Jet Veto Resummation with Jet Rapidity Cuts

Jet vetoes are widely used in experimental analyses at the LHC to distinguish different hard-interaction processes. Experimental jet selections require a cut on the (pseudo)rapidity of reconstructed jets, $|\eta_{\rm jet}| \leq \eta_{\rm cut}$. We extend the standard jet-$p_T$ (jet-veto) resummation, which implicitly works in the limit $\eta_{\rm cut}\to\infty$, by incorporating a finite jet rapidity cut. We also consider the case of a step in the required $p_T^{\rm cut}$ at an intermediate value of $|\eta| \simeq 2.5$, which is of experimental relevance to avoid the increased pile-up contamination beyond the reach of the tracking detectors. We identify all relevant parametric regimes, discuss their factorization and resummation as well as the relations between them, and show that the phenomenologically relevant regimes are free of large nonglobal logarithms. The $\eta_{\rm cut}$ dependence of all resummation ingredients is computed to the same order to which they are currently known for $\eta_{\rm cut}\to\infty$. Our results pave the way for carrying out the jet-veto resummation including a sharp cut or a step at $\eta_{\rm cut}$ to the same order as is currently available in the $\eta_{\rm cut}\to\infty$ limit. The numerical impact of the jet rapidity cut is illustrated for benchmark $q\bar q$ and $gg$ initiated color-singlet processes at NLL$'+$NLO. We find that a rapidity cut at high $\eta_{\rm cut} = 4.5$ is safe to use and has little effect on the cross section. A sharp cut at $\eta_{\rm cut} = 2.5$ can in some cases lead to a substantial increase in the perturbative uncertainties, which can be mitigated by instead using a step in the veto.


Introduction
Measurements that involve a veto on additional jets, or more generally that divide events into exclusive jet bins, play an important role at the LHC, e.g. in Higgs and diboson measurements or in searches for physics beyond the Standard Model. The jet binning differentiates between hard processes that differ in the number of hard signal jets, and hence allows one to separate signal and background processes. The separation into 0-jet and ≥ 1-jet bins also provides a model-independent way to discriminate between qq and gg initiated processes [1].
Experiments can only reconstruct jets up to some maximal pseudorapidity |η| ≤ η cut due to the range of the detector, e.g. for ATLAS and CMS η cut ∼ 4.5. In principle, the utility of the jet binning to discriminate between different hard processes increases for a tighter jet veto (smaller p cut T ). However, jets with small transverse momenta are difficult to reconstruct experimentally, especially for pseudorapidity |η| 2.5 beyond the reach of the tracking detectors, which are important to suppress the large contamination from pile up (e.g. in the jet vertex tagging algorithm used by ATLAS [26]). This is illustrated in fig. 1. As the LHC luminosity increases and pile-up conditions become harsher, the contamination from unsuppressed pile-up jets grows worse and must be avoided. One option is to increase the overall p cut T . For example, in the context of Higgs measurements, the increased pile up in Run 2 has forced raising the jet threshold from 25 GeV to 30 GeV. This however weakens the jet veto and thus reduces its utility. Alternatively, to avoid raising the jet threshold, one can consider jets only in a restricted pseudorapidity range of |η| 2.5. However, this looses the discrimination power from forward jets, which are a distinguishing feature of some processes (most notably weak-boson fusion topologies in Higgs and diboson production). The best possible option combines both approaches and performs a step-like jet selection, with a lower p cut T threshold for central jets and a somewhat higherp cut T threshold for forward jets. For example, recent ATLAS Higgs measurements [27] reconstruct jets using p cut T = 25 GeV for |η| < 2.4 andp cut T = 30 GeV for |η| > 2.4 (and no jets beyond rapidity |y| = 4.4).
A discontinuous step in the jet threshold can also pose challenges on its own, as it makes the experimental measurements more complex. Theoretically, we will see that it can complicate the resummation of logarithms in some extreme cases. An alternative to a step is to use jet vetoes that smoothly depend on the jet rapidity [16,28], providing a tighter veto at central rapidities and a looser one at forward rapidities. These rapiditydependent vetoes can also be supplemented with an additional sharp jet rapidity cut, which we briefly discuss in appendix B.
The usual jet-p T resummations [6-9, 12, 13] do not account for any jet rapidity dependence, i.e., the resummation is performed for η cut → ∞. Using parton-shower Monte Carlos, one finds that a jet rapidity cut at η cut = 4.5 has a very small numerical effect, while η cut = 2.5 has a sizable effect on the jet-p T spectrum in Higgs production (see e.g. refs. [2,6]), so it is important to properly include it in the resummation. This was already pointed out in ref. [8], where it was also speculated that a jet rapidity cut might change the resummation structure.
Our analysis in this paper fully addresses these questions by systematically incorporating the jet rapidity cut into the jet-p T resummation, including in particular the case of a step-like veto. For this purpose, we extend the formalism of refs. [8,13], which uses the framework of Soft-Collinear Effective Theory (SCET) [29][30][31][32]. To be concrete, our discussion focuses on color-singlet production, including the important cases of Higgs and Drell-Yan production. Our results for how to incorporate the η cut dependence also carry over to processes with additional signal jets in the final state to the same extent to which the usual jet-p T resummation for color-singlet production carries over to such cases [10,11].
We identify all relevant parametric regimes in the veto parameters p cut T , η cut ,p cut T , and discuss the factorization and resummation structure for each regime. We also study the relations between the different regimes and perform numerical studies to check their respective ranges of validity. An important conclusion of our analysis is that all regions of parameter space that are of phenomenological interest can be described by parametric regimes that are free of large nonglobal logarithms.
We analytically compute the η cut dependence of all ingredients at O(α s ) as well as of the dominant O(α 2 s ) corrections (those enhanced by jet-veto or jet clustering logarithms), which matches the order to which they are currently known in the η cut → ∞ limit. Our results allow for carrying out the jet-veto resummation including jet rapidity cuts to the same order as is currently available without such cuts, which for color-singlet production is NNLL +NNLO. (Reaching this level also requires the still unknown nonlogarithmic O(α 2 s ), which can be extracted numerically from the full NNLO calculation, as was done for η cut → ∞ in ref. [13]. Carrying out such an analysis is beyond the scope of this paper.) The effect of a rapidity cut for transverse momentum vetoes has also been considered independently in refs. [33,34] for dijet production, and more recently for the transverse energy event shape in Drell-Yan in ref. [35]. We compare their results to our results for the case of a sharp cut at η cut and no measurement beyond in sec. 2.5.
The paper is organized as follows: In sec. 2, we discuss the parametric regimes and corresponding effective field theory (EFT) setups for a sharp cut on reconstructed jets at η cut and no measurement beyond, as in the middle panel of fig. 1. We give the perturbative ingredients at O(α s ) and the leading small-R clustering terms at O(α 2 s ) for all partonic channels. We numerically validate the EFT setup by comparing to the relevant singular limits of full QCD, and also compare the regimes to each other and identify their respective ranges of validity. In sec. 3, we generalize the results of sec. 2 to a step in the jet veto at η cut , as in the right panel of fig. 1. In sec. 4, we illustrate the numerical impact of the rapidity cut at NLL +NLO for Drell-Yan at Q = m Z and Q = 1 TeV and for gg → H at m H = 125 GeV and gg → X at m X = 1 TeV for different values of η cut . We conclude in sec. 5. Details of our calculations can be found in app. A. In app. B, we briefly discuss how an additional sharp rapidity cut affects the description of the rapidity-dependent jet vetoes introduced in ref. [16].
2 Factorization with no constraint beyond η cut (p cut T = ∞)

