# Automated NNLL \(+\) NLO resummation for jet-veto cross sections

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## Abstract

In electroweak-boson production processes with a jet veto, higher-order corrections are enhanced by logarithms of the veto scale over the invariant mass of the boson system. In this paper, we resum these Sudakov logarithms at next-to-next-to-leading logarithmic accuracy and match our predictions to next-to-leading-order (NLO) fixed-order results. We perform the calculation in an automated way, for arbitrary electroweak final states and in the presence of kinematic cuts on the leptons produced in the decays of the electroweak bosons. The resummation is based on a factorization theorem for the cross sections into hard functions, which encode the virtual corrections to the boson production process, and beam functions, which describe the low-\(p_T\) emissions collinear to the beams. The one-loop hard functions for arbitrary processes are calculated using the MadGraph5_aMC@NLO framework, while the beam functions are process independent. We perform the resummation for a variety of processes, in particular for \(W^+ W^-\) pair production followed by leptonic decays of the \(W\) bosons.

## Keywords

Higgs Production Soft Function Virtual Correction Hard Function Beam Function## 1 Introduction

In many experimental measurements a veto on hard jets is imposed to suppress backgrounds. Such a veto is particularly useful to suppress top-quark backgrounds to processes involving \(W\) bosons, since the \(W\) bosons from the decay of the top quarks come in association with \(b\)-jets, which are rejected by the jet veto. For example, a jet veto is crucial to measure Higgs production with subsequent decay \(H\rightarrow W^+W^-\). It is imposed by rejecting events which involve jets with transverse momentum above a scale \(p_T^{{\mathrm{veto}}}\), which is typically chosen to be \(p_T^{{\mathrm{veto}}}\approx \) 20–30 GeV. Since the veto scale is much lower than the invariant mass \(Q\) of the electroweak final state, perturbative corrections to the cross section are enhanced by Sudakov logarithms of the ratio \(p_T^{{\mathrm{veto}}}/Q\). There has been a lot of theoretical progress over the past two years concerning the resummation of jet-veto logarithms in Higgs-boson production. Using the CAESAR formalism [1], these logarithms were first computed at next-to-leading logarithmic (NLL) order in [2], and this treatment was later extended to next-to-next-to-leading logarithmic (NNLL) accuracy [3]. In between these papers, an all-order factorization formula derived in soft-collinear effective theory (SCET) [4, 5, 6, 7] was proposed [8], and a resummed result which includes almost all of the ingredients required for N\(^3\)LL accuracy was presented [9]. A third group of authors performed an independent analysis in SCET [10] and also combined the results for different jet multiplicities [11, 12, 13].

The jet veto is not only necessary in \(H\rightarrow W^+ W^-\) but also in the measurement of the diboson cross section itself. The fact that LHC measurements [14, 15, 16, 17] yield values of the \(W^+ W^-\) cross section that are higher than theoretical predictions has triggered discussions as to whether this excess could be due to New Physics [18, 19, 20]. To be sure whether there indeed is an excess, it is important to have reliable theoretical predictions not only for the total cross section, for which the next-to-next-to-leading-order (NNLO) result has been obtained recently [21], but also for the cross section in the presence of experimental cuts, most importantly in the presence of a jet veto.^{1} Several recent papers have addressed this issue and have come to somewhat different conclusions. In [23], the Sudakov logarithms associated with the jet veto were resummed at NNLL accuracy. It was claimed that resummation effects increase the cross section and bring the Standard-Model prediction in agreement with the experimental measurements. On the other hand, based on a study of transverse-momentum resummation, the authors of [24] concluded that resummation effects are small for the relevant values of \(p_T^{{\mathrm{veto}}}\). Most recently, the effect of using a matched parton shower to predict the fiducial cross section, as is done in the experimental analyses, was analyzed in [25]. These authors concluded that resummation effects are small and that a fixed-order computation of the fiducial rate would lead to theoretical predictions in agreement with the measurements, but that the matched parton shower overestimates the Sudakov suppression of the rate and leads to systematically lower theoretical predictions when extrapolating back to the total rate.

