JHEP02(2018)125 X − W + Springer bW ¯ b February 7, 2018 February 9, 2018 → February 20, 2018 : : December 11, 2017 : − : e + e Revised Accepted Received Published , c Published for SISSA by https://doi.org/10.1007/JHEP02(2018)125 and P. Ruiz-Femen´ıa [email protected] b , [email protected] , T. Rauh b . 3 1711.10429 top anti-top threshold The Authors. A. Maier, Effective Field Theories, Heavy Quark Physics, Perturbative QCD, Quark c

− We determine the NNLO electroweak correction to the a , e [email protected] + e E-28049 Madrid, Spain E-mail: [email protected] Physik Department T31, Technische Universit¨atM¨unchen, James-Franck-Straße 1, 85748 Garching, Germany IPPP, Department of Physics,Durham, University DH1 of 3LE, Durham, U.K. Departamento and de Te´orica Instituto F´ısica UAM-CSIC, de Te´orica F´ısica Universidad Aut´onomade Madrid, b c a Open Access Article funded by SCOAP Keywords: Masses and SM Parameters ArXiv ePrint: production cross section nearnon-resonant the top-pair production production of threshold. theanti-top The final pair calculation state production includes as withprediction well to non-relativistic as NNNLO resummation, QCD electroweak and plusof effects elevates the NNLO in new electroweak the resonant accuracy. contributions theoretical top on We then the study top-pair the threshold scan impact at a future lepton collider. Abstract: M. Beneke, Non-resonant and electroweak NNLOthe correction to JHEP02(2018)125 34 16 21 22 24 48 ) k ( EW C σ 3 Γ 20 σ t Z ) k ( Abs, C 30 33 σ 29 10 44 – i – 5 35 conv IS σ 37 13 33 35 35 Threshold ] ] 28 48 3 threshold 50 31 49 31 QQbar 33 37 51 QQbar 1 47 31 29 13 6.3.1 Comparison6.3.2 to [ Comparison to [ 3.1.1 Finite-width3.1.2 corrections to the NNLO Mixed3.1.3 Green QCD-electroweak NNLO function, corrections in Absorptive PNREFT part3.1.4 from field renormalization, Electroweak contributions3.1.5 to the hard Initial-state matching radiation, coefficient, A.1 Non-resonant corrections A.2 Initial-state radiation A.3 Width corrections 7.1 Size of7.2 the electroweak effects Sensitivity to Standard Model parameters 6.1 Consistency checks 6.2 Implementation in 6.3 Comparison to other approaches 5.1 The automated5.2 part Endpoint-finite part of the interference contribution 3.2 The squared contribution 4.1 Absorptive contribution4.2 to the matching Endpoint coefficient divergence of the interference contribution 2.1 Resonant and2.2 non-resonant separation in Organization unstable of2.3 particle the EFT computation Implementation of a “top invariant mass cut” 3.1 Resonant electroweak effects A Implementation in 7 Discussion of results 8 Conclusions 5 Part (III) 6 Checks and implementation 4 Part (II) 3 Part (I) Contents 1 Introduction 2 Setup of the computation JHEP02(2018)125 1 54 (1). The O 345 GeV can = ≈ /v of integrated lu- s s 1 α √ − 3%. Similarly, it has ]. In addition, the top  6 1, but  v ]. About 100 fb 1 1 and 55 52 ]  s 31 MadGraph α ]. 8 of the order of the strong coupling constant collider [ , ], but the extension of this approach to hadron 7 MS mass with an experimental uncertainty of 2 – 1 – 5 − 51 / e 1 + 2) e ]. The NNNLO corrections are well behaved and the − t 23 (1 GeV) due to the limited understanding of the relation s/m O √ = ( v ] in an expansion where 9 ]. This must be compared to the ultimate precision possible at the 4 – 2 MS mass and the mass parameter in the calculation and simulation of the fi- ] for the computation of specific NNNLO ingredients. ], to which we refer for more details on the QCD aspects of the calculation. 22 – 10 11 1 is still small. This interplay between the strong Coulomb attraction and the large collisions in the boosted top regime [ A significant effort has since been invested into providing high-precision predictions The threshold region is defined as the kinematic regime where the top quarks have − See [ A.5 Calculation of the non-resonant correction A.4 Note on backwards compatibility  1 e . Thus, the top quarks are non-relativistic and are subject to the colour Coulomb s s + rized in [ The NNNLO QCD resultturbative has expansion finally up settled to the NNLOremaining issue [ scale of the uncertainty poor of convergence the of QCD the result per- is at the level of for top pair production nearteraction threshold. effects, which The have major now(NNNLO) focus been accuracy has computed [ naturally to been next-to-next-to-next-to-leadingeffective the field order strong theory in- formalism and ingredients that underlie this calculation are summa- were stable.prevents The hadronization. sizeable Therefore, top theCoulomb decay top interaction, width which threshold must dynamics caused be isα treated by governed non-perturbatively, the by while the the electroweak strongtop colour interaction coupling decay width also has first been realized in [ width, the strong couplingthe constant threshold and scan the to top varying degree Yukawa coupling of can accuracy. bea extracted small from three-velocity α interaction, that would facilitate the formation of toponium bound states if the top quarks between the nal state from which thein top the mass quantification is of directly reconstructed.e this relation There has when been the somecollider mass progress processes is requires reconstructed from the two-jettiness consideration in of additional effects [ for the construction ofminosity a spread high-energy over tenprovide center-of-mass a energies measurement distributed ofabout around the 50 MeV top-quark [ LHC, which is constrained to 1 Introduction The precision study of the top pair production threshold is among the main motivations B Implementation of the subtractions in C Further details on the comparison to [ JHEP02(2018)125 (1.3) (1.2) (1.1) , } t α NLO NNLO NNNLO EW α , √ , ]. To define the 2 and count ) coupling. For the t 29 5%. , v, v λ t EW s  ], which are cancelled 28 α , α , α NLO NNLO NNNLO 15 t 2 s ) electromagnetic correc- α α α and 3 ( EW O , v × { em 2 α collisions. The electromagnetic, v α , √ ]. s em 2 2 v , threshold [ − v α e 27 , − + , v, α v, v EW ∼ } e of the decayed top anti-top pair. An 2 s s 2 s W 26 , v , α α , α , α , v + s X } 2 3 s s s α ∼ 1α α α LO W , v + ,... s , which is always adopted in the pure QCD 2 t              π − 2 o α × { λ 4 v ) and top-quark Yukawa ( 2 × t W , v  – 2 – s k ≡ × { + m ] has demonstrated that they are as large as 10%. EW  t em v , α s α α ∼ em α 25 v v bW ¯ α α b em  v , ∼  , , α EW 3 2   n α =0 ∞ EW t k X × α em em v v m v t accounts for the phase-space suppression of the cross section em α α v α α ∼ v   2 EW NLO NNLO NNNLO t                      α 2 EW s × ∼ α α k ] stabilizes the scale uncertainty at the level of of the form  s EW EW v σ 24 α α α ... ), SU(2) electroweak (         QCD only em σ × =0 α ∞ k X v 2 EW α 2 EW + α The main result of this work is the NNLO calculation of all electroweak and non- In the present work we are concerned with electroweak effects and non-resonant pro- ∼ σ where the global factor near the threshold and the electroweakelectroweak, production Yukawa and in non-resonant terms are of the parametric form total cross section that is, anconsistent electroweak with coupling counting counts Γ ascalculation. two The powers pure of QCD calculation the up strong to coupling, NNNLO which then accounts is for all terms in the consistent with Coulomb resummationtions, and following the a inclusion of similarprecise treatment meaning as of “NNLO” fortromagnetic for the ( electroweak effects, we notepurpose that of they power counting introduce we the do elec- not distinguish between vage the precision ofthey the are prediction. required to Even obtaincontains a more divergences well-defined importantly, result, proportional as since to will theonly the pure be once QCD top-quark discussed the cross decay non-resonant below, section width production by [ itself is included [ resonant effects. We also provide an implementation of initial-state radiation in a scheme logarithmic order [ duction of the observableanalysis final of state various electroweak effectsThus, [ the full NNLO non-resonant and electroweak contributions must be included to sal- been observed that the RG-improved prediction at the (almost) next-to-next-to-leading JHEP02(2018)125 W ]. The 32 code [ we analyze the 7 bosons are always hard. terms arise from the ) denote third-order . The narrow-width ], which describes the k X ]. This cancellation has W ) 1.3 code. 34 coupled to ultrasoft radi- + 15 , /v threshold v − 33 t em α ], but only partial results are m W + 26 QQbar we describe how the calculation bW ¯ 2 threshold b QQbar after hard and soft effects have been integrated – 3 – describes a consistency check we performed for 2 6 v t m -boson lifetime being of similar size as the top lifetime, the in the other terms. W The above final state can also be produced non-resonantly, EW . Several appendices collect technical results, in particular α ]. On the resonant side, the ( ]. This result together with the NLO non-resonant and the 3 bosons as stable particles. 8 . 2 25 31 These as well as all Yukawa coupling effects have already been W v and , t 2 m em 30 , α ∼ t 27 = 1. Despite the , respectively. Section m 2 2 | 5 tb − V s | and √ 4 , = 3 E The top-pair production cross section is thus an ill-defined quantity. Instead one The outline of the paper is as follows. In section We do not distinguish We assume 2 3 the sum is well-defined and finite-width divergences must cancel [ decay width can beThus, dropped it is (expanded justified out) to in treat the the propagators, because the must consider the finalapproximation is state not applicable of since the theenergy top width decay is products not small comparedi.e. to the without top kinetic anproduction intermediate mechanisms non-relativistic cannot top be pair. distinguished physically and The must resonant be and summed. non-resonant Only in potential non-relativistic effectivedynamics field of theory slowly (PNREFT) moving [ particlesation/massless with particles three-momentum with energy out. Thewidth, computation which contains start uncancelled at NNLO divergences in proportional dimensional regularization. to the top-quark 2 Setup of the computation 2.1 Resonant and non-resonantPrecision separation calculations in of unstable top particle pair EFT production near threshold are most conveniently done our results and theimportance comparison of with the various some contributionsation. previous for We results. the conclude threshold in In scanthe section section including implementation initial-state of radi- the new results into the is split into resonant andand non-resonant the contributions, divergences such that are noinvariant cancelled mass double-counting consistently. cut. occurs We For also theseparately discuss practical finite calculation the parts, we implementation which split are ofsections the computed, an each total within cross its section own into computational three scheme, in NNNLO QCD calculation has beenNNLO made available non-resonant in the and thein this remaining work, NNLO thus electroweakQCD+Yukawa elevating and contributions the NNLO are precision EW+non-resonant. computed atelectroweak and the non-resonant The top-pair terms ellipses threshold that to remain in unknown. complete ( NNNLO the absence of phase-spacepart. suppression The and non-resonant Coulombavailable contribution resummation at is for NNLO known the [ atQED non-resonant NLO Coulomb potential. [ included up to NNNLO in [ where the first line refers to resonant and the second to non-resonant production. We note JHEP02(2018)125 , ]. 4 EW 3 EW 36 α (2.3) (2.1) (2.2) , anni- α , while ) l 35 t ( count as m O EW are the hard . ) corrections = 2 s  ) α l s α i ( ( + √ C with the process e O and − . The non-resonant ¯ tb e v E | . )] i . x 2 EW + ( X e α ) l + ( − e ∼ . | O bW ) i decays to s LO ( (0) ) cuts and are of the order σ (0) e k † corrections are of the order . Since both W ( 4 ) − t k E O ( The production operators i NNLO non-res | O bW σ ¯ 4 i + t e E/m T[ − | ) + e + s h and ( e i ) − . In unstable particle theory this contribution ], and will be reproduced in the computation + e k e ( 4 h – 4 – 27 C final state. W b NNLO res ] into Unstable Particle Effective Theory [ x h ¯ tbW σ 4 X d 10 Im − , computed directly at the threshold Z ) = − , where the off-shell s ) W final state are not possible kinematically near threshold. ∗ k l ( ( ¯ X t + t bW ¯ C decays to t ) ∼ t k bW ¯ WW ) ( + b NNLO s C ( + σ → states and produce a nearly on-shell top and anti-top quark with k,l − X − ¯ tbW e e  NNLO non-res + + σ e → e Im − final states. The imaginary part arises, for example, from the interfer- cuts as well as electroweak and ∼ e . − ) + − ¯ t s e ( W bW ¯ . Hard cuts over the + t and , where the on-shell X ¯ t t t − NNLO res bW ¯ b σ and W → + + The non-resonant part takes the form To account for the non-resonant production mechanism, one must embed the effective − This includes cutting nearly on-shell top lines in the effective theory, since the effective top propagator 4 e + bW ¯ ¯ and only contribute atpropagators NNNLO. in The the non-resonant construction contributiontion, implies are since that not any regulated the width by poles terms a would of finite-width have internal prescrip- to top becontains expanded the out. top This width and leads the to top singularities is at assumed to decay exclusively into which constitutes a NLO contributionterm to arises the from cross expanding section thetwo full-theory orders diagrams in in the expansion, theto NNLO the contribution process is given by theactual QCD It originates from cutsb over hard propagators thatThus, correspond the to leading the corrections physical are final from state e appears in the resonant term,strictly since on the the basis separation of intofor the resonant both virtuality and of non-resonant the is top done propagators, which in this example is small must consider the interactions of thematching coefficients energetic of initial-state the electrons. production operators. The and They furthermore also acquire receive an electroweak corrections imaginarytbW part from diagrams involving cutsence corresponding of to the process It is understood thatplitude the that imaginary correspond part to refers a hilate only the to incoming discontinuities ofsmall the relative forward velocity. am- Thegeneralized matrix from element QCD is to evaluated account within for PNREFT, electroweak effects appropriately and top decay. In addition one The resonant contribution has the form of the full NNLO correction in this paper. theory framework for the QCDThe result complete [ NNLO crossresonant contribution section can be written as the sum of a resonant and a non- already been demonstrated up to NNLO [ JHEP02(2018)125 with e e e e − cuts are b ¯ γ/Z bW γ/Z ¯ − t + tW b b 10 + 7 h t e e b h t 3 process, existing e e W ¯ tbW W W W γ/Z → → γ/Z b γ/Z − ν e e e b t e e 2 + t h e 4 W h b t W t W e e e e γ/Z ν e e e e γ/Z γ/Z b are equal by CP symmetry, hence we shall b b 9 ¯ , which must be regulated dimensionally. The h − 6 – 5 – t 2 t h t e e e e b m tW W W W W → γ/Z γ/Z → 2 γ/Z below and multiply the result by two in the end. ν ) e e − t b + e e e t + + W 3 1 e p h h b b ¯ tW W + W b t t W and p b e e + e γ/Z γ/Z e ν e e e e γ/Z ¯ tW → 8 5 − h h e t t b W ]. At NNLO real and virtual gluon corrections must be considered. While this + e W W 26 b b . NLO non-resonant diagrams. Symmetric diagrams and diagrams with [ 1 ν e γ/Z e In unitary gauge the NLO non-resonant contribution is given by the diagrams shown in poles cancel exactly the finite-width divergences that appear in the resonant contribu- e e / processes only consider the final state figure appears to be a standard NLO QCD correction computation to a 2 2.2 Organization of theWe computation now discuss the structureclarification of the of phase-space their endpointnon-resonant divergences diagrammatic in NNLO origin more contributions detail. allows intoseparation The several us determines separately the to organization divergence-free divide of parts, the the and actual sum this calculation. of The cross resonant sections and of the 1 tion. The computation ofthis the specific QCD prescription, correction required toin for the dimensional consistency process regularization, with is the the resonant major PNREFT calculation result of the present work. Figure 1 not displayed. phase-space boundaries ( JHEP02(2018)125 . ]. = → t 38 ) 2.3 (2.7) (2.6) (2.4) (2.5) , n ) with i , 37
p , ( 2 t b ) 2.4 , because b n −  a , p ym − b + 1 . . . i − ) ) W ], but a simpler − y 2 1 , since even the i = 1 manifest as i ) in the exponent,  0 f a , p b 39 p − ¯ t i p a ( p f 2 − ] as subtractions to b 1 , p (1 1 i p + ) 1 1 − − 27 e d + − , which stem from top- a,b ) d ( ix 2 t , p W π d ˆ g p + ( e (2 /m b p ) . ( θ X 2 ) b m ) =1 2 ix , i Y b b 1, where the integrand becomes − f 3 2 b p a , 0. b , p ! − → + + i X − + 1 =1 f ) → t a + a W p W y  + ) must not be expanded in W → − t t , p p m − =1 # ¯ ( t i . X ( 1 t b ix (1 , p − − g + − ) 2 t a i LIPS e i = ) . The Heaviside function accounts for the d t p a,b m 2 t ] the leading terms in an expansion around ( , p ix b ˆ g + − Z n ) are decomposed as follows: − – 6 – =1 27 e ¯ t /m i a X t p 2 (1 − ( ) = ( )

