Transverse-momentum resummation for vector-boson pair production at NNLL+NNLO

Journal of High Energy Physics, Aug 2015

Abstract We consider the transverse-momentum (p T ) distribution of ZZ and W + W − boson pairs produced in hadron collisions. At small p T , the logarithmically enhanced contributions due to multiple soft-gluon emission are resummed to all orders in QCD perturbation theory. At intermediate and large values of p T , we consistently combine resummation with the known fixed-order results. We exploit the most advanced perturbative information that is available at present: next-to-next-to-leading logarithmic resummation combined with the next-to-next-to-leading fixed-order calculation. After integration over p T , we recover the known next-to-next-to-leading order result for the inclusive cross section. We present numerical results at the LHC, together with an estimate of the corresponding uncertainties. We also study the rapidity dependence of the p T spectrum and we consider p T efficiencies at different orders of resummed and fixed-order perturbation theory.

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Transverse-momentum resummation for vector-boson pair production at NNLL+NNLO

Received: July Transverse-momentum resummation for vector-boson pair production at NNLL+NNLO Massimiliano Grazzini 0 1 3 4 5 Stefan Kallweit 0 1 2 4 5 Dirk Rathlev 0 1 3 4 5 Marius Wiesemann 0 1 3 4 5 Open Access, c The Authors. 0 Johannes Gutenberg University , D-55099 Mainz , Germany 1 CH-8057 Zu ̈rich , Switzerland 2 PRISMA Cluster of Excellence, Institute of Physics 3 Physik-Institut, Universita ̈t Zu ̈rich 4 [21] T. Binoth , T. Gleisberg, S. Karg, N. Kauer and G. Sanguinetti, NLO QCD corrections to 5 [56] R. Frederix and M. Grazzini , Higher-order QCD effects in the h We consider the transverse-momentum (pT ) distribution of ZZ and W +W − boson pairs produced in hadron collisions. At small pT , the logarithmically enhanced contributions due to multiple soft-gluon emission are resummed to all orders in QCD perturbation theory. At intermediate and large values of pT , we consistently combine resummation with the known fixed-order results. We exploit the most advanced perturbative information that is available at present: next-to-next-to-leading logarithmic resummation combined with the next-to-next-to-leading fixed-order calculation. After integration over pT , we recover the known next-to-next-to-leading order result for the inclusive cross section. We present numerical results at the LHC, together with an estimate of the corresponding uncertainties. We also study the rapidity dependence of the pT spectrum and we consider pT efficiencies at different orders of resummed and fixed-order perturbation theory. QCD Phenomenology; Hadronic Colliders 1 Introduction 2 3 Results Transverse-momentum resummation for vector-boson pair production Choice of the central resummation scale Inclusive transverse-momentum distribution Rapidity dependence of the transverse-momentum distribution The W +W − cross section and pT -veto efficiencies Introduction Run 1 of the Large Hadron Collider (LHC) has been a great success for the Standard Model (SM). The collected data are in good agreement with the theoretical predictions so far and led to the discovery [1, 2] of a resonance at a mass of 125 GeV, which appears to be fully consistent with the SM Higgs boson. Among the most important reactions at hadron colliders is the production of vector-boson pairs. This class of processes gives access to the vector-boson trilinear couplings which may be modified in a large set of Beyond the Standard Model (BSM) theories. Even small deviations in both the production rate and the shape of distributions could be a signal of new physics. Anomalous couplings related to vector-boson pair production have been constrained first by LEP2, and later by the Tevatron for larger invariant masses. ATLAS and CMS will continue to tighten the bounds on anomalous couplings, especially with increasing sensitivity during Run 2 of the LHC.1 On the other hand, vector-boson pair production constitutes an irreducible background to new-physics searches as well as Higgs studies. Particularly important are the off-shell the high-mass tail used to extract the width of the Higgs boson [4–6]. Furthermore, Higgs rejection through specific categories based on the transverse momenta of final-state particles, such as the classification into jet bins or in the Higgs transverse momentum and related variables. An accurate modelling of the respective observables for both signal