order 1/Λ^2 and additionally of order 1/Λ^4
to show the impact of higher order  corrections. In the top
panel the absolute corrections are shown and in the bottom panel we
show K≡/ as a representation of the perturbative
corrections.  For comparison we have also included the K-factor for
the  contribution itself in black.

The K-factor for the  contribution is not flat, and
unique   corrections are indeed important, especially for
the region of ≃ - 1. The impact of the 1/Λ^4
corrections on the K-factor is moderate in size, apart from the
first bins. This effect reduces for a smaller Wilson coefficient or a
larger scale Λ, but might have to be considered depending on
the analysis.  Differentially the corrections are an important effect
to consider, but the size of the contributions at ≃
-1 are small in comparison to the other regions. We show the same
operator contributions for  in <ref>.
Generally  corrections are sizable differentially and important
to correctly capture the shapes.







  
The imaginary part of .

While for most of the 1/Λ^2 contributions with Wilson
coefficients of one, the higher order  effects seem to be
moderate, this is not the case for the imaginary part of 's
Wilson coefficient, .  It it claimed in the
literature that the imaginary part of  does not contribute at
1/Λ^2 <cit.>.  We do not
find this to be true (with our set of cuts). For the stable-top approximation
one indeed does not have enough linearly independent four-vectors contracted with the Levi-Civita tensor,
so the contribution vanishes
<cit.>, but this no longer holds for a decaying top-quark.
The Wilson coefficient  enters the composite observable δ_- 
<cit.> as measured for example
by  <cit.>.

  While inclusively
the operator contribution for = 1 and
Λ = 1000 is tiny at 1/Λ^2, this is not
true differentially.  At  the 1/Λ^4 contributions
are small, but they are not flat, and the  contribution is
neither small nor flat.  We display these issues
in <ref>
and <ref>. <Ref> shows with the
 distribution that inclusively the contribution from
 is small at , but it is
large at   with a complex angular structure.  The 
distribution in <ref> shows that differentially in
 at ,  leads to large
contributions, where large negative contributions for < 0
cancel with positive contributions in > 0 inclusively, but
will strongly modify the  result. 
Interestingly the shift in
 here is similar in shape and size to the shift seen
in <ref> when going from an on-shell approximation to
the full off-shell result in the .







  
The sensitivity to 1/Λ^2 contributing  operators.

One way to present the sensitivity of the angular observables
to the various operators is to show normalized distributions where
the  contributions are divided out. As previous we compute
the  contribution σ^, but now we also normalize
these distributions to themselves and divide by the normalized 
distribution.  In this way one can see the shape difference of
the  contribution with respect to the . Since shape
differences in differential distributions result in the highest
discriminating power, operators with stronger shape differentials can
be constrained better.  In short, what we show in the following for a
differential distribution with respect to the variable x is

    ( 1/σ^·Δσ^/Δ x) 
    	 /
    	( 1/σ^·Δσ^/Δ x) ,

where σ^ is the  subtracted pure 
contribution, σ = σ^ + σ^.

In <ref> we show the resulting shapes for the real
and imaginary parts of the Wilson coefficients for the operators
 and  and the four quark operator . Note that
the operator  has no  contribution since it only enters
at .  Generally  and
 have the largest shape differentials. Arriving
at precise limits on Wilson coefficients is thus strongly influenced
by unique  contributions.  The corresponding distributions for
the angles ,  and  as well as
 are in included in <ref>
in <ref>
for completeness, and are maximally sensitive to the imaginary parts
of the operators.





  
The sensitivity to 1/Λ^4 contributing  operators.

We treat operators that enter at 1/Λ^4 as leading order in the
 in the sense that we do not consider possible mixing with
dimension eight operators that appear at  in .  We
include these operators to allow for a mapping onto the right-handed
anomalous coupling experimental limits, such as arise in W^' models
<cit.>.  These operators do not
interfere with the  amplitude.