is a Lorentz scalar, i.e. it only depends on scalar 2.4 instead of → ) t b ) t ix − d p b t 2 ( f e − ix X − 2 are present, but only those with δ b + f + 2 , e + , /m 2 = (1 3 2 1 are regulated dimensionally by the 2 + , / 1 dimensional phase-space integral. At NNLO endpoint- ) ¯ tW 3 g m W dt X ≥ , =1 p → − p LIPS 1 a ( a ...f − y d d 1 e + − Z f + = 1 b ) ) that the integrands ≡ Z e t → p ( a n t ) were obtained using the expansion by regions approach [ dt 2.6 t + ix ...i ( 1 g 1 + y Z LIPS i ix " . The real corrections can be brought into the same form as ( d g W 2 t 2 t π dt p 2 ) and m ( 1 Z -integration for the expanded result is trivial, y Z m LIPS /m t f ≡ d ≡ = p 2 W ∗ ( ) t ) = t m m t ( ( dimensions with up to four external legs were obtained recently [ = ix ix -dimensional Lorentz-invariant phase space for the process d y . . . f below, where the integrand d ) dt g dt g 1 1 ix It is obvious from ( The endpoint divergences originate from the region To illustrate this issue, we consider the phase-space integral of a virtual diagram such f poles. This is related to the well-known property of dimensional regularization, that it 1 y h p y Z ( Z / 1 strategy is to take thethe results complete for integrand. the endpoint The divergent integrals terms ( from [ it would spoil thethat the dimensional loop regularization integrals oftree-level in phase-space the the integration endpoint virtual is corrections divergences. divergent.general cannot Expressions be This for scalar expanded implies one-loop in integrals in The divergent integrals with which is inherited fromdivergent the integrals with 1 renders some power-like divergent integrals finite for quark propagators becoming resonant. In1 [ of the integrands The remaining Since the bottom quark masssection can be safely neglected forthe this variable calculation, for the total cross singular due to negative powers of (1 is the f optional cut on the invariant mass of the top quark as will be discussed in section where are present in addition to the usual UV andas IR singularities. products of its arguments. This allows us to define automation tools can nevertheless not be employed due to the endpoint divergences, which JHEP02(2018)125 , ) ,b (1 ix g W b g 1 h in all possible 1 ). ¯ t t ) are available up t . Thus, the integral  2.7 − W W W b b t 2 δ c 3 1 6 g ) contributions to ˆ g h b  ( O pole in ( t t t ¯ ¯ ¯ t t t / W b f 1 h . This set of endpoint divergent diagrams code. This latter contribution will be 1 ¯ t t b 2 h δ . They are computed manually by applying W W W b b 3 b b 2 1 5 -integration on the right-hand side finite and g g h t – 7 – and MadGraph 2 t t t ¯ ¯ ¯ t t t . of the series expansion in (1 ) W aut b e σ 1 a,b ( ix h g = 1 are multiplied with a 1 t ¯ t a m t t m i δ W W W b b b − a ]. This renders the 1 4 1 g g h 27 ) cuts are not displayed. ). The remaining large number of finite diagrams is computed in an g ( b ¯ 2.7 t t t ¯ ¯ ¯ t t t − ] and are shown in figures W b tW ) from [ d 27 1 0  h ( O . Gluon corrections to the tree-level diagram t ¯ t In total the NNLO non-resonant correction requires the evaluation of the order of ways. Fortunately only about 15%identified of in [ those contain endpointthe divergences. subtractions They ( have been automated fashion using suitablyreferred edited to as the automated part allows us to expand the subtractedcan expression be in the performed square numerically. bracketbecause in the Additionally, coefficients we with require the 100 diagrams obtained by attaching one gluon to the diagrams in figure is UV and IRdiagrams with finite and will be denoted as the squared contribution.where the Symmetric required diagramsto coefficients and order ˆ Figure 2 JHEP02(2018)125 / and k ) /v + − s e e α ¯ t t b ¯ t t . † ) i ( − + a 4.2 O e 4 e h b boson exchange. There W W W W Z . In addition to the end- ν ν + − int + − e e e e σ , and is denoted as the squared + ... 2 cuts are not displayed. b ¯ ¯ t t − b ¯ t t tW -channel photon and s W W a 3 ) i + − ( h b e e with – 8 – with the diagrams in the upper panel and replacing O † ) γ, Z ) i i ( , shown in figure ( 1 W W O Re + O − h e e . The remaining endpoint divergent diagrams, shown 1 h + ) ) i i ( ( Abs,bare . The lower panel sketches the respective contribution to the t ) C ¯ t O ¯ t t ) k i ( ( 0 C C W 2 v, a + − b b ¯ Σ pole and the scheme dependence cancel, see section e e = i b b ¯ / by restoring the full theory graphs in place of the effective operators, i.e. by ∼ 3 a 2 2 times the insertion of ) h ) W k i ( ( Abs,bare . It is UV and IR finite, because it includes the complete virtual, real γ, Z Abs,bare C sq iC σ σ + − v, a ¯ t t e e ) Σ i = ( , are referred to as the interference contribution Abs,bare i C . Additional endpoint singular diagrams for the NNLO non-resonant part. This set of . The middle panel shows the diagrams accounting for the bare absorptive contribution 3 = The endpoint divergent diagrams are divided into two parts. The first is given by times the insertion of the insertion of ) i ( 0 the QCD corrections tocontribution the diagram and counterterm contributions to in figure point divergences, the interference part contains UV divergences, which are cancelled by C is no double counting, because theboth two contributions are account summed, for different the momentum 1 regions. When to the matchingcross coefficients section, where theare, ladder therefore, exchanges of of thepole, gluons same which cause order. implies the a Only Coulombsymmetry) scheme as singularities the dependence in diagram ( figure of with thereplacing a finite single part. gluon We exchange obtain contains the a same 1 diagrams (up to Figure 4 Figure 3 endpoint divergent diagrams alsocontribution. contains UV Symmetric divergences diagrams and and will diagrams be with denoted as the interference JHEP02(2018)125 . ) k 3 ( C (2.8) (2.9) (2.12) (2.13) (2.10) (2.11) . Following 3 . } i aut , respectively. An σ 5 + in figure {z . (III) ia i # and h  ) and obtain 4 aut , (EP fin) int σ 2.7 1+2 σ 2) 3 ) ,  h | + , t 2 (1 ia , + − ˆ g − )  is used unless indicated otherwise. }  2 y (1 − − res, rest (EP fin) int − σ of the hard matching coefficients ) (EP fin) int σ (1 k ) ( h σ Abs,bare t + ( C 2) , + + σ ia MadGraph (1 ia g ) ˆ g {z + (II) " k ) ( Abs,bare k ( Abs,bare – 9 – dt 4 C =2 C i X 1 σ of the interference contribution is compensated by (EP div) (EP div) int int y σ σ Z ∼ cancels the endpoint divergence of the squared con- = (EP div) int 7 GeV of + = . σ 4 =2 i  | X sq are given by the diagrams in the upper and middle panel int σ + = 4 (EP div) int ∼ σ NNLO res } i res, rest σ b (EP div) int σ = σ σ contains various contributions described in detail in section m ) k ( Abs,bare (EP fin) int res, rest C NNLO non-res σ σ (I) {z . The bottom-quark mass is neglected in all contributions except σ res, rest + σ 7 sq σ ). The h | = 2.2 , which have a direct correspondence to the diagrams from the bare absorptive parts . The computations are performed in the top-quark mass pole scheme. The results 4 NNLO 1 σ ) , where the default value k ( Abs,bare aut C overview over the divergences that appearin table in the individual partsare (I), (II) then and converted (III) to isperformed given an in IR section renormalon-freeσ mass scheme for the numerical evaluations The finiteness allowscomputational us scheme. to evaluate They each will of be the computed in parts sections (I), (II) and (III) in a different where the remainder Here, we only pointtribution. out that Thus, we can now split the cross section into three separately finite parts appearing in ( of figure this observation we split the resonant contribution into two parts, Only the first two termsbrackets, contain is endpoint finite. divergences. The The endpointically, third divergences the cancel term, endpoint with enclosed divergence the in resonant square σ contribution. Specif- is endpoint-finite butcontribution UV-divergent. into the following In parts total, this allows us to split the non-resonant is endpoint-divergent but UV-finite, and the two types of divergences by performing the subtraction ( where endpoint-finite counterterm contributions contained in the automated part. We disentangle JHEP02(2018)125 (2.14) – – – – – – – ? – , ] and depends on the scaling 2 ) – –– 29 XX XX t M + ∆ t correspond to the virtual plus counterterm m we indicate contributions that are endpoint ( – – XXX XX XX XXX XXX XX XXX XX XXX XXX XX XX XX X X XX XXX ) 6 ? ≤ ¯ t UV finite IR finite EP finite ,...,g 2 t, 1 p g ( sq – 10 – ) . Thus, a loose cut never affects the resonant σ ≤ g t ) 1 t 6 Γ 2 Z ) ) k t ) Γ ( EW QED H and k ,...,h ,...,g ( conv Abs, IS σ  C QCD 1 σ a M ) δV P-wave 1 g C σ σ σ g ( sq t h σ 1 σ ∆ σ ( sq σ . σ M − 3 ,...,h t a 1 m h , respectively. With ( ( sq with respect to the power counting parameters of the EFT. The σ sq t ) sq σ aut k σ ( M ), but we also present results with loose cuts on the invariant mass Abs,bare σ res, rest (EP fin) int (EP div) int C σ X σ σ σ − W + (I) (II) (III) bW ) denotes the momentum of the (anti-) top quark. The implementation of cuts ¯ ¯ t b . are defined in section t p ( Γ . Overview over the divergences that appear in the various contributions to the cross → t t p m − e res, rest + σ e ( in the effective field theory frameworkof has the been cut discussed parameter in ∆ [ cut is termedcontribution “loose” to when the ∆ crossorder section, where the off-shellness of the tops is parametrically of σ of the top and anti-top quark where divergent by power counting, butup finite in dimensional regularization. The contributions that make 2.3 Implementation of aThe main “top result invariant of mass this cut” work is the non-resonant NNLO correction to the full cross section Table 1 section. The contributions and real contributions to JHEP02(2018)125 t - ). b p -jet b . At 2.14 b ¯ (2.15) (2.16) p + − W p what is meant consists of two = . ) was already im- S other jets and let , which minimizes to the momentum -jet and other jets. ¯ t Ji and at least one b ¯ . We then define t A p p N S ¯ − define 2.14 , A b ∈ i p -jet, X W b , + + b ¯ − + p , one W W + . p , +

− + ) ) refers to the most energetic W 2 t = b ¯ W W p p t m ( p ], but since at this order the partonic b − = p ¯ ¯ 26 A A without reference to whether it was pro- ¯ 2 t ¯ t p X )( . 2 t − t m W – 11 – , p . The momentum of the “top cluster” and the + − S Ji bosons and bottom jets have been reconstructed. In practice, p A 2 t bW = ¯ p W b ∈A ( , respectively, of the partitioning ¯ i X

¯ A A ¯ t + ≡ p -jet) in the event, b b ¯ , one simply defines ]. We also do not discuss the non-trivial combinatorial problem of χ p A ∪ − ), if the so-defined top momenta satisfy the inequality ( 40 + and W -jet ( + + b A 2.14 t W be the set of jet momenta. A partitioning of p p bW ¯ } b bosons are considered as stable particles. = such that bosons from their hadronic decay in the presence of additional jets, since in the logic A W t ¯ ...N p A W Any jet algorithm can be used to define these objects. In a second step we , 5 A event is rather arbitrary and a matter of definition subject to the previous two = 1 X , i -jet. b ¯ − Ji by the value of p W ¯ t { p + To implement the second step, assume that the event contains In the following we describe an algorithm that satisfies these requirements. The algo- Since the physical final state is We assume here that the charges of the = 5 -jet), and the remaining ones are considered to be part of the set bW ¯ b ¯ this will be inefficientconcluded although a that Monte the Carloof discrimination study for 80% between the bottom at topreconstructing and about forward-backward the asymmetry anti-bottom 60% at jets efficiency theof can ILC [ our be discussion the achieved with a purity ( and the product An event passes the cut ( “anti-top” cluster for a given partitioning are defined as If there is more than one since close to thresholdany in other an particle can event be with at a most resonant of top orderS and Γ anti-top, the momentumdisjoint of sets For the purpose of thisjets”. discussion energetic leptons We and require photonsand that are one also the among event the contains “other group exactly the one above pre-definedclusters objects define into the exactly (anti-) two clusters. top momentum. The momenta This of fulfills these the two above-mentioned requirement, requirements. Because non-resonant production isdiffer a only sub-leading by effect, small different definitions amounts. rithm is most likely notcluster optimal any and hadronic or serves partonic only event as into a the proof objects of concept. In the first step we is required at the hadroniccalled and “top the momentum” partonic should level.of be Any the sensible equal definition top up of quark toshould an an in observable also amount the lend of hypothetical itselfhand, order limit to Γ the that simple, assignment the of infrared-finiteb top top theoretical quark momentum computations. were to a the On stable final the particle. state other of It a non-resonantly produced duced through an intermediateby the top (anti-) or top anti-top,plemented momentum. it in An the is invariant NLO necessary massfinal non-resonant cut to calculation state of [ is the form always NNLO ( the final state may contain an additional gluon and a definition of the observable JHEP02(2018)125 , 2 t ym Since (2.17) . Con- t 6 Γ t m to form the b ¯ -jet and gluon ) might not be b − ¯ , W − 2.14 -jet momenta. The b W pair is larger than -jet as above, but in this and b ¯ b ¯ − , the misassignment is irrelevant t W W final state from above can be ). The other partitioning where ) and then satisfy the loose-cut , bg ¯ . (With obvious modifications the by an amount of order ) decay products are automatically 2.4 b b ¯ ¯ t 2.16 2 t ( − − 1% . This possibility is suppressed by the . t t m -jet, the set of partitionings is empty 0 W b ¯ Γ in ( t t W + , and the other, where it is not. The first, ≈ b χ m at NLO, and at NNLO the previous and p W - or 2 b 3. We can therefore safely neglect this error. b + π . + 4 0 + πR -jet. The first factor represents the suppression W – 12 – ≈ b ¯ ¯ tW × p , ) R b ¯ t ) and hence the event always passes the loose cut as ¯ t − only if the invariant mass of the m p π ( ( χ final state, which appears at NNLO in the non-resonant s t tW α p bg ¯ is the sum of the appropriate -jet. In this case the loose-cut condition ( − within an amount ¯ t b and . Hence, imposing the cut yields a single Heaviside function p 2 t 2 t ¯ t from anti-top decay, in which case the jet algorithm merges it t W m t ) means a final state that can originate from on-shell (anti-) top b ¯ ¯ t /m and ( 2 ) t t t p ) in the phase-space integral, as in ( M 2 t ∆ ym final state. One might think that the − . Here ¯ t t − t only at LO, bg 2 m ) + ¯ t ( b ¯ t p -jet rather than the is the half-opening angle of the ≡ b ¯ tW ¯ + , y R − bg ¯ Finally we discuss the Next we consider the non-resonant final state We now apply this definition to the partonic NNLO calculation. The partonic final W If the gluon is not energetic but ultrasoft with momentum of order Γ − p 6 (( part. Atapproximation. NNLO, it Hence, is the sufficient discussion to of consider the and the the resonant loose-cut condition top remains satisfied. quark decay in the tree anti-top momentum can minimize jet accidentally adds up to NNLO probability for the processfraction to where happen the in kinematic thebe requirements first neglected for place at misassignment times NNLO. are the satisfied, small phase-space and hence can (correctly) added to the top decay,correct, i.e. to possibility will almostcondition, always whenever minimize the invariant masswhere of the non-resonant θ the additional gluon jet is incorrectly combined with the non-resonant and the definition of loose cut is passedcase except the when the probability for gluonnon-resonant is this production misassigned to in to occur the the is firstan even place. additional further (gluon) If, suppressed jet, on there due the are to other two the partitionings, hand, one the suppression where jet of the algorithm gluon returns jet momentum is The numerical estimate is obtained for following discussion applies to thetop CP-conjugate decay final must state.) beis To taken NNLO in no accuracy, NLO, gluon on-shell and or may the contain gluon an is additional merged gluon. with Whenever the there of order where of energetic, large-angle gluon radiation and the second the jet area on the unit sphere. desired. However, this is notradiated always collinear true, to since the aninto energetic the gluon from topsatisfied, decay and may the be event isour missed calculation even does though both notit tops include for were the produced this kinematics resonantly. misassignment. of resonant However, top the decay, probability we for cannot such correct a misassignment is at most states are tW decay at the given order, withsider first invariant mass the within clustered into the correct parent JHEP02(2018)125 . L + − pair. (3.1) W NNLO res ]. The b ¯ p σ − 41 (( [ θ ) from the W c (¯ σ , MadGraph and the correlation Γ)] i L + E ( 0 G with overwhelming probability χ Im [ ) of the external momenta, which i i p 2 ) ( a c ) is implemented in the real-emission and Γ is the on-shell top-quark width ) is loose, the complementary-cut cross ( 0 i t C p ( m 2.14 c + 2 2 ) − v ( 0 s – 13 – C ) is fixed by the existing QCD results for √ h ). The former is evaluated in four dimensions; the ] 2 c = q 2.13 32 s E πN ). 24 contained in part (II)), and we consider them first. 1.1 0 σ ). Assuming that i p = ) ( k ( Abs,bare c C LO res − , and the same conventions must be applied. We compute this part σ σ dimensions in the naive dimensional regularization scheme (NDR). H ) is the high-energy limit of the photon-mediated muon pair produc- = d s ) = 1 ], but the generalization of this method to NNLO accuracy as discussed i (3 cross section, where the cancellation of divergences between the resonant LO p / σ ( 42 2 ). c 2 t bX ¯ b to perform a powerful check of our computation. . The electroweak NNLO corrections to the resonant part must also abide by πα − ym ) is purely non-resonant and free of endpoint divergences. It can therefore be c = 4 6.1 3.2 − W (¯ 0 2 + σ ) σ g W p For ease of reference and to set up notation, we briefly recapitulate the well-known A generalization to arbitrary cuts would affect the resonant contributions and is beyond We have not implemented other cuts, but it is in principle straightforward to do so as + b ¯ where tion cross section at leading order, Electroweak corrections to the resonantEFT cross framework section extended are from computedNNLO QCD in in the the to non-relativistic counting the scheme full ( Standard Model.expressions We for consider the LO them cross to section [ The squared contribution containedand in a part hadronic (I) tensor factorizesin into section the same leptonicthis tensor scheme (except for 3.1 Resonant electroweak effects The scheme for part (I)The as defined resonant in QCD ( crossfunction of section two factorizes top-quark currents, intolatter Π( a completely in leptonic tensor cross section with NLL accuracy nearbeen threshold presented matched to [ the NLOhere fixed order is result not have straightforward. 3 Part (I) total and non-resonant parts has already beenin taken section care of. This approach will also be exploited the scope of this work. Recently, first results of an implementation of the fully differential long as they areevaluates loose. to one A if general thementary cut cut event as is passes ¯ a the function cutsection and to zerocomputed otherwise. with We automated define NLO the partonnon-resonant comple- level contribution event with generators the such original as cut is then given by subtracting is the correct combinationHence, of up the to gluon a jetphase-space negligible momentum of error, with the the the NNLOp non-resonant non-resonant loose contribution cut as ( the Heaviside function repeated, except that now the partitioning that minimizes JHEP02(2018)125 , 2 t Z = 7 m 2 w (3.8) (3.9) (3.2) (3.3) (3.4) (3.5) (3.6) (3.7) s ( = 4 -channel 2 Z has been s s , /m Z 2 . 2 W πα/s are set by the 2 Z m m s 2 m − s n = s − χ t  s 2 w is the fermion electric v , χ, t c e and D  f a w a · , the energy-dependence e e c f 1 3 D   a − are the sine and cosine of , w E n ) σ t T · s f t ) i w = = i 2 m σ t c denotes the effective field (as n ) ) 2 − -dependence in the i a ( m i = ( 0 s c + ¯ , 2 − f e k χ, ( and x a ) k , σ ( P-wave = 2. The factor of 4 , a i k  χ, σ w ( 0 c i ≡ † † k σ s ¯  A n ). σ ψ ψ Z f · † † · 1 1 i i c c ψ ψ 3.8 n e e ¯ n 1 1 1 1 c c † † c c boson are given by e e dt 1 1 † † W W c c Z 0 ) to ( 5 5 with , a Z W γ W γ i −∞ 2 w – 14 – k k k k 3.3 n ,C s ie γ γ γ γ f 2 2 2 2 Z 2 w   c c c c e c 2 m w W W W W s ,C s 2 2 2 2 − − c c c c 2 2 Z ¯ ¯ ¯ ¯ e e e e f s 3 m boson propagators could be expanded around t T s v s s s s e πα πα πα πα − Z ) = P exp = v 4 4 4 4 s x t f ( + a i v = = = = t c e e ) ) v ≡ e v a W ( ( e − is summed from 1 to 3. Z . In the present context, the directions f ) ) O O µ i v a v k = = ( ( P-wave P-wave n ) O O v ( 0 and )) scheme. The Weinberg angle is then given by C k Z ) γ P-wave , v ( m 0 − ( C = ) is the non-relativistic top (anti-top) field and is the third component of the weak isospin of fermion , α k χ γ f -channel photon and Z 3 ( s T ψ , m The renormalization scheme for the electroweak parameters adopted here is the Note W 7 m However, we apply apropagators, which convention therefore where appears we in keep ( the full ( absorbed into the operators toP-wave render production the operators coefficients and dimensionlessthat and their because of the Wilson order cross coefficients one. sectionof will is The the constructed be as required anin below. expansion which in case Note the short-distance matching coefficients would be truly energy-independent. have been introduced to make the operatorsin invariant SCET, under collinear as gauge well transformations as the light-like vectors ¯ Here defined in soft-collinear effectivelight-like theory direction (SCET)) ofelectron an and energetic positron electron beams, respectively. moving The in collinear the electromagnetic Wilson lines with leading-order matching coefficients charge measured in units ofthe the Weinberg angle, positron respectively. charge, The and S- and P-wave production operators are given by boson. The couplings of the fermions to the where as defined below. At LO the top pair is produced via s-channel exchange of a photon or JHEP02(2018)125 ) ) ) k w b ¯ ( ( λ − L C (3.12) (3.10) (3.11) tW ( ]. Higgs refers to 8 ) and ). b α 22 E + ( . (0) and the ) w α ( 0 ¯ tW scale-dependence 3.1.5 G conv IS r , = σ µ  0 , where ) ) appears. +  2 α α ( t to the resonant cross Z O (section ) Γ k and write ( σ Abs, ) + C w mE/µ conv IS λ µ 4 σ σ − was computed in [ = − + ]. The imaginary part is split µ ) ln( (1 k 46 ( EW ˆ 2 1 ψ C − σ P-wave − of the LO QED Coulomb potential σ + λ ) defined through the photon vacuum 1 2 Γ α )) = σ µ QED F F − ( E/m δV + C C α 1 2 s ) and a contribution from field renormaliza- s σ − α α + t has been obtained in the formalism employed p from now on is denoted by cross section [ QED λ 4.1 2 – 15 – m δV − ( L from the real part of the hard matching coefficients em σ = α ) QCD + W k + λ σ ( EW λµ/ +  ], 1 C 4 is the logarithmic derivative of the gamma function. H σ (section  σ 44 bW ¯ from the NNLO resonant contributions as explained ), there are electroweak corrections to the hard matching ψ , b F ]. + = ln( 43 C λ s 22 → π ) , L k α 4 3.1.2 ( Abs,bare ) 2 t − 14 k C ), because the two parts are treated in different schemes. Partial e ]. At NNLO, top decay introduces additional contributions to ( P-wave Abs,bare m σ + C σ . Contrary to the QCD case the electroweak hard matching coeffi- 25 ), where e ]; similarly the effect σ ). [ x + Z Γ) denotes the non-relativistic zero-distance Coulomb Green function ( ) = 2 ). While there are no electroweak contributions to the non-relativistic 3.1.3 i 45 ], but they only contribute at NNNLO and will not be considered here. ψ m q , E ( + / ( 3.1.4 that only involve the top Yukawa coupling have been computed already QCD 2 t 0 α + 25 47 E , σ pole is related to a finite-width divergence, we set . The contribution 3.1.1 H G ( ) E 0 σ k 45 γ = / παe ( (section ), the remaining parts are G 4 C t ) ]. Top-pair production in a P-wave state − Z ) 2.12 ) = 34 3.1 k = ( x Abs, res, rest ( ). The electromagnetic coupling C σ ˆ ψ 2 w σ After separating In ( c When the 1 8 QED − are available [ Finally, we consider effects from initial-state radiation (ISR), to distinguish the finite-width scale-dependencedue of to the the resonant strong contribution coupling, from cf. the [ cients contain an imaginary part fromcuts cuts contribute over all to possible the finalinto states. a The bare contribution tion results for the mixed-QCD-electroweak corrections to the hard matching coefficients the bilinear part ofsection the (section PNREFT Lagrangian, whichpotential contribute at NNLO (section coefficients is given in section The pure QCD S-wavehere contribution in [ contributions up to NNNLOδV [ around ( and Furthermore, the logarithm which describes the propagation ofIt the top-anti-top is pair expressed at through LO the in variable the non-relativistic EFT. the scale dependent electromagneticpolarization, coupling which interpolates betweeninput parameter the fine-structure constant in dimensional regularization [ 1 JHEP02(2018)125 ) χ the 2 ∂ † (3.14) (3.15) (3.13) χ ), which − defines ψ 3.16 2 ∂ † ]. Electroweak ) complex. The Γ ψ χ, 49 σ χ, ] implies that this i 3.1 2)(  ψ ) contains the term 50 Γ, which  ... i X/ ... + 3.14 ... + ) + E 0 + 2 t, 2 ( ∆) ) and ( → ) t γ χ 3 t ∆) ( 0 t † m E 3 t m χ m ) can be removed by a field trans- 8 eA − m t − 8 2 − e ]). The generalization of the bilinear h ∂ 2 ψ Γ (3.16) 3.13 ( † † ∂ 48 χ ( ψ , im − Γ + 4 to the right hand side of ( + i − 2)( 2 / ∆ ψ 2 2 − 2 ∆ i Γ X/ + m ) belongs to the LO Lagrangian and must be ] t − – 16 – − 2 ... = t ∆ = m 36 ∂ 2 2 , ? 2 3.14 m ∂ 2 ) + 35 M − 0 0 + t, 0 ( ) in ( i∂ ) correction to the Green function in [ χ  γ i∂ † ( 0 † 2  ) that contain the width are of NNLO and can be treated χ χ † eA t ψ + − e 3.14 ) from the terms ( h ψ = † † E ), which has the form of a mass shift. It can be absorbed into the ( ψ ψ 0 χ † plane. We note that with this convention ( G = 2)( χ ) / 2 bilinear γ Γ ( us p − L i L ψ † ψ ) is the non-relativistic top (anti-top) field and ∆ is a hard matching coefficient. )( t χ ( m ψ ]. The absence of the Γ (8 The term ( / 50 2 Γ treated non-perturbatively. ItQCD leads contribution, to and the makes replacement thetwo argument remaining of terms the in Greenperturbatively. ( function Only two in simple ( singlethe insertions are Green required. function We denote the correction to corrections to the top-pair production crossin section near [ threshold have also been considered different convention is alsodifference adopted in there. the definition Thus, of one the must top be pole careful mass to when account comparing for their this results to ours. in the complex − definition of the polecompletes scheme the by square adding and defines a different convention used e.g. in [ where Γ is the polethe width pole of of the the top top quark propagator defined through the gauge-invariant position of where It can be determinedtheory. by In matching the pole the mass top scheme propagator we in obtain the effective theory to the full formation involving an ultrasoftpart Wilson of line the (cf. PNREFT [ Lagrangian is [ is relevant at NNLO. However,field its can contribution only vanishes, because resolveanalogy the the to multipole-expanded QCD, net the electric couplings charge in of the the Lagrangian ( top anti-top pair, which is zero. In Additional terms appear in the PNREFTand Lagrangian its due to coupling the to instability photons.multipole of the expanded The top in quark coupling the ofgluons. spatial the component, top Only just quarks the like to leading the term ultrasoft interactions photons with must be the ultrasoft 3.1.1 Finite-width corrections to the NNLO Green function, JHEP02(2018)125 + E (3.18) (3.19) (3.20) (3.21) (3.22) (3.17) → ] the top E 32 , poles. These ,   )  / ) ) λ  λ code [ 2 W σ. ) − x − /  W E (1 . − Γ ), which multiply the γ x (1 1 2 t δ e ˆ Γ ψ ψ m i − 2 w 2 3 2 µ 3.20 λ = (1 4 − , 2 ln(1 2 t X ) Γ)] + | i threshold i ) + 2 w m − with a finite part that follows ( λ 2 E + 2 t  L ( 2 w 0 λ 2 ) µ m E G )  P-wave ( , a 2 + 2 ( 0 + 2 QQbar ∂ − G 1 2 C λ ) terms of ( X Γ π 2 , + ln   δ 2 / ( + √ + ) δ F + O  W W 1 C t 2 dimensions, is given by s 2 x x π 2Γ(3 Im [ m Γ α 4 2 4 d + ) i ) 2 P-wave , v − 2 ( ) 0 W λ m = 3 a 4 – 17 – 1 + 2 x ( C 0 X 2(1 + ) ) and the one in the pure QCD result, to be added 2 F |  C )  − + -dimensional hard matching coefficient. Hence the C 2 s c d 2 E −  + , ). For consistency, the tree-level contribution to the X ) ( 3.19 0  N a 2 F ). On the whole, we obtain  ) ( ) 0 G F ( mα v C λ ( C 0 s X π C O δ W s C α 4 − s 3.20 + h 2 x denotes the scale related to the finite-width divergences as α 9 ) = 2 0 c (1 ) (1 + 2(1 m E w ) = 1 v Γ 2 ( ( t 0 ) µ ψ 0 0 s E πN ), respectively. They are given by C ( m 2 W λ h mX E x 24 G 0 + ( − XG 0 Γ × σ and , − σ G 2 2 t -dimensional tree-level expression of the top width contribute finite 0 limit of ( 2 δ = ) is the first derivative of the polygamma function. The corresponding ]. The contribution from the d ∂ (1 = ) = ) = x σ → X /m ( 2 w Γ α 0 E E δ 22 s t  / are also treated non-perturbatively through the replacement σ ( ( , 2 W ψ  ) must be treated as a ]. Γ m G G 16 0 m 14 δ 2 X ∂ 52 ) = δ = , = ) and 3.14 X x is the ) 51 δ ( d E 0 ( 1 0 W ( x Γ ψ G The LO pole width, which is required in In the implementation of the cross section in the X ) terms in the QCD corrections to the width, however, are only needed in four dimensions where they are known up δ  9 ( Γ. A subtlety arises at electroweak NNLO when this result is combined with the non- where Γ to NNLO [ to the cross section is where discussed in [ finite-width divergence contained in ( terms to the resonant partfinite from parts their are multiplication not with included theand finite-width when must 1 Γ be is added treated separately. as a four-dimensional numerical parameter, resonant contribution, which is computedthe in NNLO dimensional non-resonant regularization. contributionfrom is The expanding proportional pole diagrams to part up Γ of width to in ( O width is treated aslevel a width parameter. Γ i This implies that higher-order corrections to the tree- where NNLO contribution to the Green function is by JHEP02(2018)125 (3.26) (3.27) (3.28) (3.29) (3.23) (3.24) (3.25) (3.30) ]. The 53 ) the Green 3.27 i , ˜ m | ! ) , n k ˜ E ( n F = ]. The pole position and residue i =1 ). After applying ( ∞ k ˜ X ] 14 m | + regular 53 † . 3.26 , ) n 1 + † | 2 0 ˜ Γ n

