Chebyshev Approximations of Feynman Integrals for Collider Physics
CERN（欧洲核子研究中心）；Higgs Centre for Theoretical Physics, School of Physics and Astronomy, The University of Edinburgh（爱丁堡大学物理与天文学学院希格斯理论物理中心）；Instituto Superior Técnico (IST), Universidade de Lisboa（里斯本大学高级技术研究所）；LIP（葡萄牙粒子与核物理研究所）；Department of Physics and Astronomy, Ghent University（根特大学物理与天文学系）
We present a novel approach for solving canonical differential equations for Feynman integrals based on an approximation of the integrals with Chebyshev polynomials. By exploiting the analyticity properties of Feynman integrals, the method constructs rapidly converging polynomial approximations along a path, enabling highly efficient numerical evaluation. Moreover, we introduce an adaptive approximation method that dynamically samples to optimise convergence. We implement this framework in double-precision arithmetic and demonstrate its stability across physical phase space using a series of two-loop, five-point examples. Our proof-of-principle implementation proves competitive with state-of-the-art one-fold integral methods, while requiring little to no case-by-case intervention to handle spurious singularities.
Feynman diagrams constitute one of the essential ingredients for making precision predictions for collider experiments. Yet, while the simplest Feynman diagrams can be evaluated in terms of multiple polylogarithms -- whose properties as special functions are well understood -- more complex diagrams often involve integrals over complicated algebraic manifolds. Such diagrams already contribute at NNLO to the self-energy of the electron, $t \bar{t}$ production, $γγ$ production, and Higgs decay, and appear at two loops in the planar limit of maximally supersymmetric Yang-Mills theory. This makes t…
Two-loop Six-point Planar Massless Feynman Integrals to Higher $ε$ Orders
University of Science and Technology of China（中国科学技术大学）；Johannes Gutenberg-Universität Mainz（美因茨约翰内斯古腾堡大学）；Universitá di Bologna e INFN, Sezione di Bologna（博洛尼亚大学及意大利国家核物理研究所博洛尼部分所）；Karlsruhe Institute of Technology (KIT)（卡尔斯鲁厄理工学院）；Peng Huanwu Center for Fundamental Theory（彭桓武基础理论中心）；Center for High Energy Physics, Peking University（北京大学高能物理中心）
These lectures are intended to provide the theoretical basis of describing high-energy particle collisions at a level appropriate to graduate students in experimental high energy physics. They are supposed to be familiar with quantum electrodynamics, the concept of Feynman rules, Feynman graphs and computation of the cross section in quantum field theory.
We calculate the cross sections for the neutral Higgs-boson production at e+e- colliders in the Minimal Supersymmetric Standard Model (MSSM) using the Feynman-diagrammatic approach and the on-shell renormalization scheme. We incorporate the Higgs-boson propagator corrections, evaluated up to two-loop order, into the prediction of the cross sections for the Higgs-boson production mechanism e+e- -> hZ, hA. The propagator corrections consist of the full one-loop contribution, including the effects of non-vanishing external momentum, and at the two-loop level of the dominant corrections of O(alpha…
Two-loop corrections to scattering amplitudes are crucial theoretical input for collider physics. Recent years have seen tremendous advances in computing Feynman integrals, scattering amplitudes, and cross sections for five-particle processes. In this paper, we initiate the study of the function space for planar two-loop six-particle processes. We study all genuine six-particle Feynman integrals, and derive the differential equations they satisfy on maximal cuts. Performing a leading singularity analysis in momentum space, and in Baikov representation, we find an integral basis that puts the d…
We present compact analytic expressions for all one-loop helicity amplitudes contributing to ttbar production at hadron colliders. Using recently developed generalised unitarity methods and a traditional Feynman based approach we produce a fast and flexible implementation.