and backgrounds is crucial for such analyses. The first precise predictions for ZZ production at hadron colliders in the SM were obtained at the next-to-leading order (NLO) already more than 20 years ago for stable Z bosons [7, 8], and the leptonic decays were added in ref. [9]. The full spin correlations 1See ref. [3] and references therein. and off-shell effects at the NLO were first included in refs. [10, 11] using the corresponding one-loop helicity amplitudes [12]. An important loop-induced contribution proceeds through gluon fusion; it is enhanced by the gluon densities and was first computed for on-shell Z bosons in refs. [13, 14], while leptonic decays were included later [15–17]. All the contributions to ZZ production discussed so far are implemented in the numerical program MCFM [18]. Electroweak (EW) corrections were evaluated in ref. [19, 20], while on-shell ZZ+jet production is known through NLO QCD [21, 22]. Recently, the inclusive cross section for the production of ZZ at the next-to-next-to-leading order (NNLO) has been presented [23]. neutrinos prohibits a reconstruction of mass peaks. Therefore, a precise understanding of both the signal and the background is required. gluon-fusion component [13, 26] have been known for decades. Also in this case, spin and off-shell effects were included in the NLO prediction [10, 11] after the relevant one-loop helicity amplitudes had been computed [12]. Leptonic decays for the loop-induced gluonfusion contribution were considered in refs. [27, 28]. More recently, also interference effects with the Higgs production mode through gluon fusion were determined [29]. Analogously plemented in MCFM [18]. Furthermore, EW corrections have been evaluated [20, 30, 31]. ation with up to one jet at NLO have been presented in ref. [35]. Recently, the first NNLO Due to the very recent computation of the qq¯ → V V 0 helicity amplitudes at two-loop order [37, 38], the inclusion of the off-shell effects and the leptonic decays in the NNLO cross section is expected in the near future. In the meanwhile, also the calculation of gg → V V 0 helicity amplitudes has been performed at two-loop order [39, 40]. This renders the evaluation of NLO QCD corrections to the gluon-fusion channel feasible. The production of ZZ pairs in hadron collisions has been measured extensively at section has also been measured already at the Tevatron, see e.g. ref. [48], and at the LHC both at 7 TeV [49, 50] and 8 TeV [46, 51, 52]. The ATLAS collaboration recently reported an excess [51] with respect to the SM prediction, which has drawn a lot of attention on narios [53]. In the meanwhile, the excess has been alleviated to a significant degree by the recent NNLO computation [36]. The more recent measurement by the CMS collaboration [52] is in good agreement with the NNLO prediction. important differential observables for these processes. The pT spectrum has already been measured in the case of ZZ production [47] at the LHC. Transverse-momentum resummations the resummed computation is essentially performed up to next-to-leading logarithmic (NLL) accuracy (next-to-next-to-leading logarithmic (NNLL) effects in the Sudakov exponent are considered in refs. [55, 57, 58]) and matched to the fixed-order result up to the In this paper we consider transverse-momentum resummation for the production of result valid at large pT . Although our focus is on the inclusive pT spectrum of the ZZ the vector bosons and allows us to include their EW decays, once the helicity amplitudes are implemented.2 The pT -resummation formalism of ref. [59] is closely related to the subtraction method of ref. [63], which was used to compute the NNLO cross section for and remove all contributions with final-state bottom quarks from our computation of the tt¯ and W t production. The difference to the prediction in the five-flavour scheme (5FS), where such terms have to be consistently subtracted, has been shown to be small for the NNLO inclusive cross section [36]. Furthermore, we neglect the loop-induced gluon-fusion We note that in the CMS measurement reported in ref. [52] an approximate NNLL prediction [58] of the pT spectrum has been used to correct the spectrum from the Monte Carlo simulation. Along these lines, the computation reported in this paper will be useful in order to validate the predictions obtained from Monte Carlo simulations for both ZZ of ref. [59]. The manuscript is organized as follows. In section 2 we review the transversemomentum resummation formalism applied to vector-boson pair production. In section 3 we