Master Integrals Research Articles

Master integrals for mathcal{O} (ααs) corrections to H → ZZ*

We present analytic results for all the Feynman integrals relevant for mathcal{O} (ααs) virtual corrections to H → ZZ* decay. We use the method of differential equations to solve the master integrals while keeping the full dependence on the masses of all the particles, including those propagating in the loop. Due to the presence of four mass scales, we encounter multiple square roots. We argue that all the occurring square roots can not be rationalized at the same time as a simultaneous rationalization brings us to integrals over CY3 manifolds. Hence we rationalize only three square roots simultaneously and construct suitable ansätze to obtain dlog-forms containing the square roots after obtaining an epsilon-factorised form for the differential equations in D = 4 − 2ϵ dimensions. We present the alphabet and the analytic form of all the boundary constants that appear in the solutions of the differential equations. The results for master integrals are expressed in terms of Chen’s iterated integrals with dlog one-forms.

Two-loop master integrals for pseudo-scalar quarkonium and leptonium production and decay

We compute the master integrals relevant for the two-loop corrections to pseudo-scalar quarkonium and leptonium production and decay. We present both analytic and high-precision numerical results. The analytic expressions are given in terms of multiple polylogarithms (MPLs), elliptic multiple polylogarithms (eMPLs) and iterated integrals of Eisenstein series. As an application of our results, we obtain for the first time an analytic expression for the two-loop amplitude for para-positronium decay to two photons at two loops.

Soft integrals and soft anomalous dimensions at N3LO and beyond

We calculate soft phase-space and loop master integrals for the computation of color-singlet cross sections through N3LO in perturbative QCD. Our results are functions of homogeneous transcendental weight and include the first nine terms in the expansion in the dimensional regulator ϵ. We discuss the application of our results to the computation of deeply-inelastic scattering and e+e− annihilation processes. We use these results to compute the perturbative coefficient functions for the Drell-Yan and gluon-fusion Higgs boson production cross sections to higher orders in ϵ through N3LO in QCD in the limit where only soft partons are produced on top of the colorless final state. Furthermore, we extract the anomalous dimension of the inclusive threshold soft function and of the N-Jettiness beam and jet functions to N4LO in perturbative QCD.

Specializations of partial differential equations for Feynman integrals

Starting from the Mellin–Barnes integral representation of a Feynman integral depending on a set of kinematic variables zi, we derive a system of partial differential equations w.r.t. new variables xj, which parameterize the differentiable constraints zi=yi(xj). In our algorithm, the powers of propagators can be considered as arbitrary parameters. Our algorithm can also be used for the reduction of multiple hypergeometric sums to sums of lower dimension, finding special values and reduction equations of hypergeometric functions in a singular locus of continuous variables, or finding systems of partial differential equations for master integrals with arbitrary powers of propagators. As an illustration, we produce a differential equation of fourth order in one variable for the one-loop two-point Feynman diagram with two different masses and arbitrary propagator powers.

Evaluation of Feynman integrals with arbitrary complex masses via series expansions

We present an algorithm to evaluate multiloop Feynman integrals with an arbitrary number of internal massive lines, with the masses being in general complex-valued, and its implementation in the Mathematica package SeaSyde. The implementation solves by series expansions the system of differential equations satisfied by the Master Integrals. At variance with respect to other existing codes, the analytical continuation of the solution is performed in the complex plane associated to each kinematical invariant. We present the results of the evaluation of the Master Integrals relevant for the NNLO QCD-EW corrections to the neutral-current Drell-Yan processes. Program summaryProgram Title:SeaSydeCPC Library link to program files:https://doi.org/10.17632/4hk9rbw4y8.1Developer's repository link:https://github.com/TommasoArmadillo/SeaSydeLicensing provisions: GPLv3Programming language:MathematicaNature of problem: Solution of multi-loop Feynman integrals with an arbitrary number of internal complex-valued massive lines.Solution method:: Solution via series expansions of the systems of differential equations satisfied by the loop integrals. Analytic continuation of the solution to any point of the kinematic variables. The solution is transported from the initial to the final point, by moving in the complex plane associated to each kinematic variable. In multi-variable problems, the transport is computed one variable at the time.

Glucose-driven TOR-FIE-PRC2 signalling controls plant development.

