Resummation for rapidity distributions in top-quark pair production

Transcription

1 Nikhe IPPP/18/99 arxiv: v1 [hep-ph] 26 Nov 2018 Resummation or rapidity distributions in top-quark pair production Benjamin D. Pecjak, a Darren J. Sco, b,c Xing Wang, d and Li Lin Yang d,e, a Institute or Particle Physics Phenomenology, University o Durham, DH1 3LE Durham, UK b Institute or Theoretical Physics, University o Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands c Nikhe, Theory Group, Science Park 105, 1098 XG, Amsterdam, The Netherlands d School o Physics and State Key Laboratory o Nuclear Physics and Technology, Peking University, Beijing 871, China e Collaborative Innovation Center o Quantum Maer, Beijing, China Center or High Energy Physics, Peking University, Beijing 871, China ben.pecjak@durham.ac.uk, d.j.sco@uva.nl, x.wong@pku.edu.cn, yanglilin@pku.edu.cn Abstract: We extend our ramework or the simultaneous resummation o sot and small-mass logarithms to rapidity distributions in top quark pair production. We give numerical results or the rapidity distribution o the top quark or the anti-top quark, as well as the rapidity distribution o the t t pair, inding that resummation eects stabilize the dependence o the dierential cross sections on the choice o actorization scale. We compare our results with recent measurements at the Large Hadron Collider and ind good agreement. Our results may be useul in the extraction o the gluon parton distribution unction ro t production.

2 Contents 1 Introduction 1 2 Formalism 2 3 Numerical results 6 4 Conclusions 11 A Modiications to the coeicient unctions 12 1 Introduction Being the heaviest undamental particle in the Standard Model (SM), the top quark plays an important role in studying the spontaneous breakdown o electroweak symmetry and in searching or new physics beyond the SM. The Large Hadron Collider (LHC) is expected to produce billions o top quarks during its lietime and with such a large amount o data, precision studies in the top quark sector have become a key goal o the LHC program. Accurate measurements o the production and decay channels o the top quark allow us to probe some o the less-well-determined SM parameters, such as the CKM matrix element V tb as well as the gauge and Yukawa coupling o the top quark, which are all sensitive to new physics eects. In addition, the kinematic distributions in top quark production may be aected by the existence o new resonances, particularly in the high energy tails. Last but not least, top quark production processes are important backgrounds or many processes in and beyond the SM, which need to be careully modeled in order to extract possible new physics signals. Even i no deviations rohe SM are ound in the top quark sector, the dierential cross sections in top quark pair production can still help to constrain the parton distribution unction (PDF) o the gluon [1 3]. This is due to the act that around 90% o the t t events at the LHC come rohe gluon-initiated partonic subprocess. The precise knowledge o the gluon PDF is indispensable or Higgs physics, since at the LHC the Higgs boson is also mainly produced via the gluon-usion process. The theoretical predictions or single Higgs boson production, Higgs boson production associated with 1

