diff --git "a/3dFKT4oBgHgl3EQfQy0a/content/tmp_files/2301.11768v1.pdf.txt" "b/3dFKT4oBgHgl3EQfQy0a/content/tmp_files/2301.11768v1.pdf.txt" new file mode 100644--- /dev/null +++ "b/3dFKT4oBgHgl3EQfQy0a/content/tmp_files/2301.11768v1.pdf.txt" @@ -0,0 +1,5350 @@ +Prepared for submission to JHEP +FERMILAB-PUB-23-028-T, +IPPP/23/05 +Jet-veto resummation at N3LLp+NNLO in boson +production processes +John M. Campbell,a R. Keith Ellis,b Tobias Neumann,c Satyajit Sethd +aFermilab, PO Box 500, Batavia IL 60510-5011, USA +bInstitute for Particle Physics Phenomenology, Durham University, Durham, DH1 3LE, UK +cDepartment of Physics, Brookhaven National Laboratory, Upton, New York 11973, USA +dPhysical Research Laboratory, Navrangpura, Ahmedabad - 380009, India +E-mail: johnmc@fnal.gov, keith.ellis@durham.ac.uk, tneumann@bnl.gov, +seth@prl.res.in +Abstract: Vetoing energetic jet activity is a crucial tool for suppressing backgrounds and +enabling new physics searches at the LHC, but the introduction of a veto scale can introduce +large logarithms that may need to be resummed. We present an implementation of jet-veto +resummation for color-singlet processes at the level of N3LLp matched to fixed-order NNLO +predictions. Our public code MCFM allows for predictions of a single boson, such as Z/γ∗, +W ± or H, or with a pair of vector bosons, such as W +W −, W ±Z or ZZ. The implementation +relies on recent calculations of the soft and beam functions in the presence of a jet veto +over all rapidities, with jets defined using a sequential recombination algorithm with jet +radius R. However one of the ingredients that is required to reach full N3LL accuracy is only +known approximately, hence N3LLp. We describe in detail our formalism and compare with +previous public codes that operate at the level of NNLL. Our higher-order predictions improve +significantly upon NNLL calculations by reducing theoretical uncertainties. We demonstrate +this by comparing our predictions with ATLAS and CMS results. +arXiv:2301.11768v1 [hep-ph] 27 Jan 2023 + +Contents +1 +Introduction +1 +2 +Jet-veto factorization and resummation +2 +2.1 +The collinear anomaly coefficient and its approximations +5 +3 +Setup for phenomenology +9 +3.1 +Input parameters +9 +3.2 +Uncertainty estimates at fixed order +10 +3.3 +Uncertainty estimates at the resummed and matched level +11 +3.4 +Effects of cuts on rapidity at fixed order +13 +4 +Comparison with JetVHeto +15 +5 +Phenomenological results +17 +5.1 +Z and W production +17 +5.2 +W +W − production +21 +5.3 +W ±Z production +22 +5.4 +ZZ production +25 +5.5 +Higgs production +25 +6 +Conclusions +28 +A Reduced beam functions +30 +A.1 Structure of the two-loop reduced beam function +31 +B Definition of the beta function and anomalous dimensions +32 +B.1 +Expansion of β-function +32 +B.2 +Cusp Anomalous Dimension +34 +B.3 +Non-cusp anomalous dimension +35 +C Definitions for beam function ingredients +37 +C.1 Exponent h +37 +C.2 One loop splitting functions +38 +C.3 Two loop splitting functions +38 +C.4 P (1) ⊗ P (1) and R(1) ⊗ P (1) +41 +D Rapidity anomalous dimension +42 +D.1 dveto +2 +expansion +43 +E Renormalization Group Evolution +44 +– i – + +E.1 +Recovery of the double log formula +45 +F The hard function for the Drell-Yan process +46 +G The hard function for Higgs production +48 +G.1 Implementation of one-step procedure +48 +G.2 Implementation of the two-step procedure +50 +G.3 Assessment of the two schemes for the Higgs hard function +52 +1 +Introduction +Jet vetoing is a crucial technique in particle physics that is used primarily to suppress +backgrounds in processes involving the production of W +W − final states (e.g. directly or +in Higgs decays). By identifying and removing events that contain energetic hadronic jets +(vetoing), the impact of the dominant top-quark pair production background is reduced. +The concrete jet-veto implementation depends on factors such as the jet algorithm and its +parameters, as well as the kinematic selection cuts applied to the identified jets. For LHC +analyses, the most common jet vetoing scheme is to impose a maximum transverse momentum +cut pveto +T +on anti-kT jets. +The jet veto scale pveto +T +can induce large logarithms if it is smaller than the hard process scale +Q, which then mandates resummation. In this paper we describe a coherent implementation of +jet veto resummation in processes involving the production of a color-singlet boson (W, Z/γ∗ +and H bosons) or a pair of bosons (ZZ, W ±Z, and W +W −). Our resummation operates at +the level of N3LLp1 matched to fixed order NNLO. +We build on the pioneering work of previous studies, which have demonstrated the effectiveness +of resummation methods for a jet veto [1–5]. General purpose implementations include a +numerical approach to resummation at NNLO+NNLL [6, 7] and an automated approach to jet +veto studies at NLO+NNLL [8]. Publicly available codes operating at NNLL and addressing +the same issue are, JetVHeto [9], the code MCFM-RE [10] which is derivative of both MCFM +and JetVHeto, and MATRIX+RadISH [11]. Both JetVHeto and RadISH implement the same +analytic resummation formula of ref. [5]. +Our research extends and improves upon these earlier results through detailed phenomenological +studies of specific final states, including Higgs boson production [5, 12–14], W +W − production +[15, 16], and ZZ and W ±Z production [17]. Another important aspect of our study is the +performance of the resummation at N3LLp accuracy, which has not always been the case in +1The last missing ingredient for N3LL resummation is the exact dveto +3 +(the three-loop rapidity anomalous +dimension) which we approximate and take into account with an uncertainty estimate. We discuss this in detail +in the subsequent section. +– 1 – + +previous work. We also describe our approximation of the missing dveto +3 +that would be necessary +to reach full N3LL accuracy. Finally, we include our results in MCFM, a publicly distributed +code, which allows users to easily perform studies in practice. +Resummation of jet-veto logarithms has a close relationship with the resummation of transverse +momentum logarithms. In the latter, one is interested in transverse momenta all the way down +to zero pT , so the logarithms can be larger than in jet-veto processes where pveto +T +in the range +25 to 30 GeV is used experimentally. In this paper we explore which jet-veto processes actually +require resummation at these values of pveto +T +, supply the best predictions for those processes +where it is warranted, and confront our theoretical predictions with experimental data where +it is available. +In Section 2 we discuss the jet-veto factorization theorem including its ingredients that result +in the resummation. We describe our setup for phenomenology including our uncertainty +procedure in Section 3, compare with the public code JetVHeto in Section 4, and study the +phenomenological implications for a wide range of processes in Section 5. We conclude in +Section 6. +2 +Jet-veto factorization and resummation +We consider processes where jets have been defined using sequential recombination jet algorithms +[18] with distance measure +dij = min(k2n +Ti, k2n +Tj) +∆y2 +ij + ∆φ2 +ij +R2 +, +diB = k2n +Ti , +(2.1) +where the choice n = −1 is the anti-kT algorithm [19], n = 0 is the Cambridge-Aachen algorithm +[20, 21], and n = 1 is the kT algorithm [22, 23]. kTi denotes the transverse momentum of +(pseudo-)particle i with respect to the beam direction, and ∆yij and ∆φij are the rapidity and +azimuthal angle differences of (pseudo-)particles i and j. +To describe the resummation method we focus on the simplest case of quark-antiquark induced +Drell-Yan production of a lepton pair of invariant mass Q and rapidity y. The case of gluon +initiated processes is structurally the same, but with different ingredients that we give below +and in the appendices. In the presence of a jet veto over all rapidities we have a factorization +formula [3, 12, 13], +d2σ(pveto +T +) +dQ2dy += dσ0 +dQ2 +��CV (−Q2, µ) +��2 +× +� +Bq(ξ1, Q, pveto +T +, R, µ, ν) B¯q(ξ2, Q, pveto +T +, R, µ, ν) S(pveto +T +, R, µ, ν) +� ++ O +�pveto +T +Q +� +(2.2) +where ξ1,2 = (Q/√s) e±y and, +dσ0 +dQ2 = +4πα2 +3NcQ2s . +(2.3) +– 2 – + +Table 1: Counting of orders in the resummation, adapted from ref. [26]. The second column +indicates the nominal order when counting L⊥ ∼ 1/αs. +The third column states which +logarithms are included. The last three columns show the necessary additional anomalous +dimensions and hard function corrections in each successive order. The requisite anomalous +dimensions are provided in Appendix B. +Approximation +Nominal order +Accuracy ∼ αn +s Lk +⊥ +Γcusp +γcoll. +H +LL +α−1 +s +2n ≥ k ≥ n + 1 +Γ0 +tree +tree +NLL+LO +α0 +s +2n ≥ k ≥ n +Γ1, +γ0 +tree +N2LL+NLO +α1 +s +2n ≥ k ≥ max(n − 1, 0) +Γ2 +γ1 +1-loop +N3LL +NNLO +α2 +s +2n ≥ k ≥ max(n − 2, 0) +Γ3 +γ2 +2-loop +In this equation CV is a matching coefficient whose square is the hard coefficient function that +corrects the lowest order cross-section, see Eq. (2.3). Bq and B¯q are the quark beam functions +which describe the emission of radiation collinear to the two beam directions in the presence of +a jet veto, and S describes the emission of soft radiation in the presence of a jet veto. The +quantity ν is a supplementary scale necessitated by the rapidity divergences present in beam +and soft functions. The main process-independent ingredients are the beam and soft functions +for both incoming quarks and gluons which have been published recently at the two-loop level +[24, 25]. The hard function is process specific. We have used the existing two-loop fixed order +implementations in MCFM. +Overall the factorization theorem achieves a separation of scales. The hard function contains +logarithms of the ratio Q2/µ2, which can be minimized by setting µ2 = µ2 +h ∼ Q2. However, +inside the beam and soft functions, it is natural to choose µ = pveto +T +to avoid large logarithms. +The resummation of large logarithms is achieved by choosing µ ∼ Q in the hard function and +evolving it down to the resummation scale µ ∼ pveto +T +using the renormalization group (RG). +For the hard function the evolution is solved analytically, see Appendix E. +In RG-improved power counting the logarithms L⊥ = 2 log(µh/pveto +T +), where µh is of order Q, +are assumed to be of order 1/αs. With this definition the counting of powers of αs and of the +large logarithm L⊥ is shown in Table 1. The non-logarithmic terms that the resummation +does not provide are easily accounted for by adding the matching corrections. The matching +corrections are a finite contribution and add the effect of fixed-order corrections while removing +the logarithmic overlap through a fixed-order expansion of the resummation. +2.0.1 +Soft function +The jet veto soft function has been calculated using an exponential regulator [27] in Ref. [25]. +The calculation is divided into the sum of the soft function for a reference observable and a +correction factor, +S(pveto +T +, R, µ, ν) = S⊥(pveto +T +, µ, ν) + ∆S(pveto +T +, R, µ, ν) . +(2.4) +– 3 – + +In Ref. [25] the reference observable is the transverse momentum of the color singlet system +denoted by S⊥. S⊥ can be derived from the expression given in Refs. [28, 29] after performing +the Fourier transform to momentum space (see, for instance, the rules given in Table 1 of +Ref. [30]). ∆S depends on the jet algorithm and contributes for two or more emissions. It thus +depends only on double real emission diagrams. +2.0.2 +Refactorization and reduced beam functions +For consistency with the transverse momentum resummation framework in CuTe-MCFM [31] we +cast the factorization theorem in terms of the collinear anomaly framework. In this framework +the rapidity logarithms are exponentiated directly instead of resummed by solving rapidity RG +equations [32, 33]. For this we rewrite the square bracket in Eq. (2.2) as follows, +Bq(ξ1, Q, pveto +T +, R, µ, ν) B¯q(ξ2, Q, pveto +T +, R, µ, ν)S(pveto +T +, R, µ, ν) += +� Q +pveto +T +�−2Fqq(pveto +T +,R,µ) +e2hF (pveto +T +,µ) ¯Bq(ξ1, pveto +T +, R, µ) ¯B¯q(ξ2, pveto +T +, R, µ) . +(2.5) +The ν dependence vanishes in this combination of beam and soft functions. +We have factored out ehF/A(pveto +T +,µ) from each beam function, resulting in the reduced beam +functions ¯B. By construction hF/A are solutions of the RGE equation, +d +d ln µ hF/A(pveto +T +, µ) = 2ΓF/A +cusp(µ) ln +µ +pveto +T +− 2γq/g(µ) , +(2.6) +with boundary condition hF/A(µ, µ) = 0. The superscripts F or A signify whether the color +is treated in the fundamental (F) or adjoint (A) representation, corresponding to a quark +initiated process or a gluon initiated process, respectively. The exact form of hF/A(pveto +T +, µ), +determined by solving Eq. (2.6), is given in Appendix C.1. In terms of the reduced beam +functions the jet-vetoed cross-section is now given by, +d2σ(pveto +T +) +dQ2dy += dσ0 +dQ2 ¯H(Q, µ, pveto +T +) ¯Bq(ξ1, pveto +T +, R, µ) ¯B¯q(ξ2, pveto +T +, R, µ) + O(pveto +T +/Q) , +(2.7) +The choice of hF/A in Eq. (2.6) divides Eq. (2.2) into two separately RG invariant pieces, the +product of the two reduced beam functions ( ¯Bq ¯B¯q), and the hard function, ( ¯H) +¯H(Q, µ, pveto +T +) = +��CV (−Q2, µ) +��2 � Q +pveto +T +�−2Fqq(pveto +T +,R,µ) +e2hF (pveto +T +,µ) . +(2.8) +For quark-initiated processes the functions CV and Fqq obey the following RG equations. +d +d ln µ CV (−Q2, µ) = +� +ΓF +cusp(µ) ln −Q2 +µ2 ++ 2γq(µ) +� +CV (−Q2, µ) , +(2.9) +d +d ln µFqq(pveto +T +, R, µ) = 2ΓF +cusp(µ) . +(2.10) +– 4 – + +Eqs. (2.9) and (2.10) are structurally the same for the gluon case with different anomalous +dimensions. +The function ¯H is RG invariant due to the RGE’s satisfied by CV and Fqq and hF : +d +dµ +¯H(Q, µ, pveto +T +) = O(α3 +s) . +Consequently, the remaining product of reduced beam functions is also RG invariant, up to +the order calculated. In our case, +d +dµ +¯Bq(ξ1, pveto +T +, R, µ) ¯B¯q(ξ2, pveto +T +, R, µ) = O(α3 +s) . +(2.11) +The confirmation of Eq. (2.11) and the confirmation of the R-dependence of the collinear +anomaly given in the next section are two simple checks of the results of Refs. [24, 25]. Full +details of the formulas needed to perform this check are given in Appendix C. +If the scale pveto +T +is in the perturbative domain, the reduced beam function can be written in +terms of the matching kernels ¯I as +¯Bi(ξ, pveto +T +, R, µ) = +� +j=g,q,¯q +� 1 +ξ +dz +z +¯Iij(z, pveto +T +, R, µ) φj/P (ξ/z, µ) , +where φ denotes the usual collinear parton distribution of a parton of flavor j in a proton P. +The matching coefficients ¯I are extracted from I, the two-loop beam and soft functions of +Refs. [24, 25] as, +¯Iij(z, pveto +T +, R, µ) = e−hF/A(pveto +T +,µ) Iij(z, pveto +T +, R, µ) . +(2.12) +The coefficients in Ref. [24] are presented as a Laurent expansion in the jet radius parameter +R. Analytic expressions are presented for all flavor channels except for a set of R-independent +non-logarithmic terms which are presented as numerical grids. For our purposes we have +interpolated the numerical grids using a spline fit. We give further details on the reduced beam +functions in Appendix A. +2.1 +The collinear anomaly coefficient and its approximations +The missing ingredient for a complete N3LL resummation is the three-loop collinear anomaly +coefficient and therefore