We present self-normalized angular distributions that are divided by
the  angular distribution shape for the operators
, , and  in
<ref> 
in <ref>.  The cross sections only depend on the modulus
squared of the Wilson coefficients |C_i|^2.  Operators like 
that act like a right-handed Wtb coupling predictably create a
strong  dependence that can be distinguished from the
four-fermion operator  by comparing to .










§ CONCLUSIONS


In this study we present for the first time a full 2→ 4 off-shell
calculation of t-channel single-top-quark production at 
in , taking into account the decay of the W-boson at the
amplitude level.  We use the complex mass scheme to gauge-invariantly
include such off-shell effects in both the , and in the 
framework. We include all relevant dimension six operators that affect 
the Wtb vertex at  in .

We examine off-shell effects in the  and find significant
differences with respect to the on-shell approximation.  We also
consider effects due to the reconstruction of the W-boson and
neutrino coincident with the off-shell effects.  While sensitivity of
the reconstructed top-quark invariant mass distribution near its peak
due to soft radiation is expected for an off-shell top quark; we point
out for the first time how this carries through to certain angular
distributions used by experimental analyses.
In particular, we find that the  distribution used by some
analyses, which is constructed in the top-quark rest frame, becomes
unphysical when off-shell effects are included.  This problem is
hidden in the on-shell calculations.  Without a resummation of soft
radiation in the on-shell region, we cannot recommend the use of this
observable for precision studies.

Our results move beyond the common   picture and allow for
a fully consistent  evaluation at 1/Λ^2, as well as
partial corrections from 1/Λ^4 operators in order to compare
to the anomalous couplings picture.  We show that  
effects to the  contributions can be large in angular
distributions, and in general are not captured by a rescaling
with  K-factors.  In addition, the operators  and
 are included for the first time in the full process
at .  These operators only begin to enter at  in ,
and are important for consistent  limits on Wilson coefficients.

We present an extensive list of checks of our amplitudes, and we
successfully compare several  and  terms at 
and  with results from the literature. We are able to identify
the correct relative signs of all pieces of our calculation through
checks of gauge-invariance, and  and  finiteness.  Because
of the asymmetric nature of limits for positive and negative 
operators that arise in the  calculations, it is imperative for
experimental analyses to clearly state which sign convention is used
for both minimal coupling in the  and for the .

While we present the most complete fixed-order perturbative
calculation of the  operators, a dedicated study of parton
shower effects to observables used for  studies would be
useful.  Parton shower effects are included in
ref. <cit.>, but they do not discuss their impact on
the fixed order results.  Some observables, like the top-quark
invariant mass distribution clearly need to include all-order effects.
Our study is also performed with a massless bottom quark, but bottom-quark
mass effects could become interesting at future precision levels.

Our implementation is publicly available in -8.3 and includes
preconfigured plotting routines to reproduce all distributions in this
study.  It is our goal to allow for easy, yet precise and refined
determination of observables in the  and better constraints
on  operators.





  
Acknowledgments.

We thank Stefan Dittmaier for helpful comments on the reduction to
spinor chain master structures, John Campbell for invaluable
continuous discussion, Bogdan Dobrescu for useful discussion on model
interpretation of the  operators, Claudius Krause for helpful
discussion about the , and Mario Prausa for help on
interpreting Fermat output. We would also like to thank Cen Zhang
for helping to track down the sign discrepancy we stated in the preprint of
this study to a different sign convention used in parts of the literature.
Feynman diagrams were generated with TikZ-Feynman <cit.>. 


This work was supported by the U.S. Department of Energy under award
No. DE-SC0008347.  This document was prepared using the resources of
the Fermi National Accelerator Laboratory (Fermilab), a
U.S. Department of Energy, Office of Science, HEP User
Facility. Fermilab is managed by Fermi Research Alliance, LLC (FRA),
acting under Contract No. DE-AC02-07CH11359.








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