Feynman integrals obey linear relations governed by intersection numbers, which act as scalar products between vector spaces. We present a general algorithm for the construction of multivariate ...intersection numbers relevant to Feynman integrals, and show for the first time how they can be used to solve the problem of integral reduction to a basis of master integrals by projections, and to directly derive functional equations fulfilled by the latter. We apply it to the decomposition of a few Feynman integrals at one and two loops, as first steps toward potential applications to generic multiloop integrals. The proposed method can be more generally employed for the derivation of contiguity relations for special functions admitting multifold integral representations.
A
bstract
We elaborate on the recent idea of a direct decomposition of Feynman integrals onto a basis of master integrals on maximal cuts using intersection numbers. We begin by showing an ...application of the method to the derivation of contiguity relations for special functions, such as the Euler beta function, the Gauss
2
F
1
hypergeometric function, and the Appell
F
1
function. Then, we apply the new method to decompose Feynman integrals whose maximal cuts admit 1-form integral representations, including examples that have from two to an arbitrary number of loops, and/or from zero to an arbitrary number of legs. Direct constructions of differential equations and dimensional recurrence relations for Feynman integrals are also discussed. We present two novel approaches to decomposition-by-intersections in cases where the maximal cuts admit a 2-form integral representation, with a view towards the extension of the formalism to
n
-form representations. The decomposition formulae computed through the use of intersection numbers are directly verified to agree with the ones obtained using integration-by-parts identities.
A
bstract
The computation of Feynman integrals is often the bottleneck of multi-loop calculations. We propose and implement a new method to efficiently evaluate such integrals in the physical region ...through the numerical integration of a suitable set of differential equations, where the initial conditions are provided in the unphysical region via the sector decomposition method. We present numerical results for a set of two-loop integrals, where the non-planar ones complete the master integrals for
gg
→
γγ
and
q
q
¯
→
γγ
scattering mediated by the top quark.
A
bstract
We present the result of the quadratic-in-spin interaction Hamiltonian for binary systems of rotating compact objects with generic spins, up to N
3
LO corrections within the post-Newtonian ...expansion. The calculation is performed by employing the effective field theory diagrammatic approach, and it involves Feynman integrals up to three loops, evaluated within the dimensional regularization scheme. The gauge-invariant binding energy and the scattering angle, in special kinematic regimes and spin configurations, are explicitly derived. The results extend our earlier study on the spin-orbit interaction effects.
A
bstract
We present the result of the spin-orbit interaction Hamiltonian for binary systems of rotating compact objects with generic spins, up to N
3
LO corrections within the post-Newtonian ...expansion. The calculation is performed by employing the effective field theory diagrammatic approach, and it involves Feynman integrals up to three loops, evaluated within the dimensional regularization scheme. We apply canonical transformations to eliminate the non-physical divergences and spurious logarithmic behaviours of the Hamiltonian, and use the latter to derive the gauge-invariant binding energy and the scattering angle, in special kinematic regimes.
Understanding the nature of chemical bonding and lattice dynamics together with their influence on phonon-transport is essential to explore and design crystalline solids with ultralow thermal ...conductivity for various applications including thermoelectrics. TlInTe2, with interlocked rigid and weakly bound substructures, exhibits lattice thermal conductivity as low as ca. 0.5 W/mK near room temperature, owing to rattling dynamics of weakly bound Tl cations. Large displacements of Tl cations along the c-axis, driven by electrostatic repulsion between localized electron clouds on Tl and Te ions, are akin to those of rattling guests in caged-systems. Heat capacity of TlInTe2 exhibits a broad peak at low-temperatures due to contribution from Tl-induced low-frequency Einstein modes as also evidenced from THz time domain spectroscopy. First-principles calculations reveal a strong coupling between large-amplitude coherent optic vibrations of Tl-rattlers along the c-axis, and acoustic phonons that likely causes the low lattice thermal conductivity in TlInTe2.
At hadron colliders, the leading production mechanism for a pair of photons is from quark-antiquark annihilation at the tree level. However, due to large gluon-gluon luminosity, the loop-induced ...process gg → γγ provides a substantial contribution. In particular, the amplitudes mediated by the top quark become important at the tt¯ threshold and above. In this paper we present the first complete computation of the next-to-leading order (NLO) corrections (up to αS3) to this process, including contributions from the top quark. These entail two-loop diagrams with massive propagators whose analytic expressions are unknown and have been evaluated numerically. We find that the NLO corrections to the top-quark induced terms are very large at low diphoton invariant mass m(γγ) and close to the tt¯ threshold. The full result including five massless quarks and top quark contributions at NLO displays a much more pronounced change of slope in the m(γγ) distribution at tt¯ threshold than at LO and an enhancement at high invariant mass with respect to the massless calculation.
A
bstract
We propose a new method for the evaluation of intersection numbers for twisted meromorphic
n
-forms, through Stokes’ theorem in
n
dimensions. It is based on the solution of an
n
-th order ...partial differential equation and on the evaluation of multivariate residues. We also present an algebraic expression for the contribution from each multivariate residue. We illustrate our approach with a number of simple examples from mathematics and physics.
A
bstract
We present a detailed description of the recent idea for a direct decomposition of Feynman integrals onto a basis of master integrals by projections, as well as a direct derivation of the ...differential equations satisfied by the master integrals, employing multivariate intersection numbers. We discuss a recursive algorithm for the computation of multivariate intersection numbers, and provide three different approaches for a direct decomposition of Feynman integrals, which we dub the
straight decomposition
, the
bottom-up decomposition
, and the
top-down decomposition
. These algorithms exploit the unitarity structure of Feynman integrals by computing intersection numbers supported on cuts, in various orders, thus showing the synthesis of the intersection-theory concepts with unitarity-based methods and integrand decomposition. We perform explicit computations to exemplify all of these approaches applied to Feynman integrals, paving a way towards potential applications to generic multi-loop integrals.
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