Massless Partons Research Articles

Bootstrapping the QCD soft anomalous dimension

The soft anomalous dimension governs the infrared singularities of scattering amplitudes to all orders in perturbative quantum field theory, and is a crucial ingredient in both formal and phenomenological applications of non-abelian gauge theories. It has recently been computed at three-loop order for massless partons by explicit evaluation of all relevant Feynman diagrams. In this paper, we show how the same result can be obtained, up to an overall numerical factor, using a bootstrap procedure. We first give a geometrical argument for the fact that the result can be expressed in terms of single-valued harmonic polylogarithms. We then use symmetry considerations as well as known properties of scattering amplitudes in collinear and high-energy (Regge) limits to constrain an ansatz of basis functions. This is a highly non-trivial cross-check of the result, and our methods pave the way for greatly simplified higher-order calculations.

VINCIA for hadron colliders.

We present the first public implementation of antenna-based QCD initial- and final-state showers. The shower kernels are 2rightarrow 3 antenna functions, which capture not only the collinear dynamics but also the leading soft (coherent) singularities of QCD matrix elements. We define the evolution measure to be inversely proportional to the leading poles, hence gluon emissions are evolved in a p_perp measure inversely proportional to the eikonal, while processes that only contain a single pole (e.g., grightarrow qbar{q}) are evolved in virtuality. Non-ordered emissions are allowed, suppressed by an additional power of 1/Q^2. Recoils and kinematics are governed by exact on-shell 2rightarrow 3 phase-space factorisations. This first implementation is limited to massless QCD partons and colourless resonances. Tree-level matrix-element corrections are included for QCD up to mathcal {O}(alpha _s^4) (4 jets), and for Drell–Yan and Higgs production up to mathcal {O}(alpha _s^3) (V / H + 3 jets). The resulting algorithm has been made publicly available in Vincia 2.0.

Three-Loop Corrections to the Soft Anomalous Dimension in Multileg Scattering.

We present the three-loop result for the soft anomalous dimension governing long-distance singularities of multileg gauge-theory scattering amplitudes of massless partons. We compute all contributing webs involving semi-infinite Wilson lines at three loops and obtain the complete three-loop correction to the dipole formula. We find that nondipole corrections appear already for three colored partons, where the correction is a constant without kinematic dependence. Kinematic dependence appears only through conformally invariant cross ratios for four colored partons or more, and the result can be expressed in terms of single-valued harmonic polylogarithms of weight five. While the nondipole three-loop term does not vanish in two-particle collinear limits, its contribution to the splitting amplitude anomalous dimension reduces to a constant, and it depends only on the color charges of the collinear pair, thereby preserving strict collinear factorization properties. Finally, we verify that our result is consistent with expectations from the Regge limit.

Higgs production in bottom-quark fusion: Matching beyond leading order

We compute the total cross-section for Higgs boson production in bottom-quark fusion using the so-called FONLL method for the matching of a scheme in which the $b$-quark is treated as a massless parton to that in which it is treated as a massive final-state particle, and extend our previous results to the case in which the next-to-next-to-leading-log five-flavor scheme result is combined with the next-to-leading-order O(as^3) four-flavor scheme computation.

Master integrals for splitting functions from differential equations in QCD

A method for calculating phase-space master integrals for the decay process $1 \to n$ massless partons in QCD using integration-by-parts and differential equations techniques is discussed. The method is based on the appropriate choice of the basis for master integrals which leads to significant simplification of differential equations. We describe an algorithm how to construct the desirable basis, so that the resulting system of differential equations can be recursively solved in terms of (G)HPLs as a series in the dimensional regulator $\epsilon$ to any order. We demonstrate its power by calculating master integrals for the NLO time-like splitting functions and discuss future applications of the proposed method at the NNLO precision.

Intrinsic charm in a matched general-mass scheme

The FONLL general-mass variable-flavour number scheme provides a framework for the matching of a calculation in which a heavy quark is treated as a massless parton to one in which the mass dependence is retained throughout. We describe how the usual formulation of FONLL can be extended in such a way that the heavy quark parton distribution functions are freely parameterized at some initial scale, rather than being generated entirely perturbatively. We specifically consider the case of deep-inelastic scattering, in view of applications to PDF determination, and the possible impact of a fitted charm quark distribution on F2c is assessed.

