Momentum-space threshold resummation in tW production at the LHC
PPrepared for submission to JHEP
CERN-TH-2019-014LA-UR-19-21475TUM-HEP-1190/19
Momentum-space threshold resummation in tW production at the LHC Chong Sheng Li, a,b
Hai Tao Li, c Ding Yu Shao d and Jian Wang e,f a Department of Physics and State Key Laboratory of Nuclear Physics and Technology, PekingUniversity, Beijing 100871, China b Center for High Energy Physics, Peking University, Beijing 100871, China c Theoretical Division, Los Alamos National Laboratory,Los Alamos, NM, 87545, USA d CERN, Theoretical Physics Department, CH-1211, Geneva 23, Switzerland e Physik Department T31, Technische Universität München, James-Franck-Straße 1, D–85748 Garch-ing, Germany f School of Physics, Shandong University, Jinan, Shandong 250100, China
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We calculate the soft-gluon corrections for tW production to all orders. Thesoft limit is defined in the pair invariant mass or one particle inclusive kinematic schemes.We find that at NLO the contribution of the soft-gluon effect dominates in the total crosssection or the differential distributions. After resumming the soft-gluon effect to all ordersusing the renormalization group equation, we find that the NLO+NNLL results increasethe NLO cross sections by ∼ depending on the scheme and the collider energy.Our results are in agreement with the measurements at the 8 and 13 TeV LHC. We alsoprovide predictions for the total cross section at the 14 TeV LHC. a r X i v : . [ h e p - ph ] J un ontents The top quark, the most massive elementary particle discovered so far, is playing an im-portant role in testing the standard model (SM) and searching for new physics beyond theSM. At a hadron collider, such as the large hadron collider (LHC), the top quarks canbe produced in pairs via strong interaction or in association with a jet or a W boson viaweak interactions. The associated production with a W boson offers a particular windowto the weak interactions of the top quark and potentially can lead to a direct measurementof the CKM matrix element V tb . Besides, tW production is the second largest single topproduction channel and thus serves as an essential background in search for new physics.So far, the LHC has accumulated a large number of data, based on which the total anddifferential cross sections of this channel have been measured directly [1–7].On the other hand, the precise theoretical predictions provide a valid framework inwhich important information can be extracted from the experimental data. Since theleading-order (LO) cross section of the W boson associated production is proportionalto the strong coupling α s and | V tb | , it is crucial to understand the value of α s in order toextract V tb . The inclusion of higher order QCD corrections can help to estimate the fac-torization and renormalization scale dependence of the cross sections. The next-to-leadingorder (NLO) QCD corrections have been calculated in refs. [8–11], and the results includingdecays of the top quark and the W boson are also available [12]. The parton shower effectsin this channel have been studied in refs. [13–15].When considering the higher order corrections for tW − production, there is one sub-tlety to deal with. Due to the same tW − b final state in both the real correction to tW − and the t ¯ t production with the decay ¯ t → W − b , one needs to find a way to differentiatethe two processes or to define the tW − process properly beyond tree level. There are sev-eral proposals on the market [16–19]. Here we point out that so far we discuss the tW − – 1 –roduction in the five flavor scheme, i.e., the LO is gb → tW − . It is also possible to workin the four flavour scheme in which the LO is gg → tW − ¯ b [20, 21].The next-to-next-to-leading order (NNLO) QCD corrections to tW production are notavailable for now. Though the NNLO N -jettiness soft function, one of the ingredientsfor an NNLO calculation using N -jettiness subtraction method, has been computed inrefs. [22, 23], the two-loop virtual correction is still the bottleneck because of its dependenceon multiple scales. The soft gluon corrections near the threshold have been calculated up toNNNLO based on next-to-next-to-leading logarithms (NNLL) resummation [24–26], whichare considered as an approximation to the full higher order corrections. The three-loop softanomalous dimension for tW production was calculated by ref. [27] which can be used tostudy the full NNNLO threshold effects.In this paper, we will present the soft gluon corrections to tW production using thesoft-collinear effective theory [28–32] (see [33] for a review) which separates the hard contri-butions with the large momentum transfer and the soft gluon corrections characterized aslow energy contributions. Two different definitions of the soft limit are