Higgs transverse momentum with a jet veto: a double-differential resummation

We consider the simultaneous measurement of the Higgs ($p_t^{\tiny{\mbox{H}}}$) and the leading jet ($p_t^{\tiny{\mbox{J}}}$) transverse momentum in hadronic Higgs-boson production, and perform the resummation of the large logarithmic corrections that originate in the limit $p_t^{\tiny{\mbox{H}}}\,,p_t^{\tiny{\mbox{J}}} \ll m_{\tiny{\mbox{H}}}$ up to next-to-next-to-leading-logarithmic order. This work constitutes the first resummation for a double-differential observable involving a jet algorithm in hadronic collisions, and provides an important milestone in the theoretical understanding of joint resummations. As an application, we provide precise predictions for the Higgs transverse-momentum distribution with a veto $p_t^{\tiny{\mbox{J}}} \leq p_{t}^{\tiny{\mbox{J,v}}}$ on the accompanying jets, whose accurate description is relevant to the Higgs precision programme at the Large Hadron Collider.

The thorough scrutiny of the properties of the Higgs boson [1,2] is central to the future physics programme of the Large Hadron Collider (LHC). In the High-Luminosity run of the LHC, the experimental precision in Higgs-related measurements will increase significantly [3], hence allowing for detailed studies of the Higgs sector of the Standard-Model (SM) Lagrangian.
A full exploitation of such measurements requires an unprecedented level of precision in the theoretical description of the relevant observables. In this context, a prominent role is played by kinematic distributions of the Higgs boson and the accompanying QCD radiation, which are sensitive to potential new-physics effects, such as modifications of light-quark Yukawa couplings [4,5], or heavy new-physics states [6][7][8][9][10][11]. Experimental analyses of Higgs processes typically categorise the collected events in jet bins, according to the different number of jets -collimated bunches of hadrons in the final state -produced in association with the Higgs boson. Since the future performance of the LHC will allow for the precise measurement of kinematic distributions in different jet bins, it is paramount to achieve an accurate theoretical understanding of Higgs observables at the multidifferential level.
In this letter we consider Higgs-boson production in gluon fusion, the dominant channel at the LHC, and we focus on the Higgs transverse-momentum (p H t ) spectrum in the presence of a veto p J,v t bounding the transverse momentum p J t of the hardest accompanying jet. Veto constraints of such a kind are customarily enforced to enhance the Higgs signal with respect to its backgrounds, relevant examples being the selection of H → W + W − events from tt → W + W − bb production [12,13] or the categorisation in terms of different initial states [14].
Fixed-order perturbative predictions of the p H t spectrum in gluon fusion are currently available at nextto-next-to-leading order (NNLO) in the strong coupling α s [15][16][17][18][19] in the infinite top-mass limit, and heavyquark mass effects are known up to next-to-leading order (NLO) [20][21][22][23][24]. Fixed-order perturbation theory is, however, insufficient to accurately describe the observable considered here. When exclusive cuts on radiation are applied, it is well known that the convergence of the perturbative expansion is spoiled by the presence of log- where the Higgs mass m H represents the typical hard scale of the considered process. In this regime, such large logarithmic terms must be summed to all perturbative orders to obtain a reliable theoretical prediction. The resummation accuracy is commonly defined at the level of the logarithm of the cumulative cross section, where terms of order α n s n+1 are referred to as leading logarithms (LL), α n s n as nextto-leading logarithms (NLL), α n s n−1 as next-to-next-toleading logarithms (NNLL), and so on. The resummation of the inclusive p H t spectrum has been carried out up to high perturbative accuracy [25][26][27][28] and is currently known to N 3 LL order [29,30]. Such calculations have been combined with NNLO fixed order in refs. [29][30][31] to obtain an accurate prediction across the whole p H t spectrum. Similarly, the resummation of the jet-vetoed cross section has been achieved in refs. [32][33][34][35][36][37][38], reaching NNLL accuracy matched to N 3 LO [39].
In this work, we present the first joint resummation of both classes of logarithms, by obtaining a prediction which is differential in both p H t and p J t , and NNLL accurate in the limit p H t , p J t m H . Specifically, we integrate the double-differential distribution dσ/dp J t dp H t over p J t up to p J t = p J,v t , which results in the single-differential p H t distribution with a jet veto. The results presented here are of phenomenological relevance in the context of the Higgs physics programme at the LHC, and constitute an important milestone in the theoretical understanding of the structure of resummations of pairs of kinematic observables, which has received increasing interest lately [40][41][42]. Other kinds of joint resummations for hadronic Higgs production have been considered in the literature. Relevant examples are combined resummations of logarithms of p H t and small-x [43,44], of p H t and large-x [45][46][47][48], of small-x and large-x [49], and of p J,v and the jet radius [39].