warrants a longer discussion. This limitation has been discussed in the +literature and approximated in various ways. Here we discuss the uncertainty associated with +the approximations and how we take it into account in our phenomenological predictions. +As presented in Eq. (2.10) the collinear anomaly coefficients obey the RG equations, +d +d ln µFqq(pveto +T +, R, µ) = 2ΓF +cusp(µ) , +(2.13) +d +d ln µFgg(pveto +T +, R, µ) = 2ΓA +cusp(µ) , +(2.14) +– 5 – + +where, for example, Fqq has the expansion, +Fqq(pveto +T +, R, µ) = αs +4πF (0) +qq (pveto +T +, R, µ) + +�αs +4π +�2 +F (1) +qq (pveto +T +, R, µ) ++ +�αs +4π +�3 +F (2) +qq (pveto +T +, R, µ) + +�αs +4π +�4 +F (3) +qq (pveto +T +, R, µ) + . . . +(2.15) +While the logarithmic structure is given by the RG equations, the constant boundary parts +dveto +k +(R, B) where B = F or A need to be determined by separate calculations and are also +referred to as the rapidity anomalous dimensions in the framework of Refs. [32, 33]: +F (0) +qq (pveto +T +, R, µh) = ΓF +0 L⊥ + dveto +1 +(R, F) , +F (1) +qq (pveto +T +, R, µh) = 1 +2ΓF +0 β0L2 +⊥ + ΓF +1 L⊥ + dveto +2 +(R, F) , +F (2) +qq (pveto +T +, R, µh) = 1 +3ΓF +0 β2 +0L3 +⊥ + 1 +2(ΓF +0 β1 + 2ΓF +1 β0)L2 +⊥ ++ (ΓF +2 + 2β0dveto +2 +(R, F))L⊥ + dveto +3 +(R, F) , +F (3) +qq (pveto +T +, R, µh) = 1 +4β3 +0ΓF +0 L4 +⊥ + (ΓF +1 β2 +0 + 5 +6ΓF +0 β0β1)L3 +⊥ ++ (1 +2ΓF +0 β2 + ΓF +1 β1 + 3 +2ΓF +2 β0 + 3dveto +2 +(R, F)β2 +0)L2 +⊥ ++ (ΓF +3 + 3dveto +3 +(R, F)β0 + 2dveto +2 +(R, F)β1)L⊥ + dveto +4 +(R, F) . +(2.16) +The analogous expression for gluons (F → A) is given in Eq. (D.1). The coefficients in the +expansion of the cusp anomalous dimension, ΓF +k , are given in Appendix B.2. +For single gluon emission dveto +1 +(R, B) = 0. The function dveto +2 +is defined below in Eq. (2.17). +There is only partial information on dveto +3 +from Refs. [14, 34, 35], and we have to rely on +an approximation. +To estimate the validity of this approximation we first study similar +approximations of dveto +2 +. +The function dveto +2 +is given by [12], +dveto +2 +(R, B) = dB +2 − 32CB f(R, B) , where +dB +2 = CB +��808 +27 − 28ζ3 +� +CA − 224 +27 TF nf +� +. +(2.17) +The function f(R, B), which gives the dependence on the jet radius R, is known as an expansion +about R = 0 up to terms including R4, +f(R, B) = CB +� +− π2R2 +12 ++ R4 +16 +� ++ CA +� +cA +L ln R + cA +0 + cA +2 R2 + cA +4 R4 + . . . +� ++ TF nf +� +cf +L ln R + cf +0 + cf +2R2 + cf +4R4 + . . . +� +. +(2.18) +The terms on the first line are due to independent emission, whereas the terms on the second +and third lines are due to correlated emission [4]. The expansion coefficients are given in +Appendix D in analytic and numerical form. +– 6 – + +2.1.1 +Approximations for dveto +2 +Using Eqs. (2.17) and (D.4) we have for the gluon case in the limit nf → 0 and retaining only +logarithmic and constant terms in R, +dveto +2 +(R, A) = −32C2 +A +� +− +1 +32C2 +A +dA +2 + cA +L ln R + cA +0 +� +≃ −32C2 +A +� +− 1.096259 ln R + 0.7272641] +∼ 32C2 +A × ln +�R +2 +� +. +(2.19) +This result was used as a basis for an approximation to dveto +3 +in ref. [12]. However, the leading +color (nf = 0) approximation is rather poor. With full nf dependence, but retaining only +logarithmic and constant terms in R and setting nf = 5 we have +dveto +2 +(R, B) = 32CBCA +� +(1.096 + 0.0295nf) ln R − (0.72726 + 0.12445nf) +� +∼ 32CBCA +� +1.2435 ln +� R +2.96 +�� +. +(2.20) +In Fig. 1 we show dveto +2 +(R, A) and its approximations in units of dA +2 as a function of the jet +radius R. As a reminder, dA +2 is the non-R dependent part of d2, see Eq. (2.17). We first +compare the full result (red) with the inclusion of terms up order R2 (green). This shows +that the R expansion converges quickly and it is sufficient to consider only terms up to R4 for +practical applications. Including only the logarithm and the constant (blue) gives a reasonable +approximation for sufficiently small R, with percent-level deviations around R = 0.4. The +leading color approximation (magenta) works only crudely as a first guess and could be used +in the absence of any better estimate. +2.1.2 +The function dveto +3 +While the complete dveto +3 +is unknown so far, we can extract the leading logarithmic term +from results in the literature. Given that this approximation works reasonably well for dveto +2 +for R ∼ 0.4, it is reasonable to expect a similar behavior for dveto +3 +. We further estimate the +uncertainty associated with such an approximation. +From Eq. (2.15) the collinear anomaly coefficient at µ = pveto +T +is given by, +Fgg(pveto +T +, R, pveto +T +) = +�αs +4π +�2 +dveto +2 +(R, A) + +�αs +4π +���3 +dveto +3 +(R, A) + . . . +(2.21) +Therefore, expanding the collinear anomaly we have that +� Q +pveto +T +�−2Fgg(pveto +T +,pveto +T +) +=1 − 2 +�αs(pveto +T +) +4π +�2 +ln +� Q +pveto +T +� +dveto +2 +(R, A) +− 2 ln +�αs(pveto +T +) +4π +�3 +ln +� Q +pveto +T +� +dveto +3 +(R, A) + O(α4 +s). +(2.22) +– 7 – + +Figure 1: Approximations of dveto +2 +(R, A) +scaled by the constant dA +2 . The full result, +Eq. (2.17) is plotted in red. The approxi- +mation retaining only constant terms and +logarithms of R is shown in blue. The ap- +proximation retaining constant terms and +logarithms of R and R2 terms is shown in +green. The leading color ansatz, Eq. (2.19), +derived setting nf = 0, is 32C2 +A ln(R/2) and +is shown in magenta. The red, blue and green +curves are all plotted for nf = 5. +Figure 2: Effect of R0 variation in dveto +3 +as +given by Eq. (2.24) with nf = 5, compared to +the case dveto +3 += 0: R0 = 1 (black), R0 = 0.5 +(red, dashed), R0 = 2 (blue, dashed). +At order α3 +s the leading term in the limit R → 0 can be extracted from Eq. (C.2) of Ref. [14] +which reads, +Fcorrel +LLR,31(R) = +�αs +4π +�3 +ln +� Q +pveto +T +� +· 128CA ln2 R +R0 +× +� +1.803136C2 +A − 0.589237nf2TRCA + 0.36982CF nf2TR − 0.05893n2 +f4T 2 +R +� +. +(2.23) +Comparing the third-order coefficient in the two equations we thus have for a general color +representation +dveto +3 +(R, B) = −64CB ln2 � R +R0 +� +(1.803136C2 +A + 0.36982CF nf − 0.589237CAnf − 0.05893n2 +f) += −8.38188 × 64CB ln2 � R +R0 +� +for nf = 5 . +(2.24) +Hence, the sign of the leading term in the small R limit is known. In this limit dveto +3 +leads to an +increase in the cross-section. This approximation only gives the leading R behavior, and it has +been suggested that one may plausibly take 1 +2 < R0 < 2 as an uncertainty envelope [14]. +Since dveto +3 +enters through the collinear anomaly as an overall factor, we consider the impact of +varying R0 in Fig. 2. For typical values of pveto +T += 30 GeV (as considered in this paper for the +– 8 – + +Table 2: Input and derived parameters used for our numerical estimates. +MW +80.385 GeV +ΓW +2.0854 GeV +MZ +91.1876 GeV +ΓZ +2.4952 GeV +Gµ +1.166390 × 10−5 GeV−2 +mt +173.2 GeV +mh +125 GeV +m2 +W = M2 +W − iMW ΓW +(6461.748225 − 167.634879 i) GeV2 +m2 +Z = M2 +Z − iMZΓZ +(8315.17839376 − 227.53129952 i) GeV2 +cos2 θW = m2 +W /m2 +Z +(0.7770725897054007 + 0.001103218322282256 i) +α = +√ +2Gµ +π +M2 +W (1 − M2 +W +M2 +Z ) +7.56246890198475 × 10−3 giving 1/α ≈ 132.23 . . . +comparison with experimental studies) there is an effect of less than two percent for R = 0.4. +This is in agreement with the deviations we found for dveto +2 +for this approximation. +We take into account this variation in our uncertainty estimates, see Section 3.3. A definitive +statement on this issue will have to await an exact calculation of dveto +3 +. +3 +Setup for phenomenology +Before discussing phenomenological results, we list our input parameters, the method for +matching to fixed order, and the approach for estimating uncertainties at fixed order and at +the resummed level. +3.1 +Input parameters +The input values used in our numerical studies are shown in Table 2. +As indicated in +the table we use the complex mass scheme for the W and Z boson masses. The number +of light quarks, nf, is set equal to five, except for the case of W +W −-production where +nf = 4. We use the PDF distribution NNPDF31_nnlo_as_0118 except for W +W − where we use +NNPDF31_nnlo_as_0118_nf_4 [36]. Note that we use these NNLO parton distributions even in +our lower order predictions. +In the cases of WW and ZZ production, at O(α2 +s) the cross-section receives contributions from +processes with two gluons in the initial state. When performing the resummed calculations we +only include such contributions at NLL. However, these contributions only represent about 3% +of the cross-section for pveto +T += 10 GeV, rising to about 6–8% for pveto +T += 60 GeV. Therefore, +neglecting higher order corrections to these contributions, which are not implemented in our +code, is justified. Although only strictly true for the leading q¯q component we refer to the full +resummed calculation as N3LLp. +We match the resummation and fixed-order NkLO corrections using a naive additive scheme as +– 9 – + +follows, +σN(k+1)LL+N(k)LO(pveto +T +) = σN(k+1)LL(pveto +T +) + σ∆,k(pveto +T +) , where +(3.1) +σ∆(pveto +T +) = σNkLO(pveto +T +) − dσN(k+1)LL(pveto +T +) +���� +exp. to NkLO +. +(3.2) +The matching correction σ∆(pveto +T +) is defined as a function of pveto +T +, using the difference between +the fixed-order contribution and the resummed result expanded to the same fixed order. The +limit pveto +T +→ 0 of σ∆(pveto +T +) is finite, which also allows its use as a higher-order subtraction +scheme. +The use of a naive matching without a transition mechanism that switches off the resummation +at large pveto +T +is justified since the matching corrections for all considered cases in this paper are +small; even in the most extreme case they are less than 20%. In other words, the resummation +alone provides a good description of the cross-sections and does not need to be switched off. +Any transition function to turn off the resummation at large pveto +T +would have a very small +effect. This is in contrast to transverse-momentum resummation where a transition function is +necessary [31]. +3.2 +Uncertainty estimates at fixed order +Ultimately the resummed predictions should offer a practical advantage compared to the +fixed-order predictions. In many cases, the quantity log(Q/pveto +T +) is not very large, and it may +not seem worthwhile to use resummed results. However, as we will show, the resummation +works remarkably well on its own and has matching corrections of only up to around 20%, often +much less. The clear separation of scales and the resummation then allow for smaller and more +reliable uncertainty estimates. To set the stage, we first examine perturbative convergence and +uncertainties at fixed order for quark and gluon induced boson processes, as well as for WW +and ZZ production. +Constructing jet-vetoed cross-sections at fixed order requires the combination of different +cross-sections. However, if we naively subtract the jet cross-section from the inclusive result, it +can result in underestimated uncertainties and narrowing uncertainty bands. To avoid this, +different methods have been proposed in the literature, of which we compare the following +two. +One strategy, which we term the "two-scale" approach, is to consider the different relevant +scales Q and pveto +T +of the vetoed cross-section σ0, and include both of them in the uncertainty +estimate through a multi-point variation around both scales [8]. To compute this uncertainty, +we separately vary the renormalization scale µr and the factorization scale µf over the values +{µh, 2µh, µh/2, pveto +T +, 2pveto +T +, pveto +T +/2}, where µh depends on the process under consideration. An +estimate of the uncertainty is then obtained by adding in quadrature the maximum deviations +from µr = µf = µh, from µr and µf variation separately. +– 10 – + +Another approach, advocated by Refs. [14, 37], takes the jet-veto efficiency (JVE) as the central +quantity, which is the ratio of jet-vetoed cross-section to total cross-section. By combining the +uncertainties of these two quantities in quadrature, one obtains a more robust estimate of the +uncertainty in the jet-vetoed cross-section. This is because the uncertainties are considered +uncorrelated: the uncertainties in the jet-veto efficiency are typically due to non-cancellation +of real and virtual contributions, while those in the total cross-section are connected with large +corrections from higher orders [14]. +For our JVE approach, we follow the simplest formulation (“scheme (a)” of Ref. [14]) to compute +a JVE-based uncertainty. For this we consider variation over the scales {µh, 2µh, µh/2} of σincl +and combine in quadrature the uncertainty from the calculation of the 0-jet efficiency (σ0/σincl) +and the uncertainty from the inclusive calculation. Our final fixed-order uncertainty band is +the envelope of the two-scale and JVE approaches. +With these procedures, our fixed-order results for Z and H production are shown in Fig. 3. +For Z production we use the canonical choice µh = Q, where Q is the invariant mass in the +final state. For Higgs production we use µh = Q/2, guided by the calculation of the inclusive +cross-section where such a choice results in markedly-improved perturbative convergence. We +observe that for Z production the NNLO uncertainty band is wholly contained within the NLO +one, while for the Higgs case the bands at least overlap somewhat throughout the range. For +Higgs production following the combined two-scale and JVE approach results in a significantly +larger uncertainty at both NLO and NNLO, especially at smaller values of pveto +T +. On the other +hand, for Z production the additional uncertainty from the JVE approach is very small and +negligible at NNLO. +Predictions for WW and ZZ production (with µh = Q) are shown in Fig. 4. The limited +overlap between the NLO and NNLO bands indicates that uncertainties are underestimated, +even with the generous scale uncertainty procedure that we follow. The additional uncertainty +resulting from the JVE procedure is small, especially at NNLO, because the scale uncertainty +of the inclusive cross-sections is very small. +3.3 +Uncertainty estimates at the resummed and matched level +For our central predictions, we set the resummation and factorization scales to µ = pveto +T +and +the hard scale (corresponding to the renormalization scale) to µh = Q, where Q is the invariant +mass of the color-singlet final state. The exception is Higgs production, where we choose +µh = Q/2 as previously discussed. For the collinear anomaly coefficient dveto +3 +, we use the form +given in Eq. (2.24) [14] with R0 = 1. +Complications arising at fixed order, described in Section 3.2, are not present in the resummed +case and therefore we can follow a simpler approach where we vary all scales in our formalism +and take the envelope, as detailed below. While the matching of resummed predictions to +fixed-order could still introduce a complication, the matching corrections are not dominant. +The bulk of the cross-section comes from the resummation and it allows us to follow the simple +– 11 – + +(a) Z production using the setup of ref. [38]. +(b) H production. +Figure 3: Comparison of NLO and NNLO fixed order predictions as a function of the jet veto. +Central predictions solid, uncertainty estimates using either the two-scale approach (dotted) +or the envelope of that and the JVE approach (dashed). +(a) WW