investigated. Oneis measured by the threshold variable − z = 1 − M tW / ˆ s → , while the other is given by s = ( p + p − p t ) − M W → . In principle, these two definitions encode the same softgluon physics in the threshold limits and they only differ by power suppressed corrections.The threshold contributions up to NNLO are obtained and the resummation is achievedthrough solving the RG equations of the hard and soft functions. Our results could betaken as an important theoretical input in future experimental analyses.The paper is organized as follows. In section 2 we show the basic information aboutthe kinematics in this process and the factorization formula of the cross section in the softlimit. The numerical results and relevant discussions are then presented in section 3. Weconclude in section 4. The evolution equation of parton distribution functions (PDFs) inthe threshold limits and the analytic result of the soft function are given in the appendices. We consider inclusive stable top quark and W boson associated production at the LHC p ( P ) + p ( P ) → t ( p ) + W − ( p ) + X ( P X ) , (2.1)where X denotes all the other possible extra radiations in the final states. In the thresholdlimit at the leading power we only need to consider the partonic channel b ( p ) + g ( p ) → t ( p ) + W − ( p ) + X ( p X ) . (2.2)The corresponding LO Feynman diagrams are shown in figure 1. The partonic kinematicvariables are defined to be ˆ s = ( p + p ) , ˆ t = ( p − p ) − m t , ˆ u = ( p − p ) − m t , ˆ t W = ( p − p ) − M W , ˆ u W = ( p − p ) − M W . (2.3)– 2 – b tW − Figure 1 . LO Feynman diagrams for tW − production. The corresponding variables at hadronic level are s = ˆ s/x /x , t = ˆ t /x , u = ˆ u /x , t W = ˆ t W /x , u W = ˆ u W /x , (2.4)where x , are the Bjorken scaling variables.In the soft limit or threshold limit, the real emissions are highly constrained, only softgluons allowed in the final state X . This limit would be reached if the invariant mass of thefinal state M = (cid:112) ( p + p ) approaches the initial partonic center-of-mass energy √ ˆ s . As aresult, the variable − z ≡ − M / ˆ s → in the threshold limit , and the perturbative expan-sion of cross section contains a series of large logarithms α ns (cid:2) ln n − i (1 − z ) / (1 − z ) (cid:3) + ( i =1 , , ..., n ) , which might spoil the convergence of perturbative series. It is our purpose inthis work to study the threshold behavior and resum such large logarithms to all orders.Since the soft limit is characterized by the final-state two particles’ invariant mass, it iscalled the pair invariant mass (PIM) scheme. Besides, there is another scheme, called oneparticle inclusive (1PI) scheme, in which the soft limit is defined by partonic level s → with s ≡ ˆ s + ˆ t + ˆ u + m t − M W = ( p + p X ) − M W . The two schemes measure the softlimit in different ways and the combination of the studies in two schemes provides morecomplete information on the structure.In the rest part of this section we briefly show the factorization formula for tW produc-tion, which can be derived in a similar way used in the other processes, such as the singletop or top quark pair productions [34–38]. In the PIM scheme the cross section in thethreshold limit can be factorized to a product of the hard and the soft function [40, 41],which describes the physics at two different scales, i.e., the large hard scale and the smallsoft scale, respectively. They contain no large logarithms at their intrinsic scales ( µ h and µ s respectively) as expected, since they depend only on a single scale there. The resummationof all the large logarithms caused by soft gluon effects is achieved by evolving the two func-tions from the intrinsic scales to a common factorization scale using their renormalization In the central-of-mass frame the energy of the soft radiation is E g ≈ M (1 − z ) / √ z . This factorization is carried out at the leading power of the threshold variable. At next-to-leadingpower, the threshold factorization becomes more complicated; see the recent paper [39]. – 3 –roup (RG) equations. The RG improved differential cross section can be written as d σ PIM dM d cos θ = λ / πsM (cid:88) ij (cid:90) τ dzz (cid:90) z dxx f i/p ( x, µ f ) f j/p ( z/x, µ f ) H ( µ h ) U PIM ( µ h , µ s , µ f ) × z − η (1 − z ) − η ˜ s PIM (cid:18) ln M (1 − z ) zµ s + ∂ η , µ s (cid:19) e − γ E η Γ(2 η ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) η =( C A + C F ) a γ cusp ( µ s ,µ f ) , (2.5)where polar angle of top quark θ is defined in the center-of-mass frame, λ = (1 − m t / ˆ s − M W / ˆ s ) − m t M W / ˆ s , τ = M /s and the other kinematic variables have already beendefined in eq. (2.3). For convenience we suppress the dependence of hard function H , softfunction ˜ s PIM and evolution factor U PIM on the kinematic variables. ˜ s PIM denotes the softfunction defined