We present the calculation of the next-to-next-to-leading order (NNLO) zero-jettiness beam and soft functions, up to the second order in the expansion in the dimensional regularization parameter $ε$. These higher order terms are needed for the computation of the next-to-next-to-next-to-leading order (N$^3$LO) zero-jettiness soft and beam functions. As a byproduct, we confirm the $O(ε^0)$ results for NNLO beam and soft functions available in the literature.
We discuss next-to-next-to-leading order QCD results for Higgs, W-boson and Z-boson production in association with a jet in hadronic collisions, obtained using the recently developed N-jettiness subtraction method.
We calculate the soft function for the global event variable 1-jettiness at next-to-next-to-leading order (NNLO) in QCD. We focus specifically on the non-Abelian contribution, which, unlike the Abelian part, is not determined by the next-to-leading order result. The calculation uses the known general forms for the emission of one and two soft partons and is performed using a sector-decomposition method that is spelled out in detail. Results are presented in the form of numerical fits to the 1-jettiness soft function for LHC kinematics (as a function of the angle between the incoming beams and …
The $N$-jettiness subtraction has proven to be an efficient method to perform differential QCD next-to-next-to-leading order (NNLO) calculations in the last few years. One important ingredient of this method is the NNLO soft function. We calculate this soft function for one massive colored particle production at hadron colliders. We select the color octet and color triplet cases to present the final results. We also discuss its application in NLO and NNLO differential calculations.
We present a subtraction method utilizing the N-jettiness observable, Tau_N, to perform QCD calculations for arbitrary processes at next-to-next-to-leading order (NNLO). Our method employs soft-collinear effective theory (SCET) to determine the IR singular contributions of N-jet cross sections for Tau_N -> 0, and uses these to construct suitable Tau_N-subtractions. The construction is systematic and economic, due to being based on a physical observable. The resulting NNLO calculation is fully differential and in a form directly suitable for combining with resummation and parton showers. We exp…
We present a general framework for the calculation of soft functions for SCET_I observables through next-to-next-to-leading order (NNLO) in the strong coupling constant. As an example of our formalism we show how it can be used to obtain the complete NNLO soft function for the $N$-jettiness event shape variable. We present numerical results for two examples with phenomenological impact: the one-jettiness soft function for both electron-proton and proton-proton collisions.
We use the recently proposed jettiness-subtraction scheme to provide the complete calculation of Higgs boson production in association with a jet in hadronic collisions through next-to-next-to-leading order in perturbative QCD. This method exploits the observation that the $N$-jettiness event-shape variable completely describes the singularity structure of QCD when final-state colored particles are present. Our results are in agreement with a recent computation of the $gg$ and $qg$ partonic initial states based on sector-improved residue subtraction. We present phenomenological results for bot…
We revisit the next-to-next-to-leading order (NNLO) calculation of the Higgs boson+1~jet production process, calculated in the $m_t \to \infty$ effective field theory. We perform a detailed comparison of the result calculated using the jettiness slicing method, with published results obtained using subtraction methods. The results of the jettiness calculation agree with the two previous subtraction calculations at benchmark points. The performance of the jettiness slicing approach is greatly improved by adopting a definition of 1-jettiness that accounts for the boost of the Born system. Nevert…
The virtuality-dependent beam function is a universal ingredient in the resummation for observables probing the virtuality of incoming partons, including N-jettiness and beam thrust. We compute the gluon beam function at two-loop order. Together with our previous results for the two-loop quark beam function, this completes the full set of virtuality-dependent beam functions at next-to-next-to-leading order (NNLO). Our results are required to account for all collinear ISR effects to the N-jettiness event shape through N^3LL order. We present numerical results for both the quark and gluon beam f…
We discuss peculiarities that arise in the computation of real-emission contributions to observables that contain Heaviside functions. A prominent example of such a case is the zero-jettiness soft function in SCET, whose calculation at next-to-next-to-next-to-leading order in perturbative QCD is an interesting problem. Since the zero-jettiness soft function distinguishes between emissions into different hemispheres, its definition involves $θ$-functions of light-cone components of emitted soft partons. This prevents a direct use of multi-loop methods, based on reverse unitarity, for computing …
