30 research outputs found
Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph
We consider the calculation of the master integrals of the three-loop massive
banana graph. In the case of equal internal masses, the graph is reduced to
three master integrals which satisfy an irreducible system of three coupled
linear differential equations. The solution of the system requires finding a matrix of homogeneous solutions. We show how the maximal cut can be
used to determine all entries of this matrix in terms of products of elliptic
integrals of first and second kind of suitable arguments. All independent
solutions are found by performing the integration which defines the maximal cut
on different contours. Once the homogeneous solution is known, the
inhomogeneous solution can be obtained by use of Euler's variation of
constants.Comment: 39 pages, 3 figures; Fixed a typo in eq. (6.16
Adaptive Integrand Decomposition in parallel and orthogonal space
We present the integrand decomposition of multiloop scattering amplitudes in
parallel and orthogonal space-time dimensions, , being
the dimension of the parallel space spanned by the legs of the
diagrams. When the number of external legs is , the corresponding
representation of the multiloop integrals exposes a subset of integration
variables which can be easily integrated away by means of Gegenbauer
polynomials orthogonality condition. By decomposing the integration momenta
along parallel and orthogonal directions, the polynomial division algorithm is
drastically simplified. Moreover, the orthogonality conditions of Gegenbauer
polynomials can be suitably applied to integrate the decomposed integrand,
yielding the systematic annihilation of spurious terms. Consequently, multiloop
amplitudes are expressed in terms of integrals corresponding to irreducible
scalar products of loop momenta and external momenta. We revisit the one-loop
decomposition, which turns out to be controlled by the maximum-cut theorem in
different dimensions, and we discuss the integrand reduction of two-loop planar
and non-planar integrals up to legs, for arbitrary external and internal
kinematics. The proposed algorithm extends to all orders in perturbation
theory.Comment: 64 pages, 4 figures, 8 table
Master integrals for the NNLO virtual corrections to scattering in QED: the planar graphs
We evaluate the master integrals for the two-loop, planar box-diagrams
contributing to the elastic scattering of muons and electrons at
next-to-next-to leading-order in QED. We adopt the method of differential
equations and the Magnus exponential series to determine a canonical set of
integrals, finally expressed as a Taylor series around four space-time
dimensions, with coefficients written as combination of generalised
polylogarithms. The electron is treated as massless, while we retain full
dependence on the muon mass. The considered integrals are also relevant for
crossing-related processes, such as di-muon production at -colliders,
as well as for the QCD corrections to -pair production at hadron
colliders.Comment: published version, 39 pages, 7 figures, 3 ancillary file
Two-loop master integrals for the leading QCD corrections to the Higgs coupling to a pair and to the triple gauge couplings and
We compute the two-loop master integrals required for the leading QCD
corrections to the interaction vertex of a massive neutral boson , e.g.
or , with a pair of bosons, mediated by a quark
doublet composed of one massive and one massless flavor. All the external legs
are allowed to have arbitrary invariant masses. The Magnus exponential is
employed to identify a set of master integrals that, around space-time
dimensions, obey a canonical system of differential equations. The canonical
master integrals are given as a Taylor series in , up to
order four, with coefficients written as combination of Goncharov
polylogarithms, respectively up to weight four. In the context of the Standard
Model, our results are relevant for the mixed EW-QCD corrections to the Higgs
decay to a pair, as well as to the production channels obtained by
crossing, and to the triple gauge boson vertices and .Comment: 42 pages, 5 figures, 2 ancillary file
On the maximal cut of Feynman integrals and the solution of their differential equations
The standard procedure for computing scalar multi-loop Feynman integrals
consists in reducing them to a basis of so-called master integrals, derive
differential equations in the external invariants satisfied by the latter and,
finally, try to solve them as a Laurent series in , where
are the space-time dimensions. The differential equations are, in general,
coupled and can be solved using Euler's variation of constants, provided that a
set of homogeneous solutions is known. Given an arbitrary differential equation
of order higher than one, there exist no general method for finding its
homogeneous solutions. In this paper we show that the maximal cut of the
integrals under consideration provides one set of homogeneous solutions,
simplifying substantially the solution of the differential equations.Comment: 25 pages, v2 minor typos fixed and references added, version accepted
for publication on NP
Off-shell Currents and Color-Kinematics Duality
We elaborate on the color-kinematics duality for off-shell diagrams in gauge
theories coupled to matter, by investigating the scattering process , and show that the Jacobi relations for the kinematic numerators
of off-shell diagrams, built with Feynman rules in axial gauge, reduce to a
color-kinematics violating term due to the contributions of sub-graphs only.
