Single-top-quark production in the t -channel at NNLO
FFERMILAB-PUB-20-608-T, IIT-CAPP-20-05
Single-top-quark production in the t -channel at NNLO
John Campbell , Tobias Neumann , and Zack Sullivan Fermilab, PO Box 500, Batavia, Illinois 60510, USA Department of Physics, Illinois Institute of Technology, Chicago, Illinois 60616, USA Department of Physics, Brookhaven National Laboratory, Upton, New York 11973, USA
We present a calculation of t -channel single-top-quark production and decayin the five-flavor scheme at NNLO . Our results resolve a disagreement betweentwo previous calculations of this process that found a difference in the inclusivecross section at the level of the
NNLO coefficient itself. We compare in detail withthe previous calculations at the inclusive, differential and fiducial level including b -quark tagging at a fixed scale µ = m t . In addition, we advocate the useof double deep inelastic scattering ( DDIS ) scales ( µ = Q for the light-quarkline and µ = Q + m t for the heavy-quark line) that maximize perturbativestability and allow for robust scale uncertainties. All NNLO and
NLO ⊗ NLO contributions for production and decay are included in the on-shell and vertex-function approximation. We present fiducial and differential results for a varietyof observables used in Standard Model and Beyond Standard Model analyses, andfind an important difference between the
NLO and
NNLO predictions of exclusive t + n -jet cross sections. Overall we find that NNLO corrections are crucial for aprecise identification of the t -channel process. Contents
1. Introduction 22. Calculation 4
NLO ⊗ NLO interference contributions . . . . . . . . . . . . . . . . . . . . . . 12
3. Comparison with previous results 13
4. Results 20
5. Conclusions 28A. Additional fiducial distributions 30 a r X i v : . [ h e p - ph ] F e b . Introduction Top quarks play a special role in the Standard Model ( SM ). They stand out from the otherquarks by virtue of their mass, significantly larger than their peers, and by the fact thatthey decay before they are able to form bound states. The former property ensures that topquarks are primarily responsible for the production of Higgs bosons at hadron colliders, bymediating the loop-induced coupling gg → h , and positions them as potential portals to so-farundiscovered extensions of the SM . The latter enables both a clean theoretical descriptionof top-quark processes and, on the experimental side, their identification through decaysinto well-measured objects. Therefore the study of the production and decay of top quarksis a cornerstone of the current and future physics program of the Large Hadron Collider( LHC ).The primary mechanism for producing top quarks is through the strong interaction, resultingin the creation of a top quark-antiquark pair. However, a single top quark may also beproduced through a weak interaction involving a bottom quark. In fact, the t -channel singletop production processes, represented at leading order by, q + ( − ) b → q (cid:48) + ( − ) t , (1)and first observed at the Tevatron [1, 2], are responsible for approximately of all top-quarkproduction at the LHC . Despite this production mode proceeding through weak couplings, therate is large due to its t -channel nature and the fact that it is kinematically favored comparedto the pair-production process. The channel of course provides a useful probe of the topquark itself, with measurements of the top-quark mass [3] and polarization [4], detailed testsof the Standard Model at the differential level [5, 6], as well as constraints on anomalous W tb couplings [7–9]. In addition, it can also provide valuable information on the elements of theproduction mechanism: the bottom quark parton distribution function (
PDF ) and the
CKM matrix element, V tb . Indeed, measurements of V tb have been made at both the Tevatron [10]and the LHC [11–14].In order to turn experimental observations into precision measurements it is essential tohave theoretical calculations with small residual uncertainties. One of the largest sources oftheoretical uncertainty results from the truncation of the perturbative expansion at a fixedorder and is usually estimated by scale variation. Generically, this scale variation is expectedto decrease order-by-order, with percent-level uncertainties only expected when going tothe current gold-standard for precision measurements, next-to-next-to-leading order ( NNLO ).For top-quark processes another consideration that complicates the perturbative calculationstems from the fact that the top quark decays, t → W + b . Accounting for such a decay, atleading order, does not present a tremendous complication – in fact, using the spinor helicityapproach, expressions for relevant matrix elements are extremely compact [15]. However, oncestrong-coupling corrections are included one should, in principle, account for the effects ofvirtual radiation that connects strongly-interacting particles in the ‘production’ and ‘decay’elements of the process.The earliest results for t -channel single top production that included corrections to next-to-leading order ( NLO ) in the strong coupling were computed for a stable top quark, that is,for exactly the processes shown in eq. (1) [16–18]. These were soon extended beyond thecase of the inclusive cross section, to also describe fully-differential measurements of thisprocess [19–21]. Somewhat later the decay of the top quark was also included at the same2rder, with calculations performed in a factorized approach in which the top quark remainson its mass shell and radiation that connects production and decay is neglected [22–24].Although this is an inherent approximation, for sufficiently inclusive observables such off-shelleffects are small, of the order of Γ t /m t [25, 26]. For a precision prediction of the invariantmass of the top quark it is clearly essential to move away from this approximation and, ingeneral, more differentially such effects are of increasing importance. Calculations of off-shelleffects were first performed in an effective field theory approach, valid in the region of theresonance [27–30], and subsequently computed exactly [31–33].The first steps towards including the effects of NNLO corrections to this process were taken inrefs. [34, 35], in which the top-quark decay was computed at this order. The corresponding
NNLO corrections for the hadroproduction of a stable top quark in this channel were subse-quently computed [36] and, finally, the full corrections in both the production and decay stageswere included in the calculation presented in refs. [37, 38]. The results of these calculationsindicate that, at the inclusive level, the effect of the
NNLO corrections is small, leading toa change in the cross section of only a few percent. However, the effects of the correctionscan be larger in the fiducial volume [37, 38]. Importantly, a comparison between the twogroups – at the level of inclusive cross sections for a stable top quark – revealed that the twocalculations disagree at the 1% level [37], that is, at almost the same level as the effect ofthe
NNLO corrections themselves. Recently, the
NNLO calculation of refs. [37, 38] has beenused to study the top-quark mass determination [39] and differences between flavor schemes[40].The importance of this production mode demands that the existing theoretical calculationsbe scrutinized, cross-checked and any discrepancies understood. To this end, in this paper wepresent a full re-calculation of the
NNLO QCD corrections to this process, including all theeffects of radiation in top-quark production and decay at this order. Although the calculationnecessarily shares some elements and methodology with previous work, it has been performedfrom scratch from ingredients that have been, where possible, verified against independentcomputations. In order to ensure the maximum value of the calculation, it is performedusing the framework of the
MCFM package [41–43] and will be distributed publicly. The aimof this approach is to ensure that the latest possible theoretical information can always beincorporated in future experimental analyses, incorporating any demands on input parameterssuch as the top-quark mass and
PDF s. The outline of the paper is as follows. In section 2 we describe the setup of our calculation:outlining the general structure, discussing the role of a strict fixed-order expansion thatbecomes relevant when including the top-quark decay and introducing the distinctive choiceof factorization and renormalization scales for this process. We also describe our prescriptionfor b -quark tagging, which at NNLO needs special attention. We then present details of all thenecessary ingredients for the three independent
NNLO sub-calculations entering our results,as well as of the
NLO ⊗ NLO interference contributions, and describe the checks that wehave performed to validate our results. In section 3 we compare in-depth with the previousstable-top calculation and the on-shell calculation with decay. Last, in section 4 we discussfully inclusive cross sections, cross sections with fiducial cuts, and the relevant differentialdistributions used in experimental analyses. We furthermore consider
NNLO results for angularobservables in the top-quark rest frame, angles that are used for searches of physics beyond For a recent example where theoretical input was limited to the NLO level, justified by the lack of suchflexibility, see ref. [6]. H D
W tu db bν e + Figure 1.: Feynman diagram for t -channel production in the on-shell approximation with blobsdenoting vertex corrections on the light line (L), on the heavy line in production(H) and in decay (D).the Standard Model and which are sensitive to W tb -vertex modifications in the productionand decay stages.
2. Calculation
Our calculation of
NNLO QCD corrections to t -channel single-top-quark production and decayis performed in the five-flavor scheme, a → scattering process at Born level that is depictedschematically in fig. 1. In order to render the calculation tractable we perform the calculationin the on-shell approximation, neglecting contributions from radiation that connects theproduction and decay stages such as the diagram shown in fig. 2 (left). For a large classof observables this amounts to discarding off-shell effects that are of order Γ t /m t [25, 26].We furthermore neglect a class of two-loop box diagrams that connect the light-quark andheavy-quark lines, that are suppressed by a color factor of /N c , see fig. 2 (right). The
NNLO corrections therefore consist of
NNLO vertex corrections on the light-quark line and the heavy-quark line in the production part of the process, as well as
NNLO vertex corrections in thedecay of the top quark, as indicate in fig. 1. In the literature this is also known as the structurefunction approximation. Of course, in addition to second-order corrections to each vertex, at
NNLO we must also include one-loop times one-loop interference contributions.We calculate each of the three
NNLO vertex corrections in fig. 1 using slicing-based subtractionsderived from different factorization theorems. The interference contributions are calculatedin a mixed slicing-dipole subtraction scheme. While we use the same
NNLO factorizationtheorem to assemble the corrections on the heavy line in production and decay as in a previouscalculation [37, 38], we have performed a series of exhaustive analytical and numerical checksto ensure a correct cancellation of IR divergences between all amplitudes. Our implementationof each contribution is, of course, completely independent and all amplitudes have either beencalculated from scratch or obtained from different sources. With this we can guarantee a fullyindependent cross-check of previous results. The calculational details for the three NNLO vertex corrections and the
NLO ⊗ NLO interference are given in the next four subsections, butwe first discuss some general aspects of the calculation. While there has been progress on the reduction of these two-loop integrals to a basis of master integrals[44], the solution of the integrals themselves is still an open issue. tu db bν e + W tu db bν e + Figure 2.: Example Feynman diagrams for one-loop off-shell contributions (left) and color-suppressed interference contributions (right) in t -channel production. Fixed-order expansion.