in the Laplace space. The evolution factor embodies RG running fromhard and soft scale to factorization scale, which is expressed as U PIM ( µ h , µ s , µ f ) = (cid:18) M µ h (cid:19) ( C A + C F ) a γ cusp ( µ s ,µ h ) exp (cid:104) C A + C F ) S ( µ h , µ s )+ a γ h ( µ s , µ h ) + 2 a γ φq ( µ s , µ f ) + 2 a γ φg ( µ s , µ f ) (cid:105) . (2.6)where the definitions of function S and a γ and all relevant anomalous dimensions, e.g., γ φ q ,can be found in appendix A of ref. [34]. The hard anomalous dimension specific to the tW process is given by [42, 43] γ h = 2 (cid:0) γ Q + γ q + γ g (cid:1) − C A γ cusp ln m t ( − ˆ s )ˆ u − (cid:18) C F − C A (cid:19) γ cusp ln m t ( − ˆ s )ˆ t . (2.7)In the 1PI scheme, the soft radiations are characterized via the threshold variable s ≈ p · k with k the sum of all the momenta of the soft final state [35, 38, 44, 45]. Theanalysis is simplified by going to the rest frame of the inclusive final state W + X where | (cid:126)p W | = O ( s /M W ) . In this frame the energy of the soft radiation is E g ≈ s / M W . In the1PI scheme, the RG improved cross section is d σ dp T dy = 116 πs (cid:88) ij (cid:90) x min1 dx x (cid:90) s max4 ds s + M W − m t − x t f i/p ( x , µ f ) f j/p ( x , µ f ) × H ( µ h ) U ( µ h , µ s , µ f )˜ s ( ∂ η , µ s ) 1 s (cid:18) s M W µ s (cid:19) η e − γ E η Γ(2 η ) , (2.8)where the integration range is defined as x min1 = ( M W − m t − u ) / ( s + t ) , s max4 = m t − M W + u + x ( s + t ) , and momentum fraction x is defined via x and s as x = ( s + M W − m t − x t ) / ( u + x s ) . The Mandelstam variables are related to the top quark’stransverse momentum p T and rapidity y via t = −√ s m ⊥ e − y , and u = −√ s m ⊥ e y , with– 4 – ⊥ = (cid:113) p T + m t . Here the 1PI evolution factor has the form as U ( µ h , µ s , µ f ) = (cid:18) M µ h (cid:19) ( C A + C F ) a γ cusp ( µ s ,µ h ) exp (cid:34) C A + C F ) S ( µ h , µ s ) + a γ h ( µ s , µ h )+ 2 a γ φq ( µ s , µ f ) + 2 a γ φg ( µ s , µ f ) + a γ cusp ( µ s , µ f ) (cid:32) C A ln M W µ s (cid:0) ˆ t W (cid:1) + C F ln M W µ s (cid:0) ˆ u W (cid:1) (cid:33)(cid:35) . (2.9)The difference between the RG factors in eq. (2.5) and eq. (2.8) arises from the RG equationof the PDFs and soft function. We present the RG evolution of the PDFs in the PIM and1PI schemes in appendix A. The two-loop anomalous dimensions that govern the evolutionof the hard and soft functions and thus determine the scale dependent part are derived fromthe general structure of the anomalous dimension [42, 43]. The scale independent part ofthe hard function has been obtained at NLO using modified MadLoop [46] which makesuses of Ninja [47], CutTools [48] and OneLOop [49] packages. We have computed one-loopsoft function analytically, which is shown in appendix B. Combining all the ingredientstogether, we have checked the RG invariance dd ln µ (cid:0) f i/p ⊗ f j/p ⊗ H ⊗ S PIM , (cid:1) = 0 (2.10)in both of the kinematic schemes.The NLO and NNLO leading power contributions are obtained by setting the scales µ h , µ s , µ f in Eqs. (2.5) and (2.8) equal. In this way, for PIM scheme we capture all thethreshold logarithms α s [ln n (1 − z ) / (1 − z )] + with n = 1 , at NLO and α s [ln n (1 − z ) / (1 − z )] + with n = 3 , , , at NNLO, as well as the scale dependent logarithms predicted by eq. (2.5).The similar procedure can be applied to obtain the threshold enhanced logarithms for 1PIscheme. In the following calculations the approximate NNLO (aNNLO) cross section isdefined as dσ (aNNLO) = dσ (NNLO leading) + dσ (NLO) − dσ (NLO leading) , (2.11)where the NLO power suppressed terms in − z or s have been included to give moreprecise results. We can also match the resummed prediction to the fixed order result by dσ (NLO + NNLL) = dσ (NNLL) + dσ (NLO) − dσ (NLO leading) (2.12)with the NNLL result given by eq. (2.5) and eq. (2.8). To perform the numerical calculation, the input parameters are set as m t = 173 . GeV, Γ t =1 . GeV, M W = 80 . GeV, α = 1 / . and the Fermi-constant G F = 1 . × − GeV − . For the LO and NLO calculations we use the CT14 LO and NLO PDF sets [50] asprovided by the LHAPDF library [51], respectively. The aNNLO and resummed predictions– 5 – pb] PIM 1PIb-veto DS DR b-veto DS DRLO . +5% − . +5% − NLO bg . − . − NLO leading . − . − NLO . +0% − . +2% − . +2% − . − . +1% − . +2% − power corr. − . +0 . − . − . +0 . − . − . +0 . − . − . +0 . − . − . +0 . − . − . +0 . − . Table 1 . The fixed-order total cross section and the power corrections for tW − production with √ s = 8 TeV. The power corrections are defined as dσ (NLO) − dσ (NLO leading) . The LO results aredifferent in two schemes due to the different choice of the factorization scale. The scale uncertaintiesare shown. are obtained using CT14 NNLO PDF sets. For fixed-order calculations the renormalizationscale is set to be the same as the factorization