In the present paper, we present an automated method to perform resummations for arbitrary vector-boson-production processes involving jet vetoes. Instead of computing resummed cross sections analytically, on a case-by-case basis, we obtain them in an automated way using the MadGraph5_aMC@NLO framework [26]. Our method yields results which are accurate at NNLL and are matched to NLO fixed-order results. Such an automated procedure is obviously much more efficient and less error prone than computing the ingredients by hand or extracting them from the literature. Most importantly, our approach allows us to also include the decay of the vector bosons, along with cuts on the leptons in the final state.

We have implemented two different methods to perform the resummation. The first one is based on reweighting tree-level events generated by MadGraph. It yields jet-veto cross sections accurate at NNLL order. The event weight includes universal resummation factors as well as the process-specific one-loop virtual corrections, which are computed using Madgraph5_aMC@NLO. In the second method, we modify the NLO fixed-order computation in such a way that the end result is accurate at both NNLL and NLO. In this second method not only the hard function, which encodes the virtual corrections, but also the beam functions, which describe the emissions at small transverse momentum, are computed by Madgraph5_aMC@NLO.

Our paper is organized as follows. We start in Sect. 2 by reviewing the resummation formula for cross sections in the presence of a jet veto. We also discuss non-perturbative corrections and point out that they could be sizable, similar in magnitude as the recently calculated NNLO corrections. We then explain in Sect. 3 how the automated resummation can be implemented in the Madgraph5_aMC@NLO framework. In Sect. 4 we use our method to compute cross sections for different boson-production processes and discuss in detail the scale and scheme choices and the resulting theoretical uncertainties. We compare our resummed predictions to fixed-order results for the cross sections, to the results obtained from a matched parton shower, and to the NNLL results of [23]. We also match our resummed result to fixed-order NLO predictions. The relevant matching corrections turn out to be very small, which indicates that the bulk of the NLO result is already captured by the factorization formula evaluated with NNLL accuracy. This remains true after imposing cuts on the leptonic final state in the decays of the electroweak bosons. We compare predictions for the final states \(Z\), \(W^+W^-\), and \(W^+W^-W^+\) and consider ratios of cross sections, which have small uncertainties if they are properly defined. We then discuss the implications of our results on the value of the \(W^+W^-\) cross section and conclude in Sect. 5.

## 2 Factorization theorem for jet-veto cross sections

## 3 Automated resummation

We now explain how to automate the resummation by suitably modifying existing fixed-order results. We shall employ two different resummation schemes. In Scheme A, we work with tree-level events obtained from MadGraph5_aMC@NLO [26]. We supply the beam functions from explicit calculations but compute the hard functions automatically and then reweight the events to achieve the resummation. In Scheme B, we use MadGraph5_aMC@NLO in fixed-order mode and compute the NLO cross section with a jet veto. To achieve the resummation we subtract the logarithmically enhanced pieces from the fixed-order cross section and multiply them back in resummed form. In this second scheme, both the hard functions and the beam functions are computed using MadGraph5_aMC@NLO. The second scheme is more convenient for practical computations but limited to NNLL order, while the first scheme allows (in principle) for arbitrary accuracy of the resummation.

### 3.1 Scheme A: NNLL from reweighting born-level events

### 3.2 Scheme B: NNLL \(+\) NLO with automated computation of the beam functions and matching corrections

In the reweighting scheme discussed above, we use MadGraph5_aMC@NLO to compute the hard functions but supply the beam functions from an explicit calculation. One can go even further and also compute the beam functions and the matching corrections automatically and in a single step. This is done by first factoring out the hard corrections and then performing a NLO run in the presence of the jet veto. An advantage of this second approach is that the beam functions are computed on the fly and it is therefore easy to use different PDF sets without any need to recompute the beam functions. A slight disadvantage is that one has to run MadGraph5_aMC@NLO in NLO mode. One can thus no longer work with events and will have to perform a new run when changing the cuts. However, if the matching is included in Scheme A described above, then a NLO run is needed also in this case. Note also that Scheme B only works at NNLL accuracy, while Scheme A allows for arbitrary precision if the necessary reweighting factor is supplied.

