We present a fully analytic calculation of the leading-order one-loop amplitude for triple Higgs production via gluon fusion, gg -> HHH, retaining full dependence on the mass of the heavy quark circulating in the loop. This amplitude provides a direct probe of the triple and quartic Higgs self-couplings, the measurement of which is a central goal of current and future colliders. The amplitude can be presented in compact form thanks to the use of analytic reconstruction techniques, based on finite-field and p-adic evaluations, multivariate partial fraction decompositions, and primary decompositions to identify common numerator factors. Although full analytic results are given in the text and in the supplementary material, the main thrust of this paper is to further test and illustrate these analytic reconstruction techniques in a concrete physical example. Our results provide a compact and efficient representation of the matrix element for this process, enabling evaluations that are more than an order of magnitude faster than existing numerical alternatives. Full analytic control of the leading-order, loop-induced amplitude is an important step towards handling more complex 2-loop or real-radiation corrections to this and related processes.

We present a systematic decomposition of QCD splitting functions into scalar dipole radiators and pure splitting remainders up to second order in the strong coupling. The individual components contain terms that are formally sub-leading in soft or collinear scaling parameters, but well understood and universal due to their origin in scalar QCD. The multipole radiator functions which we derive share essential features of the known double-soft and one-loop soft gluon currents, and are not based on kinematical approximations.

We introduce a local infrared subtraction method for next-to-next-to-leading order QCD calculations in color singlet decays, with counterterms based on scalar radiators and pure splitting functions. Overlapping singularities in the multipole radiation pattern are disentangled by partial fractioning, and the kinematics mapping corresponds to iterated next-to-leading order kinematics. We verify that the double-real remainder to e^+e^-→ qq̅ is rendered finite in the single and double unresolved limits and investigate the numerical convergence of the Monte-Carlo integral. We compute the phase-space integrals of the scalar counterterms in the back-to-back configuration, both analytically and with the help of numerical techniques based on sector decomposition.

We present an analytic reconstruction of one-loop amplitudes for the process 0→q qttH . Our calculation is a novel use of analytic reconstruction, retaining explicit covariance in the massive spin states through the massive spinor-helicity formalism. The analytic reconstruction relies on embedding the massive five-point kinematics in a fully massless eight-point phase space while still building a minimal ansatz directly in the five-point phase space. In order to obtain compact analytic expressions it is necessary to identify suitable partial fraction decompositions and extract common numerator factors, which we achieve through careful inspection of limits in which pairs of denominators vanish. We find that the resulting amplitudes are more numerically efficient than ones computed using automatic methods but that the gains are not as significant as in the massless case, at least at present. The method opens the door to applications at two-loop order, where numerical efficiency and improvements in the reconstruction methodology are more crucial, especially with regards to the number of free parameters in the ansatz.

The level 3 case for Ramanujan-type series has been considered as the most mysterious and the most challenging, out of all possible levels for Ramanujan-type series. This motivates the development of new techniques for constructing Ramanujan-type series of level 3. Chan and Liaw introduced an alternating analogue of the Borwein brothers’ identity for Ramanujan-type series of level 3; subsequently, Chan, Liaw, and Tian formulated another proof of the Chan–Liaw identity, via the use of Ramanujan’s class invariant. Using the elliptic lambda function and the elliptic alpha function, we prove, via a limiting case of the Kummer–Goursat transformation, a new identity for evaluating the summands for alternating Ramanujan-type series of level 3, and we apply this new identity to prove three conjectured formulas for quadratic-irrational, Ramanujan-type series that had been discovered via numerical experiments with Maple in 2012 by Aldawoud. We also apply our identity to prove a new Ramanujan-type series of level 3 with a quartic convergence rate and quartic coefficients.

We present an analytic reconstruction of one-loop amplitudes for the process $$ 0\to \overline{q} qt\overline{t}H $$ 0 → q ¯ qt t ¯ H . Our calculation is a novel use of analytic reconstruction, retaining explicit covariance in the massive spin states through the massive spinor-helicity formalism. The analytic reconstruction relies on embedding the massive five-point kinematics in a fully massless eight-point phase space while still building a minimal ansatz directly in the five-point phase space. In order to obtain compact analytic expressions it is necessary to identify suitable partial fraction decompositions and extract common numerator factors, which we achieve through careful inspection of limits in which pairs of denominators vanish. We find that the resulting amplitudes are more numerically efficient than ones computed using automatic methods but that the gains are not as significant as in the massless case, at least at present. The method opens the door to applications at two-loop order, where numerical efficiency and improvements in the reconstruction methodology are more crucial, especially with regards to the number of free parameters in the ansatz.

