Do we need N3^3LO Parton Distributions?

We discuss the uncertainty on processes computed using next-to-next-to leading (NNLO) parton distributions (PDFs) due to the neglect of higher order perturbative corrections in the PDF determination, in the specific case of Higgs production in gluon fusion. By studying the behaviour of the perturbative series for this process, we show that this uncertainty is negligible in comparison to the theoretical uncertainty on the matrix element. We then take this as a case study for the use of the Cacciari-Houdeau method for the estimate of theoretical uncertainties, and show that the method provides an effective way of treating theoretical uncertainties on the matrtix element and the PDF on the same footing.Comment: 10 pages 5 figures. Final version, to be published in Phys. Lett. B. Comparison with top production (figs 4-5) added. Several typos corrected and references updated. Grant info adde

A Subleading Operator Basis and Matching for gg→Hgg \to H

The Soft Collinear Effective Theory (SCET) is a powerful framework for studying factorization of amplitudes and cross sections in QCD. While factorization at leading power has been well studied, much less is known at subleading powers in the $\lambda\ll 1$ expansion. In SCET subleading soft and collinear corrections to a hard scattering process are described by power suppressed operators, which must be fixed case by case, and by well established power suppressed Lagrangians, which correct the leading power dynamics of soft and collinear radiation. Here we present a complete basis of power suppressed operators for $gg \to H$, classifying all operators which contribute to the cross section at $\mathcal{O}(\lambda^2)$, and showing how helicity selection rules significantly simplify the construction of the operator basis. We perform matching calculations to determine the tree level Wilson coefficients of our operators. These results are useful for studies of power corrections in both resummed and fixed order perturbation theory, and for understanding the factorization properties of gauge theory amplitudes and cross sections at subleading power. As one example, our basis of operators can be used to analytically compute power corrections for $N$-jettiness subtractions for $gg$ induced color singlet production at the LHC.Comment: v2. JHEP version. Minor clarifications and typos fixe

Subleading Power Factorization with Radiative Functions

The study of amplitudes and cross sections in the soft and collinear limits allows for an understanding of their all orders behavior, and the identification of universal structures. At leading power soft emissions are eikonal, and described by Wilson lines. Beyond leading power the eikonal approximation breaks down, soft fermions must be added, and soft radiation resolves the nature of the energetic partons from which they were emitted. For both subleading power soft gluon and quark emissions, we use the soft collinear effective theory (SCET) to derive an all orders gauge invariant bare factorization, at both amplitude and cross section level. This yields universal multilocal matrix elements, which we refer to as radiative functions. These appear from subleading power Lagrangians inserted along the lightcone which dress the leading power Wilson lines. The use of SCET enables us to determine the complete set of radiative functions that appear to $\mathcal{O}(\lambda^2)$ in the power expansion, to all orders in $\alpha_s$. For the particular case of event shape observables in $e^+e^-\to$ dijets we derive how the radiative functions contribute to the factorized cross section to $\mathcal{O}(\lambda^2)$.Comment: 62 pages + appendices, many pretty and colorful figures. v2: journal versio

A Subleading Power Operator Basis for the Scalar Quark Current

Factorization theorems play a crucial role in our understanding of the strong interaction. For collider processes they are typically formulated at leading power and much less is known about power corrections in the $\lambda\ll 1$ expansion. Here we present a complete basis of power suppressed operators for a scalar quark current at $\mathcal{O}(\lambda^2)$ in the amplitude level power expansion in the Soft Collinear Effective Theory, demonstrating that helicity selection rules significantly simplify the construction. This basis applies for the production of any color singlet scalar in $q\bar{q}$ annihilation (such as $b \bar b \to H$). We also classify all operators which contribute to the cross section at $\mathcal{O}(\lambda^2)$ and perform matching calculations to determine their tree level Wilson coefficients. These results can be exploited to study power corrections in both resummed and fixed order perturbation theory, and for analyzing the factorization properties of gauge theory amplitudes and cross sections at subleading power.Comment: 41 pages + Appendices. 3 tables. v2: text changes. arXiv admin note: text overlap with arXiv:1703.0340

A Subleading Power Operator Basis for the Scalar Quark Current

Factorization theorems play a crucial role in our understanding of the strong interaction. For collider processes they are typically formulated at leading power and much less is known about power corrections in the $\lambda\ll 1$ expansion. Here we present a complete basis of power suppressed operators for a scalar quark current at $\mathcal{O}(\lambda^2)$ in the amplitude level power expansion in the Soft Collinear Effective Theory, demonstrating that helicity selection rules significantly simplify the construction. This basis applies for the production of any color singlet scalar in $q\bar{q}$ annihilation (such as $b \bar b \to H$). We also classify all operators which contribute to the cross section at $\mathcal{O}(\lambda^2)$ and perform matching calculations to determine their tree level Wilson coefficients. These results can be exploited to study power corrections in both resummed and fixed order perturbation theory, and for analyzing the factorization properties of gauge theory amplitudes and cross sections at subleading power.Comment: 41 pages + Appendices. 3 tables. v2: text changes. arXiv admin note: text overlap with arXiv:1703.0340

