Non-singlet splitting functions at four loops
We present a complete calculation of the non-singlet splitting functions in four-loop quantum chromodynamics.
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arXiv subjects
Vasily Sotnikov
Publications and source records attributed to Vasily Sotnikov.
How to choose a good rational basis for elliptic Feynman integrals?
Representing multi-loop scattering amplitudes as linear combinations of multivalued transcendental functions with process-dependent rational coefficients has long been understood to be advantageous. In general, these transcendental functions satisfy a system of differential equations with coupled homogeneous blocks. When these coupled blocks can be removed through algebraic basis transformations, the relation between the rational and algebraic bases is universal and minimal. Here, we ask whether an analogous universal and minimal relation exists when decoupling requires transformations involving complete elliptic integrals. We elaborate on the method proposed in ref. arXiv:2504.20897, in which we suggested that a basis constructed using an elliptic generalization of leading singularities may provide an answer to this question. We extend this analysis to the off-diagonal blocks of the rational differential equations satisfied by the bases obtained through this construction. We find that the $ε$ dependence of these blocks can be organized into a universal structure that is preserved under the decoupling transformation used to express the solutions in terms of iterated integrals.
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Two-loop leading-color QCD corrections for Higgs plus two-jet production in the heavy-top limit
We present the leading-color two-loop QCD corrections for Higgs-boson production in association with two jets through gluon fusion in the heavy-top effective theory. We provide analytic expressions for the finite remainders of the helicity amplitudes, written in terms of one-mass pentagon functions with spinor-helicity coefficients. These expressions are obtained by reconstructing the amplitudes from numerical finite-field samples computed within the numerical unitarity framework. The reconstruction is made possible by several advances in exploiting the analytic structure of the amplitudes, which both reduce the number of required samples and lead to compact representations. In particular, we introduce a new algorithm for multivariate partial fraction decomposition, based on a generic bivariate slice and a simplified treatment of ideal intersections. Using the resulting analytic expressions, we provide an efficient and stable implementation of their numerical evaluation, ready for phenomenological applications. Finally, we study the singularity structure of the remainders and confirm the existence of a threshold at non-degenerate physical momentum configurations, usually associated with massive virtual particle exchanges.
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The four-loop non-singlet splitting functions in QCD
The scale evolution of parton distributions is governed by splitting functions. We compute the four-loop splitting functions in perturbative QCD that control the evolution of quark non-singlet distributions. We confirm previous partial results and obtain, for the first time, fully analytic expressions for all non-singlet contributions at this order. These allow us to extract the analytic form of the four-loop virtual and rapidity anomalous dimensions entering logarithmic resummation. We provide precise numerical representations of the splitting functions suitable for parton evolution.
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The Four-Jet Rate in Electron-Positron Annihilation at Order $α_s^4$
We compute for the first time the production rate for four jets in electron-positron annihilation at next-to-next-to-leading order. Our calculation exhibits the highest final-state jet multiplicity considered at this perturbative accuracy to date. The cancellation of infrared singularities is achieved in the antenna subtraction scheme, relying particularly on generalized antenna functions. The evaluation of the two-loop virtual corrections is enabled by the construction of a new basis of transcendental special functions tailored to four-particle decay kinematics. Our results are compared with LEP data, finding improved agreement with respect to the next-to-leading order calculation. In the region where perturbative predictions are most reliable, we observe a significant reduction of theory uncertainties, which now fall below the experimental ones.
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Elliptic leading singularities and canonical integrands
In the well-studied genus zero case, bases of $\mathrm{d}\log$ integrands with integer leading singularities define Feynman integrals that automatically satisfy differential equations in canonical form. Such integrand bases can be constructed without input from the differential equations and without explicit involvement of dimensional regularization parameter $ε$. We propose a generalization of this construction to genus one geometry arising from the appearance of elliptic curves. We argue that a particular choice of algebraic one-forms of the second kind that avoids derivatives is crucial. We observe that the corresponding Feynman integrals satisfy a special form of differential equations that has not been previously reported, and that their solutions order by order in $ε$ yield pure functions. We conjecture that our integrand-level construction universally leads to such differential equations.