We present an event generator for the process $pp\to \ellν_\ellγ$ at next-to-next-to-leading order (NNLO) in QCD and matched to the PYTHIA8 parton shower. The calculation makes use of the GENEVA framework, which combines a resummed calculation obtained via Soft-Collinear Effective Theory (SCET) with the fixed-order result. We validate the NNLO accuracy of our results by comparing predictions for inclusive quantities with an independent fixed-order calculation, and then present the resummed 0-jettiness spectrum at next-to-next-to-leading logarithmic (NNLL$'$) accuracy. Finally, we compare our p…
We present the implementation of several color-singlet final-state processes at Next-to-Next-to Leading Order (NNLO) accuracy in QCD to the publicly available parton-level Monte Carlo program MCFM. Specifically we discuss the processes $pp\rightarrow H$, $pp\rightarrow Z$, $pp\rightarrow W$, $pp\rightarrow HZ$, $pp\rightarrow HW$ and $pp\rightarrowγγ$. Decays of the unstable bosons are fully included, resulting in a flexible fully differential Monte Carlo code. The NNLO corrections have been calculated using the non-local $N$-jettiness subtraction approach. Special attention is given to the nu…
In this paper we present a next-to-next-to-leading order (NNLO) QCD calculation of the processes $pp\rightarrow l^+l^-γ$ and $pp\rightarrow ν\barνγ$ that we have implemented in MCFM. Our calculation includes QCD corrections at NNLO both for the Standard Model (SM) and additionally in the presence of $Zγγ$ and $ZZγ$ anomalous couplings. We compare our implementation, obtained using the jettiness slicing approach, with a previous SM calculation and find broad agreement. Focusing on the sensitivity of our results to the slicing parameter, we show that using our setup we are able to compute NNLO c…
The $N$-jettiness observable $\mathcal{T}_N$ provides a way of describing the leading singular behavior of the $N$-jet cross section in the $τ=\mathcal{T}_N/Q \to 0$ limit, where $Q$ is a hard interaction scale. We consider subleading power corrections in the $τ\ll 1$ expansion, and employ soft-collinear effective theory to obtain analytic results for the dominant $α_s τ\lnτ$ and $α_s^2 τ\ln^3τ$ subleading terms for thrust in $e^+e^-$ collisions and $0$-jettiness for $q\bar q$-initiated Drell-Yan-like processes at hadron colliders. These results can be used to significantly improve the numeric…
We present an automated framework for the calculation of beam functions that describe collinear initial-state radiation at hadron colliders at next-to-next-to leading order (NNLO) in perturbation theory. By exploiting the infrared behaviour of the collinear matrix elements, we factorise the phase-space singularities with suitable observable-independent parametrisations. Our numerical approach applies to a large class of collider observables, and as a check of its validity, we compute the quark beam functions for transverse-momentum resummation and N-jettiness, which are known analytically at t…
We present the implementation of several processes at Next-to-Next-to Leading Order (NNLO) accuracy in QCD in the parton-level Monte Carlo program MCFM. The processes treated are $pp\to H$, $W^\pm$, $Z$, $W^\pm H$, $ZH$, $W^\pmγ$, $Zγ$ and $γγ$ and, for the first time in the code, $W^+W^-$, $W^\pm Z$ and $ZZ$. Decays of the unstable bosons are fully included, resulting in a flexible fully differential Monte Carlo code. The NNLO corrections have been calculated using two non-local slicing approaches, isolating the doubly unresolved region by cutting on the zero-jettiness, ${\cal T}_0$, or on $q…
We present the NNLL$'$ resummed $2$-jettiness distribution for decays of the Standard Model Higgs boson to a $b\bar{b}$-quark pair and to gluons. The calculation exploits a factorisation formula derived using Soft-Collinear Effective Theory, in which large logarithms of the $2$-jettiness are resummed by renormalisation group evolution of the hard, soft and jet contributions to the differential decay rate. We match the resummed predictions to the fixed-order NNLO result using the GENEVA framework, extending the validity of the results to all values of the resolution variable and providing a ful…
We calculate the beam function for longitudinally-polarized quarks through next-to-next-to-leading order (NNLO) in QCD perturbation theory. This is the last missing ingredient needed to apply the factorization theorem for the $N$-jettiness event-shape variable in polarized collisions through the NNLO level. We present all technical details of our derivation. As a by-product of our calculation we provide the first independent check of the previously-obtained unpolarized quark beam function. We anticipate that our result will have phenomenological applications in describing data from polarized c…
We present a subtraction scheme for computing jet cross sections in electron-positron annihilation at next-to-next-to-leading order accuracy in perturbative QCD. In this second part we deal with the regularization of the real-virtual contribution to the NNLO correction.