H ( i mn E 2 ˜ δ ∗ n i h

is the bound-state energy assuming stable , = − ψ ) is exponentially growing at the same rate as n ) − | = ) k n 1 0 10 n i ( n 0 ( n i − ˜ E n ( E E n E | n – 18 – n X (0) n | = ψ =0 ) interactions. ∞ ψ ˜ = k X m − n h THT l n ). The implications have been discussed in [ ,H E 1 = + i l = →E n ) = n | E 3.14 0 E → n ( ) ˜ ∗ E n ¯ t t is the total inclusive width of the bound state. In accordance ψ E = ) ( n 0 i G ( Γ n δ n | ψ H ) implies that the state = 2Γ+ ]. This implies, that the completeness relation must be modified. We n 53 3.25 . Assuming that the Hamiltonian transforms as ∗ n E ) or electroweak (e.g. = ) have the following perturbative expansion n ˜ E ggg describe the effects of time dilatation on the top decays due to the residual movement The eigenstates of a non-Hermitian operator do not form an orthogonal basis of the In the numerical evaluation we resum the perturbative corrections to the would-be 3.28 is decaying, which facilitates the normalization ( We do not distinguish between bound states and continuum states, since this is irrelevant for the n → i 10 Γ ¯ t n discussion. which generalizes the expressionof for ( the QCD result [ The property ( | function takes the following form near the poles which implies that the completeness relation takes the form with under time reversal, andnormalized that such the that eigenvalues they are form non-degenerate, a bi-orthogonal the set eigenstates [ can be t Hilbert space [ consider the sets of right and left eigenstates of the non-Hermitian Hamiltonian,top quarks where and Γ with the earlier discussion, the top-quarkδ width Γ is treated as aof parameter. the The top corrections quarks and the annihilation of the would-be toponium state through strong (e.g. toponium bound-state poles. Duea to non-Hermitian the Lagrangian, instability cf. of ( positions the top of quarks, the we would-be are toponium dealing poles with are the complex eigenvalues JHEP02(2018)125 ). x ( (3.35) (3.36) (3.31) (3.32) (3.37) (3.38) (3.34) (0) n but not ψ . ] we apply (0) n E ) = 32 x , ( ˜ (0) n !# + regular ψ # ... expanded + ,

Γ ) ! i 0 2 ( ) − E ˜ ∗ n E ψ 2 F − ) − C 2 0 n (1) 2 n 2 s ( ! E n E , n α (0) 4 Γ (3.33) n t ). Since it is non-Hermitian, it also E , i ψ 2 F E ( 3 m 2 . − − C

], respectively. Using this input we can (1) t n 2 s n  , − 3.36 Γ + E 4 α i t − m 25 F (0) n 2 3 Γ 2 2 = E ) C m Γ (1) / n s n 4 0 − − Γ = ) ). In the actual implementation [ E ( 2 i − α † 0 E t E ˜ ∗ n = = = – 19 – (1) H − n ψ ) alone in the electroweak contributions, but to ] and [ m (1) − − n ) has the form of a mass shift and therefore leads ). ) 3.34 F E − 0 F  n 14 (2) (2) n (2) n (0) ( n n ( (0) E 0 + n Γ

1 π F n E ) to the position of the pole, while it does not affect E G 3.14 E 3.37 Γ H Γ Γ ) accounts for time dilatation, which reduces the total ψ δ ( δ (2) δ n = = " E 1 + 3.35 2 " | 3.14 n (0) − ) 2 n X | E 0 E ) ( (2) 0 n term in ( − ( ) + (0) F n 2. At LO we have made use of the relation 2 / E ψ (0) n

(0) ) n | ( k ψ E term in ( ( n | + G Γ 2 . ∂ → iδ (0) (0) n Γ ) to the Green function ) i i − = n E →E | ) ( k ), but contain higher-order poles E 3.19 ( n G ) E E 3.28 ( = G ) -independent correction ( k n ( n E The corrections to the bound states from QCD effects as well as the QED Coulomb The results for the non-relativistic Green function in perturbation theory do not take where the expanded term hasthe the pole form resummation ( procedurethe entire to current correlation functioncontributions. of vector and axial vector currents in the pure QCD and Higgs potentials canresum be the found higher-order in poles [ in the expanded Green function by the replacement As discussed above, the Γ to an the residue. The width of the would-bemakes toponium the residues resonance complex by due ( to ( This allows us tocontribution read ( off the NNLO correction to the bound state parameters from the the form ( This holds, because the non-Hermitian part of the LO Hamiltonian is proportional to thethe identity eigenstates operator and thus only affects the eigenvalues and with the LO expressions JHEP02(2018)125 0 . The → = 0 5 q are shown in vertex correction α OS EW s ¯ t . 1) α α γt s q − α t Z ( 0 → = 0 + q + t OS QCD potential at order W potential at order 1) q q 2 2 − q q / b t / Z ( + vertex correction and the QCD t – 20 – ¯ t gt + ) the PNREFT Lagrangian contains potential inter- + t Z 3.14 q q + are hard, while the momenta of the external particles can be . Contributions to the 1 vanishes. We have not drawn the four diagrams that involve soft 5 t 6 + t Figure 6 γ q . Cancellation of the electroweak , where the momenta of the external top quarks are potential and the loop mo- 6 t The potentials are determined in the matching procedure between NREFT and PN- vertex corrections to thevanish tree-level in potentials, dimensional because regularization. theseinsertions Last, corrections of but are the not scaleless one-looppotential, least and because there corrections are these to no coefficients theQCD-electroweak contributions vanish potentials hard as from appear matching argued at coefficients above. NNLO. in We conclude the that tree-level no mixed REFT. The diagrams thatfigure contribute to thementum is 1 soft. Theexactly contributions opposite of to the thosethe first of diagrams and the in second third figure diagram and are fourth identical, diagram, and which are implies that the sum of of the QCD vertex,loop and momenta vice in versa. figure either soft, The potential or relevant ultrasoft diagramsthe and must vertices are be are shown expanded effectively in outcontribution evaluated of figure is at the exactly loop zero cancelled integral. external by Therefore the momentum, on-shell and external the field corresponding renormalization factor. In addition to theactions. kinetic terms We ( nowThe show that construction there of are PNREFTscale, no proceeds which mixed by yields QCD-electroweak first NREFT,the corrections integrating first and at out step, NNLO. then fluctuations one integrating must at consider out the electroweak fluctuations hard corrections at to the the hard soft matching scale. coefficients In 3.1.2 Mixed QCD-electroweak NNLO corrections in PNREFT in the on-shell scheme. The vector current itself is conserved and therefore not renormalized. Figure 5 JHEP02(2018)125 ] ]. 54 54 (3.40) (3.39) and/or µ I ..., is the gluon loop correc- + k + π α 4 . bW  EW ) k α ( , where Abs | k iC t | , i.e. at NNLO. We have Z / dimensions in accordance + 0 ) d EW Γ k ) ( Abs, α k s ( C in momentum space. Thus, it EW α σ C 2 t √  + /m  vanish. potential at order EW 5 ... 2 t α + scales as γ, Z /m µ vH (2) I c 2 t 2 compared to the LO colour Coulomb potential y – 21 – 3 v + + 2 ∼  become complex at NNLO due to s ) π 2 t Z ) α 4 k (  /m C 2 (2) v q , which corresponds to the cut to the right of the gluon of c ( µ . Their result for I + . × and is proportional to 5  ) 7 s s . Contributions to the 1 π ] that they vanish at NLO. In their calculation, the authors of [ ) and, thus, it has to be determined in α 4 /α ) 54  vanish. In our approach, where loop integrals are strictly expanded on the other hand, is contained in part (II) and therefore has to be k comes from the region of hard loop momentum, where no inverse ( Abs,bare ) 3.12 EW µ (1) v µ C k c J ( I α + C Figure 7 in ( t 1+ can appear because the external momenta are expanded out. Therefore, the ,Z  | ) ) k k k ( ( 0 Abs | res, rest C C σ = = ) ) Finally, we comment on the so-called ‘jet-jet’ interactions that were considered in [ Furthermore, we demonstrate that the potential does not receive any pure electroweak k k ( ( Abs C C are expanded as tions. The imaginary partsection contributes to the finite-widthwith divergence of the the scheme resonant usedcontribution cross to to evaluate the othercomputed in components a of different scheme. partwe Since (I). find the The two it parts bare convenient are to absorptive treated separate using them different in conventions, notation. The matching coefficients up to NNLO already proven that there areelectroweak also corrections no to ‘jet-jet’ interactions the at QCD NNLO vertex by in demonstrating3.1.3 figure that Absorptive part fromThe field hard renormalization, matching coefficients according to the scalingto of the the subgraph momentumpowers regions, of the onlyabsence non-vanishing of contribution any ‘jet-jet’implies interactions that at corrections NLO can first is appear a at matter the of relative order simple power counting, which first consider the subgraph the third diagram in figure three-momentum, which isby either explicit potential computation or thatits ultrasoft CP all in conjugate NLO their corrections case. that involve They the then subgraph show is suppressed by ( and only contributes at NNNLO. These are corrections involvingwas gluon demonstrated emission in from [ the final-state bottom quarks and it NNLO corrections from thehard exchange of and Z therefore bosons. haveThis We to implies, count integrate the that out massbecome all the of local Z the interactions in Z boson that PNREFT. bosons in arediagrams The as shown Z-boson the mediated in exchange hard by figure potential matching the corresponds to to Z NREFT. the boson full-theory in the full theory JHEP02(2018)125 has ) ) k (3.44) (3.43) (3.45) (3.41) (3.42) 2 (  C ) k ( EW + 2 C ,  ) of the hard i σ 5 . )) i ,  − ). We therefore i )) 3.40  − (2 Γ) i − W (1 ) + 3.1.5 x 2 (1 boson is of NNLO and 2 W )  x + E 2 W Z (  x ) − , w or ) ( 0 −  (1 ) + 2 ) are included. ( γ 2  G ) + 2 2 W h h 2 O  x 2 )) )) 3.43 − + vertices contains divergences that divergence in the real part of the Re   . From the field renormalization, ] − i . − (1 t ]. We have repeated the calculation / − − 46 e ) ) + 2 (1 k W 2 ,Z + 4.1 -channel ( WZ 46  (1 (1 x ) s W ]. The box diagram with a photon and a C ) and ( ( Ze W W Abs x from [ , does not yield a finite-width divergence 58 + 2 x x + )  C , separately finite. There is no contribution ) k  ( 3.42 EW a 5 )(1 + ( and 0 )  C k )(1 + conv C ( IS 2 QED − −  σ e bW,abs V,A C – 22 – 2 + )(1 + 2 )(1 + 2 − (2 + C t ) − = W W  ) d W γe ,Z ( 0 and x x d x ) ( 0 2 t ) ))(3 (3 Γ k 2 ) v Γ ( − − EW ). ( 2 w k Abs π + m ) parts of ( W α ( EW π s  4 C  C t x − C ( ) e )(1 )(1 σ v 3.39 − 2 = − O ( 0 W W ) C − x x (4 will be given in section h )(2 e 2 v,a ( c e Abs ) − − 2 w ],  s ]. Pure QED corrections have been neglected there. Therefore, C 2 w 2 w e 46 s c c αN e − 57 ) (1 + – k (4 (4 12 t ( (1 Abs,bare ) with the 2 w 2 w e 0 55 C 2 W 2 w σ αs αs s x (1 + 4 ] differs from ( t t to render both, h 2 3.10 = m m × − +2 46 t Z conv IS = = ) σ k t t ( Abs, C ,Z ,Z σ ) ) a v ( ( Abs Abs C C The bare part boson is considered to be a non-QED correction to the photon-exchange contribution assign it to from the box diagram involving twoof photons, the since top only pair its through interference the withthe vector the correlator component production of of the threeZ vector currents vanishes [ been computed in [ we split The hard QED vertex correction to the cancel among initial-state radiation (ISR) contributions (see section Green function ( 3.1.4 Electroweak contributions to theThe real hard part matching of coefficient, the electroweak contributions to the NNLO matching coefficients The contribution to the NNLO cross section is given by where the finite terms from the multiplication of the 1 we obtain where the normalization factorcoefficients is in [ necessary because the definition of the hard matching at NNLO. Thus, it is notmatching necessary coefficient to is split it available as inof well. four The the dimensions absorptive [ part individual ( contributionsreproduce the in result the of schemes [ described above. In four dimensions we The real part of the electroweak corrections, JHEP02(2018)125 0 ], 0) α , 60 2 t → (3.49) (3.46) (3.47) (3.48) m α (4 appears, 2 α AA R /µ , ], i.e. . In the limits ) coincide with α 0) 57 , µ , – 2 t 3.47 55 Γ)] m i (4 . This scheme suffers + 0 ) differ, because we use , α AA R E α , ) and ( ( Π ) 0 → 3.47 2 α t 2 t 0 G e ) as a dimensionful quantity α e m 3.46 s

