32 research outputs found
AZURITE: An algebraic geometry based package for finding bases of loop integrals
No full textFor any given Feynman graph, the set of integrals with all possible powers of the propagators spans a vector space of finite dimension. We introduce the package Azurite (A ZUR ich-bred method for finding master I nTE grals), which efficiently finds a basis of this vector space. It constructs the needed integration-by-parts (IBP) identities on a set of generalized-unitarity cuts. It is based on syzygy computations and analyses of the symmetries of the involved Feynman diagrams and is powered by the computer algebra systems Singular and Mathematica. It can moreover analytically calculate the part of the IBP identities that is supported on the cuts. In some cases, the basis obtained by Azurite may be slightly overcomplete
Azurite: An algebraic geometry based package for finding bases of loop integrals
No full textFor any given Feynman graph, the set of integrals with all possible powers of the propagators spans a vector space of finite dimension. We introduce the package Azurite (A ZUR ich-bred method for finding master I nTE grals), which efficiently finds a basis of this vector space. It constructs the needed integration-by-parts (IBP) identities on a set of generalized-unitarity cuts. It is based on syzygy computations and analyses of the symmetries of the involved Feynman diagrams and is powered by the computer algebra systems Singular and Mathematica. It can moreover analytically calculate the part of the IBP identities that is supported on the cuts. In some cases, the basis obtained by Azurite may be slightly overcomplete.</span
Complete integration-by-parts reductions of the non-planar hexagon-box via module intersections
No full textWe present the powerful module-intersection integration-by-parts (IBP) method, suitable for multi-loop and multi-scale Feynman integral reduction. Utilizing modern computational algebraic geometry techniques, this new method successfully trims traditional IBP systems dramatically to much simpler integral-relation systems on unitarity cuts. We demonstrate the power of this method by explicitly carrying out the complete analytic reduction of two-loop five-point non-planar hexagon-box integrals, with degree-four numerators, to a basis of 73 master integrals
Complete sets of logarithmic vector fields for integration-by-parts identities of Feynman integrals
No full textIntegration-by-parts identities between loop integrals arise from the vanishing integration of total derivatives in dimensional regularization. Generic choices of total derivatives in the Baikov or parametric representations lead to identities which involve dimension shifts. These dimension shifts can be avoided by imposing a certain constraint on the total derivatives. The solutions of this constraint turn out to be a specific type of syzygies which correspond to logarithmic vector fields along the Gram determinant formed of the independent external and loop momenta. We present an explicit generating set of solutions in Baikov representation, valid for any number of loops and external momenta, obtained from the Laplace expansion of the Gram determinant. We provide a rigorous mathematical proof that this set of solutions is complete. This proof relates the logarithmic vector fields in question to ideals of submaximal minors of the Gram matrix and makes use of classical resolutions of such ideals.</p
Topics in perturbation theory : From IBP identities to integrands
No full textIn this thesis we present different topics in perturbation theory. We start by introducing the method of integration by parts identities, which reduces a generic Feynman integral to a linear combination of a finite basis of master integrals. In our analysis we make use of the Baikov representation as this form gives a nice framework for generating efficiently the identities needed to reduce integrals. In the second part of the thesis we briefly explain recent developments in the integration of Feynman integrals and present a method to bootstrap the value of p-integrals using constraints from certain limits of conformal integrals. We introduce also another method to obtain p-integrals at l-loops by cutting vacuum diagrams at l+1-loops. In the last part of the thesis we present recent developments in N=4 SYM to compute structure constants. We use perturbation theory to obtain new results that can be tested against this new conjecture. Moreover we use integrability based methods to constrain correlation function of protected operators
Two-loop integral reduction from elliptic and hyperelliptic curves
No full textISSN:1126-6708ISSN:1029-8479ISSN:1029-847
Inelastic Exponentiation and Classical Gravitational Scattering at One Loop
No full textWe calculate the inelastic one-loop amplitude for the scattering of
two point-like, spinless objects with generic masses involving the additional
emission of a single graviton. We focus on the near-forward, or classical,
limit. Our results include the leading and subleading orders in the soft-region
expansion, which captures all non-analytic contributions in the transferred
momentum and in the graviton's frequency. This allows us to check the first
constraint arising from the inelastic exponentiation put forward in Refs.
2107.12891, 2112.07556, 2210.12118 and to calculate the one-loop matrix
element of the -operator, linked to the -matrix by , showing
that it is real, classical and free of infrared divergences. We discuss how our
results feature in the calculation of the corrections to the
asymptotic waveform.Comment: 33 pages + refs. v2: Includes ref. to the addendum arXiv:2312.14710
by the same author
Bohr space in six dimensions
No full textAbstractA conformal factor in the Bohr model embeds Bohr space in six dimensions, revealing the O(6) symmetry and its contraction to the E(5) at infinity. Phenomenological consequences are discussed after the re-formulation of the Bohr Hamiltonian in six dimensions on a five sphere
Konishi OPE coefficient at the five loop order
No full textAbstract We use the method of asymptotic expansions to study the OPE limit of a fourpoint function of protected operators in N = 4 SYM. We use a new method for evaluating the resulting propagator-type integrals and then extract the OPE coefficient with Konishi at the five loop order