In this paper, we propose a new method for evaluating scalar one-loop Feynman integrals in generalized D-dimension. The calculations play an important building block for two-loop and higher-loop corrections to the processes at future colliders such as the Large Hadron Collider (LHC) and the International Linear Collider (ILC). In this method, scalar one-loop N-point functions will be presented as the one-fold Mellin-Barnes representation of (N-1)-point ones with shifting space-time dimension. This representation offers a clear advantage that we can construct recursively the analytic expression…
A recently derived approach to the tensor reduction of 5-point one-loop Feynman integrals expresses the tensor coefficients by scalar 1-point to 4-point Feynman integrals completely algebraically. In this letter we derive extremely compact algebraic expressions for the contractions of the tensor integrals with external momenta. This is based on sums over signed minors weighted with scalar products of the external momenta. With these contractions one can construct the invariant amplitudes of the matrix elements under consideration, and the evaluation of one-loop contributions to massless and ma…
Instituto de F\' sica Corpuscular, Universitat de Val\` e ncia -- Consejo Superior de Investigaciones Cient\' ficas, Parc Cient\' fic, E-46980 Paterna, Valencia, Spain
The analytic integration and simplification of multi-loop Feynman integrals to special functions and constants plays an important role to perform higher order perturbative calculations in the Standard Model of elementary particles. In this survey article the most recent and relevant computer algebra and special function algorithms are presented that are currently used or that may play an important role to perform such challenging precision calculations in the future. They are discussed in the context of analytic zero, single and double scale calculations in the Quantum Field Theories of the St…
Tremendous ongoing theory efforts are dedicated to developing new methods for QCD calculations. Qualitative rather than incremental advances are needed to fully exploit data still to be collected at the LHC. The maximally supersymmetric Yang-Mills theory (${\mathcal N}=4$ sYM) shares with QCD the gluon sector, which contains the most complicated Feynman graphs, but at the same time has many special properties, and is believed to be solvable exactly. It is natural to ask what we can learn from advances in ${\mathcal N}=4$ sYM for addressing difficult problems in QCD. With this in mind, we revie…
An approach for an effective computer evaluation of one-loop multi-leg diagrams is proposed. It's main feature is the combined use of several systems - DIANA, FORM and MAPLE. As an application we consider the one-loop correction to Higgs production in e+ e- to nu anti-nu H, which is important for future e+ e- colliders. To improve the stability of numerical evaluations a non-standard basis of integrals is introduced by transforming integrals to higher dimensions.
With a key improvement, the auxiliary mass flow method is now able to compute Feynman integrals encountered in cutting-edge collider processes. We have successfully applied it to compute some integrals involved in two-loop electroweak corrections to $e^+e^-\to HZ$, two-loop QCD corrections to $3j$, $W/Z/H+2j$, $t\bar{t}H$ and $4j$ production at hadron colliders, and three-loop QCD corrections to $t\bar{t}$ production at hadron colliders, all of which are crucial for precision frontier in collider physics in the following decade. Our results are important building blocks and benchmarks for futu…
We present standard model predictions for a four fermion reaction e+ e- -> mu+ mu- bar_b b being one of the best detection channels of a low mass Higgs boson produced through the Higgsstrahlung mechanism at a linear collider. We include leading virtual and real QED corrections to all the lowest order Feynman diagrams of e+ e- -> mu+ mu- bar_b b and the complete O(alpha) electroweak corrections to the Z-Higgs production and to the Z and Higgs boson decay widths in the double pole approximation.
There are steady advances in the calculation of electroweak corrections to massive scattering problems at colliders, from the very beginning in the nineteen seventies until contemporary developments. Recent years brought a remarkable progress due to new calculational technologies. This was motivated by demands from phenomenological applications at particle accelerators: higher multiplicities of the final states, extreme kinematics, need of higher precision and thus of higher orders in perturbation theory. We describe selected contributions from the project "Massive particle production" of Sond…
In this paper, we study systematically scalar one-loop two-, three-, and four-point Feynman integrals with complex internal masses. Our analytic results presented in this report are valid for both real and complex internal masses. The calculations are then implemented into a {\tt Mathematica} (version $9$) package and {\tt FORTRAN} program. Our program is cross-checked numerically with {\tt LoopTools} (version $2.14$) in real as well as complex internal masses. We find a perfect agreement between our results and {\tt LoopTools} for all cases. Additionally, this work is applied for evaluating s…
We compute several total and differential cross sections relevant to top and WWZ physics at future $e^+e^-$ colliders taking into account the full set of Feynman diagrams for six fermion final states. We also include in our calculations initial state radiation and beamstrahlung effects, and the most important QCD corrections in an approximate (naive) form. We compare such a complete approach with "production x decay" approximation and we suggest that in many physical studies the former is needed.