Overview of parametric regimes
We consider exclusive 0-jet cross sections, where the veto is applied by identifying jets with radius R (the details of the jet-clustering algorithm are not relevant at the order we are working) and cutting on the transverse momentum p jet T of the leading jet within The resulting constraints on the rapidities and transverse momenta of initial-state radiation (ISR) are displayed as black lines in fig. 2. We can identify two distinct power-counting parameters that govern the typical angular size of energetic collinear ISR with energy E ∼ Q, where Q is the momentum transferred in the hard interaction: First, the p T of the emissions is constrained by p T < p cut T for |η| < η cut , corresponding to a maximum opening angle Second, the p T of an energetic emission at rapidity η is parametrically p T ∼ Qe −|η| . The rapidity cut removes the first constraint for |η| > η cut . Hence, if η cut is central enough, emissions beyond η cut can reach a characteristic p T Qe −|ηcut| , corresponding to a maximum opening angle There are three parametric regimes for p cut T /Q and e −ηcut , which are illustrated in fig. 2 for η cut = 2.5. The thick black lines show the veto for different values of p cut T /Q. The thick gray curve shows the relation p T /Q = e −|η| , while the thin gray lines show the values of η cut and p T /Q = e −ηcut .
The first parametric regime is p cut T /Q e −ηcut . As we will demonstrate in sec. 2.2, in this regime effects due to the rapidity cut are power suppressed by Qe −ηcut /p cut T . Hence, they can be treated as a fixed-order power correction to the standard jet-veto resummation, which implicitly works in the limit η cut = ∞. For Higgs measurements with p cut T = 25 GeV, η cut = 4.5, Q ≡ m H = 125 GeV, this parametric assumption is well justified, as m H e −ηcut /p cut T ∼ 5%. For heavier final states and/or more central rapidity cuts the relevant parametric regime is p cut T /Q ∼ e −ηcut . This is the case for example for Q = 1 TeV and η cut = 4.5 or Q = 125 GeV and η cut = 2.5 at p cut T = 25 GeV. In sec. 2.3, we show that in this regime the rapidity cut effects must be treated as a leading-power correction, and that they can be seamlessly incorporated into the existing jet-veto resummation without rapidity cut. We will see that they affect only the boundary terms in the resummed cross section, but not the anomalous dimensions and evolution factors. Hence, they start contributing at NLL or NNLL.
Finally in sec. 2.4, we discuss the parametric regime p cut T /Q e −ηcut . This case is conceptually interesting, since logarithms of the ratio of scales Qe −ηcut and p cut T appear, changing the logarithmic structure already at leading-logarithmic (LL) order. In addition, formally large nonglobal logarithms of the same ratio appear. This regime is of very limited phenomenological relevance for typical jet-binned analyses at the LHC. For example, for η cut = 2.3 corresponding to e −ηcut = 0.1, it would require an extremely tight jet veto p cut T 0.1 Q, which is unrealistic as it would leave almost no signal in the 0-jet cross section. For the purpose of explicitly probing this regime experimentally, one could lower η cut 1.0 − 1.5, such that the jet veto only acts on radiation in the very central region.

Regime 1: p cut
T /Q e −ηcut (standard jet veto resummation) As usual, the scaling of the modes in the EFT follows from the nontrivial constraints imposed on emissions by the measurement. Soft emissions at central rapidities are always restricted by the jet veto. Collinear emissions with energy ∼ Q and rapidity η have a transverse momentum ∼ Qe −|η| and are constrained by the measurement if Qe −|η| ∼ p cut T , which determines their scaling. Since Qe −ηcut p cut T , these collinear modes are parametrically not forward enough to be sensitive to the rapidity cut, such that the description of their dynamics is simply governed by the power counting in p cut T /Q. The relevant EFT modes in this regime are thus the same as for a jet veto without any rapidity cut, Here and below, we give the scaling of momenta in terms of light-cone components defined as (with n ≡ n a ,n ≡ n b ), In addition, there are the usual inclusive collinear modes that describe the initial protons at the scale Λ QCD , and which are not specific to our discussion here. In principle, we can consider collinear emissions that are forward enough to resolve rapidities η ∼ η cut , n a -collinear (η cut ): p µ ∼ Qe −2ηcut , Q, Qe −ηcut , n b -collinear (η cut ): p µ ∼ Q, Qe −2ηcut , Qe −ηcut .
(2.6) However, since Qe −ηcut p cut T , these emissions have too little transverse momentum to be affected by the jet veto, and are therefore unconstrained and integrated over without requiring additional modes in the EFT. To explicitly see that the η cut dependence is power suppressed, note that the full jet-veto measurement for the collinear modes contains a θ function which thus only induces power corrections in Qe −ηcut /p cut T . Therefore, at leading order in the power expansion, 1 we recover the factorization for the 0-jet cross section with η cut = ∞ [7,8,13], The hard function H κ contains the short-distance matrix element for producing a colorsinglet final state and depends on the hard kinematic phase space Φ, which encodes e.g. the total rapidity Y and invariant mass Q of the color-singlet final state. The soft function S κ encodes soft radiation restricted by p cut T . The partonic channels is denoted by κ and is implicitly summed over (if necessary). The beam functions B a,b are forward proton matrix elements of collinear SCET fields and encode the perturbative collinear ISR constrained by p cut T as well as the unconstrained ISR below that scale down to the nonperturbative scale of the PDFs [4]. In eq. (2.8), they are evaluated at ω a,b = Qe ±Y . They are given by a convolution of perturbative matching coefficients I ij , which encode the p cut T constraint, and the standard inclusive quark and gluon PDFs f j , As discussed in detail in ref. [13], all logarithms of the ratio p cut T /Q in eq. (2.8) are resummed by evaluating each of the hard, beam, and soft functions at their characteristic virtuality and rapidity scales, and evolving them to common scales µ, ν using renormalization group (RG) evolution. The power corrections in eq. (2.8) can be included at fixed order in α s by matching the resummed result to the corresponding fixed-order result in full QCD. The O(Qe −ηcut /p cut T ) corrections stop being suppressed for large Q, small p cut T , or central η cut . In the next section, we show that they can be incorporated into the beam functions in eq. (2.9).