3 a jet and Higgs boson pair production can be improved by incorporating the t t data in the PDF it. To match the high precision o the LHC experiments, it is necessary to have theoretical predictions with equally high or even higher accuracy. For dierential cross sections in top quark pair production, the best ixed-order results are the next-to-nexo-leading order (NNLO) ones given in [4]. Combining with the resummation ramework developed in [5, 6] at next-to-next-to-leading logarithmic accuracy (NNLL ), the works [7, 8] presented predictions or the t t invariant mass distribution and the top quark transverse momentum distribution up to the NNLO+NNLL precision. In this short article, we extend the resummation ramework to the rapidity distribution o the top/anti-top quark, as well as the rapidity distribution o the t t pair. We present NNLO+NNLL predictions or these observables at the LHC with a center-o-mass energy s = 13 TeV. The paper is organized as ollows. In section 2 we briely review the ormalism or the resummation o threshold logarithms and or the joint resummation o threshold and small-mass logarithms. We then discuss the modiications necessary to our resummation ormalism or the computation o rapidity distributions. In section 3, we present numerical predictions, with emphasis on the sensitivity o the results to scale choices and to PDFs. We conclude in section 4. In appendix A we discuss a ew dierences between the ormulas used in this work and in [7, 8] due to a recent calculation o the NNLO sot unction or top quark pair production [10], and show the numerical impact o these corrections. 2 Formalism In this section, we briely review the resummation ramework established in [5, 6, 8], and discuss its extension to describe rapidity distributions in t t production. We consider inclusive top quark pair production at the LHC p(p 1 ) + p(p 2 ) t(p 3 ) + t(p 4 ) + X, (2.1) where the dierential cross section with respect to the t t invariant mass M t t = (p 3 +p 4 ) 2 and the scaering angle θ in the center-o-mass rame can be wrien as d 2 σ(τ) dm t t d cos θ = 8πβ t dzdx 1 dx 2 δ(τ zx 1 x 2 ) i/p (x 1, µ ) j/p (x 2, µ ) C ij (z, µ ). 3sM t t ij (2.2) Here we have deined s = (P 1 + P 2 ) 2, τ = Mt t 2/s, and β t = 1 4m 2 t /Mt t 2. The hardscaering kernel C ij (z, µ ) and the PDFs i/p (x, µ ) depend on the actorization scale 2

4 µ. The sum runs over all parton species i, j = q, q, g. Note that we have suppressed the dependence o C ij on the kinematic variables M t t, and cos θ or convenience. As in [8], we transoro Mellin space, where the convolution in eq. (2.2) becomes a product which we write as d 2 σ(n) dm t t d cos θ = 8πβ t 3sM t t ij i/p (N, µ ) j/p (N, µ ) c ij (N, µ ), (2.3) where the unctions with tildes are the Mellin transorms o the corresponding momentumspace unctions in eq. (2.2), and N is the Mellin moment variable. In general, the hard-scaering kernels C ij (z, µ ) or c ij (N, µ ) can be calculated in perturbation theory as a series in the strong coupling α s. However, in certain kinematic conigurations, the dierential cross sections are dominated by sot emissions. In such cases, the hard-scaering kernels in the lavor diagonal channels (i.e., ij = q q, gg) are enhanced by Sudakov double logarithms ln 2 N at each order in the strong coupling α s. The methods to resuhese logarithms to all orders in α s or t t production are well-known [5, 9]. The resummation is based on the actorization o the cross section in the limit where inal state gluons are sot and leads to the ormula [ c ij (N, M t t,, cos θ, µ ) = Tr H m ij (M t t,, cos θ, µ ) s m ij ( ) ] ln M t t 2, M t t, m N 2 µ 2 t, cos θ, µ + O ( ) 1, (2.4) N or the hard-scaering kernel where the massive hard unction Hij m is available up to NLO [5], and the massive sot unction s m ij has been calculated up to NNLO [10], though we use only the NLO result or consistency with the hard unction in this work. Together with the anomalous dimensions governing the renormalization group evolution o these two unctions, resummation has been perormed at NNLL accuracy in [5]. Besides sot logarithms, in the high energy regime when M t t, small-mass logarithms ln( /M t t) may also become important. In [6], a ramework to simultaneously resum sot and small-mass logarithms was developed. The actorization ormula or the hard-scaering kernel needed to achieve this resummation is given by 1 ( c ij (N, µ ) = Tr [H ij (M t t, cos θ, µ ) s ij ln M ) ] t t 2, M t t, cos θ, µ N 2 µ 2 ( CD(m 2 t, µ ) s 2 D ln m ) ( ) ( ) t 1 mt, µ + O + O, (2.5) Nµ N M t t 1 For simplicity, we have ignored the n h -dependent contributions roop quark loops. 3