report our numerical results, starting with remarks on the choice of the resummation scale in section 3.1. In section 3.2 we present our numerical predictions for the inclusive pT spectrum and study the ensuing uncertainties. In section 3.3 we analyse the behaviour of the spectrum at different rapidities of the vector-boson pair. In section 3.4 we investigate pT efficiencies at different orders in resummed and fixed-order perturbation theory. In section 4 we summarize our results. Transverse-momentum resummation for vector-boson pair production In this section we recall the main points of the transverse-momentum resummation formalism we use in this paper. For a more detailed discussion the reader is referred to refs. [59, 64, 65]. 2The analogous computations for Higgs, single vector-boson production, and diphoton production are presented in refs. [60], [61] and [62], respectively. We consider the inclusive hard-scattering process h1(P1) + h2(P2) → V (p3) + V 0(p4) + X , where the collision of the two hadrons h1 and h2 with momenta P1 and P2 produces the two vector bosons of momenta p3 and p4. In the center-of-mass frame the momentum of The kinematics of the vector bosons is fully determined by the vector-boson pair mopendent variables that specify the angular distribution of the vector bosons with respect According to the QCD factorization theorem, the differential cross section can be writ(y, pT , M, s) = The rapidity yˆ and the center-of-mass energy sˆ of the partonic scattering process are related to the corresponding hadronic variables y and s by yˆ = y − 2 sˆ = x1x2s . When the transverse momentum pT of the vector-boson pair is of the same order as the invariant mass M , the QCD perturbative expansion is controlled by a single expansion M the convergence of the perturbative expansion is spoiled by the presence of large logarithmic terms of the The resummation is performed at the level of the partonic cross section, which is dM 2dp2T dyˆ dM 2dp2T dyˆ dM 2dp2T dyˆ The first term on the right-hand side of eq. (2.4) contains all the logarithmically-enhanced contributions at small pT and has to be evaluated by resumming them to all orders. The second term is instead free of such contributions and can thus be evaluated at fixed order in perturbation theory. running coupling in the MS renormalization scheme. Resummation is based on the factorization of soft and collinear radiation and is viable in the impact-parameter (b) space, where the kinematical constraint of momentum conservation and the factorization of the phase space can be consistently taken into account [66–68]. Using the Bessel transformation between the conjugate variables pT and b, the resummed component is expressed as [59, 68] dM 2dp2T dyˆ M 2 Z ∞ where J0(x) is the 0-order Bessel function. In the case of fully inclusive pT resummation, the rapidity dependence is integrated out in eq. (2.5). In that case it is convenient to consider rapidity dependence in the resummed cross section we take the ‘double’ (N1, N2) Mellin V V 0 dz1 z1N1−1 dz2 z2N2−1W V V 0 (b, yˆ, M, sˆ; αS, μ2R, μ2F ) , and organize the structure of WV V 0 in the following exponential form [64], W(N1,N2)(b, M ; αS, μ2R, μ2F ) = H(N1,N2) M ; αS, M 2/μ2R, M 2/μ2F , M 2/Q2 V V 0 V V 0 where we have defined the logarithmic expansion parameter L as L = ln and b0 = 2e−γE (γE = 0.5772 . . . is the Euler number). The scale Q appearing in eqs. (2.7), (2.8), named resummation scale in ref. [59], parametrizes the arbitrariness in the resummation procedure, and has to be chosen of the order of the hard scale M . Variations of Q around its reference value can be exploited to estimate the size of yet uncalculated V V 0 parameter b, and therefore includes all the perturbative terms that behave as constants as b → ∞. It can thus be expanded in powers of αS = αS(μ2R): H(N1,N2)(M, αS; M 2/μ2R, M 2/μ2F , M 2/Q2) V V 0 includes the complete dependence on b and, in particular, it contains all the terms that (2) where the term L g(1) collects the LL contributions, the function g(N1,N2) includes the NLL contributions, g((N3)1,N2) controls the NNLL terms and so forth. The resummation of the large logarithmic terms carried out in eq. (2.7), after transforming back to pT space, allows us to obtain a well behaved transverse-momentum spectrum as pT → 0. However, the logarithmic expansion parameter L in eq. (2.8) is divergent as b → 0. This implies that the resummation produces higher-order contributions also in the high-pT region, which is conjugated to b → 0 after Fourier transformation. In this region the