( ) s, µ 0 ( 2 α Im [ π e ) is defined in the scheme of [ α µ AA 2 α ]. The matching coefficients given i ) and ( ( AA R + ) , 57 a Π ( EW AA s, µ – ]. The explicit factor 1 ]. t ( (2) vH (2) vH 3.46 C e Π c c ) 55 e 55 57 a 2 α AA R 2 2 t t s ( 0 2 2 ]. The subtraction terms are present be- µ α s y y π e C 4 57 − π π α α + 4 4 ) − s . ) ) ) ) ) ( v a – 23 – v is the fine-structure constant and Π ( ( v a ( 0 0 EW ( ( ] converges to the scheme of [ 0 0 0 from [ C C C AA C C ) 60 α 2 t 2 t v − − ( 0 currents and the photon self energy, which yield e e AA R . Thus, the only pure QED effects are the hard photon ], 8 8 C ¯ t ) is taken from [ h k − − 57 ) = Π ( WZ c Zt = 1) = 1) 2 α C = = ] and the photon self-energy is contained in the QED con- AA ν ν s ( ( αN 25 s, µ and ) ) ( ew ew a v V A 12 ¯ t ( ( QED QED (0), and the self-energy terms in ( C C in [ 0 AA R C C γt 2 t 2 t σ AA Π 2 2 0 0 the scheme of [ H , ). 0 m m α α = σ 2 t 4 4 Π ) ]. In this scheme the renormalized photon self-energy takes the form m k 3.47 ( EW = = 4 → 60 C ) ) 2 α σ v a = 1) is given in [ → ). Corrections that involve Higgs couplings to gauge bosons or Goldstone ( ( WZ WZ /µ s C C ν ) ( ] defines the photon vacuum polarization Π 2 α 3.46 µ ) from [ 55 ] are expressed in terms of the fine-structure constant ( ], the renormalized photon self-energy Π ew V,A α 57 C µ 59 0 and AA ( – α The cross section receives the contribution We note that the photon self-energy terms in ( 55 → ≡ α from the electroweakoperators. corrections to the hard matching coefficients of the production because [ and does not imply aµ power-dependence of the crossand section Π on the scale each other and with the respective expression in [ α where the bare self energy Π in [ from a large spurious dependencethe on self-energy the corrections light fermion tois masses, the expressed that matching cancels in explicitly coefficients termsconstant. with when of Therefore, the a we fine-structure less write constant infrared-dependent the cross definition section of in the terms electroweak of coupling the running on-shell coupling separately as part of tribution ( bosons remain in ( a renormalization scheme which is different from [ where coincides with the expressioncause for Higgs effects Π which only involve the top Yukawa coupling have already been included As in [ and will be discussed below. The non-QED contributions are vertex correction to the and is therefore already part of JHEP02(2018)125 . . , i ], t )  ¯ 3 n e , m 28 ] at v,a i ( 8 n to all (3.50) (3.51) ∼ . C k k  . Virtual )) 2 2 e -dimensional 8 6 π d 7 s/m − ]. In fact, the . ] by substituting ln( = 1) and positron + 8 29 29 Γ)] α , i i 11 2 t ) from the 2  28 + m µ − 4 E ( 0  G cross section. We extend the 2 ] in DIS kinematics. We only 6 + 3 ln ¯ π t t 2 t 61 2 ] Im [ m µ 8 + 4 ) 2 2) familiar from SCET, where − a , ( γ/Zee  3 C ) + ln a = 1 − ( 0 i  2 2 ( C  2 t 2 ] contain an extra factor (1 v + − m µ t 4  29 – 24 – , = 4 dimensions as described at the beginning of section m = 2 defined by the electron (  ) d i 28 v  ∼ ( γ/Zee ¯ n 0 · i C ⊥ ) i i conv IS v − 3 + 2 ln n ( 0 σ 2 , k 2 t  2 C µ [ v pair production near threshold in [ m  1 t c 4 m + − W s πN 2  ∼ 2  48 π . The correction to the cross section from hard ISR is k  α 0 4 · , momentum regions, and in addition two hard-collinear momentum − ) 2 σ π i α v 4 W t v,a ) = , n ( 0 + t 12, and by adapting the different tree-level process. m C v,a / m ( 0 2  (H) IS ∼ bW ¯ π C σ b ∼ k − vertex correction which contributes to the hard matching coefficients of the electron beam. No ultrasoft corrections that couple to the collinear and k 4 + · 1 = Re = i n n ) → − γ/Zee v,a ( γ/Zee The contributions from the ultrasoft momentum region are shown in figure When the electron mass is neglected the ISR contribution involves the hard, Compared to our results the expressions in [ C fin) , = 2) momentum. Real collinear emission is kinematically forbidden in the resonant part, 11 p,LR (1 i and outgoing electron vanishes,direction because it isnon-relativistic proportional sector to occur the at square NNLO, of because the the light-like spin leading sum over ultrasoft the initial photon state coupling which we to treat in the final state ultrasoft corrections are scaleless. The diagram with the photon attached to incoming As it should be, thisvector agrees current with to the SCET, QCD firstkept analogue the obtained of real the in hard part, this matching because context coefficient the in of imaginary [ the part comes from cuts that do not correspond to QED We find because it carries awayVirtual a collinear hard corrections momentum areand fraction, we scaleless. are which left pushes Hence, withhard the the the photon hard top hard-collinear cannot and be pair regions ultrasoft exchanged off-shell. contributions.would vanish between also We the [ push evaluate incoming the these and top separately. outgoing pair A electrons, off-shell. since Thus this the only correction from the hard region is the c and ultrasoft, regions, ¯ are pairs of light-like vectors( with treatment of ISR toby NNLO+LL QED accuracy radiation below.effects at The from NNNLO the non-resonant hard and part momentumclosely is will region, only follows all not affected contributions the be areequations one universal considered. in and for this our section With treatment can the often exception be obtained of directly the from those in [ In this section wecount take all into account corrections effects thatstates from involve relative QED an initial-state to additional radiation. the photonthe attached LO As leading only such cross logarithmic we to section. order,orders, the whereas i.e. external ISR later summing works was corrections concentrated already on of the considered the ‘partonic’ in form the ( 1980s [ 3.1.5 Initial-state radiation, JHEP02(2018)125 ), . x (   ) (3.52) (3.53) LL ee k ) is cru- − 2 e  Γ − + i e e s/m 1+2 ) are defined. + ( in the “parent k x n ( E ( = 0. Subtracting ln xp 0 LL ee e and reads k . G +1 ) ) are summed to all m 8 n s ) 2 e 0 2 dk α ). At higher orders it k x ∞ 1 0 Z − s/m x 3.1 ( (   n E pole cancels, but a collinear ( 0 ln Im 2 conv n G  σ / α
) ]. First, we need to convert the 2 E γ x e ( 29 2 . Thus, the contribution to the cross , µ LL ee 28 )Γ ) − + ) is given by ( 1  e e s 3.1.1 x ( π ( − 2 √ LL ee ), which correspond to a minimal subtraction conv / 8 Γ – 25 – σ − − + 2 e e Γ(1 3.52  dx 1 π α 0 Z 4 1 i 2 ) k ) and ( ). We observe that the 1 dx a ( 0 1 ) 0 0 Z C . The cross section with resummed ISR from the electron and k 3.51 3.52 p + − ) = 2 ) s E v ( ( ( 0 0 C ) and ( G ISR h ). They can be resummed into an electron distribution function Γ σ 2 e c 3.51 s πN s/m 24 0 + − σ e e ], where LL implies that all terms of the form . Ultrasoft photon corrections to the resonant cross section. The symmetric diagrams, ) terms that appear in the convolution of the structure functions with the LO cross = 65 α – ( At LO, the ‘partonic’ cross section The divergence is regularized by the non-zero electron mass, which in turn yields large O 62 (US) IS σ have to adapt thescheme, results to ( the conventionalThis scheme procedure in which has thedimensional been structure regulator described of functions the in Γ collinear divergencesthe detail to a in finite [ electron mass regulator. Then, orders. The resummation ofcial the for next-to-leading the logarithms precision (NLL) presently program unknown at at a this order. future lepton collider, butdepends the on structure the scheme functions used to are dependence regularize and cancels subtract in the collinear the divergence. convolution The with scheme the structure functions. This implies that we Expressions for the structurein function [ at leading-logarithmic (LL) accuracy can be found logarithms ln( which describes the probabilityelectron” of with finding momentum an electronpositron with is momentum given by When the smallby electron the mass sum is ofdivergence neglected, ( remains, the because the photonthis cross radiation divergence section defines corrections is a not are scheme-dependent infrared ‘partonic’ given safe cross for section. section from the ultrasoft region is due to the right diagram in figure Figure 8 obtained by the interchange of electrons and positrons, are notthe shown. final state vanishes, as discussed in section JHEP02(2018)125 . in    ) k (3.56) (3.58) (3.54) (3.55) (3.59) (3.57) 1+2 − /k  , Γ . i )  )] 1+2 + k k Γ)] E i , + 12 − ( 2 0 + , v 2 Γ 3 e e i G π + E m m a ( + ] dk + 0 k ∼ ∼ [ E 2 e ∞ G 2 0 Z ( µ ⊥ ⊥ 0 m , . i i   ) ) G  Im [ k ( Im O −   Im [ , k
+ 6 ln 2 t a + 2 e 2 ( E Γ)] dk 2 e , k t v 2 γ m θ i m + t 2 e 2 e e 4 µ ∞ m a m 2 0 1 ] Z k + m m m (0)  2 µ k [  f vertex correction diagram and gives E ln 2 t ∼ ∼  ( 2 e − 0   k k m ) ) + m G 2 t 2 e · · 2 t 4 + 2 ln 2 k k i i ( ( a m m  m  δ γ/Zee f Im [ 2 e    2 2 i – 26 – µ ) dk m 8 ln − 2 ). The remaining singularities cancel in the sum  ) ) , n  2 e ∞ ( a 2 0 2 Z ( 0 , n v π 8Γ( t t + 4 ln C α /m a/µ = 4 2 t π ( (SC) m m IS α  + i 4 ) + σ m 2 2 t − 6 + 4 ln 2 a k 2 e ) ∼ ∼ i ] ) ( a 2  m + v ( k = 0 ) m ( k k f , respectively. The soft-collinear region contributes in the 0 [ 4 a  1 4 · · C (  0 C v  i i  h dk C 2 e 2 + (US) + IS n n . As before, the left diagram vanishes, and one finds c 1+2 µ ∞ σ 2 2 m 0 + Z 4 ) k 8  v 2 s + ( πN 0  ) ∼ v C ( 0 π 24 2 4 + 6 ln α h 4 C k 0 c (HC) IS h − poles cancel in the sum of the hard and hard-collinear, and ultrasoft σ × σ c 2 s πN = 3 / π + and s 4 πN 24 2 e  introduces the additional momentum regions (H) IS 0 24 m 0 is arbitrary and we have introduced the modified plus-distribution σ σ + (HC) IS 2 0 soft-collinear: ¯ v σ hard-collinear: ¯ σ ∼ t = = 2 a > m = IS k The first point is accomplished by noting that the presence of the additional scale σ  (SC) e IS σ We obtain where The collinear 1 and soft-collinear contributions, separately.through The the collinear large sensitivity logarithmsover is ln(4 all instead regions. expressed Tothe make distribution the sense: cancellation explicit, one can expand the factor 1 The hard-collinear contribution comes from with diagrams shown in figure double counting. m section have to be subtracted from the fixed order NNLO partonic cross section to avoid JHEP02(2018)125 − 1), E − , ) (3.63) (3.64) (3.61) (3.62) (3.60) = 2 e ) t )] , m k ) s/m 2 ) x − − and neglected − Γ i t Γ)] i )(ln( 1 i xs  . + m ] + x − + √  k [ ) = α/π  E  ) ln(1 E x ( 2 a ( 2 t x 0 2 e π 0 ) − 3 m G 1 is important, − = 2( x m ) G 4 ( ) + 5 + x (1 − . 2 H →  ) δ x ) Im [ β ) for the LO Green function 2 Im [ x x ln − α ) ln = (  + 3 (1 + h dk 2 − t + 5) (1 S 3.10 x O β π + m β 0 α x ] Z (1 4 + 2 1 4 2 x  ) ln(1 = 2 4 ) + ). We also set 2 − x − 3 ) π x x a ( (1 + 7 4 1 ( 0 exp [1 ) + 6 ( − − 1 2  β C 2 Γ)] + 8 x  i (1 + − β/ i LL(1) ee + ≡ t ) + 12 ln ) + 1 ) 2 x )] x β ) m . x ) ln( ( v k E − 2 2 ) + 4(1 + ( 0 − ( π = − x α x ) + Γ 0 4 − – 27 – C   x k )  (1 2 e G i 6 Li Γ 2 ln( s c 2  + 3 i ] with − 2 x m ) β a ) 2 + x ) subtracted, resulting in ] ( + 0 65 s 
) ln(1 x 15 x πN x (1 k x β C δ 6 Im [ [ E ln − 24 ) − in the argument of the logarithm. The initial-state QED ( ( 3.63 h ) + 0 E 0 1 x 2 t (1 + β × (1 + 8 2 γ 1 + 3 ) = σ G π ) ln( ) α m 1 2 24 2 x   v 1 2 4 3 2 ( ( 0 x 1  2 3 − ) = C − Im [ LL ee β β → h x 3 8 x −→ ) with ( Γ xs c and 4 ( ( 2! 3! ) dk 1 (39 − t Γ(1 + 1 1 x 2 3 s 1 4 ( 1 m s 4 4 LO 0 πN Z 3.59 σ 8 exp ) -dependence cancels. +12 (1 + + − − + 24 a x ( 0 LL(1) ee ( ) the non-relativistic Green function was evaluated at σ × and we have substituted Γ ) = xs x ( ( LL(1) ee ... ) = Γ s LO LL ee ( σ ) term in the convolution of the leading order partonic cross section with the Γ ) + dx α x 1 ( conv IS 0 Z − O σ 2 We determine the subtraction terms by expanding the convolution of the LO cross (1 t function is given by ( where in m the difference between correction to the partonic cross section in the conventional scheme for the electron structure The structure functions is The perturbative expansion of the structure function can be written as For the determination of the subtraction term only the limit can be used. The section with the structurethe function electron structure in function the from couplinggiven [ by constant. We take the expression for which is finite, such that the four-dimensional expression ( JHEP02(2018)125 ) ], R # t # 27 2.7 v R t R t L t a v (3.66) (3.65) ) con- v ]. The ). The L t  L t ( a ], Feyn- ), which a 70 2 W 2.7 O x 66 W dimensions W x 2 x 2 2.13 R t − d − v 1 1 1154 L t , the subtracted in ( v b outside the non- − contains an addi- 1 ) ln ) ln ] must be interpreted sq t h W ) b W σ W x 2 27 W 1 m x W x − x h W 1 x − 2 in [ x ) 239 ln µ − was taken from [ ,
W − b x 2 W 6(1 + 2 1 E < 6(1 ) to the cross section in this 4.2 x h are evaluated in − 18 + 36 ) 18(1 + 2 623 − ] and terms from the W H + 2 W 3.64 − W x ) integral contains all terms with x 27 )(1 + 2 x . ∗ W W 4 t x W x F 2 x − − π C 1 4 s 5 (or − 1 + )(1 + 2 α 11 + 16 t c + ) ln W 3 ). With the exception of N x + R t 2 W 0 y a − x R t Γ 144(1 − L t t − a v – 28 – 2 W L t m x + 5 ) terms, that cause numerical instabilities in the v . 2(1 W t = x . The computer programs Package-X [ W 2 W W x − − x x H 2 1 + 15 2 ] quoted below and in section − N 1 27 W ) ln x ] have been employed for certain steps of the computation. 2 ) ) ln W ). The numerical 69 x W − W 2 and ln(1 x 192(2 + 2 1 x W 2.7 − t " x " H H − 6(1 N N 1 (1 + 2 2 + 4 − √ + + 18(1 + 2 / − ] . The latter encapsulate the dependence of the squared contribution ] ) , , W 27 27 ,b x W ] must be identified with is the contribution from the first term on the right-hand side of ( (1 ix x 28 g 27 b a from [ − from [ ] and LoopTools [ (EP fin) (EP fin) | 1 1 | in [ a b 2 + 3 17 12 1 1 H H x x 68 (EP fin) 1 ) analytically and used them as further subtractions. , − + − + H H t H 67 = = and − The contributions corresponding to the second term on the right-hand side of ( b a w Note that in the expressions from [ 1 1 µ 12 H H prefactor is defined as as where we obtain in (1 are given by thetributions sum to of ˆ the respectiveon expressions the in computational [ scheme and are therefore specified below. In the notation of [ positive integer or half-integer powersintegrands of (1 were all obtainedtional in numerical analytical angular form. integral.and The will The integrand not expressions for be forlarities given the explicitly. involving integrands 1 The are numericalevaluation rather integrals of are some lengthy diagrams. plagued As by a integrable remedy, singu- we computed additional terms in the expansion is given byCalc the [ diagrams inThe figure result for theindividual scalar diagram four-point contributions integral to inand the the hadronic written diagram tensor in the form ( relativistic regime. Thescheme photon is radiation finite contribution and ( free of large logarithms3.2 of the electron mass. The squared contribution In this section we discuss the calculation of the squared contribution The imaginary part of the Green function is neglected for JHEP02(2018)125 , d