We present a subtraction scheme for computing jet cross sections in electron-positron annihilation at next-to-next-to-leading order accuracy in perturbative QCD. In this first part we deal with the regularization of the doubly-real contribution to the NNLO correction.
We present results for the Higgsstrahlung process within the GENEVA Monte Carlo framework. We combine the fully differential NNLO calculation with the higher-order resummation in the 0-jettiness resolution variable (beam-thrust). The resulting parton-level events are further showered and hadronised by Pythia8. The beam-thrust resummation is carried out to NNLL$'$ accuracy, which consistently incorporates all singular virtual and real NNLO corrections. It thus provides a natural perturbative connection between the NNLO calculation and the parton shower regime, including a systematic assessment …
$N$-jettiness subtractions provide a general approach for performing fully-differential next-to-next-to-leading order (NNLO) calculations. Since they are based on the physical resolution variable $N$-jettiness, $\mathcal{T}_N$, subleading power corrections in $τ=\mathcal{T}_N/Q$, with $Q$ a hard interaction scale, can also be systematically computed. We study the structure of power corrections for $0$-jettiness, $\mathcal{T}_0$, for the $gg\to H$ process. Using the soft-collinear effective theory we analytically compute the leading power corrections $α_s τ\lnτ$ and $α_s^2 τ\ln^3τ$ (finding par…
We present a next-to-next-to-leading order (NNLO) QCD calculation of the bottom-induced contributions to the production of a Higgs boson plus a jet, i.e. the process $p p \rightarrow H +j$ to $\mathcal{O}(y_b^2 α_s^3)$. We work in the five-flavor scheme (5FS) in which the bottom quark mass is retained only in the coupling to the Higgs boson. Our calculation uses $N$-jettiness slicing to regulate infrared divergences, allowing for fully-differential predictions for collider observables. After extensively validating the methodology, we present results for the 13 TeV LHC. Our NNLO predictions sho…
We evaluate all phase space master integrals which are required for the total cross section of generic 2 -> 1 processes at NNLO as a series expansion in the dimensional regulator epsilon. Away from the limit of threshold production, our expansion includes one order higher than what has been available in the literature. At threshold, we provide expressions which are valid to all orders in terms of Gamma functions and hypergeometric functions. These results are a necessary ingredient for the renormalization and mass factorization of singularities in 2 -> 1 inclusive cross sections at NNNLO in QC…
We investigate QCD amplitudes with massive quarks computed in the four-dimensional helicity scheme (FDH) and dimensional reduction at NNLO and describe how they are related to the corresponding amplitudes computed in conventional dimensional regularization. To this end, the scheme dependence of the heavy quark and the velocity-dependent cusp anomalous dimensions is determined using soft-collinear effective theory. The results are checked against explicit computations of massive form factors in FDH at NNLO. Our results complete the description of the scheme dependence of QCD amplitudes at NNLO.
We compute the one-loop QCD amplitude for the process gg-->Q\bar{Q} in dimensional regularization through order ε^2 in the dimensional regulator and for arbitrary quark mass values. This result is an ingredient of the NNLO cross-section for heavy quark production at hadron colliders. The calculation is performed in conventional dimensional regularization, using well known reduction techniques as well as a method based on recent ideas for the functional form of one-loop integrands in four dimensions.