fixed-order cross section is perfectly viable and any resummation effect is necessarily artificial. To reduce the impact of such contributions, the logarithmic variable L is replaced by [59] L˜ ≡ ln The variables L and L˜ are equivalent when Qb 1 but they have a very different behaviour as b → 0. When Qb 1, L˜ → 0 and G(N1,N2) → 1. Moreover, since the behaviour of the allows us to enforce a unitarity constraint such that the fixed-order prediction is recovered upon integration over pT . A well known property of the formalism of ref. [59] is that the process dependence (as well as the factorization scale and scheme dependence) is fully encoded in the hard function HV V 0 . In other words, the functions g(i) are universal: they depend only on the channel in which the process occurs at Born level (qq¯ annihilation in the case of vectorof the universal perturbative coefficients A(q1), A(q2), A(q3), B˜q(1,N) , B˜q(2,N) . In particular, the LL function g(1) depends on the coefficient A(q1), the NLL function g(N1,N2) also depends on A(q2) (2) and B˜q(1) [69] and the NNLL function g(N1,N2) also depends on A(q3) [70] and B˜q(2,N) [71–73]. (3) The hard coefficients H coefficients H V V 0 depend on the process we want to consider. The first order V V 0,(1) are known since long time [72, 73]: they can be obtained from the coefficient H V V 0,(2). one-loop scattering amplitudes qq¯ → V V 0 by using a process independent relation. By (2) for single Higgs [74] and vector-boson [75] production, exploiting the expressions of H amplitude for qq¯ → V V 0 is the only process-dependent information needed to obtain the We now turn to the finite component of the transverse-momentum spectrum, i.e. the logarithmic terms in the small-pT region, it can be evaluated by truncating the perturbative series at a given fixed order. In practice, the finite component is computed starting from the customary fixed-order (f.o.) perturbative truncation of the partonic cross section and subtracting the expansion of the resummed cross section in eq. (2.5) at the same perturbative order: dM 2dp2T dyˆ dM 2dp2T dyˆ dM 2dp2T dyˆ At least formally, this matching procedure between resummed and finite contributions guarantees to achieve a uniform theoretical accuracy over the entire range of transverse momenta. At large values of pT , the resummation procedure cannot improve the fixedorder result, and the resummation (and matching) procedure is eventually superseded by the customary fixed-order calculations. (2) In summary, the inclusion of the functions g(1), g(N1,N2), H V V 0,(1) in the resummed us to perform the resummation at NLL+NLO accuracy. This is the theoretical accuracy of the calculations of refs. [55, 56]. Including also the functions g((N3)1,N2) and H the recently computed two-loop amplitudes for qq¯ → V V 0, and the process independent relation of ref. [65], we are now able to present the complete result for the transverse V V 0,(2), together momentum distribution of the vector-boson pair up to NNLL+NNLO accuracy. We point out that the NNLL+NNLO (NLL+NLO) result includes the full NNLO (NLO) perturbative contribution in the entire pT range. In particular, the NNLO (NLO) result for the double differential cross section at NNLL+NNLO (NLL+NLO) accuracy. We conclude this section by adding a few comments on the way in which our calculation is actually performed. The practical implementation of eq. (2.4) is done in the numerical program Matrix,4 which is an extension of the numerical program applied in the NNLO calculations of refs. [23, 36, 76, 77] and based on a combination of the qT -subtraction formalism [63] with the Munich5 code. Since already performed within the qT -subtraction formalism, the extension of these calculations to compute the resummed cross section is conceptually quite straightforward, and is obtained by replacing the hard-collinear terms in the fixed-order computation by the proper all-order resummation formula of eq. (2.5). This procedure is the same that was applied to perform the NLL+NLO calculations for production of ref. [60]. To obtain the numerical results presented here, the resummed component of eq. (2.5) is evaluated with an extension of the numerical program used for the calculation of Higgs production [60], based on the earlier computations of refs. [59, 64]. The hard-collinear 4Matrix is the abbreviation of “Munich Automates qT subtraction and Resummation to Integrate X-sections”, by M. Grazzini, S. Kallweit, D. Rathlev, M. Wiesemann. In preparation. 