4 W 4.2 )) (4.2) (4.1) 2 W x (3.67) x 4 − 2.13 and − 3 W bosons are x . W 4 4.1 Z x ) terms in the  − ( , ) O 2 W x  (EP fin) 1 4 + 4 x ix ), 3 W H H − x is the fermion charge ) + + 28 ] 3.67 f 3 W + 8 e Re ( is given by the diagrams 27 x W ) 2 W x 2 R x + 2 m from [ | , ) 2 W x i k − R e ( = 0. In ( 1 Abs,bare ) + 36 x 5 + 44 are shown in section 2 t a 3 γ C H W L e , to the non-resonant part is W m ) x a − m i x = = 0, where W g − + )(4 x W x , γ f poles from the endpoint divergence and, 1 ) 2 L x R e k a ix ( (EP div) − Abs,bare int H v m h / 21 C σ L e (1 σ v − − h and 2 t ) + ) + 4 + 48 and (4 f m – 29 – W t 2 W e e x (4 ) arctanh(1 x − 2 2 − W ) . We define the scheme as follows: the coefficients ) γ,Z x = − k (EP div) 2 w int ( Abs,bare W = c 4 σ − C γ f X x W (4 v σ x 2 w L,R
− c 5 s do not contain 1 2 W n in four dimensions are (1 π − 2 2 )(4 x e e α (1 W  − 2 2 x 2 w π ) 1 s 8 ) ) W k − ( Abs,bare x W W (1 + C x x = − 4 w W − s ix x (1 t σ 24 (1 e 12 ). The photon mass obviously vanishes, − − + − how the interference contribution must be treated to achieve consistency with = 3.2 is a symmetry factor, that is either two for diagrams which are symmetric with are calculated in four dimensions, but the loop integrations in the third row of s , i.e. the ones related to the non-relativistic Green function, are performed in 4.2 n 4 Our results for The contribution of an individual diagram ) ) v ( k Abs,bare ( Abs,bare C dimensions. The Diracsection algebra is completely treatedthis in scheme. four dimensions. We describe in 4.1 Absorptive contribution toThe the bare matching absorptive coefficient partshown of in the the matching secondC coefficients row of figure figure as for part (I).Dirac We algebra can however and simplifyscheme one the and of computation the of the computation this loop of respectively. part integrations by in performing four the dimensions. The details of this It would be a natural choice to use the same scheme for part (II), given by (see ( leptonic tensor have been discarded,checked as explicitly discussed that at IR the and beginningsquared UV of contribution. this divergences section. cancel in We have the sum over the diagrams in4 the Part (II) where respect to thecut. cut, or The four photonmeasured for couplings in diagrams are units of whichgiven the are by positron ( not charge, symmetric and couplings with of respect the to fermions the to therefore, no terms of this type are present, i.e. The other diagrams in figure JHEP02(2018)125 ) d ]. w 2)  , 2
27 /s (1 ia  (4.5) (4.6) (4.7) (4.3) (4.4) g w 4 W − c  x 3 W , 2 t (3 − x ) . , i.e. the 2 w − / m R b µ  4 4 a 3 W + 8 follow from Γ) x i + a 4 2 W 3 ). We evaluate ) is not affected R x b + h − ) terms in ˆ v R t 2 + ln  ( 4.4 E ( 4.4 v 2 W L t ( − ) v + 36 x O vertex, and and w ( 0 L  WW a W .   I 2 t 2 G x )  2 w + 28  h e  m 1 µ 2 2 Z a WW 4 2 t W 3 2 w − − M + m x ]. For each of the diagrams µ 4 3 e W 2 . 3 ln − v 27 x  i ( s ) t −  v ) + 4 + 48 8 Re W 5 + 44 ln x  i 2 W ) + is given by ) ) x 2 W 8 + 3 ln and the contributions from the corre- e − W . By introducing the factor 3 2 a x W e W  4 − x s a t x 2 x , in four dimensions. The Dirac algebra h − 3 e 2 W )  a h a x 4 − ( ) − Abs,bare  W 2 + 12 h 2 W C x s 1 W
x ) (1 (1 + 2 5 − x 2 a 2 h and 3 ( 2 W 0 )(1 + 2 ) ) α 2 w − a and W – 30 – x 2 − C s 2 W W W ) arctanh(1 x π a (1 h x 23 x 5 x + 3 W 2 W H (1 + 2 ) h x x − − − − ), we adapted the expression to the scheme described N W W s 1 2 w − x W (1 x c n ] we obtain 4.4 ) x 2 w 2 + 5 W v 36(1 + 36 2 − with a photon attached to the (4 c ( + 27 Abs,bare also in the scheme of part (I) and found perfect agreement x ]
a H H 3 C − (1 3 ) )(4 27 t 2 W N N v 1 h π e ( 0 x W  + + ) C x ) 2 w = 4. − ] ] h k s from [ = 4. The contribution of W ( s Abs,bare | ) 1 c 27 27 x a s n C 4 W n (1 + s σ − x pole. This term carries the dependence on the computational scheme αN W ) also contains an endpoint-finite term from the W x from [ from [ ∆ − | | (1 12 x and the left loop in / 4 w s 2.9 a a s 0 2 3 × n a (1 12 vertex. The endpoint divergent contributions of σ 2 ) with H H 24 − − h ======1 for diagram , we treat the loop contained in the corresponding diagram in figure 3.67 3 WWZ ) L k WW a ( Abs,bare (EP div) 4 I ) a a C To verify that the treatment of the scheme is consistent we computed the finite sum σ a (EP div) (EP div) 3 2 ( Abs,bare σ H H C with symmetry factor of the contributions from thesponding diagrams diagrams in equation ( with for the right loop in is also done indimensions. four In dimensions, the notation but of the [ remaining loop integrations are performed in The expression ( multiplying the 1 and must, therefore, be treatedit in using the same the scheme expansionin as by figure the contribution regions ( approach described in [ above, which involves four-dimensional Dirac algebra.by The loose contribution ( cuts. 4.2 Endpoint divergenceThe of the endpoint-divergent part interference contribution of the interference contribution has been computed in [ We recall that at LO therelativistic Dirac regime structure of and the only top yieldsin anti-top a pair front becomes prefactor of trivial 3 the in Green the function non- in ( The contribution to the cross section is given by JHEP02(2018)125 ] is )- )-   74 the 2 2 (5.1) − − C (EP fin) int σ ] to absorb the ( , 73 )-dimensional and quan- 2  , which is evaluated with R 2 aut + − σ i ) terms [ ¯ l D ( 2 i R N Q term can be written as a sum over l ¯ terms, the UV counterterms and the d 2 d 2 R R Z -dimensional tensor integrals. The ( terms. = d 2 i . This fixes the scheme in which ) ¯ l ¯ R D uses rational ( pole and the finite part. Afterwards, only four- – 31 – i , which are relevant to our definition of the com- ¯ 5.1 N Q / terms, the l ¯ d 2 d R Z MadGraph , quantities with a bar are (4 MadGraph 2 i contains the automated part in section m C ≡ pole, the 1 aut − 2 σ 2 , since it avoids the more complicated integration of the left loop / ) . + i a 4 p h ]. The IR singularities in the real corrections are subtracted before the 5.2 + MadGraph l ¯ are all contained in the 72 , (EP fin) int = ( σ 71 i D dimensions. . The automated part is UV divergent and therefore scheme dependent. The d of the interference contribution. We first describe the implementation of the MadGraph in coefficient of the 1 dimensional operations arefour-dimensional performed, born i.e. amplitude and the the multiplication four-dimensional with phase-space integration. the conjugated 4-dimensional coefficients multiplying dimensional parts related to thein implementation of the ’t Hooft-Veltman scheme [ FKS subtraction terms are written as separate lists, each of them containing the where tities without are 4-dimensional. The non- always done in four dimensions. dimensional parts ofdecomposition the takes numerators. the form For a given diagram with amplitude method [ phase-space integration and the subtractionspace terms of are the then real integrated emission overIR the and phase added singularities to that the arise virtual in corrections, the where they loop cancel integrals. the The phase-space integration is then a The amplitudes for the non- In the virtual corrections, The subtraction of IR singularities is performed automatically using the FKS 4 3. 2. 1. h (EP fin) int computed in section 5.1 The automatedWe part first recall some aspectsputational of scheme. The part (III) MadGraph divergence and the corresponding schemeσ dependence cancel with theautomated endpoint-finite part part in tation, especially for in 5 Part (III) with the results presented above. Applying the scheme of part (II) simplifies the compu- JHEP02(2018)125 = 2 2 and |A| , which is given and the 1 including 3 grid. More 2 b aut . Since the have to be σ + 2 ¯ tW A.5 to obtain the au- . → are then performed . Furthermore, due − Threshold e + e (EP fin) int (EP fin) int σ σ MadGraph MadGraph QQbar . The contribution t in the squared amplitude m ∗ j poles vanish for a number of phase A i = 2 / A s to generate an integration grid from four from below. We remove the contribution √ , the diagrams shown in figures t 1 and 1 m h 2 – 32 – must be computed. We choose to only subtract / = 2 ]. Therefore, we can safely switch it off. The code 4. Following the discussion of the items s , . The evaluation of the automated part in the code MadGraph 75 3 √ , B (EP fin) int has to be computed by using dimensional regularization σ = 2 i by editing the code generated by parts, the UV counterterms and the FKS subtraction terms. checks if the 1 (EP fin) int with 3 2 σ is defined, we never have to modify the construction of an amplitude R ia h aut σ relies on a precomputed grid as described in appendix is rather small, we do not aim for more precision than about 10% in the MadGraph contributions. Further details on the implementation and modifications in aut σ terms of − are provided in appendix 2 . All the contributions associated with the diagrams in figure ∗ j R bW ¯ terms from figure t A Threshold i 2 A R Finally, we have to deactivate some checks inside the code, that are invalidated by the In the following, we describe the steps we performed in , but we have to remove certain contributions i,j i iterations with 15000 pointsusing six per iterations integration with channel 100000precise and points results for perform are each possible the pointgeneration at actual of of the integration the the cost grid. of a considerably increased computing time for the for the MadGraph QQbar contribution automated part. Thereach resulting this error target of precision the we set cross up section is less than one per mille. To squared absolute value ofpoints. an The amplitude positivity and, ofbut these thus, expressions is no is only longer not used necessary positive ascan to an for now make internal be all the check evaluated code phase-space [ directlyby run the at properly, difference the of threshold fixed-order runs at NLO and LO, multiplied by a factor two to account have addressed this issue witha in minimal two subtraction ways of —The the by UV minimal deactivating divergence the subtraction — check waswith and or also found by the performing agreement endpoint-finite used of part toto both of verify the approaches. the the subtractions, interference the cancellation contribution tree-level of cross the section UV and the divergence real corrections are no longer the from the endpoint divergentnon- born diagram modifications. space points. Here, this is not the case because the automated part is UV divergent. We modifying the process generation,squared because amplitude. the We automated thereforeQCD part first corrections. does generate By not the not correspondenergy full invoking insertions to the process are complex a treated masscenter-of-mass perturbatively. scheme, energies we Hence, approaching make the sure cross that section the diverges self- rapidly for the non- above, this implies that for the tensor integrals.in All other four steps dimensions. in the computation of tomated part in the scheme defined above. It is obvious that this cannot be achieved by A P removed, i.e. also the There is however andetermines ambiguity the in scheme the in subtraction which of the contributions in figure Given the way that JHEP02(2018)125 , t 1 ). M ). t → 5.1 (6.2) 4. In 2.10 M ) , ∗ ∆ ( 3 . at NLO t c , (¯ 1 h 4.2 = 2 i non-res as subtractions σ t ). We performed , shows our semi- -integration. The − t )] with 9 t 6.2 in section M ia ∆ h (6.1) c ) (¯ t ) in the presence of the 1 t M h 1 of the M ∆ σ (EP div) int ∆ c as a one-dimensional angular → (¯ c ). The points show the same σ code, described in section − (¯ t a ) int 4 t 6.1 σ h . The contribution from diagram M ), with the exception of the contri- ∆ ), which originate from the non-res c ) + ) that appears in ( (¯ σ t t 2.11 2.7 M M aut MadGraph ∆ ∆ σ c [ c (¯ (¯ MadGraph 1 . Here, we have rearranged the contributions − sq h ) σ 9 σ t ) is finite. Therefore, we can evaluate it using i − M – 33 – p is neglected, as is done in our calculation. ) terms in ( ( ) + ∆ ), allows us to perform a very powerful numerical t t t b i c ) ) using (¯ M p M M ,b m ( ∆ ). The comparison for various values of the cut ∆ ∆ t ∆ (1 analytically and for ix 6.2 c t c c g (¯ M (¯ a M 3 1 ∆ − ∆ h non-res c aut ) obtained by means of ( c , h σ σ t σ ( a 2 M ≡ = ), because our edited h ∆ ) = 1 t ) c i t (¯ p M it must be evaluated by taking only the tensor integrals of the non-res ( M ∆ t σ c ∆ (¯ M c check 5.1 − (¯ ) parts of the ˆ ∆ σ  c code. On the other hand, it can be obtained from our result by taking ( is given by applying the same prefactors and symmetry factors as for ), denoted by check O check σ dimensions and then performing all other steps in the computation strictly σ ) is the contribution to the non-resonant part from the diagram non-res t d 2.14 σ M ∆ (EP fin) int c we use additional terms in the expansion of the amplitudes in 1 σ (¯ MadGraph 1 ) in the presence of the complementary cut. The line in figure h 3.2 σ 1 The result of our check is shown in figure is included in 1 the same check for theparticular, individual this contributions provides from thepart very diagrams (III) welcome has reassurance beenagreement, that treated if consistently. the the bottom-quark Within scheme mass estimated dependence numerical within errors we find good (figure analytical result for quantity determined by evaluating ( h corresponds to the combination where the difference numerically tests the whole non-resonantbutions result from in the ( region and are independent of the value of theas cut, follows, i.e. are not present in Having performed the computation ofant the mass non-resonant cut part ( inconsistency the check. presence of The thecomplementary invari- non-resonant cut cross ¯ section unedited 6 Checks and implementation 6.1 Consistency checks integral. We refrainsection from giving the lengthyto results deal with for integrable the divergencesresult that integrands. appear for in As the described limit the in endpoint divergent part of the interference contribution We recall that the endpoint-finiteAs part of detailed the in interference section contributionvirtual has loop the in form ( in four dimensions.integrands for Within the this diagrams scheme, we have determined the endpoint subtracted 5.2 Endpoint-finite part of the interference contribution JHEP02(2018)125 can . The 7 GeV 2 . ) ). The t = 4 6.1 M b ∆ 507. ■ ■ ● ● runs with the . m − 170 . An updated t threshold 132 m is related to large ■ ■ ● ● / ( t ≤ M . = 1 ■ 2 t ■ ● ● 4.7 GeV 0.1 GeV p α QQbar 160 MadGraph ) in pb, given by the sum = = . t b b M A m m ∆ c (¯ ■ ■ ■ ● ● ● 118 and 150 . check σ ) = 0 t code and the standard deviation of ten Z ■ ■ ● ● M m 140 ) for ( Δ s α 6.1 - , t , the squared and the interference contributions. Z m 1 m MadGraph h ]. A summary of the code changes and some 130 = – 34 – 32 [ µ ) obtained from the difference of ■ ■ ● ● Threshold 120 6.2 https://qqbarthreshold.hepforge.org/ 4.7 GeV 0.1 GeV = = = 173 GeV, threshold QQbar 1 GeV. The error bars in the lower plot are obtained by adding the t b b . code in quadrature, while the estimated uncertainty of an individual 110 m m m , = 0 ■ ● b QQbar m https://www.hepforge.org/downloads/qqbarthreshold/ ■ ■ ● ● 100 4 1 MadGraph - MadGraph

10 1.04 1.00 0.96 0.92

0.010 0.001 Ratio 0.100

M check t pb ] [ ) ( Δ σ c results have been obtained for the default value of the bottom-quark mass . Consistency check for various values for the complementary cut All of the aforementioned2 NNLO of corrections the have publiccode been code examples implemented for in the thebe new new downloaded functions version from areonline provided manual in is appendix available under default values of 6.2 Implementation in MadGraph and a negligible value standard deviation of tenruns runs of of the the edited run unedited is ignored. Thecancellations increase between the of results the of relative the uncertainty edited for and large unedited values code. of For ∆ this check, we have used the Figure 9 line in the upperof panel the is contributions our from semi-analytical theThe result tree-level points ( diagram give theunedited same and quantity edited ( code. The lower panel shows the same results normalized to ( JHEP02(2018)125 , t is 2 Λ t /m / 2 t /m m W m can be un- t − is thus a con- ]. We briefly t /m 31 expansion of the below. Because = 1 /m 2 / 14 1 ρ ρ ] and [ 50 , where 2 / 1 ρ ]. 27 ] was reproduced by combining a deep 26 ] within the so-called phase-space match- is not small in reality, one may hope that ρ 50 – 35 – . On the other hand, the endpoint-divergent terms t Γ ], aims at the computation of the non-resonant con- 31 non-resonant terms are obtained in the PSM approach |  t t p | /m ] ] 31 50 were determined in [ ) of the full non-resonant result, namely the terms of order 2 t t ]. The Λ m M expansion we refer the reader to [ log Λ. The latter correspond to the endpoint-divergent terms computed 27 ∆ 0 t total cross section near the top anti-top threshold.  − Λ) , upon going to the (infrared) on-shell limit within a distance regulated by t /m t t / X t Γ M − M m ∆ ∆ W t |  + m t 2  2 ], which give the approximate result labelled “aNNLO” in figure m t bW ¯ Our concern here is the computation of the first two terms in the The agreement between the PSM result and the full non-resonant computation of the ≡ b Λ and ( 27 − / 2 t t 2 p with sufficiently many termsIndeed, in the the exact expansion, NLO a non-resonantexpansion result good with from approximation Pad´eapproximants. [ might be obtained. NNLO non-resonant contribution. In the present notation, the first term in the expansion 6.3.2 Comparison to [ Another approach, introduced intribution [ to thetreated total as cross a section small in parameter. an Even expansion though in | Λ. The fact that bothsequence of expansions the provide cancellation the of sameand the divergent dependence non-resonant terms on in regions. the Λ cut-off Forterms Λ the in that limitations the separates on Λ the resonant the PSM result to provide higher-order invariant mass of thetively top acts and as anti-top a quarkamplitude, regulator that has of is the been obtained ultraviolet implemented. assuminginto singular on-shell the behaviour Therefore top off-shell of quarks, limit, Λ the i.e. when effec- are for resonant the obtained part latter from is of the further non-resonant the part taken of the amplitude, that assumes off-shell tops with non-analytic terms in the expansionderstood in as the a invariant-mass cut consequenceloop parameter of Λ momentum the [ cancellation ofby singularities computing between the adjacent ultraviolet regions behaviour of of the resonant amplitude where the cut on the (Λ m in [ of the “not-too-loose” cutthe condition, the PSM approach does not allow the calculation of 6.3.1 Comparison to [ The leading NNLO non-resonant contributionsing for the Γ case of “not-too-loose”ing cuts (PSM) satisfy- approach. This result captures the first terms in the expansion in Λ While a completethe calculation top-pair of threshold NNLO ascorrections reported electroweak have here and been has non-resonant never evaluatedcomment contributions been on in these done to certain approximations before, and approximations NNLO their in non-resonant limitations [ here. 6.3 Comparison to other approaches JHEP02(2018)125 , / . and (6.5) (6.3) (6.4) . We  a ) 3.1.2 1 ρ C h √ . We note ( t but referred Z . O ] and in the b ) 1  k 30 ( Abs, h C σ . Our result for this a + 4 ln 2 1 ], refers to a different | h Γ 54 i ] already from the leading t differs by a factor of two, + 4 ln 2 + 1 + | + 31 m | Γ Γ i E i | ρ t . Each of the two terms contains + + as described in appendix ln m in the sum of all contributions sq E b  E ) from this diagram explicitly, and σ | | 1 ρ that contribute to F h /ρ in ) gives the sum of all contributions at π s C ). However, as was already mentioned b ] there is a non-vanishing contribution 13 4 t s 2 ln α 1 Z ( α 6.3 h  C 31 ) O k 0 F ( Abs, ) terms from the results in [ Γ π C C t ρ and 4 s σ /ρ m a α s 1 – 36 – + i α 0 h 2 ( ) sq Γ a t ρ O σ ( 0 correspond to the same diagrams but taking the loop momenta m b , such that ( C 1 ] originates only from diagram 2 i h / ) in appendix + 2 1 31 ) ρ 2 a ) ( 0 C.5 , called “jet-jet” contribution in [ and v ) contribution to the total cross section reads b ( 0 C a 1 1 C /ρ h ), ( h + h s 2 ] does not distinguish between what we call non-resonant and c ) from [ α ) ( v C.4 ( 31 0 O s 6.3 πN ] did not actually attempt the calculation of diagram C h 24 31 c 0 σ expansion. s πN ) for the individual results). This can be traced to the cancellation of = 2 / ) 24 ). 1 0 /ρ C.7 ρ ] reads explicitly s /ρ σ α s ], contrary to the statement made in [ ( at the same order. We computed the 31 )–( α = O ( b 1 σ 30 nr ] O pole, which cancels in the sum. This holds separately for the two diagrams C.4 , , ] to claim that it must not contribute. However, as already discussed in section h . enhanced term. Furthermore, the coefficient of ln The authors of [ It is instructive to construct the 31 a divergences and the scaling of the leading momentum regions that contribute to the plus their corresponding resonant counterparts [ The resonant counterparts of 27 (1) 1 54 / h b σ 13 1 / /ρ order in the non-relativistic expansion,work namely as NLO, the and non-renormalization is of reflected the in coupling the of present the frame- topin quark the top to anti-top a loops in potentialin the gluon potential region, and and keeping only the NNLO term of the self-energy insertion therefore disagree with the NNLO non-resonantterm result given in in the [ to [ the purported vanishing of Note the absence(see of ( a1 logarithmic dependence on 1 which is related to the contribution of the diagram in [ from find that the complete a 1 h that the leading term ( diagram including its resonantconstant counterpart term indeed +1 agrees (see with ( the above except for the present paper. We find that they arise only from in part (I) and specifically from diagrams pole. It appearsabsorptive that matching coefficient [ contributionthe to expansion the of resonant the part diagramorder and in directly constructs It is immediately clear fromfrom this ours, expression that in the which meaning case of the “non-resonant” non-resonant is contribution different is analytic in energy and has a 1 given in [ JHEP02(2018)125 (7.1) ] to the 3%. The 22 ,  can be found , b 1 944 between 50 GeV . h expansion is con- r 2 385 GeV ] and will combine µ 128 . / 4 1 / ρ = 350 GeV = 80 ], but briefly recapitulate ) = 1 w 6%. Overall, the effect of 25 W . We have already discussed . Z 3 m 11 ( − 0 . ] is in progress [ , m term arises from a momentum region 32 [ /ρ , µ , α . 1876 GeV . – 37 – ] and the running electroweak coupling is taken 1184 . 76 3.1.4 threshold 2% directly below the peak, where the uncertainty . = 80 GeV = 91 5 ]. For the numerical evaluation we adopt the input 8%. In the remaining regions it is about ) = 0 r . Z  Z 3 8 GeV. The size of the uncertainty band is somewhat ] concluded that the statistical uncertainties of the top . collider can be very small in realistic running scenarios. 59  m , 3 ( , QQbar − s 2 e 32 + , e = 341 14 s also shows the net effect of all the corrections discussed above, ]. , α √ ]. Further details on the comparison and diagram , µ , m 10 25 31 ]. The result is shown by the grey hatched band labelled “QCD” in 9 5 GeV . 33 GeV . . is the top-quark PS mass [ C = 171 = 125 GeV = 1 t , which is spanned by variation of the renormalization scale PS t ], see the discussion in section H Γ PS t m 60 m 10 m The size of the individual contributions is shown in figure To avoid the IR renormalon ambiguities, we exclusively employ the PS shift (PSS) increase of thealready scale observed uncertainty in is [ mainly due tothe the Higgs, QED Higgs Coulomb and potential NLOthe non-resonant insertion corrections results in as [ here was to give a complete overview over the non-QCD correction up to NNLO. the non-QCD corrections isis to observed make below the thethe resonance peak, non-resonant more contribution where pronounced. dominate theto the The absorptive non-QCD overlap largest parts corrections. at effect of Here,increased the around the matching and bands coefficients now cease and reachesestimate up for to the QCD result is figure and 350 GeV. Figure excluding ISR, which willthe be height considered of below. thepeak peak These the non-QCD and cross effects move section slightly it is increase towards slightly smaller decreased center-of-mass by energies. about Above 3 the from [ 7.1 Size ofWe the define electroweak a effects reference QCDS-wave prediction result by adding of the [ small P-wave contribution [ where statistical and systematic experimental errors with theory uncertainties. mass scheme defined invalues [ Recent experimental studies [ threshold scan at aThus, future when discussing the impactsection, of we the focus electroweak on andtheory the non-resonant theoretical prediction corrections uncertainties. available in in An this experimental analysis based on the structed from momentum regions,that the was leading missed in 1 [ in appendix 7 Discussion of results the Coulomb potential by electroweak effects. Moreover, when the JHEP02(2018)125 at NNLO and H σ 348 348 8%, depending on − 346 346 QCD full ] ] GeV GeV [ [ 344 344 = 80 GeV. s s r µ – 38 – full 342 342 QCD is present. At NNNLO, there is also a contribution from the local (2) vH 340 340 c