To derive the main result of this letter, it is instructive to first consider the standard transverse-momentum resummation [50,51], starting with a description of the effects that enter at NLL in a toy model with scaleindependent parton densities. The core of the inclusive p H t resummation lies in the description of soft, collinear radiation emitted at disparate angles off the initial-state gluons. Observing that in such kinematic configurations each emission is independent of the others, one obtains the following formula in impact-parameter (b) space where σ 0 denotes the Born cross section, and [dk i ]M 2 (k i ) is the phase space and squared amplitude for emitting a parton of momentum k i . The exponential factor in eq. (1) encodes in a factorised form the kinematic constraint δ 2 ( p H t − n i=1 k t,i ), while the −1 term in the round brackets arises because, by unitarity, virtual corrections come with a weight opposite to that of the real emissions, but do not contribute to p H t . The factorisation of the phase-space constraint allows for an exact exponentiation of the radiation in eq. (1), leading to the well known formula of refs. [50,51].
In order to include the constraint due to a veto on accompanying jets, let us first consider the effect of a jet algorithm belonging to the k t -type family. 1 Owing to the strong angular separation between the emissions, the clustering procedure at NLL will assign each emission to a different jet [32]. Therefore, imposing a veto p J,v t on the resulting jets corresponds to constraining the real radiation with an extra factor Plugging the above equation into eq. (1) leads to where the radiator S NLL reads [32] 1 An example within this family is the anti-kt algorithm [52].
To evaluate the above integral, we can perform the integration over the rapidity of the radiation k and obtain with where α CMW s (k t ) indicates the strong coupling expressed in the CMW scheme [53][54][55], which ensures the correct treatment of non-planar soft corrections at NLL accuracy in processes with two hard emitters, and β 0 is the first coefficient of the QCD beta function. The azimuthal integral of eq. (4) leads to In the first integral, we exploit the large-b property [29,34] J with b 0 = 2e −γ E , to recast eq. (6) as where α s ≡ α s (µ R ) (with µ R being the renormalisation scale), L = ln(m H b/b 0 ), and the g i functions are those used in the standard p H t resummation [56]. The procedure that led to eq. (3) can be used to extend the above result to higher logarithmic orders. The crucial observation is that, as already stressed, in impactparameter space the measurement function for p H t is entirely factorised, resulting in a phase factor e i b· kt for each emission k. This implies that the jet-veto constraint Θ(p J,v t − p J t ) can be included by implementing the jetveto resummation [34] at the level of the b-space integrand, namely directly in impact-parameter space. We now describe the derivation of the NNLL result.
Starting from eq. (3), the first step is to promote the R NLL (k t ) function that appears in the radiator S NLL to NNLL. The corresponding expression is given in refs. [29,34], and leads to The above step assumes that the veto on the radiation is encoded in a phase-space constraint of the type (2). While this approximation is correct at NLL, where the jet algorithm does not recombine the emissions with one another, it fails beyond this order. Specifically, at NNLL, at most two soft emissions can become close in angle, and therefore may get clustered into the same jet (whose momentum is defined according to the so-called E-scheme, where the four momenta of the constituents are added together). The configurations in which the resulting cluster is the leading jet are not correctly described by the constraint in (2). In order to account for this effect, one has to include a clustering correction [34] in impact parameter space, that reads where k t,ab = k t,a + k t,b and k t,ab is its magnitude. The constraint J ab (R) = Θ R 2 − ∆η 2 ab − ∆φ 2 ab restricts the phase space to the region where the recombination between the two emissions takes place. Here R is the jet radius and ∆η ab and ∆φ ab are the pseudo-rapidity and azimuthal separation between the two emissions, respectively. We observe that eq. (10) differs from the corresponding clustering correction for the standard jet-veto resummation [34] by the factor e i b· k t,ab , which accounts for the p H t constraint in impact-parameter space. Eq. (10) describes the clustering correction due to two independent soft emissions. A similar correction arises when the two soft emissions k a , k b are correlated, i.e. their squared matrix element cannot be factorised into the product of two independent squared amplitudes. The contribution of a pair of correlated emissions is accounted for in the CMW scheme for the strong coupling that was already used in the NLL radiator (4). However, such a scheme is obtained by integrating inclusively over the correlated squared amplitudeM 2 (k a , k b ), given in ref. [57]. While this inclusive treatment is accurate at NLL, at NNLL one needs to correct for configurations in which the two correlated emissions are not clustered together by the jet algorithm. This amounts to including a correlated correction [34] of the form The corrections (10) and (11) describe the aforementioned effects for a single pair of emissions. At NNLL, all remaining emissions can be considered to be far in angle from the pair k a , k b , and therefore they never get clustered with the jets resulting from eqs. (10), (11). As a final step towards a NNLL prediction, one must account for non-soft collinear emissions off the initialstate particles. Since a k t -type jet algorithm never clusters the soft emissions discussed above with non-soft collinear radiation, the latter can be conveniently handled by taking a Mellin transform of the resummed cross section. In Mellin space, the collinear radiation gives rise to the scale evolution of the parton densities f (µ) and of the collinear coefficient functions C(α s ). The latter, as well as the hard-virtual corrections H(α s ), must be included at the one-loop level for a NNLL resummation. The equivalent of the clustering and correlated corrections for hard-collinear radiation enters only at N 3 LL, and therefore is neglected in the following.