Since the top quark can be treated as a quasi-stable particle to agood approximation, the differential cross section d σ ( α s ) can be factorized into a productionpart d σ production ( α s ) and a decay part dΓ t ( α s ) / Γ t ( α s ) . In the decay corrections the numeratoris the differential decay width and the denominator is its integrated expression, such that theintegration of the decay fully inclusively leads to no change in the cross section. Throughoutthis paper we assume that the top-quark decays 100% of the time to a W boson and a b quark.While it is possible to keep the full expression of the width in the denominator, it is customaryto expand the whole cross section d σ ( α s ) in α s to avoid higher-order contributions fromthe expansion of the width in the denominator and the factorized contributions of theindividual vertex corrections. Consequently, we expand the fully differential cross section d σ ( α s ) = d σ production ( α s ) ⊗ dΓ t ( α s ) / Γ t ( α s ) consistently in α s to obtain cross sections at LO ( α s ), NLO ( α s ) and NNLO ( α s ). The symbol ⊗ denotes that production and decay aretaken fully at the amplitude level including all spin correlations. Denoting with d σ ( j ) thecontribution of order α js in production and with dΓ ( k ) t the contribution of order α ks in decay,the fixed-order expansion is explicitly given by: d σ LO = 1Γ (0) t d σ (0) ⊗ dΓ (0) t , d σ δ NLO = 1Γ (0) t (cid:34) d σ (1) ⊗ dΓ (0) t + d σ (0) ⊗ (cid:32) dΓ (1) t − Γ (1) t Γ (0) t dΓ (0) t (cid:33) (cid:35) , d σ δ NNLO = 1Γ (0) t (cid:34) d σ (2) ⊗ dΓ (0) t + d σ (1) ⊗ (cid:32) dΓ (1) t − Γ (1) t Γ (0) t dΓ (0) t (cid:33) + d σ (0) ⊗ (cid:32) dΓ (2) t − Γ (2) t Γ (0) t dΓ (0) t − Γ (1) t Γ (0) t (cid:32) dΓ (1) t − Γ (1) t Γ (0) t dΓ (0) t (cid:33)(cid:33) (cid:35) , (2)where Γ ( l ) t are the integrated decay corrections of order α ks . By construction, the expressionsin parentheses vanish upon fully inclusive integration over the decay5 ouble deep inelastic scattering ( DDIS ) scales.
Our implementation allows for differentfactorization and renormalization scales in light-line corrections, heavy-line corrections inproduction, and corrections in decay. For our approximation in terms of vertex correctionsthe natural choice to minimize scale-dependent logarithms for fully inclusive results is totake Q as the central scale on the light-quark line, Q + m t on the heavy-quark line inproduction, and m t in the decay, where Q is the positive squared momentum of the W boson.We abbreviate this set of scale choices as DDIS (double deep inelastic scattering) scales.The motivation for considering
DDIS scales stems from the observation that single-top-quark production in our approximation is double deep inelastic scattering with a light andheavy quark. When
PDF s are extracted from
DIS measurements, one therefore expects ourperturbative results to be stable across orders with
DDIS scales. That is, strictly speaking,the cross sections at LO , NLO and
NNLO should agree within
PDF uncertainties. Therefore,at least in principle, using
DDIS scales can be exploited as a constraint for
PDF s. At
NNLO ,off-shell effects and color-suppressed contributions from two-loop box diagrams break thisproperty, see fig. 2, but such effects are estimated to be small, at least inclusively. b -quark tagging. Tagging a specific quark flavor in jets, say a b quark, raises the questionof infrared safety. Throughout NLO there are no issues, but at
NNLO a large-angle softgluon splitting g → b ¯ b can lead to b -quarks being clustered into different jets, violating flavorinfrared safety. Typical jet algorithms only ensure infrared safety at the event momentumlevel, and this situation breaks infrared safety at the jet flavor level. A general solution is toemploy a flavor-jet algorithm for infrared-safe predictions [45, 46], which ensures that such asoft splitting would recombine to a flavorless jet.Experimentally such a flavor-jet algorithm is not used in single-top-quark analyses and we donot adopt such an algorithm in this study. Although it could, in principle, be interesting toconsider its impact in the future, the difference is likely to be negligible due to the smallnessof the g → b ¯ b contributions, see below. For now we adopt the strategy used in ref. [38], whichalso enables a more transparent comparison with the results presented in that reference. Theprocedure is as follows: for the NNLO vertex corrections in production the g → b ¯ b splittingdoes not come with another b quark at the same vertex. The flavor in the pair can thereforebe ignored on the premise that it gives a tiny contribution. In the decay the situation withan additional b quark from the top-quark decay arises and the flavor is ignored for the b ¯ b pairwith the smaller invariant mass, which is most likely the pair originating from the g → b ¯ b splitting.The authors of ref. [38] check the infrared safety of this approach numerically by evaluatingthe infrared subtraction slicing-cutoff dependence. They find that when no flavors are ignoredthere is indeed a difference, but it is tiny. We do confirm these findings, and note that theminuscule size of this effect, unless one probes tiny slicing cutoffs to enhance it, confirmsthat the flavor contribution of the g → b ¯ b contribution is negligible. Therefore, using a fullflavor-jet algorithm does not seem necessary in practice.We furthermore adopt the following choice of the b -jet definition for the remainder of this paper.Namely, we define the b -number of a jet to be the sum of the b -numbers of its constituentpartons, where the b -number of a b and ¯ b quark are +1 and − , respectively. A jet withnon-zero b -number is termed a b -jet and all other jets are light jets.6 .1 Top-quark-production corrections on the light line Common ingredients.
Before detailing the individual
NNLO calculations, we first summarizesome ingredients shared by all parts. For the calculation of the
NNLO vertex corrections,the above-cut slicing contributions are obtained by performing
NLO calculations of partonicprocesses corresponding to Born configurations plus an additional parton. Importantly, thesehave to be numerically stable in the doubly-singular limits. The necessary tree-level andone-loop amplitudes for these calculations are assembled from existing amplitudes in
MCFM for one-loop corrections [22] and tree-level results in the four-flavor scheme [47]. We usedipole subtractions to combine them into
NLO calculations for massless [48] and massive [49]particles. The extension to limit the size of subtractions provides an additional check of acorrect implementation (“alpha independence”) and can improve numerical stability. Theseextensions have been worked out for both massless [50, 51] and massive [22, 52, 53] partons.Note that to retain a proper definition of the
DDIS scales, the dipoles in our
NLO calculationsmust be chosen with some care so that no initial-initial dipoles transform momenta on bothlines.