Fermionic Glauber Operators and Quark Reggeization

We derive, in the framework of soft-collinear effective field theory (SCET), a Lagrangian describing the $t$-channel exchange of Glauber quarks in the Regge limit. The Glauber quarks are not dynamical, but are incorporated through non-local fermionic potential operators. These operators are power suppressed in $|t|/s$ relative to those describing Glauber gluon exchange, but give the first non-vanishing contributions in the Regge limit to processes such as $q\bar q \to gg$ and $q\bar q \to \gamma \gamma$. They therefore represent an interesting subset of power corrections to study. The structure of the operators, which describe certain soft and collinear emissions to all orders through Wilson lines, is derived from the symmetries of the effective theory combined with constraints from power and mass dimension counting, as well as through explicit matching calculations. Lightcone singularities in the fermionic potentials are regulated using a rapidity regulator, whose corresponding renormalization group evolution gives rise to the Reggeization of the quark at the amplitude level and the BFKL equation at the cross section level. We verify this at one-loop, deriving the Regge trajectory of the quark in the $3$ color channel, as well as the leading logarithmic BFKL equation. Results in the $\bar 6$ and $15$ color channels are obtained by the simultaneous exchange of a Glauber quark and a Glauber gluon. SCET with quark and gluon Glauber operators therefore provides a framework to systematically study the structure of QCD amplitudes in the Regge limit, and derive constraints on higher order amplitudes.Comment: 31 pages, many figure

The Four-Loop Rapidity Anomalous Dimension and Event Shapes to Fourth Logarithmic Order

We obtain the quark and gluon rapidity anomalous dimension to fourth order in QCD. We calculate the N$^3$LO rapidity anomalous dimensions to higher order in the dimensional regulator and make use of the soft/rapidity anomalous dimension correspondence in conjunction with the recent determination of the N$^4$LO threshold anomalous dimensions to achieve our result. We show that the results for the quark and gluon rapidity anomalous dimensions at four loops are related by generalized Casimir scaling. Using the N$^4$LO rapidity anomalous dimension, we perform the resummation of the Energy-Energy Correlation in the back-to-back limit at N$^4$LL, achieving for the first time the resummation of an event shape at this logarithmic order. We present numerical results and observe a reduction of perturbative uncertainties on the resummed cross section to below 1%.Comment: 5 pages, 3 figures, 2 ancillary files. v2: corrected typo in tabled values from ref [28] in eq. 10 and 11. Analytic formulae, EEC section and ancillary files unchange

Subleading Power Resummation of Rapidity Logarithms: The Energy-Energy Correlator in N=4\mathcal{N}=4 SYM

We derive and solve renormalization group equations that allow for the resummation of subleading power rapidity logarithms. Our equations involve operator mixing into a new class of operators, which we term the "rapidity identity operators", that will generically appear at subleading power in problems involving both rapidity and virtuality scales. To illustrate our formalism, we analytically solve these equations to resum the power suppressed logarithms appearing in the back-to-back (double light cone) limit of the Energy-Energy Correlator (EEC) in $\mathcal{N}$=4 super-Yang-Mills. These logarithms can also be extracted to $\mathcal{O}(\alpha_s^3)$ from a recent perturbative calculation, and we find perfect agreement to this order. Instead of the standard Sudakov exponential, our resummed result for the subleading power logarithms is expressed in terms of Dawson's integral, with an argument related to the cusp anomalous dimension. We call this functional form "Dawson's Sudakov". Our formalism is widely applicable for the resummation of subleading power rapidity logarithms in other more phenomenologically relevant observables, such as the EEC in QCD, the $p_T$ spectrum for color singlet boson production at hadron colliders, and the resummation of power suppressed logarithms in the Regge limit.Comment: 32 pages, a small number of figures. v2: fixed minor typos, journal versio

Transverse momentum dependent PDFs at N3^3LO

We compute the quark and gluon transverse momentum dependent parton distribution functions at next-to-next-to-next-to-leading order (N$^3$LO) in perturbative QCD. Our calculation is based on an expansion of the differential Higgs boson and Drell-Yan production cross sections about their collinear limit. This method allows us to employ cutting edge techniques for the computation of cross sections to extract the universal building blocks in question. The corresponding perturbative matching kernels for all channels are expressed in terms of simple harmonic polylogarithms up to weight five. As a byproduct, we confirm a previous computation of the soft function for transverse momentum factorization at N$^3$LO. Our results are the last missing ingredient to extend the $q_T$ subtraction methods to N$^3$LO and to obtain resummed $q_T$ spectra at N$^3$LL$^\prime$ accuracy both for gluon as well as for quark initiated processes.Comment: 12 pages + appendices, 4 awesome figures, important ancillary files. v2: journal versio