In this paper we present a Next-to-Next-to Leading Order (NNLO) calculation of the production of a Higgs boson in association with a massive vector boson. We include the decays of the unstable Higgs and vector bosons, resulting in a fully flexible parton-level Monte Carlo implementation. We also include all $\mathcal{O}(α_s^2)$ contributions that occur in production for these processes: those mediated by the exchange of a single off-shell vector boson in the $s$-channel, and those which arise from the coupling of the Higgs boson to a closed loop of fermions. We study final states of interest f…
So far, the use of different variants of dimensional regularization has been investigated extensively for two-loop virtual corrections. We extend these studies to real corrections that are also required for a complete computation of physical cross sections at next-to-next-to-leading order. As a case study we consider two-jet production in electron-positron annihilation and describe how to compute the various parts separately in different schemes. In particular, we verify that using dimensional reduction the double-real corrections are obtained simply by integrating the four-dimensional matrix …
We present results for the 2-jettiness differential distribution for boosted top quark pairs produced in $e^+e^-$ collisions in the peak region accounting for QCD large-logarithm resummation at next-to-next-to-next-to-leading logarithmic (N$^3$LL) order and fixed-order corrections to matrix elements at next-to-next-to-leading order (NNLO) calculated in the framework of soft-collinear effective theory and boosted heavy quark effective theory. Electroweak and finite-width effects are included at leading order. We study the perturbative convergence of the cross section in the pole and MSR mass sc…
We investigate the regularization-scheme dependence of scattering amplitudes in massless QCD and find that the four-dimensional helicity scheme (FDH) and dimensional reduction (DRED) are consistent at least up to NNLO in the perturbative expansion if renormalization is done appropriately. Scheme dependence is shown to be deeply linked to the structure of UV and IR singularities. We use jet and soft functions defined in soft-collinear effective theory (SCET) to efficiently extract the relevant anomalous dimensions in the different schemes. This result allows us to construct transition rules for…
We present the first calculation of direct photon production at next-to-next-to leading order (NNLO) accuracy in QCD. For this process, although the final state cuts mandate only the presence of a single electroweak boson, the underlying kinematics resembles that of a generic vector boson plus jet topology. In order to regulate the infrared singularities present at this order we use the $N$-jettiness slicing procedure, applied for the first time to a final state that at Born level includes colored partons but no required jet. We compare our predictions to ATLAS 8 TeV data and find that the inc…
We present a fully-differential calculation of the $H\rightarrow b\overline{b}$ decay at next-to-next- to-next-to-leading order (N$^3$LO) accuracy. Our calculation considers diagrams in which the Higgs boson couples directly to the bottom quarks, i.e. the perturbative order we consider is $\mathcal{O}(α_s^3y_b^2)$. In order to regulate the infrared divergences present at this order we use the Projection-to-Born technique coupled with N-jettiness slicing. After validating our methodology at next-to-next-to-leading order (NNLO) we present exclusive jet rates and differential distributions for je…
We consider Higgs boson production at hadron colliders in the gluon fusion channel and compute higher order terms in the regularization parameter $ε$. In particular, the next-to-next-to-leading order cross section is evaluated including order $ε$ terms. These results are used to compute all convolutions with the splitting functions entering the next-to-next-to-next-to-leading order cross section.