To implement (20) in MadGraph5_aMC@NLO we have directly modified its Fortran code by including the logarithmically enhanced terms. The expanded logarithmically enhanced terms, i.e. the second term on the right-hand side of (19), is similar to the compensating Sudakov factor introduced in the FxFx merging prescription, see (2.46) of [34], and it is therefore implemented at the same place in the code. In MadGraph5_aMC@NLO each real-emission phase-space configuration has corresponding Born kinematics defined by the FKS mapping [35]. Therefore we can always compute the prefactor \(P_{ij}\) using Born kinematics, and it can multiply the complete reduced cross section, including the real-emission contributions. In order to improve the run time, the time-consuming one-loop matrix elements are computed only once for each phase-space configuration, cached in memory, and used also for the (expanded) hard function. However, compared to normal running of MadGraph5_aMC@NLO, we cannot reduce the number of calls to the virtual corrections by using suitable approximations of it, as described in Sect. 2.4.3 of [26], because the reduced cross section is multiplied by them, resulting in positive feedback loops in setting up the approximations. When running MadGraph5_aMC@NLO in fNLO mode, setting the parameter ickkw in the run_card.dat to -1 turns on in the inclusion of the logarithmically enhanced terms and sets the hard and soft scales to \(Q\) and \(p_T^{{\mathrm{veto}}}\) (given by the ptj parameter in the run_card.dat), respectively. Hard and soft scale variations, as well as PDF uncertainties, can be computed at minimal CPU costs by reweighting [36]. This addition to the MadGraph5_aMC@NLO will become public with the next release of the code.

## 4 Phenomenological results

We now proceed to give numerical results for different electroweak-boson-production cross sections. Before presenting our final results, we discuss a variety of issues such as the proper choice of matching and factorization scales, the size of the matching corrections and the difference between the two resummation schemes discussed in the previous section. We then present results for the \(W^+W^-\) cross section as well as the cross section including the decay of the \(W\) bosons with cuts on the final-state leptons. Since the published measurements [14, 15] were taken at \(\sqrt{s} = 7\,\mathrm{TeV}\), we will present our results for this center-of-mass energy. For the electroweak parameters we use MadGraph5 default values, in particular \(\alpha _\mathrm{em}=1/132.5\), \(G_F=1.166\times 10^{-5}\,\mathrm{GeV}^{-2}\), \(M_W=80.42\,\mathrm{GeV}\), and \(M_Z=91.19\,\mathrm{GeV}\).

In all of our results below, we work with the MSTW2008 NNLO PDF set and its associated value \(\alpha _s(M_Z)=0.1171\) [37]. The choice of a NNLO PDF set seems appropriate, because we believe that the resummation captures the most important part of the NNLO corrections. In order to illustrate the size of the higher-order terms captured by resummation, we will also evaluate the NLO corrections using the NNLO PDF set, which increases the NLO prediction at \(p_T^{{\mathrm{veto}}}= 20\, \mathrm{GeV}\) by about 2 % in the case of \(W^+W^-\) production. We will define our jets using the anti-\(k_T\) algorithm with a jet radius of \(R=0.4\). The only quantity sensitive to the jet radius at NNLL \(+\) NLO accuracy is the anomaly exponent \(F_{ij}(p_T^{{\mathrm{veto}}},\mu ,R)\), and it is the same for all \(k_T\)-style clustering algorithms. As the default scheme for our plots we use Scheme A, since it is easier to disentangle and discuss the individual ingredients of the calculation (NLL versus NNLL resummation, matching to fixed-order perturbation theory) in this scheme. However, we find that both schemes give almost indistinguishable numerical results at NNLL \(+\) NLO level.

### 4.1 Resummed results and choice of the hard scale

In Fig. 2 we show the results for the resummed \(Z\)-boson and \(W^+ W^-\)-pair-production cross sections at \(\sqrt{s}=7\) TeV, obtained with \(n_f=5\) light quark flavors and jet radius parameter \(R=0.4\). Here and below the two scales \(\mu \) and \(\mu _h\) are varied independently by factors of 2 about their default values \(\mu =p_T^{{\mathrm{veto}}}\) and \(\mu _h=Q\), where \(Q\) is the invariant mass of the electroweak final state, i.e. \(Q^2=M_Z^2\) for \(Z\)-boson production and \(Q^2=(q_{1}+q_{2})^2\) for the \(W^+ W^-\) final state (defined on an event-by-event basis). The resulting uncertainties are then added quadratically. In addition to the standard scale choice \(\mu _h^2\approx Q^2\) we consider using an imaginary value for the hard matching scale, such that \(\mu _h^2\approx -Q^2\). The corresponding results are shown on the right-hand side of Fig. 2. For comparison, we also show the NLO fixed-order results, which will be discussed in more detail in the next section. In all cases, we observe that going from NLL to NNLL accuracy improves the stability of the predictions significantly. Also, the NNLL bands are closer to the fixed-order NLO results than the NLL bands.