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Compact Two-Loop QCD Corrections for $Vjj$ Production in Proton Collisions
We present compact two-loop QCD corrections in the leading-color approximation for the production of an electroweak vector boson, $V = \{W^{\pm}, Z,γ^\star\}$, in association with two light jets ($Vjj$) at hadron colliders. Leptonic decays of the electroweak boson are included at the amplitude level. Working in the analytic-reconstruction approach, we develop two techniques to build compact partial-fraction forms for individual rational functions. One approach exploits their analytic structure. In the other, we iteratively construct subtraction terms that match the rational functions in singular limits. Moreover, we show how the singular behavior of the rational functions can be systematically used to find a more compact basis of the space that they span. We apply our techniques to the $Vjj$ amplitudes, yielding a representation that is three orders of magnitude smaller than previous results. We then use these compact expressions to provide an efficient and stable C++ numerical implementation suitable for phenomenological applications.
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Two-Loop Five-Point Two-Mass Planar Integrals and Double Lagrangian Insertions in a Wilson Loop
We consider the complete set of planar two-loop five-point Feynman integrals with two off-shell external legs. These integrals are relevant, for instance, for the calculation of the second-order QCD corrections to the production of two heavy vector bosons in association with a jet or a photon at a hadron collider. We construct pure bases for these integrals and reconstruct their analytic differential equations in canonical form through numerical sampling over finite fields. The newly identified symbol alphabet, one of the most complex to date, provides valuable data for bootstrap methods. We then apply our results to initiate the study of double Lagrangian insertions in a four-cusp Wilson loop in planar maximally supersymmetric Yang-Mills theory, computing it through two loops. We observe that it is finite, conformally invariant in four dimensions, and of uniform transcendentality. Furthermore, we provide numerical evidence for its positivity within the amplituhedron region through two loops.
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Next-to-next-to-leading order event generation for Z-boson production in association with a bottom-quark pair
We consider the production of a Z boson decaying to leptons in association with a bottom-quark pair in hadronic collisions. For the first time, we compute predictions at next-to-next-to-leading order (NNLO) in QCD, and we combine them with the all-orders radiative corrections from a parton-shower simulation (NNLO+PS). Our method represents the first approach to NNLO+PS event generation applicable to processes featuring a colour singlet and a heavy-quark pair in the final state. The novel two-loop corrections are computed for massless bottom quarks, and the leading mass corrections are restored through a small-mass expansion. The calculation is carried out in the four-flavour scheme, and we find that the sizeable NNLO QCD corrections lift the long-standing tension between lower-order predictions in four- and five-flavour schemes. Our predictions are compared to a CMS measurement for Z boson plus b-jet production, achieving an excellent description of the data.
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Double-Virtual NNLO QCD Corrections for Five-Parton Scattering: The Gluon Channel
We compute the two-loop helicity amplitudes for the scattering of five gluons, including all contributions beyond the leading-color approximation. The analytic expressions are represented as linear combinations of transcendental functions with rational coefficients, which we reconstruct from finite-field samples obtained with the numerical unitarity method. Guided by the requirement of removing unphysical singularities, we find a remarkably compact generating set of rational coefficients, which we are able to display entirely in the manuscript. We implement our results in a public code, which provides efficient and reliable numerical evaluations for phenomenological applications.
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Double-Virtual NNLO QCD Corrections for Five-Parton Scattering: The Quark Channels
We complete the computation of two-loop helicity amplitudes required to obtain next-to-next-to-leading order QCD corrections for three-jet production at hadron colliders, including all contributions beyond the leading-color approximation. The analytic expressions are reconstructed from finite-field samples obtained with the numerical unitarity method. We find that the reconstruction is significantly facilitated by exploiting the overlaps between rational coefficient functions of quark and gluon processes, and we display their compact generating sets in the appendix of the paper. We implement our results in a public code, and demonstrate its suitability for phenomenological applications.