1.2 1.1 1.0 0.9 1.3 0.8 0.6 0.4 0.2 0.0 1.0

full pb μ σ σ 80GeV ) = ( / ] [ X σ , and make the peak more pronounced. The comparison of the dashed s √ . The cross section in pure QCD (grey hatched band) and including the electroweak the same time thethe Higgs value corrections of increaseand the solid cross curves section demonstrates byfor that 3 correctly the capturing inclusion the ofreliable energy the measurement NNNLO dependence of corrections the of top is the Yukawa important Higgs coupling. effects, which is crucial for a In the top-panel weNNNLO. show At the NNLO relative effectmatching there coefficient of is the an HiggsHiggs almost contribution potential, constant which relative modifiesof shift, the the because potential, position the only of binding the the energy peak. hard- is increased Due and to the the peak attractive is nature shifted to the left. At Figure 10 and non-resonant corrections (redvariation. hatched band). The The upper bandsnormalized panel represent to the shows the uncertainty the full from one cross scale for section the in central pb scale and the lower panel shows the results JHEP02(2018)125 348 348 348 res. - res. 346 346 346 - 8% and a shift of the peak towards − ] ] ] QED V NLO non NNLO non GeV GeV GeV δ NNLO EW [ [ [ 5% depending on the energy. Including the 344 344 344 . s s s 1 − – 39 – NNLO Higgs NNNLO Higgs 342 342 342 , are shown in the middle panel. The dashed line cor- t Z 340 340 340 )

k

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(

Abs,bare+

QCD QCD EW QCD full QCD EW QCD + + + + + + 80GeV μ σ σ μ σ σ μ σ σ 80GeV 80GeV H H H H ) = ( / ) = ( / ) = ( / C σ , ) k ( EW C σ , decreases the cross section by 0 . Relative corrections to the cross section by adding Higgs (top), electroweak (middle) Γ σ Γ σ , The remaining electroweak contributions to the ‘partonic’ resonant cross section, QED δV σ responds to the correctiontherefore leads from to an the increase QEDsmaller of center-of-mass the Coulomb cross energy. potential section The by only.bution solid 2 line It shows is the attractive full correction. and The width contri- Figure 11 and non-resonant (bottom) effects cumulatively. JHEP02(2018)125 . ] w µ 12 30 , (7.2) 27 = 350 GeV). The w captures the ‘large’ µ t m ], where an enhancement contained in the full cross ∼ 30 v , w , 27 µ } v = 350 GeV is kept fixed. The t w 15%. res µ σ w m {z − 22% for low center-of-mass energies, µ ⊂ | − dependence is of NNNLO, where the + ln w t } µ w m µ 3)% effect near and above the peak and only {z non-res . σ 0 | ⊂ − – 40 – only include the variation of the renormalization 2 = ln . v (0 10 ln is very mild as discussed below and, thus, mainly the  w ⊃ µ full = 30 GeV) compared to the present ( σ cancels exactly between all contributions of a given order. We w µ w = 350 GeV for the finite-width scale is chosen near the hard scale µ w µ 3%. The absorptive part of the matching coefficient multiplies the real part dependence of the resonant cross section and the full cross section in figure . 3 8% below the peak. The remaining w . − µ 1  4)% near and above the peak and reaches up to The central value We recall that the bands in figure The lower panel illustrates the behaviour of the non-resonant contribution to the total − are introduced by the separation into different momentum regions and are therefore (3 3% near and above the peak. However, it becomes even more important below the peak, w where the first logarithm totribution the and right the of thelogarithms second equality present one sign at of is NNNLO partin in the of the the resonant. the resonant non-resonant non-resonant part con- Choosing part and renders small. the logarithms While contained the NNNLO resonant contributions are already µ spurious in nature. Explicitly, somesection of are the split ‘large’ logarithms as ln follows a mild full QCD corrections, but onlycancellation a is few achieved. electroweak effects are included and thereforeto no make full the corresponding logarithms in the non-resonant part small. The logarithms of show the For the resonant-only cross section,larger it than is mild the near renormalization andis scale above greatly the dependence reduced peak, below for butconsidered the the is in peak. full significantly the cross plots The only section, yields sensitivity where a to the variation between 20 and 700 GeV size of the individual contributionsand is the affected one — most from notably the the absorptive non-resonant part correction of thescale hard between matching coefficients. 50 GeVdependence and on 350 the GeV, scale while the scale 40% of the NLO result.of This is the in negative contrast to non-resonant theThe correction findings apparent from of discrepancy [ an is approximate entirely explainedfor NNLO by the result the was finite-width very different observed. scale choicedependence made ( of in the [ full result on where the cross section is small and modified bycross up section. to Its absolute sizeis is given nearly by energy-independent. the Thus,− “inverse” the of shape the of resonant the crosswhere curves the section. resonant cross At section NLO, becomes the small. effect The is NNLO of corrections the compensate order about of about of the non-relativistic Greenthe function, point where which the has imaginaryThus, a part the has broad absolute its peak, maximal contribution roughly slope, has− on centered only top around a of mild a energy smooth dependence background. and is of the order of real part of the electroweak matching coefficient leads to an almost constant relative shift JHEP02(2018)125 normalized w µ 700 700 700 = 344 GeV (middle panel) and ], since in these papers the NNLO 600 600 600 s 30 √ , 27 500 500 500 resonant full resonant full resonant full ] ] ] 400 400 400 GeV GeV GeV 344GeV 348GeV 340GeV [ [ [ w w w = = = μ μ μ s s s 300 300 300 – 41 – 200 200 200 = 340 GeV (top panel), s 100 100 100 √

1.01 1.00 0.99 1.00 0.99 1.04 1.03 1.02 1.10 1.05 1.00 0.95 1.03 1.02 1.01 1.25 1.20 1.15

μ σ σ μ σ σ μ σ σ ) = ( / ) = ( / ) = ( / w X X w X X w X X 350GeV 350GeV 350GeV = 350 GeV for w 14 µ . Dependence of the resonant-only and full cross section on the scale The same argument motivated the different choice made in [ = 348 GeV (bottom panel). 14 s partially known, the NNNLOtional non-resonant limits. corrections are Thus our beyondcontributions. scale the choice present minimizes computa- the uncertainty from the missingresonant NNNLO electroweak contribution was not available. Figure 12 to the√ one at JHEP02(2018)125 ) is 2.14 between 20 and w 348 µ res. - 346 around 4 GeV the NNLO non- t 30 GeV have only a mild influence resonant 5) GeV reduce the cross section by M , ], which includes only the endpoint- NLO non ] ≥ 10 + 27 , t GeV 20 M [ , 344 s , where we have varied 13 – 42 – dependence of the resonant contribution at this = (30 . The dependence on the cut defined in ( t w µ M 2.3 342 = 80 GeV. The bands are the envelope of the values obtained by can be used to estimate the size of the missing NNNLO non- 0, for comparison. It describes the dependence on the cut full r w µ µ → t 340 M 084) pb. We observe that for ∆ .

0

, where the dotted and solid lines denote the NLO and NNLO non- 0.90 0.85 1.20 1.15 1.10 1.05 1.00 0.95

= 350 GeV.

,

μ σ σ ) = ( / full w X 350GeV w 14 µ 037 . 0 , shows the approximate NNLO result [ between 20 and 700 GeV. The results have been normalized to the full cross section for . The resonant plus NLO non-resonant (grey hatched band) and full cross section (red 014 is potentially less reliable, because the leading NNNLO terms might not cause w 14 . dependence. This would be similar to the situation at NLO, where the leading non- µ 0 , 13 w We discussed the possibility of imposing loose cuts, which affect only the non-resonant Variation of the scale µ 007 . in figure divergent terms as ∆ very well, since the-0.004 endpoint-divergent pb terms for are the full most crossabsence sensitive section of to and any up it, to cuts but -0.013 the pb it including exact is invariant result mass shifted cuts. corresponds by In to the a 46% correction with respect to the shown in figure resonant contribution. Veryon loose the cuts cross with section.(0 ∆ Tighter cuts ∆ resonant contribution becomes ascut large is as loose the is NLO no one. longer Here, appropriate the and assumption our that description the breaks down. The dashed line resonant effect arises, yetorder there at is all, no since theis divergence purely from linear. factorizing resonant and non-resonant contributions part of the cross section, in section one and much narrower. Thus,a the chosen reasonable range estimate of of themissing the finite-width scale NNLO NNNLO variation non-resonant non-resonant provides correction. correctionfigure However, based an on estimate the ofany width the of the “full” (red) band in resonant corrections. Theand corresponding full bands cross for section the700 GeV. are resonant We plus shown observe NLO in that non-resonant figure the inner (red) band is entirely contained in the outer (grey) Figure 13 hatched band) are shownvarying for the central scale JHEP02(2018)125 ] and in the 27 348 ]. The full cross section NNLO 27 0 + shows the partonic cross 50 15 NLO NNLO aNNLO NLO 346 ) is a small effect of the order with ISR with ISR the approximation is much better. ] t 3.64 m ] 20 GeV  [ t t GeV [ M 344 M s Δ ∆ – 43 – 10  t ISR ) is obtained by convoluting only the leading order ‘partonic’ without 0 342 5 . W m − t 340 m 0.00 0.15 0.05 0.10

- - - =

0.2 0.0 1.0 0.8 0.6 0.4

nr pb 350GeV σ μ pb 80GeV ] )[ = ( μ σ w ] )[ = t ( X to the partonic cross section from ( M and its convolution with the electron structure functions. The QED con- ). The dotted line shows the result at NLO and the dashed line the NNLO correction conv IS . The effect of initial-state QED radiation on the cross section. The dotted curve σ . The dependence of the non-resonant contribution to the cross section on the invariant 2.14 conv σ We finally discuss the effects of initial-state QED radiation, which have so far only range of loose, but not too loose cuts Γ been taken into accountsection in the experimental studies.tribution Figure convoluting the full ‘partonic’ cross(see text). section with The the dashed linecross structure (with section functions ISR with with the different structure systematics functions and adding the full ‘partonic’approximate corrections NNLO on result. top. We note, however, that for the scale choice of [ Figure 15 shows the full result without ISR. The solid band (with ISR) is the envelope of results obtained by Figure 14 mass cut ( (without the NLOdenotes terms). the approximate The NNLO sumcorresponds result of to (without ∆ both the NLO is terms) drawn from as [ a solid line. The red dot-dashed line JHEP02(2018)125 is 0 (7.3) ) of the 44%. The 3.53 , ) − estimate the s ) and a purely 2 x 18 1 x 3.60 ( 1% near and above . and LO σ ) 17 2 , , the PNREFT and unstable- ] of the sensitivity of the x = 328 GeV, or an alterna- s ( = 320 GeV) = 0 pb and our √ s 16 25 ) for the different subtraction LL ee s , √ 4% above the peak and reaches 9 . √ )Γ ( 1 For both extrapolations we con- 3.64 σ x ( ) in the structure function and ac- 15 2 e LL ee Γ 2 s/m dx 1 ) ln( 0 Z 1 = 340 GeV. – 44 – α/π dx s 1 ]. We either use the shape of the LO cross section √ 0 Z 32 = (2 [ ) + ]. β s ( 8% at 36 , . LO 35 σ ) with the structure functions as defined in ( − Threshold ) s ( 3.53 LO correction by order one, the difference between ISR and ISR k conv QQbar σ ) = = 328 GeV. Numerically, we find a small difference of 0 collider processes, but beyond the scope of this work. We further note that at s ( s − 0 = 328 GeV, rescaled to match the full result at ). The difference is formally a NLL effect and provides a rough estimate of the 3)%. The convolution, however, reduces the cross section by 28 . e √ 1 s + 1% in the region where the slope is large. ISR 44%. It also leads to a significant modification of the shape. The peak is shifted by . e √ σ 3.63 − − 6 We see, as it is of course expected, that ISR is a huge effect, reducing the cross section For comparison we furthermore show as the dashed line the expression . While the grid could technically be extended to smaller values of 15 (0 sensitivity by comparing theterms of effects the of relative variation parameter to variations a to reference crossparticle the section. EFT scale breaks uncertainty downmatching far in the below EFT the descriptionfor threshold. to a the single-particle Improving fixed-order resonance the calculation in accuracy of [ in the this full region non-resonant process would as require discussed 7.2 Sensitivity toSince Standard the Model parameters non-QCD effectscross computed section in we provide thistop an paper threshold update scan cause of to substantial Standard the corrections Model discussion to parameters. in the [ Figures of the convergence andenergy remaining uncertainty, which isthe of level universal of importance NNLO for electroweakemployed accuracy high- for the the partonic electron cross structureapplied section function, in depends and experimental a on studies phenomenological the in convolution scheme as an often unspecified scheme is no longer adequate. with the full partonic result. by 28 almost 200 MeV to the40%. right and smeared This out emphasizes considerably. the Its height need is for reduced a by about full NLL treatment of ISR and a proper analysis where the ISR resummationmation is modifies only applied a to N formally the a LO NLO cross section. effect. Since This the emphasizes LL that resum- it is mandatory to perform the convolution term ( overall size of NLL ISRup corrections. to It 2 amounts to about 1 tive implementation that interpolatesresult linearly at between the peak, which goessider the up convolution to ( 0 LL approximation where wecordingly set modify the non-logarithmic ISR contribution ( black band is spannedfull by NNNLO four QCD different implementations plusvolves of NNLO an the extrapolation EW convolution of cross ( grids the section available cross with in section the forbelow structure energy functions. values outside This of in- the range of the − JHEP02(2018)125 % for the 25 +20 − 348 348 346 346 +100 MeV -100 MeV -100 MeV PS t PS t ) up to variations of the top mass (top panel) t Γ m 344 344 m 7.1 s[GeV] s[GeV] 60 MeV for the top-quark width,  – 45 – 342 342 -50 MeV +50 MeV PS PS t t +100 MeV t m m Γ we expect the threshold scan to be sensitive to variations of 17 340 340 ], such as a small reduction of the height of the peaks present in the and

0.95 0.90 0.85 1.15 1.10 1.05 1.00 0.95 0.90 0.85 1.15 1.10 1.05 1.00

25 μ σ σ μ σ σ ) = ( / ) = ( / ISR ISR ISR ISR 80GeV 80GeV , 16 9 ). The prediction is normalized to the full cross section. . The cross section for the input values ( 10 40 MeV for the top-quark PS mass,  From figures All electroweak and non-resonant effects discussed in this paper are included in the produced by the convolutionconvolution with of the the luminosity cross function, sectionsensitivity we with to the expect the collider-specific that parameters, beam the either. function additional will not dilute the about figures, in particularwith also respect the to ISR [ top-mass corrections. variation curves near There 344.5 GeV, areconclusions but small remain the essence quantitative unchanged. of differences thediscussed It results above is and does the especially associated not noteworthy degrade that the the sensitivity. huge Since ISR the correction bulk of the ISR correction is Figure 16 and top width (bottom(cf. panel) figure is shown in comparison the uncertainty band from scale variation JHEP02(2018)125 = SM t y ), when Z m ( s , where α SM t y t κ 348 348 = t y for the position and height 346 346 18 )=0.1204 )=0.1164 =1.5 =0.5 t t Z Z κ κ (m (m s s α α 344 344 s[GeV] s[GeV] . – 46 – 16 0015 for the strong coupling constant . =1.2 342 342 0 t κ  340 340

0.95 0.90 0.85 1.15 1.10 1.05 1.00 0.95 0.90 0.85 1.15 1.10 1.05 1.00

μ σ σ μ σ σ

) = ( / ) = ( / ISR ISR ISR ISR 80GeV 80GeV for the energy dependence and in figure 17 . The upper (lower) panel show the effects of variations of the Yukawa coupling (strong is the Standard Model value. The predictions are normalized to the full cross section and /v t m 2 several parameters can beconcerns disentangled the from Yukawa their and energyseen the dependence, strong in which coupling, figure particularly whereof variations the lead peak to in similarwhich the include effects cross experimental as section. uncertainties as This well. needs to be addressed within realistic simulations, top-quark Yukawa coupling and only a single parameter is variedwidth at a of time. the band These for numbers the areand parameter obtained requiring variation from with that comparing the the the one former fromleaves is the open larger theoretical uncertainty, than the the question latter for of a how sufficient range well in the energy. corrections This from the simultaneous variation of Figure 17 coupling) on√ the cross section. The Yukawa coupling isthe parametrized uncertainty as band is the same as in figure JHEP02(2018)125 X − ], the 001 in W . 9 0 + collider. scale de-  − w bW e ¯ b µ + ) by e Z → ] has motivated 9 m − ( ]) or the one-loop s e α + 47 e , 45 344.2 )=0.1164 Z )=0.1174 Z (m s α (m s α ● ● 344.1 ● ● =1.2 t ● ● ●●●● κ =0.5 t κ =1.5 ● ● t s[GeV] κ =0.8 t ● ● κ – 47 – ] with the NNNLO fixed-order calculation [ 344.0 24 3% accuracy estimated for the pure QCD calculation )=0.1194  Z (m )=0.1204 s Z α (m s 343.9 α ]. in the QCD part [