The use of an imaginary value of the hard matching scale \(\mu _h\) has been advocated in the context of Higgs production, because it maps the relevant hard function onto the space-like gluon form factor [38, 39]. This Euclidean quantity shows a much better perturbative behavior than the time-like form factor, which suffers from large numerical corrections \(\sim \) \((\alpha _s\pi ^2)^n\) due to imaginary parts from Sudakov double logarithms, which arise in time-like kinematics. The same arguments apply to the case of \(Z\) production. In [23], the choice \(\mu _h^2<0\) was applied to \(W^+ W^-\) pair production, and it was argued that this leads to a significant enhancement of the cross section, bringing the theoretical prediction in agreement with LHC measurements. Indeed, one can observe from Fig. 2 that the resummed results for the cross sections obtained with \(\mu _h^2<0\) are significantly larger than those obtained with the standard choice \(\mu _h^2>0\). For \(W^+ W^-\) production with \(p_T^{{\mathrm{veto}}}=25\) GeV, the increase in the central value of the NNLL \(+\) NLO cross section is about 4.8 % (which is of the same order as the recently calculated NNLO corrections [21]).^{2} We stress, however, that in the case of multi-particle final states such as \(W^+W^-\) the hard function depends on several kinematic scales (\(\hat{s}\) and \(\hat{t}\) in the present case), some of which are time-like and some of which are space-like. Unfortunately, it is impossible to adopt a suitable choice of the hard matching scale, which would map the hard function onto a Euclidean quantity, such that all \((\alpha _s\pi ^2)^n\) terms can be resummed by means of RG evolution equations. It is therefore not clear whether the convergence of the perturbation series can be improved by using the choice \(\mu _h^2<0\). This problem was discussed in detail in the context of Higgs plus jet production in [40]. Even though the convergence in the right panels of Fig. 2 looks somewhat better than in the plots shown on the left, we have decided to adopt the conventional prescription \(\mu _h>0\) for the hard matching scale. Perhaps a more conservative way to assess the scale uncertainty would be to allow for arbitrary complex scale choices \(Q/2<|\mu _h|<2Q\) and then give the resulting uncertainty, as was recently proposed in [41].

For Higgs production, the resummation of jet-veto logarithms was performed to higher accuracy by including the two-loop hard and beam functions as well as the RG evolution factor at approximate N\(^3\)LL order [9]. The only missing ingredients for full N\(^3\)LL \(+\) NNLO accuracy are the three-loop anomaly exponent and the four-loop cusp anomalous dimension, whose effects have been estimated and included in the error budget. It was observed in this reference that the two-loop beam functions decrease the cross section, and we expect a similar effect in the present case. In the future, it should be possible to reach the same level of accuracy also for \(W^+ W^-\) production and related processes. The corresponding two-loop hard functions can be extracted from the two-loop virtual corrections, which have recently been obtained in [21, 42]. The product of beam functions integrated over rapidity could be extracted numerically from NNLO fixed-order codes for \(Z\)-boson production such as [43, 44], following the procedure employed in [9]. This is sufficient to obtain the inclusive \(W^+ W^-\) cross section, while a two-loop computation of the beam functions would be required for more exclusive cross-section predictions. Once (approximate) N\(^3\)LL+NNLO predictions for the \(W^+ W^-\) cross sections are available, the above-mentioned ambiguities, related to the choice of the hard matching scale, will be reduced significantly.