production using the setup of ref. [39]. +(b) ZZ production using the setup of ref. [40]. +Figure 4: Comparison of NLO and NNLO fixed order predictions as a function of the jet veto. +Central predictions solid, uncertainty estimates using either the two-scale approach (dotted) +or the envelope of that and the JVE approach (dashed). +– 12 – + +procedure of varying all scales in the naively obtained (without JVE) jet-veto cross-section +too. +The small and narrowing uncertainty bands at fixed order would typically appear in regions +where the resummation is found to be dominant, i.e. where fixed-order contributes very little +through the matching corrections. In practice we observe that the size of uncertainties are +overall uniform in both the resummation and large pveto +T +fixed-order regions, as can be seen in all +of our following predictions. This supports the conclusion that our procedure is sufficient. +Overall, our procedure for estimating uncertainties is as follows. +1. For the resummation (fixed-order) parts we vary both the resummation (factorization) +and hard (renormalization) scales by a factor of two about their central values, adding +the excursions in quadrature to obtain the total scale uncertainty. +2. For the resummation we re-introduce the rapidity scale in Eq. (2.5) by re-writing the +collinear anomaly factor as follows [12, 41]: +� Q +pveto +T +�−2Fii(pveto +T +,R,µ) += +�Q +ν +�−2Fii(pveto +T +,R,µ)� ν +pveto +T +�−2Fii(pveto +T +,R,µ) +. +(3.3) +For ν ∼ pveto +T +the second factor can be expanded since it does not contain a large logarithm. +We vary the rapidity scale ν in the range [pveto +T +/2, 2pveto +T +] for gluon-initiated processes +and in the range [pveto +T +/6, 6pveto +T +] for quark-initiated processes. The large variation for +quark-initiated processes ensures overlapping uncertainty bands at NNLL and N3LLp; +this is achieved by the range given above, as demonstrated explicitly in Sections 4 and 5. +3. The parameter R0 in dveto +3 +is varied between 0.5 and 2. +We first combine the scale uncertainties (1 and 2) in quadrature and then, to obtain our total +uncertainty, add the variation of R0 (3) linearly. +3.4 +Effects of cuts on rapidity at fixed order +The usual jet veto resummation described so far imposes no cut on the jet rapidity. This is in +contrast to experimental analyses, see Table 3, which impose such a cut because of limited +detector acceptance and to diminish the effect of pileup. Ref. [42] identifies three different +regimes, depending on pt, Q and ycut. +• For pveto +T +/Q ≫ exp(−ycut) standard jet veto resummation should apply, effects due to +the rapidity cut are corrections power suppressed by Q exp(−ycut)/pveto +T +. +• For pveto +T +/Q ∼ exp(−ycut) the effects of a rapidity cut must be treated as a leading power +correction. +• For pveto +T +/Q ≪ exp(−ycut) the logarithmic structure is changed already at leading log +level, and non-global logarithms appear. +– 13 – + +Table 3: Jet rapidity cuts applied in the experimental studies examined later in this paper. +Process +Ref. +ycut +Higgs +– +no study +Z (CMS) +[38] +2.4 +W (ATLAS) +[43] +4.4 +WW (CMS) +[39] +4.5 +WZ (ATLAS) +[44] +4.5 +WZ (CMS) +[45] +2.5 +ZZ (CMS) +– +no study +(a) Z production following the setup of ref. [38]. +(b) H production. +Figure 5: Effect of the jet rapidity cut at NNLO with pveto +T += 30 GeV. +We estimate the practical impact of experimentally used jet rapidity cuts at fixed order. +Including the rapidity cut in the resummation requires large changes and ingredients, which +are also only available a low order so far [42]. +The effect of the jet rapidity cut for the Z and Higgs production cases is illustrated in Fig. 5. +These calculations are performed at NNLO for pveto +T += 30 GeV. The rapidity cut plays a bigger +role for Higgs production: for example for ycut = 2.5 the cross-section is 11% larger than +the result with no rapidity cut, compared to only 2% for Z production. This is due to the +larger logarithm (log(mH/pveto +T +)/ log(mZ/pveto +T +) ≈ 1.28) and the larger color prefactor (CA/CF += 2.25) in Higgs production. However, for ycut = 4.5 the effect of the rapidity cut is negligible +in both cases. +– 14 – + +(a) WW production. +(b) ZZ production. +Figure 6: Effect of the jet rapidity cut at NNLO with pveto +T += 30 GeV. +The corresponding results for diboson processes are shown in Fig. 6. In this case, the disparity +between Q and pveto +T +is much larger, so the rapidity cut can play a crucial role, although the +effect is still not as important as for Higgs production. For ycut = 2.5 the WW and ZZ +cross-sections 4% larger than the results with no rapidity cut, and the effect of ycut = 4.5 is +negligible. +4 +Comparison with JetVHeto +While jet-veto resummed phenomenology has been extensively studied in the literature, the +only public codes that permit detailed predictions use JetVHeto or RadISH. +For jet-veto +resummation RadISH implements the analytic JetVHeto resummation formula [5]. The codes +rely on the formalism of the CAESAR approach [4, 46] extended to NNLL [5]. An extension +of the RadISH code has been used to perform joint jet-veto and boson transverse momentum +resummation [47]. +For our comparisons we use RadISH version 3.0.0 [48, 49] and JetVHeto version 3.0.0 [5, 14, 37] +including small-R resummation [4, 35] as part of MCFM-RE [16]. Both codes operate at the +level of NNLL and we have checked that they give indeed the same results. +In our comparison, we would like focus on the differences in the resummation part, since +the fixed-order part is identical in each calculation. +We explore how central values and +uncertainties compare at NNLL to our results and in how far N3LLp results improve the +perturbative convergence. However, the matching to fixed-order is handled differently in each +formalism. Different matching schemes (e.g. additive or multiplicative schemes of various +– 15 – + +types) probe higher-order effects. It has also been advocated to match at the level of jet-veto +efficiencies [14]. Fortunately, matching corrections are generally small for jet-veto scales of +30 to 40 GeV for all considered boson and di-boson processes. We therefore focus on the +resummation in our comparison. +The JetVHeto formalism considers three scales µR, µF and Q that are all similar in magnitude +to the hard scale. To ensure that the resummation switches off for pveto +T +≳ Q, the resummed +logarithms are modified through the prescription log(Q/pveto +T +) → 1/p log((Q/pveto +T +)p + 1). For +JetVHeto p has a default value of 5 [14], while for RadISH the default choice is 4. For comparison +purposes we use p = 5 in both cases. It is evident that for sufficiently small pveto +T +the precise +value of p does not matter. Changing this parameter has a similar effect to turning off the +resummation with a transition function. In principle this demands a fully matched calculation, +but the matching corrections of our considered cases are small and we have checked that the +effect of changing p to 3 or 4 is subleading compared to the scale uncertainties. Here we focus +on those scale uncertainties. +In ref. [14] it has been argued that the Q should be varied by a factor of 3 +2 around its central +value, based on new insights from convergence at N3LO for Higgs production. For simplicity, +we use a more conservative variation by a factor of two. We independently vary µR, µF and Q +by a factor of two around a central scale of mℓℓ for Z-boson production and around mH/2 +for Higgs production. Our uncertainty bands for this comparison are obtained by taking the +envelope of these results. +Z-boson production +For the comparison of Z production we choose a central hard scale of mℓℓ with results shown +in Fig. 7. We find that our MCFM NNLL central values have only marginal compatibility with +our JetVHeto uncertainty estimates, despite having the same logarithmic order. This indicates +that the JetVHeto uncertainties (as estimated according to our procedure just described) do +not fully account for the higher-order corrections. On the other hand, our uncertainties at +NNLL are larger, leading to an overall agreement between the two methods. +At N3LLp uncertainties decrease dramatically compared to NNLL, but they are quite asymmetric, +which suggests that a symmetrization of uncertainties may be necessary in this case. We also +observe that without the large uncertainties at NNLL, there would be no overlap between the +N3LLp results and NNLL. This highlights the importance of carefully estimating and comparing +uncertainties to accurately assess the compatibility of different methods and results. +H-boson production +In our study of Higgs production, we choose a central hard scale of mH/2 and show results in +Fig. 8. All results are computed in the mt → ∞ theory and rescaled by a factor of 1.0653 to +account for finite top-quark mass effects, see Eq. (G.5). +– 16 – + +qq → Z → e+e−, s = 13 TeV, µh = me+e− +200 +400 +600 +800 +10 +20 +30 +40 +pt +veto [GeV] +σveto [pb] +RadISH/JetVHeto/MCFM−RE NNLL +MCFM NNLL +MCFM N3LLp +Figure 7: Comparison of JetVHeto NNLL resummation with our NNLL and N3LLp results for +Z production with cuts as in Table 4. +The Higgs case is distinct from Z production since it is gluon-gluon initiated instead of +quark-initiated. In this case, our predictions agree well with the JetVHeto results, but our +uncertainties at NNLL are again much larger. +Note that we vary the JetVHeto scale Q by a factor of two, while the JetVHeto authors vary by +a factor of 3/2 in the Higgs case. This difference in the amount of variation may require some +tuning in our formalism, at least at the NNLL level. However, the perturbative convergence is +again excellent with small uncertainties at N3LLp and central predictions that agree well with +NNLL. +5 +Phenomenological results +In this section, we present the results of our phenomenological studies, which are based +on the uncertainty procedure, matching to fixed-order, and input parameters described in +Section 3. We compare our findings with experimental results from the literature and discuss +their implications. +5.1 +Z and W production +The process of Z production has already been extensively studied in the literature, thus +enabling a variety of cross-checks of our calculation. The implementation of the hard function +and its evolution has been verified by comparison with the explicit results given in Table 1 of +ref. [50]. The full machinery of the resummation and matching procedure can also be compared +with the results of ref. [5], with which we find excellent agreement within uncertainties, see +also Section 4. +– 17 – + +gg → H, s = 13.6 TeV, µh = mH 2 +20 +30 +40 +20 +30 +40 +50 +pt +veto [GeV] +σveto [pb] +RadISH/JetVHeto/MCFM−RE NNLL +MCFM NNLL +MCFM N3LLp +Figure 8: Comparison of JetVHeto NNLL resummation with our NNLL and N3LLp results for +non-decaying H production. +Table 4: Cuts used in the analysis of Z production, adapted from ref. [38]. +lepton cuts +ql1 +T > 30 GeV, ql2 +T > 20 GeV, |ηl| < 2.4 +lepton pair mass +71 GeV < ml−l+ < 111 GeV +jet veto +anti-kT , R = 0.4, 0-jet events only +We first investigate the impact of choosing a time-like hard scale in the resummed result for +Z production. Previous work has shown that choosing a space-like hard scale (µ2 +h = Q2) +can lead to significant corrections in the perturbative expansion of some processes, while a +time-like hard scale (µ2 +h = −Q2) can resum certain π2 contributions [51] using a complex +strong coupling. +For this comparison we consider purely resummed results at NNLL and N3LLp, only considering +uncertainties originating from scale variation (items 1 and 2 of our uncertainty procedure in +Section 3.3). We consider the process pp → Z/γ∗ → ℓ−ℓ+, i.e. a final state of definite lepton +flavor. We use the same set of cuts and vetoes as in the √s = 13 TeV CMS analysis [38], but +extend the veto to jets of all rapidities, rather than only those with |y| < 2.4. This difference, +and the effect of matching to NNLO, is discussed in detail in Section 5.1.1. +Our results are shown in Fig. 9a as a function of the value of the jet veto. We observe that +the results do not depend strongly on the choice of hard scale, with a difference of about 4% +at NNLL and only 1% at N3LLp. This indicates that resumming the π2 terms results in only +– 18 – + +(a) Predictions are computed using a central choice +for the hard scale given by either µ2 +h = Q2 or +µ2 +h = −Q2. The lower panel shows the ratio of +the result for µ2 +h = −Q2 to the one for µ2 +h = Q2. +(b) Predictions and CMS measurement as ratio to +matched result. +Figure 9: Comparison of NNLL and N3LLp predictions for Z production as a function of the +jet veto, using the setup of ref. [38] (central predictions solid, uncertainty estimate according +to the text, dashed). +a small enhancement of the cross-section for W and Z production. Based on these findings, +we use the space-like hard scale (µ2 +h = Q2) in our subsequent studies of Z and W boson +production, as it is the more commonly used choice in the literature. +5.1.1 +CMS Z production +As previously mentioned, the CMS measurement we are comparing to includes a jet rapidity cut +of |y| < 2.4. To assess the importance of this restriction, we first compare the NNLO predictions +with and without the rapidity cut, as a function of the jet veto value. This comparison, shown +in Table 5, helps us better understand the limitations of our analysis. +We use the quantity ϵ(pveto +T +) to quantify the increase in the cross-section when the rapidity cut +is applied, defined as +ϵ(pveto +T +) = σ0−jet(ycut = 2.4) +σ0−jet(no ycut) +− 1 . +(5.1) +The experimental measurement we are comparing to uses a jet veto of pveto +T += 30 GeV, for which +the rapidity cut has only a 3% effect on the cross-section. This suggests that our calculation +with an all-rapidity jet veto is appropriate for comparing to the experimental measurement. +However, as pveto +T +decreases, the impact of the rapidity cut becomes more significant, until at +– 19 – + +Table 5: The Z + 0-jet cross-section prediction at NNLO (µ = Q), with and without a jet +rapidity cut. +pveto +T +[GeV] +5 +10 +20 +30 +40 +σ0−jet(no ycut) [pb] +140 +347 +539 +627 +675 +σ0−jet(ycut = 2.4) [pb] +242 +411 +569 +643 +685 +ϵ +0.73 +0.18 +0.06 +0.03 +0.01 +qq → Z → l+l−, s = 13 TeV, CMS cuts, arXiv:2205.02872 +618 ± 17 pb +592−13 ++9 pb +600 +650 +700 +750 +800 +CMS +NLO +NNLL +NNLL+NLO +NNLO +N3LLp +N3LL+NNLO +σveto [pb] +Figure 10: Comparison of Z-boson jet-vetoed predictions with the CMS [38] 13 TeV measure- +ment. Shown are results at fixed-order, purely resummed and matched. +pveto +T += 5 GeV it is no longer appropriate to neglect the rapidity cut. This is consistent with the +arguments of Ref. [42], which suggest that the standard jet veto resummation formalism should +suffice as long as ln(Q/pveto +T +) ≪ ycut. In our case, ln(Q/pveto +T +) ranges from 0.8 to 2.9 for pveto +T +from 40 down to 5 GeV, so the standard jet veto resummation should be appropriate, albeit +with sizeable power corrections, for ycut = 2.4 except for the smallest values of pveto +T +. +We now turn to a comparison with the CMS result [38], which uses a jet threshold of 30 GeV. +Our comparison with fixed-order, purely resummed and matched predictions is shown in Fig. 10. +We find that the fixed-order and resummed