scale. It is natural to set the default hardscale to be the invariant mass of the top quark and W boson, i.e., µ h = M , where the hardfunction contains no large logarithms. The soft scale is chosen numerically according to thecriterion that the perturbative series of the soft function are well behaved [41]. Explicitly,we find that the ratio of the soft and hard scale is . ∼ . in the PIM scheme and . ∼ . in the 1PI scheme at the 8 TeV LHC. The default factorization scale has been chosen to be µ f = M and µ f = m t + M W in the PIM and the 1PI schemes, respectively. The final scaleuncertainties are evaluated by varying these scales by a factor of two independently.As discussed in the introduction there are several methods to deal with the problemof the interference between the real corrections to tW production and t ¯ t calculation [12,13, 16–19] at NLO, such as Diagram Removal (DR), Diagram Subtraction (DS) and b-jettransverse momentum veto. The differences between these schemes have been discusseda lot before; see e.g. refs. [13, 14]. Since they are only relevant in the power suppressedchannels , we will not repeat the discussion about their difference in this paper. We noticethat in refs. [25, 26] where the higher order threshold corrections were studied, the powercorrections as well as certain leading power logarithmic independent terms are not taken intoaccount. In the rest of this section we only show the predictions of the process pp → tW − .The total cross section for tW − and ¯ tW + can be obtained by doubling the results, asdemonstrated in ref. [13]. The NLO cross sections in the b-jet veto scheme are evaluatedusing MCFM [12] with p b − jet T < GeV. The cross sections in the DS and DR schemes arecalculated by POWHEG-BOX [14, 52].Before presenting the resummed result, we firstly investigate the contribution of theleading power terms. From table 1 we can see that the NLO corrections are sizeable,enhancing the LO result by ∼ depending on the different methods to isolate the tW process. These NLO corrections get contributions from all the bg , gg and qq (cid:48) channels,though at LO only bg channel exists. Among them, the bg channel dominates or even The problem of the interference exists only in the gg → tW − ¯ b or q ¯ q → tW − ¯ b channel which is atsubleading power near the threshold. – 6 – Figure 2 . The factorization scale dependence of the cross sections in the PIM scheme for tW − production with √ s = 8 TeV. The NLO result is obtained in the DS scheme. The plots are shownin the region / < µ f /M < . surpasses the NLO corrections, as indicated in table 1 too. Moreover, the leading powerterms of the bg channel can approximate the total result of the bg channel very well, thedifference being only and in the PIM and 1PI schemes, respectively. Since the leadingpower terms can be obtained from the resummed results as discussed above in eq.(2.11),they can be calculated up to higher orders in α s , namely beyond NLO. These make up amajor part of the full NNLO corrections and can be taken as an approximation of the latter.The quality of the approximation could be estimated by looking at the power corrections.The NNLO results are still unavailable, so we study the NLO ones which are shown in table1 as well. It is ready to see that they are negative and around − ∼ − in PIMand − ∼ − in 1PI kinematic scheme depending on the methods to deal with theinterference problem. The contributions of the higher order (in α s ) power corrections canbe obtained by calculating the full NNLO QCD corrections or by making use of the next-to-leading power factorization and resummation, both of which are difficult at the momentand beyond the scope of this paper.Although the usual way to evaluate the scale uncertainty is to vary the scales by afactor of two, it is also interesting to investigate the factorization scale dependence in alarger region. From the figure 2, we can see that the ratio of the NLO over LO result isinsensitive to the factorization scale, always in the region (1 . , . , when it is variedfrom M/ to M . This means that there is no clear choice of the factorization scale toensure fastest convergence. Moreover, we find that the bg channel is very sensitive to thefactorization scale when it is smaller than M/ . In order to avoid such a dependence, wehave chosen the default factorization scale at M . It can also be seen that the NLO leadingpower terms dominate the bg channel over a large region.Then we turn to the differential cross sections. We show the tW invariant mass distri-butions in the PIM scheme in figure 3 and the top quark p T distributions in the 1PI schemein figure 4. We show results at both the 8 TeV and 13 TeV LHC. It can be seen that the– 7 – ������������������� ��� ��� ��� ��� ��� ������������������ �������������������� ��� ��� ��� ��� ��� ������������������ Figure 3 . Invariant