We compute several total and differential cross sections relevant to top, WWZ and Higgs physics at future $e^+e^-$ colliders taking into account the full set of Feynman diagrams for six fermion final states. We examine in particular charged current processes, in which final particles cannot be formed by three Z's decay. We include in our calculations initial state radiation and beamstrahlung effects, and the most important QCD corrections in an approximate (naive) form. We also compare this complete approach with production $\times$ decay approximation.
Using an effective field theory approach for higher-spin fields, we derive the interactions of colour singlet and electrically neutral particles with a spin higher than unity, concentrating on the spin-3/2, spin-2, spin-5/2 and spin-3 cases. We compute the decay rates and production cross sections in the main channels for spin-3/2 and spin-2 states at both electron-positron and hadron colliders, and identify the most promising novel experimental signatures for discovering such particles at the LHC. The discussion is qualitatively extended to the spin-5/2 and spin-3 cases. Higher-spin particles…
The success of the experimental program at the Tevatron re-inforced the idea that precision physics at hadron colliders is desirable and, indeed, possible. The Tevatron data strongly suggests that one-loop computations in QCD describe hard scattering well. Extrapolating this observation to the LHC, we conclude that knowledge of many short-distance processes at next-to-leading order may be required to describe the physics of hard scattering. While the field of one-loop computations is quite mature, parton multiplicities in hard LHC events are so high that traditional computational techniques be…
WPHACT 2.0 is the new fully massive version of a MC program and unweighted event generator which computes all Standard Model processes with four fermions in the final state at e^+ e^- colliders. The program can now generate unweighted events for any subset of all four fermion final states in a single run, by making use of dedicated pre-samples which can cover the entire phase space. Improvements with respect to WPHACT 1.0 include the Imaginary Fermion Loop gauge restoring scheme, new phase space mappings, a new input system, the possibility to compute subsets of Feynman diagrams and options fo…
We study higgs physics at future $e^+e^-$ colliders taking into account the full set of Feynman diagrams for six fermion final states, which are produced for higgs masses near or above the two $W$'s threshold. In particular we examine events where one isolated lepton or two isolated leptons of different flavours signal the presence of two $\rm W^*$'s. For these charged current processes, a detailed analysis of the relevance of irreducible and QCD background shows that appropriate cuts are generally sufficient to deal with it in the case of reconstructed or missing higgs mass distributions. The…
Modern particle physics is increasingly becoming a precision science that relies on advanced theoretical predictions for the analysis and interpretation of experimental results. The planned physics program at the LHC and future colliders will require three-loop electroweak and mixed electroweak-QCD corrections to single-particle production and decay processes and two-loop electroweak corrections to pair production processes, all of which are beyond the reach of existing analytical and numerical techniques in their current form. This article presents a new semi-numerical approach based on diffe…
Low energy phenomenology of the unparticle physics associated with an exact scale invariant sector possessing a non-trivial infrared fixed point at a higher energy scale is explored for both electron-positron and hadronic colliders. Feynman rules for a spin 0, 1 or 2 unparticle coupled to a variety of standard model gauge invariant operators that are relevant to many low energy processes involving either real emissions of unparticles or their virtual propagator effects are presented. Missing energy and/or recoil mass distributions of the unparticle in the associated production of unparticle to…
We review the phenomenology of models with flat, compactified extra dimensions where all of the Standard Model fields are allowed to propagate in the bulk, known as Universal Extra Dimensions (UED). UED make for an interesting TeV-scale physics scenario, featuring a tower of Kaluza-Klein (KK) states approximately degenerate in mass at the scale set by the inverse size of the compactification radius. KK parity, the four-dimensional remnant of momentum conservation in the extra dimensions, implies two basic consequences: (1) contributions to Standard Model observables arise only at loop level, a…