5Munich is the abbreviation of “MUlti-chaNnel Integrator at Swiss (CH) precision”—an automated parton level NLO generator by S. Kallweit. In preparation. coefficients are obtained by exploiting the implementation of the corresponding virtual respectively, and the knowledge of the collinear coefficients relevant to quark-initiated processes [75].6 The finite component of eq. (2.12) is obtained from an NLO calculation of V V 0+jet, computed with the Munich code, which provides a fully automated implementation of the Catani-Seymour dipole formalism [80, 81] as well as an interface to the one-loop generator OpenLoops [82] to obtain all required (spin and color-correlated) tree-level and one-loop amplitudes. For the numerically stable evaluation of tensor integrals we rely on the Collier library [83], which is based on the Denner-Dittmaier reduction techniques [84, 85] and the scalar integrals of [86]. To deal with problematic phase-space points, OpenLoops provides a rescue system using the quadruple-precision implementation of the OPP method in CutTools [87], involving scalar integrals from OneLOop [88]. In this section we present our results for the resummed transverse-momentum distributions NLL+NLO, and discuss the corresponding theoretical uncertainties. Additionally, we also study the rapidity dependence of the pT cross section as well as the pT -veto efficiency. split off all contributions related to bottom-quark final states in order to remove the tt¯ and W t contamination from our computation. This is straightforward in the 4FS, because the W +W −b¯b process is separately finite. We consider proton-proton collisions at √ central resummation scale Q0 is discussed in the next subsection. Choice of the central resummation scale As discussed in section 2, the resummation scale Q is the scale entering the large logarithmic terms we are resumming (see eq. (2.8)), and it plays the role of the scale up to which resummation is effective. In on-shell Higgs [59] and vector-boson [90] production, the scale of the scale lead to a worse matching at high pT . The natural extension of this choice for 6We note that an independent computation of these coefficients in the framework of Soft Collinear Effective Theory has been presented in refs. [78, 79]. s = 8 TeV. The central values of the factorM(W+W-) [GeV] and NNLO (blue, solid). the hardness of the process. This is indeed the choice that was adopted in the calculations of refs. [55, 56]. peaked in the threshold region, and that it quickly decreases as MW W increases. As a We can compare the transverse-momentum distributions obtained with a dynamical bands are obtained by varying the resummation scale around the central value by a factor of two. Considering the ratio of the central curves for pT . 250 GeV, the differences between a fixed and a dynamical scale are extremely small and remain at the 1-2% level over the whole range. In this region of transverse momenta the uncertainty bands obtained with the uncertainties, it appears to be the more conservative choice. Therefore, we can conclude that either choice of the resummation scale is perfectly valid and indeed consistent with each other as expected from the discussion of the invariant mass distribution. Looking further at the comparison of the high-pT tails in figure 2 (pT & 250 GeV), we observe a very well known feature [59, 60, 90–92] of the applied matching procedure, namely the fact that for large values of the resummation scale the fixed-order cross section (black dotted curve) is not recovered in the tail of the distribution. It is important to recall that transverse-momentum resummation is supposed to improve the perturbative expansion in the low-pT region. At large pT , any large dependence on the resummation scale is necessarily artificial and an unwanted remnant of the matching procedure. This NNLL+NNLO dyn. Qres NNLL+NNLO fixed Qres NNLO are obtained by variation of the resummation scales in the numerator by a factor of two around the central scale. For reference, we show the fixed-order NNLO curve with the same normalization. figure 2. With this choice, in fact, the resummed result loses predictivity, as its uncertainty of the resummed prediction. scale in what follows. Inclusive transverse-momentum distribution We now present our resummed predictions for the inclusive transverse-momentum spectrum of the vector-boson pair and compare them with the corresponding fixed-order results. the ZZ case. For completeness, we provide the corresponding reference prediction with uncertainties for ZZ below. Before presenting our resummed predictions, we recall the well known fixed-order retogether with their perturbative uncertainties. The uncertainty bands are obtained by varyThe lower inset shows the same results normalized to the central NLO curve. The NNLO region where pT . 