Regime 2: p cut
T /Q ∼ e −ηcut (η cut dependent beam functions) In this regime, the scaling of soft and collinear modes is unchanged from the previous case. However, the characteristic rapidity of the collinear modes now coincides parametrically with η cut , i.e., Thus, collinear emissions resolve the rapidity cut, and are constrained by the jet veto for |η| < η cut , while for |η| > η cut they are unconstrained. As a result, the cross section factorizes at leading power as The beam functions now explicitly depend on both p cut T and η cut , while the hard and soft functions are unchanged (with their characteristic scales still given by eq. (2.10)). The RG consistency of the cross section fixes the anomalous dimensions of the beam function in terms of those for the soft and hard functions. Thus, the η cut dependence cannot change the renormalization of the beam function, i.e., where the anomalous dimensions are the same as in the η cut → ∞ limit [8,13], and η i Γ in the resummed rapidity anomalous dimension is given by Hence, the η cut effects do not affect the RG evolution, but only change the beam function boundary conditions, and therefore first appear at NLL . In analogy to eq. (2.9) the beam functions can be factorized into collinear matching coefficients, which now also depend on η cut , and the PDFs. We write the matching coefficients as the sum of the usual η cut -independent matching coefficients plus a correction term that encodes the η cut dependence, I ij (p cut T , η cut , R, ω, z, µ, ν) = I ij (p cut T , R, ω, z, µ, ν) + ∆I ij (p cut T , η cut , R, ω, z, µ, ν) . (2.16) The η cut -independent I ij are given in app. A.2, and in the following we focus on the ∆I ij . Consistency between the cross sections in eqs. (2.8) and (2.12) implies that ∆I ij vanishes as η cut → ∞. Specifically, defining which is simply the statement from the previous subsection that the η cut effects are power suppressed in ζ cut for ζ cut 1. In fact, ∆I ij vanishes altogether for z > ζ cut /(1 + ζ cut ), which can be seen from purely kinematic considerations as follows: For the n-collinear sector the term ∆I ij accounts for the case where at least one jet with p jet T ≥ p cut T and η jet ≥ η cut is reconstructed (and no jet with η jet < η cut ). For R 1 all radiation in this jet has η ≥ η cut , as well. Thus, contributions to ∆I ij can only appear if where the second equality follows from the jets being massless for R 1. Rewriting this in terms of momentum fractions p − k = z k P − n = z k ω/z yields, with k z k + z = 1 and P − n the momentum of the initial state proton, The second inequality follows from all reconstructed n-collinear jets having η k > η cut . This implies that eq. (2.18) is trivially satisfied since the domain of integration in z scales as x ≤ z ζ cut . Hence ∆I ij is parametrically important for ζ cut ∼ z ∼ 1, but vanishes in the threshold limit z → 1. This leads to an additional numerical suppression due to the falloff of the PDFs towards larger partonic momentum fractions. The RGE of ∆I ij follows from the beam-function RGE eq. (2.13) and the analogue of the matching onto the PDFs in eq. (2.9). It is given by (with the remaining arguments of ∆I ij understood) The Mellin convolution ⊗ z is defined as and 2P ij (α s , z) is the standard PDF anomalous dimension with respect to µ, Note that the RGE in eq. (2.21) does not mix ∆I ij with I ij and therefore does not change the ζ cut scaling in eq. (2.18). Solving eq. (2.21) order by order in perturbation theory, we find the following structure through two loops: ij is the boundary condition of the RGE at µ = p cut T , ν = ω, and the required anomalous dimension coefficients are collected in app. A.1. By dimensional analysis and boost invariance, ∆I (n) ij can only depend on ζ cut = ωe −ηcut /p cut T in addition to R and z.
In app. A.3 we determine the one-loop contribution ∆I (1) ij , which has the simple form with the one-loop splitting functions P (0) ij (z) as given in eq. (A.6). The correction vanishes at the kinematic threshold encoded in the overall θ-function, which also cuts off the singular distributions in P (0) ij (z) at z = 1. The Mellin convolutions of ∆I (1) ik ⊗ z P (0) kj appearing in the coefficient of ln(µ/p cut T ) in ∆I (2) ij (z) are given in app. A.6. While the computation of the full two-loop contribution ∆I (2) ij is beyond the scope of this paper, we analytically compute its leading contribution in the small-R limit, which contains a clustering logarithm of R. We write the full two-loop result as ∆I (2) ij (ζ cut , R, z) = ln R ∆I (2.26) In the limit R 1, we exploit that for the emission of two close-by collinear partons with relative rapidity ∆η ∼ R, the collinear matrix element factorizes into two sequential collinear splittings at the scale µ ∼ p cut T and µ ∼ p cut T R, respectively. This allows us to evaluate the coefficient of ln R in a generic two-loop beam function as a convolution of a primary on-shell emission and (the anomalous dimension of) the semi-inclusive jet function of ref. [36]. Specifically, for the case of ∆I where the coefficient functions c R,cut ij are given by depending on whether the primary emission we split is a gluon (first line) or a quark (second line). Their explicit expressions read The coefficients c R ij in eq. (2.27) are the (in principle known) coefficients of ln R in the η cut -independent two-loop beam function [13,19], which we also verified. 2 They satisfy and are given by Our general setup for computing the small-R clustering contributions implies that the coefficient of the ln R terms of the two-loop rapidity anomalous dimension must be equal to c R gg = c R qq , in agreement with the corresponding result given in refs. [8,13]. In addition, it also applies to the leading ln 2 R and ln R terms in the beam functions for rapidity dependent jet vetoes in ref. [28], with which we agree as well.
The R-independent term ∆I (2,c) ik (ζ cut , z) and the O(R 2 ) terms in eq. (2.26) are currently unknown. Their contribution to the cross section can in principle be obtained numerically from the singular limit of the full-theory calculation at O(α 2 s ), as was done for the corresponding η cut -independent pieces in ref. [13].
Numerical validation To validate our results numerically and highlight the differences in the singular behaviour for regimes 1 and 2, we consider the fixed O(α s ) p jet T spectrum, dσ/dp jet T , where p jet T is the transverse momentum of the leading jet within |η jet | < η cut . Its relation to the jet veto cross section with a jet rapidity cut is simply At leading power in p jet T /Q, we obtain it by taking the derivative with respect to p cut T of either eq. (2.12), retaining the exact dependence on η cut in the beam functions (regime 2), or of eq. (2.8), incurring power corrections in Qe −ηcut /p jet T (regime 1). The numerical results for all singular spectra are obtained with the help of SCETlib [37]. The O(α s ) spectra in full QCD are obtained from MCFM 8.0 [38][39][40].
As representative gluon-induced processes, we consider gluon-fusion Higgs production gg → H at m H = 125 GeV in the infinite top-mass limit, rescaled with the exact LO top-mass dependence for m t = 172.5 GeV (rEFT). In addition, we consider gluon fusion to a hypothetical heavy color-singlet scalar X, gg → X, mediated by the contact operator We always choose m X = 1 TeV, Λ = 1 TeV, and divide the cross section by |C X | 2 . To the order we are working, this is equivalent to setting C X ≡ 1, since C X only starts to run at O(α 2 s ). 3 For quark-induced processes we consider Drell-Yan pp → Z/γ * → + − at the Z 3 In MCFM 8.0 we mock up this process using a standard-model Higgs with mH = 1 TeV and manually account for the nonzero one-loop contribution from integrating out the top quark in the SM, which differs from our choice of CX = 1 + O(α 2 s ) for the effective coupling of X to gluons. We also checked the results against the native gg → X support of SusHi 1.6.1 [41][42][43][44]. Here we set all scales to µ FO = m H , m X , or Q, respectively. We use PDF4LHC nnlo 100 [45][46][47][48][49][50] NNLO PDFs with α s (m Z ) = 0.118 throughout.
In fig. 3, we compare the regime 2 and regime 1 leading-power (singular) results for dσ/dp jet T at fixed p jet T as a function of η cut for gg → H and Drell-Yan. The regime 1 result (dashed blue) does not depend on η cut , while the regime 2 result (solid orange) decreases as η cut becomes more central. The difference between the two (dotted green) has the expected behaviour, vanishing as Qe −ηcut /p jet T for η cut → ∞. We observe that regime 1 is applicable beyond η cut 4, where the difference to regime 2 is suppressed by an order of magnitude.
Another check is provided by comparing the regime 1 and regime 2 singular results to the full QCD result, which is shown in figs. 4 and 5 for gluon-fusion and Drell-Yan. For η cut = 2.5 (left panels), it is clear that regime 1 (dashed gray) fails to describe the singular limit of full QCD, with their difference (dotted gray) diverging for p jet T → 0 like an inverse power of p jet T as expected. While the singular mismatch becomes less pronounced for η cut = 4.5 (right panels), the uncanceled singular contributions are still clearly visible in the difference. On the other hand, regime 2 (dashed blue) correctly reproduces the singular limit p jet T → 0, with the difference (dotted green) vanishing like a power of p jet T as it must.
This provides a strong check of the intricate p cut T dependence encoded in our O(α s ) results for ∆I ij . (The power corrections in e −ηcut , which are present in regime 2, drop out when taking the derivative of the fixed-order cumulant with respect to p cut T .) Note that at m X = 1 TeV or Q = 1 TeV, the fixed-order spectrum is completely dominated by the rapidity-cut dependent singular result up to p jet T 100 GeV. Hence, the resummation should provide a significant improvement over the fixed-order result for typical p cut T ∼ 50 GeV, which we will indeed find in sec. 4.