Nutrients and energy have emerged as central modulators of developmental programmes in plants and animals1-3. The evolutionarily conserved target of rapamycin (TOR) kinase is a master integrator of nutrient and energy signalling that controls growth. Despite its key regulatory roles in translation, proliferation, metabolism and autophagy2-5, little is known about how TOR shapes developmental transitions and differentiation. Here we show that glucose-activated TOR kinase controls genome-wide histone H3 trimethylation at K27 (H3K27me3) in Arabidopsis thaliana, which regulates cell fate and development6-10. We identify FERTILIZATION-INDEPENDENT ENDOSPERM (FIE), an indispensable component of Polycomb repressive complex 2 (PRC2), which catalyses H3K27me3(refs.6-8,10-12), as a TOR target. Direct phosphorylation by TOR promotes the dynamic translocation of FIE from the cytoplasm to the nucleus. Mutation of the phosphorylation site on FIE abrogates the global H3K27me3 landscape, reprogrammes the transcriptome and disrupts organogenesis in plants. Moreover, glucose-TOR-FIE-PRC2 signalling modulates vernalization-induced floral transition. We propose that this signalling axis serves as a nutritional checkpoint leading to epigenetic silencing of key transcription factor genes that specify stem cell destiny in shoot and root meristems and control leaf, flower and silique patterning, branching and vegetative-to-reproduction transition. Our findings reveal a fundamental mechanism of nutrient signalling in direct epigenome reprogramming, with broad relevance for the developmental control of multicellular organisms.

Higgs boson contribution to the leading two-loop Yukawa corrections to gg → HH

We analytically compute two-loop Yukawa corrections to Higgs boson pair production in the high-energy limit. Such corrections are generated by an exchange of a Higgs boson between the virtual top quark lines. We propose two approaches to obtain expansions of the massive two-loop box integrals and show that precise results are obtained for transverse momenta of the Higgs bosons above about 150 GeV. We discuss in detail the computation of all 140 master integrals and present analytic results.

Landau discriminants

Scattering amplitudes in quantum field theories have intricate analytic properties as functions of the energies and momenta of the scattered particles. In perturbation theory, their singularities are governed by a set of nonlinear polynomial equations, known as Landau equations, for each individual Feynman diagram. The singularity locus of the associated Feynman integral is made precise with the notion of the Landau discriminant, which characterizes when the Landau equations admit a solution. In order to compute this discriminant, we present approaches from classical elimination theory, as well as a numerical algorithm based on homotopy continuation. These methods allow us to compute Landau discriminants of various Feynman diagrams up to 3 loops, which were previously out of reach. For instance, the Landau discriminant of the envelope diagram is a reducible surface of degree 45 in the three-dimensional space of kinematic invariants. We investigate geometric properties of the Landau discriminant, such as irreducibility, dimension and degree. In particular, we find simple examples in which the Landau discriminant has codimension greater than one. Furthermore, we describe a numerical procedure for determining which parts of the Landau discriminant lie in the physical regions. In order to study degenerate limits of Landau equations and bounds on the degree of the Landau discriminant, we introduce Landau polytopes and study their facet structure. Finally, we provide an efficient numerical algorithm for the computation of the number of master integrals based on the connection to algebraic statistics. The algorithms used in this work are implemented in the open-source Julia package Landau.jl available at https://mathrepo.mis.mpg.de/Landau/.

Reduction with degenerate Gram matrix for one-loop integrals

An improved PV-reduction (Passarino-Veltman) method for one-loop integrals with auxiliary vector R has been proposed in [1, 2]. It has also been shown that the new method is a self-completed method in [3]. Analytic reduction coefficients can be easily produced by recursion relations in this method, where the Gram determinant appears in denominators. The singularity caused by Gram determinant is a well-known fact and it is important to address these divergences in a given frame. In this paper, we propose a systematical algorithm to deal with this problem in our method. The key idea is that now the master integral of the highest topology will be decomposed into combinations of master integrals of lower topologies. By demanding the cancellation of divergence for obtained general reduction coefficients, we solve decomposition coefficients as a Taylor series of the Gram determinant. Moreover, the same idea can be applied to other kinds of divergences.

Master integrals for mixed QCD-QED corrections to charged-current Drell-Yan production of a massive charged lepton

The master integrals for the mixed QCD-QED two-loop virtual corrections to the charged-current Drell-Yan process qoverline{q} ′ → ℓν are computed analytically by using the differential equation method. A suitable choice of master integrals makes it successful to cast the differential equation system into the canonical form. We keep the dependence on charged lepton mass in the building of differential equations and then expand the system against the ratio of small charged lepton mass to large W-boson mass. In such a way the final results will contain large logarithms of the form log( {m}_{ell}^2 / {m}_W^2 ). Finally, all the canonical master integrals are given as Taylor series around d = 4 spacetime dimensions up to order four, with coefficients expressed in terms of Goncharov polylogarithms up to weight four.