We present an analytic reconstruction of one-loop amplitudes for the process 0 -> q (q) over bar qt (t) over bar >H. Our calculation is a novel use of analytic reconstruction, retaining explicit covariance in the massive spin states through the massive spinor-helicity formalism. The analytic reconstruction relies on embedding the massive five-point kinematics in a fully massless eight-point phase space while still building a minimal ansatz directly in the five-point phase space. In order to obtain compact analytic expressions it is necessary to identify suitable partial fraction decompositions and extract common numerator factors, which we achieve through careful inspection of limits in which pairs of denominators vanish. We find that the resulting amplitudes are more numerically efficient than ones computed using automatic methods but that the gains are not as significant as in the massless case, at least at present. The method opens the door to applications at two-loop order, where numerical efficiency and improvements in the reconstruction methodology are more crucial, especially with regards to the number of free parameters in the ansatz.

We introduce a Combinatorial Hopf Algebra (CHA) with bases indexed by the partition diagrams indexing the bases for partition algebras. By analogy with the operation H alpha H beta=H alphabeta for the complete homogeneous basis of the CHA NSym given by concatenating compositions alpha and beta, we mimic this multiplication rule by setting H pi H rho=H pi circle times rho for partition diagrams pi and rho and for the horizontal concatenation pi circle times rho of pi and rho. This gives rise to a free, graded algebra ParSym , which we endow with a CHA structure.

The pair production of Higgs bosons at the LHC can give information about the triple Higgs boson coupling. We perform an analytic one-loop calculation of the amplitudes for a pair of Higgs bosons in association with three partons, retaining the exact dependence on the quark mass circulating in the loop. These amplitudes constitute the real radiation corrections in the calculation of Higgs boson pair production at next-to-leading order in the strong coupling. The results of an analytic generalised-unitarity computation are simplified via analytic reconstruction in spinor variables. Compact ansätze for kinematic pole residues are iteratively fitted via p-adic evaluations near said poles and subtracted until no pole remains. A new ansatz construction is introduced to minimally parametrise coefficients of amplitudes with multiple massive external legs. The simplified expressions are faster to evaluate than automatic codes and can lead to more stable results near singular regions.

Diboson processes are one of the most accessible and stringent probes of the Standard Model's electroweak gauge structure at the LHC. They will be probed at the percent level at the high-luminosity LHC, challenging current theory predictions. We present transverse momentum resummed calculations at N3LL+NNLO for the processes $ZZ$, $WZ$, $WH$ and $ZH$, compare our predictions with most recent LHC data and present predictions at 13.6 TeV including theory uncertainty estimates. For $W^+W^-$ production we further present jet-veto resummed results at N3LLp+NNLO. Our calculations will be made publicly available in the upcoming MCFM release and allow future analyses to take advantage of improved predictions.

Vetoing energetic jet activity is a crucial tool for suppressing backgrounds and enabling new physics searches at the LHC, but the introduction of a veto scale can introduce large logarithms that may need to be resummed. We present an implementation of jet-veto resummation for color-singlet processes at the level of N$^3$LL$_\text{p}$ matched to fixed-order NNLO predictions. Our public code MCFM allows for predictions of a single boson, such as $Z/\gamma^*$, $W^{\pm}$ or $H$, or with a pair of vector bosons, such as $W^+W^-$, $W^{\pm} Z$ or $ZZ$. The implementation relies on recent calculations of the soft and beam functions in the presence of a jet veto over all rapidities, with jets defined using a sequential recombination algorithm with jet radius $R$. However one of the ingredients that is required to reach full N$^3$LL accuracy is only known approximately, hence N$^3$LL$_\text{p}$. We describe in detail our formalism and compare with previous public codes that operate at the level of NNLL. Our higher-order predictions improve significantly upon NNLL calculations by reducing theoretical uncertainties. We demonstrate this by comparing our predictions with ATLAS and CMS results.

We introduce a method that is based on Fourier series expansions related to Jacobi elliptic functions and that we apply to determine new identities for evaluating hyperbolic infinite sums in terms of the complete elliptic integrals $K$ and $E$. We apply our method to determine generalizations of a family of $\text{sech}$-sums given by Ramanujan and generalizations of a family of $\text{csch}$-sums given by Zucker. Our method has the advantage of producing evaluations for hyperbolic sums with sign functions that have not previously appeared in the literature on hyperbolic sums. We apply our method using the Jacobian elliptic functions $\text{dc}$ and $\text{nc}$, together with the elliptic alpha function, to obtain new closed forms for $q$-digamma expressions, and new closed forms for series related to discoveries due to Ramanujan, Berndt, and others.