results differ by only a few percent, indicating +that resummation is not necessary for this value of the jet veto. This is because the quantity +ln(MZ/pveto +T +) = 1.1 is not large enough to require resummation. The CMS measurement yields +a cross-section of 618 ± 17 pb, while our best prediction is 592+9 +−13 pb. +We study the production of Z bosons as a function of the jet veto in Fig. 9b. We observe +that the difference between the resummed and central fixed-order results is small, even for the +smallest values of pveto +T +considered. However, the uncertainties in the fixed-order prediction are +larger across the whole range, particularly for small pveto +T +. For values of pveto +T +in the range of 20 +to 40 GeV, which are of practical interest, the N3LLp uncertainty is smaller than the NNLO +uncertainty by about a factor of 1.5. +5.1.2 +ATLAS W production +We now perform a comparison with √s = 8 TeV ATLAS data on W production [43]. For +this study, jets were identified using the anti-kT algorithm with R = 0.4 and must satisfy +– 20 – + +qq' → W± → eν, s = 8 TeV, ATLAS cuts, arXiv:1711.03296 + 4.72 ± 0.3 nb +4.71−0.1 ++0.07 nb +4.0 +4.5 +5.0 +5.5 +6.0 +ATLAS +NLO +NNLL +NNLL+NLO +NNLO +N3LLp +N3LL+NNLO +σveto [nb] +Figure 11: Comparison of W-boson jet-vetoed predictions with the ATLAS [43] 8 TeV mea- +surement. Shown are results at fixed-order, purely resummed and matched. +pT > 30 GeV and |y| < 4.4. We have checked at fixed order that this large rapidity cut has a +negligible impact of a few per mille, i.e. results are unchanged within the numerical precision +to which we work. +Summing over both W charges and including only the decay into electrons we compare our +predictions in Fig. 11. We show results at fixed order, at the resummed level, and at the +matched level. The effect of matching is large and we thus conclude that this value for the jet +veto is outside the sensible range for a purely resummed result, unlike for the Z study in the +previous subsection. +We observe excellent agreement with the theoretical prediction, albeit with a larger experimental +uncertainty. +The experimentally measured cross-section is 4.72 ± 0.30 nb while our best +prediction is 4.71+0.07 +−0.10 nb. Since this measurement corresponds to an integrated luminosity of +only 20 fb−1 it is clear that the high-luminosity LHC will eventually be able to provide a much +keener test of perturbative QCD in this process. +5.2 +W +W − production +Experimental studies of WW production were performed by both ATLAS [52, 53] and CMS [39, +54]. Here we focus on the CMS analysis of ref. [39] since it provides a measurement of the 0-jet +cross-section as a function of the jet pT veto. This cross-section measurment corresponds to +a sum over both electron and muon decays of the W bosons, which we denote by the label +pp → W −W + → 2ℓ2ν. In order to account for this in our calculation, we compute the result +for pp → e−µ+¯νeνµ at NNLO and multiply it by the factor that accounts exactly for all lepton +combinations through NLO. The impact of ZZ contributions in the same-flavor case results in +a slight enhancement over the naïve factor of four. We find that, independent of the value of +the jet veto in the range that we consider, this factor is equal to 4.15. +The CMS analysis only imposes a jet rapidity cut of ycut = 4.5, so our expectation is that the +standard jet veto resummation formalism should be appropriate for pveto +T +values between 60 +and 10 GeV, since in this case the logarithm of the ratio of Q to pveto +T +are in the range of 1.3 +to 3.1. This expectation is supported by the NNLO analysis in Table 6, which shows only a +– 21 – + +small 2% effect from the rapidity cut for pveto +T += 10 GeV (and none for values above that). +Unlike the processes considered so far, Q is no longer set by a resonance mass but is instead a +distribution with a peak slightly above the 2MW threshold. For illustration, we have used an +average value of Q ∼ 220 GeV. +We first fix the value of pveto +T += 30 GeV and study the sensitivity of the pure fixed-order and +resummed calculations to the jet-clustering parameter R. The results are shown in Fig. 12a. At +NLO, there is at most one additional parton, so the NLO result does not depend on the value of +R. However, the NNLL result exhibits a mild dependence on R, which is most noticeable in the +size of the uncertainties. These uncertainties are much larger for smaller values of R, as was +previously observed and discussed in the context of Higgs production in Ref. [12]. At NNLO, +the fixed-order calculation becomes sensitive to the value of R, although the dependence is very +small. At N3LLp, the dependence is reduced compared to NNLL, especially at small R. Overall, +these results suggest that the jet-clustering parameter has a mild effect on the predictions of +the fixed-order and resummed calculations for WW production. We have not investigated the +effect of small R resummation [14] on these results. +In Fig. 12b, we extend our previous analysis of the jet-veto dependence of WW production, +which was presented in Ref. [55]. The effect of matching is substantial for values of pveto +T +greater +than 20 GeV, so for typical jet vetoes in the range of 20 to 40 GeV, matched predictions are +important. We find that the fixed-order description is only capable of providing an adequate +result for the highest value of pveto +T +studied here. A comparison with the CMS measurement +shows better agreement with the matched resummed calculation, although the experimental +uncertainties are still substantial, corresponding to an integrated luminosity of 36 fb−1. +We eagerly anticipate a measurement with more statistics in order to hone this comparison. +Future measurements with higher precision and larger data samples will provide a more +stringent test of the theoretical predictions and help to refine our understanding of WW +production at the LHC. +5.3 +W ±Z production +5.3.1 +ATLAS +For W ±Z production, we first compare our results with an analysis from the ATLAS collabora- +tion at √s = 13 TeV [44]. The 0-jet cross-section is measured with jets defined by the anti-kT +Table 6: The pp → W −W + → 2ℓ2ν+0-jet cross-section at NNLO, with and without a jet +rapidity cut. +pveto +T +[GeV] +10 +25 +30 +35 +45 +60 +σ0−jet(no ycut) [fb] +535 +963 +1004 +1054 +1145 +1237 +σ0−jet(ycut = 4.5) [fb] +548 +963 +1004 +1054 +1145 +1237 +ϵ +0.02 +0.00 +0.00 +0.00 +0.00 +0.00 +– 22 – + +(a) Jet radius R dependence of fixed-order and +purely resummed results. +(b) Predictions and CMS measurement as a ratio +to the matched result. +Figure 12: Comparison of NNLO, N3LLp and matched N3LLp+NNLO results for W +W − +production. +qq' → W±Z, s = 13 TeV, ATLAS cuts, arXiv:1902.05759 + 31 ± 2.5 fb +29.7−1.2 ++0.9 fb +27 +30 +33 +36 +ATLAS +NLO +NNLL +NNLL+NLO +NNLO +N3LLp +N3LL+NNLO +σveto [pb] +Figure 13: Comparison of W ±Z jet-vetoed predictions with the ATLAS 13 TeV measurement +[44]. Shown are results at fixed order, purely resummed and matched. +algorithm with pT > 25 GeV, |y| < 4.5, and R = 0.4. +Since ln(Q/pveto +T +) = 2.3 (for pveto +T += 25 GeV, using an average Q of about 240 GeV), we expect +that standard jet veto resummation should be applicable in this case, since ycut = 4.5. We +have checked that the effect of the rapidity cut is at the per mille level, which is less than our +numerical precision. +The ATLAS result is presented for a single leptonic channel and summed over both W charges. +The corresponding theoretical predictions at fixed order, at the resummed level, and at the +matched level are shown in Fig. 13. +– 23 – + +qq' → W±Z, s = 13 TeV, CMS cuts, arXiv:2110.11231 + 166 ± 6 fb +128 ± 8 fb, ycut < ∞ +120 +140 +160 +180 +CMS +NLO +NNLL +NNLL+NLO +NNLO +N3LLp +N3LL+NNLO +σveto [pb] +Figure 14: Comparison of W ±Z jet-vetoed predictions with the CMS [45] 13 TeV measurement. +Shown are results at fixed-order, purely resummed and matched, all without a rapidity cut. +Overall, the measurement is in good agreement with both the N3LLp+NNLO and NNLO +predictions, within the mutual uncertainties. Only a more precise measurement would be +able to definitively support the need for resummation in this case. Since the ATLAS analysis +includes only 36 fb−1 of data, it is likely that a more precise measurement will be possible in +the near future. +5.3.2 +CMS +We now contrast the ATLAS study of the W ±Z process with one from CMS [45]. In the +CMS study, jets are defined by the anti-kT algorithm with pT > 25 GeV, |y| < 2.5, and +R = 0.4. +To assess the applicability of the jet-rapidity inclusive resummation framework, we must com- +pare ln(Q/pveto +T +) = 2.3 with ycut = 2.5. This suggests that the standard jet veto resummation +formalism may not be appropriate in this case, and that the use of ycut-dependent beam +functions [42] may be necessary to provide a reliable theoretical prediction. Despite this, we +still pursue the comparison here, without using ycut-dependent beam functions, to examine +the limitations of our approach. +The CMS result for W ±Z production is presented after summing over all lepton flavors and +both W charges. On the theoretical side, we perform a similar analysis, but ignore same-flavor +effects that only enter at the 2% level. To construct the jet-vetoed cross-section for the CMS +measurement, we combine the differential results in Figure 14(c) of Ref. [45] with the inclusive +cross-sections reported in Table 6 of the same reference. Our results are shown in Fig. 14. +We find that neither the resummed prediction nor the NNLO one are in good agreement +with the CMS data, even when the NNLO calculation takes the jet rapidity cut into account +(increasing the NNLO result from 128 fb to 137 fb). This suggests that resummation is required +in this case, and that the use of ycut-dependent beam functions is necessary to provide a reliable +theoretical prediction. Overall, these results highlight the importance of using appropriate +resummation techniques to accurately predict W ±Z production at the LHC with a small jet +rapidity cut. +– 24 – + +lepton cuts +ql1 +T > 20 GeV, ql2 +T > 10 GeV, +ql3,4 +T +> 5 GeV, |ηl| < 2.5 +lepton pair mass +60 GeV < ml−l+ < 120 GeV +jet veto +anti-kT , R = 0.5 +Table 7: Fiducial cuts used for the ZZ analysis, taken from the CMS study in Ref. [40]. +Table 8: The ZZ + 0-jet cross-section at NNLO (µ = Q), with and without a jet rapidity cut. +pveto +T +[GeV] +10 +20 +30 +40 +50 +60 +σ0−jet(no ycut) [fb] +13.3 +21.5 +25.8 +28.4 +30.3 +31.6 +σ0−jet(ycut = 4.5) [fb] +13.4 +21.5 +25.8 +28.4 +30.3 +31.6 +σ0−jet(ycut = 2.5) [fb] +14.9 +22.4 +26.3 +28.8 +30.6 +31.8 +ϵ(ycut = 4.5) +0.01 +0.00 +0.00 +0.00 +0.00 +0.00 +ϵ(ycut = 2.5) +0.12 +0.04 +0.02 +0.01 +0.01 +0.01 +5.4 +ZZ production +In the absence of jet-vetoed cross-sections for comparison, we use the cuts from a recent CMS +study [40] to investigate our theoretical predictions for ZZ production as a function of pveto +T +. +In the results that follow we consider a sum over Z decays into both electrons and muons, +which we denote by pp → ZZ → 4 leptons, and apply the cuts shown in Table 7. +We expect that standard jet veto resummation should provide good predictions for ycut = 4.5, +since ln(Q/pveto +T +) is in the range of 1.4 to 3.2 for pveto +T +values between 60 and 10 GeV, using an +average Q of about 240 GeV. For ycut = 2.5, we expect larger rapidity effects for the smallest +values of pveto +T +. This is supported by our analysis in Table 8, which shows only a very small +(1%) effect from a rapidity cut of ycut = 4.5 for pveto +T += 10 GeV (and no effect for higher values). +Even for ycut = 2.5, the rapidity cut has a relevant effect only for pveto +T +values below 30 GeV, +and is mostly insignificant beyond that. +Fig. 15a shows a comparison of the dependence on pveto +T +for purely-resummed results at two +different logarithmic orders. The central predictions are very similar at NNLL and N3LLp +and are consistent within uncertainties for all values of pveto +T +. Fig. 15b compares the matched +N3LLp+NNLO and NNLO results. The NNLO prediction has large uncertainties over the whole +range of pveto +T +and only overlaps with N3LLp+NNLO around 40 GeV and higher. The difference +between the central resummed and fixed-order results is significant (around 10%) for typical +values of pveto +T +around 30 GeV. For most relevant values of pveto +T +at the LHC, resummation is +clearly important for providing a precision prediction for this process. +5.5 +Higgs production +For gluon fusion Higgs production an important topic is the inclusion of finite top-quark mass +effects. Although at NNLO these could be included exactly [56, 57], the mass effects are not +relevant in the jet-vetoed case [58] at the current level of precision. A simple overall one-loop +– 25 – + +(a) Purely resummed results. +(b) Ratio to matched result. +Figure 15: Comparison of NNLO, N3LLp and matched N3LLp+NNLO results for ZZ production +as a function of the jet veto. +rescaling factor that takes into account the full mass dependence is sufficient to introduce mass +effects into mt → ∞ EFT predictions. In the resummation formalism, the coefficient for the +matching of Higgs production in QCD onto SCET can be calculated in two ways, referred to as +one-step and two-step procedures. +5.5.1 +One-step and two-step schemes +The one-step procedure is based on the observation that the ratio mH/mt is not large in a +logarithmic sense (c.f. ρ = m2 +H/m2 +t ≈ 1/2 and αs log 1/ρ ≈ 0.07). This procedure matches the +full QCD result, typically obtained at higher orders as an expansion in the parameter r, onto +SCET at the scale µh ∼ mH. In this way, terms of order ρ are retained, but logs of mt/mH +are neglected. +In the two-step procedure outlined in Refs. [59–62], the top quark is first integrated out at a +scale µt ≊ mt, and then the QCD effective Lagrangian is matched onto the SCET at a scale +µh ≊ mH. Running between µt and µh allows one to sum logarithms of mt/mH, and finite +top-mass effects are included by scaling the result by a correction factor obtained at leading +order (an increase with respect to the EFT result by a factor of 1.0653, see Eq. (G.5)). Terms +enhanced by powers of mH/mt are thus only included in an approximate fashion at NLO and +beyond. The one-step procedure is described in detail in Appendix G.1 and the two-step +procedure is described in Appendix G.2. +We compare the numerical difference between the one- and two-step schemes, computed at +√s = 13.6 TeV and for R = 0.4 in Fig. 16a. Guided by fixed-order results, and in accord with +– 26 – + +(a) Results in the one- or two-step scheme. The +lower panel shows the ratio of the one-step to the +two-step result. +(b) Results using a central scale of either µ2 +h = Q2 +or µ2 +h = −Q2. The lower panel shows the ratio of +the result for µ2 +h = −Q2 to the one for µ2 +h = Q2. +Figure 16: Comparison of NNLL and N3LLp predictions for Higgs production at √s = 13.6 TeV +as a function of the jet veto. +previous studies of this process [14], we set the hard (renormalization) scale using µh = Q/2. +We observe that the one-step scheme results in a cross-section that is about 1.7–2.3% larger at +NNLL and only 1.6% larger at N3LLp. This small difference occurs if one works rigorously at +a fixed order of αs. Working at a fixed order in αs in the component parts of the two-step +scheme can lead to larger