Regime 3: p cut
The hierarchy p cut T Qe −ηcut (with e −ηcut 1) exhibits different features than the regimes discussed before. The typical transverse momentum for emissions with |η| > η cut is parametrically Qe −|η| , indicated by the horizontal gray line in fig. 2, which is now much larger than for the strongly constrained emissions at |η| < η cut . While the soft modes at central rapidities are not affected, there are now two types of collinear modes at forward rapidities with |η| ∼ η cut , and analogously for the n b -collinear sector. The collinear and soft-collinear modes have the same angular resolution and only differ in their energy. This makes their all-order factorization challenging and leads to the appearance of nonglobal logarithms ln(Qe −ηcut /p cut T ) starting at O(α 2 s ). Their factorization and resummation requires the marginalization over all possible configurations of energetic collinear emissions, involving soft-collinear matrix elements with a separate Wilson line along each individual energetic collinear emission, see e.g. refs. [51][52][53][54].
Since this regime has no immediate phenomenological relevance, we will not carry out this complete procedure but restrict ourselves to the configuration with soft-collinear Wilson lines along n andn, i.e, along the two main collinear emitters. This is sufficient for the LL resummation, for isolating the nonglobal effects, and for discussing the relation to the other regimes. Our discussion here is in close analogy to the regime 3 in the factorization of the exclusive jet mass spectrum with small jet radius R in ref. [55], where the rapidity cut e −ηcut here takes the role of R there. 4 The factorized cross section takes the form The initial-state collinear functions B i encode the contributions of both soft-collinear and energetic collinear modes. They are related to the η cut dependent beam functions B i in eq. (2.12) by an expansion in the limit p cut T /(ωe −ηcut ) 1, Without further factorization, B i contains large unresummed Sudakov double logarithms α n s ln 2n (p cut T /ωe −ηcut ). To resum the leading double logarithms, we can decompose B i as The function B (cut) i mainly describes contributions from the energetic collinear modes. It was dubbed "unmeasured" beam function in refs. [33,34], in analogy to the unmeasured jet function [56]. At one loop its matching coefficients account for an energetic collinear emission with |η| > η cut . They are calculated in app. A.3 and read are the color-stripped LO splitting functions given in eq. (A.7), and the flavor structure is trivial, As argued in ref. [33] the results are directly related to the matching coefficients for fragmenting jet functions in ref. [57]. The function S (cut) i in eq. (2.37) mainly describes contributions from soft-collinear modes. At one loop it accounts for a soft-collinear emission that couples eikonally to the incoming collinear parton i. The emission is constrained to p T < p cut T for |η| < η cut by the jet veto, and is unconstrained for |η| > η cut . Using the η regulator [58,59] it is given by where C i = C F for an incoming quark or antiquark and C A for an incoming gluon. We checked explicitly that the above results obey the consistency constraint in eq. (2.36). For this purpose, one has to note that eq. (2.25) becomes distribution valued in (1 − z) when taking the limit ζ cut 1. At two loops S (cut) i contains a ln R enhanced term. Focusing on the constant terms not predicted by the RG evolution, we have an unknown two-loop constant. The coefficient of ln R is obtained by expanding the ln R coefficient in the η cut dependent beam function [see eqs. (2.27) and (A.21)] to leading power in 1/ζ cut . In the limit ζ cut 1, the sum I The anomalous dimensions of B where the coefficients of the cusp anomalous dimension follow from our explicit one-loop calculation. Consistency with eq. (2.14) implies All of the above noncusp anomalous dimensions vanish at one loop. The canonical scales for B With these choices and the anomalous dimensions in eq. (2.43) one may resum logarithms of e ηcut , p cut T /Q to any logarithmic order, and at LL also logarithms of p cut term in eq. (2.37) contains nonglobal logarithms of the form α n s ln n (p cut T /Qe −ηcut ). A boost by η cut translates the measurement into two hemispheres with one loose (η > η cut ) and one tight constraint (η < η cut ) on emissions. The nonglobal structure in such a scenario is well understood [60]. Depending on the desired accuracy, the NGLs may be included at fixed order via B (NG) i as indicated in eq. (2.37), or (partially) summed using more steps in a dressed parton expansion [53].
Note that beyond one loop there is some freedom in the choice of measurement that defines the B We stress that the result eq. (2.42) for the ln R coefficient in the two-loop soft-collinear function is, however, still unique. This is because a ln R contribution to B (NG) requires a collinear parton in the unconstrained region to emit a soft-collinear gluon into the constrained region, which then undergoes a further collinear splitting. This is only possible starting at O(α 3 s ).
Numerical validation To illustrate the numerical relevance of regime 3, we again consider the fixed O(α s ) p jet T spectrum. In regime 2, it is given to leading power in p jet T /Q by the derivative of eq. (2.12), while in regime 3, it is given to leading power in p jet T /(Qe −ηcut ) by the derivative of eq. (2.35).
In fig. 6 we compare the two results for η cut = 2.5. In regime 3, the 0-jet cross section at O(α s ) contains only single logarithms of p cut T , because the double logarithms cancel between the soft and soft-collinear functions. For this reason, the dashed-blue regime 3 spectrum with respect to ln p jet T is just a constant. The exact regime 2 result (solid orange) becomes well approximated by the further factorized regime 3 expression for p jet T → 0, with their difference (dotted green) behaving like a power in p jet T . This provides a strong check of the regime 3 ingredients, more precisely, of the p cut T dependence encoded in the soft-collinear function. (Since the beam function in regime 3 is independent of p cut T , it drops out when computing the fixed-order spectrum. ) We also observe that for gg → H and Drell-Yan at Q = m Z , the regime 3 limit is applicable only at very small p jet T 1 GeV and already at p jet T ∼ 10 − 20 GeV the power corrections with respect to regime 2 are of the same size as the full regime 2 result. This means that one would have to turn off the additional regime 3 resummation above this region. For gg → X with m X = 1 TeV and Drell-Yan at Q = 1 TeV, the canonical regime 3 resummation region, i.e., the region where the regime 3 singular corrections clearly dominate, extends up to p jet one would typically also apply higher jet thresholds to retain enough signal in the 0-jet bin. Realistically, one would not go below p cut T ∼ 0.1Q, which means one never enters the limit where the regime 3 resummation is necessary. This of course does not exclude the possibility that measurements designed to probe simultaneously very high Q and very low p jet T could benefit from the regime 3 resummation. To explicitly explore this regime experimentally, the best option is to restrict the jet veto to the very central region with η cut ∼ 1 − 1.5.

Comparison to the literature
Jet vetoes in a restricted rapidity range were already encountered in ref. [33] for the case of dijet production. Without spelling it out explicitly, ref. [33] used a factorization for the regime 3 hierarchy p cut T Qe −ηcut Q, but did not distinguish between the soft and soft-collinear modes necessary in this regime. As a result, parametrically large rapidity logarithms ln e ηcut were not captured, which are relevant starting at NLL. The numerical results in ref. [33] were obtained for Q ∼ 1 TeV, η cut = 5, and p cut T = 20 GeV, which rather corresponds to the opposite regime 1, p cut T Qe −ηcut . The difference between regimes 1 and 3 already matters at LL.
In ref. [34], the soft and soft-collinear modes in regime 3 are distinguished and the presence of nonglobal logarithms in this regime is recognized. Their factorization for dijet production is carried out at a level analogous to ours in the previous subsection. That is, at NLL and beyond it only captures logarithms of "global" origin, but does not capture nonglobal logarithms that are parametrically of the same size. Our results for the oneloop quark matching coefficients in eq. (2.38) and the one-loop soft-collinear function in eq. (2.40) agree with ref. [34] [see their eqs. Ref. [34] does not consider regime 2 as a separate parametric regime. Instead, it attempts to extend the validity of the regime 3 factorization into regime 2. This is done by effectively adding the regime 2 nonsingular corrections appearing in eq. (2.36) to the unmeasured beam functions. Since some of the regime 3 modes become redundant in regime 2, this also requires them to account for a nontrivial soft-collinear zero bin. At fixed order, the sum of all their contributions must reproduce our result for the regime 2 beam function; in app. A.5 we check that this is indeed the case for the quark matrix elements given in ref. [34]. As we have seen in fig. 6, outside the canonical regime 3, there are large cancellations between the terms that are singular in the regime 3 limit and the remaining regime 2 nonsingular contributions. This means that the distinction between these contributions becomes arbitrary in regime 2 and that they must not be treated differently, as otherwise one risks inducing large miscancellations. (This is completely analogous to the situation when matching to full QCD, in which case the p cut T resummation must be turned off when entering the fixed-order region at large p cut T to properly recover the full-QCD result.) In particular, in regime 2 all contributions that belong to the full η cut -dependent regime 2 beam function must be evaluated at a common scale µ p cut T and evolved together according to eq. (2.13). This is not the case in ref. [34], where individual contributions to the regime 2 beam function are evaluated at different scales throughout (µ cut B and µ cut S in our notation). Recently, the setup of ref. [34] was applied in ref. [35] to the case of transverse energy E T in a restricted rapidity range in Drell-Yan. In ref. [35], profile scales are used to combine regimes 3 and 1, requiring that asymptotically µ (cut) B = µ (cut) S in the regime 1 limit E T Qe −ηcut . While this can alleviate the issue raised above, formally this relation must be satisfied already in regime 2 for E T ∼ Qe −ηcut .
As we have seen in sec. 2.3, there is no need to distinguish collinear and soft-collinear modes in regime 2. Since for jet-veto analyses regimes 1 and 2 are the phenomenologically relevant ones, doing so unnecessarily complicates the description. Recovering the NNLL structure in regime 2 [see eq. (2.24)] based on regime 3 would be quite challenging due to the intricate nonglobal structure in regime 3. Our dedicated treatment of regime 2 makes the absence of nonglobal logarithms manifest, avoiding the associated complications, and automatically ensures the correct treatment of the regime 2 nonsingular terms. Furthermore, it shows how regime 2 generalizes the well-understood regime 1, and as we will see in the next section allows for the generalization to a step in the jet veto.
Concerning regime 1, ref. [35] also gave an argument that regime 1 holds up to power corrections in Qe −ηcut /E T , which was more intricate due to immediately comparing regime 1 to regime 3. The power suppression of η cut effects at sufficiently large η cut was also pointed out briefly in a somewhat different context in ref. [61].
3 Generalization to a step in the jet veto at η cut