300 GeV. This implies that, in this region of transverse momenta, the ito 1 a as described in the text. Lower inset: results normalized to the NLO prediction at central values of size of the band obtained through scale variations at NLO definitely underestimates the theoretical uncertainty. We now move on to the resummed results. In figure 4 (a) the NLL+NLO spectrum is compared to the fixed-order NLO result and to the finite component of the resummed cross section (see eq. (2.4)) in the region between 0 and 80 GeV. As expected, the NLO diverges component contributes less than 1% in the peak region, where the result is dominated NLL+NLO result normalized to NLO. In figure 4 (b) the region between 80 and 400 GeV is displayed. We see that even at large values of pT the NLL+NLO resummed result does not match very well the fixed-order NLO result, with a difference of about 5%. The analogous results at NNLL+NNLO are shown in figure 5. The NNLO has an unphysical (divergent) behaviour as pT → 0, whereas the resummed spectrum is well behaved, with a slightly harder peak with respect to the NLL+NLO. The finite component paring the right panels of figure 4 and figure 5, we see that the quality of the matching at high pT is significantly improved when going from NLL+NLO to NNLL+NNLO, and we find that this behaviour is indeed preserved up to very high transverse momenta. The NNLL+NNLO result thus gives a prediction with uniform accuracy from small to very large transverse momenta and, in fact, provides a sufficiently large region where a hard switching to the fixed-order result is feasible. We point out that, thanks to our unitarity constraint, both at NLL+NLO and at NNLL+NNLO the integral of the resummed spectrum region and (b) at high transverse momenta. The NLL+NLO result (red, dashed) is compared to the fixed-order NLO prediction (grey, dash-dotted) and to the finite component of eq. (2.4) (magenta, dash-double dotted). The lower insets show the NLL+NLO result normalized to NLO. ito 1 a t io 1 t low-pT region and (b) at high transverse momenta. The NNLL+NNLO result (red, dashed) is compared to the fixed-order NNLO prediction (grey, dash-dotted) and to the finite component of eq. (2.4) (magenta, dash-double dotted). The lower insets show the NNLL+NNLO result normalized is in excellent agreement with the respective total cross sections; the differences are at the few-permille level. We now turn to the scale uncertainties of our resummed results. We start our discussion by separately considering factorization and renormalization scale variations. In figure 6 we compare the NLL+NLO (red, dashed) and NNLL+NNLO (blue, solid) predictions with their cases, the bands are obtained by varying the factorization (renormalization) scale by a N to 1 N to 1 toN 1 N to 1 NNLL+NNLO (blue, solid); thick lines: central scale choices; bands: uncertainty due to (left) factor of two around its central value, while keeping the other scales at their default values. First of all, we notice that when going from NLL+NLO to NNLL+NNLO the pT spectrum becomes harder. Comparing with the results of ref. [55], where the NNLL resummation spectrum is a combined effect of both features, i.e. NNLL resummation and NNLO matching at high pT . We note that neither in the case of the factorization scale, nor in the case of the renormalization scale, the NLL+NLO and NNLL+NNLO bands overlap. Actually, in the case of the factorization scale, there is no reduction in scale dependence when going from NLL+NLO to NNLL+NNLO, and the uncertainty slightly increases with the perturbative order, even if it is always well below 10%, except at very low pT . The renormalization scale dependence instead exhibits the expected reduction when going from NLL+NLO to NNLL+NNLO. In figure 7 we present our resummed predictions with uncertainty bands obtained from find that at NNLL+NNLO the resummation scale uncertainty is reduced roughly by a factor of two in the region of transverse momenta considered in the figure. respectively, with an estimate of their full perturbative uncertainty. In order to obtain a which is the same as applied in figure 3 and figure 7 (left), has the purpose of avoiding large logarithmic contributions from the evolution of parton densities. Analogously, the between the NNLL+NNLO and NLL+NLO predictions is driven by the NNLO effects, which increase the NLO result