0.58 0.56 0.64 0.62 0.60

32

t ISR [ pb ] [ σ ], the present work completed the calculation of NNLO electroweak correc- E/m 25 . The effect of a variation of the Yukawa coupling (strong coupling) on the position threshold Despite the level of sophistication already achieved, further improvement could be these additions factorization-scheme independent.pendence To of cancel the the finite-width NNNLOsame QCD accuracy, which result appears completely, prohibitive theof at QED non-resonant present. initial-state part radiation Finally, is a with next-to-leadingeral consistent needed prerequisite logarithmic implementation to for accuracy accurate the seems predictions to of be scattering a at gen- a future high-energy considered or might belogarithms of necessary, such asinclusion the of already combination known ofcorrection NNNLO the to electroweak the NNLL corrections Higgs (see potential summation (a [ of N4LO effect) together with the terms required to make NNNLO QCD plus NNLOradiation electroweak in accuracy, a including scheme fordeed consistent non-negligible the with compared first to one-loop the time QEDand initial-state corrections. are therefore The essential for newfrom accurate effects the top are and threshold. Standard in- They ModelQQbar have parameter been determinations implemented in the new version 2 of the public code the consideration of non-QCDcorrection. effects While of Higgs/top-Yukawa potentially couplingobtained similar effects in size [ up to totions the the and third third-order in order QCD particular wereprocess the already near NNLO the non-resonant top-pair contribution production to threshold. the This elevates the theoretical prediction to quadrature, while the inner bar shows only the latter contribution. 8 Conclusions The recent advance in the QCD calculation of the top anti-top threshold [ Figure 18 and height of thethe peak theoretical is indicated uncertainty by withby the adding the red the default (green) uncertainties line input from and the parameters. points. renormalization The scale The black and outer cross variation error represents of bar is obtained JHEP02(2018)125 ] NLO } . code is aNLO, aNNLO { Mathematica > − ). ; }} , order, opt); } NNLO Mathematica . The equivalent in ] to draw Feynman diagrams. AM was sup- mt , width https://qqbarthreshold.hepforge.org/ NLO , a { True 80 a NNLO – 48 – , > , order , } {{ − } w can be downloaded from threshold mu, mu function now includes the NNLO non-resonant contribution { QQbar mt , width { threshold ] and JaxoDraw [ ExceptContributions [nonresonant s , , 79 } ( ResonantOnly > xsection − QQbar NLO = 1 . 0 ; NNLO = 0 . 0 ; . In the following, we summarize the changes and give code examples https://www.hepforge.org/downloads/qqbarthreshold/ option. For example, true a a mu, muw to { only xsection(sqrt threshold sqrts , Contributions ] t t b a r options opt; opt.contributions.nonresonant = TTbarXSection [ const double const double As before, the completeresonant nonresonant contribution can be disabled by setting the option will calculate the cross sectionand with the the NLO NNLO non-resonant correction correction multiplied multiplied by by a a A.1 Non-resonant corrections By default, the ttbar to the cross section.the The contributions NLO and NNLO corrections can be controlled individually with for new functions. An updated online manual is available under A Implementation in The new NNLO corrections haveQQbar been implemented in the new version 2 of the public code Agencia Estatal de Investigacion” (AEI),(FEDER) the through EU the “Fondo Europeo project decelencia FPA2016-78645-P, Desarrollo and Severo Regional” from Ochoa SEV-2016-0597”. the grant MBWilhelm and “IFT Leibniz TR Centro programme have de of been the Ex- supported Deutschecluster by Forschungsgemeinschaft (DFG), the of and Gottfried the excellence DFG “Origin and Structure of the Universe.” We are grateful to B.V. Chokouf´eNejad, C. Shtabovenko, Degrande, F. R. Frederix, Simon, O.made Mattelaer, and use J. of J.F. Piclum, Axodraw vonported [ Soden-Fraunhofen by for a helpful Europeangrant discussions. agreement Union number COFUND/Durham We 267209. Junior PRF Research acknowledges Fellowship financial under support EU from the “Spanish Acknowledgments aNLO = 1 . 0 ; aNNLO = 0 . 0 ; JHEP02(2018)125 (A.3) (A.2) in the , s , alpha) min 1 β x t − . The function β ) . log(sqrt /s ) = 1 min t 2 ( x − function in the header , x as ISR ) 591736 pb after initial state x = (1 . ( . ) (A.1) L ) β max = (330 GeV) = 0 t − xs 1 ( β x/y σ ) ( min x ) the cross section after initial-state threshold x LL ee conv − σ )Γ ) 3.53 y (1 ) covers the whole energy range from zero x ( ( QQbar LL ee L A.2 ) = = 344 GeV, including all known perturbative provides an integrate Γ t ( dx from ( s ¯ – 49 – y ) and a cut-off 1 L dy t √ 0 ( ee 1 Z ¯ y L , Z
and write the cross section as s ) = in Mathematica. The default setting for this option is ) s β t ) = ( ) threshold ( x x if (and only if) the logarithmically enhanced component of x ( ISR L − σ True 1. In order to eliminate this divergence, we can change the )). The non-logarithmic correction can be included with the conv true QQbar > ) + 1] is available in ; σ → = (1 ) − /s t t ). In the following we choose 3.64 x 2 e ( ¯ L in Mathematica. m true 7.1 dt (see ( [log( , which can be used to compute the convolution integral as shown below. max t 0 /π Z ) conv IS is the partonic cross section including the non-logarithmic initial-state radiation α σ const = µ ( ) = s α conv ( 2 σ . It should be set to − The following C++ code prints the cross section Finally, version 2 of A further, purely numerical problem arises from the integrable divergence of the lu- In principle, the convolution integral in ( After defining the luminosity function ISR σ = options opt; opt . ISR false radiation for a center-of-masscorrections: energy of with the modified luminosity function β in C++ and ISRLog integrate.hpp minosity function for integration variable to to the nominal center-of-mass energy.valid However, in our prediction the for vicinity thesection cross becomes of section negligible. the is This only threshold.integral implies (cf. that section Sufficiently we below can introduce the a threshold lower the cut-off actual cross in C++ and ISRConst the initial-state radiation is also included via convolution with the luminosity function. Here, correction option setting with the electron structureradiation functions is given Γ by Initial-state radiation requires thefunctions. computationally Therefore, expensive this convolution correction with is not structure included automatically. A.2 Initial-state radiation JHEP02(2018)125 , QQt : : N3LO, } n ’ ; \ ’ mZ ) ; << s ) ; PS , width grid.tsv”); function(t, beta); mt max ) s q r t { ∗ , s } s , QQt::alpha { min , beta ) ; width t ) x x s e c t i o n ( function .hpp” l u m i n o s i t y − – 50 – 3 3 0 . / ( s q r t log ( s q r t g r i d . hpp” ∗ grid.tsv”]; o p t i o n s ( ) ; mu, mu { double directory() + ”ttbar s , std::pow(t, 1/beta); threshold ; ” t t b a r s = 3 4 4 . ; − width = 350.; ; PS = 1 7 1 . 5 ; s q r t <> min = 3 3 0 . ∗ s q r t width = 1.33; x mu mt mu = 8 0 . ; true > sigma = QQt:: ttbar x = 1 L = QQt:: modified threshold/constants .hpp” threshold/integrate .hpp” threshold/structure threshold/xsection .hpp” threshold/load max = std::pow(1 beta = QQt::ISR t QQt:: integrate(integrand , 0, t > integrand = [=]( sigma ; grid(QQt:: grid ∗ double double double double double double QQt = QQbar << L const = iostream cmath { ” QQbar ” QQbar ” QQbar ” QQbar ” QQbar < < std:: sqrt(x) opt [”QQbarThreshold ‘” ]; ); const double const double const double return main ( ) ; } constexpr constexpr constexpr constexpr constexpr constexpr std : : cout opt . ISR const double const auto const double namespace QQt::options opt = QQt::top QQt : : l o a d sqrts = 344.; order = ”N3LO” ; int beta = ISRLog[sqrts , alphamZ]; width = 1.33; LoadGrid[ GridDirectory } muWidth = 3 5 0 . ; The corresponding Mathematica code is Needs mtPS = 171.5; mu = 8 0 . ; #include #include #include #include #include #include #include JHEP02(2018)125 , ) Γ ep σ 3.35 beta) ∗ and , vwidth2 conv IS 16 σ , order , } function can )), and width function(t, 2 3.17 ), only residue can now take both is very close to the lumi- 3.12 59169 pb, i.e. by less than β (eq. ( . 2 option will produce results mtPS , width { → structure = 0 . In version 2, the correction function, which calculates the width in Mathematica). , β σ } width2 width , order, opts) } )), v threshold 3.18 )) can be controlled individually with the ∗ with modified (TTbarWidth mu, muWidth 3.22 m, width ) can be computed with the new function { { QQbar – 51 – (eq. ( ). The corresponding Mathematica functions ), ( are similarly extended. Finally, the toponium 3.36 3.37 3.19 kinetic s q r t s , ∗ , order, opts) } level (n, mu, width function(t, beta) True > energy m, width − { } and TTbarResidue tˆ(1/beta)] options v − luminosity [ [1 . xmin)ˆbeta ; ISRConst Sqrt − )). The respective Mathematica contribution names are vwidthkinetic t , 0 , tmax ], ModifiedLuminosityFunction[t , beta] TTbarXSection [ { [ width(n, mu, 3.21 . This observation can be used to somewhat accelerate the computation of the convo- To incorporate the width corrections to the quarkonium energy levels from ( ] 4 NIntegrate proportional to the top-quark width is included by default in the prediction for the This new function should not be confused with the older top − 16 Γ ]; Print xmin = (330./sqrts)ˆ2; tmax = (1 Disabling the new corrections inthat version 2 are via similar, the but contributions difference. not identical to version 1 of the code.of the There top are quark itself two as causes opposed for to the the width of a toponium bound state. TTbarEnergyLevel width including thettbar corrections in ( A.4 Note on backwards compatibility the functionthe ttbar massnow and take width bothtions to as arguments. the arguments. wave In functions this from Similarly, way, ( this the function ttbar includes the width correc- are not already availableσ in version 1cross of section. Itsnew components contributions (cf. ( (eq. ( and widthep lution at the cost of accuracy. A.3 Width corrections Among the resonant NNLO electroweak corrections listed in ( Numerically, the structure function under thenosity replacement function. The same holdstuting for modified the modified versions of bothin functions. the Indeed, example substi- changes the10 result for the cross section to JHEP02(2018)125 . . t ),  Γ ) W 0 ) x σ (A.4) (A.5) a ) ( Abs of this is then − r C µ ) w (1 ( in C++ a 3 and ex- s / ( . µ 0 ) , 0 α ) is required C y > + − A.4 ) v . The remaining W grid.tsv NLO ( non-res Abs x directory σ = (1 C 1 ) 0 2.3 w v Γ ( 0 y Γ δ C ( 15 or − . To obtain their contri- . t t 0  m t m ) 2 t < 2 w Γ α µ m = π µ nonresonant ( 4 W w α x internally uses interpolation of a ) parametrize the automated and log µ 3 . The last term in ( w and therefore implicitly contains the + log NNLO log , y t W x + Σ ( 2 , the calculational scheme for the NNLO ) a ( 0,P-wave is the NLO QCD correction to the top-quark has to cancel exactly against the dependence threshold C 1 4.1 w manual Γ manual ) to subtract this double counting. + 01 are not supported. Furthermore, the default – 52 – δ µ . 0 and + Σ A.4 QQbar < 2 ) P-wave , v w ( 0 ) and Σ y , where 3.1.3 w C , y + in Mathematica. The coordinates of the grids are given by 2 automated ) W a Σ NLO non-res x ( 0 -channel propagators in Σ ( σ  ) t s C 0 Γ + Γ 0 2 / σ ) 1 ) v ( r 0 Γ µ δ C automated (  s t α . The numerical effect of this change is of the order of a few per mille for , accounting for variations of the top-quark mass, and m w 2 t = F µ . s Z C /m c m N 2 W 3 NNLO non-res m σ and = Note that highly unphysical top-quark masses, i.e. Second, in contrast to the original code, version 2 now captures the full dependence First, as detailed in sections = . The NLO non-resonant part is proportional to Γ W log t W and we must include the last term in ( tremely tight invariant mass cuts values are assumed form the remaining Standard Model parameters, such as the values of in the resonant cross section.energy Like dependence in of the the resonant part,because we therefore we do have not expressed expandΓ the out the non-resonant cross sectionNNLO in correction terms ( of thewidth. all-order The width same contribution appears in the NNLO calculation of the non-resonant part where the coefficient of the logarithm reads Σ As mentioned before, the dependence on The complete NNLO correction togiven by the non-resonant cross section for arbitrary x which covers changes intwo the grid invariant entries mass Σ cutthe discussed manual in part of section bution the to non-resonant the cross cross section section, these for entries have to be multiplied by a factor of The dynamic numeric evaluationally of prohibitively the expensive. NNLO non-resonantprecomputed Hence, corrections grid. is computation- Forinternal reference grid purposes, is a providedand copy the in variable the GridDirectory directory given by the function grid the (new and old) NNLO electroweak corrections are sizeable. on the scale low energies and significantly less than one perA.5 mille in the Calculation peak of region. the non-resonant correction electroweak corrections to the resonantdictions cross section for has the been cross changed. sectiondiffer at Consequently, between pre- or the beyond NNLO twodifferences that versions, are include typically even electroweak less if corrections than will ization all 1%, scale but new and can corrections far amount below to are the almost disabled. threshold, 10% for where The a the small cross numerical renormal- section is already very small and JHEP02(2018)125 w × y ) with from W x w , a grid y − will sup- ]. In or- container. (1 ] 78 [ − 1 in the absence False ≤ True > t Cuba Docker , where the phase threshold yw entry.tar.gz ≤ > − automated 1 is only computed once > < ], and option to QQbar grid automated xw → 77 [ Σ < t virtualisation software. Af- − . The entry for some Verbose =1 nonres w 2.3 Verbose , it can be imported with image as described above and load- y