Overview of parametric regimes
We now generalize our results to the experimentally relevant scenario of the step-like jet veto illustrated in the right panel of fig. 1. Here, jets with p jet T > p cut T are vetoed if |η jet | < η cut , while for |η jet | > η cut the veto is loosened to p jet T >p cut T > p cut T . The 0-jet cross section is thus defined by the following measurement: There are now three relevant power-counting parameters p cut T /Q,p cut T /Q, and e −ηcut with four distinct parametric regimes (assuming p cut T ≤p cut T ), illustrated in fig. 7: • p cut T /Q p cut T /Q ∼ e −ηcut (collinear NGLs, top right), • p cut T /Q ∼p cut T /Q e −ηcut (soft-collinear step, bottom left), • p cut T /Q p cut T /Q e −ηcut (soft-collinear NGLs, bottom right).
We discuss each of them in turn in the following subsections. For p cut T /Q ∼ e −ηcut (top left) the only relevant case isp cut T ∼ p cut T , leading to a modified measurement on the collinear modes, a collinear step, compared to the case without a step (p cut T = p cut T ). For p cut T /Q e −ηcut , we have to distinguish three cases depending onp cut T . Keeping p cut T ∼ e −ηcut implies the hierarchy p cut T /Q p cut T /Q ∼ e −ηcut (top right). Here, the mode setup is the same as for regime 3 without step (corresponding top cut T = ∞). As in that case, the large difference in the constraints on collinear radiation above and below η cut gives rise to collinear NGLs.
Forp cut T /Q e −ηcut , we can then have either p cut T /Q ∼p cut T /Q e −ηcut (bottom left) or p cut T /Q p cut T /Q e −ηcut (bottom right). For the former, the standard jet veto factorization is recovered except that there are additional soft-collinear modes that resolve the shallow step at η cut . For the latter, the steep step p cut T p cut T at η cut gives rise to two distinct sets of soft-collinear modes with parametrically large soft-collinear NGLs between them.

p cut
T /Q ∼p cut T /Q ∼ e −ηcut (collinear step) We first note that the hierarchy p cut T /Q ∼ e −ηcut p cut T /Q is effectively equivalent to the case without any jet veto beyond η cut (regime 2 in sec. 2.3). Since collinear emissions with |η| > η cut cannot resolve the loose veto atp cut T , its effect is suppressed by 1/p cut T and vanishes forp cut T → ∞. The first nontrivial hierarchy is p cut T /Q ∼p cut T /Q ∼ e −ηcut , illustrated in the top left panel of fig. 7. In this regime, the required modes are the same as in regime 2 in sec. 2.3. The collinear radiation resolves the step at η cut while soft emissions are insensitive to it, leading to a generalization of eq. (2.12), with the beam functions now additionally depending onp cut T . In analogy to eq. (2.16) we write the modified beam function matching coefficients as I ij (p cut T ,p cut T , η cut , R, ω, z, µ, ν) = I ij (p cut T , R, ω, z, µ, ν) + ∆I ij (p cut T ,p cut T , η cut , R, ω, z, µ, ν) . 3) The first term on the right-hand side is again the matching coefficient for a single veto at p cut T without any rapidity dependence. The second term is the correction due to the step in the jet veto at |η| = η cut , which vanishes for p cut T =p cut T . The correction is again renormalized according to eq. (2.21), which as before follows from RG consistency. In particular, its two-loop structure predicted by the RGE is the same as in eq. (2.24), where the finite terms now depend on two dimensionless ratios, The one-loop and ln R enhanced two-loop finite terms in ∆I ij can be written in terms of the results in eqs. (2.25) and (2.27) as ∆I (1) ij (ζ cut ,ζ cut , z) = ∆I since for a single (primary) n a -collinear emission at (η, p T ) the measurement function for the step correction can be rewritten as Due to the presence of correlated emissions with rapidities smaller and larger than η cut at two loops, this decomposition no longer applies for the full two-loop finite term ∆I (2,c) ij , which therefore needs to be determined separately.
This regime is free of nonglobal logarithms and is of direct phenomenological interest. The parametric assumptions are satisfied e.g. for high-mass searches, Q 300 GeV, a realistic rapidity cut η cut = 2.5, and veto parameters p cut T = 25 GeV,p cut T = 50 GeV, which clearly warrant resummation of logarithms of p cut T /Q ∼p cut T /Q.

Numerical validation
We now check that the factorized 0-jet cross section in eq. (3.2) reproduces the singular limit of full QCD. For this purpose, we construct an observable that simultaneously forces p cut T → 0 andp cut T → 0 as it approaches its singular limit. Following the rapidity-dependent jet vetoes in ref. [16], we define i.e., we can express the step veto by ordering the jets with respect to their weighted transverse momenta, where for η > η cut the corresponding step weight function f step (η) is given by the ratio of veto parameters, The differential spectrum in T step is then related to the jet-vetoed cross section with a step by the relation In fig. 8 we compare dσ(ρ, η cut )/dT step at fixed O(α s ) in full QCD to the singular spectrum predicted by eq. (3.2) as well as the standard factorization eq. (2.8) without a step for gg → H (left panel) and Drell-Yan at the Z pole (right panel). The singular result using the fullp cut T and η cut dependent beam functions (dashed blue) correctly reproduces the singular behaviour of full QCD (solid orange) in the limit T step → 0, with the difference to the full QCD spectrum (dotted green) vanishing like a power in T step as it should. On the other hand, the standard factorization without step (dashed gray) does not reproduce the correct singular behavior of full QCD, with the difference (dotted gray) diverging for T step → 0. Note that the mismatch here is reduced compared to thep cut T = ∞ case shown in figs. 4 and 5, owing to the larger phase space available to unconstrained radiation at |η| > η cut forp cut T = ∞. have the same renormalization as those in eq. (2.37), i.e., the additional dependence onp cut T does not change their renormalization. The associated matching coefficients at one loop are given by subtracting the correction term ∆I (1) ij in eq. (2.25), which accounts for an emission with η > η cut and p T >p cut T , from the coefficient I (cut) ij in eq. (2.38), which accounts for any emission with η > η cut without constraints from a jet veto, such that
The soft-collinear function S i encodes the actual step at η cut and is defined by the measurement eq. (3.1). Forp cut T = p cut T there is no step in the jet veto and S i has to vanish. The RG consistency of the cross section implies that its µ anomalous dimension vanishes in general, while its resummed ν anomalous dimension is given by (3.14) It does not depend on µ at all, as required by exact path independence in the (µ, ν) plane. Note that the beam functions in eq. (3.13) depend onp cut T (rather than p cut T ) because collinear radiation is too forward to be constrained by the tighter central veto. This is reflected in the somewhat curious rapidity anomalous dimension of S i in eq. (3.14), which accounts for the mismatch between the logarithms of p cut T andp cut T generated by the soft and beam rapidity evolution, respectively. Solving eq. (3.14) order by order in α s we find the following very simple structure of the soft-collinear function through two loops: (3.16) It is straightforward to check that the one-loop finite term vanishes (see app. A.4), The two-loop finite term is a generic function of the dimensionless ratio ρ and the jet radius parameter R, which must satisfy S i,2 (ρ = 1, R) = 0. As usual, we can decompose it according to its R dependence as where c R ii is given by eq. (2.31) and C i = C F (C A ) for i = q (g). The coefficient of ln R at this order is completely determined by the R dependence of the noncusp rapidity anomalous dimensions in eq. (3.14). The full two-loop finite term S i,2 (ρ, R) could readily be obtained numerically using the methods of refs. [62,63], which would enable the full NNLL resummation.
This regime is again free of nonglobal logarithms and hence can easily be applied to phenomenological studies. It can be used to supplement the EFT setup from sec. 3.2, which enables the resummation of logarithms of the ratio p cut T /Q ∼p cut T /Q, with an additional resummation of logarithms of the ratio p cut T /Qe −ηcut ∼p cut T /Qe −ηcut by choosing the canonical scales Numerical Validation To validate our setup in this regime, we exploit that eq. (3.13) provides a refactorization of the collinear step in eq. (3.2), where I ij (p cut T ,p cut T , η cut , R, ω, z, µ, ν) = S i (p cut T ,p cut T , η cut , R, µ, ν) I ij (p cut T , R, ω, z, µ, ν) In particular, eq. (3.13) must reproduce eq. (3.2) up to power corrections in p cut T /Qe −ηcut andp cut T /Qe −ηcut . We can test this numerically using the T step observable defined in sec. 3.2, which simultaneously probes both classes of power corrections. In fig. 9, we show the fixed O(α s ) T step spectra for the collinear step (solid orange) and soft-collinear step (dashed blue). In all cases their difference (dotted green) vanishes like a power in T step .
The additional resummation using the soft-collinear step may be applicable up to values of p cut T = 20 GeV (p cut T = 80 GeV) for Q ∼ 100 GeV (Q = 1 TeV), for the choice of ρ = 2, η cut = 2.5 displayed in fig. 9. This can be read off from the relative size of leading-power (soft-collinear step) and subleading power (difference) contributions, which leave some room where resummation in the leading-power cross section can improve the prediction. We find a slightly larger potential resummation region than for the analogous refactorization in thep cut T = ∞ case, where an earlier onset of the power corrections was observed in fig. 6.