differences, as described in more detail in Appendix G.3. +5.5.2 +Time-like vs. space-like µ2 +h +We now study the impact of choosing a time-like hard scale for the calculation of the Higgs +cross-section. To do this, we compare µ2 +h = (Q/2)2 (the space-like scale) with µ2 +h = −(Q/2)2 +(the time-like scale). The use of a time-like hard scale allows us to resum certain π2 terms, by +employing a complex strong coupling [51]. For this comparison, we consider purely resummed +results at NNLL and N3LLp accuracy. +Results are shown in Fig. 16b, for the two-step scheme computed at √s = 13.6 TeV with +R = 0.4. We observe that at NNLL, the resummation of the π2 terms significantly enhances +the cross-section by 17%. However, at N3LLp accuracy, this resummation only leads to a small +increase of 2% in the cross-section. +Results for the matched vetoed cross-section are shown in Fig. 17. +After matching, we +observe substantial agreement between the NNLO and N3LLp+NNLO calculations within +– 27 – + +Figure 17: Comparison of NNLL, N3LLp and N3LLp+NNLO predictions for Higgs production +at √s = 13.6 TeV as a function of the jet veto. +uncertainties. The central predictions differ by about 5% across the range, but the uncertainties +are substantially smaller in the resummed calculation. +6 +Conclusions +We have presented a comprehensive study of jet-veto resummation in the production of color +singlet final states using the most up-to-date theoretical ingredients and achieving N3LLp +accuracy. Our implementation in MCFM improves upon previous public NNLL calculations +by reducing theoretical uncertainties, as demonstrated by comparisons with ATLAS and CMS +results. Once the one remaining theoretical element, dveto +3 +, becomes available, it will be simple +to upgrade our predictions to full N3LL accuracy.2 +The primary motivation for this work comes from the need for reliable and accurate predictions +of jet-veto cross-sections in processes such as Higgs boson and W +W − production, which are +commonly used to study new physics at the LHC. In these processes, the imposition of a jet +veto is often necessary to suppress backgrounds and enhance sensitivity to new physics signals. +Experimental results going beyond these two processes are much less frequent. We encourage +the experimental collaborations to consider measurements of more Standard Model processes +with a jet veto, as larger data samples become available, to better understand the dependence +of these processes on the jet veto parameters pveto +T +and R. +2We have shown that the effect of including gluon-induced process in W +W − and ZZ production is +numerically a small effect, so that NLL accuracy is sufficient for these sub-processes. +– 28 – + +In addition to providing improved predictions for jet-veto cross-sections, our work also serves +as a valuable tool for testing and validation of general purpose shower Monte Carlo programs. +Our code allows for a detailed investigation of the dependence on the jet parameters pveto +T +and +R, providing a benchmark for assessing the logarithmic accuracy and reliability of Monte Carlo +simulations in this important class of processes. +Our analysis shows that at the currently experimentally used values of pveto +T +in W and Z +production, the logarithms are not large enough to justify the use of jet-veto resummation. +In these cases, fixed-order perturbation theory, which can be used to give the results with +a jet veto over a limited range of rapidities, is simpler and sufficient. We have also found +that attempts to resum π2 terms using a timelike renormalization point have little numerical +importance at N3LLp if the pveto +T +scale is around 20 to 30 GeV. +The production of a Higgs boson is an exception among single-boson processes. In this case, +the combination of larger corrections from color factors and slightly larger values of the scale +(mH) appearing in the jet veto logarithms make resummation an important tool for improving +the accuracy of predictions. In the appendix we have investigated the differences between the +one-step and two-step procedures for calculating the hard function at the scale of pveto +T +. We +find agreement within 2% of these two approaches. +The W +W − production process, where the jet veto has experimental importance, requires +both resummation and matching to NNLO. For the ZZ process resummation is mandatory +but the matching to fixed order is less important. Although this reflects the expectation that +the resummed prediction is more accurate for systems of higher invariant mass, these findings +depend on the exact nature of the cuts for each process. Our work provides a comprehensive +theoretical framework for studying jet vetoes in vector boson pair processes, and as data +becomes available, a comparative experimental study would be of great interest and could help +to validate our theoretical predictions. +Acknowledgments +RKE would like to thank Simone Alioli, Thomas Becher, Andrew Gilbert, Pier Monni and +Philip Sommer for useful discussions. In addition, RKE would like to thank TTP in Karlsruhe +for hospitality during the drafting of this paper. TN would like to thank Robert Szafron +for useful discussions. SS is supported in part by the SERB-MATRICS under Grant No. +MTR/2022/000135. This manuscript has been authored by Fermi Research Alliance, LLC +under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of +Science, Office of High Energy Physics. This research used resources of the Wilson High- +Performance Computing Facility at Fermilab. This research also used resources of the National +Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office +of Science User Facility located at Lawrence Berkeley National Laboratory, operated under +Contract No. DE-AC02-05CH11231 using NERSC award HEP-ERCAP0021890. +– 29 – + +A +Reduced beam functions +We have used the two loop beam function in the presence of a jet veto calculated in Ref. [24]. +Their calculation, together with the corresponding soft function [25] has been performed in +SCET using the exponential rapidity regulator [27]. The beam function for quark initiated +processes in the presence of a jet veto has also been presented in Mellin space in Ref. [63]. +The calculation in Ref. [24] has a perturbative expansion, +Iij = +∞ +� +k=0 +�αs +4π +�k +I(k) +ij . +(A.1) +The beam functions with a jet veto are decomposed into a reference observable, the beam +function for the transverse momentum of a color singlet observable and a remainder term +accounting for the effects of jet clustering, +Iij(x, Q, pveto +T +, R; µ, ν) = I⊥ +ij(x, Q, pveto +T +; µ, ν) + ∆Iij(x, Q, pveto +T +, R; µ, ν) . +(A.2) +Since the divergence structure of the reference observable is the same as the beam function +with a jet veto, ∆Iij can be calculated in four dimensions. Results for the reference observable +are available in Refs. [64, 65]. +The reduced beam function kernels ¯I as used in our setup are extracted from the coefficient I +as +¯Iij(z, pveto +T +, R, µ) = e−hA(pveto +T +,µ) Iij(z, pveto +T +, R, µ) . +(A.3) +They similarly follow a perturbative expansion +¯Iik(z, pveto +T +, R, µ) = δik δ(1−z)+ αs +4π +¯I(1) +ik (z, pveto +T +, µ)+ +�αs +4π +�2 ¯I(2) +ik (z, pveto +T +, R, µ)+O(α3 +s) . (A.4) +Contributions at order αs +The αs contributions to ¯I were first obtained in Refs. [3, 50] and read, +¯Iij(z, pveto +T +, R, µ) = δ(1 − z) δij + αs +4π +� +−2P (1) +ij (z) L⊥ + R(1) +ij (z) +� ++ O(α2 +s) , +(A.5) +where L⊥ = 2 ln(µ/pveto +T +). R is the jet measure used in Eq. (2.1) and R(1)(z) is a remainder +function given below. At this order there is no dependence on the jet radius, R. +Throughout this paper we expand in powers of αs/(4π). The one exception to this rule are +the perturbative DGLAP splitting functions, +Pij(z) = αs +2πP (1)(z) + +�αs +2π +�2 +P (2)(z) + . . . +(A.6) +– 30 – + +Explicit expressions for P (1) and P (2) are given in Appendices C.2 and C.3. The remainder +functions at order αS are [66] +R(1) +qq (z) = CF +� +2(1 − z) − π2 +6 δ(1 − z) +� +, +R(1) +qg (z) = 4TF z(1 − z) , +R(1) +gg (z) = −CA +π2 +6 δ(1 − z) , +R(1) +gq (z) = 2CF z . +(A.7) +where CA = 3, CF = 4 +3, TF = 1 +2. +Contributions at order α2 +s +At order α2 +s we have +¯I(2) +ik (z, pveto +T +, R, µ) = +� +2P (1) +ij (x) ⊗ P (1) +jk (y) − β0P (1) +ik (z) +� +L2 +⊥ ++ +� +− 4P (2) +ik (z) + β0R(1) +ik (z) − 2R(1) +ij (x) ⊗ P (1) +jk (y) +� +L⊥ + R(2) +ik (z, R) . +(A.8) +In this equation ⊗ represents a convolution, +f(x) ⊗ g(y) = +� 1 +0 +dx +� 1 +0 +dyf(x) g(y) δ(z − xy) = +� 1 +z +dy +y f(z/y) g(y) . +(A.9) +Explicit expressions for P (1) and P (2) are given in Appendices C.2 and C.3. The expressions +for P (1) ⊗ P (1), R(1) ⊗ P (1) are given in appendix C.4. +The results from Refs. [24, 25] recast in the language of reduced beam functions allow us to +extract R(2) +ik (z, R). We have checked that the reduced beam functions have the form predicted +by Eqs. (A.5) and (A.8). In addition, we have confirmed the known results for the α2 +s R- +dependent contribution to the collinear anomaly exponent. The result for the collinear anomaly +exponent is given in Section 2.1. +A.1 +Structure of the two-loop reduced beam function +While a numerical evaluation of the analytical formulas for the reduced beam functions is +possible, we choose to perform a spline interpolation for improved numerical efficiency. +The reduced beam functions contain distributions of the following structure, +¯I(2) +ij (z, pveto +T +, R, µ) = ¯I(2) +ij,−1(pveto +T +, R, µ) δ(1 − z) + ¯I(2) +ij,0(pveto +T +, R, µ) D0(1 − z) ++ ¯I(2) +ij,1(pveto +T +, R, µ) D1(1 − z) + ¯I(2) +ij,2(z, pveto +T +, R, µ) , +(A.10) +where, +D0(1 − z) = +1 +[1 − z]+ +, +D1(1 − z) = +�ln(1 − z) +(1 − z) +� ++ +. +(A.11) +¯I(2) +ij,2(z, pveto +T +, R, µ) contains terms which are regular at z = 1. +– 31 – + +The analytic results for the beam function of Ref. [24] are presented as a power series in +R up to powers of R8. The functions themselves contain powers of 1/(1 − z)n, in certain +cases up to n = 7 or 8. However, these singularities at z = 1 are fictitious as can be seen by +explicit expansion. The beam functions require special treatment in this region for numerical +stability. +The dominant region in the convolution of the function ¯I with the parton distributions is +precisely the region z ∼ 1. If we assume a parton distribution f(x) ∼ 1/x we have, +¯I ⊗ f = +� 1 +x +dz +z +¯I(z) f(x/z) ∼ 1 +x +� 1 +x +dz ¯I(z) , +(A.12) +showing that all regions of z contribute equally to the integral. However if, as expected, the +parton distribution function falls off more rapidly as x → 1, say f(x) ∼ (1 − x)n/x, +¯I ⊗ f = +� 1 +x +dz +z +¯I(z) f(x/z) ∼ 1 +x +� 1 +x +dz ¯I(z) (1 − x/z)n . +(A.13) +Thus, it is precisely the large values of z which are crucial for the integral. In other words, +the parton shower process is dominated by cascade from nearby values of x. Larger cascades +from more distant points are suppressed by the fall-off of the parton distributions. In view of +the importance of the region z = 1, for numerical stability we perform an expansion about +z = 1. +The absolute value of R(2) for the various parton transitions is shown in Fig. 18. Individual +R-dependent terms contain expressions of the form R2n/(1 − z)k where k can be a high power. +However, the singularity at z = 1 is only apparent. The resultant limiting forms obtained by +series expansion about z = 1 are shown by the dashed lines in the figures. In practice, we +switch to the expanded form at z = 0.9, although the figures demonstrate that the expanded +forms are accurate down to much smaller values of z. +B +Definition of the beta function and anomalous dimensions +The coefficients βn, ΓA +n and γg +n have perturbative expansions in powers of the renormalized +coupling. Details are presented below. +B.1 +Expansion of β-function +The beta function is defined as, +dαs(µ) +d ln µ += β(µ) = −2αs(µ) +∞ +� +n=0 +βn +�αs +4π +�n+1 += −2αs(µ) αs(µ) +4π +� +β0 + β1 +αs(µ) +4π ++ β2 +�αs(µ) +4π +�2 ++ β3 +�αs(µ) +4π +�3 ++ . . . +� +. +(B.1) +– 32 – + +(a) gg case. +(b) qq case +(c) gq case. +(d) qg case +(e) ¯qq case. +(f) q′q case. +Figure 18: Absolute value of R(2) for jet measure R = 0.5. The ¯q′q case is the same as the +q′q case. The sign of the contribution in the various regions is indicated. +The coefficients of the MS β function to four loops are [67–69], +β0 = 11 +3 CA − 4 +3 TF nf , +– 33 – + +β1 = 34 +3 C2 +A − +�20 +3 CA + 4CF +� +TF nf , +β2 = 2857 +54 C3 +A + +� +C2 +F − 205 +18 CF CA − 1415 +54 C2 +A +� +2TF nf + +�11 +9 CF + 79 +54 CA +� +4T 2 +F n2 +f , +β3 = C4 +A +�150653 +486 +− 44 +9 ζ3 +� ++ C3 +ATF nf +� +−39143 +81 ++ 136 +3 ζ3 +� ++C2 +ACF TF nf +�7073 +243 − 656 +9 ζ3 +� ++ CAC2 +F TF nf +� +−4204 +27 ++ 352 +9 ζ3 +� ++46C3 +F TF nf + C2 +AT 2 +F n2 +f +�7930 +81 ++ 224 +9 ζ3 +� ++ C2 +F T 2 +F n2 +f +�1352 +27 +− 704 +9 ζ3 +� ++CACF T 2 +F n2 +f +�17152 +243 ++ 448 +9 ζ3 +� ++ 424 +243CAT 3 +F n3 +f + 1232 +243 CF T 3 +F n3 +f ++dabcd +A +dabcd +A +NA +� +−80 +9 + 704 +3 ζ3 +� ++ nf +dabcd +F +dabcd +A +NA +�512 +9 +− 1664 +3 +ζ3 +� ++n2 +f +dabcd +F +dabcd +F +NA +� +−704 +9 ++ 512 +3 ζ3 +� +. +(B.2) +For the normalization of the SU(N) generators, the conventions of Refs. [69, 70] are, +dabcd +A +dabcd +A +NA += N2(N2 + 36) +24 +, +dabcd +F +dabcd +A +NA += N(N2 + 6) +48 +, +dabcd +F +dabcd +F +NA += N4 − 6N2 + 18 +96N2 +, +NA = N2 − 1 , +NF = N. +(B.3) +Numerical values for the β-function coefficients are, +β0 = 11 − 2 +3 nf , +β1 = 102 − 38 +3 nf , +β2 = 2857 +2 +− 5033 +18 nf + 325 +54 n2 +f , +β3 = 149753 +6 ++ 3564ζ3 − +�1078361 +162 ++ 6508 +27 ζ3 +� +nf + +�50065 +162 ++ 6472 +81 ζ3 +� +n2 +f ++ 1093 +729 n3 +f . +(B.4) +B.2 +Cusp Anomalous Dimension +The cusp anomalous dimension depends on the label B which takes the two values, B = A, F +for gluons and quarks, respectively. Its perturbative expansion is, +ΓB +cusp(µ) = +∞ +� +n=0 +ΓB +n +�αs +4π +�n+1 +. +(B.5) +– 34 – + +The coefficients up to four loops are [71, 72], +ΓB +0 = 4CB , +(B.6) +ΓB +1 = 16CB +� +(CA +�67 +36 − π2 +12 +� +− 5 +9nfTF +� +, +(B.7) +ΓB +2 = 64CB +� +C2 +A +�11ζ3 +24 + 245 +96 − 67π2 +216 + 11π4 +720 +� ++ nfTF CF +� +ζ3 − 55 +48 +� ++ nfTF CA +� +−7ζ3 +6 − 209 +216 + 5π2 +54 +� +− 1 +27(nfTF )2 +� +, +(B.8) +ΓB +3 = 256CB +� +C3 +A +�1309ζ3 +432 +− 11π2ζ3 +144 +− ζ2 +3 +16 − 451ζ5 +288 + 42139 +10368 − 5525π2 +7776 ++ 451π4 +5760 − 313π6 +90720 +� ++ nfTF C2 +A +� +−361ζ3 +54 ++ 7π2ζ3 +36 ++ 131ζ5 +72 +− 24137 +10368 + 635π2 +1944 − 11π4 +2160 +� ++ nfTF CF CA +�29ζ3 +9 +− π2ζ3 +6 ++ 5ζ5 +4 − 17033 +5184 + 55π2 +288 − 11π4 +720 +� ++ nfTF C2 +F +�37ζ3 +24 − 5ζ5 +2 + 143 +288 +� ++ (nfTF )2CA +�35ζ3 +27 − 7π4 +1080 − 19π2 +972 + 923 +5184 +� ++ (nfTF )2CF +� +−10ζ3 +9 ++ π4 +180 + 299 +648 +� ++ (nfTF )3 +� +− 1 +81 + 2ζ3 +27 +� � ++ 256dabcd +B +dabcd +A +NB +�ζ3 +6 − 3ζ2 +3 +2 ++ 55ζ5 +12 − π2 +12 − 31π6 +7560 +� ++ 256nf +dabcd +B +dabcd +F +NB +�π2 +6 − ζ3 +3 − 5ζ5 +3 +� +. +(B.9) +In addition to the relations in Eq. (B.3) we need the related quantities, +dabcd +F +dabcd +A +NF += (N2 − 1)(N2 + 6) +48 +, +dabcd +F +dabcd +F +NF += (N2 − 1)(N4 − 6N2 + 18) +96N3 +. +(B.10) +B.3 +Non-cusp anomalous dimension +The non-cusp anomalous dimension has the expansion, +γq,g(µ) = +∞ +� +n=0 +γq,g +n +�αs +4π +�n+1 +. +(B.11) +– 35 – + +We take the coefficients up to three loops from ref. [73] Eq. I.4, +γq +0 = −3CF , +(B.12) +γq +1 = C2 +F +� +2π2 − 3 +2 − 24ζ3 +� ++ CF CA +� +26ζ3 − 961 +54 − 11π2 +6 +� ++ CF TF nf +�130 +27 + 2π2 +3 +� +, +(B.13) +γq +2 = C3 +F +� +− 29 +2 − 3π2 − 8π4 +5 +− 68ζ3 + 16π2 +3 +ζ3 + 240ζ5 +� ++ C2 +F CA +� +− 151 +4 ++ 205π2 +9 ++ 247π4 +135 +− 844 +3 ζ3 − 8π2 +3 ζ3 − 120ζ5 +� ++ CF C2 +A +� +− 139345 +2916 +− 7163π2 +486 +− 83π4 +90 ++ 3526 +9 +ζ3 − 44π2 +9 +ζ3 − 136ζ5 +� ++ C2 +F TF nf +�2953 +27 +− 26π2 +9 +− 28π4 +27 ++ 512 +9 ζ3 +� ++ CF CATF nf +� +− 17318 +729 ++ 2594π2 +243 ++ 22π4 +45 +− 1928 +27 ζ3 +� ++ CF T 2 +F n2 +f +�9668 +729 − 40π2 +27 +− 32 +27ζ3 +� +. +(B.14) +From ref. [74], Eq A5 we take, +γg +0 = −β0 , +(B.15) +γg +1 = C2 +A +�11π2 +18 +− 692 +27 + 2ζ3 +� ++ CATF nf +�256 +27 − 2π2 +9 +� ++ 4CF TF nf += C2 +A +� +2ζ3 − 59 +9 +� ++ CAβ0 +�π2 +6 − 19 +9 +� +− β1 , +(B.16) +γg +2 = C3 +A +� +− 97186 +729 ++ 6109π2 +486 +− 319π4 +270 ++ 122 +3 ζ3 − 20π2 +9 +ζ3 − 16ζ5 +� ++ C2 +ATF nf +�30715 +729 +− 1198π2 +243 ++ 82π4 +135 + 712 +27 ζ3 +� ++ CACF TF nf +�2434 +27 +− 2π2 +3 +− 8π4 +45 − 304 +9 ζ3 +� +− 2C2 +F TF nf + CAT 2 +F n2 +f +� +− 538 +729 + 40π2 +81 +− 224 +27 ζ3 +� +− 44 +9 CF T 2 +F n2 +f . +(B.17) +Primary references for the calculation of these coefficients can be found in Ref. [73]. +We now present results for γS and γt which are needed for the implementation of the two-step +calculation of the hard function for Higgs boson production. Following Ref. [61] we have, for +the first three expansion coefficients of the anomalous dimension γS that enters the evolution +– 36 – + +equation of the hard matching coefficient CS (see also [59, 60]), +γS +0 = 0 , +(B.18) +γS +1 = C2 +A +� +−160 +27 + 11π2 +9 ++ 4ζ3 +� ++ CATF nf +� +−208 +27 − 4π2 +9 +� +− 8CF TF nf , +(B.19) +γS +2 = C3 +A +�37045 +729 ++ 6109π2 +243 +− 319π4 +135 ++ +�244 +3 +− 40π2 +9 +� +ζ3 − 32ζ5 +� ++ C2 +ATF nf +� +−167800 +729 +− 2396π2 +243 ++ 164π4 +135 ++ 1424 +27 ζ3 +� ++ CACF TF nf +�1178 +27 +− 4π2 +3 +− 16π4 +45 +− 608 +9 ζ3 +� ++ 8C2 +F TF nf ++ CAT 2 +F n2 +f +�24520 +729 ++ 80π2 +81 +− 448 +27 ζ3 +� ++ 176 +9 CF T 2 +F n2 +f . +(B.20) +The function γt is given by, +γt(αs) = α2 +s +d +dαs +� +β(αs) +α2s +� += −2β1 +�αs +4π +�2 +− 4β2 +�αs +4π +�3 +− 6β3 +�αs +4π +�4 ++ O(α5 +s) . +(B.21) +As shown in Eq. (G.22) µ independence provides the constraint, +2γg(αs) = γt(αs) + γS(αs) + β(αs)/αs , +(B.22) +leading to the simple relationship between the coefficients in γg and γS, +γS +0 = 2γg +0 + 2β0 , γS +1 = 2γg +1 + 4β1 , γS +2 = 2γg +2 + 6β2, γS +3 = 2γg +3 + 8β3 . +(B.23) +C +Definitions for beam function ingredients +C.1 +Exponent h +We define the auxiliary functions hB for B = F, A which, when combined with the hard +function and the collinear anomaly factor, will yield a renormalization group invariant hard +function. hF/A is defined to satisfy the RGE equation, +d +d ln µ hF/A(pveto +T +, µ) = 2 ΓF/A +cusp(µ) ln +µ +pveto +T +− 2 γq/g(µ) , +(C.1) +The factor h removes logarithms from the beam function and has a perturbative expansion in +terms of the renormalized coupling, +hB(pveto +T +, µ) = αs +4πhB +0 + +�αs +4π +�2 +hB +1 + +�αs +4π +�3 +hB +2 + +�αs +4π +�4 +hB +3 + . . . . +(C.2) +– 37 – + +Thus for the particular case B = F we have that, +hF +0 (pveto +T +, µ) = 1 +4ΓF +0 L2 +⊥ − γq +0L⊥ , +hF +1 (pveto +T +, µ) = 1 +12ΓF +0 β0L3 +⊥ + 1 +4(ΓF +1 − 2γq +0β0)L2 +⊥ − γq +1L⊥ , +hF +2 (pveto +T +, µ) = 1 +24ΓF +0 β2 +0L4 +⊥ + ( 1 +12ΓF +0 β1 + 1 +6ΓF +1 β0 − 1 +3γq +0β2 +0)L3 +⊥ ++ (1 +4ΓF +2 − 1 +2γq +0β1 − γq +1β0)L2 +⊥ − γq +2L⊥ , +hF +3 (pveto +T +, µ) = + 1 +40ΓF +0 β3 +0L5 +⊥ + ( 5 +48ΓF +0 β0β1 + 1 +8ΓF +1 β2 +0 − 1 +4γq +0β3 +0)L4 +⊥ ++ ( 1 +12ΓF +0 β2 + 1 +6ΓF +1 β1 + 1 +4ΓF +2 β0 − 5 +6γq +0β0β1 − γq +1β2 +0)L3 +⊥ ++ (1 +4ΓF +3 − 1 +2γq +0β2 − γq +1β1 − 3 +2γq +2β0)L2 +⊥ − γq +3L⊥ , +(C.3) +where L⊥ = 2 ln(µ/pveto +T +). The corresponding result for B = A, q = g, (i.e. for incoming gluons) +is given by a similar expression mutatis mutandis. The expansion coefficients of the β-function, +ΓF/A +cusp and γq/g, used in Eq. (C.3), are as given in Appendices B.1,B.2 and B.3. +C.2 +One loop splitting functions +The one-loop DGLAP splitting functions as defined in [75] are +P (1) +qq (z) = CF +�1 + z2 +1 − z +� ++ +, +(C.4) +P (1) +qg (z) = TF +� +z2 + (1 − z)2� +, +(C.5) +P (1) +gg (z) = 2CA +� +z +(1 − z)+ ++ 1 − z +z ++ z(1 − z) +� ++ β0 +2 δ(1 − z) , +(C.6) +P (1) +gq (z) = CF +1 + (1 − z)2 +z +, +(C.7) +C.3 +Two loop splitting functions +Now we turn to the two-loop anomalous dimensions that contribute at sub-leading log level to +the transitions between parton types. In the quark sector there are four independent transitions +that we must produce values for (viz. q′ ← q,¯q′ ← q,q ← q and ¯q ← q). They are expressed in +terms of four functions, +P (2) +q′q = P S(2) +qq +, P (2) +¯q′q = P S(2) +¯qq +, P (2) +qq = P V (2) +qq ++ P S(2) +qq +, P (2) +¯qq = P V (2) +¯qq ++ P S(2) +¯qq +. +(C.8) +– 38 – + +At next-to-leading order, the functions P S +qq and P S +¯qq are non-zero, but we have the additional +relation, P S +qq = P S +¯qq. To facilitate the presentation we define the auxiliary functions, +pqq(z) = +2 +1 − z − 1 − z , p(r) +qq (z) = −1 − z , +(C.9) +pqg(z) = z2 + (1 − z)2 , +(C.10) +pgq(z) = 1 + (1 − z)2 +z +, +(C.11) +pgg(z) = +1 +1 − z + 1 +z − 2 + z(1 − z), p(r) +gg (z) = 1 +z − 2 + z(1 − z) . +(C.12) +The two valence functions needed for the quark sector are, [76–78], +P V (2) +qq +(z) = C2 +F +� +− +� +2 ln z ln(1 − z) + 3 +2 ln z +� +pqq(z) +− +� +3 +2 + 7 +2z +� +ln z − 1 +2(1 + z) ln2 z − 5(1 − z) +� ++CF CA +� +(1 + z) ln z + 20 +3 (1 − z) + +� +1 +2 ln2 z + 11 +6 ln z +� +pqq(z) ++ +� +67 +18 − π2 +6 +�� +1 +(1 − z)+ ++ p(r) +qq (z) +�� +−CF TF nf +� +4 +3(1 − z) + 2 +3pqq(z) ln z + 10 +9 +� +1 +(1 − z)+ ++ p(r) +qq (z) +�� ++ +� +C2 +F +� +3 +8 − π2 +2 + 6ζ3 +� ++ CF CA +� +17 +24 + 11π2 +18 +− 3ζ3 +� +−CF TF nf +� +1 +6 + 2π2 +9 +�� +δ(1 − z) , +(C.13) +P V (2) +¯qq +(z) = CF +� +CF − CA +2 +�� +2pqq(−z)S2(z) + 2(1 + z) ln z + 4(1 − z) +� +, +(C.14) +and for the singlet function we have, +P S(2) +qq += CF TF +� +20 +9z − 2 + 6z − 56 +9 z2 + (1 + 5z + 8 +3z2) ln z − (1 + z) ln2 z +� +. +(C.15) +The other three transitions are simply given by, +P (2) +qg = CF TF +� +2 − 9 +2z − (1 +2 − 2z) ln z − (1 +2 − z) ln2 z + 2 ln(1 − z) +– 39 – + ++ +� +ln2 +� +1 − z +z +� +− 2 ln +� +1 − z +z +� +− π2 +3 + 5 +� +pqg(z) +� ++CATF +� +91 +9 + 7 +9z + 20 +9z + +� +68 +3 z − 19 +3 +� +ln z +−2 ln(1 − z) − (1 + 4z) ln2 z + pqg(−z)S2(z) ++ +� +− 1 +2 ln2 z + 22 +3 ln z − ln2(1 − z) + 2 ln(1 − z) + π2 +6 − 109 +9 +� +pqg(z) +� +, +(C.16) +P (2) +gq (z) = C2 +F +� +− 5 +2 − 7z +2 + +� +2 + 7 +2z +� +ln z − +� +1 − 1 +2z +� +ln2 z +− 2z ln(1 − z) − +� +3 ln(1 − z) + ln2(1 − z) +� +pgq(z) +� ++CF CA +� +28 +9 + 65 +18z + 44 +9 z2 − +� +12 + 5z + 8 +3z2 +� +ln z ++(4 + z) ln2 z + 2z ln(1 − z) + S2(z)pgq(−z) ++ +� +1 +2 − 2 ln z ln(1 − z) + 1 +2 ln2 z + 11 +3 ln(1 − z) + ln2(1 − z) − π2 +6 +� +pgq(z) +� ++CF TF nf +� +− 4 +3z − +� +20 +9 + 4 +3 ln(1 − z) +� +pgq(z) +� +, +(C.17) +P (2) +gg (z) = CF TF nf +� +− 16 + 8z + 20 +3 z2 + 4 +3z − (6 + 10z) ln z − (2 + 2z) ln2 z +� ++CATF nf +� +2 − 2z + 26 +9 +� +z2 − 1 +z +� +− 4 +3(1 + z) ln z +−20 +9 +� +1 +(1 − z)+ ++ p(r) +gg (z) +�� ++C2 +A +� +27 +2 (1 − z) + 67 +9 +� +z2 − 1 +z +� +− +� +25 +3 − 11 +3 z + 44 +3 z2 +� +ln z ++4(1 + z) ln2 z + 2pgg(−z)S2(z) ++ +� +ln2 z − 4 ln z ln(1 − z) +� +pgg(z) + +� +67 +9 − π2 +3 +�� +1 +(1 − z)+ ++ p(r) +gg (z) +�� ++ +� +C2 +A +�8 +3 + 3ζ3 +� +− CF TF nf − 4 +3CATF nf +� +δ(1 − z) . +(C.18) +– 40 – + +The function S2(z) is defined by +S2(z) = +� +1 +1+z +z +1+z +dy +y ln +�1 − y +y +� +. +(C.19) +In terms of the dilogarithm function +Li2(z) = − +� z +0 +dy +y ln(1 − y) , +(C.20) +we have +S2(z) = −2 Li2(−z) + 1 +2 ln2 z − 2 ln z ln(1 + z) − π2 +6 . +(C.21) +C.4 +P (1) ⊗ P (1) and R(1) ⊗ P (1) +We give here expressions for the convolutions of functions appearing in the beam functions. +The convolutions are defined as in Eq. (A.9). Similar expressions have been given in [1, 12] +The convolutions of the one-loop DGLAP kernels from Eqs. (C.4) are, +P (1) +qq +⊗ P (1) +qg = CF TF +� +2z − 1 +2 + (2z − 4z2 − 1) ln z + (2 − 4z(1 − z)) ln(1 − z) +� +, +(C.22) +P (1) +qg +⊗ P (1) +gg = CATF +� +2(1 + 4z) ln z + 4 +3z + 1 + 8z − 31 +3 z2� ++ +� +2CA ln(1 − z) + β0 +2 +� +P (1) +qg (z) , +(C.23) +P (1) +gq ⊗ P (1) +qq = C2 +F +� +2 − 1 +2z + (2 − z) ln z +� ++ 2CF P (1) +gq (z) ln(1 − z) +� +, +(C.24) +P (1) +gg ⊗ P (1) +gq = CACF +� +8 + z + (4z3 − 31) +3z +− 4(1 + z + z2) +z +ln z +� ++ +� +2CA ln(1 − z) + β0 +2 +� +P (1) +gq (z) , +(C.25) +P (1) +qg +⊗ P (1) +gq = CF TF +� +2(1 + z) ln z + 1 − z + 4 +3 +(1 − z3) +z +� +, +(C.26) +P (1) +qq +⊗ P (1) +qq = C2 +F +� +8 +�ln(1 − z) +(1 − z) +� ++ − 4(1 + z) ln(1 − z) − 2(1 − z) ++ +� +3 + 3z − +4 +(1 − z) +� +ln z +� ++ 3CF P (1) +qq (z) − C2 +F (9 +4 + 4ζ2)δ(1 − z) , +(C.27) +P (1) +gg ⊗ P (1) +gg = 4C2 +A +� +2 +�ln(1 − z) +(1 − z) +� ++ + 2((1 − z) +z ++ z(1 − z) − 1) ln(1 − z) + 3(1 − z) +− ( +1 +1 − z + 1 +z − z2 + 3z) ln z − 11(1 − z3) +3z +� ++ β0P (1) +gg (z) − (β2 +0 +4 + 4C2 +Aζ2)δ(1 − z) . +(C.28) +The convolutions of lowest order DGLAP kernels, Eq. (C.4) with the one-loop finite terms in +the beam functions, Eq. (A.7) are, +R(1) +gg ⊗ P (1) +gg = −CAζ2P (1) +gg (z) , +(C.29) +– 41 – + +R(1) +gq ⊗ P (1) +qg += 2CF TF +� +(1 − z)(1 + 2z) + 2z ln z +� +, +(C.30) +R(1) +qq ⊗ P (1) +qq += CF +� +CF (1 − z)(4 ln(1 − z) − 2 ln z − 1) − ζ2P (1) +qq (z) +� +, +(C.31) +R(1) +qg ⊗ P (1) +gq = −4CF TF +� +1 + z ln z − (1 + 2z3) +3z +� +, +(C.32) +R(1) +qg ⊗ P (1) +gg = −CATF (16z ln z − 68 +3 z2 + 20z + 4 − 4 +3z ) ++ (2CA ln(1 − z) + β0 +2 )R(1) +qg (z) , +(C.33) +R(1) +qq ⊗ P (1) +qg += CF TF (2z2 + 2z − 4 − (2 + 4z) ln z) − CF ζ2P (1) +qg (z) , +(C.34) +R(1) +gq ⊗ P (1) +qq += −C2 +F (2z ln z − 4z ln(1 − z) − z − 2) , +(C.35) +R(1) +gg ⊗ P (1) +gq = −CAζ2P (1) +gq (z) . +(C.36) +D +Rapidity anomalous dimension +Solving the collinear anomaly RG equation (Eq. (2.13)) as an expansion in αs (Eq. (2.15)) we +have that, +F (0) +gg (pveto +T +, µh) = ΓA +0 L⊥ + dveto +1 +(R, A) , +F (1) +gg (pveto +T +, µh) = 1 +2ΓA +0 β0L2 +⊥ + ΓA +1 L⊥ + dveto +2 +(R, A) , +F (2) +gg (pveto +T +, µh) = 1 +3ΓA +0 β2 +0L3 +⊥ + 1 +2(ΓA +0 β1 + 2ΓA +1 β0)L2 +⊥ ++ (ΓA +2 + 2β0dveto +2 +(R, A))L⊥ + dveto +3 +(R, A) , +F (3) +gg (pveto +T +, µh) = 1 +4β3 +0ΓA +0 L4 +⊥ + (ΓA +1 β2 +0 + 5 +6ΓA +0 β0β1)L3 +⊥ ++ (1 +2ΓA +0 β2 + ΓA +1 β1 + 3 +2ΓA +2 β0 + 3dveto +2 +(R, A)β2 +0)L2 +⊥ ++ (ΓA +3 + 3dveto +3 +(R, A)β0 + 2dveto +2 +(R, A)β1)L⊥ + dveto +4 +(R, A) . +(D.1) +where L⊥ = 2 ln(µh/pveto +T +). The corresponding result for Fqq is given in Eq. (2.16). Because +Fgg appears in the exponent, we see that dveto +1 +contributes in NLL, dveto +2 +in NNLL, and dveto +3 +in +N3LL. +– 42 – + +D.1 +dveto +2 +expansion +The expansion coefficients for dveto +2 +, which is defined in Eq. (2.18), are given by [4, 5, 12], +cA +L = 131 +72 − π2 +6 − 11 +6 ln 2 = −1.096259 , +cA +0 = −805 +216 + 11π2 +72 ++ 35 +18 ln 2 + 11 +6 ln2 2 + ζ3 +2 = 0.6106495 , +cA +2 = +1429 +172800 + π2 +48 + 13 +180 ln 2 = 0.263947 , +cA +4 = − 9383279 +406425600 − +π2 +3456 + +587 +120960 ln 2 = −0.0225794 , +cA +6 = +74801417 +97542144000 − +23 +67200 ln 2 = 5.29625 · 10−4 , +cA +8 = − +50937246539 +2266099089408000 − +π2 +24883200 + +28529 +1916006400 ln 2 = −1.25537 · 10−5 , +cA +10 = +348989849431 +243708656615424000 − +3509 +3962649600 ln 2 = 8.18201 · 10−7 . +(D.2) +and +cf +L = −23 +36 + 2 +3 ln 2 = −0.1767908 , +cf +0 = 157 +108 − π2 +18 − 8 +9 ln 2 − 2 +3 ln2 2 = −0.03104049 , +cf +2 = 3071 +86400 − +7 +360 ln 2 = 0.0220661 , +cf +4 = − +168401 +101606400 + +53 +30240 ln 2 = −4.42544 · 10−4 , +cf +6 = +7001023 +48771072000 − +11 +100800 ln 2 = 6.79076 · 10−5 , +cf +8 = − +5664846191 +566524772352000 + +4001 +479001600 ln 2 = −4.20958 · 10−6 , +cf +10 = +68089272001 +83774850711552000 − +13817 +21794572800 ln 2 = 3.73334 · 10−7 , +(D.3) +We see that for values of the jet radius R < 1 the terms c6, c8 and c10 can be dropped. +For the gluon case the expansion of the function in numerical form is, +f(R, A) = − (1.0963 CA + 0.1768 TF nf) ln R + (0.6106 CA − 0.0310 TF nf) ++ (−0.5585 CA + 0.0221 TF nf) R2 ++ (0.0399 CA − 0.0004 TF nf) R4 + . . . , +(D.4) +whereas for the quark case we have +f(R, F) = − (1.0963 CA + 0.1768 TF nf) ln R + (0.6106 CA − 0.0310 TF nf) ++ (−0.8225 CF + 0.2639 CA + 0.0221 TF nf) R2 ++ (0.0625 CF − 0.02258 CA − 0.0004 TF nf) R4 + . . . . +(D.5) +– 43 – + +E +Renormalization Group Evolution +The evolution equation matching for a generic hard matching coefficient C has the form, +d +d ln µ ln C(Q2, µ) = +� +Γcusp(αs(µ)) ln Q2 +µ2 + γ(αs(µ)) +� +. +(E.1) +Following ref. [26] the solution to the evolution equation Eq. (E.1) is, +C(Q2, µ) = exp [2S(µh, µ) − aγ(µh, µ)] +�Q2 +µ2 +h +�−aΓ(µh,µ) +C(Q2, µh) , +(E.2) +ln C(Q2, µ) = 2S(µh, µ) − aγ(µh, µ) − aΓ(µh, µ) ln +�Q2 +µ2 +h +� ++ ln C(Q2, µh) , +(E.3) +where µh ∼ Q is a hard matching scale at which the Wilson coefficient C is calculated using +fixed-order perturbation theory. The Sudakov exponent S and the exponents aγ, aΓ are the +solutions to the auxiliary differential equations, +d +d ln µ S(ν, µ) = −Γcusp +� +αs(µ) +� +ln µ +ν , +(E.4) +d +d ln µ aΓ(ν, µ) = −Γcusp +� +αs(µ) +� +, +(E.5) +d +d ln µ aγ(ν, µ) = −γ +� +αs(µ) +� +. +(E.6) +with the boundary conditions S(ν, ν) = aΓ(ν, ν) = aγ(ν, ν) = 0 at µ = ν. Differentiating +Eq. (E.3) we recover Eq. (E.1). +The solutions to the evolution equation are conveniently expressed in terms of the running +coupling, +aΓ(ν, µ) = − +αs(µ) +� +αs(ν) +dα Γcusp(α) +β(α) +, +(E.7) +S(ν, µ) = − +αs(µ) +� +αs(ν) +dα Γcusp(α) +β(α) +α +� +αs(ν) +dα′ +β(α′) . +(E.8) +Substituting the values for the beta function coefficients in the MS scheme given in Appendix B.1 +and the values for cusp anomalous dimension given in Appendix B.2 into Eq. (E.7) we +obtain, +aΓ(µh, µ) = aΓ +0 + aΓ +1 + aΓ +2 + aΓ +3 , +(E.9) +– 44 – + +where the coefficients in the expansion are, +aΓ +0 = Γ0 ln(r) +2β0 +, +r = αs(µ)/αs(µh) , +(E.10) +aΓ +1 = αs(µh)(r − 1)(β0Γ1 − β1Γ0) +8πβ2 +0 +, +(E.11) +aΓ +2 = α2 +s(µh)(r2 − 1) +� +−β0β1Γ1 + β0(β0Γ2 − β2Γ0) + β2 +1Γ0 +� +64π2β3 +0 +, +(E.12) +aΓ +3 = −α3 +s(µh) +� +r3 − 1 +� +× +� +β2 +0(−β0Γ3 + β2Γ1 + β3Γ0) − β0β2 +1Γ1 + β0β1(β0Γ2 − 2β2Γ0) + β3 +1Γ0 +� +384π3β4 +0 +. +(E.13) +The solution for aγ follows from the one for aΓ by making the replacement Γk → γk. The +non-cusp anomalous dimensions γ are given in Appendix B.3. +Evaluating Eq. (E.8) to obtain the evolution for S we get, +S(µh, µ) = S0 + S1 + S2 . +(E.14) +with, +S0 = +1 +8β3 +0 +� +8πβ0Γ0(r + r(− ln(r)) − 1) +αs(µh)r ++ 2(r − 1)(β1Γ0 − β0Γ1) ++ ln(r)(2β0Γ1 + β1Γ0 ln(r) − 2β1Γ0) +� +, +(E.15) +S1 = −αs(µh) +32πβ4 +0 +� +2 ln(r) +� +−β0β1Γ1r + β0β2Γ0 + β2 +1Γ0(r − 1) +� ++ (r − 1) +� +−β0β1Γ1(r − 3) + β0(β0(r − 1)Γ2 − β2Γ0(r + 1)) + β2 +1Γ0(r − 1) +� +� +,(E.16) +S2 = α2 +s(µh) +256π2β5 +0 +� +2 ln(r) +� +β1r2 � +−β0β1Γ1 + β0(β0Γ2 − β2Γ0) + β2 +1Γ0 +� +− Γ0 +� +β2 +0β3 − 2β0β1β2 + β3 +1 +� � ++ (r − 1) +� +β2 +0(2(β0(r + 1)Γ3 − 2β2Γ1) − β3Γ0(r + 1)) + β0β2 +1Γ1(r + 5) ++ β0β1(β2Γ0(r + 5) − 3β0(r + 1)Γ2) − 4β3 +1Γ0 +�� +. +(E.17) +E.1 +Recovery of the double log formula +As we have seen S satisfies a RGE given by Eq. (E.4) with a solution given by Eq. (E.8). The +leading term in S0, Eq. (E.15) is +S0 ≈ +πΓ0 +β2 +0αs(µh) +� +1 + ln +�1 +r +� +− 1 +r +� +, +(E.18) +– 45 – + +where r = αs(µ)/αs(µh). In this form the presence of a double log is