5 where the massless hard unction H ij and the massless sot unction s ij are deined in the limit 0, and are available up to NNLO [11, 12]. The logarithms o are absorbed into the two unctions C D and s D, which are related to collinear and sotcollinear emissions rohe inal state top quarks and are also available up to NNLO [6]. 2 These ingredients allow resummation in this boosted-sot limit to be perormed at NNLL order. To obtain the resummed hard-scaering kernels, one has to derive and solve the RG equations or each o the matching unctions in eqs. (2.4) and (2.5). This allows each unction to be evaluated at a scale which rees it o large logarithms, and resums them in an evolution kernel. We denote the scales at which to evaluate the matching unctions H (m) ij, s (m) ij, C D, and s D by µ h, µ s, µ dh, and µ ds respectively. Given the resummed hard-scaering kernels, a remaining diiculty in evaluating the ormula (2.3) or the dierential cross section is that the Mellin-space PDFs i/p (N, µ ) are not readily available rom program packages such as LHAPDF [13]. It is possible to perorhe Mellin-inversion on the kernel c ij (N, µ ), and carry out the convolution with the x-space PDFs as in, e.g. [14, 15]. However, this method suers rom numerical instabilities due to the act that the resummed kernel C ij (z, µ ) is ill-behaved around the singular point z = 1. An alternative method described in [16] amounts to approximating the parton luminosity unction L ij (N, µ ) i/p (N, µ ) j/p (N, µ ) (2.6) by an analytic expression with ied coeicients. Briely speaking, one can approximate the x-space version o the parton luminosity unction L ij (ξ, µ ) dx 1 dx 2 δ(ξ x 1 x 2 ) i/p (x 1, µ ) j/p (x 2, µ ) (2.7) by a linear combination o Chebyshev polynomials with µ -dependent coeicients. The above ormula can be numerically evaluated using inputs rom LHAPDF and the coeicients can then be ied using standard methods. The Mellin transorm o the Chebyshev polynomials can then be calculated analytically, which inally leads to an expression or the N-space luminosity unction L ij (N, µ ). Note that the it has to be done or each distinct value o µ, which leads to computational overheads i µ depends on the integration variables. In [7, 8], we combined the above treatment o the parton luminosity unction with the resummation ramework in [5, 6] to produce phenomenological predictions or the 2 Note that the coeicient unctions C D, s D and s ij used in this work are slightly dierent to those obtained in [6, 12]. We explain the details in appendix A. 4

6 t t invariant mass distribution as well as the top-quark transverse momentum (p T ) distribution. The transverse momentum o the top quark is obtained in the sot limit via the relation p T = M t tβ t sin θ/2. Since this relation does not involve the variables x 1, x 2 and z in eq. (2.2), the same luminosity unction L ij (N, µ ) can be used or a given p T phase-space point without diiculty. However, to extend our ramework to rapidity distributions, we need to slightly modiy the treatment o the PDFs, which we discuss in the ollowing. We will be concerned with two kinds o rapidities: the rapidity o the t t pair Y t t, and the rapidity o the top quark or the anti-top quark y t/ t. In the sot limit we can express these as Y t t = 1 2 ln x 1 x 2, ŷ = 1 2 ln 1 + β t cos θ 1 β t cos θ, y t/ t = Y t t ± ŷ, (2.8) where ŷ is the rapidity o the top quark in the partonic center-o-mass rame. We start by rewriting eq. (2.2) as d 3 σ(τ) = 8πβ ( t dzdξdx 1 dx 2 δ(τ zξ) δ(ξ x 1 x 2 ) δ Y t t 1 dm t t d cos θ dy t t 3sM t t 2 ln x ) 1 x 2 ij i/p (x 1, µ ) j/p (x 2, µ ) C ij (z, µ ), (2.9) and use the last two delta unctions to integrate over x 1 and x 2 to arrive at d 3 σ(τ) = 8πβ t dz dξ δ(τ zξ) C ij (z, µ ) dm t t d cos θ dy t t 3sM t t ij ( ) ( ) i/p ξe Y t t, µ j/p ξe Y t t, µ. (2.10) Using techniques employed previously or the Drell-Yan process [17], we deine a rapiditydependent parton luminosity unction as ( ) ( ) L ij (ξ, Y t t, µ ) i/p ξe Y t t, µ j/p ξe Y t t, µ, (2.11) and its Mellin transorm 3 L ij (N, Y t t, µ ) 1 0 dξ ξ N 1 L ij (ξ, Y t t, µ ). (2.12) We can now perorm a Mellin transorm on eq. (2.10) to arrive at d 3 σ(n) dm t t d cos θ dy t t 3 This unction is called L rap (N, 1 2 ) in [16]. = 8πβ t L ij (N, Y t t, µ ) c ij (N, µ ). (2.13) 3sM t t ij 5