After repeating the same procedure outlined for the soft radiation we obtain the main result of this letter, namely the NNLL master formula for the p H t spectrum with a jet veto p J,v t , differential in the Higgs rapidity y H : , where x 1,2 = m H / √ s e ±y H , and M gg→H is the Born matrix element. The ν subscripts denote the Mellin transform, while the latin letters represent flavour indices, and the sum over repeated indices is understood. Here Γ ν and Γ (C) ν are the anomalous dimensions describing the scale evolution of the parton densities and coefficient functions, respectively. The contours C 1 and C 2 lie parallel to the imaginary axis to the right of all singularities of the integrand. The path-ordering symbol P has a formal meaning, and encodes the fact that the evolution operators are matrices in flavour space. All the ingredients of eq. (12) are given in ref. [56]. The multi-differential distribution dσ/dp J t dy H d 2 p H t is simply obtained by taking the derivative of eq. (12) in p J,v t . All integrals entering the above formula are finite in four dimensions and can be evaluated numerically to very high precision. We point out that, similarly to the standard p H t resummation [28,29], the result in eq. (12) can also be deduced directly in momentum space, without resorting to an impact-parameter formulation. The momentum-space approach is particularly convenient for computational purposes, in that it gives access to dif- ferential information on the QCD radiation, thereby enabling an efficient Monte Carlo calculation. Therefore, we adopt the latter method for a practical implementation of eq. (12). The relevant formulae are detailed in ref. [56], and implemented in the RadISH program.  Figure 1 shows eq. (12) integrated over the rapidity of the Higgs boson y H and over the p H t azimuth, as a function of p H t and p J,v t . We observe the typical peaked structure along the p H t direction, as well as the Sudakov suppression at small p J,v t . The two-dimensional distribution also features a Sudakov shoulder along the diagonal p H t ∼ p J,v t , which originates from the sensitivity to soft radiation beyond LO [61]. Eq. (12) provides a resummation of the logarithms associated with the shoulder in the regime p H t ∼ p J,v t m H , which can be appreciated by the absence of an integrable singularity in this region.
To verify the correctness of eq. (12), we perform a number of checks. As a first observation, we note that in the region p J,v t m H , the terms F clust and F correl vanish by construction and, as expected, one recovers the NNLL resummation for the inclusive p H t spectrum. Conversely, considering the limit p H t m H (i.e. small b), eq. (12) reproduces the standard NNLL jet-veto resummation of ref. [34] as detailed in ref. [56]. As a further test, we expand eq. (12) to second order in α s relative to the Born, and compare the result with an O(α 2 s ) fixed-order calculation for the inclusive production of a Higgs boson plus one jet [62][63][64], with jets defined according to the anti-k t algorithm [52]. In particular, to avoid the perturbative  of two about the central value µ R = µ F = m H , while keeping 1/2 ≤ µ R /µ F ≤ 2. Moreover, for central µ R and µ F scales, we vary the resummation scale by a factor of two around Q = m H /2, and take the envelope of all the above variations. Figure 3 compares the NNLL+NLO prediction to the NLL+LO, and to the fixed-order NLO result. The integral of the NNLL+NLO (NLL+LO) distribution yields the corresponding jet-vetoed cross section at NNLL+NNLO (NLL+NLO) [34].