obscured. We can easily +recover the double log by retaining only the leading terms. The leading expression for r is +given by solving the equation for the beta function, +1 +r = 1 − αs(µh) +2π +β0 ln +�µh +µ +� +, +(E.19) +S0 ≈ +πΓ0 +β2 +0αs(µh) +�αs(µh) +2π +β0 ln +�µh +µ +� ++ ln +� +1 − αs(µh) +2π +β0 ln +�µh +µ +��� +. +(E.20) +Expanding for small αs(µh) ln(µh/µ) we get, +S(µh, µ) ≈ −Γ0 +2 +αS(µh) +4π +ln2 �µh +µ +� +. +(E.21) +This gives the expected log squared with a negative sign. +F +The hard function for the Drell-Yan process +The form factors of the vector current have been presented several places in the literature [79–84]. +The bare form factor is given as, +F q,bare(q2, µ2) = 1 + +�αbare +s +4π +� +(∆)ϵFq +1 + +�αbare +s +4π +�2 +(∆)2ϵFq +2 + O(α3 +s) , +(F.1) +where, +∆ = 4πe−γE +� +µ2 +−q2 − i0 +� +. +(F.2) +In the following we will drop 4πe−γE, so that all poles should be understood in the MS sense. +The values found for the bare coefficients are, +Fq +1 = CF +� +− 2 +ϵ2 − 3 +ϵ + ζ2 − 8 + ϵ +�3ζ2 +2 + 14ζ3 +3 +− 16 +� ++ ϵ2 +�47ζ2 +2 +20 ++ 4ζ2 + 7ζ3 − 32 +�� ++ O(ϵ3) , +(F.3) +Fq +2 = C2 +F +� +2 +ϵ4 + 6 +ϵ3 − 1 +ϵ2 +� +2ζ2 − 41 +2 +� +− 1 +ϵ +�64ζ3 +3 +− 221 +4 +� +− +� +13ζ2 +2 − 17ζ2 +2 ++ 58ζ3 − 1151 +8 +�� ++ CF CA +� +− 11 +6ϵ3 + 1 +ϵ2 +� +ζ2 − 83 +9 +� +− 1 +ϵ +�11ζ2 +6 +− 13ζ3 + 4129 +108 +� ++ +�44ζ2 +2 +5 +− 119ζ2 +9 ++ 467ζ3 +9 +− 89173 +648 +�� ++ CF nf +� +1 +3ϵ3 + 14 +9ϵ2 + 1 +ϵ +�ζ2 +3 + 353 +54 +� ++ +�14ζ2 +9 +− 26ζ3 +9 ++ 7541 +324 +�� ++ O(ϵ) . +(F.4) +– 46 – + +The renormalized form factor can then be written as, +F q(µ2, q2, ϵ) = 1 + +�αs(µ) +4π +� +F q +1 (µ2, q2, ϵ) + +�αs(µ) +4π +�2 +F q +2 (µ2, q2, ϵ) + O(α3 +s) . +(F.5) +where, +F q +1 (µ2, q2, ϵ) = ∆ϵFq +1 , +F q +2 (µ2, q2, ϵ) = ∆2ϵFq +2 − β0 +ϵ ∆ϵFq +1 . +(F.6) +In the full theory the matrix element between on-shell massless quark and gluon states, after +charge renormalization is given by F q(µ2, q2, ϵ). Charge renormalization has removed the UV +poles, but the renormalized form factor still contains IR poles. +The matrix element in the effective theory involves only scaleless, dimensionally regulated +integrals and hence is equal to zero. This vanishing can be interpreted as a cancellation between +ultra-violet and infrared poles: +1 +ϵIR +− +1 +ϵUV +. +(F.7) +After matching, the IR poles in the on-shell matrix element are effectively transformed into UV +poles and need to be renormalized as follows, +CV (αs(µ2), µ2, q2) = lim +ϵ→0 +� +ZV (ϵ, µ2q2) +�−1 F q(µ2, q2, ϵ) , +ln +� +CV (αs(µ2), µ2, q2) +� += ln +� +Fq(µ2, q2, ϵ) +� +− ln +� +ZV (ϵ, µ2, q2) +� +. +(F.8) +The renormalization constant, ZV contains only pure pole terms, +ln ZV (ϵ, µ2, q2) = +�αs +4π +� +� +− ΓF +0 +2ϵ2 + 1 +2ϵ +� +ΓF +0 L + 2γq +0 +�� ++ +�αs +4π +�2 +� +3ΓF +0 β0 +8ϵ3 +− 1 +ϵ2 +�ΓF +0 β0 +4 +L − CF +� +CA(16 +9 + ζ2 +� ++ 4 +9nf) +� ++ 1 +4ϵ +� +ΓF +1 L + 2γq +1 +�� +, (F.9) +where L = ln((−q2 − i0)/µ2). +The matching coefficients have a perturbative expansion in terms of the renormalized cou- +pling, +CV (αs(µ2), µ2, q2) = 1 + +∞ +� +n=1 +�αs(µ2) +4π +�n +CV +n (µ2, q2). +(F.10) +The matching coefficients, which are known to two loop order [85, 86] (and beyond [84]) for +Drell-Yan production, can be obtained from Eq. (F.8): +CV +1 = CF +� +− L2 + 3L − 8 + ζ2 +� +, +(F.11) +– 47 – + +CV +2 = C2 +F +�1 +2L4 − 3L3 + +�25 +2 − ζ2 +� +L2 + +� +− 45 +2 + 24ζ3 − 9ζ2 +� +L ++ 255 +8 +− 30ζ3 + 21ζ2 − 83 +10ζ2 +2 +� ++CF CA +�11 +9 L3 + +� +− 233 +18 + 2ζ2 +� +L2 + +�2545 +54 +− 26ζ3 + 22 +3 ζ2 +� +L +− 51157 +648 ++ 313 +9 ζ3 − 337 +18 ζ2 + 44 +5 ζ2 +2 +� ++ CF nf +� +− 2 +9L3 + 19 +9 L2 + +� +− 209 +27 − 4 +3ζ2 +� +L + 4085 +324 + 2 +9ζ3 + 23 +9 ζ2 +� +, +(F.12) +where L = ln((−q2 − i0)/µ2). CV satisfies the renormalization group equation, +d +d ln µ ln[CV (αs(µ2), µ2, q2)] = ΓF +cusp(µ) ln +�−q2 − i0 +µ2 +� ++ 2γq(µ) , +(F.13) +with the anomalous dimensions as given in Appendix B.2 and Appendix B.3. +The derivation of the hard function for boson pair processes has been described in Ref. [87]. +G +The hard function for Higgs production +G.1 +Implementation of one-step procedure +The one-step procedure [1, 13] is based on the observation that the ratio mt/mH is not large. +For an on-shell Higgs boson the parameter, m2 +H/m2 +t ≈ 1 +2 whereas αs ln(m2 +t /m2 +H) ≈ 0.65αs, +indicating that power corrections should be more important than resumming logarithms. The +matching is performed at a scale µh by integrating out the top quark and all gluons and light +quarks with off-shellness above µh. +The hard Wilson coefficient so defined satisfies the RGE, +µ d +dµ ln CH(m2 +t , q2, µ2) = ΓA +cusp(αs(µ)) ln −q2 − i0 +µ2 ++ 2γg[αs(µ)] , +(G.1) +where Γcusp and γg are given in Eqs. (B.5) and (B.11). As a consequence of Eq. (G.1) the +Wilson coefficient has the following structure, +CH(m2 +t , q2, µ2 +h) = αs(µh)F H +0 +� q2 +4m2 +t +�� +1 + αs(µh) +4π +� +CH +1 +�−q2 − i0 +µ2 +h +� ++ F H +1 +� q2 +4m2 +t +�� ++ +� +αs(µh) +(4π) +�2� +CH +2 +�−q2 − i0 +µ2 +h +, q2 +4m2 +t +� ++ F H +2 +� q2 +4m2 +t +��� +, +(G.2) +The finite terms can be derived from Ref. [88], +F H +0 (z) = 3 +2z − 3 +2z +���1 − 1 +z +��� +� +arcsin2(√z) , +0 < z ≤ 1 , +ln2[−i(√z + √z − 1)] , +z > 1 , +(G.3) +– 48 – + +≈ 1 + 7z +30 + 2z2 +21 + 26z3 +525 + 512z4 +17325 + O(z5), +z < 1 . +(G.4) +For the values of mt and mH in Table 2, +|F H +0 (z0)|2 = 1.0653 , +z0 = m2 +H +4m2 +t +. +(G.5) +The coefficients CH +1 and CH +2 are fixed by the Eq. (G.1). +CH +1 (L) = CA +� +−L2 + π2 +6 +� +, +(G.6) +CH +2 (L, z) = 1 +2C2 +AL4 + 1 +3CAβ0L3 + CA +�� +−4 +3 + π2 +6 +� +CA − 5 +3β0 − F1(z) +� +L2 ++ +��59 +9 − 2ζ3 +� +C2 +A + +�19 +9 − π2 +3 +� +CAβ0 − F1(z)β0 +� +L . +(G.7) +where z = q2/4/m2 +t and L = ln[(−q2 − i0)/µ2 +h]. +The full analytic mt dependence of the virtual two-loop corrections to gg → H in terms of +harmonic polylogarithms were obtained in Refs. [89–91]. For our purposes the results expanded +in m2 +H/m2 +t from Refs. [88, 92, 93] will be sufficient. +The functions F H +1 (z), F H +2 (z) which, +together with F H +0 (z) in Eq. (G.4) encode the mt dependence of the hard Wilson coefficient in +Eq. (G.2). Following the procedure described in Appendix F they are easily extracted from +Ref. [88], +F H +1 (z) = +� +5 − 38 +45 z − 1289 +4725 z2 − 155 +1134 z3 − 5385047 +65488500 z4� +CA ++ +� +−3 + 307 +90 z + 25813 +18900 z2 + 3055907 +3969000 z3 + 659504801 +1309770000 z4� +CF + O(z5) +(G.8) +F H +2 (z) = +� +7C2 +A + 11CACF − 6CF β0 +� +ln(−4z − i0) + +� +−419 +27 + 7π2 +6 ++ π4 +72 − 44ζ3 +� +C2 +A ++ +� +−217 +2 +− π2 +2 + 44ζ3 +� +CACF + +�2255 +108 + 5π2 +12 + 23ζ3 +3 +� +CAβ0 − 5 +6CATF ++ 27 +2 C2 +F + +�41 +2 − 12ζ3 +� +CF β0 − 4 +3CF TF ++ z +� +C2 +A +�11723 +384 ζ3 − 404063 +14400 − 223 +108 ln(−4z − i0) − 19 +135π2� ++ CF CA +�2297 +16 ζ3 − 1099453 +8100 +− 242 +135 ln(−4z − i0) − 953 +540π2 + 28 +15π2 ln 2 +� ++ C2 +F +�13321 +96 +ζ3 − 36803 +240 ++ 7 +3π2 − 56 +15π2 ln 2 +� ++ CF +�77 +12ζ3 − 4393 +405 − 7337 +2700β0 + 39 +10 ln(−4z − i0)β0 + 28 +45π2 + 7 +15π2β0 +� ++ CA +� 77 +384ζ3 − 64097 +129600 − 269 +75 β0 + 2 +15 ln(−4z − i0) − 31 +180 ln(−4z − i0)β0 +�� ++ z2� +C2 +A +�110251 +9216 ζ3 − 3084463261 +254016000 − 2869 +4536 ln(−4z − i0) − 1289 +28350π2� +– 49 – + ++ CF CA +�2997917 +23040 ζ3 − 55535378557 +381024000 +− 18337 +28350 ln(−4z − i0) − 128447 +113400π2 + 1714 +1575π2 ln 2 +� ++ C2 +F +�36173 +192 ζ3 − 95081911 +453600 ++ 857 +630π2 − 3428 +1575π2 ln 2 +� ++ CA +� 265053121 +1524096000 − 16177 +92160ζ3 − 45617 +47250β0 + 16 +315 ln(−4z − i0) − 623 +5400 ln(−4z − i0)β0 +� ++ CF +�21973 +7680 ζ3 − 8108339 +1555200 − 509813 +3969000β0 − 8 +15 ln(−4z − i0) + 29147 +18900 ln(−4z − i0)β0 ++ 1714 +4725π2 + 857 +3150π2β0 +�� ++ O(z3) . +(G.9) +We can assess the quality of the expansion in z by numerical evaluation, +CH(m2 +t , q2, q2) = αs(q)F0(z) +� +1 + 15.9348αs +4π(1 + 0.0158(8z) + .00098312(8z)2) ++ 97.0371 +�αs +4π +�2 +(1 + 0.1883(8z) + 0.0120(8z)2) ++ 143.466 +�αs +4π +�2 ln(−8z − i0) +π +(1 + 0.0288(8z) + 0.001462(8z)2) +� +. +(G.10) +In the vicinity of the Higgs boson pole (8z ≈ 1) subsequent terms in the z expansion are +expected to contribute below the percent level. +G.2 +Implementation of the two-step procedure +In the two-step procedure of Refs. [59–62] one first integrates out the top quark at a scale +µt ≊ mt and subsequently matches from the QCD effective Lagrangian onto SCET at µh ≊ mH. +Running between µh and µt allows one to sum logarithms of mt/mH, but one neglects power +of mH/mt. +G.2.1 +Ct(m2 +t , µ2 +t ) +For a heavy top quark the effective Lagrangian for the production of a top quark is given +by, +Leff = Ct(m2 +t , µ2 +t ) H +v +αs(µ2 +t ) +12π +Gµν aGµν +a , +(G.11) +where v ≈ 246 GeV is the Higgs boson vacuum expectation value. The hard matching scale +µt at which the Wilson coefficient can be computed perturbatively is of order mt. The short +distance coefficient Ct(m2 +t , µ2) obeys the RGE, +d +d ln µCt(m2 +t , µ2) = γt(αs) Ct(m2 +t , µ2), +γt(αs) = α2 +s +d +dαs +�β(αs) +α2s +� +. +(G.12) +The expressions for the short-distance coefficient Ct(m2 +t , µ2 +t ) at NNLO is, +Ct(m2 +t , µ2 +t ) = 1 + αs(µt) +4π +Ct +1 + +�αs(µt) +4π +�2 +Ct +2(m2 +t , µ2 +t ) + . . . , +(G.13) +where (c.f. Eq. (12) of Ref. [61]), +Ct +1 = 5CA − 3CF +– 50 – + +Ct +2(m2 +t , µ2 +t ) = 27 +2 C2 +F + +� +11 ln m2 +t +µ2 +t +− 100 +3 +� +CF CA − +� +7 ln m2 +t +µ2 +t +− 1063 +36 +� +C2 +A +−4 +3CF TF − 5 +6CATF − +� +8 ln m2 +t +µ2 +t ++ 5 +� +CF TF nf − 47 +9 CATF nf . (G.14) +The evolution of these coefficients to the resummation scale µ is described in Appendix A of +Ref. [3]. The solution to the evolution equation Eq. (G.12) for Ct at scale µ is, +Ct(m2 +t , µ2) = β(αs(µ)) +α2s(µ) +α2 +s(µt) +β(αs(µt)) Ct(m2 +t , µ2 +t ) . +(G.15) +The result at NNLO for the square of the coefficient function is, +� +Ct(m2 +t , µ2) +�2 = 1 + +�αs +4π +�� +2Ct +1 + 2(rt − 1)β1 +β0 +� ++ +�αs +4π +�2� +(Ct +1)2 + 2Ct +2(m2 +t , µ2 +t ) + (2β2β0 + β2 +1) +β2 +0 +(rt − 1)2 ++ 2(2β2β0 + 2β1β0Ct +1 − β2 +1) +β2 +0 +(rt − 1) +� +, +(G.16) +where rt = αs(µ)/αs(µt). This extends the NLO result in Eq. (2) of Ref. [3]. +G.2.2 +CS(−q2, µh) +CS is the Wilson coefficient matching the two gluon operator in Eq. (G.11) to an operator +in SCET in which all the hard modes have been integrated out. The result for the matching +coefficient CS from Eqs.(16) and (17) of Ref. [61]. It is given by, +CS(−q2, µ2 +h) = 1 + +∞ +� +n=1 +CS +n (L) +�αs(µ2 +h) +4π +�n +. +(G.17) +The coefficient CS obeys the renormalization equation, +d +d ln µ CS(−q2 − iϵ, µ2) = +� +ΓA +cusp(αs) ln −q2 − iϵ +µ2 ++ γS(αs) +� +CS(−q2 − iϵ, µ2) , +(G.18) +with L = ln(−q2 − i0)/µ2 +h and γS is given in Eq (B.20). +The logarithmic terms are determined by Eq. (G.18). The full results for the one- and two-loop +coefficients are, +CS +1 = CA +� +− L2 + π2 +6 +� +, +(G.19) +CS +2 = C2 +A +�L4 +2 + 11 +9 L3 + +� +− 67 +9 + π2 +6 +� +L2 + +�80 +27 − 11π2 +9 +− 2ζ3 +� +L ++ 5105 +162 + 67π2 +36 ++ π4 +72 − 143 +9 ζ3 +� ++ CF TF nf +� +4L − 67 +3 + 16ζ3 +� +– 51 – + ++ CATF nf +� +− 4 +9 L3 + 20 +9 L2 + +�104 +27 + 4π2 +9 +� +L − 1832 +81 +− 5π2 +9 +− 92 +9 ζ3 +� +. +(G.20) +The full result for the renormalization group invariant hard function in the two-step scheme +is, +¯H(mt, mH, pveto +T +) = +� αs(µ) +αs(pveto +T +) +�2 +(Ct(m2 +t , µ))2 ��CS(−m2 +H, µ) +��2 +× +� mH +pveto +T +�−2Fgg(pveto +T +,µ) +e2hA(pveto +T +,µ) . +(G.21) +The µ-independence of this hard function can be used to constrain γS, +d +d ln µ +¯H(mt, mH, pveto +T +) = 0 . +(G.22) +Using Eqs. (B.1,G.12,G.18,2.13,C.1) we can derive the relation between the collinear anomalous +dimensions, +2γg(αs) = γt(αs) + γS(αs) + β(αs)/αs . +(G.23) +This relation could be cast in a more transparent form by noting that the quantity (αsCS) +obeys a similar evolution equation to Eq. (G.18), +d +d ln µ +� +αs(µ)CS(−m2 +H − iϵ, µ2) +� += +αs(µ) +� +ΓA +cusp(αs) ln −m2 +H − iϵ +µ2 ++ γS(αs) +� +CS(−m2 +H − iϵ, µ2) + β(αs)CS(−m2 +H − iϵ, µ2) += +� +ΓA +cusp(αs) ln −m2 +H − iϵ +µ2 ++ γS′(αs) +� � +αs(µ)CS(−m2 +H − iϵ, µ2) +� +, +(G.24) +but with anomalous dimension γS′(αs) = γS(αs) + β(αs)/αs. We then have the relation +2γg(αs) = γt(αs) + γS′(αs). This indicates that after the second matching, the evolution +down to a lower scale satisfies the same renormalization equation in both the one-step and the +two-step schemes. +G.3 +Assessment of the two schemes for the Higgs hard function +The two schemes for the calculation of the hard function have application in jet veto resum- +mation but also in the resummation of the Higgs boson transverse momentum. A complete +discussion of the error budget for Higgs boson production including scale dependence, parton +distribution dependence, the influence of loops of b-quarks and electroweak corrections is +beyond the scope of this paper. Here we shall simply compare and contrast the one-step and +the two-step scheme, in the Higgs on shell region where m2 +H ≈ m2 +t /2. +It is easy to check the internal consistency of the two schemes in the limit where we drop +terms of order q2/(4m2 +t ). Setting z = 0 in Eq. (G.2) and evaluating all coefficient functions at +– 52 – + +a common scale µ, we have that, +αs(µ) Ct(m2 +t , µ2) CS(−q2, µ2) = CH(m2 +t , q2, µ2)z=0 + O(α4 +s) . +(G.25) +We can test this equivalence numerically. We start by fixing µ2 = q2 and consider the quantities +that enter the calculation of the cross-section, i.e. the square of the absolute values. In the +two-step scheme we have, +|Ct(m2 +t , q2)|2 = 1 + 0.1957 + 0.0204 , +|Cs(−q2, q2)|2 = 1 + 0.6146 + 0.2155 , +(G.26) +where the second and third terms represent the O(αs) and O(α2 +s) terms respectively, evaluated +using αs(q2) = 0.1118. In the one-step case we get, +|CH +z=0(m2 +t , q2, q2)/αs(q)|2 = 1 + 0.8104 + 0.3563 . +(G.27) +Performing a strict fixed-order truncation of the product of the two-step result we have, +� +|Ct(m2 +t , q2)|2|Cs(−q2, q2)|2� +expanded = 1 + 0.8104 + 0.3563 , +(G.28) +which is in perfect agreement with the one-step case. This indicates that the numerical +implementation of the two procedures is correct. If we instead evaluate the product after the +individual expansions have been performed, a choice of equal formal accuracy, we have, +|Ct(m2 +t , q2)|2 +expanded |Cs(−q2, q2)|2 +expanded = 1 + 0.9306 + 0.2953 . +(G.29) +This results in a significant difference. We therefore work with with the strict fixed-order +truncation throughout this paper. +We now restore the z-dependence in F H +1 +and F H +2 +in Eq. (G.2), but still keep z = 0 in the +overall factor F H +0 (z). We then find that the ratio of the one-step to the two-step becomes +1.0028 at NLO and 1.0053 at NNLO, i.e. these corrections are very small. Now we allow the +matching scale for the top quark, µt to take its natural value, µt = mt and find one/two-step +ratios of 1.0054 at NLO and 1.0073 at NNLO, again a small effect. Finally, we reinstate the +hard evolution down to the resummation scale and find that the ratio of the one-step to the +two-step (at pveto +T += 25 GeV) is 1.0177 at NLO and 1.0125 at NNLO. The cumulative effect at +this point is noticeable but still small. However, we note that we have so far kept z = 0 in +the overall factor F H +0 (z). The one-step procedure is recovered by re-instating F H +0 (z). This +implies that, in order to obtain the level of agreement quoted above between the two schemes, +the overall factor of F H +0 (z) must also be applied to give a modified version of the two-step +scheme. Neglecting this step would result in a significant difference, since |F H +0 (z)|2 = 1.0653 +see Eq.