Higgs production in bottom-quark fusion in a matched scheme

We compute the total cross-section for Higgs boson production in bottom-quark fusion using the so-called FONLL method for the matching of a scheme in which the b-quark is treated as a massless parton to that in which it is treated as a massive final-state particle. We discuss the general framework for the application of the FONLL method to this process, and then we present explicit expressions for the case in which the next-to-next-to-leading-log five-flavor scheme result is combined with the leading-order O(αs2) four-flavor scheme computation. We compare our results in this case to the four- and five-flavor scheme computations, and to the so-called Santander matching.

N-jettiness subtractions for NNLO QCD calculations

We present a subtraction method utilizing the N -jettiness observable, T 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 T N → 0, and uses these to construct suitable T 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 explain in detail the application to processes with an arbitrary number of massless partons at lepton and hadron colliders together with the required external inputs in the form of QCD amplitudes and lower-order calculations. We provide explicit expressions for the T N -subtractions at NLO and NNLO. The required ingredients are fully known at NLO, and at NNLO for processes with two external QCD partons. The remaining NNLO ingredient for three or more external partons can be obtained numerically with existing NNLO techniques. As an example, we employ our results to obtain the NNLO rapidity spectrum for Drell-Yan and gluon-fusion Higgs production. We discuss aspects of numerical accuracy and convergence and the practical implementation. We also discuss and comment on possible extensions, such as more-differential subtractions, necessary steps for going to N3LO, and the treatment of massive quarks.

Transport coefficients of heavy quarks aroundTcat finite quark chemical potential

The interactions of heavy quarks with the partonic environment at finite temperature $T$ and finite quark chemical potential $\mu_q$ are investigated in terms of transport coefficients within the Dynamical Quasi-Particle model (DQPM) designed to reproduce the lattice-QCD results (including the partonic equation of state) in thermodynamic equilibrium. These results are confronted with those of nuclear many-body calculations close to the critical temperature $T_c$. The hadronic and partonic spatial diffusion coefficients join smoothly and show a pronounced minimum around $T_c$, at $\mu_q=0$ as well as at finite $\mu_q$. Close and above $T_c$ its absolute value matches the lQCD calculations for $\mu_q=0$. The smooth transition of the heavy quark transport coefficients from the hadronic to the partonic medium corresponds to a cross over in line with lattice calculations, and differs substantially from perturbative QCD (pQCD) calculations which show a large discontinuity at $T_c$. This indicates that in the vicinity of $T_c$ dynamically dressed massive partons and not massless pQCD partons are the effective degrees-of-freedom in the quark-gluon plasma.

Non-planar master integrals for the production of two off-shell vector bosons in collisions of massless partons

We present the calculation of all non-planar master integrals that are needed to describe production of two off-shell vector bosons in collisions of two massless partons through NNLO in perturbative QCD. The integrals are computed analytically using differential equations in external kinematic variables and expressed in terms of Goncharov polylogarithms. These results provide the last missing ingredient needed for the computation of two-loop amplitudes that describe the production of two gauge bosons with different invariant masses in hadron collisions.

Complete Nagy-Soper subtraction for next-to-leading order calculations in QCD

We extend the Helac-Dipoles package with the implementation of a new subtraction formalism, first introduced by Nagy and Soper in the formulation of an improved parton shower. We discuss a systematic, semi-numerical approach for the evaluation of the integrated subtraction terms for both massless and massive partons, which provides the missing ingredient for a complete implementation. In consequence, the new scheme can now be used as part of a complete NLO QCD calculation for processes with arbitrary parton masses and multiplicities. We assess its overall performance through a detailed comparison with results based on Catani-Seymour subtraction. The importance of random polarization and color sampling of the external partons is also examined.

Nagy-Soper subtraction scheme for multiparton final states

In this work, we present the extension of an alternative subtraction scheme for next-to-leading order QCD calculations to the case of an arbitrary number of massless final-state partons. The scheme is based on the splitting kernels of an improved parton shower and comes with a reduced number of final state momentum mappings. While a previous publication including the setup of the scheme has been restricted to cases with maximally two massless partons in the final state, we here provide the final state real emission and integrated subtraction terms for processes with any number of massless partons. We apply our scheme to three jet production at lepton colliders at next-to-leading order and present results for the differential C parameter distribution.

Helicity-dependent showers and matching with VINCIA

We present an antenna-shower formalism that includes helicity dependence for massless partons. The formalism applies to both traditional (global) showers and to sector-based variants. We combine the shower with VINCIA's multiplicative approach to matrix-element matching, generalized to operate on each helicity configuration separately. The result is a substantial gain in computational speed for high parton multiplicities. We present an implementation of both sector and global showers, with min and max variations, and helicity-dependent tree-level matching applied for vector bosons or Higgs decay to q qbar plus up to 4 gluons and for Higgs decay to up to 5 gluons.