mass distributions in the PIM scheme for tW − production. In the upper plots,the black lines represent the NLO cross section from bg channel while the blue and red lines are theNLO leading and NNLO leading predictions, respectively. In the bottom plots, we show the ratioof NLO (NNLO) leading over NLO bg by blue (red) lines. ������������������������ �� ��� ��� ������������������ ���������������������������� � �� ��� ��� ��� ������������������ Figure 4 . Top quark p T distributions in the 1PI scheme for tW − production. The color scheme isthe same as figure 3. – 8 –eading power terms are dominant in all the invariant mass or the top p T regions, as inthe case of total cross sections. The NNLO leading terms increase the NLO leading crosssection by about in most of the region.Now we present in table 2 the aNNLO and NLO+NNLL result defined in eq. (2.11)and eq. (2.12), respectively. The NLO+NNLL (aNNLO) predictions increase the NLOtotal cross section by ∼ ( ∼ ) depending on the collider energy and thethreshold variable, but with larger scale uncertainties. These large uncertainties are mainlyfrom the variation of the factorization scale µ f . At first sight, this is unexpected since wehave checked the scale independence near the threshold analytically in eq. (2.10). However,this is based on the assumption x , → as discussed in appendix A. When the kinematicsis far away from the threshold limit, this assumption is not valid. The very small scaleuncertainties of the NLO results seem like a coincidence because the NLO contributionsfrom gg and qq (cid:48) channels are negative while the contributions from bg channel are positive.Meanwhile they display an opposite behavior under the scale variation; see table 1. Ourresummed result or its expansion in α s improves only the result in bg channel. It would beinteresting to investigate whether the scale cancellation among different channels happensat higher orders. From table 2 we also find that the total cross sections in the PIM and 1PIscheme are compatible. And the resummed cross sections in PIM kinematics have smallerscale uncertainties.Lastly, we compare the theoretical results with the measurements of the total crosssection for tW − and ¯ tW + production at the LHC in figure 5. After considering the largeexperimental uncertainties, the NLO+NNLL predictions are in good agreement with thedata at the 8 TeV and 13 TeV LHC. We also give the predictions at the 14 TeV LHC. We have investigated the soft-gluon resummation for tW production in the framework ofsoft-collinear effective theory. We considered the two different definitions of the thresholdlimit, − M / ˆ s → and s → , corresponding to the PIM and 1PI kinematic schemes,respectively. We briefly discussed the factorization and resummation formalism in bothkinematic schemes. In addition, we have calculated the hard function and soft function at[pb] PIM 1PI √ s . +5% − . +5% − . +5% − . +3% − NLO . +2% − . +1% − . +1% − . +1% − aNNLO . +4% − . +5% − . +6% − . +7% − NLO+NNLL . +7% − . +7% − . +12% − . +16% − aNNLO/NLO 1.16 1.13 1.12 1.09(NLO+NNLL)/NLO 1.15 1.12 1.17 1.13 Table 2 . Total cross sections for tW − production in PIM and 1PI schemes. The NLO cross sectionsare calculated using DS scheme. – 9 – ◆◆◆ ◆◆ ◆◆ ◆◆◆◆ ◆◆ ◆◆ ◆◆◆◆ �������������� Figure 5 . Comparison between measured cross section for tW − and ¯ tW + production at theLHC [2, 3, 6, 7] and RG-improved predictions. NLO. Expanding the resummed formula in α s gives the leading power terms of the fixed-order results. We found that the NLO leading power contribution is a good approximationto the bg channel at NLO not only for the total cross sections but also for the differentialdistributions. After resumming the soft gluon effects to all orders using renormalizationgroup equation, we find that the NLO+NNLL results increase the NLO cross sections byabout 15(12)% in PIM and 17(13)% in 1PI scheme at the 8(13) TeV LHC, but with largeuncertainties which is mostly generated by varying the factorization scale. We comparedwith the data at the 8 and 13 TeV LHC and found good agreement within uncertainties.We provide the prediction for the 14 TeV LHC.In future, we can obtain more precise predictions for the tW process by includinghigher order hard and soft functions in the resummation formalism or by calculating the fullNNLO corrections. The latter may be achieved making use of the N -jettiness subtractionmethod [53–55]. The NNLO beam function [56, 57] and N -jettiness soft function [23] forthis process have been computed. The only missing part is the two-loop hard function,which requires a huge amount of work. We defer this study to future work. Acknowledgements
We would like to thank Shi Ang Li for the contribution in the early stage of this work.C.S.L. was supported by the National Nature Science foundation of China, under Grants No.11875072. H.T.L. was supported by the Los Alamos National Laboratory LDRD program.J.W. was supported by the BMBF project No. 05H15WOCAA and 05H18WOCA1.– 10 –