p cut
T /Q p cut T /Q e −ηcut (soft-collinear NGLs) For this hierarchy (bottom right panel of fig. 7), two types of soft-collinear modes arise, n a -soft-collinear (p cut T ): p µ ∼ p cut T (e −ηcut , e ηcut , 1) , n a -soft-collinear (p cut T ): p µ ∼p cut T (e −ηcut , e ηcut , 1) , (3.21) and analogously for the n b -soft-collinear sectors, which are both parametrically distinct from the energetic collinear modes. Compared to the regime p cut T ∼p cut T Qe −ηcut there are now parametrically large logarithms ln(p cut T /p cut T ) in the soft-collinear function S i in eq. (3.13). The cross section can be written as in eq. (3.13), where the soft-collinear function is refactorized as . Evolving the two soft-collinear functions to separate renormalization scales µ S,1 = p cut T , ν S,1 = p cut T e ηcut and µ S,2 =p cut T , ν S,2 =p cut T e ηcut resums Sudakov logarithms of p cut T /p cut T , but does not account for the nonglobal logarithms of the same ratio in S (NG) i .

Numerical results
In sec. 2 we discussed in detail how to incorporate the jet rapidity cut into the resummed 0jet cross section. In particular, in the regime p cut T /Q ∼ e −ηcut (regime 2), the dependence on η cut is incorporated into the resummation via the RG evolution of the η cut dependent beam functions. In this section, we illustrate these results by presenting numerical predictions for the resummed cross section at NLL +NLO.
In sec. 4.1, we outline how the resummed results are combined with the full QCD results, as well as our estimation of perturbative uncertainties. In sec. 4.2, we assess the impact of the additional perturbative ingredients by comparing the different treatments of η cut . In sec. 4.3, we show the predictions for selected η cut as a function of p cut T . In the following, we consider the four cases of gluon-fusion Higgs production gg → H at m H = 125 GeV, gluon fusion to a generic heavy scalar gg → X with m X = 1 TeV, and Drell-Yan production at Q = m Z and Q = 1 TeV, with the same setup and inputs as described in sec. 2.3. The numerical results for the resummed predictions for all processes are obtained from our implementation in SCETlib [37]. The NLO results in full QCD are obtained from MCFM 8.0 [38][39][40].

Fixed-order matching and perturbative uncertainties
The resummed cross section obtained from eq. (2.12) describes the 0-jet cross section up to power corrections in p cut T /Q, which become relevant when p cut T ∼ Q. We account for them by the usual additive matching, Here, σ res 0 is the resummed singular cross section obtained from eq. (2.12), σ sing 0 is its fixed-order expansion, and σ FO 0 is the fixed-order result in full QCD. By construction, the difference in square brackets is nonsingular and vanishes as p cut T → 0, η cut → ∞ and can therefore be included at fixed order even at small p cut T . The dominant corrections at small p cut T are resummed in σ res 0 . At large p cut T , fixed-order perturbation theory is the appropriate description, so eq. (4.1) should recover σ FO 0 . This is achieved by turning off the resummation in σ res 0 as a function of p cut T , and by constructing σ res 0 such that it precisely reproduces σ sing when the resummation is fully turned off.
To smoothly turn off the resummation as we approach p cut T → Q, we use profile scales [64,65], following the setup developed in ref. [13]. We stress that the profile scales for regime 2 are in one-to-one correspondence with the standard treatment in regime 1, since both regimes have the same RG structure. Similarly, our treatment of perturbative uncertainties is based on profile scale variations following ref. [13]. We distinguish an overall yield uncertainty ∆ µ0 , which is determined by a collective variation of all scales up and down, and a resummation (jet bin migration) uncertainty ∆ resum from varying individual scales in the beam and soft functions. For the gluon-induced processes, we follow ref. [66] and include an additional uncertainty ∆ ϕ from varying the complex phase of the hard scale, which was not considered in ref. [13]. The total uncertainty is then obtained by considering the different uncertainty sources as independent, and hence uncorrelated, and adding them in quadrature,