For the light-quark line, which represents
DIS -like jet production, we compute the
NNLO corrections using a factorization theorem in the 1-jettiness variable τ l for massless partons[54]. This approach has been first used in ref. [55] for a calculation of W +jet at NNLO andthe formalism including all necessary ingredients at
NLO has been presented in full generalityin ref. [56].At
NNLO we have to take into account contributions with one and two extra emissionsin addition to the parton already present at Born level. We denote with M the numberof (light-line) partonic emissions p µk , k = 1 , . . . , M and the initial state light-quark (beam)momentum as q b . Then our 1-jettiness variable is defined as τ l = 1 Q M (cid:88) k =1 min { n b p k , n j p k } (3)where n b is the normalized beam direction q µb and n j is a normalized jet direction. The scale Q is introduced to make τ l dimensionless, as discussed further below. Obtaining the jetdirection n j from an explicit minimization of τ l for one and two additional emissions has beendiscussed in refs. [54, 56]. For example for M = 2 the result is τ l = 1 Q min { min { E − p z , E − p z } , E + E − | (cid:126)p + (cid:126)p |} , (4)effectively partitioning the phase space into extra emission clustered with the beam, or togetherwith the other final state emission. Here p zk is the z -component of p µk , E k is the energy of p µk ,and (cid:126)p k is the spatial component of p µk . The case of M = 3 results in a similar partitioning,where now all extra emissions can either be together, or with the beam, or one with the beamand one with the other emission, in various combinations. For further details, see ref. [56] orour implementation in MCFM .For the plots presented in this study we choose Q = 1 GeV . An alternative choice is Q ∝ (cid:112) Q , where Q is the squared W boson momentum transfer. We find that at √ s = 13 TeV the choice of Q ∼ (cid:112) Q / has similar numerical stability as Q = 1 GeV andis subject to power corrections that are of the same magnitude, as may be anticipated from7 .1 Top-quark-production corrections on the light line the explicit minimization indicated in eq. (4). We do not explore this difference any furtherhere.Instead of the explicit minimization of τ l , in principle any infrared-safe Born projection canbe used to obtain the jet axis n j . For example one could determine it from the jet clusteringalgorithm, as has previously been done in MCFM . Any difference in the Born projectionresults in different power-suppressed terms in the factorization theorem, but does not modifyits singular structure [54]. With this phase-space partitioning, one can write down the -jettiness factorization theoremintegrated over τ l as a subtraction scheme as d σ a,b ( τ l < τ cut l ) = d σ Born f b ( x b , µ ) (cid:90) τ cut l d τ (cid:48) l (cid:90) d k j (cid:90) d k s (cid:90) d t a B a ( t a , x a , µ ) J ( k j , µ ) × S ( k s , µ ) ⊗ H ( µ ) δ (cid:18) τ (cid:48) l − t a E a − k j E j − k s (cid:19) , (5)where we denote color correlations between hard function H and soft function S with ⊗ .Parton flavors are labeled as a and b and the fully differential Born-level cross section d σ Born has to be understood without
PDF s.The beam functions B a have been calculated up to NNLO in ref. [58] (for
NLO see refs. [59, 60])and previously implemented in
MCFM [61]. The jet function J has been calculated in ref. [56].The hard function H corresponds to a crossed version of the NNLO (two loop) q ¯ q → V formfactor [62]. Lastly, we use the NNLO soft function S from ref. [63] (calculated at NLO inref. [64]) as used in ref. [65] for H +jet production, see also ref. [66]. f b ( x b , µ ) are the usual PDF s with parton momentum fraction x b of parton flavor b for the b -quark line.We perform the integrations and distributional convolutions over τ (cid:48) l , k s and k j in eq. (5)explicitly in Laplace space and implement the result as the below-cut contribution. The full NNLO result is obtained by adding the
NLO calculation with an additional jet in the presenceof a small slicing cutoff τ l and considering either the limit τ l → and/or a sufficiently smallvalue of τ l such that power-suppressed corrections of O ( τ l log k τ l ) to the factorization theoremare negligible.The matrix elements entering the calculation of the above-cut contribution are easily computed.We have recycled results from existing calculations of W + jet production at NLO [67] thatare implemented in
MCFM , attaching a current W → t ¯ b , in order to obtain the relevantamplitudes. As a cross-check, the matrix elements have been separately evaluated using theRecola package [68] to ensure the correctness of all contributions.In fig. 3 we show the τ cut l -dependence of the NNLO corrections for the leading jet pseudorapiditydistribution as a fraction of the LO distribution. We use the anti- k T jet clustering with R = 0 . .These are calculated for a stable top quark, that is ignoring the b -quark in the decay as apotential jet contributor. The absolute distribution (not shown here) therefore peaks around η = ± . , as its dominant contribution is from the forward jet on the light line.Decreasing the slicing cutoff τ cut l exponentially by factors of 10, we see that the differencebetween τ cut l = 0 . and τ cut l = 0 . is negligible. Even for τ cut l = 0 . the difference is In ref. [57] it has been found numerically for Z +jet production at NLO that the explicit minimizationreduces power-suppressed corrections relative to a jet axis obtained from the anti- k T jet algorithm. .2 Top-quark-decay corrections −0.050.000.05 −5.0 −2.5 0.0 2.5 5.0 η j,1 NN L O / L O r a t i o d σ d η j , τ lcut Figure 3.:
NNLO corrections from the light-line production vertex for the leading jet pseu-dorapidity distribution, relative to LO . The τ cut l -dependence of the corrections isshown in the range . – . .small, especially in the central region. This demonstrates the successful cancellation of IR divergences through application of the slicing subtractions. Note that the corrections in thecenter are negative, while they are positive in the tails. In the absence of any cuts this leadsto strong cancellations and a small NNLO correction to the inclusive cross section.
For the top-quark decay we make use of the factorization theorem described in refs. [69–73]for the observable τ d = p X /m t , where p X is the sum of all final-state parton momenta in the decay. The respective factorizationtheorem for corrections in decay and integrated over τ d reads d σ ab ( τ d < τ cut d ) = d σ Born f a ( x a , µ ) f b ( x b , µ ) × H ( x, µ ) (cid:90) τ cut d d τ d (cid:90) d m (cid:90) d k s J ( m , µ ) S ( k s , µ ) δ (cid:18) τ d − m + 2 E b k s m t (cid:19) , (6)where x = Q /m t > , E b = ( m t − Q ) / (2 m t ) is the b -quark energy, and Q = m W foran on-shell W -boson. We also allow for the generation of the W -boson decay distributedaccording to a Breit-Wigner peak with Q (cid:54) = m W .We use the bare soft function S bare from ref. [74] and the bare jet function J bare from ref. [75].We combined these with the bare hard function H bare , which has been independently calculatedin four different references [76–79] . We perform all integral convolutions in Laplace-space andfind successful cancellation of all IR poles between J bare , S bare and H bare . We furthermorechecked that we can individually reproduce the renormalized results of the soft ( S ) and jetfunctions ( J ) as given in refs. [74, 75]. The hard function is meanwhile also available with full bottom-quark mass dependence through ref. [80]. .2 Top-quark-decay corrections Table 1.: Comparison between our calculation of the second-order correction to the top-quarkwidth ( δ (2) ), as a percentage of the LO result of . ( m t = 172 . ), andprevious computations of the same quantity from Gao et al. [34] ( δ (2) GLZ ) and Bloklandet al. [86] ( δ (2) BCST ). µ δ (2) BCST δ (2) GLZ δ (2) m t − . − . − . m t − . − . − . m t / − . − . − . For the hard function we successfully checked the agreement between all four calculationsusing the Mathematica package
HPL [81] for the evaluation of harmonic polylogarithms.We converted the hard function into an expression using only harmonic polylogarithms, sothat our hard function implementation can be reused for the production part based on theautomatic analytic continuation in the hplog Fortran code [82].The matrix elements entering the above-cut calculation have been obtained by crossingfrom the virtual amplitudes used in the computation of
W t production at NLO [52] andthe real-radiation amplitudes employed in the four-flavor calculation of t -channel single-topproduction [83]. As for the corrections on the light-line, we cross-checked all amplitudes by anumerical comparison with Recola [68]. We then set up the NLO above-cut calculation usingdipole subtractions. For the case of q → qg splitting with the massive top-quark spectatorwe have used the dipole expression from ref. [22], also presented in ref. [84] . The dipoleexpression for the g → gg splitting with the massive top spectator is taken from ref. [84]. Calculation of the top-quark decay width.