The two-loop QCD corrections to vector boson pair production at hadron colliders involve a new class of Feynman integrals: two-loop four-point functions with two off-shell external legs. We describe their reduction to a small set of master integrals by solving linear relations among them. We then use differential equations in the external invariants to compute all master integrals that are relevant to planar Feynman amplitudes. Our results are expressed analytically in terms of generalised harmonic polylogarithms. The calculation relies heavily on techniques that exploit the algebraic structur…
We present results of a calculation of the cross section for the production of a charged Higgs boson in association with an electron and a neutrino at electron-positron colliders (e+ e- -> H+ e- nu_e-bar, H- e+ nu_e). We study predictions for the cross section in the Minimal Supersymmetric Standard Model (MSSM) and the Two Higgs Doublet Model (THDM), highlighting possible differences. The process is effectively loop-induced in both models. Hence, the cross section is expected to be strongly model-dependent. Most notably, due to the presence of superpartners, the MSSM amplitude contains Feynman…
One of the main challenges in obtaining predictions for collider experiments from perturbative quantum field theory, is the direct evaluation of the Feynman integrals it gives rise to. In this chapter, we review an alternative bootstrap method that instead efficiently constructs physical quantities by exploiting their analytic structure. We present in detail the setting where this method has been originally developed, six- and seven-particle amplitudes in the large-color limit of $\mathcal{N}=4$ super Yang-Mills theory. We discuss the class of functions these amplitudes belong to, and the stro…
We consider Feynman integrals with algebraic leading singularities and total differentials in $ε\,\mathrm{d}\ln$ form. We show for the first time that it is possible to evaluate integrals with singularities involving unrationalizable roots in terms of conventional multiple polylogarithms, by either parametric integration or matching the symbol. As our main application, we evaluate the two-loop master integrals relevant to the $αα_s$ corrections to Drell-Yan lepton pair production at hadron colliders. We optimize our functional basis to allow for fast and stable numerical evaluations in the phy…
The evaluation of higher-loop Feynman integrals is at the core of the quest to reduce the uncertainty of theoretical predictions and match experimental data from the LHC and future colliders. pySecDec is a program to evaluate such integrals numerically based on the sector decomposition approach; its new release version 1.5 introduces features significantly improving its performance: automatic adaptive evaluation of weighted sums of integrals (e.g. amplitudes) and asymptotic expansion in kinematic ratios. Here we briefly review both, illustrating the expected performance benefits.
An explicit investigation about the equal-mass two-loop sunrise Feynman graph is performed. Such perturbative amplitude is related with many important physical process treated in the standard model context. The background of this investigation is an alternative strategy to handle with the divergences typical of perturbative solutions of quantum field theory. Since its proposition, the mentioned method was exhaustively used to calculate and manipulate one-loop Feynman integrals with a great success. However, the great advances in precision of experimental data collected in particle physics coll…
We present version 3.4 of the CalcHEP software package which is designed for effective evaluation and simulation of high energy physics collider processes at parton level. The main features of CalcHEP are the computation of Feynman diagrams, integration over multi-particle phase space and event simulation at parton level. The principle attractive key-points along these lines are that it has: a) an easy startup even for those who are not familiar with CalcHEP; b) a friendly and convenient graphical user interface; c) the option for a user to easily modify a model or introduce a new model by eit…
On the Numerical Integration of One-Loop Cosmological Collider Signals
Institute for Advanced Study（高等研究院）；MIT Center for Theoretical Physics - a Leinweber Institute, Massachusetts Institute of Technology（麻省理工学院理论物理中心-莱因韦伯研究所）；Center for Cosmology and Particle Physics, New York University（纽约大学宇宙学与粒子物理中心）；Perimeter Institute for Theoretical Physics（围道理论物理研究所）