We present for first time the computation at large $q_T$ (or small $b_T$) matching coefficients of Transverse Momentum Dependent Parton Distribution Function (TMDPDF) for linearly polarised to the integrated gluon distribution at next-to-next-to leading order (NNLO). The computation is performed using the modified $δ$-regulator for rapidity divergences and dimensional regularization. This TMDPDF matriz element presnts phenomenological interest in two kind of processes. The factorization of the Higgs production transverse momentum ($q_T$) distribution through gluon-gluon fusion and the quarkoni…
We present a detailed comparison of the fixed-order predictions computed by four publicly available computer codes for Drell-Yan processes at the LHC and Tevatron colliders. We point out that while there is agreement among the predictions at the next-to-leading order accuracy, the predictions at the next-to-next-to-leading order (NNLO) differ, whose extent depends on the observable. The sizes of the differences in general are at least similar, sometimes larger than the sizes of the NNLO corrections themselves. We demonstrate that the neglected power corrections by the codes that use global sli…
We show at NNLO that the soft factors for double parton scattering (DPS) for both integrated and unintegrated kinematics, can be presented entirely in the terms of the soft factor for single Drell-Yan process, i.e. the transverse momentum dependent (TMD) soft factor. Using the linearity of the logarithm of TMD soft factor in rapidity divergences, we decompose the DPS soft factor matrices into a product of matrices with rapidity divergences in given sectors, and thus, define individual double parton distributions at NNLO. The rapidity anomalous dimension matrices for double parton distributions…
A general subtraction scheme, STRIPPER (SecToR Improved Phase sPacE for real Radiation), is derived for the evaluation of next-to-next-to-leading order (NNLO) QCD contributions from double-real radiation to processes with at least two particles in the final state at leading order. The result is a Laurent expansion in the parameter of dimensional regularization, the coefficients of which should be evaluated by numerical Monte Carlo integration. The two main ideas are a two-level decomposition of the phase space, the second one factorizing the singular limits of amplitudes, and a suitable parame…
We present an analytical calculation of the two-loop QCD corrections to the electromagnetic form factor of heavy quarks. The two-loop contributions to the form factor are reduced to linear combinations of master integrals, which are computed through higher orders in the parameter of dimensional regularization epsilon=(4-D)/2. Our result includes all terms of order epsilon at two loops and extends the previous literature. We apply the exponentiation of the heavy-quark form factor to derive new improved three-loop expansions in the high-energy limit. We also discuss the implications for predicti…
We compute the ${\cal O}(α^2α_s^2)$ perturbative corrections to inclusive jet production in electron-nucleon collisions. This process is of particular interest to the physics program of a future Electron Ion Collider (EIC). We include all relevant partonic processes, including deep-inelastic scattering contributions, photon-initiated corrections, and parton-parton scattering terms that first appear at this order. Upon integration over the final-state hadronic phase space we validate our results for the deep-inelastic corrections against the known next-to-next-to-leading order (NNLO) structure …
We discuss the prescription for the Dirac matrix gamma_5 in dimensional regularization used in most second- and third-order QCD calculations of collider cross sections. We provide an alternative implementation of this approach that avoids the use of an explicit form of gamma_5 and of its (anti-) commutation relations in the most important case of no more than one gamma_5 in each fermion trace. This treatment is checked by computing the third-order corrections to the structure functions F_2 and g_1 in charged-current deep-inelastic scattering with axial-vector couplings to the W-bosons. We deri…
We revisit the calculation of perturbative quark transverse momentum dependent parton distribution functions and fragmentation functions using the exponential regulator for rapidity divergences. We show that the exponential regulator provides a consistent framework for the calculation of various ingredients in transverse momentum dependent factorization. Compared to existing regulators in the literature, the exponential regulator has a couple of advantages which we explain in detail. As a result, the calculation is greatly simplified and we are able to obtain the next-to-next-to-leading order …
We describe how NNLO final state quark-pair corrections are computed in FDR by directly enforcing gauge invariance and unitarity in the definition of the regularized divergent integrals. The whole procedure is strictly four-dimensional and renormalization is performed by simply fixing bare parameters in terms of physical measurements. We give details of our approach and demonstrate how virtual and real contributions can be merged together without relying on dimensional regularization. As an example, we recompute the ${ H} \to b \bar b + {jets}$ and $γ^\ast \to {jets}$ inclusive rates at the NN…