Our calculation can be used to compute the
NNLO QCD corrections to the top-quark width. Such a calculation has previously beenperformed numerically in refs. [34, 35] (
GLZ ), as well as analytically as an expansion in m W /m t [85, 86] ( BCST ). We write the perturbative corrections to the top-quark widthas Γ t = Γ (0) t (1 + δ (1) + δ (2) ) , where the relative corrections of order α ks are written as δ ( k ) ≡ Γ ( k ) t / Γ (0) t and Γ (0) t is the LO decay width. In order to demonstrate the correctnessof our calculation, in fig. 4 we show the τ cut d -dependence of the NNLO corrections δ (2) forthe choice µ = m t fig. 4. We observe a good convergence of the slicing procedure to theasymptotic value, finding no significant dependence around τ cut d = 10 − .A comparison of our calculation with previous results for the top-quark width is shown intable 1, demonstrating excellent agreement. The previous approximate results are confirmedat the 0.5% level for the second-order coefficient, corresponding to a difference of at most . % in the total width. The comparison is performed with m t = 172 . , but the relativesize of the corrections is somewhat insensitive to the precise top-quark mass [34]. We note that eq. 5 in ref. [84] contains a typo, where it should read η(cid:15) (1 − z ) instead of y(cid:15) (1 − z ) , where η is a parameter that distinguishes between the CDR and FDH schemes. .3 Top-quark-production corrections on the heavy line −2.24−2.18−2.12−2.09−2.06 10 − − τ dcut Γ ( ) Γ ( ) i n pe r c en t Figure 4.: τ cut d -dependence of the top-quark decay width NNLO corrections. Error-bars indi-cate numerical integration uncertainties. The solid line (with dashed band) indicatesthe fitted result (and uncertainties), which agrees with the result using τ cut d = 10 − . For the heavy line production corrections we make use of the factorization theorem given inref. [87] using the observable τ h [54], τ h = 2 p X p b m t − q , where p X is the sum of the additionally radiated partons, p b is the b -quark beam momentumand q is the momentum-squared of the t − channel W boson ( q < ).The factorization theorem in τ h , integrated over τ h , reads d σ ab ( τ h < τ cut h ) = d σ Born f a ( x a , µ ) × H ( x + i , µ ) (cid:90) τ cut h d τ h (cid:90) d t (cid:90) d k s B b ( x b , t, µ ) S ( k s , µ, µ F ) δ (cid:18) τ h − t + 2 k s E b m t − q (cid:19) , (7)where x = q /m t . E b = ( m t − q ) / (2 m t ) is the b -quark beam energy in the top-quark restframe. We reuse the renormalized soft function S from the top-quark decay [74]. This ispossible when the soft function is defined in the top-quark rest frame as above and has beendiscussed in ref. [38]. We also reuse the beam functions B b [54, 59, 60] implemented throughref. [61]. The hard function H is reused from the top-quark decay, but requires an analyticcontinuation. Since we implemented it in terms of harmonic polylogarithms, we can usethe automatic analytic continuation of the hplog Fortran code [82]. We have verified thatthese analytically continued numerical results agree with an explicit analytic continuation inMathematica using the HPL package [81]. Lastly, we have explicitly performed the integralsand distributional convolutions in Laplace space to obtain the assembled result in termsof log k ( τ h ) contributions. As for the other contributions, we analytically verified that therenormalization scale dependence vanishes to the corresponding order, or in other words,that all poles cancel between bare hard, soft and beam function. We also find full agreementbetween our τ h -subtraction results at NLO and a calculation using dipole subtractions [22].The above-cut contributions are obtained by crossing from the matrix elements used in thedecay corrections. 11 .4 NLO ⊗ NLO interference contributions −0.08−0.06−0.04−0.020.000.020.040.060.08 −5.0 −2.5 0.0 2.5 5.0 η j,1 NN L O / L O r a t i o d σ d η j , τ hcut Figure 5.:
NNLO corrections from the heavy-line production vertex for the leading jet pseudo-rapidity distribution, relative to LO . The (automatically) fitted result coincideswithin numerical uncertainties with the τ cut h = 10 − result.We show the NNLO corrections to the leading jet pseudorapidity distribution as a fraction ofthe LO result in fig. 5, as well as their dependence on τ cut h . As for the light-line corrections,this distribution is calculated for a stable top quark, thereby ignoring the b -quark in the decayas a jet contributor. Comparing the results for the exponentially decreasing slicing cutoffs τ cut h ,we see that τ cut h = 10 − and τ cut h = 10 − differ only marginally within remaining numericaluncertainties. Even τ cut h = 10 − differs only in the most central bins. This demonstratesthe successful cancellation of IR divergences. We conclude that τ cut h = 0 . constitutes asufficiently small cutoff for reliable results. Once more there are strong cancellations, thatlead to a small correction to the inclusive cross section, since the corrections are small andpositive in the central region and larger and negative towards the tails. NLO ⊗ NLO interference contributions
Apart from two-loop vertex corrections,
NLO ⊗ NLO contributions also arise at
NNLO fromproduction corrections times decay corrections (light ⊗ decay, heavy ⊗ decay) and light-linecorrections times heavy-line corrections (light ⊗ heavy).We implemented all of these contributions in a mixed scheme where one part of the NLO cal-culation is handled with dipole subtractions and the other half with
NLO slicing subtractions,as described for the individual
NNLO vertex corrections above. This makes the implemen-tation within our existing infrastructure easier. Each of the three
NLO ⊗ NLO calculations(light ⊗ decay, heavy ⊗ decay, light ⊗ heavy) consists of four contributions, categorized intoreal ⊗ real, (RR), real ⊗ virtual (RV), virtual ⊗ real (VR), and virtual ⊗ virtual (VV). The firstpart is always calculated with dipole subtractions and denotes the real emission contributionswith dipole subtractions (real), and virtual loop corrections with integrated dipoles (virtual).The second part is calculated using the discussed factorization theorems and denotes theabove-cut emission (real), and the below-cut loop contribution (virtual) from the integratedfactorization theorem.For each of the three NLO ⊗ NLO calculations we checked cancellation of poles for the12ipole subtractions through the alpha parameter [22, 50–53] for the RV+VV and RR+VRcontributions separately. We furthermore checked for VV+VR and RV+RR separately thatremaining τ cut effects for our default choice of τ cut = 10 − are negligible at the per-mille leveland that the asymptotic approach is logarithmic as predicted by the factorization theorems.Therefore all four pieces of each NLO ⊗ NLO contribution are checked through a chain oftests.We show the differential τ cut -dependence for all three interference contributions in fig. 6 forthe leading jet pseudorapidity distribution at √ s = 13 TeV using anti- k T jet clustering with R = 0 . . We indeed find that for τ cut = 10 − residual power corrections are negligible at thesingle per-mille level.While one might naively expect that the factorization between production and decay allowsan implementation of NNLO interference corrections as the product of
NLO corrections inproduction and decay, at the level of differential cross sections, this is not true. This propertyonly holds fully inclusively, and serves as another check of the implementation, see eq. (2). Forinstance, the light ⊗ decay interference contribution ( σ L × D ) is equal to the light-line production NLO coefficient ( σ (1) L ) multiplied by the NLO width correction factor, Γ (1) / Γ (0) = − . .As an example, at √ s = 13 TeV we have, σ L × D = − ± , σ (1) L × Γ (1) Γ (0) = −
155 fb . (8)Furthermore, the heavy ⊗ decay interference ( σ H × D ) is also in excellent agreement, σ H × D = 645 ± , σ (1) H × Γ (1) Γ (0) = 645 fb . (9)
3. Comparison with previous results
In section 2 we have presented extensive cross-checks of all parts of our calculation, rangingfrom the amplitudes, to analytical checks of the factorization theorems and their ingredients,to the numerical implementation. Thus, having confidently assessed the validity of ourcalculation, we now compare against the two previous calculations of this process. Like ourcalculation, these are also performed in the approximation of an on-shell top quark, but theprevious results disagree on the size of the
NNLO corrections.
We first compare with the
NNLO results computed by Brucherseifer, Caola and Melnikov(
BCM ) [36]. This calculation is performed in the stable-top-quark approximation and thereforedoes not include the decay of the top quark. To compare with their results we adopt √ s = 8 TeV , m t = 173 . , m W = 80 .
398 GeV , G F = 1 .
166 39 × − GeV − and usethe MSTW2008 PDF set [88] that corresponds to the order of the calculation. The resultingcomparison between our calculation and the results of ref. [36] is shown in table 2, wherewe tabulate cross sections for four different minimum top-quark transverse momenta ( p T ),including uncertainties due to scale variation.13 .1 Stable top quark L × H interference −0.050.000.05 −5.0 −2.5 0.0 2.5 5.0 η j,1 NN L O / L O r a t i o d σ d η j , τ hcut L × D interference −0.010−0.0050.0000.0100.0050.0100.015 −5.0 −2.5 0.0 2.5 5.0 η j,1 NN L O / L O r a t i o d σ d η j , τ dcut H × D interference −0.010−0.0050.0000.0100.0050.015 −5.0 −2.5 0.0 2.5 5.0 η j,1 NN L O / L O r a t i o d σ d η j , τ dcut Figure 6.:
NNLO corrections to the leading jet pseudorapidity distribution, relative to LO ,from light ⊗ heavy interference (upper panel), light ⊗ decay interference (centerpanel) and heavy ⊗ decay interference (lower panel).14 .1 Stable top quark Table 2.: Comparison with fully inclusive top-quark production results from Brucherseifer,Caola, Melnikov in ref. [36]. Cross-sections in picobarns. Scale uncertainties insuper- and subscript are from simultaneous variation of µ R = µ F = m t by a factorof two and one half, respectively. p top T, min σ BCMLO σ LO ± . σ BCMNLO σ NLO ± . σ BCMNNLO σ NNLO ± .