7 The new luminosity unction L ij (N, Y t t, µ ) can be approximated by an analytic expression using the same techniques as beore. The ormula (2.13) can be used to calculate single-variable dierential cross sections, and can also be used to calculate double dierential cross sections where two kinematic variables are measured simultaneously (see, e.g., [18]). In this paper, we will only study the rapidity distributions, and leave the double dierential cross sections to uture work. It is straightorward to obtain the rapidity distribution o the t t pair by integrating over M t t and cos θ rom eq. (2.13). The integration ranges are given by 1 cos θ 1, 2 M t t s/ cosh(y t t). (2.14) It is also easy to obtain the rapidity distribution o the top quark via a change o variables, which leads to d 3 σ(n) = 8πβ t L ij (N, y t ŷ, µ ) c ij (N, µ ), (2.15) dm t t d cos θ dy t 3sM t t ij where on the right-hand side, ŷ should be expressed as a unction o M t t and cos θ through eq. (2.8). A similar change o variables can be used to calculate the rapidity distribution o the anti-top quark. Phenomenologically, one is oten interested in the average o the y t distribution and the y t distribution, i.e. dσ dy 1 2 ( dσ dy t + dσ yt=y dy t y t=y ). (2.16) In the next section, we will present our predictions or the Y t t distribution and the y distribution. Finally, to obtain precision predictions, it is necessary to combine the resummed results with ixed order ones whenever possible. The ixed order part accounts or ormally power-suppressed terms, which can be important numerically. This matching procedure was described in detail in [8], and we reer the interested reader to that article. 3 Numerical results We now use the ormalism introduced in the last section to produce numerical results relevant or LHC experiments at s = 13 TeV. Throughout this section we set the top quark mass to. For both the Y t t distribution and the y distribution, the deault actorization scale is chosen to be µ de = H T /4 ollowing [4], while the 6

8 deault values o the other matching scales are set as µ de h = and µ de µ de dh constant and H T is deined by = H T /2, µ de s = H T / N, ds = / N ollowing [8]. Here N Ne γ E with γ E denoting Euler s H T p 2 T,t + m2 t + p 2 T, t + m2 t. (3.1) We compare predictions using two dierent PDF sets: CT14 [19] and NNPDF3.1 [20], obtained rom LHAPDF with α s (m Z ) = We match our resummed predictions to ixed-order results at NLO and NNLO. The NLO distributions [21 23] are generated using MCFM [24], while the NNLO ones [4] are obtained rom astnlo [25, 26] using the tables available with [27]. Note that the astnlo tables do not provide scale variations and as a result, we only provide the central values or the NNLO(+NNLL ) predictions. On the other hand, or the NLO+NNLL predictions, we estimate the perturbative uncertainties by varying each o the scales individually up and down by a actor o two and combining the resulting variations o the dierential cross sections in quadrature. All our predictions are calculated using NNLO PDFs. Only when we show the NLO predictions or comparison do we use NLO PDFs. In igure 1 we compare our resummed predictions or the Y t t distribution to the ixed-order results at NLO and NNLO. We show the results with the CT14 PDF sets here. The let plot shows a comparison between the NLO and NLO+NNLL predictions, where in the lower panel we show the ratio deined by Ratio = dσ dσ NLO (µ de ). (3.2) We see that the two results are rather similar, with the scale dependence slightly reduced by the resummation. On the right plot o igure 1, we compare the central values o the NNLO and the NNLO+NNLL predictions by displaying their ratios to the NLO result (the K-actors). The NLO+NNLL result is also shown (with uncertainty) as a reerence. We ind that matching our resummation to NNLO compared to NLO predictions increases the central value by about 8%. We also ind that the central values o the NNLO and NNLO+NNLL results are close to each other. This implies that with the choice µ = H T /4 in ixed-order calculations, higher order corrections beyond NNLO are well under control, consistent with the behavior o the M t t and p T distributions studied in [8]. In igure 2 we compare the theoretical predictions to recent experimental measurement rohe CMS collaboration using 35.8 b 1 o data [18]. The let and the right plots show predictions produced with the CT14 and the NNPDF3.1 PDF sets, respectively. The CMS results are given in terms o the dierential production rate in the 7