Fully differential next-to-next-to-leading order calculations require a method to cancel infrared singularities. In a previous publication, I discussed the general setup for the subtraction method at NNLO. In this paper I give all subtraction terms for electron-positron annihilation associated with one-loop amplitudes with one unresolved parton. These subtraction terms are integrated within dimensional regularization over the unresolved one-particle phase space. The results can be used with all variants of dimensional regularization (conventional dimensional regularization, the 't Hooft-Veltma…
We compute the relation between the quark mass defined in the minimal modified $\MSbar$ scheme and the mass defined in the ``Regularization Invariant" scheme (RI), up to the NNLO order. The RI scheme is conveniently adopted in lattice QCD calculations of the quark mass, since the relevant renormalization constants in this scheme can be evaluated in a non-perturbative way. The NNLO contribution to the conversion factor between the quark mass in the two schemes is found to be large, typically of the same order of the NLO correction at a scale $μ\sim 2$ GeV. We also give the NNLO relation between…
We present the full next-to-next-to-leading order (NNLO) coefficient functions for the polarized cross section $dΔσ/dQ$ for the Drell-Yan process $p + p\to l^+l^- + 'X'$. Here $'X'$ denotes any inclusive hadronic state and $Q$ represents the invariant mass of the lepton pair. All QCD partonic subprocesses have been included provided the lepton pair is created by a virtual photon, which is a valid approximation for $Q<50$ GeV. Unlike the differential distribution w.r.t. transverse momentum the dominant subprocess for the integrated cross section is given by $q+\bar q \to γ^* + 'X'$ and its high…
In differential measurements at a hadron collider, collinear initial-state radiation is described by process-independent beam functions. They are the field-theoretic analog of initial-state parton showers. Depending on the measured observable they are differential in the virtuality and/or transverse momentum of the colliding partons in addition to the usual longitudinal momentum fraction. Perturbatively, the beam functions can be calculated by matching them onto standard quark and gluon parton distribution functions. We calculate the inclusive virtuality-dependent quark beam function at NNLO, …
We present complete analytical ${\mathcal O}(ε^2)$ results on the one-loop amplitudes relevant for the NNLO quark-parton model description of the hadroproduction of heavy quarks as given by the so-called loop-by-loop contributions. All results of the perturbative calculation are given in the dimensional regularization scheme. These one-loop amplitudes can also be used as input in the determination of the corresponding NNLO cross sections for heavy flavor photoproduction, and in photon-photon reactions.
We compute the next-to-next-to-leading order (NNLO) soft and virtual QCD corrections for the partonic cross section of colourless-final state processes in hadronic collisions. The results are valid to all orders in the dimensional regularization parameter $\ep$. The dependence of the results on a particular process is given through finite contributions to the one and two-loop amplitudes. To evaluate the accuracy of the soft-virtual approximation we compare it with the full NNLO result for Drell-Yan and Higgs boson production via gluon fusion. We also provide a universal expression for the hard…
We compute the effective hamiltonian for non-leptonic |Delta F| = 1 decays in the standard model including next-to-next-to-leading order QCD corrections. In particular, we present the complete three-loop anomalous dimension matrix describing the mixing of current-current and QCD penguin operators. The calculation is performed in an operator basis which allows to consistently use fully anticommuting gamma_5 in dimensional regularization at an arbitrary number of loops. The renormalization scheme dependences and their cancellation in physical quantities is discussed in detail. Furthermore, we de…
In this paper we illustrate the simplifications produced by FDR in NNLO computations. We show with an explicit example that - due to its four-dimensionality - FDR does not require an order-by-order renormalization and that, unlike the one-loop case, FDR and dimensional regularization (DR) generate intermediate two-loop results which are no longer linked by a simple subtraction of the ultraviolet (UV) poles in epsilon. Our case study is the two-loop amplitude for H -> gamma gamma, mediated by an infinitely heavy top loop, in the presence of gluonic corrections. We use this to elucidate how gaug…
All (un)polarized transverse momentum dependent functions (TMDs), both distribution and fragmentation functions, are defined with the same universal soft function, which cancels spurious rapidity divergences within an individual TMD and renders them well-defined hadronic quantities. Moreover, it is independent of the kinematics, whether it is Drell-Yan, deep inelastic scattering or $e^+e^-\rightarrow 2$ hadrons. In this paper we provide this soft function at next-to-next-to-leading order (NNLO), necessary for the calculation of all TMDs at the same order, and to perform the resummation of larg…
The 3S1-3D1 mixing angle for nucleon-nucleon scattering, epsilon_1, is calculated to next-to-next-to-leading order in an effective field theory with perturbative pions. Without pions, the low energy theory fits the observed epsilon_1 well for momenta less than $\sim 50$ MeV. Including pions perturbatively significantly improves the agreement with data for momenta up to $\sim 150$ MeV with one less parameter. Furthermore, for these momenta the accuracy of our calculation is similar to an effective field theory calculation in which the pion is treated non-perturbatively. This gives phenomenologi…