10 GeV 53 . +3 . − . . +3 . − . . +1 . − . . +1 . − . . +0 . − . . + . − .
20 GeV 46 . +2 . − . . +2 . − . . +1 . − . . +1 . − . . +0 . − . . + . − .
40 GeV 33 . +1 . − . . +1 . − . . +0 . − . . +0 . − . . +0 . . . + . + .
60 GeV 22 . +1 . − . . +0 . − . . +0 . . . +0 . − . . − . . . + . + . Figure 7.: τ cut dependence of NNLO light, heavy, interference and summed contributionsrelative to the
NLO part for the fully inclusive result in table 2. The dotted arearepresents the
NNLO coefficient with an absolute uncertainty of ± . to obtainthe result from Brucherseifer, Caola, Melnikov (BCM) [36].While we find agreement at the per-mille level throughout NLO , we find discrepancies at thelevel of about to . percent for the NNLO results. The calculational uncertainties for bothour and the
BCM results are about . , which is a relative uncertainty of two per-mille. Afirst re-calculation of single-top-quark production in refs. [37, 38] by Berger, Gao, Zhu (
BGZ )has reported a similarly-sized difference to the
BCM calculation. While these authors didnot perform a direct comparison with the
BCM results, we will demonstrate in section 3.2 acomparison with the results of
BGZ that finds agreement at
NNLO at a much greater level(within a few per-mille), even for fiducial and differential results.The numerical integration uncertainties and residual systematic uncertainty from our slicingcutoff τ cut are at the per-mille level and small compared to the discrepancy we find. Wefirst demonstrate this in the fully inclusive case ( p top T, min = 0 ) in fig. 7. We show the τ cut dependence of the heavy-line contribution, light-line contribution, interference contribution We assume that for
BCM results the uncertainties are one in the last significant digit of the quoted result. .1 Stable top quark BCM light sumheavy interference0 50 100 150 200 0 50 100 150 200 − − − − p Ttop [GeV] d σ dp T t op NN L O c oe ff i c i en t / N L O τ cut Figure 8.: τ cut dependence of NNLO light, heavy, interference and summed contributionsrelative to the
NLO part for the top-quark p T distribution as input to table 2. Theblack points with error bars represent the reconstructed results from Brucherseifer,Caola, Melnikov (BCM) [36], see table 2, with an absolute uncertainty of ± . .and the sum, relative to the NLO part of the
NNLO result. The black dotted area representsthe (reconstructed)
NNLO coefficient of
BCM shown in table 2 with an absolute uncertaintyof ± . . Our numerical integration uncertainty is indicated by the error bars. All resultshave been obtained using the multi- τ cut sampling in MCFM -9 with a minimum τ cut of − .Since the largest uncertainties are for the light-line contributions, the figure also shows aseparate re-calculation of the light-line corrections with a nominal τ cut l value of . . Thesehave a significantly smaller integration uncertainty but agree perfectly with results calculatedwith a nominal value of τ cut = 0 . . Overall, with τ cut = 0 . we are already well withinthe asymptotic regime where results are precise at the per-mille level. The discrepancy withthe BCM result is clearly visible and cannot be explained by calculational uncertainties or adifference in the
NLO contribution.We next consider the top-quark p T distribution which, when integrated or summed over bins,reproduces the numbers in table 2 that require a minimum top-quark p T . Also here we haveto carefully inspect the τ cut dependence, since it may manifest in a different way dependingon the kinematics. For this we present top-quark p T -differential results in fig. 8 for τ cut valuesvarying over three orders of magnitude. The results are presented normalized to the NLO partof the distribution, that is to the
NLO cross section calculated with
NNLO PDF s, to emphasizethe relative size of the
NNLO contributions. Just as in the fully inclusive case it is evidentthat with τ cut = 0 . the asymptotic regime is reached for all contributions, since it agrees16 .2 Top quark production and decay Table 3.: Comparison with fully inclusive anti-top-quark production results from Brucherseifer,Caola, Melnikov in ref. [36]. Cross-sections in picobarns. Scale uncertainties insuper- and subscript from simultaneous variation of µ R = m t and µ F = m t by afactor of two and one half, respectively. p anti-top T, min σ BCMLO σ LO ± . σ BCMNLO σ NLO ± . σ BCMNNLO σ NNLO ± .
10 GeV 29 . +1 . − . . +1 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − .
20 GeV 24 . +1 . − . . +1 . − . . +0 . − . . +0 . − . . − . − . . +0 . − .
40 GeV 17 . +0 . − . . +0 . − . . +0 . . . +0 . . . − . . . + . + .
60 GeV 10 . +0 . − . . +0 . − . . +0 . . . +0 . . . − . . . − . + . well within our numerical uncertainties (small error bars) with the τ cut = 0 . result. Forcomparison with BCM we have reconstructed their corresponding results for the first threebins from table 2 and assume an absolute uncertainty of ± . .Our results show a definitive difference with the BCM results at both the differential andinclusive level. At the total level the discrepancy is about one percent. The discrepancywith the
NNLO coefficient itself is 100%: For fully inclusive top-quark production our
NNLO coefficient is − . , while the BCM coefficient is just − . . At the differential level thediscrepancy is also large. For example for the top-quark p T distribution the difference in thefirst bin from to
20 GeV is six percent relative to
NLO . For the last reconstructed bin for p T >
60 GeV we agree within mutual uncertainties. We observe similarly-sized discrepanciesfor an anti-top quark, as detailed in table 3.
We now turn to a comparison with the re-calculation by Berger, Gao, Zhu (
BGZ ) [37, 38],who also include the top-quark decay and provide fully fiducial predictions and differentialresults with which to compare.We first compare with inclusive cross sections (table 1 in ref. [38]), at √ s = 7 TeV and √ s = 14 TeV, for t and ¯ t . This comparison is performed with m t = 172 . , m W =80 .
385 GeV , G F = 1 .
166 379 × − GeV − and the CT14nnlo PDF set [89], irrespective ofthe order of the calculation that is considered. At
NNLO we obtain all results with a nominal τ cut = 0 . but ensure through the multi- τ cut sampling in MCFM -9 that the residual τ cut dependence is small compared to our quoted combined integration and residual τ cut uncertainty. The perturbative truncation uncertainties from scale variation are obtained byvarying µ R = µ F = m t simultaneously by a factor of two and one half, respectively, so by atwo-point variation. We do this to compare with the results in ref. [38], but note that witha six-point variation, where µ R and µ F are varied independently, the uncertainties increaseat both NLO and
NNLO . The results in table 4 show that, for both the central values andscale variations, the two calculations are in agreement at the level of less than three per-millein the
NNLO cross sections. Since our overall numerical and calculational precision is at thelevel of two to three per-mille, we consider this to be a excellent agreement, which is furthersupported by the following fiducial comparison.17 .2 Top quark production and decay
Table 4.: Comparison with fully inclusive production results from Berger, Gao, Zhu [37, 38].Scale uncertainties in super- and subscript from simultaneous variation of µ R = m t and µ F = m t by a factor of two and one half, respectively. σ BGZLO σ LO σ BGZNLO σ NLO ± . σ BGZNNLO σ NNLO top . +5 . − . . +5 . − . . +2 . − . . +2 . − . . +1 . − . . +1 . − . anti-top . +5 . − . . +5 . − . . +2 . − . . +2 . − . . +1 . − . . +1 . − .
14 TeV top . +8 . − . +8 . − . . +3 . − . . +2 . − . . +1 . − . . +1 . − . anti-top . +8 . − . +8 . − . . +3 . − . . +3 . − . . +1 . − . . +1 . − . η lj,1 d σ d η lj , [ pb ] LONLONNLOLO BGZNLO BGZNNLO BGZ
Figure 9.: Leading jet pseudorapidity distribution in stable top-quark production at LO , NLO and
NNLO using
NNLO CT14 PDF s. Our cross sections are compared with theresults in ref. [38] (
BGZ ).We also successfully compared at the differential level for the fully inclusive process (stabletop quark). For example, we show the leading jet pseudorapidity distribution in fig. 9 whereagreement within numerical uncertainties can be seen. Although not illustrated explicitly here,the same holds true for the individual components of the calculation: heavy-line corrections,light-line corrections and light ⊗ heavy interference contributions. Comparison with fiducial results.
We now compare with the fiducial results in ref. [38](
BGZ ), where we adopt m t = 173 . and the kinematical cuts are summarized in ta-ble 5.The results of this comparison are presented in table 6. Through NNLO we find agreement forall contributions within mutual uncertainties, where we assume an uncertainty of one in thefinal digit of the
BGZ results. In addition to this we have performed a check of the individualcontributions to the top-quark production number of − .
24 pb in table 6 from corrections to We thank Jun Gao for providing the corresponding division of the
NNLO result. .2 Top quark production and decay Table 5.: Kinematical cuts at √ s = 13 TeV for comparison with BGZ , ref. [38]. The top-quarkmass is . .Lepton cuts p lT >
30 GeV , | η l | < . anti- k T jet clustering p jT >
40 GeV , R = 0 . , | η j | < jet requirements exactly two jets; with least one b jet and p T -leading b jet | η | < . .Table 6.: Comparison with fiducial results in ref. [37, 38]. Cross sections are given in pb. σ BGZLO σ LO σ BGZNLO σ NLO σ BGZNNLO σ NNLO top total .
067 4 .
07 2 .
95 2 .
94 2 .
70 2 . production − . − . − . − . decay − . − . − . − . prod. × decay +0 .
12 +0 . anti-top total .
45 2 .
45 1 .
78 1 .
78 1 .
62 1 . production − . − . − . − . decay − . − . − . − . prod. × decay +0 .
07 +0 . the light-line, heavy-line and heavy-light interference. For the light-line contribution we find − ± , in agreement with −
100 fb ( BGZ ), and the interference contribution is identical, ± compared to
135 fb ( BGZ ). However, the heavy-line production contribution differsslightly: we find − ± , to be compared with −
273 fb ( BGZ ). While this tension isrelatively large on the
NNLO heavy-line production coefficient itself, it is only a per-mille leveleffect on the full
NNLO result and does not affect the overall level of agreement. Subsequentcommunication regarding this comparison revealed a small error in the earlier publishedcalculation that has now been identified and this small discrepancy is now understood. In contrast to our fiducial results that we present in the next section, the authors of ref. [38]use
NNLO PDF s throughout their study, and so also for
NLO predictions. They furthermoreestimate scale uncertainties using a two-point variation ( µ R = µ F = k · m t , k = / , ) anda quadrature procedure from production and decay contributions. Their scale uncertaintiesof +4 . − . and +1 . − . at NLO and
NNLO , respectively, increase to +6 . − . at NLO and +2 . − . at NNLO when a standard six-point scale variation (defined later, in section 4) is used. Theseuncertainties are mostly driven by the renormalization scale variation. We thank the authors of Ref. [38] for providing the detailed breakdown discussed here, and for clarifyingthis point. When using
DDIS scales, the
NLO result and uncertainties change slightly to . +6 . − . , but the NNLO scale
NNLO PDF s leads to
NNLO corrections of about − relative to NLO , see table 6,and the two-point quadrature scale-variation procedure leads to relatively small uncertainties,such that one observes a large gap between the
NLO and
NNLO predictions. This largedifference shrinks to about − when NLO PDF s are used consistently at
NLO , and one findsagreement within scale uncertainties. The consistent use of
PDF s is crucial for the t -channelsingle-top-quark process. In fact, the authors speculate about the smallness of the totalinclusive NNLO corrections due to
DIS data used for
PDF fits and that the t -channel processmimics double deep inelastic scattering. To fully exploit this property, it is essential that theorder of the PDF s is consistent with the hard scattering cross section order. This propertytransfers also to the fiducial region to some extent, but depends on the inclusiveness of thecuts. The double
DIS aspect can be maximally exploited in the factorized vertex-correctionapproach that we work with, namely by using
DDIS scales, where one then expects a maximumof perturbative stability of predictions. This point will be discussed further in the followingsection.