(G.5). +Our overall conclusion on the two schemes is in line with the known result that Higgs boson +production has substantial corrections. Accounting for the most important mass effects by +– 53 – + +rescaling the two-step result by the exact result at leading order, the one-step procedure +gives a larger result than the two-step procedure for pveto +T += 25 GeV at the level of 1.3%. +Any substantial difference between the two methods beyond this level is most likely due to +uncontrolled higher order effects. +References +[1] C.F. Berger, C. Marcantonini, I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, Higgs +Production with a Central Jet Veto at NNLL+NNLO, JHEP 04 (2011) 092 [1012.4480]. +[2] I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, N-Jettiness: An Inclusive Event Shape to +Veto Jets, Phys. Rev. Lett. 105 (2010) 092002 [1004.2489]. +[3] T. Becher and M. Neubert, Factorization and NNLL Resummation for Higgs Production with a +Jet Veto, JHEP 07 (2012) 108 [1205.3806]. +[4] A. Banfi, G.P. Salam and G. Zanderighi, NLL+NNLO predictions for jet-veto efficiencies in +Higgs-boson and Drell-Yan production, JHEP 06 (2012) 159 [1203.5773]. +[5] A. Banfi, P.F. Monni, G.P. Salam and G. Zanderighi, Higgs and Z-boson production with a jet +veto, Phys. Rev. Lett. 109 (2012) 202001 [1206.4998]. +[6] S. Kallweit, E. Re, L. Rottoli and M. Wiesemann, Accurate single- and double-differential +resummation of colour-singlet processes with MATRIX+RADISH: W +W − production at the +LHC, JHEP 12 (2020) 147 [2004.07720]. +[7] E. Re, L. Rottoli and P. Torrielli, Fiducial Higgs and Drell-Yan distributions at N3LL′+NNLO +with RadISH, 2104.07509. +[8] T. Becher, R. Frederix, M. Neubert and L. Rothen, Automated NNLL + NLO resummation for +jet-veto cross sections, Eur. Phys. J. C 75 (2015) 154 [1412.8408]. +[9] A. Banfi, P.F. Monni, G.P. Salam and G. Zanderighi, “JetVHeto.” +https://jetvheto.hepforge.org/, 2016. +[10] L. Arpino, A. Banfi, S. Jäger and N. Kauer, “MCFM-RE.” +https://github.com/lcarpino/MCFM-RE, 2019. +[11] L.R. Stefan Kallweit, Emanuele Re and M. Wiesemann, “MCFM-RE.” +https://matrix.hepforge.org/matrix+radish.html, 2020. +[12] T. Becher, M. Neubert and L. Rothen, Factorization and N 3LLp+NNLO predictions for the +Higgs cross section with a jet veto, JHEP 10 (2013) 125 [1307.0025]. +[13] I.W. Stewart, F.J. Tackmann, J.R. Walsh and S. Zuberi, Jet pT resummation in Higgs production +at NNLL′ + NNLO, Phys. Rev. D 89 (2014) 054001 [1307.1808]. +[14] A. Banfi, F. Caola, F.A. Dreyer, P.F. Monni, G.P. Salam, G. Zanderighi et al., Jet-vetoed Higgs +cross section in gluon fusion at N3LO+NNLL with small-R resummation, JHEP 04 (2016) 049 +[1511.02886]. +[15] S. Dawson, P. Jaiswal, Y. Li, H. Ramani and M. Zeng, Resummation of jet veto logarithms at +N3LLa + NNLO for W +W − production at the LHC, Phys. Rev. D 94 (2016) 114014 +[1606.01034]. +[16] L. Arpino, A. Banfi, S. Jäger and N. Kauer, BSM WW production with a jet veto, JHEP 08 +(2019) 076 [1905.06646]. +– 54 – + +[17] Y. Wang, C.S. Li and Z.L. Liu, Resummation prediction on gauge boson pair production with a +jet veto, Phys. Rev. D 93 (2016) 094020 [1504.00509]. +[18] G.P. Salam, Towards Jetography, Eur. Phys. J. C 67 (2010) 637 [0906.1833]. +[19] M. Cacciari, G.P. Salam and G. Soyez, The anti-kt jet clustering algorithm, JHEP 04 (2008) 063 +[0802.1189]. +[20] Y.L. Dokshitzer, G.D. Leder, S. Moretti and B.R. Webber, Better jet clustering algorithms, +JHEP 08 (1997) 001 [hep-ph/9707323]. +[21] M. Wobisch and T. Wengler, Hadronization corrections to jet cross-sections in deep inelastic +scattering, in Workshop on Monte Carlo Generators for HERA Physics (Plenary Starting +Meeting), pp. 270–279, 4, 1998 [hep-ph/9907280]. +[22] S. Catani, Y.L. Dokshitzer, M.H. Seymour and B.R. Webber, Longitudinally invariant Kt +clustering algorithms for hadron hadron collisions, Nucl. Phys. B 406 (1993) 187. +[23] S.D. Ellis and D.E. Soper, Successive combination jet algorithm for hadron collisions, Phys. Rev. +D 48 (1993) 3160 [hep-ph/9305266]. +[24] S. Abreu, J.R. Gaunt, P.F. Monni, L. Rottoli and R. Szafron, Quark and gluon two-loop beam +functions for leading-jet pT and slicing at NNLO, 2207.07037. +[25] S. Abreu, J.R. Gaunt, P.F. Monni and R. Szafron, The analytic two-loop soft function for +leading-jet pT , 2204.02987. +[26] T. Becher, M. Neubert and B.D. Pecjak, Factorization and Momentum-Space Resummation in +Deep-Inelastic Scattering, JHEP 01 (2007) 076 [hep-ph/0607228]. +[27] Y. Li, D. Neill and H.X. Zhu, An exponential regulator for rapidity divergences, Nucl. Phys. B +960 (2020) 115193 [1604.00392]. +[28] A.A. Vladimirov, Correspondence between Soft and Rapidity Anomalous Dimensions, Phys. Rev. +Lett. 118 (2017) 062001 [1610.05791]. +[29] Y. Li and H.X. Zhu, Bootstrapping Rapidity Anomalous Dimensions for Transverse-Momentum +Resummation, Phys. Rev. Lett. 118 (2017) 022004 [1604.01404]. +[30] G. Billis, M.A. Ebert, J.K.L. Michel and F.J. Tackmann, A toolbox for qT and 0-jettiness +subtractions at N3LO, Eur. Phys. J. Plus 136 (2021) 214 [1909.00811]. +[31] T. Becher and T. Neumann, Fiducial qT resummation of color-singlet processes at N3LL+NNLO, +JHEP 03 (2021) 199 [2009.11437]. +[32] J.-Y. Chiu, A. Jain, D. Neill and I.Z. Rothstein, A Formalism for the Systematic Treatment of +Rapidity Logarithms in Quantum Field Theory, JHEP 05 (2012) 084 [1202.0814]. +[33] J.-y. Chiu, A. Jain, D. Neill and I.Z. Rothstein, The Rapidity Renormalization Group, Phys. Rev. +Lett. 108 (2012) 151601 [1104.0881]. +[34] S. Alioli and J.R. Walsh, Jet Veto Clustering Logarithms Beyond Leading Order, JHEP 03 (2014) +119 [1311.5234]. +[35] M. Dasgupta, F. Dreyer, G.P. Salam and G. Soyez, Small-radius jets to all orders in QCD, JHEP +04 (2015) 039 [1411.5182]. +[36] NNPDF collaboration, Parton distributions from high-precision collider data, Eur. Phys. J. C 77 +(2017) 663 [1706.00428]. +[37] A. Banfi, P.F. Monni and G. Zanderighi, Quark masses in Higgs production with a jet veto, +JHEP 01 (2014) 097 [1308.4634]. +– 55 – + +[38] CMS collaboration, Measurement of differential cross sections for the production of a Z boson in +association with jets in proton-proton collisions at √s = 13 TeV, 2205.02872. +[39] CMS collaboration, W+W− boson pair production in proton-proton collisions at √s = 13 TeV, +Phys. Rev. D 102 (2020) 092001 [2009.00119]. +[40] CMS collaboration, Measurements of pp → ZZ production cross sections and constraints on +anomalous triple gauge couplings at √s = 13 TeV, Eur. Phys. J. C 81 (2021) 200 [2009.01186]. +[41] P. Jaiswal and T. Okui, Reemergence of rapidity-scale uncertainty in soft-collinear effective +theory, Phys. Rev. D 92 (2015) 074035 [1506.07529]. +[42] J.K.L. Michel, P. Pietrulewicz and F.J. Tackmann, Jet Veto Resummation with Jet Rapidity Cuts, +JHEP 04 (2019) 142 [1810.12911]. +[43] ATLAS collaboration, Measurement of differential cross sections and W +/W − cross-section +ratios for W boson production in association with jets at √s = 8 TeV with the ATLAS detector, +JHEP 05 (2018) 077 [1711.03296]. +[44] ATLAS collaboration, Measurement of W ±Z production cross sections and gauge boson +polarisation in pp collisions at √s = 13 TeV with the ATLAS detector, Eur. Phys. J. C 79 (2019) +535 [1902.05759]. +[45] CMS collaboration, Measurement of the inclusive and differential WZ production cross sections, +polarization angles, and triple gauge couplings in pp collisions at √s = 13 TeV, JHEP 07 (2022) +032 [2110.11231]. +[46] A. Banfi, G.P. Salam and G. Zanderighi, Principles of general final-state resummation and +automated implementation, JHEP 03 (2005) 073 [hep-ph/0407286]. +[47] P.F. Monni, L. Rottoli and P. Torrielli, Higgs transverse momentum with a jet veto: a +double-differential resummation, Phys. Rev. Lett. 124 (2020) 252001 [1909.04704]. +[48] W. Bizon, P.F. Monni, E. Re, L. Rottoli and P. Torrielli, Momentum-space resummation for +transverse observables and the Higgs p⊥ at N3LL+NNLO, JHEP 02 (2018) 108 [1705.09127]. +[49] P.F. Monni, E. Re and P. Torrielli, Higgs Transverse-Momentum Resummation in Direct Space, +Phys. Rev. Lett. 116 (2016) 242001 [1604.02191]. +[50] T. Becher, M. Neubert and D. Wilhelm, Electroweak Gauge-Boson Production at Small qT : +Infrared Safety from the Collinear Anomaly, JHEP 02 (2012) 124 [1109.6027]. +[51] V. Ahrens, T. Becher, M. Neubert and L.L. Yang, Origin of the Large Perturbative Corrections to +Higgs Production at Hadron Colliders, Phys. Rev. D 79 (2009) 033013 [0808.3008]. +[52] ATLAS collaboration, Measurement of the W +W − production cross section in pp collisions at a +centre-of -mass energy of √s = 13 TeV with the ATLAS experiment, Phys. Lett. B 773 (2017) +354 [1702.04519]. +[53] ATLAS collaboration, Measurement of fiducial and differential W +W − production cross-sections +at √s = 13 TeV with the ATLAS detector, Eur. Phys. J. C 79 (2019) 884 [1905.04242]. +[54] CMS collaboration, Search for anomalous triple gauge couplings in WW and WZ production in +lepton + jet ev ents in proton-proton collisions at √s = 13 TeV, JHEP 12 (2019) 062 +[1907.08354]. +[55] J.M. Campbell, R.K. Ellis, T. Neumann and S. Seth, Transverse momentum resummation at +N3LL+NNLO for diboson processes, 2210.10724. +[56] R. Bonciani, V. Del Duca, H. Frellesvig, M. Hidding, V. Hirschi, F. Moriello et al., +– 56 – + +Next-to-leading-order QCD Corrections to Higgs Production in association with a Jet, +2206.10490. +[57] M. Czakon, R.V. Harlander, J. Klappert and M. Niggetiedt, Exact Top-Quark Mass Dependence +in Hadronic Higgs Production, Phys. Rev. Lett. 127 (2021) 162002 [2105.04436]. +[58] T. Neumann and M. Wiesemann, Finite top-mass effects in gluon-induced Higgs production with +a jet-veto at NNLO, JHEP 11 (2014) 150 [1408.6836]. +[59] A. Idilbi, X.-d. Ji and F. Yuan, Transverse momentum distribution through soft-gluon +resummation in effective field theory, Phys. Lett. B 625 (2005) 253 [hep-ph/0507196]. +[60] A. Idilbi, X.-d. Ji, J.-P. Ma and F. Yuan, Threshold resummation for Higgs production in effective +field theory, Phys. Rev. D 73 (2006) 077501 [hep-ph/0509294]. +[61] V. Ahrens, T. Becher, M. Neubert and L.L. Yang, Renormalization-Group Improved Prediction +for Higgs Production at Hadron Colliders, Eur. Phys. J. C 62 (2009) 333 [0809.4283]. +[62] S. Mantry and F. Petriello, Factorization and Resummation of Higgs Boson Differential +Distributions in Soft-Collinear Effective Theory, Phys. Rev. D 81 (2010) 093007 [0911.4135]. +[63] G. Bell, K. Brune, G. Das and M. Wald, The NNLO quark beam function for jet-veto +resummation, 2207.05578. +[64] M.-X. Luo, X. Wang, X. Xu, L.L. Yang, T.-Z. Yang and H.X. Zhu, Transverse Parton +Distribution and Fragmentation Functions at NNLO: the Quark Case, JHEP 10 (2019) 083 +[1908.03831]. +[65] M.-X. Luo, T.-Z. Yang, H.X. Zhu and Y.J. Zhu, Transverse Parton Distribution and +Fragmentation Functions at NNLO: the Gluon Case, JHEP 01 (2020) 040 [1909.13820]. +[66] T. Becher and M. Neubert, Drell-Yan Production at Small qT , Transverse Parton Distributions +and the Collinear Anomaly, Eur. Phys. J. C 71 (2011) 1665 [1007.4005]. +[67] O.V. Tarasov, A.A. Vladimirov and A.Y. Zharkov, The Gell-Mann-Low Function of QCD in the +Three Loop Approximation, Phys. Lett. B 93 (1980) 429. +[68] S.A. Larin and J.A.M. Vermaseren, The Three loop QCD Beta function and anomalous +dimensions, Phys. Lett. B 303 (1993) 334 [hep-ph/9302208]. +[69] T. van Ritbergen, J.A.M. Vermaseren and S.A. Larin, The Four loop beta function in quantum +chromodynamics, Phys. Lett. B 400 (1997) 379 [hep-ph/9701390]. +[70] T. van Ritbergen, A.N. Schellekens and J.A.M. Vermaseren, Group theory factors for Feynman +diagrams, Int. J. Mod. Phys. A 14 (1999) 41 [hep-ph/9802376]. +[71] J.M. Henn, G.P. Korchemsky and B. Mistlberger, The full four-loop cusp anomalous dimension +in N = 4 super Yang-Mills and QCD, JHEP 04 (2020) 018 [1911.10174]. +[72] A. von Manteuffel, E. Panzer and R.M. Schabinger, Cusp and collinear anomalous dimensions in +four-loop QCD from form factors, Phys. Rev. Lett. 124 (2020) 162001 [2002.04617]. +[73] T. Becher, A. Broggio and A. Ferroglia, Introduction to Soft-Collinear Effective Theory, vol. 896, +Springer (2015), 10.1007/978-3-319-14848-9, [1410.1892]. +[74] T. Becher and M. Neubert, On the Structure of Infrared Singularities of Gauge-Theory +Amplitudes, JHEP 06 (2009) 081 [0903.1126]. +[75] G. Altarelli and G. Parisi, Asymptotic Freedom in Parton Language, Nucl. Phys. B 126 (1977) +298. +– 57 – + +[76] G. Curci, W. Furmanski and R. Petronzio, Evolution of Parton Densities Beyond Leading Order: +The Nonsinglet Case, Nucl. Phys. B 175 (1980) 27. +[77] W. Furmanski and R. Petronzio, Singlet Parton Densities Beyond Leading Order, Phys. Lett. B +97 (1980) 437. +[78] R.K. Ellis, W.J. Stirling and B.R. Webber, QCD and collider physics, vol. 8, Cambridge +University Press (2, 2011), 10.1017/CBO9780511628788. +[79] G. Kramer and B. Lampe, Two Jet Cross-Section in e+ e- Annihilation, Z. Phys. C 34 (1987) +497. +[80] T. Matsuura and W.L. van Neerven, Second Order Logarithmic Corrections to the Drell-Yan +Cross-section, Z. Phys. C 38 (1988) 623. +[81] T. Matsuura, S.C. van der Marck and W.L. van Neerven, The Calculation of the Second Order +Soft and Virtual Contributions to the Drell-Yan Cross-Section, Nucl. Phys. B 319 (1989) 570. +[82] S. Moch, J.A.M. Vermaseren and A. Vogt, Three-loop results for quark and gluon form-factors, +Phys. Lett. B 625 (2005) 245 [hep-ph/0508055]. +[83] S. Moch, J.A.M. Vermaseren and A. Vogt, The Quark form-factor at higher orders, JHEP 08 +(2005) 049 [hep-ph/0507039]. +[84] T. Gehrmann, E.W.N. Glover, T. Huber, N. Ikizlerli and C. Studerus, Calculation of the quark +and gluon form factors to three loops in QCD, JHEP 06 (2010) 094 [1004.3653]. +[85] A. Idilbi and X.-d. Ji, Threshold resummation for Drell-Yan process in soft-collinear effective +theory, Phys. Rev. D 72 (2005) 054016 [hep-ph/0501006]. +[86] A. Idilbi, X.-d. Ji and F. Yuan, Resummation of threshold logarithms in effective field theory for +DIS, Drell-Yan and Higgs production, Nucl. Phys. B 753 (2006) 42 [hep-ph/0605068]. +[87] J.M. Campbell, R.K. Ellis and S. Seth, Non-local slicing approaches for NNLO QCD in MCFM, +JHEP 06 (2022) 002 [2202.07738]. +[88] J. Davies, F. Herren and M. Steinhauser, Top Quark Mass Effects in Higgs Boson Production at +Four-Loop Order: Virtual Corrections, Phys. Rev. Lett. 124 (2020) 112002 [1911.10214]. +[89] M. Spira, A. Djouadi, D. Graudenz and P.M. Zerwas, Higgs boson production at the LHC, Nucl. +Phys. B 453 (1995) 17 [hep-ph/9504378]. +[90] R. Harlander and P. Kant, Higgs production and decay: Analytic results at next-to-leading order +QCD, JHEP 12 (2005) 015 [hep-ph/0509189]. +[91] C. Anastasiou, S. Beerli, S. Bucherer, A. Daleo and Z. Kunszt, Two-loop amplitudes and master +integrals for the production of a Higgs boson via a massive quark and a scalar-quark loop, JHEP +01 (2007) 082 [hep-ph/0611236]. +[92] R.V. Harlander and K.J. Ozeren, Top mass effects in Higgs production at next-to-next-to-leading +order QCD: Virtual corrections, Phys. Lett. B 679 (2009) 467 [0907.2997]. +[93] A. Pak, M. Rogal and M. Steinhauser, Virtual three-loop corrections to Higgs boson production in +gluon fusion for finite top quark mass, Phys. Lett. B 679 (2009) 473 [0907.2998]. +– 58 – +