Splitting Functions In QCD Research Articles

Four-loop splitting functions in QCD – The quark-to-gluon case

We present the even-N moments N≤20 of the fourth-order (N3LO) contribution Pgq(3)(x) to the quark-to-gluon splitting function in perturbative QCD. These moments, obtained by analytically computing off-shell operator matrix elements for a general gauge group, agree with all known results, in particular with the moments N≤10 derived before from structure functions in deep-inelastic scattering. Using the new moments and the available endpoint constraints, we construct approximations for Pgq(3)(x) which improve upon those obtained from the lowest five even moments. The remaining uncertainties of this function are now practically irrelevant at momentum fractions x>0.1. The resulting errors of the convolution of Pgq at N3LO with a typical quark distribution are small at x≳10−3 and exceed 1% only at x≲10−4 for a strong coupling αs=0.2. The present results for Pgq(3)(x) should thus be sufficient for most collider-physics applications.

Event generator for jet tomography in electron-ion collisions

We develop the first event generator, the electron-heavy-ion-jet-interaction-generator (eHIJING), for the jet tomography study of electron-ion collisions. In this generator, energetic jet partons produced from the initial hard scattering undergo multiple collisions with the nuclear target. The collision rate is proportional to the transverse-momentum-dependent (TMD) gluon density in the nucleus, which is given by a simple model inspired by the physics of gluon saturation. Medium-modified QCD splitting functions within the higher-twist (HT) and generalized higher-twist (GHT) frameworks are utilized to simulate parton showering in the nuclear medium that takes into account the non-Abelian Landau-Pomeranchuck-Midgal interference effect. Employing eHIJING, we revisit hadron production in semi-inclusive deep inelastic scattering (SIDIS) as measured by EMC, HERMES, and recent CLAS experiments. eHIJING with both GT and GHT frameworks gives reasonably good descriptions of these experimental data. Predictions for experiments at the future electron-ion colliders are also provided. It is demonstrated that future measurements of the transverse momentum broadening of single hadron spectra can be used to map out the two-dimensional kinematic (Q2, xB) dependence of the jet transport coefficient q^ in cold nuclear matter. Published by the American Physical Society 2024

The 1 → 3 massive splitting functions from QCD factorization and SCET

Splitting functions are universal functions describing the collinear dynamics of gauge theories, and as such are crucial ingredients for a wide variety of calculations in perturbative QCD. We present analytic results for the triple collinear splitting functions in QCD with a single massive parton. We derive the splitting functions using two distinct methods; first by expanding the squared matrix elements in the collinear limit, and secondly by using soft-collinear effective theory with massive quarks. We find agreement between these two approaches, providing a strong check of our results. Additionally, we also check all iterated and soft limits of our results, finding agreement with predictions from factorization. Our results provide an important ingredient for higher order perturbative calculations involving massive partons, and for the description of the collinear dynamics of heavy flavor jets.

Additional moments and x-space approximations of four-loop splitting functions in QCD

We have extended our previous computations of the even-N moments of the flavour-singlet four-loop splitting functions to N=12 for the pure-singlet quark case and N=10 for all other cases. These results, obtained using physical quantities in inclusive deep-inelastic scattering, have been and will be used to validate conceptionally much more challenging determinations of these splitting functions from off-shell operator matrix elements (OMEs). For the quark-gluon and gluon-gluon splitting functions, which have yet to be computed to higher N using OMEs, we construct approximations based on our moments and endpoint constraints, where we present new large-x results for the gluon-gluon case. These approximations facilitate an approximate N3LO evolution of parton distributions which are sufficiently accurate outside the region of small momentum fractions x.

The soft drop momentum sharing fraction zg beyond leading-logarithmic accuracy

Grooming techniques, such as soft drop, play a central role in reducing sensitivity of jets to e.g. underlying event and hadronization at current collider experiments. The momentum sharing fraction zg, of the two branches in a jet that pass the soft drop condition, is one of the most important observables characterizing a collinear splitting inside the jet, and directly probes the QCD splitting functions. In this work, we present a factorization framework that enables a systematic calculation of the corresponding cross section beyond modified leading-logarithmic (MLL) accuracy, showing that this measurement is not only sensitive to the QCD charge but also the spin of the parton that initiates the jet. Our results at next-to-leading logarithmic (NLL′) accuracy include non-global logarithms, and provide a first meaningful assessment of the perturbative uncertainty. We present a comparison to the available experimental data from ALICE, ATLAS, and STAR and find excellent agreement.

Low moments of the four-loop splitting functions in QCD

We have computed the four lowest even-N moments of all four splitting functions for the evolution of flavour-singlet parton densities of hadrons at the fourth order in the strong coupling constant αs. The perturbative expansion of these moments, and hence of the splitting functions for momentum fractions x≳0.1, is found to be well behaved with relative αs-coefficients of order one and sub-percent effects on the scale derivatives of the quark and gluon distributions at αs≲0.2. More intricate computations, including other approaches such as the operator-product expansion, are required to cover the full x-range relevant to LHC analyses. Our results are presented analytically for a general gauge group for detailed checks and validations of such future calculations.