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All Two-Loop Feynman Integrals for Five-Point One-Mass Scattering
We compute the complete set of two-loop master integrals for the scattering of four massless particles and a massive one. Our results are ready for phenomenological applications, removing a major obstacle to the computation of complete next-to-next-to-leading order (NNLO) QCD corrections to processes such as the production of a $H/Z/W$ boson in association with two jets at the LHC. Furthermore, they open the door to new investigations into the structure of quantum-field theories and provide precious analytic data for studying the mathematical properties of Feynman integrals.
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Complete $N_f^2$ contributions to four-loop pure-singlet splitting functions
The scale evolution of parton distributions is determined by universal splitting functions. As a milestone towards the computation of these functions to four-loop order in QCD, we compute all contributions to the pure-singlet quark-quark splitting functions that involve two closed fermion loops. The splitting functions are extracted from the pole terms of off-shell operator matrix elements, and the workflow for their calculation is outlined. We reproduce known results for the non-singlet four-loop splitting functions and validate our new pure-singlet results against fixed Mellin moments.
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The $N_f \,C_F^3$ contribution to the non-singlet splitting function at four-loop order
We report a new result for the $N_f \,C_F^3$ contribution to the four-loop anomalous dimensions of non-singlet, twist-two operators in Quantum Chromodynamics. This result is obtained through computations of off-shell operator matrix elements. Employing integration-by-parts reductions and differential equations with respect to a tracing parameter allowed us to derive analytic results valid for arbitrary Mellin moment $n$.
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Two-Loop QCD Corrections for Three-Photon Production at Hadron Colliders
We complete the computation of the two-loop helicity amplitudes for the production of three photons at hadron colliders, including all contributions beyond the leading-color approximation. We reconstruct the analytic form of the amplitudes from numerical finite-field samples obtained with the numerical unitarity method. This method requires as input surface terms for all relevant five-point non-planar integral topologies, which we obtain by solving the associated syzygy problem in embedding space. The numerical samples are used to constrain compact spinor-helicity ansätze, which are optimized by taking advantage of the known one-loop analytic structure. We make our analytic results available in a public C++ library, which is suitable for immediate phenomenological applications. We estimate that the inclusion of the subleading-color contributions will decrease the size of the two-loop corrections by about 30% to 50%, and the NNLO cross sections by a few percent, compared to the results in the leading-color approximation.
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Pentagon Functions for One-Mass Planar Scattering Amplitudes
We present analytic results for all planar two-loop Feynman integrals contributing to five-particle scattering amplitudes with one external massive leg. We express the integrals in terms of a basis of algebraically-independent transcendental functions, which we call one-mass pentagon functions. We construct them by using the properties of iterated integrals with logarithmic kernels. The pentagon functions are manifestly free of unphysical branch cuts, do not require analytic continuation, and can be readily evaluated over the whole physical phase space of the massive particle production channel. We develop an efficient algorithm for their numerical evaluation and present a public implementation suitable for direct phenomenological applications.
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Pentagon Functions for Scattering of Five Massless Particles
We complete the analytic calculation of the full set of two-loop Feynman integrals required for computation of massless five-particle scattering amplitudes. We employ the method of canonical differential equations to construct a minimal basis set of transcendental functions, pentagon functions, which is sufficient to express all planar and nonplanar massless five-point two-loop Feynman integrals in the whole physical phase space. We find analytic expressions for pentagon functions which are manifestly free of unphysical branch cuts. We present a public library for numerical evaluation of pentagon functions suitable for immediate phenomenological applications.
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Triphoton production at hadron colliders in NNLO QCD
We present next-to-next-to-leading-order (NNLO) QCD corrections to the production of three isolated photons in hadronic collisions at the fully differential level. We employ qT subtraction within MATRIX and an efficient implementation of analytic two-loop amplitudes in the leading-colour approximation to achieve the first on-the-fly calculation for this process at NNLO accuracy. Numerical results are presented for proton-proton collisions at energies ranging from 7 TeV to 100 TeV. We find full agreement with the 8 TeV results of arXiv:1911.00479 and confirm that NNLO corrections are indispensable to describe ATLAS 8 TeV data. In addition, we demonstrate the significance of NNLO corrections for future precision studies of triphoton production at higher collision energies.
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