4. Results
In this section we examine both the fully inclusive cross sections and the differential dis-tributions relevant to experimental analyses. Stability of inclusive cross sections betweenperturbative orders is a signature of t -channel single-top-quark production, particularly whencalculated with the DDIS scales, as it is effectively undoing the
DIS fits to the
PDF s [90]. Thedifferential distributions are used for a variety of experimental signatures as both a signalprocess and as a background to any process that includes W + jets, e.g. W H or supersymmetry.We describe below large differences between
NLO and
NNLO in key distributions that areused to distinguish signal from background.We begin by presenting fully inclusive results for and
14 TeV proton-proton collisions(
LHC ) and for .
96 TeV proton-antiproton collisions (Tevatron) in table 7. We show resultsusing fixed scales µ R = µ F = m t for comparison with other results, and using the DDIS scales. Scale uncertainties are obtained using a six-point scale variation by evaluating thecross section for the scale choices ( k F · µ F ; k R · µ R ) , where µ F and µ R are the central scalesand ( k F ; k R ) ∈ { (2 , , (0 . , . , (2 , , (1 , , (0 . , , (1 , , (1 , . } . Subsequently maximum and minimum values are taken as the envelope. Our default
PDF set in this section is
CT14 [89] and we use it at a consistent order together with the partoniccross section order. That is, we use
NLO PDF s for our
NLO prediction and
NNLO PDF s forour
NNLO prediction.
PDF uncertainties are given by evaluating the 56 eigenvector members(using
DDIS scales). While there are noticeable differences in the inclusive cross section between
DDIS scales and µ = m t at LO , the differences are within scale uncertainties at the LHC. At the Tevatron thelarge difference is due to cancellations between corrections on the light-quark line and thebottom-quark line, and the LO scale uncertainties grow to ± when varied independently[19]. Overall we see that the NNLO corrections are almost zero and that
NLO and
NNLO uncertainties increase somewhat more to . +5 . − . . There are no such eigenvectors at LO and we therefore do not show PDF uncertainties at LO . .1 Fiducial and differential cross sections Table 7.: Fully inclusive results in pb for pp at and
14 TeV ( LHC ), as well as p ¯ p at .
96 TeV (Tevatron) with scales µ R = µ F = m t and DDIS scales and using
CT14PDF s. Uncertainties next to the cross section in super- and subscript are from asix-point scale variation, while
PDF uncertainties are below. pp
14 TeV pp .
96 TeV ¯ pp top anti-top top anti-top t + ¯ tσ µ = m t LO . +7 . − . . +7 . − . . +10 . − . . +10 . − . . +0 . − . σ DDIS LO . +6 . − . . +7 . − . . +9 . − . . +10 . − . . − . − . σ µ = m t NLO . +3 . − . . +3 . − . . +3 . − . . +3 . − . . +3 . − . σ DDIS
NLO . +3 . − . . +3 . − . . +3 . − . . +3 . − . . +3 . − . PDF +1 . − . PDF +2 . − . PDF +1 . − . PDF +1 . − . PDF +4 . − . σ µ = m t NNLO . +1 . − . . +1 . − . . +1 . − . . +1 . − . . +2 . − . σ DDIS
NNLO . +1 . − . . +1 . − . . +1 . − . . +1 . − . . +1 . − . PDF +1 . − . PDF +1 . − . PDF +1 . − . PDF +1 . − . PDF +3 . − . results overlap well within the small scale uncertainties of ∼ . PDF uncertainties are at asimilar level, but larger for the Tevatron.For the Tevatron it is noteworthy that when using the
DDIS scales and
CT14 PDF s the crosssection varies by 10–15% between LO and NLO [90]. This large difference, which should vanishfor pure
DIS data and fully correlated fits, shrinks to a few percent between
NLO and
NNLO .Whether this is an artifact of these particular
PDF fits or is a sign of a deeper problem withthe way
PDF s are parameterized is material for a dedicated study [91].
The key to understanding signals and backgrounds in t -channel single-top-quark analyses arethe W + n -jet exclusive cross sections. The -jet cross section with exactly one b -quark tagis enriched with the t -channel signal, while the -jet cross section with one or two b -quarktags is dominated by background processes like t ¯ t . It is essential that precise predictionsare available for both the signal region of t -channel single-top-quark production, but also forthe background region, which subsequently constraints the t ¯ t contribution in final fits. Inpractice, the final extraction is done using a multivariate analysis with various discriminatorobservables, see e.g. ref. [92].Our initial acceptance cuts are inclusive and defined in table 8. We require at least one b -tagged jet and one non- b -tagged (light) jet in order to study the effect of the NNLO correctionson jet counting. In the results presented below we reconstruct the top-quark momentumby adding the exact W -boson momentum and the momentum of the b -jet with the largest p T . We are particularly interested in reconstructed t + n -jet observables. In the following j b .1 Fiducial and differential cross sections Table 8.: Our fiducial cuts at √ s = 13 TeV .Lepton cuts p lT >
25 GeV , | η l | < . anti- k T jet clustering p jT >
30 GeV , R = 0 . , | η j | < . jet requirements at least one non- b (light) jet and at least one b jetdenotes a b -tagged jet and j l a light-quark jet (not b tagged), while j without any subscriptslabels any kind of jet.For our set of cuts we find that, when PDF s are evaluated at the same order as the matrixelement correction, the total fiducial
NNLO prediction σ NNLO = 5 .
75 pb agrees with the
NLO prediction σ NLO = 5 .
65 pb within less than two percent (see table 9). The stability acrossorders is due to a consistent use of
PDF orders. We also see a significant reduction in thescale uncertainty of the individual exclusive tj and tjj channels from ∼ – at NLO to ∼ – % at NNLO . The prediction for tjjj is a LO prediction with correspondingly largeruncertainties. The tjjj exclusive channel uncertainty dominates the semi-inclusive tj l fiducialuncertainty at NNLO , which is therefore not much smaller than the
NLO uncertainty.However, while the semi-inclusive cross section is stable with small uncertainties, there isa large 20% shift of events from the t + 2 -jet bin to the t + 3 -jet bin (along with a smallreduction of the cross section in the t + 1 -jet signal bin) when moving from NLO to NNLO .This is very significant because the t + 2 -jet bin is precisely the bin on which the cut is madeto separate signal from background. In practice, the experimental data in each of the jet binsis normalized to the relative fraction predicted by theory, and the absolute cross section isfloated to match the inclusive data. This means that the while the signal prediction in tj goes down by 5% at NNLO , too many events are being cut if the
NLO normalization is usedfor the tjj bin. Hence, the net signal predicted by
NNLO will increase compared to
NLO , andthe tjj background to other physics processes is smaller than expected by
NLO . Differential cross sections.