Large-nf contributions to the four-loop splitting functions in QCD

We have computed the fourth-order nf2 contributions to all three non-singlet quark–quark splitting functions and their four nf3 flavour-singlet counterparts for the evolution of the parton distributions of hadrons in perturbative QCD with nf effectively massless quark flavours. The analytic form of these functions is presented in both Mellin N-space and momentum-fraction x-space; the large-x and small-x limits are discussed. Our results agree with all available predictions derived from lower-order information. The large-x limit of the quark–quark cases provides the complete nf2 part of the four-loop cusp anomalous dimension which agrees with two recent partial computations.

The three-loop splitting functions in QCD: The helicity-dependent case

We present the next-to-next-to-leading order (NNLO) contributions to the main splitting functions for the evolution of longitudinally polarized parton densities of hadrons in perturbative QCD. The quark–quark and gluon–quark splitting functions have been obtained by extending our previous all Mellin-N calculations to the structure function g1 in electromagnetic deep-inelastic scattering (DIS). Their quark–gluon and gluon–gluon counterparts have been derived using third-order fixed-N calculations of structure functions in graviton-exchange DIS, relations to the unpolarized case and mathematical tools for systems of Diophantine equations. The NNLO corrections to the splitting functions are small outside the region of small momentum fractions x where they exhibit a large double-logarithmic enhancement, yet the corrections to the evolution of the parton densities can be unproblematic down to at least x≈10−4.

Fragmentation in jets at NNLO

Beam and jet functions in Soft-Collinear Effective Theory describe collinear initial- and final-state radiation (jets), and enter in factorization theorems for N-jet production, the Higgs pT spectrum, etc. We show that they may directly be calculated as phase-space integrals of QCD splitting functions. At NLO all computations are trivial, as we demonstrate explicitly for the beam function, the transverse-momentum-dependent beam function, the jet function and the fragmenting jet function. This approach also highlights the role of crossing symmetry in these calculations. At NNLO we reproduce the quark jet function and calculate the fragmenting quark jet function for the first time. Here we use two methods: a direct phase-space integration and a reduction to master integrals which are computed using differential equations.

The quark beam function at two loops

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, which is relevant for any observables probing the virtuality of the incoming partons, including N-jettiness and beam thrust. For such observables, our results are an important ingredient in the resummation of large logarithms at N3LL order, and provide all contributions enhanced by collinear t-channel singularities at NNLO for quark-initiated processes in analytic form. We perform the calculation in both Feynman and axial gauge and use two different methods to evaluate the discontinuity of the two-loop Feynman diagrams, providing nontrivial checks of the calculation. As part of our results we reproduce the known two-loop QCD splitting functions and confirm at two loops that the virtuality-dependent beam and final-state jet functions have the same anomalous dimension.

Evolution of minimum-bias parton fragmentation in nuclear collisions

Minimum-bias fragment distributions (FDs) are calculated by folding a power-law parton energy spectrum with parametrized fragmentation functions (FFs) derived from ${e}^{+}$-${e}^{\ensuremath{-}}$ and $p$-$\overline{p}$ collisions. Substantial differences between measured ${e}^{+}$-${e}^{\ensuremath{-}}$ and $p$-$\overline{p}$ FFs suggest that FF ``universality'' may not be a valid assumption. The common parton spectrum is constrained by comparison with a $p$-$p$ ${p}_{t}$ spectrum hard component. Changes in FFs due to parton ``energy loss'' or ``medium modification'' are modeled by altering FF parametrizations consistent with rescaling QCD splitting functions. In-vacuum and in-medium FDs are compared with spectrum hard components from $200$-GeV Au-Au collisions for several centralities. The reference for all nuclear collisions is the FD derived from in-vacuum ${e}^{+}$-${e}^{\ensuremath{-}}$ FFs. The hard component for $p$-$p$ and peripheral Au-Au collisions is found to be strongly suppressed for smaller fragment momenta, consistent with the FD derived from in-vacuum $p$-$\overline{p}$ FFs. At a particular centrality the Au-Au hard component transitions to enhancement at smaller momenta and suppression at larger momenta, consistent with FDs derived from in-medium ${e}^{+}$-${e}^{\ensuremath{-}}$ FFs. Fragmentation systematics suggest that QCD color connections change dramatically in more-central $A$-$A$ collisions. Observed parton and hadron spectrum systematics are inconsistent with saturation-scale arguments used to support assumptions of parton thermalization.

A new method for calculating differential distributions directly in Mellin space

We present a new method for the calculation of differential distributions directly in Mellin space without recourse to the usual momentum-fraction (or z-) space. The method is completely general and can be applied to any process. It is based on solving the Integration-by-Parts identities when one of the powers of the propagators is an abstract number. The method retains the full dependence on the Mellin variable and can be implemented in any program for solving the IBP identities based on algebraic elimination, like Laporta. General features of the method are: 1) faster reduction, 2) smaller number of master integrals compared to the usual z-space approach and 3) the master integrals satisfy difference instead of differential equations. This approach generalizes previous results related to fully inclusive observables like the recently calculated three-loop space-like anomalous dimensions and coefficient functions in inclusive DIS to more general processes requiring separate treatment of the various physical cuts. Many possible applications of this method exist, the most notable being the direct evaluation of the three-loop time-like splitting functions in QCD.