Comparing different treatments of the jet rapidity cut
It is interesting to consider the impact of the additional perturbative ingredients in the η cut dependent beam function on the prediction, e.g. compared to treating the rapidity cut effects purely at fixed order. In figs. 10 and 11, we plot the results for fixed p cut T as a function of η cut starting at η cut = ∞ on the left and decreasing toward the right. The corresponding values of the Qe −ηcut scale are shown at the top.
Our result for the 0-jet cross section using the matching in eq. (4.1) is shown as orange bands. We refer to this prediction as NLL (η cut )+NLO(η cut ), because both the NLL resummed singular cross section and the fixed-order matching are exact in η cut . To highlight the effect of the additional η cut dependence in the regime 2 beam function, we consider two more alternative treatments of η cut . For the regime 1 result, shown by the blue bands and denoted by NLL (∞)+NLO(η cut ), the η cut dependence in the resummed cross section is dropped, σ 0 (p cut T , η cut ) = σ res 0 (p cut T , ∞) + σ FO 0 (p cut T , η cut ) − σ sing 0 (p cut T , ∞) . The resummation then only acts on the singular cross section for η cut = ∞, while all η cut effects are included purely at fixed order via the matching term in square brackets. Note that the matching term is now no longer nonsingular, i.e., it no longer vanishes like a power in p cut T as p cut T → 0, as we saw in figs. 4 and 5. The plain fixed-order calculation without any resummation, is denoted by NLO(η cut ) and shown by the gray bands. In this case, the uncertainties are evaluated using the ST procedure [3]. We first consider gluon-fusion Higgs production shown in the left panel of fig. 10, where we set p cut T = 30 GeV. The NLO(η cut ) prediction (gray band) exhibits a slight, physical rise in the cross section as η cut decreases towards the right. This is not surprising as at fixed order, decreasing η cut simply amounts to accumulating the squared LO 1 matrix element over a larger part of phase space. The rise is less pronounced than for the resummed results (orange and blue bands), but still compatible with them within each others' uncertainties. Comparing NLL (η cut )+NLO(η cut ) (orange) to NLL (∞)+NLO(η cut ) (blue) we find that the additional tower of logarithms predicted by NLL (η cut ) on top of the fixed NLO η cut dependence barely affects the central value of the prediction down to η cut = 2. This is perhaps not surprising since Qe −ηcut is at most half of p cut T , which means we are not far from regime 1. However, we do observe a noticeable increase in the perturbative uncertainty estimate. This is mainly due to the resummation uncertainty, which is reasonable: ∆ resum probes the unknown higher-order finite terms (the RGE boundary condition) and is therefore sensitive to a change of the beam function boundary condition by the η cut correction ∆I (1) ij (see sec. 2.3). On the other hand, ∆I (1) ij must be large enough to accommodate -up to power corrections -the fixed-order difference to η cut = ∞ (roughly 2 pb at η cut = 2.5, as can be read off from the gray line), so we expect an impact on ∆ resum of similar size. Hence, the conclusion is not that the NLL (∞)+NLO(η cut ) result is more precise, but rather that its uncertainty is potentially underestimated because it cannot capture the η cut dependence.
In the right panel of fig. 10, we show the same results for a hypothetical color-singlet scalar resonance gg → X at m X = 1 TeV using p cut T = 50 GeV. [The dimension-five operator mediating the production of X is given in eq. (2.33).] The NLO(η cut ) result (gray) is now off by a large amount already at η cut = ∞, where it is not covered by the resummed predictions. This is expected because the high production energy of 1 TeV implies we are deep in the resummation region, even for the larger value of p cut T = 50 GeV. The central values of the two resummed treatments start to differ below η cut = 3 or above Qe −ηcut 50 GeV, where we are now fully in regime 2. However, the main difference is again the larger and likely more reliable uncertainty estimate in the NLL (η cut ) prediction.
In fig. 11 we show the analogous results for Drell-Yan production at Q = m Z using p cut T = 20 GeV (left panel) and Q = 1 TeV using p cut T = 25 GeV (right panel). For better readability, these results are normalized to the resummed 0-jet cross section at η cut = ∞. While all predictions agree in the slope of the cross section with respect to η cut , the NLO(η cut ) result has a constant offset and an unrealistically small uncertainty estimate. At the lower Q ∼ 100 GeV, we find practically no difference between the NLL (η cut ) and NLL (∞) calculations, so here the effects of the jet rapidity cut can safely be included via the fixed-order matching corrections to the regime 1 resummation. At higher production energies, the intrinsic NLL (η cut ) ingredients become more relevant, similar to gluonfusion, as shown by the increasing uncertainty estimates as η cut decreases. Note that below η cut = 2.5, Qe −ηcut 80 GeV becomes large compared to this choice of p cut T = 25 GeV, so resumming logarithms of p cut T /(Qe −ηcut ) using the regime 3 factorization given in sec. 2.4 might help reduce the uncertainties.

Resummed predictions with a sharp rapidity cut
Here, we compare predictions for different values of η cut as a function of p cut T . Our working order is NLL (η cut )+NLO(η cut ) in the notation of the previous section, which from now on we simply refer to as NLL +NLO, i.e., the η cut dependence is always included in the resummation. We stress that the differences we observe between predictions in this subsection are physical differences due to the different jet rapidity cuts, and not due to different theoretical treatments as in the previous subsection.
In fig. 12 and table 1 we present results for gg → H. Going from η cut = ∞ to η cut = 4.5 we find a 1% increase of the cross section for the typical values of p cut T = 25 GeV and 30 GeV. At η cut = 2.5 the increase becomes more sizable, 14% (19%) for p cut T = 30 GeV (25 GeV). The differences vanish as the cross section saturates around p cut T ∼ 100 GeV. The analogous results for gg → X for m X = 1 TeV are shown in fig. 13 and table 2. At such a high hard scale, the uncertainties for η cut = 2.5 become essentially beyond control for very tight vetoes p cut T 25 GeV, which would make an additional resummation of ln p cut T /(Qe −ηcut ) as outlined in sec. 2.4 necessary. As we will see in the next subsection, this effect can be tamed by replacing the sharp rapidity cut by a step in the jet veto. However, for any choice of η cut the cross section is very strongly Sudakov suppressed for such small values of p cut T . At more realistic values of the veto, the jet rapidity cut for η cut = 2.5  Figure 13. 0-jet cross section σ 0 (p cut T , η cut ) for gg → X for m X = 1 TeV at NLL +NLO for different values of η cut . The bands indicate the total uncertainty ∆ µ0 ⊕∆ ϕ ⊕∆ res . The absolute cross section is shown on the left. On the right, the same results are shown as the percent difference relative to the 0-jet cross section at η cut = ∞.  compared to η cut = ∞ still leads to a sizable increase of 20% (5%) for p cut T = 50 GeV (p cut T = 100 GeV). In contrast, the effect for η cut = 4.5 is very small.  Figure 14. The 0-jet cross section dσ 0 (p cut T , η cut )/dQ for Drell-Yan production at the Z pole Q = m Z (top row) and at Q = 1 TeV (bottom row) at NLL +NLO for different values of η cut . The bands indicate the total uncertainty ∆ µ0 ⊕ ∆ res . The absolute cross section is shown on the left. On the right, the same results are shown as the percent difference relative to the 0-jet cross section at η cut = ∞.
The results for Drell-Yan production are given in fig. 14 and table 3. For Q = m Z (top rows), we find a 5 − 7% increase in the cross section at η cut = 2.5 for p cut T = 20 − 25 GeV. Here the uncertainty for η cut = 2.5 is under good control even down to p cut T ∼ 10 GeV. For Q = 1 TeV (bottom rows), the cross section for η cut = 2.5 increases by 14% (4%) for p cut T = 25 GeV (50 GeV) compared to η cut = ∞. The Sudakov suppression in the cross section and the accompanying increase in the uncertainties at small p cut T are weaker than for gg → X due to the smaller color factor (C F vs. C A ) in the Sudakov exponent, although they are still substantial for a quark-induced process. The effect of the rapidity cut at η cut = 4.5 is negligible.

Resummed predictions with a step in the jet veto
In the previous subsection we have seen that a sharp rapidity cut at η cut = 2.5 can lead to a substantial loss of precision in the theory predictions, especially for gluon-induced processes and at high production energies.
In fig. 15 we show the resummed 0-jet cross section for gg → H and gg → X with a step in the jet veto at η cut = 2.5 as a function of the second jet veto parameterp cut T that is applied beyond η cut . The central jet veto below η cut is fixed to p cut T = 25 GeV. On the left of the plotp cut T = p cut T , which is equivalent to having no rapidity cut, in which case the uncertainties are well under control. In the limitp cut T → ∞ (towards the right) the step becomes a sharp cut, corresponding to the results of the previous subsection. While the step in the jet veto still leads to an increase in the uncertainties, this can now be controlled by the choice ofp cut T . At this order, a small step from p cut T = 25 GeV top cut T = 30 GeV only leads to a small increase in uncertainty. For a larger step top cut T = 50 GeV = 2p cut T , the uncertainties already increase substantially but are still much smaller than for a sharp cut.