9 dσ/dy NLO NLO+NNLL' de µ = H T /4 CT14_(n)nlo_as_0118 Ratio = NLO+NNLL'(NLO)/NLO K Factor de µ = H T /4 CT14_(n)nlo_as_0118 NLO K Factor = dσ/dσ NLO+NNLL' NNLO+NNLL' NNLO Ratio Y Y Figure 1. Let: the Y t t distribution at NLO and NLO+NNLL with scale uncertainties; Right: the ratios (K-actors) o NNLO and NNLO+NNLL to NLO with central values only, and NLO+NNLL with scale uncertainties. Results are obtained using the CT14 PDF sets. dσ/d Y 150 NNLO+NNLL' NNLO CMS(l+j) NLO+NNLL' -1 LHC 13TeV CMS 35.8b de µ = H T /4 CT14_nnlo_as_0118 dσ/d Y 150 NNLO+NNLL' NNLO CMS(l+j) NLO+NNLL' -1 LHC 13TeV CMS 35.8b de µ = H T /4 NNPDF31_nnlo_as_ Ratio Y Ratio Y Figure 2. Comparison between theoretical predictions and experimental data rohe CMS collaboration [18] or the Y t t distribution in the lepton+jets channel. The let and right plots use CT14 and NNPDF3.1 PDF sets, respectively. In the lower panels we normalize everything to the NNLO central values. lepton+jets channel. In order to compare with that, we must rescale our results by the branching ratio o the t t pair decaying into the b blνjj inal state, where l = e, µ. We assume the top quarks decay exclusively to a boom quark and a W boson, and extract the theoretical predictions or the braching ratios o the W lν decay and the W q q decay rohe PDG review [28]. The rescaling actor or our predictions is thereore given by Br(t t b blνjj) = (3.3) 8

10 dσ/dy NLO NLO+NNLL' de µ = H T /4 CT14_(n)nlo_as_0118 Ratio = NLO+NNLL'(NLO)/NLO K Factor de µ = H T /4 CT14_(n)nlo_as_0118 NLO K Factor = dσ/dσ NLO+NNLL' NNLO+NNLL' NNLO Ratio y y Figure 3. Let: the y distribution at NLO and NLO+NNLL with scale uncertainties; Right: the ratios (K-actors) o NNLO and NNLO+NNLL to NLO with central values only, and the NLO+NNLL with scale uncertainties. Results are obtained using the CT14 PDF sets. dσ/d y 150 NNLO+NNLL' NNLO CMS(l+j) NLO+NNLL' -1 LHC 13TeV CMS 35.8b de µ = H T /4 CT14_nnlo_as_0118 dσ/d y 150 NNLO+NNLL' NNLO CMS(l+j) NLO+NNLL' -1 LHC 13TeV CMS 35.8b de µ = H T /4 NNPDF31_nnlo_as_ Ratio y Ratio y Figure 4. Comparison between theoretical predictions and the experimental data rohe CMS collaboration [18] or the y distribution in the lepton+jets channel. The let and right plots use CT14 and NNPDF3.1 PDF sets, respectively. In the lower panels we normalize everything to the NNLO central values. Ater the rescaling, the theoretical predictions are in excellent agreement with data. However, we ind that the predictions rohe two PDF sets have slightly dierent shapes. Especially in the tail region (large Y t t ), the CT14 PDFs tend to predict a higher production rate than the NNPDF3.1 PDFs. It is possible to exploit this region to gain urther inormation about the parton distributions, which we leave or uture investigation. 9