### 4.2 Fixed-order results and matching

In order to obtained the best possible predictions, we need to match our resummed results for the cross sections with fixed-order expressions at NLO. The scale dependence of the NLO expression for the for the \(W^+ W^-\)-production cross section at \(\sqrt{s}=7\) TeV is shown in the left panel in Fig. 3. We set the factorization and renormalization scales equal (\(\mu =\mu _r=\mu _f\)) and vary them from \(\mu =p_T^{{\mathrm{veto}}}/2\) up to \(\mu =2Q\). This is a much larger scale variation than is usually considered, but this wider range seems appropriate since the problem at hand involves physics at both scales. For comparison, we also show the bands one would obtain from a variation of \(\mu \) by a factor of 2 around either a high default value \(\mu =Q\) or a low default value \(\mu =p_T^{{\mathrm{veto}}}\). Our broad scale variation is obviously more conservative, since it covers both options. Nevertheless, fixed-order computations usually adopt the high scale \(\mu =Q\) as the default value, and from Fig. 2 it appears that such a choice indeed leads to smaller higher-order corrections. A similar behavior is found for all cases studied in this paper. The invariant-mass distribution of the \(W\)-boson pair is shown in the right panel of Fig. 3. Defining the average hard scale \(\tilde{Q}\) by the median value of this distribution, one obtains \(\tilde{Q}=222\,\mathrm{GeV}\). This value will be useful in our phenomenological discussion below.

### 4.3 Multiple bosons and cross-section ratios

We are now ready to present our final results for a couple of interesting production cross sections involving multiple electroweak gauge bosons. In Fig. 6, we show predictions for the \(Z\)-, \(W^+ W^-\)-, and \(W^+ W^- W^\pm \)-production cross sections at the LHC with \(\sqrt{s}=7\) TeV; it would be straightforward to rerun our code at different values of the center-of-mass energy. In each case, we present our resummed and matched predictions at NLL \(+\) NLO and NNLL \(+\) NLO accuracy and compare them with the fixed-order NLO prediction. Notice that the value of the cross section drops by about a factor \(10^3\) with each additional boson. The triple-boson production cross section is tiny, but it constitutes a background to Higgs production in association with a \(W^\pm \) and subsequent decay \(H\rightarrow W^+ W^-\). The fact that we can obtain predictions for three-boson final states without any additional effort nicely demonstrates the power of our automated resummation scheme.

We find that the scale uncertainties of our NNLL \(+\) NLO predictions for \(W^+ W^-\) and \(W^+ W^- W^\pm \) production are estimated to be of similar size, while we obtain a much smaller uncertainty for the case of \(Z\)-boson production. This small scale variation should perhaps be taken with a grain of salt. At larger \(p_T^{{\mathrm{veto}}}\) values, our resummed cross section becomes similar to the fixed-order result, and its scale variation is similar to the scale variation of the fixed-order cross section obtained by performing a correlated scale variation with \(\mu _r=\mu _f\). An independent variation of \(\mu _r\) and \(\mu _f\), which is standard practice in fixed-order computations, would give an uncertainty that is twice as large. On the other hand, we have checked that the known NNLO corrections for \(Z\)-boson production are indeed compatible with our small uncertainty band. It is also interesting to note that for \(W^+ W^-\) production the scale uncertainties of the fixed-order prediction obtained from correlated and independent variations of \(\mu _r\) and \(\mu _f\) are found to be of similar size.

We also observe that the scale uncertainties of the fixed-order NLO predictions at small \(p_T^{{\mathrm{veto}}}\) values strongly increase with the number of produced bosons. This is not surprising if we consider the relevant scale ratio \(\tilde{Q}/p_T^{{\mathrm{veto}}}\), which governs the size of Sudakov logarithms. Using the median value \(\tilde{Q}\) of the invariant-mass distribution to estimate the hard scale, we find \(\tilde{Q}=M_Z\) for \(Z\) production, \(\tilde{Q}\approx 2.8\,M_W\) for \(W^+ W^-\) production, and \(\tilde{Q}=5.7\,M_W\) for \(W^+ W^- W^\pm \) production. In all cases, the three-momenta at which the bosons are produced scale with the boson mass, but the average scale increases with the number of the produced bosons. Note that after the resummation of Sudakov logarithms has been performed, the width of the uncertainty bands is only weakly dependent on the veto scale.