Consider the integer sequences (F-[root n] : n is an element of N-0) and (F-[log2n] : n is an element of N), letting [x] denote the integer part of a nonnegative value x, and where F-n denotes the nth Fibonacci number for a nonnegative integer n. We apply an Abel-type summation lemma to prove explicit evaluations for Sigma(m)(n=1) F-[root n] and Sigma(m)(n=1) F-[log2n] for a natural number m. We then apply this summation lemma to determine an analytical formula for Sigma(m)(n=1) F-[n/s], letting s denote a natural number parameter, and we demonstrate how our method may be applied to evaluate sums of the form Sigma(m)(n=1) F-[r root n/s] s k for integers r >= 2 and s >= 1. We also consider the problem of evaluating finite sums of expressions of the form F-[log2( n/s)] for a natural number s. Much of our work is closely connected with evaluations for Fibonacci sums of the form S(t, m) = Sigma(m)(n=1) n(t)F(n), where t is a nonnegative integer.

Drell-Yan production is one of the precision cornerstones of the LHC , serving as calibration for measurements such as the W -boson mass. Its extreme precision at the level of 1% challenges theory predictions at the highest level. We present the first independent calculation of Drell-Yan production at order α 3 s in transverse-momentum ( q T ) resummation improved perturbation theory. Our calculation reaches the state-of-the-art through inclusion of the recently published four loop rapidity anomalous dimension and three loop massive axial-vector contributions. We compare to the most recent data from CMS with fiducial and differential cross-section predictions and find excellent agreement at the percent level. Our resummed calculation including the matching to Z +jet production at NNLO is publicly available in the upcoming CuTe-MCFM 10.3 release and allows for theory-data comparison at an unprecedented level.

A “harmonic variant” of Zeilberger’s algorithm is utilized to improve upon the results introduced by Wang and Chu [ Ramanujan J. 52 (2020) 641–668]. Wang and Chu’s coefficient-extraction methodologies yielded evaluations for Ramanujan-like series involving summand factors of the form H 3 n +3 H n H (2) n +2 H (3) n , where H n denotes a harmonic number and H ( x ) n is a generalized harmonic number. However, it is unclear as to how Wang and Chu’s techniques could be applied to improve upon such results by separately evaluating the series obtained upon the expansion of the summands according to the terms of the factor H 3 n +3 H n H (2) n +2 H (3) n . In this note, we succeed in applying Zeilberger’s algorithm toward this problem, providing explicit evaluations for the series with a factor of the form H (3) n obtained from the aforementioned expansion. Our approach toward generalizing Zeilberger’s algorithm to non-hypergeometric expressions may be applied much more broadly. The series obtained by replacing H (3) n with H (2) n were highlighted as especially beautiful motivating examples in Wang and Chu’s article. These H (2) n -series motivate our main results, which are natural higher-order extensions of these H (2) n -series.

We outline a new technique for the fully-differential matching of final-state parton showers to NNLO calculations, focussing here on the simplest case of leptonic collisions with two final-state jets. The strategy is facilitated by working in the antenna formalism, making use of NNLO antenna subtraction on the fixed-order side and the sector-antenna framework on the shower side. As long as the combined real-virtual and double-real corrections do not overcompensate the real-emission term in the three-jet region, negative weights can be eliminated from the matching scheme. We describe the implementation of all necessary components in the VINCIA antenna shower in PYTHIA 8.3.

We present compact analytic results for tree-level amplitudes containing a tt pair accompanied by up to four massless partons, tt gg , tt ggg , tt gggg , ttqq , ttqqg , ttqq gg and ttqqq^'q^' . The results, obtained using BCFW on-shell recursion, are based both on previous published results and on the new calculations performed in this paper. These amplitudes are sufficient to calculate the production of a tt pair and zero, one, or two light parton jets, with the option to include the tree-level decays t → bνe+ and t → be − ν efficiently. Our results are part of the NNLO corrections to tt production including the decay correlations for on-shell top quarks.

In these proceedings, we apply the recently developed S-ACOT-MPS factorization scheme at the next-to-leading order to prompt charm production at hadron colliders.It provides a good agreement with experimental data on charm meson production measured by LHCb at 7 and 13 TeV.The low-p T data are on the margins of the theoretical error bands, emphasizing the importance of including contributions beyond the next-to-leading order.

In these proceedings, we apply the recently developed S-ACOT-MPS factorization scheme at the next-to-leading order to prompt charm production at hadron colliders. It provides a good agreement with experimental data on charm meson production measured by LHCb at 7 and 13 TeV. The low-$p_T$ data are on the margins of the theoretical error bands, emphasizing the importance of including contributions beyond the next-to-leading order.