RG equation of the PDFs near the threshold
In the threshold limit x , → , the DGLAP evolution for the PDFs can be written as[58, 59] dd ln µ f i/p ( x, µ ) = (cid:90) y dzz (cid:20) C i γ cusp ( α s )(1 − z ) + + 2 γ φ i ( α s ) δ (1 − z ) (cid:21) f i/p ( x/z, µ ) , (A.1)where the quadratic Casimir operator C i for the quark is C q = C F , and for the gluon is C g = C A . In the threshold limit s → the evolution equations for PDFs are dd ln µ f q/p ( x ( s ) , µ ) = 2 C F γ cusp ( α s ) (cid:90) s ds (cid:48) f q/p ( x ( s (cid:48) ) , µ ) − f q/p ( x ( s ) , µ ) s − s (cid:48) + (cid:20) C F γ cusp ( α s ) ln s − ˆ u W + 2 γ φ q ( α s ) (cid:21) f q/p ( x ( s ) , µ ) ,dd ln µ f g/p ( x ( s ) , µ ) = 2 C A γ cusp ( α s ) (cid:90) s ds (cid:48) f g/p ( x ( s (cid:48) ) , µ ) − f g/p ( x ( s ) , µ ) s − s (cid:48) + (cid:20) C A γ cusp ( α s ) ln s − ˆ t W + 2 γ φ g ( α s ) (cid:21) f g/p ( x ( s ) , µ ) . (A.2)A similar derivation for t ¯ t and single top production can be found in refs. [35, 38]. B Soft function
The NLO soft function can be written as ˜ s NLO ( L, µ s ) = α s π (cid:20) − C A I − (2 C F − C A ) I − C A I + C F I (cid:21) , (B.1)For convenience we evaluate soft integral I ij in the position space, and then transform theminto Laplace space as I ij ( L ) = − (4 πµ ) (cid:15) π − (cid:15) v i · v j (cid:90) d d k e − ik x v i · kv j · k (2 π ) δ ( k ) θ ( k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L →− L , (B.2)with L = ln (cid:0) − µ x e γ E / (cid:1) , and v i are normalized momenta fulfilling on-shell conditionsas v = v = 0 and v = 1 .In the PIM kinematics the full set of integrals can be found in section III of ref. [60]. Inthe 1PI kinematics the integral I can be obtained from eq. (21) of ref. [35] by replacing allthe kinematics variables to the ones related to W . The integrals I , I and I are morecomplicated, and have been first calculated in this paper. The non-vanishing integrals arecollected below I = − (cid:18) L + ln ˆ sM W ˆ t W ˆ u W (cid:19) − π − (cid:18) − ˆ sM W ˆ t W ˆ u W (cid:19) ,I = − (cid:18) L + 2 ln M W ˆ t m t ˆ u W (cid:19) − π − (cid:18) − M W m t x tW ˆ t ˆ u W (cid:19) − (cid:18) − M W x tW m t ˆ t ˆ u W (cid:19) ,I = − L − β t β W β t + β W ln x tW ,I = I (ˆ t → ˆ u , ˆ u W → ˆ t W ) , (B.3)– 11 –ith β t = (cid:113) − m t ˆ s/ ( m t − M W + ˆ s ) , β W = (cid:113) − M W ˆ s/ ( m t − M W − ˆ s ) , x tW = √ x t x W and x i = (1 − β i ) / (1 + β i ) . In the limit of M W → m t , the integrals reproducethose for t ¯ t production in ref. [35]. References [1]
ATLAS collaboration, G. Aad et al.,
Evidence for the associated production of a W bosonand a top quark in ATLAS at √ s = 7 TeV , Phys. Lett.
B716 (2012) 142–159, [ ].[2]
ATLAS collaboration, G. Aad et al.,
Measurement of the production cross-section of a singletop quark in association with a W boson at 8 TeV with the ATLAS experiment , JHEP (2016) 064, [ ].[3] ATLAS collaboration, M. Aaboud et al.,
Measurement of the cross-section for producing aW boson in association with a single top quark in pp collisions at √ s = 13 TeV with ATLAS , JHEP (2018) 063, [ ].[4] ATLAS collaboration, M. Aaboud et al.,
Measurement of differential cross-sections of asingle top quark produced in association with a W boson at √ s = 13 TeV with ATLAS , Eur.Phys. J.
C78 (2018) 186, [ ].[5]
CMS collaboration, S. Chatrchyan et al.,
Evidence for associated production of a single topquark and W boson in pp collisions at √ s = 7 TeV , Phys. Rev. Lett. (2013) 022003,[ ].[6]
CMS collaboration, S. Chatrchyan et al.,
Observation of the associated production of a singletop quark and a W boson in pp collisions at √ s = , Phys. Rev. Lett. (2014) 231802,[ ].[7]
CMS collaboration, A. M. Sirunyan et al.,
Measurement of the production cross section forsingle top quarks in association with W bosons in proton-proton collisions at √ s = 13 TeV , JHEP (2018) 117, [ ].[8] W. T. Giele, S. Keller and E. Laenen, QCD corrections to W boson plus heavy quarkproduction at the Tevatron , Phys. Lett.
B372 (1996) 141–149, [ hep-ph/9511449 ].[9] S. Zhu,
Next-to-leading order QCD corrections to bg → tW − at CERN large hadron collider , Phys. Lett.
B524 (2002) 283–288, [ hep-ph/0109269 ].[10] Q.-H. Cao,
Demonstration of One Cutoff Phase Space Slicing Method: Next-to-Leading OrderQCD Corrections to the tW Associated Production in Hadron Collision , .[11] P. Kant, O. M. Kind, T. Kintscher, T. Lohse, T. Martini, S. Mölbitz et al., HatHor for singletop-quark production: Updated predictions and uncertainty estimates for single top-quarkproduction in hadronic collisions , Comput. Phys. Commun. (2015) 74–89, [ ].[12] J. M. Campbell and F. Tramontano,
Next-to-leading order corrections to Wt production anddecay , Nucl. Phys.