by about 30%. For ZZ production the uncertainties have essentially the same pattern in the smallis entirely driven by the resummation-scale dependence. As previously pointed out, this behaviour is not particularly worrying since, in the large-pT region, the resummed results should be replaced by the corresponding fixed-order prediction. Also in the ZZ case the large enhancement of the NNLL+NNLO distribution in the high-pT tail stems from the fixed-order cross section. Rapidity dependence of the transverse-momentum distribution inclusive in the kinematics of the vector-boson pair. Our numerical program, however, allows us to compute arbitrary observables that are differential with respect to the V V 0 N to 1 N to 1 N to 1 N to 1 and Q variations obtained as described in the text; thin lines: borders of bands. (b) detail of the low-pT region. In the following we study the behaviour of the transverse-momentum spectrum in different rapidity regions of the vector-boson pair. In figure 10 we study the shape of the NNLL+NNLO transverse-momentum distribution, i.e. normalized such that its integral yields one, for |y| < 0.5 (red, solid), 0.5 < |y| < 1 (blue, dashed), 1 < |y| < 2 (black, dotted), 2 < |y| < 3 (magenta, dash-dotted) and 3 < |y| (orange, dash-double dotted). The right panel shows the same results normalized to the fully inclusive distribution. We clearly see that the pT shapes become softer as the rapidity increases. In the central region (|y| < 2) the distributions are still quite insensitive to the specific value of the rapidity and only slightly harder than the inclusive spectrum. In the forward rapidity region, on the other hand, the shapes become increasingly softer. T /dp1.5 t) T /dp0.5 t) 1 < |y| < 2 (black, dotted), 2 < |y| < 3 (magenta, dash-dotted), 3 < |y| (orange, double-dash dotted); and (b) the shape-ratio with respect to the inclusive result. The observed pattern can be understood in the following way: rapidity and transverse momentum are two not completely independent phase-space variables. Indeed, they affect their mutual upper integration bounds. At higher rapidities the kinematically allowed range of transverse momenta is reduced: this squeezes the pT spectrum which consequently becomes softer. This effect has been observed also in previous studies in the case of Higgs boson production [64]. The W +W − cross section and pT -veto efficiencies signature appears in many new physics scenarios [53]. The inclusion of the recently computed NNLO corrections [36] considerably reduces the significance of the excess. However, particular attention must be payed to the modelling of the jet veto [52, 58, 93] when extrapolating from the fiducial region to obtain the inclusive cross section. Effects of jet veto resummation have been considered in refs. [94, 95], though still matching to the fixed-order In this paper we are dealing with transverse-momentum spectra, and we perform a resummation on a different variable with respect to the jet pT . However, the vector-boson resummed and fixed-order perturbation theory. We define the pT -veto efficiency as In figure 11 we show (pvTeto) at the NNLL+NNLO (blue, solid), approximate NNLL+NLO (magenta, dash-double dotted), NLL+NLO (red, dashed), NNLO (black, dotted) and NLO (grey, dash-dotted). The lower inset shows the same curves normalized to our reference o it 1 a pT veto [GeV] NLL+NLO (red, dashed), NNLL+NNLO (blue, solid), NLO (grey, dash-dotted), NNLO (black, dotted), approximate NNLL+NLO (magenta, dash-double dotted); thick lines: central scale choices; bands: uncertainty due to combined scale variations; thin lines: borders of bands. prediction at NNLL+NNLO. Our approximate NNLL+NLO is obtained by simply adding the g(3) function in the Sudakov exponent in eq. (2.10) at NLL+NLO, and corresponds to the approximation considered in refs. [55, 58]. For reference, the corresponding numerical values of the efficiencies are given in table 1 mation, factorization and renormalization scales as in figure 8. The first thing we observe is that the NLO result appears to be well above the others and cannot be really considered a reliable prediction for the efficiency. This is because it is essentially a LO prediction at the fixed-order NLO and NNLO predictions diverge and cannot be trusted. Comparing further the fixed-order results among each other and the resummed results among each other, we observe that higher-order corrections in fixed-order and resummed perturbation theory reduce the pT -veto efficiency. Both effects can be easily understood in the light of the