We now move on to discuss differential distributions. For allthe following plots we show absolute distributions with scale uncertainties in the upper panel,while in the lower panel we show the ratios
NNLO / NLO and
NLO / NLO to study perturbativestability across orders. For the ratios we show the scale uncertainties from variation in thenumerator only. Perturbative stability across orders can therefore be judged by examiningthe extent to which the uncertainty bands overlap.We begin our discussion with the charged lepton pseudorapidity distribution shown in fig. 10at LO , NLO and
NNLO . This distribution is mostly kinematically driven and, as expected,the
NNLO corrections are consistent with zero to within a few percent fully differentially. Thedifferences in the central prediction between
DDIS scales and fixed scale m t are small at NNLO ,but we observe noticeably larger scale uncertainties using the
DDIS scales, allowing for arobust overlap of predictions from LO through NNLO . The relatively large scale uncertaintiesat
NNLO , compared to
NLO , were already visible in the tj l inclusive cross section in table 9and are an artifact of the specific set of rather inclusive cuts. Compare this, for example, with Generally, if
NNLO PDF s were used with
NLO matrix elements, the difference would increase by a fewpercent. .1 Fiducial and differential cross sections Table 9.: Cross sections for the production of a top quark ( W plus p T -leading b -jet) andadditional jets using DDIS scales, in pb. The cross sections are given in the fiducialregion as in table 8, but apart from the inclusive tj l row the jet requirements areadjusted accordingly. j b denotes a b -tagged jet and j l a light-quark jet (not b tagged).Contributions with more than one additional b -tagged jet are negligible and omitted.Uncertainties from a six point scale variation are given in super- and subscript.Percentages in parenthesis give the fraction with respect to the individual inclusivejet category. Numbers in bold font in each column add up to the tj l inclusive resultwithin numerical uncertainties. LO NLO NNLO tj l inclusive . +9 . − . +3 . − . . +3 . − . tj . +9 . − . +10 . − . (100%) . +7 . − . (100%) tj l . +9 . − . +13 . − . (86%) . +8 . − . (87%) tj b — . +14 . − . (14%) . +2 . − . (13%) tjj — . +10 . − . (100%) . +4 . − . (100%) tj l j l — . +10 . − . (38%) . +4 . − . (40%) tj b j l — . +15 . − . (62%) . +4 . − . (60%) tjjj — — . +27 . − . (100%) tj l j l j l — — . +20 . − . (17%) tj b j l j l — — . +30 . − . (83%)23 .1 Fiducial and differential cross sections Figure 10.: Pseudorapidity distribution of the charged lepton with
DDIS scales (left) and fixedscale m t (right).the fully inclusive uncertainties in table 7, which are much smaller at the level of and thelarge uncertainty decrease in the individual jet cross sections in table 9. Overall, the largescale uncertainties are also consistent with the observation that the NNLO effects are almostas large as the
NLO effects.For all other distributions we find similar central results between
DDIS and m t scales. Since DDIS scales are the most consistent with the DIS nature of the t -channel process we onlyshow the DDIS results with their more robustly estimated scale uncertainties.Other standard observables to consider are the top-quark transverse momentum and rapidity.We show these distributions in fig. 11. The
NNLO rapidity corrections are small and consistentwith zero within scale uncertainties. The
NNLO transverse momentum corrections are, onthe other hand, sizable and correct
NLO results by up to +15% at
200 GeV to
250 GeV .Kinematically, the top quark p T is driven by recoil from additional jets beginning at an NLO calculation. Therefore, one expects to see large effects at
NNLO , which we observe here andwhich are outside scale uncertainty bands. The kinematic jet recoil threshold around
30 GeV is also significantly stabilized at
NNLO by the additional radiation. This threshold region hadto be addressed previously by a parton shower or partial resummation. t -channel signal and background. In addition to jet counting, the most important discrim-inatory observable for the signal is the light-quark jet pseudorapidity, which in t -channelproduction has its distinctive peak in the forward direction. In fig. 12 we present the ( p T )leading light-jet transverse momentum and pseudorapidity distributions. NNLO correctionsdrive the transverse momentum slightly harder in the tail but have little effect in the peakregion. The shape of the pseudorapidity distribution changes noticeably, with negative correc-24 .1 Fiducial and differential cross sections d σ dp T t op [f b / G e V ] LONLONNLO0.81.01.21.4 0 50 100 150 200 250 p Ttop [GeV] r a t i o t o N L O d σ d y t op [ pb ] LONLONNLO0.80.91.01.1 −2 0 2 y top r a t i o t o N L O Figure 11.: Top-quark transverse momentum distribution (left) and rapidity distribution(right).tions of in the very forward region and positive corrections of in the central region,leaving the peak region with corrections of just a few percent.The leading b -jet distributions shown in fig. 13 are relevant for the top-quark reconstruction andenter directly in our previously shown top-quark distributions. For the transverse momentumwe observe zero corrections in the peak region but positive corrections of at large p T driven by additional recoil available at NNLO . The pseudorapidity distribution receives onlypositive corrections of − in the relevant peak region.Another important discriminatory observable is the angle between the lepton and the leadinglight-quark jet in the top-quark rest frame, which we discuss later.Moving to the relevant distributions for a proper background estimation, we show the sub-leading light-quark jet and subleading b -tagged jet transverse momentum and pseudorapiditydistributions in figs. 14 and 15, respectively. Since these distributions enter for the firsttime at NLO in our five-flavor scheme calculation, the α s corrections at NNLO are large andsignificant. And while these distributions could be obtained from an
NLO event generator witha Born process of single-top-quark production plus an additional jet ( W +3-jet production), thecalculation within our NNLO framework allows for a correct normalization of these backgroundcontributions to the signal contribution.In appendix A we present additional distributions for the lepton transverse momentum,see fig. 17, and the b -quark-lepton system invariant mass and transverse momentum, seefig. 18. 25 .1 Fiducial and differential cross sections d σ dp T lj , [f b / G e V ] LONLONNLO0.91.01.11.2 50 100 150 200 250 p Tlj,1 [GeV] r a t i o t o N L O d σ d η lj , [ pb ] LO NLO NNLO0.80.91.01.11.21.3 −2.5 0.0 2.5 η lj,1 r a t i o t o N L O Figure 12.: p T -leading light jet transverse momentum distribution (left) and pseudorapiditydistribution (right).Figure 13.: p T -leading b -tagged jet transverse momentum distribution (left) and pseudorapiditydistribution (right). 26 .1 Fiducial and differential cross sections d σ dp T lj , [f b / G e V ] NLONNLO1.01.21.41.61.82.02.2 50 100 150 200 250 p Tlj,2 [GeV] r a t i o t o N L O d σ d η lj , [f b ] NLO NNLO123−5.0 −2.5 0.0 2.5 5.0 η lj,2 r a t i o t o N L O Figure 14.: p T -subleading light jet transverse momentum distribution (left) and pseudorapiditydistribution (right). d σ dp T b , [f b / G e V ] NLONNLO0.81.01.21.4 30 50 100 150 200 250 p Tb,2 [GeV] r a t i o t o N L O Figure 15.: p T -subleading b -tagged jet transverse momentum distribution (left) and pseudora-pidity distribution (right). 27 d σ d c o s ( θ l , z ) [ pb ] LONLONNLO0.91.01.11.21.3 −1.0 −0.5 0.0 0.5 1.0 cos ( θ l,z ) r a t i o t o N L O Figure 16.: Angular distribution for cos θ l,z .. Angular observables in the top-quark rest frame.
In addition to assessing the impact ofhigher order corrections on the usual kinematical distributions such as invariant masses,transverse momenta and rapidities, it is important to also consider their effect on angularcorrelations. The angle between the leading non- b jet and the lepton from the top-quarkdecay in the top-quark rest frame [21] exhibits a strong correlation [93, 94] and is one of thekey observables used to identify the t -channel process. Our NNLO results for this observable cos( θ l,z ) are shown in fig. 16 and we find that NNLO corrections are consistent with zero forthe bulk region. At large angles ( cos θ l,z ∼ − ) the additional radiation at NNLO becomesimportant and the corrections are significant, as expected.Several other angular distributions measured in the top-quark or W -boson rest frame areexpected to be modified by non-standard model physics effects [95, 96]. Such angles areconstructed to be sensitive to new physics in the production and decay stages of the top quark,respectively. We present NNLO results for two such sets of angles in appendix A and find that
NNLO corrections are mostly consistent with zero in the bulk regions within a few percentof scale uncertainties. For the study of anomalous couplings and
SMEFT contributions thisperturbative stability at
NNLO is important, since such analyses are performed at
NLO . Infact, the inclusion of off-shell effects at
NLO has previously been found to be equally or moreimportant for some angular distributions [33].