B726 (2005) 109–130, [ hep-ph/0506289 ].[13] S. Frixione, E. Laenen, P. Motylinski, B. R. Webber and C. D. White,
Single-tophadroproduction in association with a W boson , JHEP (2008) 029, [ ].[14] E. Re, Single-top Wt-channel production matched with parton showers using the POWHEGmethod , Eur. Phys. J.
C71 (2011) 1547, [ ]. – 12 –
15] T. Ježo, J. M. Lindert, P. Nason, C. Oleari and S. Pozzorini,
An NLO+PS generator for t ¯ t and W t production and decay including non-resonant and interference effects , Eur. Phys. J.
C76 (2016) 691, [ ].[16] T. M. P. Tait,
The tW − mode of single top production , Phys. Rev.
D61 (1999) 034001,[ hep-ph/9909352 ].[17] A. Belyaev and E. Boos,
Single top quark tW + X production at the CERN LHC: A Closerlook , Phys. Rev.
D63 (2001) 034012, [ hep-ph/0003260 ].[18] C. D. White, S. Frixione, E. Laenen and F. Maltoni,
Isolating Wt production at the LHC , JHEP (2009) 074, [ ].[19] F. Demartin, B. Maier, F. Maltoni, K. Mawatari and M. Zaro, tWH associated production atthe LHC , Eur. Phys. J.
C77 (2017) 34, [ ].[20] R. Frederix,
Top Quark Induced Backgrounds to Higgs Production in the
W W ( ∗ ) → llνν Decay Channel at Next-to-Leading-Order in QCD , Phys. Rev. Lett. (2014) 082002,[ ].[21] F. Cascioli, S. Kallweit, P. Maierhöfer and S. Pozzorini,
A unified NLO description oftop-pair and associated Wt production , Eur. Phys. J.
C74 (2014) 2783, [ ].[22] H. T. Li and J. Wang,
Next-to-Next-to-Leading Order N -Jettiness Soft Function for OneMassive Colored Particle Production at Hadron Colliders , JHEP (2017) 002,[ ].[23] H. T. Li and J. Wang, Next-to-next-to-leading order N -jettiness soft function for tW production , Phys. Lett.
B784 (2018) 397–404, [ ].[24] N. Kidonakis,
Single top production at the Tevatron: Threshold resummation and finite-ordersoft gluon corrections , Phys. Rev.
D74 (2006) 114012, [ hep-ph/0609287 ].[25] N. Kidonakis,
Two-loop soft anomalous dimensions for single top quark associated productionwith a W − or H − , Phys. Rev.
D82 (2010) 054018, [ ].[26] N. Kidonakis,
Soft-gluon corrections for tW production at N LO , Phys. Rev.
D96 (2017)034014, [ ].[27] N. Kidonakis,
Soft anomalous dimensions for single-top production at three loops , .[28] C. W. Bauer, S. Fleming and M. E. Luke, Summing Sudakov logarithms in B → X s γ ineffective field theory , Phys. Rev.
D63 (2000) 014006, [ hep-ph/0005275 ].[29] C. W. Bauer, S. Fleming, D. Pirjol and I. W. Stewart,
An Effective field theory for collinearand soft gluons: Heavy to light decays , Phys. Rev.
D63 (2001) 114020, [ hep-ph/0011336 ].[30] C. W. Bauer and I. W. Stewart,
Invariant operators in collinear effective theory , Phys. Lett.
B516 (2001) 134–142, [ hep-ph/0107001 ].[31] C. W. Bauer, D. Pirjol and I. W. Stewart,
Soft collinear factorization in effective field theory , Phys. Rev.
D65 (2002) 054022, [ hep-ph/0109045 ].[32] M. Beneke, A. P. Chapovsky, M. Diehl and T. Feldmann,
Soft collinear effective theory andheavy to light currents beyond leading power , Nucl. Phys.
B643 (2002) 431–476,[ hep-ph/0206152 ].[33] T. Becher, A. Broggio and A. Ferroglia,
Introduction to Soft-Collinear Effective Theory , Lect.Notes Phys. (2015) pp.1–206, [ ]. – 13 –
34] V. Ahrens, A. Ferroglia, M. Neubert, B. D. Pecjak and L. L. Yang,
Renormalization-GroupImproved Predictions for Top-Quark Pair Production at Hadron Colliders , JHEP (2010)097, [ ].[35] V. Ahrens, A. Ferroglia, M. Neubert, B. D. Pecjak and L.-L. Yang, RG-improvedsingle-particle inclusive cross sections and forward-backward asymmetry in t ¯ t production athadron colliders , JHEP (2011) 070, [ ].[36] H. X. Zhu, C. S. Li, J. Wang and J. J. Zhang, Factorization and resummation of s-channelsingle top quark production , JHEP (2011) 099, [ ].[37] J. Wang, C. S. Li, H. X. Zhu and J. J. Zhang, Factorization and resummation of t-channelsingle top quark production , .[38] J. Wang, C. S. Li and H. X. Zhu, Resummation prediction on top quark transversemomentum distribution at large p T , Phys. Rev.