We observe a good perturbative convergence for the resummed predictions to the left of the peak, where logarithmic corrections dominate. Above p H t ∼ 10 GeV, the NNLL+NLO prediction differs from the NLL+LO due to the large NLO K factor in the considered process. The residual perturbative uncertainty in the NNLL+NLO distribution is of O(10%) for p H t p J,v t . The comparison to the NLO fixed order shows the importance of resummation across the whole p H t region, and a much reduced sensitivity to the Sudakov shoulder 2 at p H t ∼ p J,v t . In this letter we have formulated the first doubledifferential resummation for an observable defined through a jet algorithm in hadronic collisions. As a case study, we considered the production of a Higgs boson in gluon fusion with transverse momentum p H t in association with jets satisfying the veto requirement p J t ≤ p J,v t . In the limit p H t , p J,v t m H , we performed the resummation of the large logarithms ln(m H /p H t ), ln(m H /p J,v t ) up to NNLL, resulting in an accurate theoretical prediction for this physical observable. As a phenomenological application, we presented matched NNLL+NLO results at the LHC. Our formulation can be applied to the production of any colour-singlet system, and it is relevant in a number of phenomenological applications that will be explored in future work.
We would like to thank Andrea Banfi and Gavin Salam for stimulating discussions on the subject of this letter, and Emanuele Re and Giulia Zanderighi for constructive comments on the manuscript. We are very grateful to Alexander Huss for kindly providing us with a cross check of our results with the NNLOJET program. The work of PM has been supported by the Marie Sk lodowska Curie Individual Fellowship contract number 702610 Resum-mation4PS. LR is supported by the ERC Starting Grant REINVENT (714788), and acknowledges the CERN Theoretical Physics Department for hospitality and support during part of this work, and the CINECA award under the ISCRA initiative for the availability of the highperformance computing resources needed for this work.

SUPPLEMENTAL MATERIAL
We here provide supplemental formulae that complete the discussions and results of the letter.

Explicit resummation formulae
In the present section we report the explicit expressions for the resummation functions g 1 , g 2 and g 3 computed in [25]. We report the results after the introduction of a resummation scale Q, as described in [25,34], that allows for an assessment of the size of subleading logarithmic corrections. With this convention, and a slight abuse of notation, we redefine L = ln(Qb/b 0 ), and λ = α s β 0 L. Here α s denotes α s (µ R ), Q is the resummation scale, of the order of the hard scale m H , while µ R and µ F denote the renormalisation and factorisation scales, respectively. The Sudakov radiator S then reads (the formulae in the letter correspond to setting Q = m H ) with and The g i functions read The coefficients of the QCD beta function up to three loops read and, for Higgs-boson production in gluon fusion, the coefficients A (i) and B (i) entering the above formulae are [72,73] (in units of α s /(2π)) We finally report the expressions for the collinear coefficient function C(α s (µ)) and the hard-virtual term H(µ) in eq. (12): where Here d B is the α s power of the LO cross section (d B = 2 for Higgs production). The coefficient H (1) encodes the pure hard virtual correction to the leading-order process gg → H, and in the MS scheme it is given by The anomalous dimensions Γ ν and Γ (C) ν in eq. (12) are defined as where andP ij is the perturbative expansion of the regularised splitting function (see e.g. ref. [74]). Finally, we report the explicit formulae for the clustering (10) and correlated (11) corrections used in the main result of the letter. We find where in the last step we have introduced the resummation scale Q and neglected corrections beyond NNLL. Similarly, within the same approximation, for the correlated corrections we find a. NLL formula At NLL, the measurement function for the pair of observables under consideration for a state with n emissions reads Θ(p J,v t − max{k t,1 , . . . , k t,n })Θ(p H,v t − | k t,1 + · · · + k t,n |) .
Following ref. [28], we single out the emission with the largest transverse momentum k t,1 , and express the NLL cross section as −e −RNLL(Lt,1) L NLL (µ F e −Lt,1 ) Θ p H,v t − | k t,1 + · · · + k t,n+1 | , (34) where L t,1 ≡ ln(Q/k t,1 ), and the factor L NLL reads where we introduced the explicit x dependence of the parton densities for later convenience. We also introduced the measure dZ defined as with 1 an infrared, constant, resolution parameter that allows for a numerical evaluation of eq. (34) in four space-time dimensions. We stress that the dependence on entirely cancels in eq. (34) for sufficiently small values: in practice we set = e −20 . We also introduced the quantity [28] R (k t,1 ) ≡ − d dL t,1 (L t,1 g 1 (α s L t,1 )) = 4C A α s π L t,1 (1 − 2β 0 α s L t,1 ) .