### 4.4 Experimental cuts

- 1.
lepton \(p_T>20\,\mathrm{GeV}\),

- 2.
leading lepton \(p_T>25\,\mathrm{GeV}\),

- 3.
lepton pseudorapidity \(\eta _e<1.37\) or \(1.52<\eta _e<2.47\),

- 4.
dilepton invariant mass \(m_{e^+e^-}>15\,\mathrm{GeV}\) and also \(|m_{e^+e^-}-m_Z|>15\,\mathrm{GeV}\).

The experimental analysis [14] imposes a few additional cuts, in particular a minimum total transverse momentum of the two charged leptons \(p_T^{e^+e^-}\,>30 \, \mathrm{GeV }\) and minimum requirements on the missing transverse momentum \(p_{T, \text {Rel}}^{\nu \bar{\nu }}\,>45\, \mathrm{GeV}\).^{3} The cut \(p_T^{e^+e^-}\,>30 \, \mathrm{GeV }\) is somewhat problematic for the theoretical analysis, especially when it is applied to predict the \(Z\)-boson background to \(W^+ W^-\) production. The difficulty is that we must make sure that the leptonic cuts do not (strongly) affect the hadronic final state. In the case of \(Z\) production the \(p_T^{e^+e^-}\) is equal (and opposite) to the transverse momentum \(p_T^X\) of the hadronic final state. Imposing a lower bound on \(p_T^{e^+e^-}\) is the same as imposing a lower bound on \(p_T^X\). This interferes with the jet-veto cut which at NLO corresponds to an upper cut on \(p_T^X\). The factorization formula in [8] does allow for additional cuts on \(p_T^X\) in the presence of the jet veto, but the relevant beam functions would be more complicated than those needed without such cuts. For the \(W^+ W^-\)-production process, the quantity \(p_T^{e^+e^-}\) is not directly related to \(p_T^X\) because of the presence of the neutrinos, but the corresponding cut still affects the low-\(p_T^X\) region.

### 4.5 Difficulties associated with photons

Our framework cannot immediately be applied to processes involving photons. The reason is that photons are massless particles and have hadronic substructure. At high energies, a photon thus needs to be treated as a photon jet, or more precisely a photon surrounded by some hadronic radiation. In fact, many photon-isolation requirements necessitate fragmentation functions. This can be avoided using the photon isolation proposed by Frixione [47], but also in this case the photon has a partonic content and a proper description needs to take into account partons emitted collinear to the photon. This implies that our factorization theorem does not apply, since it assumes that all energetic radiation is collinear to the beam. The photon isolation introduces new small scales to the problem (e.g. the hadronic energy around the photon), which give rise to additional large logarithms not associated with the jet veto.

## 5 Conclusion

Higher-order logarithmic resummations in collider physics, both in SCET and using traditional methods, are typically done on a case-by-case basis, similar to the way fixed-order calculations were performed a few years ago. In the meantime, several groups have automated NLO computations in a variety of computer codes. This automation saves time, reduces the possibilities for mistakes and offers the flexibility to also study effects beyond the Standard Model. It is desirable to have the same level of automation for higher-order resummations of large logarithmic corrections. In the present paper, we have achieved this goal for electroweak-boson-production cross sections in the presence of a jet veto, at NNLL \(+\) NLO accuracy. This combination is natural because in the Sudakov region, where \(\ln (Q/p_T^{{\mathrm{veto}}})\sim 1/\alpha _s\), NNLL logarithmic terms have the same parametric scaling as NLO corrections in a region where there are no large logarithms. In contrast, taking resummation effects into account using a parton shower gives a lower parametric accuracy, and the unitarization inherent in the shower approach can sometimes be problematic. In the case of the jet-veto cross section for \(W^+ W^-\) production, for example, unitarization leads to cross sections that are systematically lower than the NNLL \(+\) NLO results.

Resummations are relevant in kinematical configurations which are close to the Born-level kinematics and can therefore be obtained by reweighting Born-level cross sections with appropriate factors. The most complicated ingredient for NNLL resummations are the one-loop hard functions, which encode the virtual corrections. Their computation has been automated, and we use the MadGraph5_aMC@NLO framework to obtain the hard function required for our analysis. We have also presented a modified scheme, in which the beam functions accounting for collinear emissions and the matching onto fixed-order results is automated and performed using existing fixed-order codes. This is possible, because the hard function and the resummation of large logarithms are just overall factors in the differential cross section.