11 dσ/dy NLO(µ de =H T /4) NLO(µ de =M /2) dσ/dy NLO+NNLL'(µ de =H T /4) NLO+NNLL'(µ de =M /2) CT14_nlo_as_0118 CT14_nnlo_as_ Y Y dσ/dy NLO(µ de =H T /4) NLO(µ de =M /2) dσ/dy NLO+NNLL'(µ de =H T /4) NLO+NNLL'(µ de =M /2) CT14_nlo_as_0118 CT14_nnlo_as_ y y Figure 5. Comparison between the two deault choices or the actorization scale, µ de H T /4 and µ de = M t t/2. = We now turn to the average rapidity distribution deined in eq. (2.16). Figure 3 shows predictions or the y distribution in a orm analogous to igure 1. We observe behavior similar to that o the Y t t distribution: that the resummation eects reduce the perturbative uncertainties, and that the NNLO+NNLL results are close to the NNLO ones. We then compare the various theoretical predictions to the CMS data in igure 4. As beore, we need to rescale the theoretical results by the actor in eq. (3.3). We again ind a good overall agreement except or the last bin, where the theoretical predictions tend to overestimate the cross section. We also see that the predictions rohe two PDF sets are slightly dierent here, hinting that the y distribution may also be used to improve the PDF iing. One o the important indings o [8] is that resummation eects stabilize the dependence o the dierential cross sections on the choice o the actorization scale µ. Given 10

12 the new results in this work, it is interesting to do the same comparison or the rapidity distributions. In igure 5, we present the NLO and the NLO+NNLL predictions with two deault choices or µ : µ de = H T /4 and µ de = M t t/2, with the other scales chosen as beore. We see that or both the Y t t distribution and the y distribution, the NLO+NNLL results exhibit smaller sensitively to the choice o µ de than the NLO ones. This is similar in spirit to the conclusion o [8]. Finally, it would be interesting to compare the scale dependence o the NNLO and the NNLO+NNLL results. We have all ingredients in our resummation ormula ready or the NNLO+NNLL matching including scale variations, and it will be straightorward to combine them with the NNLO results rom [4]. Based on the experience gained rohe studies o dierential distributions in [8], we would expect that the NNLO+NNLL rapidity distributions are close to the NNLO ones or the choice µ de = H T /4, and that they are less sensitive to the dierent choices o the deault actorization scale. We hope to validate these expectations in the uture. 4 Conclusions In this work we have extended the resummation ramework or top quark pair production to the rapidity distribution o the t t system (Y t t), as well as the rapidity distributions o the top quark and the anti-top quark (and their average y ). Predictions have been presented both at NLO+NNLL and NNLO+NNLL accuracy, and are compared to recent experimental measurements in the lepton + jets channel by the CMS collaboration with good agreement. We ind that the resummation eects are mild with the deault scale choice µ de = H T /4, showing that higher order corrections beyond NNLO are under control. The ixed-order results are, however, more sensitive to the choice o µ. When µ de = M t t/2, or example, the NLO predictions are much lower than the results obtained with µ de = H T /4. In contrast, the NLO+NNLL results are aected much less by parametric changes o the deault actorization scale. In the comparison with experimental data, we ind that the theoretical predictions in the boosted regions (high Y t t or high y ) exhibit some dierences when dierent PDF sets are used. This region can thus be used to constrain the gluon PDF in the uture, which is indispensable or improving the precision o the theoretical prediction or Higgs boson production. In appendix A, we have presented some slight dierences in the coeicient unctions used in this article compared to those used in earlier works. We show that the modiications have minimal impact on the M t t and p T distributions, and thereore the results o the earlier works are not altered. 11