The three-loop splitting functions in QCD

We have computed the next-to-next-to-leading-order (NNLO) contributions to the evolution of unpolarized parton distributions in perturbative QCD [1,2]. Here we briefly recall why this huge computation was necessary and outline how it was performed. We discuss the end-point limits of our results which include the three-loop cusp anomalous dimensions of the Wilson lines. The numerical impact of the new contributions is illustrated.

Two-loop splitting functions in QCD

We present the universal two-loop splitting functions that describe the limits of two-loop $n$-point amplitudes of massless particles when two of the momenta are collinear. To derive the splitting amplitudes, we take the collinear limits of explicit two-loop four-point helicity amplitudes computed in the 't Hooft-Veltman scheme. The $g \to gg$ splitting amplitude has recently been computed using the unitarity sewing method and we find complete agreement with the results of Ref. \cite{Bern:2lsplit}. The two-loop $q \to qg$ and $g \to q\bar q$ splitting functions are new results. We also provide an expression for the two-loop soft splitting function.

The three-loop splitting functions in QCD: the singlet case

We compute the next-to-next-to-leading order (NNLO) contributions to the splitting functions governing the evolution of the unpolarized flavour-singlet parton densities in perturbative QCD. The exact expressions are presented in both Mellin- N and Bjorken- x space. We also provide accurate parametrizations for practical applications. Our results agree with all partial results available in the literature. As in the non-singlet case, the correct leading logarithmic predictions for small momentum fractions x do not provide good estimates of the respective complete splitting functions. We investigate the size of the corrections and the stability of the NNLO evolution under variation of the renormalization scale. The perturbative expansion appears to converge rapidly at x≳10 −3. Relatively large third-order corrections are found at smaller values of x.

The three-loop splitting functions in QCD: theBsinglet case

We compute the next-to-next-to-leading order (NNLO) contributions to the splitting functions governing the evolution of the unpolarized flavour-singlet parton densities in perturbative QCD. The exact expressions are presented in both Mellin- N and Bjorken- x space. We also provide accurate parametrizations for practical applications. Our results agree with all partial results available in the literature. As in the non-singlet case, the correct leading logarithmic predictions for small momentum fractions x do not provide good estimates of the respective complete splitting functions. We investigate the size of the corrections and the stability of the NNLO evolution under variation of the renormalization scale. The perturbative expansion appears to converge rapidly at x ≳10 −3 . Relatively large third-order corrections are found at smaller values of x .

The three-loop splitting functions in QCD: the non-singlet case

We compute the next-to-next-to-leading order (NNLO) contributions to the three splitting functions governing the evolution of unpolarized non-singlet combinations of quark densities in perturbative QCD. Our results agree with all partial results available in the literature. We find that the correct leading logarithmic (LL) predictions for small momentum fractions x do not provide a good estimate of the respective complete results. A new, unpredicted LL contribution is found for the colour factor d abc d abc entering at three loops for the first time. We investigate the size of the corrections and the stability of the NNLO evolution under variation of the renormalization scale. Except for very small x the corrections are found to be rather small even for large values of the strong coupling constant, in principle facilitating a perturbative evolution into the sub-GeV regime.

Improved approximations for the three-loop splitting functions in QCD

We update our approximate parametrizations of the three-loop splitting functions for the evolution of unpolarized parton densities in perturbative QCD. The new information taken into account is given by the additional Mellin moments recently calculated by Retey and Vermaseren. The inclusion of these constraints reduces the uncertainties of our approximations considerably and extends their region of applicability by about one order of magnitude to lower momentum fractions x.

Higher order corrections in perturbative quantum chromodynamics

We present some techniques which have been developed recently or in the recent past to compute Feynman graphs beyond one-loop order. These techniques are useful to compute the three-loop splitting functions in QCD and to obtain the complete second order QED corrections to Bhabha scattering.

Light-cone gauge and the calculation of the two-loop splitting functions

We present calculations of next-to-leading order QCD splitting functions, employing the light-cone gauge method of Curci, Furmanski, and Petronzio (CFP). In contrast to the `principal-value' prescription used in the original CFP paper for dealing with the poles of the light-cone gauge gluon propagator, we adopt the Mandelstam-Leibbrandt prescription which is known to have a solid field-theoretical foundation. We find that indeed the calculation using this prescription is conceptionally clear and avoids the somewhat dubious manipulations of the spurious poles required when the principal-value method is applied. We reproduce the well-known results for the flavour non-singlet splitting function and the N_C^2 part of the gluon-to-gluon singlet splitting function, which are the most complicated ones, and which provide an exhaustive test of the ML prescription. We also discuss in some detail the x=1 endpoint contributions to the splitting functions.