D87 (2013) 034030, [ ].[39] M. Beneke, A. Broggio, M. Garny, S. Jaskiewicz, R. Szafron, L. Vernazza et al.,
Leading-logarithmic threshold resummation of the Drell-Yan process at next-to-leading power , .[40] A. Idilbi and X.-d. Ji, Threshold resummation for Drell-Yan process in soft-collinear effectivetheory , Phys. Rev.
D72 (2005) 054016, [ hep-ph/0501006 ].[41] T. Becher, M. Neubert and G. Xu,
Dynamical Threshold Enhancement and Resummation inDrell-Yan Production , JHEP (2008) 030, [ ].[42] A. Ferroglia, M. Neubert, B. D. Pecjak and L. L. Yang, Two-loop divergences of scatteringamplitudes with massive partons , Phys. Rev. Lett. (2009) 201601, [ ].[43] A. Ferroglia, M. Neubert, B. D. Pecjak and L. L. Yang,
Two-loop divergences of massivescattering amplitudes in non-abelian gauge theories , JHEP (2009) 062, [ ].[44] E. Laenen, G. Oderda and G. F. Sterman, Resummation of threshold corrections for singleparticle inclusive cross-sections , Phys. Lett.
B438 (1998) 173–183, [ hep-ph/9806467 ].[45] T. Becher and M. D. Schwartz,
Direct photon production with effective field theory , JHEP (2010) 040, [ ].[46] V. Hirschi, R. Frederix, S. Frixione, M. V. Garzelli, F. Maltoni and R. Pittau, Automation ofone-loop QCD corrections , JHEP (2011) 044, [ ].[47] T. Peraro, Ninja: Automated Integrand Reduction via Laurent Expansion for One-LoopAmplitudes , Comput. Phys. Commun. (2014) 2771–2797, [ ].[48] G. Ossola, C. G. Papadopoulos and R. Pittau,
CutTools: A Program implementing the OPPreduction method to compute one-loop amplitudes , JHEP (2008) 042, [ ].[49] A. van Hameren, C. G. Papadopoulos and R. Pittau, Automated one-loop calculations: AProof of concept , JHEP (2009) 106, [ ].[50] S. Dulat, T.-J. Hou, J. Gao, M. Guzzi, J. Huston, P. Nadolsky et al., New parton distributionfunctions from a global analysis of quantum chromodynamics , Phys. Rev.
D93 (2016) 033006,[ ].[51] A. Buckley, J. Ferrando, S. Lloyd, K. Nordström, B. Page, M. Rüfenacht et al.,
LHAPDF6:parton density access in the LHC precision era , Eur. Phys. J.
C75 (2015) 132, [ ]. – 14 –
52] S. Alioli, P. Nason, C. Oleari and E. Re,
A general framework for implementing NLOcalculations in shower Monte Carlo programs: the POWHEG BOX , JHEP (2010) 043,[ ].[53] J. Gao, C. S. Li and H. X. Zhu, Top Quark Decay at Next-to-Next-to Leading Order in QCD , Phys. Rev. Lett. (2013) 042001, [ ].[54] J. Gaunt, M. Stahlhofen, F. J. Tackmann and J. R. Walsh,
N-jettiness Subtractions forNNLO QCD Calculations , JHEP (2015) 058, [ ].[55] R. Boughezal, C. Focke, X. Liu and F. Petriello, W -boson production in association with ajet at next-to-next-to-leading order in perturbative QCD , Phys. Rev. Lett. (2015) 062002,[ ].[56] J. R. Gaunt, M. Stahlhofen and F. J. Tackmann,
The Quark Beam Function at Two Loops , JHEP (2014) 113, [ ].[57] J. Gaunt, M. Stahlhofen and F. J. Tackmann, The Gluon Beam Function at Two Loops , JHEP (2014) 020, [ ].[58] G. P. Korchemsky and G. Marchesini, Structure function for large x and renormalization ofWilson loop , Nucl. Phys.
B406 (1993) 225–258, [ hep-ph/9210281 ].[59] S. Moch, J. A. M. Vermaseren and A. Vogt,
The Three loop splitting functions in QCD: TheNonsinglet case , Nucl. Phys.
B688 (2004) 101–134, [ hep-ph/0403192 ].[60] H. T. Li, C. S. Li and S. A. Li,
Renormalization group improved predictions for t ¯ tW ± production at hadron colliders , Phys. Rev.
D90 (2014) 094009, [ ].].