13 Acknowledgements This work was supported in part by the National Natural Science Foundation o China under Grant No and D.J.S. is supported under the ERC grant ERC-STG A Modiications to the coeicient unctions The coeicients C D and s D appearing in the actorization ormula (2.5) were extracted in [6] and subsequently used in [7, 8]. However, it was noted in [6] that there exists an inconsistency in the literature which may lead to a shit o a constant term at order αs 2 between C D and s D. This shit has indeed been conirmed by the explicit calculation o the NNLO massive sot unction s m ij in [10], or which the small-mass actorization can be validated by taking the limit M t t. Another by-product o the calculation in [10] is the extraction o a purely-imaginary o-diagonal contribution to the massless sot unction s ij, which had previously been omied in [12]. This contribution arises rom gluon exchanges among 3 Wilson lines, and is important or both the massive and massless sot unctions to satisy their RG equations. For these reasons, the coeicient unctions s ij, C D and s D used in this work dier slightly to those used in [7, 8]. The purpose o this appendix is to present these modiications and to assess their numerical impact on the M t t and p T, distributions. The matching unctions C D and s D admit a perturbative expansion in α s which we write as C D = 1 + α s 4π C(1) D s D = 1 + α s 4π s(1) D ( + αs ) 2 (2) C D 4π +, ( + αs 4π The modiication concerns the NNLO coeicients C (2) D. We denote the expres-, while the correct ones used sions o these coeicients used in [7, 8] by C (2) D,old in this work by C (2) D,new and s(2) D,new ) 2 s (2) D +. (A.1) and s(2) D,old. They are related by and s(2) D C (2) D,new = C(2) D,old 4π2 C A C F, s (2) D,new = s(2) D,old + 4π2 C A C F. (A.2) Another modiication concerns the NNLO massless sot unction s (2) ij. Again we denote the old and the new expressions with corresponding subscripts. They are related by s (2) ij,new = s(2) ij,old + s(2) ij,3w. (A.3) 12

14 observable bin old new [580,620] GeV pb pb M t t [2500,3000] GeV pb pb [50,] GeV pb pb p T, [800,900] GeV pb pb Table 1. Comparison o the numerical results computed with the old (used in [7, 8]) and the new (used in this work) coeicient unctions s ij, C D and s D. We show the central values with the deault scale choices given in [8]. The results here are obtained using the NNPDF3.0 PDF sets with α s (M Z ) = [29]. The subscript 3w makes it clear that this contribution arises rom correlations among 3 Wilson lines, which is non-vanishing in the virtual-real diagrams at NNLO. The sot unction is a Hermitian matrix in color space, and thereore it is enough to give the non-zero entries in the upper-right part. For the q q channel, we have ( ) [ s (2) L 2 ( q q,3w = 32iπ H0 (x t )+H 1 (x t ) ) +L ( ) H 0,0 (x t )+H 1,0 (x t ) H 1,1 (x t ) H 2 (x t )+ζ where + H 1,2 (x t ) + H 2,0 (x t ) + 2H 2,1 (x t ) + H 0,0,0 (x t ) + 2H 1,0,0 (x t ) ] + 2H 1,1,0 (x t ) + H 1,1,1 (x t ) + 2H 3 (x t ) + π2 3 H 0(x t ) + π2 2 H 1(x t ) 2ζ 3, (A.4) ( ) M 2 x t = (1 β t cos θ)/2, L = ln t t. (A.5) N 2 µ 2 The contribution or the gg channel is given by ( ) s (2) gg,3w = 9 ( ) s (2) q q,3w. (A.6) In the above ormulas we have set the number o colors N c = 3 and the number o light quarks N l = 5. The H i,j,.. unctions denote harmonic polylogarithms with weights speciied by the indices {i, j,..}. We now turn to assess the numerical impact o these modiications. Since the rapidity distributions have never been computed with the old coeicients, we only need to compare the M t t and p T, distributions. In table 1, we list the central values or the integrated cross section in two sample bins or each o these two distributions. The two bins are representative or the un-boosted and the boosted region, respectively. In igure 6, we show the two distributions with uncertainty bands relecting the scale variations. From both the table and the plots, we ind that the numerical dierences 13

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