Conclusion
We have developed a systematic framework to seamlessly incorporate a cut on the rapidity of reconstructed jets, |η jet | < η cut , into the theoretical description of jet-vetoed processes at the LHC. We have shown that the standard jet-veto resummation, which neglects the rapidity cut, is correct up to power corrections of O(Qe −ηcut /p cut T ), with Q the hard-interaction scale and p cut T the jet veto cut. We calculated the necessary η cut -dependent corrections at one loop as well as all logarithmic contributions to them at two loops (including both small-R clustering logarithms and all jet-veto logarithms predicted by the RGE; see sec. 2.3). The remaining ingredients required for a full NNLL analysis with η cut effects are finite nonlogarithmic pieces that could be either calculated explicitly or extracted numerically from the full-QCD results, which we leave to future work. In addition, we considered for the first time the case of a step in the jet veto, i.e., an increase in the veto parameter top cut T > p cut T beyond η cut , and showed how to similarly incorporate it into the jet-veto resummation (see sec. 3.2).
We also considered the jet veto cross section in the limit p cut T Qe −ηcut , corresponding to either very tight vetoes or very central rapidity cuts (see sec. 2.4). In this regime, the jet-veto resummation becomes impaired by the presence of nonglobal logarithms, requiring a refactorization of the cross section. However, we have argued that this parametric region will most likely not play a role for typical jet binning analyses at the LHC. If experimentally necessary, it can be avoided by replacing the sharp rapidity cut by a moderate step in the jet veto, which is free of nonglobal logarithms (see sec. 3.4).
There are several important outcomes of our analysis. First, a jet rapidity cut at very forward rapidities due to the finite detector acceptance, η cut 4.5, is theoretically safe and unproblematic. In contrast, restricting the jet veto to the more central region, with a sharp rapidity cut at the end of the tracking detectors, η cut 2.5, leads to an increase in the perturbative uncertainties (which may not be captured if the jet rapidity cut is not included in the resummation). This loss in theoretical precision can become particularly severe for gluon-induced processes and for processes at high scales. It can however be mitigated by replacing the sharp rapidity cut by a moderate step in the jet veto. We expect this to be a generic feature that also holds at higher orders. It will be interesting to extend our resummed predictions to the next order (NNLL ) to confirm this as well as to reduce the overall size of the theoretical uncertainties. We encourage our experimental colleagues to take full advantage of such step-like jet vetoes in order to benefit from suitably tight jet vetoes at central rapidities, while avoiding the increased pile-up contamination in the forward region.

A Perturbative ingredients
We collect known results for required anomalous dimensions in app. A.1 and for the standard p cut T beam function without a jet rapidity cut in app. A.2. In app. A.3 we provide some details on the computation of the one-loop beam function matching coefficients in eqs. (2.25) and (2.38). In app. A.4 we compute the soft-collinear functions given in eqs. (2.40) and (3.15). In app. A.5 we compare to the one-loop results of ref. [34]. In app. A.6 we discuss the Mellin convolutions required in the two-loop η cut dependent beam function in eq. (2.24).

A.1 Anomalous dimensions
We expand the β function of QCD as with the one-loop and two-loop coefficients in the MS scheme given by The cusp and all noncusp anomalous dimensions γ(α s ) are expanded as The coefficients of the MS cusp anomalous dimension through two loops are Γ q n = C F Γ n , Γ g n = C A Γ n , (for n = 0, 1, 2) , The PDF anomalous dimension in eq. (2.23) is expanded as Note that we expand the PDF anomalous dimension in α s /(4π) and not α s /(2π) as is often done. The one-loop coefficients of the PDF anomalous dimension read in terms of the standard color-stripped one-loop QCD splitting functions The two-loop coefficients were calculated in refs. [67][68][69]. They can be decomposed as where explicit expressions for the P 1 functions on the right-hand side can be found in appendices A of refs. [70,71]. [Note that in refs. [70,71] the superscript "1" here is written as "(1)" there, and the PDF anomalous dimension is expanded there in α s /(2π), which is already accounted for by the overall factors of 4 on the right-hand side of eq. (A.8).] Explicit results for the Mellin convolutions of two color-stripped leading-order splitting functions can also be found there. The coefficients of the noncusp beam anomalous dimension are [13,24] γ q B 0 = 6C F , γ q B 1 = C F (3 − 4π 2 + 48ζ 3 )C F + −14 + 16(1 + π 2 ) ln 2 − 96ζ 3 C A The coefficients of the rapidity noncusp anomalous dimension depend on the jet radius R. They read [13] γ i ν 0 (R) = 0 , (A.10) − (1 + π 2 ) ln 2 + ζ 3 C A + 4 9 + π 2 12 − 5 3 ln 2 β 0 + C i 2 (R) .
Here C i = C F (C A ) for i = q (g) and C i 2 (R) is the clustering correction due to the jet algorithm relative to a global E T veto, as computed in refs. [8,13], The small-R clustering coefficient c ii = c gg = c qq is given in eq. (2.31).

A.6 Mellin convolutions in the two-loop rapidity dependent beam function
The PDF and beam function RGEs together predict Mellin convolutions of the following form in the two-loop matching kernels eq. (2.24) for the rapidity dependent beam function: The relevant partonic channels read, leaving all arguments implicit, [∆I (1) ⊗ P (0) ] q i q j = δ ij 8C 2 F P w qq ⊗ z P qq + 8T F C F P w qg ⊗ z P gq = [∆I (1) ⊗ P (0) ]q iqj , [∆I (1) ⊗ P (0) ] q i g = 8C F T F P w qq ⊗ z P qg + 8T F C A P w qg ⊗ z P gg + β 0 2 P w qg = [∆I (1) ⊗ P (0) ]q i g , where n f is the number of light quark flavors. Here we introduced a shorthand for weighted color-stripped splitting functions that depend on ζ cut in addition to z, The Mellin convolutions P w ik ⊗ z P kj are straightforward to evaluate analytically, but the resulting expressions are lengthy. They are available from the authors upon request.

B Jet rapidity cuts in T B and T C vetoes
Here we comment on how the factorization setup for the smoothly rapidity dependent jet vetoes introduced in ref. [16] is modified when an additional sharp jet rapidity cut is introduced. The restriction on reconstructed jets reads in this case max k∈jets: |η k |<ηcut | p T,k | f (η k ) < T cut , (B.1) where f (η)e |η| → 1 for η → ±∞. Examples are the beam thrust veto with f (η) = e −|η| and the C-parameter veto with f (η) = 1/(2 cosh η). The discussion of an additional sharp rapidity cut largely parallels the case of the p cut T veto in sec. 2. We again distinguish three hierarchies between T cut /Q and e −ηcut , where now T cut /Q replaces p cut T /Q as the characteristic angular size of collinear radiation constrained by the jet veto. The hierarchy T cut /Q e −ηcut (regime 1) reduces to the factorization for η cut → ∞ [8,16,28], up to power corrections of O(e −ηcut Q/T cut ).
For T cut /Q ∼ e −ηcut (regime 2) the relevant EFT modes scale as soft: p µ ∼ (T cut , T cut , T cut ) , n a -collinear: p µ ∼ T cut , Q, T cut Q ∼ Qe −2ηcut , Q, Qe −ηcut , n b -collinear: p µ ∼ Q, T cut , T cut Q ∼ Q, Qe −2ηcut , Qe −ηcut . (B. 2) The factorized 0-jet cross section reads σ 0 (T cut , η cut , R, Φ) = H κ (Φ, µ) B a (T cut , η cut , R, ω a , µ) B b (T cut , η cut , R, ω b , µ) S κ (T cut , R, µ) 3) The beam and soft function are different from the p cut T veto. The rapidity cut again affects only the beam functions without changing their RG structure or anomalous dimension. In analogy to eq. (2.16) we can write the matching coefficients as I ij (T cut , η cut , R, ω, z, µ) = I ij (ωT cut , R, z, µ) + ∆I ij (T cut , η cut , R, ω, z, µ) , (B.4) where the first term on the right-hand side is the η cut → ∞ matching coefficient as calculated to two loops in ref. [28], which only depends on the boost-invariant product ωT cut . The correction ∆I ij vanishes for ωe −2ηcut T cut and at one loop is given by ∆I ij (T cut , η cut , R, ω, z, µ) = α s (µ) 4π θ ωe −2ηcut ωe −2ηcut + T cut − z P The contributions from these modes can be encoded in a function B i which can be refactorized in analogy to eq. (2.37) to resum Sudakov logarithms of T cut e 2ηcut /Q, B i (T cut , η cut , R, ω, z, µ) = B Here, the η cut dependent and T cut independent piece B The B (NG) i piece, which contains nonglobal logarithms starting at O(α 2 s ), is again different from the one in eq. (2.37). We verified that, up to power corrections, the explicit oneloop expressions in eqs. (2.38) and (B.8) reproduce the sum of eq. (B.5) and the matching coefficients without a rapidity cut given in app. B of ref. [16].