Such anomaly vanishes when the four particles connected by the Jacobi relation
are on their mass shell with vanishing squared momenta, being either external
or cut particles, where the validity of the color-kinematics duality is
recovered. We discuss the role of the off-shell decomposition in the direct
construction of higher-multiplicity numerators satisfying color-kinematics
identity in four as well as in dimensions, for the latter employing the
Four Dimensional Formalism variant of the Four Dimensional Helicity scheme. We
provide explicit examples for the QCD process .Comment: Accepted version for publication in PLB. Manuscript extended: 19
pages, 15 figures; C/K duality for tree-level amplitudes in dimensional
regularization added; references added; title modifie
Adaptive Integrand Decomposition
We present a simplified variant of the integrand reduction algorithm for
multiloop scattering amplitudes in dimensions, which
exploits the decomposition of the integration momenta in parallel and
orthogonal subspaces, , where is the
dimension of the space spanned by the legs of the diagrams. We discuss the
advantages of a lighter polynomial division algorithm and how the orthogonality
relations for Gegenbauer polynomilas can be suitably used for carrying out the
integration of the irreducible monomials, which eliminates spurious integrals.
Applications to one- and two-loop integrals, for arbitrary kinematics, are
discussed.Comment: Conference Proceedings, Loops and Legs in Quantum Field Theory, 24-29
April 2016, Leipzig, German
Cutting Feynman Amplitudes: from Adaptive Integrand Decomposition to Differential Equations on Maximal Cut
In this thesis we discuss, within the framework of the Standard Model (SM) of particle physics, advanced methods for the computation of scattering amplitudes at higher-order in perturbation theory. We offer a new insight into the role played by the unitarity of scattering amplitudes in the theoretical understanding and in the computational simplification of multi-loop calculations, at both the algebraic and the analytical level.
On the algebraic side, generalized unitarity can be used, within the integrand reduction method, to express the integrand associated to a multi-loop amplitude as a sum of fundamental, irreducible contributions, yielding to a decomposition of the amplitude as a linear combination of master integrals. In this framework, we propose an adaptive formulation of the integrand decomposition algorithm, which systematically adjusts to the kinematics of the individual integrands the dimensionality of the momentum space, where unitarity cuts are performed. This new formulation makes the integrand decomposition method, which in the past played a key role in streamlining one-loop computations, an efficient tool also at multi-loop level. We provide evidence of the generality of the proposed method by determining a universal parametrization
of the integrand basis for two-loop amplitudes in arbitrary kinematics and we illustrate its technical feasibility in the first automated implementation of the analytic integrand decomposition at one- and two-loop level.
On the analytic side, we discuss the role of maximal-unitarity for the solution of differential equations for dimensionally regulated Feynman integrals. The determination of the analytic expression of the master integrals as a Laurent expansion in the dimensional regulating
parameter requires the knowledge of the solutions of the homogeneous part of their differential equations at d=4. In all cases where Feynman integrals fulfil genuine first-order differential equations with a linear dependence on d, the corresponding homogeneous solutions can be determined through the Magnus exponential expansion.
In this work we apply the latter to two-loop corrections to several SM processes such as the Higgs decay to weak vector bosons, H → WW, triple gauge couplings ZWW and γ∗WW and to the elastic scattering μe → μe in quantum electrodynamics.
In some cases, the inadequacy of the Magnus method hints at the presence of master integrals that obey higher-order differential equations, for which no general theory exists. In this thesis we show that maximal-cuts of Feynman integrals solve, by construction, such homogeneous equations regardless of their order and complexity. Hence, whenever a Feynman integral obeys an irreducible higher-order differential equation, the computation of its maximal-cut along independent contours provides a closed integral representation of the full set of independent homogeneous solutions. We apply this strategy to the two-loop elliptic integrals that appear in heavy-quark mediated corrections to gg → gg and gg → gH as well as to the three-loop massive banana graph, which constitute the first example of Feynman integral that obeys a third-order differential equation.
In the light of the results presented in this thesis, generalized unitarity emerges as a powerful tool not only for handling the algebraic complexity of perturbative calculations but also for investigating the nature of new classes of mathematical functions encountered in particle physics
Master integrals for the NNLO virtual corrections to scattering in QCD: the non-planar graphs
We complete the analytic evaluation of the master integrals for the two-loop
non-planar box diagrams contributing to the top-pair production in the
quark-initiated channel, at next-to-next-to-leading order in QCD. The integrals
are determined from their differential equations, which are cast into a
canonical form using the Magnus exponential. The analytic expressions of the
Laurent series coefficients of the integrals are expressed as combinations of
generalized polylogarithms, which we validate with several numerical checks. We
discuss the analytic continuation of the planar and the non-planar master
integrals, which contribute to in QCD, as well as
to the companion QED scattering processes and .Comment: 1+26 pages, 4 figures, 1 table, 3 ancillary files. v2: references
added, text partly reworded, results unmodifie
Exact Top Yukawa corrections to Higgs boson decay into bottom quarks
In this letter we present the results of the exact computation of
contributions to the Higgs boson decay into bottom quarks that are proportional
to the top Yukawa coupling. Our computation demonstrates that approximate
results already available in the literature turn out to be particularly
accurate for the three physical mass values of the Higgs boson, the bottom and
top quarks. Furthermore, contrary to expectations, the impact of these
corrections on differential distributions relevant for the searches of the
Higgs boson decaying into bottom quarks at the Large Hadron Collider is rather
small.Comment: 6 pages, 4 figure