We have used our method to perform a detailed analysis of resummation effects for the \(W^+ W^-\)-pair-production cross section, for which experimental measurements found a slight excess compared to theoretical predictions based on NLO computations matched to parton showers. We observe that the NLO result with a high value of the renormalization and factorization scales \(\mu _r \sim \mu _f \sim Q\) is in good agreement with the NNLL \(+\) NLO resummed predictions, while the results obtained with a matched parton shower are systematically lower. This effect, together with the positive NNLO corrections to the total rate which are now known, helps to bring the Standard Model prediction into better agreement with the measurements. It would be important to include the two-loop virtual corrections into the resummation and to also compute and include two-loop beam functions. This improvement, which is beyond the scope of the present work, would lead to very precise predictions, which could be directly compared with the experimental results. This level of accuracy has already been achieved in Higgs production by extracting the beam functions numerically. It was found that the two-loop corrections to the beam functions were sizable, because they are enhanced by logarithms of the jet radius. In Higgs production, the NNLO corrections to the hard function increase the cross section, while the two-loop beam functions lower it. We expect the same behavior for the \(W^+ W^-\) case, and it will be interesting to see the combined effect of these improvements on the final predictions. Also, at NNLO the \(gg\) channel starts to contribute to \(W^+ W^-\) production and could give rise to important corrections. Since this channel has already been implemented into the MadGraph5_aMC@NLO framework, it will be straightforward to perform the corresponding resummation using our method.

It would also be interesting to generalize our methods to processes with jets in the final state. In addition to hard, beam, and soft functions, these processes involve jet functions describing the energetic final-state radiation. Furthermore, the hard function then has a non-trivial color structure. Existing programs which compute virtual corrections for NLO processes currently only supply squared matrix elements summed over colors, but they can be modified to provide the color information needed for SCET-based resummation. This color structure is then contracted with the color structure of the soft function after RG evolution. The soft, beam, and jet functions will in general need a separate calculation. However, since the jet and beam functions are two-point functions and the soft function is given by a single emission from eikonal lines, these computations are much simpler than full-fledged real-emission computations and could be automated as well. We are confident that such automated resummations will become available in the future and provide higher-order logarithmic resummations for a much wider range of observables.

## Footnotes

- 1.
Preliminary NNLO results for the rate in the presence of cuts were presented at a recent conference [22].

- 2.
There is an ambiguity when choosing \(\mu _h^2<0\) related to the fact that the running coupling \(\alpha _s(\mu ^2)\) has a cut along the negative \(\mu ^2\) axis. One can either choose the default matching scale above or below the cut, \(\mu _h^2=-Q^2\pm i\epsilon \). Our values for the cross section obtained within the MadGraph5_aMC@NLO framework correspond to the principal-value prescription, while the authors of [23] adopt the default choice \(\mu _h^2=-Q^2-i\epsilon \). At NNLL order, the latter choice yields a result that is 2 % higher (at \(p_T^{{\mathrm{veto}}}=25\,\mathrm{GeV}\)) than that obtained with the principal-value prescription. This difference would be reduced at higher orders. A detailed numerical comparison with [23] further revealed that their implementation of the beam functions was incorrect. After correcting this, our results are in agreement.

- 3.
The exact definition of \(p_{T, \text {Rel}}^{\nu \bar{\nu }}\) is more involved; see [14].

## Notes

### Acknowledgments

We thank Prerit Jaiswal and Takemichi Okui for discussions and for their help in performing a detailed numerical comparison and tracking down the sources of the numerical differences between our results and those obtained in [23]. T. Becher and L. Rothen are supported by the Swiss National Science Foundation (SNF) under Grant 200020_153294 and acknowledge support from the Munich Institute for Astro- and Particle Physics (MIAPP) and from the Mainz Institute for Theoretical Physics (MITP). The research of M. Neubert is supported by the Advanced Grant EFT4LHC of the European Research Council, the Cluster of Excellence *Precision Physics, Fundamental Interactions and Structure of Matter* (PRISMA-EXC 1098), and Grant 05H12UME of the German Federal Ministry for Education and Research.

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