On Series and Integral Representations of Some NRQCD Master Integrals

We consider new ways of obtaining series and integral representations for master integrals arising in the process of matching of QCD to NRQCD. The latter results are exact in space-time dimension $d$. In addition, we discuss series expansion of the obtained results at fixed values of $d$.

A semi-numerical method for one-scale problems applied to the $\overline{\rm MS}$-on-shell relation

We discuss a practical approach to compute master integrals entering physical quantities which depend on one parameter. As an example we consider four-loop QCD corrections to the relation between a heavy quark mass defined in the \overline{\rm MS} and on-shell scheme in the presence of a second heavy quark.

Feynman integrals for binary systems of black holes

The initial phase of the inspiral process of a binary black-hole system can be described by perturbation theory. At the third post-Minkowskian order a two-loop double box graph, known as H-graph, contributes. In this talk we report how all master integrals of the H-graph with equal masses can be expressed up to weight four in terms of multiple polylogarithms. We also discuss techniques for the unequal mass case. The essential complication (and the focus of the talk) is the occurrence of several square roots.

Multiscale pentagon integrals to all orders

We present analytical results for one-loop five-point master integrals with up to three off-shell legs. The method of canonical differential equations along with the Simplified Differential Equations approach is employed. All necessary boundary terms are given in closed form, resulting to solutions in terms of Goncharov Polylogarithms of arbitrary weight. Explicit results up to weight six will be presented.

Recent progress on two-loop massless pentabox integrals with one off-shell leg

Analytic expressions in terms of polylogarithmic functions for all three families of planar two-loop five-point Master Integrals with one off-shell leg are presented. The Simplified Differential Equations approach is the only known way to fulfil this task due to its unique factorisation property of the symbols of the canonical differential equation. The results are relevant to the study of many 2\to 32→3 scattering processes of interest at the LHC.

Master integrals contributing to two-loop leading colour QCD helicity amplitudes for top-quark pair production in the gluon fusion channel

We discuss the analytic structure of the master integrals relevant for computing the complete set of leading-colour analytic helicity amplitudes for top-quark pair production via gluon fusion at two loops in QCD. This includes corrections due to massive fermion loops which give rise to integrals involving elliptic curves. We also elaborate on the structure of singularities that play an important role in the numerical evaluation of the iterated integrals and their analytic continuation.

One-loop QCD helicity amplitudes for pp → toverline{t}j to O(ε2)

We compute helicity amplitudes for the one-loop QCD corrections to top-quark pair production analytically in terms of a set of uniformly transcendental master integrals. We provide corrections up to O(ε2) in the dimensional regulator for the first time which are relevant at NNLO. Four independent pentagon integral topologies appear in the complete description of the colour structure for which we provide numerical solutions using canonical form differential equations and the method of generalised power series expansions. Analytic forms of the boundary values are obtained in all cases except one where we find a one-dimensional integral representation.

Addendum to: Towards next-to-next-to-leading-log accuracy for the width difference in the Bs− {overline{B}}_s system: fermionic contributions to order (mc/mb)0 and (mc/mb)1

We calculate the three-loop master integrals of Ref. [1] [arXiv:1709.02160] in analytic form. This allows us to present the fermionic contributions to the $\Delta B=2$ Wilson coefficients of the $B$-$\bar B$ decay matrix in next-to-next-to-leading order of QCD with full analytic dependence on the mass of the charm quark in the fermionic loops.

Quark and Gluon Form Factors in Four-Loop QCD.

We compute the photon-quark and Higgs-gluon form factors to four-loop order within massless perturbative quantum chromodynamics. Our results constitute ready-to-use building blocks for N^{4}LO cross sections for Drell-Yan processes and gluon-fusion Higgs boson production at the LHC. We present complete analytic expressions for both form factors and show several of the most complicated master integrals.

Reduction of general one-loop integrals using auxiliary vector

As a key method to deal with loop integrals, Integration-By-Parts (IBP) method can be used to do reduction as well as establish the differential equations for master integrals. However, when talking about tensor reduction, the Passarino-Veltman (PV) reduction method is also widely used for one-loop integrals. Recently, we have proposed an improved PV reduction method, i.e., the PV reduction method with auxiliary vector R, which can easily give analytical reduction results for any tensor rank. However, our results are only for integrals with propagators with power one. In this paper, we generalize our method to one-loop integrals with general tensor structures and propagators with general powers. Our ideas are simple. We solve the generalised reduction problem by combining differentiation over masses and proper limit of reduction with power-one propagators. Finally, we demonstrate our method with several examples. With the result in this paper, we have shown that our improved PV-reduction method with auxiliary vector is a self-completed reduction method for one-loop integrals.