results presented up to now. As seen in figure 3, the inclusion of the NNLO corrections make the pT distribution harder. Furthermore, resummation effects generally harden the spectrum. A qualitatively similar result is obtained when going from NLL+NLO to NNLL+NNLO (see figure 8). It is interesting to compare the approximated NNLL+NLO result with the NNLO and NNLL+NNLO predictions. For values of pvTeto ∼ 25 − 30 GeV we see that the approximated result is in between the NNLO one and our best NNLL+NNLO prediction. This means −19% −12% −9.2% −7.3% −5.9% −4.9% −4.2% −3.6% pveto [GeV] approx. NNLL+NLO −9.6% −6.2% −4.8% −3.9% −3.4% −3.0% −2.7% −2.5% that the effect of NNLL resummation obtained by the inclusion of the g(3) function in the Sudakov exponent in eq. (2.10) is quantitatively important. Nonetheless, the efficiency obtained within this approximation is still about 5% higher than the NNLL+NNLO prediction. We also notice that in this region of pvTeto, the NNLO and NLL+NLO results differ by less than 1%. Comparing the NNLL+NNLO and NLL+NLO results, we find that they are compatible within the corresponding uncertainties. out by the CMS collaboration [52]. The result shows good agreement with the NNLO prediction of ref. [36]. The corresponding analysis, however, is based on a reweighting procedure Pythia6 [97] were reweighted by using the calculation of ref. [58], which corresponds to our NNLL+NLO approximation, and includes neither the second-order hard-collinear coef W W,(2) in eq. (2.9), nor the NNLO matching. The results in figure 11 show that the NNLL+NNLO pT -veto efficiency is lower than the efficiency obtained with the approximated NNLL+NLO calculation. As a consequence, a reweighting to the full NNLL+NNLO predicIn this paper we have studied the transverse-momentum distribution of vector-boson pairs in hadronic collisions. We presented a computation of the pT spectrum in which the logarithmically enhanced contributions at small pT are resummed up to NNLL accuracy together with a study of their perturbative uncertainties. We found that, up to relatively large transverse momenta, when scale variations are momenta the fixed-order result in the tail of the distribution is nicely recovered with the fixed-scale choice. Our new NNLL+NNLO results significantly reduce the theoretical uncertainties obtained through scale variations compared to lower orders in both the peak region and the tail of the distribution. We have also studied the rapidity dependence of the resummed transverse-momentum distribution. The rapidity dependence at NNLL+NNLO is quite flat in the central region (|y| . 2), but signals a substantially softer spectrum in the forward region. Due to phasespace suppression, the effect on the inclusive transverse-momentum distribution is very moderate though. Finally, we have studied the pT -veto efficiency at different orders in resummed and fixed-order perturbation theory. Both NNLL resummation and the NNLO effects turned out to be important to obtain an accurate prediction for this quantity. We observed that to the approximate NNLL+NLO calculation used in the CMS analysis of ref. [52]. This result suggests that our NNLL+NNLO predictions will be useful to validate the transversemomentum spectra obtained from Monte Carlo event generators, similarly to what was done for the NNLL+NNLO calculation of ref. [59] in the case of Higgs boson production. In this paper we considered the pT spectrum of stable vector-boson pairs. Exploiting the two-loop helicity amplitudes for qq¯ → V V 0 → 4 leptons [37, 38] will allow us to extend the calculation to include the leptonic decay of the vector bosons and off-shell effects. The computation of the transverse-momentum spectrum with realistic experimental cuts will then become possible. We would like to thank Andreas von Manteuffel and Lorenzo Tancredi for providing us with their private code to evaluate the helicity-averaged on-shell V V 0 amplitudes in the equalmass case. We would like to thank Giancarlo Ferrera for comments on the manuscript. This research was supported in part by the Swiss National Science Foundation (SNF) under contracts CRSII2-141847, 200021-156585 and by the Research Executive Agency (REA) of the European Union under the Grant Agreement number PITN-GA-2012-316704 Open Access. 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Massimiliano Grazzini, Stefan Kallweit, Dirk Rathlev, Marius Wiesemann. Transverse-momentum resummation for vector-boson pair production at NNLL+NNLO, Journal of High Energy Physics, 2015, 154, DOI: 10.1007/JHEP08(2015)154