5. Conclusions
We have presented a calculation of → (five-flavor scheme) t -channel single-top-quarkproduction and decay at NNLO . Due to the special importance of the top quark in theStandard Model, and the availability of large, high-quality data sets from the LHC experiments,28redictions at this level of sophistication are mandatory for future analyses. Moreover, suchcalculations should be cross-checked carefully and made available publicly in order to facilitateany such studies.Our calculation is performed in the on-shell and vertex-function approximation that allowsfor a factorization into
NNLO corrections in production on the light-quark line, heavy-quarkline and corrections in decay. We include all
NNLO vertex corrections through three
NNLO calculations but also the
NLO ⊗ NLO interference contributions between light line and heavyline, as well as production and decay at the amplitude level, thereby preserving all spincorrelations between production and decay. We performed extensive checks of all componentsof our calculation, and at all stages of the assembly. This includes analytical checks of allingredients entering the factorization theorems used for our
NNLO subtractions, comparisonsof individual amplitudes with the numerical one-loop library Recola, and verification of ournumerical implementations of dipole subtractions and slicing subtractions at the per-millelevel.With the results of our calculation we have been able to scrutinize the results of the stabletop-quark calculation [36] and the results with decay in refs. [37, 38]. We find full agreementwith the latter calculation at the per-mille level, but in the former case find differences of100% on the total
NNLO coefficient and up to six percent in the full
NNLO top-quark p T distribution. Our calculation therefore resolves this discrepancy and, furthermore, validatesthe first implementation of differential NNLO corrections in the decay [34, 37, 38].A special focus of our implementation is to keep full flexibility for dynamic factorization andrenormalization scales in all parts of the calculation. One is then able to exploit that single-top-quark production is like double deep inelastic scattering (
DDIS ) in the approximationthat we use, and set the scales as they are used in the fitting of
PDF s from light-quark andheavy-quark
DIS data. While the effect of using
DDIS scales compared to using a fixed scale m t seems small at NLO and
NNLO for the
PDF set considered here, it can serve as a consistencycheck and constraint of
PDF s that we intend to pursue in a subsequent study.We presented total cross sections at the fully inclusive level including scale uncertaintiesand
PDF uncertainties for the and
14 TeV
LHC and for proton-antiproton collisionsat the Tevatron for both a fixed central scale as well as
DDIS scales. We studied fiducialcross sections at the total and the differential level in detail, presenting standard kinematicaldistributions used for single-top-quark analyses. We estimate scale uncertainties using astandard six-point variation and
DDIS central scales. With this prescription we find thatuncertainties are about a factor of two larger than in the previous calculation [38], whichemployed a different prescription, but our choice better represents the differential variationorder-by-order.Overall we find that
NNLO corrections are crucial for a precision identification of the t -channelprocess, whose primary discriminatory observable is the leading light-quark jet pseudorapidity.This observable receives shape corrections of up to 20% at NNLO . Higher-order effects arealso important for the background categories of W +3-jet production where corrections are50–100%. We also showed NNLO predictions for certain sets of angles defined in the top-quarkrest frame that are sensitive to new physics in the production and decay vertices. We find that
NNLO corrections for these angles are small and that therefore these angles are kinematicallyrobust and can be taken into account at
NLO at the current level of precision for the searchfor new physics. 29n the future it could be interesting to investigate if the use of
DDIS scales can provide auseful constraint on
PDF s. The general relevance of t -channel single-top-quark productionfor PDF fits has been studied before in ref. [97]. To further improve the accuracy of ourpredictions,
NLO off-shell effects implemented in
MCFM [33] could be incorporated with areweighting procedure. Additionally, one could furthermore work towards the inclusion ofthe /N c color-suppressed effects from two-loop box diagrams, which should be feasible withmodern loop-calculation techniques.Our calculation will be made publicly available in an upcoming version of MCFM to be usefuldirectly for the
LHC precision physics program.
Acknowledgments
We would like to thank Jun Gao for providing information regarding thecalculation presented in ref. [38]: for supplying us with more detailed cross section numbers forcomparison, that allowed for the exhaustive cross-check detailed in table 6 and the surroundingdiscussion, as well as for clarifying the details of the scale variation procedure. We furthermorethank Jean-Nicolas Lang for providing us with Recola 4-flavor and 5-flavor scheme modelfiles.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 ismanaged by Fermi Research Alliance, LLC (FRA), acting under Contract No. DE-AC02-07CH11359. Tobias Neumann is supported by the United States Department of Energy underGrant Contract DE-SC0012704. The numerical calculations reported in this paper wereperformed using the Wilson High-Performance Computing Facility at Fermilab.
A. Additional fiducial distributions
In this Appendix we provide supplementary figures for select kinematic distributions andangular correlations using the fiducial cuts of table 8 in section 4.1.In fig. 17 we present the
NNLO transverse momentum and pseudorapidity distributions for thelepton from the top-quark decay. The
NNLO corrections in the pseudorapidity distributionare at the level of a few percent but still consistent with zero within scale uncertainties, asmay be expected from our total fiducial cross section in table 9. The transverse momentumdistribution on the other hand receives sizable corrections in the tail, but is perturbativelystable in the peak region. The corrections in the tail are large because the transverse recoil isdriven by the subleading jet in the decay, entering for the first time at
NLO .Combinations of the leading b -jet and lepton system are relevant for the top-quark massmeasurement in t -channel production [3, 98] since they allow one to circumvent the neutrinoreconstruction. We show the bl transverse momentum and invariant mass distribution infig. 18 and find that while NNLO corrections are small in the peak regions, they are importantotherwise. The large perturbative corrections to the transverse momentum distribution followdirectly from the stiffening of the lepton transverse momentum at
NNLO that is included inthis variable.We also present
NNLO results for two sets of angular observables that are sensitive tomodifications from new physics in the top-quark production and decay stages, respectively.To describe the first set of angles we introduce a coordinate system that uses the direction30 .010.1110100 50 100 150 200 250 d σ dp T l [f b / G e V ] LONLONNLO0.91.01.11.21.3 50 100 150 200 250 p Tl [GeV] r a t i o t o N L O d σ d η l [ pb ] LO NLO NNLO0.900.951.001.051.10 −2 −1 0 1 2 η l r a t i o t o N L O Figure 17.: Positron transverse momentum distribution (left) and pseudorapidity distribution(right).
110 50 100 150 200 250 d σ dp T b l [f b / G e V ] LONLONNLO1.01.11.21.3 50 100 150 200 250 p Tbl [GeV] r a t i o t o N L O d σ d m b l [f b / G e V ] LONLONNLO0.81.01.21.4 0 50 100 150 200 m bl [GeV] r a t i o t o N L O Figure 18.: Leading b -tagged jet plus lepton transverse momentum distribution (left) andinvariant mass distribution (right).31f the leading light-quark jet (cid:126)s t in the top-quark rest frame to define the z -axis, ˆ z . Theleading light-quark jet is identified at LO with the quark radiated on the light-quark line andis referred to as the spectator quark in the literature. This identification is only well-definedat LO , of course. The direction orthogonal to the plane made by the spectator quark andthe initial-state light-quark defines the y -axis, ˆ y . The initial-state light-quark direction, ( (cid:126)p q ),is chosen by selecting the beam direction whose rapidity has the same sign as that of thespectator jet. Finally, the coordinate system is completed by defining ˆ x such that the systemis right-handed [95]. Thus we have, ˆ z = (cid:126)s t | (cid:126)s t | , ˆ y = (cid:126)s t × (cid:126)p q | (cid:126)s t × (cid:126)p q | , ˆ x = ˆ y × ˆ z . (10)The (cosines of the) angles of the lepton with respect to these axes are referred to as cos θ l,x , cos θ l,y , cos θ l,z . Generically, these angles are sensitive to operators that modify theproduction of the top quark.The second system uses as its first axis the direction of the W -boson in the top-quark restframe, ˆ q . The second axis ˆ N is orthogonal to the plane defined by ˆ q and the leading light-quarkjet in the top-quark rest frame (cid:126)s t . As before, the system is completed by the requirement ofright-handedness, thus defining ˆ T [96]: ˆ q = (cid:126)q | (cid:126)q | , ˆ N = (cid:126)s t × (cid:126)q | (cid:126)s t × (cid:126)q | , ˆ T = ˆ q × ˆ N . (11)The angles between the lepton in the W -boson rest frame and these three axes define thequantities cos θ ∗ l , cos θ Nl and cos θ Tl . Finally, we construct two more angles based on theprojections of the lepton in the W boson rest frame onto the ˆ N - ˆ T plane. The angle betweenthis projection and the ˆ N and ˆ T axes define cos φ N and cos φ T , respectively. This second setof angles is particularly sensitive to modifications to top-quark decay from physics beyondthe Standard Model.As has been noted before [33], the neutrino reconstruction has a noticeable impact on mostof these observables, since they are constructed in the top-quark rest frame which has has adirect dependence on the neutrino four momentum. Here we do not consider the neutrinoreconstruction, but leave studying such effects for a future publication.It was previously observed [21] that, after cuts, going from LO to NLO had little effect on SM angular distributions like cos θ l,z that are used to measure t -channel single-top-quarkproduction, see also fig. 16. We are now able to to quantify these findings at NNLO for thefull set of observables defined above.We present the results for cos θ l,x , cos θ l,y , cos θ l,z in figs. 16 and 19, for cos θ Nl , cos θ Tl and cos θ ∗ l in figs. 20 and 21, and cos φ N and cos φ T in fig. 22. We find that all angular observablesconsidered here are perturbatively stable and NNLO corrections in the bulk are mostlyconsistent with zero within scale uncertainties.Also off-shell effects have been considered before at
NLO [33] and were found to be relativelysmall and uniform at the few percent level, except for effects of up to ∼ in cos θ l,x and cos θ ∗ l . These effects are therefore equally or more important than NNLO on-shell corrections.Note that the cos θ Nl distribution has been found to be unstable in fixed-order perturbationtheory due to sensitivity to soft radiation [33], but using the on-shell approximation we donot expose this sensitivity. 32 .02.53.03.5 −1.0 −0.5 0.0 0.5 1.0 d σ d c o s ( θ l , x ) [ pb ] LO NLO NNLO0.900.951.001.051.101.15 −1.0 −0.5 0.0 0.5 1.0 cos ( θ l,x ) r a t i o t o N L O d σ d c o s ( θ l , y ) [ pb ] LONLONNLO0.900.951.001.051.101.15 −1.0 −0.5 0.0 0.5 1.0 cos ( θ l,y ) r a t i o t o N L O Figure 19.: Angular distributions for cos θ l,x (left) and cos θ l,y (right). d σ d c o s ( θ l * ) [ pb ] LONLONNLO0.91.01.11.2 −1.0 −0.5 0.0 0.5 1.0 cos ( θ l* ) r a t i o t o N L O Figure 20.: Angular distribution for cos θ ∗ l .33
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