e n t r y Docker FastJet nnlo ], 41 [ g r i d Docker automated entry.tar.gz – 53 – . Setting the = Σ } aMC@NLO g r i d n o n r e s 1 the calculation of the automated contribution can = 1, and derive the entries for all other values of ∼ w manual y Σ w automated , y n o n r e s is restricted to the complementary region 0 MadGraph5 Mathematica package distributed with automated and downloading the image Σ . { W i n n l o automated x it amaier/nnlo replaced by the respective values for the parameters introduced above. Σ − − Docker 1 is then given by Σ and yw QQbarGridCalc < w In principle it is possible to compute an entirely new grid with the Especially for large values For convenience we provide a Mathematica interface to the calculation of the NNLO Our code for producing the NNLO non-resonant grid depends on a number of soft- . In this way, the numerically problematic endpoint region < y w docker run docker load QQbarCalcNNLONonresonantGridEntry[xw, yw, exploiting complementary cuts as0 discussed in section space integral in y for each value of final cross section.entries One several way times and to average reduce over this the results. error furtherIn would practice be it is to computationallyof calculate much an more the efficient invariant grid to mass calculate cut, Σ i.e. for fail, in which casenot a a slight severe change problem inregion. as the The the input precision of automated parameters the contributioncan automated may calculation becomes easily help. is essentially not deviate very In constant from high,non-resonant practice, in and the contribution the this this values ones itself obtained is in isand not the logarithmic very reference contributions, large grid this and by translates typically around dominated to 10%. by an the error Since manual of the at NNLO most one per mille in the which returns a list press intermediate output. non-resonant grid entries. After importinging the the entry can be computed with with xw The last linereference of grid. the output corresponds to a grid entry in the same format as in the and run with ware packages, including der to facilitateprovide reproducing an our image resultster that without installing can having to behttps://www.hepforge.org/downloads/qqbarthreshold/ install executed all using dependencies, the we JHEP02(2018)125 1 in in aMC and . The . The HEL 4 in the MG5 , pole and 1.f in figure COEFS / 1 = 3 or comment coefs.f h . The and which is matrix.f uvct where the corre- LOOP loop 4 by modifying the and BORN h1bcd 1.f loop with I ,J for the virtual correc- can be identified from calls ,..., from the code in the di- . To maintain separate IR 1 born.f for the loop diagrams corre- compute h = 1 wmtbx matrix COEFS accordingly. This is done by helas i . The subtraction is achieved 1.ps mp in 1.f in MadGraph 1 at NLO in QCD is created in the epem and with − 003.f LOOP aMC@NLO 2.5.0.beta2 MadGraph ia matrix born.ps sf 1.f bW ¯ h we remove the respective terms in the t b DCONJG(AMP(J)) 2 ∗ → To this end, we identify the corresponding and [QCD] . The same changes are applied to the mul- 3 denotes the finite part, the 1 wmtbx/V0 ampb − , d e 17 2 1 . construction , and − , + – 54 – MadGraph5 e , h in figure c epem 1 . The interference of a given one-loop diagram i = 1 calls in g coef , h 002.f 3 b wmtbx/ 1 matrix.ps t b˜ w h in sf as 3 and 4 using and > helas b IDs of the counterterm diagrams is more complicated since 2 epem , matrix.f loop − by entering the commands . / refers to the code directory. COEFS ∼ 1.f 001.f born MadGraph Madgraph sf /SubProcesses/P0 values can be determined from in prompt. First, we subtract the diagram . LOOP b ∼ 00i . We create a copy called CREATE SF , where the first index K . We note that one should avoid first adding and then subtracting the terms generate e+ e output TBWsubtractions /SubProcesses/P0 TBWsubtractions pole, are defined in > > construction ∼ 2 MadGraph5 hel.f / aMC aMC Last but not least one needs to modify the counterterms given in In the folder To subtract the real corrections Here and in the following coef 17 identification of the they are notAMPL(K,I) drawn but must bethe inferred 1 from the code. The counterterm amplitudes out the calls.the Again graphical the representation relevanttiple diagram precision numbers version in ofmp the virtual corrections given in polynomial.f modified by removinglows the us interference to with removecalls the the to diagrams CREATE tree-level diagramssponding 3 to and the left-hand 4. sides of This the al- cuts. We either add the suffix h1bcd functions B tions we firstfunction apply the MATRIX usualwith subtractions the for tree-level the diagrams squared is tree-level amplitude evaluated to by the the function CREATE squared real amplitude givensponding by the set of function I MATRIX ,J finiteness of the realterms and in virtual the contributions files weremoving also the have terms to containing edit the the product FKS subtraction of the tree-level amplitudes 3 and 4 in the diagram numbers in by removing the terms proportionalsquared to matrix AMP(I) element. Thisborn affects the functionto BORN avoid numerical instabilities since these contributions are divergent at threshold. in the rectory We briefly describe theand implementation all of the the diagrams in subtractionscode figures of for the the diagram computationdirectory of the process B Implementation of the subtractions in MG5 MG5 JHEP02(2018)125 1 ia h . The UVCT MadGraph -terms and comes from 2 b 1 R h contributions of 4 in the function CALLS matrix.ps , 2 the R = 3 5.1 loop J at NLO by comparing the ¯ t and the t to allow for negative values of g ] and its connection to diagram 1 → 28–31 and versions also require modifications 31 ] , h − , f e 1 31 + e , h e -boson exchange already in the process 1 16–23 , h at leading order. A similar argument was Z b 12 MadGraph 1 generated with the older MadGraph version , h 31 in the subroutine HELAS we have to deactivate an internal , + 3. As discussed in section code is shipped with the grid generation routines – 55 – expansion of [ / 2 = 11 5.1 / ¯ tbW . We have checked our procedure by applying similar 1 wmtbx/BinothLHA.f = 29 . Older ρ . → − with I epem e MadGraph with I + e then ]. Then we show that the leading-order term in Threshold ]. with a factor 2 matrix.f 30 singular.f , 31 27 1.f . 0 . d0 ) loop -terms and mass renormalization counterterms which are attributed QQbar 2 l t R in uvct 4 by modifying the interference of the divergent part of the wave function DCONJG(AMP(J)) ... /SubProcesses/P0 ∗ ( wgt . ∼ ,..., calls i f endif = 2 . Furthermore, we have verified that an analogous set of changes correctly removes -boson exchange contribution to the process i /SubProcesses/fks Our modified version of the Following the discussion in section In addition we can implement the minimal subtraction of the UV divergences in Z . First we explain why the cancellation of finite-width and endpoint divergences requires helas ∼ b 1 h a non-vanishing contribution from diagram already put forward in [ a loop-momentum region whichtop is EFT not formulated among in those [ considered to construct the unstable C Further details on theWe comparison extend to in thiscross appendix [ section the at discussion leading about order the in discrepancy the with the result for the in the new version of modifications to the process 2.4.3 the results to the onesgeneration. obtained by excluding the in in the file the squared Born amplitude. code run without producing error messages. This is done by removing the code block finite parts of the wave functiontion renormalization contributions are for not the modified. interference contribu- which Alternatively, one yields can the simply same deactivate results. the check for UV finiteness consistency check for the positivity of the squared real amplitude to make the modified SLOOPMATRIX with renormalization of the tree-level diagramsplicitly, we 3 multiply and AMPL(2,I) 4 within the other tree-level diagrams. Ex- to the loop diagramssecond file in contains the the same multiplicativetree-level wave order diagram. function in renormalization which counterterms Tothe for they subtract remaining the appear diagrams the in in diagrams AMPL(K,I) the squared contribution we remove the terms proportional to first file contains JHEP02(2018)125 , t Γ Z i 19 ) 19 + k , and ( Abs, E C /ρ 0 Γ in the Γ ... → ∼ loop is hard ' E 2 t ) + m W bW x − − 2 p top-quark self-energy is (1 / 0 divergence from the real bW ). Therefore, the resonant / are given by the diagrams s 5 b α 1 p h h + behaves as Γ and p t a Z as was shown by explicit computation 1 contains a ) b h k 1 t ( Abs, h Z C ) k ( Abs, C and – 56 – σ a + 1 OS EW ) t h δZ 1 2 ( vertex. The latter contribution is exactly cancelled by + ¯ tg p t ) scaling, while the loop momentum in the p ), once the symmetric diagrams are considered. In particular, 3.44 p ( , where the loop-momenta in the top anti-top loops are expanded divergences, that are cancelled with endpoint divergences from the t h Z 19 ) is equal to minus one half of the diagram with the field renormalization of /ρ k 0 ( Abs, b 1 Γ C OS EW ). It can be easily checked that ) h σ t × δZ , and the dots stand for other resonant contributions at NNLO and higher orders. 1 2 ), because upon expanding out the external (potential) momenta from the ( 3.44 t / Z . Three-loop cut diagrams generating the NNLO resonant corrections related to the s ) 19 ) and α k ( Abs, = C σ The NNLO resonant corrections related to 3.22 ( ). It should be understood that in the diagram with the self-energy insertion in the Γ h written in ( that the contribution to the crosspart section of the Green function.contain Therefore both resonantcorresponding diagrams in non-resonant the diagrams first line of figure of figure self-energy and vertex loops,the these on-shell diagrams scheme are for equivalentcounterpart zero to of transfered the momentum renormalized (see vertexthe figure in top quark leaving the production vertex, which is proportional to the coefficient one-loop self-energy. Weand have isolated split it that such contributionquark that in leaving one the half the second oftop-quark production line it field of vertex, can entering figure and be the the attributed the electroweak to other correction field half renormalization to to of the the the Coulomb top renormalization potential of (third the diagram in the second line limit gives the top-quarkpotential width, region; such which terms isin are a the already non-relativistic LO accounted propagator). contributionσ for since The by diagram the with replacement the the cut latter self-energy arises contributes from to field renormalization due the absorptive part of the electroweak displayed in figure according to the potential( ( propagator one has to consider only the NNLO piece (the cut self-energy in the on-shell Figure 19 top-quark instability. The top anti-tophard. loops are The potential, symmetric whereas cut the yield diagrams are not displayed. In the second line we display the terms that JHEP02(2018)125 , , ] . 2 2 b and 20 20 1 k (C.1) (C.2) (C.3) , that h 18 2 t − ] discuss m 0 2 2 k 31 t ρ in the hard, m 2 ) 2 (plus left-right  / − 1 2 η 5 ] ][ ρ g 2 2 2 , it corresponds to − ) k 2 η . → k 20 , −  and − v  ]. As already explained 0 2 − b k 1 is considered, while it is 1 ... t 54 ) to the cross section ... k h  Γ (1 a ( m 1 2 /ρ − 2 + ) h q 1 ][2 + 4 s  2 2 ][ 2 1 k k 2 − α η k ) + 8 + 2  1 + − + 2 ×  1 − k q 2 q  2 0 η   0 1 l ) Γ (2 3 −  2 k  3 Γ( t l − k − 2 ( ) −
2 m ) 2 t  − k − 2 t 2 − 1 m − k 2 − m 2 ) l 2 ][ 1 2 ρ 0 2 1 ρ l Γ (2 E b + k W γ E Γ q 2 ][( γ e 1 . – 57 – 2 e
− term arises from the bottom propagators, appears in the ) (4  ) 0 1 1 2 (4  l 4 t ρ k k + t 3 ρ 2 t − 3 1 2 − + 6 1 m ρm iπ m l k iπ 4 η expansion can also be obtained upon application of the [2 ( 1 1 Γ . We obtain for the relevant scalar integrals in this region: 2 k k ρ − − ρ l − must vanish quoting results from [ t d d/ iπ + − 2 b d 6 ) 1, 1 + 2 m 1 q 2 q 2 iπ e 0 , , where the h q l t −  ∼ 6 ]. Let us consider the forward scattering diagram in figure  Z ][( l ]. With the momentum assignment of figure /m = 0 − 2 t 2 0 2 η )Γ( +6 η Γ 2 38 31  k η discussed above, it can be shown that the leading order contribution d/ , d 3 − ρm and b [0] = [real] [1] = [real] ∝ 1 d iπ 1 37 ) I I ρ 2+2 t − 2 h t ρ ρ + 2 η m Z m (2 0 2 × l 2 1 − π t ] only the non-resonant diagram analogue to 2 w ∼ k d/ , the results from the latter refer to the vanishing of resonant contributions η d i m 31 . Forward-scattering diagram whose imaginary part is related to cut diagrams d iπ k 1) α/s [2 Γ (1 , 6.3 − 2 × × Z ρ t = ( ]. In [ m ] = The dominant terms in the A factor 30 η ∼ [ Figure 20 18 I 0 i Both scalar integrals produce a contribution of order Γ The relevant cases are which is related to comes from an additional regionwas with not parametrically considered smaller in virtuality [ ork order method of regions [ whose imaginary part correspondssymmetric to ones), the since no sum other ofthe cuts are cut contributions kinematically diagrams from possible. the Thesoft, authors regions of potential that [ and are ultrasoft obtained by region. replacing However, for the case of the diagram in figure in [ argued that the analogue to in section related to the top-quarkis instability of at NNLO. NLO, while the contribution under discussion here JHEP02(2018)125 , is 2 0 ρ and (C.8) (C.4) (C.5) (C.6) (C.7) ' ρ t 2 dimen- ) m d W , ∼ ]. x , i  − k | , Γ (1 2 i SPIRE ρ . ∼ t + IN 2 w ) m t ][ , µ E 20 agrees with the result | t − ∼ = 4 result used in the  . arXiv:1506.05992 0 i m ) d 2 t 4 k , 2 w suppression. Hence when Γ F /ρ expansion discussed above. µ m 2 s  π non-res C , ρ α 4 s ρ b , (  1 3 was missed in the hadronic tensor α O h . / 2 ln ) σ 0   /ρ 2 Γ − + ln 2 t ρ ρ − 2 w   2 t m and 2 µ arXiv:1303.3758 -integration (1 2 t i ρ m [ ρ 2 w 2 w 2 2 t 2 t ) ) µ  µ a m ( m 0 [0] term has no /ρ ln s I non-res C  ,  α a 3 2 ( ) compared to the 1 + ln ], which identified that the leading-order term Top quark mass measurements at and above  O h ]: a factor (3 − 2 σ ) − and their resonant counterparts at the leading 30 − 30 – 58 – v (2013) 2530 ( 0 b 1 C h h c + 3 ln 2 C 73 2 ln 2 + 2 ln + 2 ln 2 divergence, which are the properties that are needed to , s 7 6 and ) πN − 7 3 ), which permits any use, distribution and reproduction in / a /ρ − 24 7 3 1 s res − , ]. 0 α  h a ( 1 σ +  1 2 1 O h  1 − − = σ [1]. The numerator of the    1 2 I SPIRE adopting the momentum assignment of figure N Study of top quark pair production near threshold at the ILC − N N N IN Eur. Phys. J. ) below, both terms contribute at order 1 [ 2 t , CC-BY 4.0 = = = = originates from the region in the C.8 /m Physics Case for the International Linear Collider ) ) This article is distributed under the terms of the Creative Commons b 2 2 1 )–( /ρ /ρ ) k ) h s s res res , , α α /ρ /ρ b a C.4 ( ( ∼ s s 19 non-res non-res 1 1 , , ]. α O O α h h ) b ] if one takes into account that the leptonic tensor was treated in a ( ( t 1 1 σ σ O h O h : 30 − σ σ ρ , is consistent with the findings of [ SPIRE ρ t IN arXiv:1310.0563 [ threshold at CLIC of diagram For completeness, we finally provide results for the individual contributions to the written in eq. (3) therein. m T. Horiguchi et al., K. Fujii et al., K. Seidel, F. Simon, M. Tesar and S. Poss, We take the opportunity to correct a typo in [ ρ a 19 1 ∼ [3] [1] [2] numerator that comes along extracted as in ( H Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. References The sum of the non-resonant contributions given in [ sions there, which introducespresent a work. factor (1 with cross section from diagrams order in cancel the finite-width divergence atWe the also leading note order that in al the non-vanishing contribution from thein region where (1 the imaginary part contains a 1 JHEP02(2018)125 , ]. heavy ]. (2015) ) Top 2 Phys. ] R SPIRE , (2016) ) 115 − q IN SPIRE [ e ( IN + -Wave Pair m Phys. Rev. e 117 ( Phys. Rev. Lett. ][ Phys. Rev. Lett. S (2005) 67 , / , , 3 s α ]. ]. Production and Decay ]. B 714 SPIRE SPIRE Phys. Rev. Lett. IN IN arXiv:1611.03399 [ SPIRE , [ ]. ][ IN arXiv:1312.4791 NonAbelian Phys. Rev. Lett. Analytic three-loop static hep-ph/0106135 , ][ , ]. [ Three-loop matching of the vector Nucl. Phys. SPIRE , IN ]. SPIRE ][ (1986) 157 IN Annihilation Three-loop static potential ]. in German, MSc Thesis, RWTH Aachen ][ − ]. ]. 2 e SPIRE ]. meson at third order in QCD + IN /m B 181 ) e (2002) 091503 2 s SPIRE ][ S arXiv:1608.02603 α PoS(ICHEP2016)872 IN SPIRE SPIRE arXiv:1401.3004 [ , – 59 – SPIRE [ ][ IN IN Υ(1 IN D 65 ][ ][ ]. ]. Third-order Coulomb corrections to the S-wave Green Third-order correction to top-quark pair production near Third-order correction to top-quark pair production near Reducing the Top Quark Mass Uncertainty with Jet , in preparation. Static QCD potential at three-loop order arXiv:1705.07135 Phys. Lett. [ Colliders , SPIRE SPIRE arXiv:0804.4004 − [ IN IN Power counting in dimensionally regularized NRQCD Threshold Behavior of Heavy Top Production in e ]. (2016) 054029 ][ ][ Phys. Rev. + (1987) 525 [ (2014) 034027 e , arXiv:0911.4742 Ultrasoft contribution to heavy-quark pair production near threshold [ hep-ph/9707313 46 [ (2017) 151 D 94 SPIRE arXiv:1401.3005 arXiv:0911.4335 D 89 IN (2008) 143 [ [ 10 Leptonic decay of the ][ B 668 (1998) 413 Heavy quark potential at order JETP Lett. JHEP Impact of Theory Uncertainties on the Precision of the Top Quark Mass in a (2010) 112002 Phys. Rev. , , ]. Phys. Rev. , arXiv:1506.06864 arXiv:1608.01318 , [ [ 104 D 57 (2010) 112003 (2014) 151801 SPIRE hep-ph/0501289 IN 104 Lett. potential quark anti-quark potential 112 threshold II. Potential contributions Phys. Lett. University, Aachen Germany (2003). Rev. function, energy levels and[ wave functions at the origin threshold I. Effective theory set-up and matching coefficients current Production Cross Section Near Threshold192001 in Collisions Next-to-Next-to-Next-to-Leading Order QCD Prediction for the Top Antitop Grooming Properties of Ultraheavy Quarks Threshold Scan at Future [ Quark Mass Calibration for232001 Monte Carlo Event Generators A.V. Smirnov, V.A. Smirnov and M. Steinhauser, R.N. Lee, A.V. Smirnov, V.A. Smirnov and M. Steinhauser, B.A. Kniehl, A.A. Penin, M. Steinhauser and V.A. Smirnov, M. Beneke et al., C. Anzai, Y. Kiyo and Y. Sumino, M. Beneke and Y. Kiyo, S. W¨uster, M. Beneke, Y. Kiyo and K. Schuller, M. Beneke, Y. Kiyo and K. Schuller, M. Beneke, Y. Kiyo and K. Schuller, P. Marquard, J.H. Piclum, D. Seidel and M. Steinhauser, M.E. Luke and M.J. Savage, M. Beneke, Y. Kiyo, P. Marquard, A. Penin, J. Piclum and M. Steinhauser, A. Andreassen and M.D. Schwartz, I.I.Y. Bigi, Y.L. Dokshitzer, V.A. Khoze, J.H. Kuhn and P.M. Zerwas, V.S. Fadin and V.A. Khoze, M. Butenschoen, B. Dehnadi, A.H. Hoang, V. Mateu, M. Preisser and I.W. Stewart, F. Simon, [9] [6] [7] [8] [5] [4] [20] [21] [17] [18] [19] [15] [16] [13] [14] [10] [11] [12] JHEP02(2018)125 , ]. ]. ] , ]. D ] ]. SPIRE SPIRE IN IN ] (2009) 1 SPIRE (2011) 076 ][ ][ IN SPIRE 12 ][ IN (2012) 034 Phys. Rev. ][ , B 807 01 arXiv:1402.1123 [ JHEP ]. , arXiv:0707.0773 ]. JHEP [ ]. (2010) 186 , Four-fermion production ]. hep-ph/0312331 SPIRE hep-ph/0401002 arXiv:1506.06865 [ Nucl. Phys. [ IN [ , SPIRE SPIRE ][ IN arXiv:1605.03010 Effective theory approach to Effective theory calculation of IN B 840 ]. SPIRE arXiv:1312.4792 (2008) 89 [ (2014) 097501 ][ ][ [ IN ][ SPIRE (2004) 205 -representation for scalar one-loop (2015) 180 D 89 ε B 792 IN (2004) 011602 Dominant NNLO corrections to (2016) 96 ][ ]. Higgs effects in top anti-top production near Nucl. Phys. (2014) 414 Near-threshold production of heavy quarks with Electroweak non-resonant NLO corrections to 93 , B 686 B 899 209 hep-ph/9903260 General [ Top quark production near threshold and the top collisions: Endpoint-singular terms – 60 – SPIRE hep-ph/9707481 hep-ph/0001286 Phys. Rev. [ B 880 IN [ , − Nucl. Phys. e , ][ P-wave contribution to third-order top-quark pair arXiv:1309.6323 + [ e Next-to-next-to-leading order nonresonant corrections to The top-antitop threshold at the ILC: NNLL QCD Nucl. Phys. Nucl. Phys. hep-ph/9711391 (1999) 137 , Asymptotic expansion of Feynman integrals near threshold (2000) 3 , [ Threshold production of unstable top ]. ]. ]. ]. Phys. Rev. Lett. (1998) 428 resonance region 2 , ¯ t Nucl. Phys. t 64 , (2014) 121 B 454 Effective field theory for ultrasoft momenta in NRQCD and NRQED SPIRE SPIRE SPIRE SPIRE 05 IN IN IN IN (1998) 321 in the Top-anti-top pair production close to threshold: Synopsis of recent NNLO arXiv:1307.4337 ][ ][ ][ ][ annihilation First estimate of the NNLO nonresonant corrections to top-antitop b Comput. Phys. Commun. ¯ [ b , − − e JHEP + , W Phys. Lett. e B 522 Foundation and generalization of the expansion by regions + , ]. ]. ]. W Eur. Phys. J. direct threshold , → − (2013) 054011 e SPIRE SPIRE SPIRE + arXiv:1111.2589 IN arXiv:1110.1970 IN arXiv:0807.0102 arXiv:1004.2188 IN Nucl. Phys. [ quark mass unstable particle production resonant high-energy scattering QQbar Nucl. Phys. Proc. Suppl. threshold production at lepton colliders [ [ near the W pair[ production threshold four-fermion production near the W-pair[ production threshold [ threshold top-pair production from 88 threshold in [ e production near threshold results uncertainties M. Beneke and V.A. Smirnov, B. Jantzen, J. Bl¨umlein,K.H. Phan and T. Riemann, M. Beneke, A.P. Chapovsky, A. Signer and G. Zanderighi, M. Beneke, A.P. Chapovsky, A. Signer and G. Zanderighi, M. Beneke, Y. Kiyo, A. Maier and J. Piclum, A. Pineda and J. Soto, M. Beneke, A. Signer and V.A. Smirnov, P. Ruiz-Femen´ıa, A.A. Penin and J.H. Piclum, S. Actis, M. Beneke, P. Falgari and C. Schwinn, B. Jantzen and P. Ruiz-Femen´ıa, M. Beneke, P. Falgari, C. Schwinn, A. Signer and G. Zanderighi, M. Beneke, A. Maier, J. Piclum and T. Rauh, M. Beneke, B. Jantzen and P. Ruiz-Femen´ıa, A.H. Hoang et al., A.H. Hoang and M. Stahlhofen, M. Beneke, J. Piclum and T. Rauh, [37] [38] [39] [35] [36] [32] [33] [34] [30] [31] [29] [27] [28] [25] [26] [23] [24] [22] JHEP02(2018)125 ]. , B ]. ]. (2014) B 368 Phys. SPIRE ] , ]. IN 07 (2000) Phys. Lett. SPIRE ][ , SPIRE IN Nucl. Phys. IN ][ SPIRE , ][ D 61 JHEP IN [ (1937) 125 , ]. Nucl. Phys. , 51 ]. (2010) 014005 vertex at the top quark Second order QCD ¯ t SPIRE (2011) 414 γt IN arXiv:1510.01063 [ Phys. Rev. D 82 SPIRE [ arXiv:1005.1756 , [ IN hep-ph/0605227 ]. ]. [ ][ ]. Electroweak Corrections on the Phys. Rev. hep-ph/9906273 corrections are trivial B 842 , [ ]. s hep-ph/9911490 α , ]. SPIRE SPIRE SPIRE (2016) 227 IN Phys. Rev. IN IN SPIRE [ , ][ IN ][ (2006) 197 SPIRE [ (2011) 034029 Phase Space Matching and Finite Lifetime corrections to the Nucl. Phys. IN ) ]. , s arXiv:1602.08035 ][ (1999) 114015 270-272 Top quark anomalous couplings at the ]. , hep-ph/0604104 αα (1987) 18 B 757 D 84 [ ( – 61 – SPIRE O Threshold resummation for pair production of IN D 60 (1972) 114 SPIRE ][ IN ]. B 281 Non-Hermitian Hamiltonians, Decaying States and ][ C 6 ]. hep-ph/0412258 arXiv:0810.1597 Top near threshold: All Two loop QCD corrections to top quark width [ [ Phys. Rev. Electroweak absorptive parts in NRQCD matching conditions Complete Higgs mass dependence of top quark pair threshold Nucl. Phys. On electroweak matching conditions for top pair production at , SPIRE ]. ]. , (2006) 034002 s Top quark threshold and radiative corrections IN Pole Mass, Width and Propagators of Unstable Fermions arXiv:0801.0669 Phys. Rev. SPIRE α [ ][ IN ), ][ Nucl. Phys. SPIRE SPIRE Effective field theory approach to pionium Phys. Rev. D 74 , W b IN IN (2009) 038 , Nucl. Part. Phys. Proc. hep-ph/9302311 ]. ][ ][ , → [ (2005) 074022 01 hep-ph/9806244 t Top Physics in WHIZARD The automated computation of tree-level and next-to-leading order [ Γ( A Symmetry Theorem in the Positron Theory SPIRE Perturbative heavy quark-anti-quark systems (2008) 116012 D 71 IN Phys. Rev. JHEP [ , , ]. ]. hep-ph/9905543 (1994) 217 [ D 77 arXiv:1405.0301 (1999) 520 [ SPIRE SPIRE IN arXiv:1002.3223 arXiv:1007.5414 IN (1992) 38 threshold [ B 324 Toponium Resonance 544 corrections to Perturbation Theory Rev. Effects for Top-Pair Production Close[ to Threshold threshold coloured heavy (s)particles at hadron[ colliders production to order alpha Phys. Rev. 079 114027 [ International Linear Collider differential cross sections and their matching to parton shower simulations Feynman integrals A.H. Hoang and C.J. Reißer, W.H. Furry, K. Melnikov and O.I. Yakovlev, B. Grzadkowski, J.H. K¨uhn,P. Krawczyk and R.G. Stuart, R.J. Guth and J.H. K¨uhn, K.G. Chetyrkin, R. Harlander, T. Seidensticker and M. Steinhauser, M.M. Sternheim and J.F. Walker, A.H. Hoang, C.J. Reisser and P. Ruiz-Femen´ıa, A. Czarnecki and K. Melnikov, M. Beneke, P. Falgari and C. Schwinn, B.A. Kniehl and A. Sirlin, D. Eiras and M. Steinhauser, A.H. Hoang and C.J. Reisser, Y. Kiyo, D. Seidel and M. Steinhauser, J. Reuter et al., M. Beneke, D. Eiras and J. Soto, E. Devetak, A. Nomerotski and M. Peskin, J. Alwall et al., [57] [58] [54] [55] [56] [52] [53] [50] [51] [48] [49] [45] [46] [47] [42] [43] [44] [40] [41] JHEP02(2018)125 Nucl. Nucl. , ] ]. Phys. , ]. , NLO QCD (2012) ]. B 434 Comput. ]. , Nuovo Cim. SPIRE ]. SPIRE , IN [ IN C 72 SPIRE SPIRE Acta Phys. Polon. ][ , IN IN ]. Comput. Phys. SPIRE ][ ]. ][ , IN (2003) 151 Phys. Lett. [ , arXiv:0908.4272 SPIRE [ pair production in SPIRE IN Single Photon Annihilation [ IN B 653 ]. ]. − W [ Eur. Phys. J. e , (1994) 45 ]. + hep-ph/9602351 e , hep-ph/9807565 83 SPIRE [ SPIRE (2009) 003 IN IN ]. SPIRE arXiv:1411.3132 ]. 10 ]. ][ arXiv:1503.01469 ][ [ Automation of next-to-leading order (1994) 4837 IN (1991) 345 [ Nucl. Phys. using soft collinear effective theory , ]. 64 1 SPIRE A 9 The bottom-quark mass from non-relativistic SPIRE On the Rational Terms of the one-loop SPIRE (1999) 153 JHEP IN → New Developments in FeynCalc 9.0 IN IN , ][ x SPIRE ][ (2015) 42 ][ 118 IN – 62 – (2015) 276 (1985) 466 [ FastJet User Manual ][ ]. Three jet cross-sections to next-to-leading order FEYN CALC: Computer algebraic calculation of 41 197 B 891 arXiv:0802.1876 ]. Regularization and Renormalization of Gauge Fields arXiv:1601.01167 [ Automatized one loop calculations in four-dimensions and SPIRE ]. [ Standard model predictions for IN Comput. Phys. Commun. On Radiative Corrections to , ][ SPIRE Int. J. Mod. Phys. ]. cross-sections and distributions hep-ph/0309176 , ]. SPIRE hep-ph/0404043 IN hep-ph/9512328 [ [ IN [ ][ [ Nucl. Phys. (2008) 004 Comput. Phys. Commun. (2016) 432 , WW SPIRE arXiv:1107.4683 Axodraw , SPIRE [ IN 05 production in hadron collisions IN ][ 207 Sov. J. Nucl. Phys. Deep inelastic scattering as Electroweak effective couplings for future precision experiments (2005) 78 ¯ tH , Comput. Phys. Commun. t Leading logarithmic calculations of QED corrections at LEP (1996) 399 , (1972) 189 Package-X: A Mathematica package for the analytic calculation of one-loop JHEP A Quark mass definition adequate for threshold problems hep-ph/9804241 (2003) 114019 , [ 168 (2011) 31 CUBA: A Library for multidimensional numerical integration Comput. Phys. Commun. ]. , B 467 B 44 (1992) 135 [ arXiv:1111.6097 D 68 [ SPIRE IN hep-ph/0211352 at High-Energy Rev. 1896 Commun. Phys. (1998) 115 computations in QCD: The[ FKS subtraction amplitudes corrections to [ Phys. Feynman amplitudes Phys. Commun. D-dimensions electron-positron collisions integrals B 23 sum rules at NNNLO C 034S1 M. Cacciari, G.P. Salam and G. Soyez, T. Hahn, J.A.M. Vermaseren, G. ’t Hooft and M.J.G. Veltman, R. Frederix, private communication. M. Beneke, R. Frederix, S. Frixione, F. Maltoni and T. Stelzer, G. Ossola, C.G. Papadopoulos and R. Pittau, W. Beenakker, S. Dittmaier, M. Kr¨amer,B. M. Pl¨umper, Spira and P.M. Zerwas, S. Frixione, Z. Kunszt and A. Signer, V. Shtabovenko, R. Mertig and F. Orellana, T. Hahn and M. P´erez-Victoria, W. Beenakker et al., H.H. Patel, R. Mertig, M. B¨ohmand A. Denner, E.A. Kuraev and V.S. Fadin, M. Skrzypek, W. Beenakker and A. Denner, F. Jegerlehner, A.V. Manohar, M. Beneke, A. Maier, J. Piclum and T. Rauh, [77] [78] [79] [74] [75] [76] [72] [73] [70] [71] [68] [69] [65] [66] [67] [62] [63] [64] [60] [61] [59] JHEP02(2018)125 (2009) 180 Comput. Phys. Commun. , JaxoDraw: A Graphical user interface for – 63 – ]. SPIRE IN ][ arXiv:0811.4113 [ drawing Feynman diagrams. Version 2.01709 release notes D. Binosi, J. Collins, C. Kaufhold and L. Theussl, [80]