Antenna showers with hadronic initial states

We present an antenna shower formalism including contributions from initial-state partons and corresponding backwards evolution. We give a set of phase-space maps and antenna functions for massless partons which define a complete shower formalism suitable for computing observables with hadronic initial states. We focus on the initial-state components: initial–initial and initial–final antenna configurations. The formalism includes comprehensive possibilities for uncertainty estimates. We report on some preliminary results obtained with an implementation in the Vincia antenna-shower framework.

Antenna subtraction with massive fermions at NNLO: double real initial-final configurations

We derive the integrated forms of specific initial-final tree-level four-parton antenna functions involving a massless initial-state parton and a massive final-state fermion as hard radiators. These antennae are needed in the subtraction terms required to evaluate the double real corrections to $t\bar{t}$ hadronic production at the NNLO level stemming from the partonic processes $q\bar{q}\to t\bar{t}q'\bar{q}'$ and $gg\to t\bar{t}q\bar{q}$.

Random polarizations of the dipoles

We extend the dipole formalism for massless and massive partons to random polarisations of the external partons. The dipole formalism was originally formulated for spin-summed matrix elements and later extended to individual helicity eigenstates. For efficiency reasons one wants to replace the spin sum by a smooth integration over additional variables. This requires the extension of the dipole formalism to random polarisations. In this paper we derive the modified subtraction terms. We only modify the real subtraction terms, the integrated subtraction terms do not require any modifications.

The infrared structure of gauge theory amplitudes in the high-energy limit

We develop an approach to the high-energy limit of gauge theories based on the universal properties of their infrared singularities. Our main tool is the dipole formula, a compact ansatz for the all-order infrared singularity structure of scattering amplitudes of massless partons. By taking the high-energy limit, we show that the dipole formula implies Reggeization of infrared-singular contributions to the amplitude, at leading logarithmic accuracy, for the exchange of arbitrary color representations in the cross channel. We observe that the real part of the amplitude Reggeizes also at next-to-leading logarithmic order, and we compute the singular part of the two-loop Regge trajectory, which is universally expressed in terms of the cusp anomalous dimension. Our approach provides tools to study the high-energy limit beyond the boundaries of Regge factorization: thus we show that Reggeization generically breaks down at next-to-next-to-leading logarithmic accuracy, and provide a general expression for the leading Reggeization-breaking operator. Our approach applies to multiparticle amplitudes in multi-Regge kinematics, and it also implies new constraints on possible corrections to the dipole formula, based on the Regge limit.

Numerical NLO QCD calculations

We present an algorithm for the numerical calculation of one-loop QCD amplitudes. The algorithm consists of subtraction terms, approximating the soft, collinear and ultraviolet divergences of one-loop amplitudes and a method to deform the integration contour for the loop integration into the complex space. The algorithm is formulated at the amplitude level and does not rely on Feynman graphs. Therefore all required ingredients can be calculated efficiently using recurrence relations. The algorithm applies to massless partons as well as to massive partons.

Thermodynamics of quasiconformal theories from gauge/gravity duality

We use gauge/gravity duality to study the thermodynamics of a generic almost conformal theory, specified by its beta function. Three different phases are identified, a high temperature phase of massless partons, an intermediate quasi-conformal phase and a low temperature confining phase. The limit of a theory with infrared fixed point, in which the coupling does not run to infinity, is also studied. The transitions between the phases are of first order or continuous, depending on the parameters of the beta function. The results presented follow from gauge/gravity duality; no specific boundary theory is assumed, only its beta function.

Infrared Singularities and Soft Gluon Resummation with Massive Partons

Infrared divergences of QCD scattering amplitudes can be derived from an anomalous dimension matrix, which is also an essential ingredient for the resummation of large logarithms due to soft gluon emissions. We report a recent analytical calculation of the anomalous dimension matrix with both massless and massive partons at two-loop level, which describes the two-loop infrared singularities of any scattering amplitudes with an arbitrary number of massless and massive partons, and also enables soft gluon resummation at next-to-next-to-leading-logarithmic order. As an application, we calculate the infrared poles in the q qbar -> t tbar and gg -> t tbar scattering amplitudes at two-loop order.