New results for the epsilon-expansion of certain one-, two- and three-loop Feynman diagrams

Abstract

For certain dimensionally-regulated one-, two- and three-loop diagrams, problems of constructing the epsilon-expansion and the analytic continuation of the results are studied. In some examples, an arbitrary term of the epsilon-expansion can be calculated. For more complicated cases, only a few higher terms in epsilon are obtained. Apart from the one-loop two- and three-point diagrams, the examples include two-loop (mainly on-shell) propagator-type diagrams and three-loop vacuum diagrams. As a by-product, some new relations involving Clausen function, generalized log-sine integrals and certain Euler--Zagier sums are established, and some useful results for the hypergeometric functions of argument 1/4 are presented.

49 pages, LaTeX, two figures (one in eps format), v3 is journal version; in v3 the typos in Egs.(2.17), (4.16), (B.12) corrected; in v4 the typos in Egs.(2.7),(4.10),(4.25),(B.11) corrected (see p.30 of present version)

References in corpus (3)

• The Standard Model Effective Potential at Two Loops
• Application and explicit solution of recurrence relations with respect to space-time dimension
• A class of series acceleration formulae for Catalan's constant

Cited by in corpus (121)

• Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals
• From polygons and symbols to polylogarithmic functions
• HypExp 2, Expanding Hypergeometric Functions about Half-Integer Parameters
• Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes
• Massive Feynman diagrams and inverse binomial sums
• Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials
• Two-loop amplitudes and master integrals for the production of a Higgs boson via a massive quark and a scalar-quark loop
• Feynman Diagrams and Differential Equations
• Analytic Results for Virtual QCD Corrections to Higgs Production and Decay
• An algebraic/numerical formalism for one-loop multi-leg amplitudes
• Algebraic Relations Between Harmonic Sums and Associated Quantities
• Extension of HPL to complex arguments
• MS Versus Pole Masses of Gauge Bosons II: Two-Loop Electroweak Fermion Corrections
• Expansion around half-integer values, binomial sums and inverse binomial sums
• Three-mass triangle integrals and single-valued polylogarithms
• High-precision epsilon expansions of single-mass-scale four-loop vacuum bubbles
• Analytical Result for Dimensionally Regularized Massive On-Shell Planar Double Box
• Analytic results for the planar double box integral relevant to top-pair production with a closed top loop
• MS vs. Pole Masses of Gauge Bosons: Electroweak Bosonic Two-Loop Corrections
• Iterated Binomial Sums and their Associated Iterated Integrals
• O(αα_s) correction to the pole mass of the t-quark within the Standard Model
• Azurite: An algebraic geometry based package for finding bases of loop integrals
• A New Hypergeometric Representation of One-Loop Scalar Integrals in $d$ Dimensions
• Gauss hypergeometric function: reduction, epsilon-expansion for integer/half-integer parameters and Feynman diagrams
• High-precision e-expansions of massive four-loop vacuum bubbles
• Two-loop sunset diagrams with three massive lines
• Multi-loop Feynman integrals and conformal quantum mechanics
• From multiple unitarity cuts to the coproduct of Feynman integrals
• ep-Finite Basis of Master Integrals for the Integration-By-Parts Method
• Master Integrals For Massless Two-Loop Vertex Diagrams With Three Offshell Legs
• RationalizeRoots: Software Package for the Rationalization of Square Roots
• Subleading Power Resummation of Rapidity Logarithms: The Energy-Energy Correlator in $\mathcal{N}=4$ SYM
• Two-loop matching factors for light quark masses and three-loop mass anomalous dimensions in the RI/SMOM schemes
• Towards all-order Laurent expansion of generalized hypergeometric functions around rational values of parameters
• All order epsilon-expansion of Gauss hypergeometric functions with integer and half/integer values of parameters
• Finding new relationships between hypergeometric functions by evaluating Feynman integrals
• Differential reduction of generalized hypergeometric functions from Feynman diagrams: One-variable case
• Gluon-fusion Higgs production in the Standard Model Effective Field Theory
• Hypergeometric representation of the two-loop equal mass sunrise diagram
• Evaluating `elliptic' master integrals at special kinematic values: using differential equations and their solutions via expansions near singular points
• About higher order epsilon-expansion of some massive two- and three-loop master-integrals
• Two-loop QED radiative corrections to the decay pi0 -> e+ e- : The virtual corrections and soft-photon bremsstrahlung
• HYPERDIRE: HYPERgeometric functions DIfferential REduction: MATHEMATICA based packages for differential reduction of generalized hypergeometric functions: $F_D$ and $F_S$ Horn-type hypergeometric functions of three variables
• LSJK - a C++ library for arbitrary-precision numeric evaluation of the generalized log-sine functions
• Counting the number of master integrals for sunrise-type Feynman diagrams via Mellin-Barnes representation
• On the all-order epsilon-expansion of generalized hypergeometric functions with integer values of parameters
• Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order epsilon-expansion of generalized hypergeometric functions with one half-integer value of parameter
• Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Volume I
• Two-loop diagrams in non-relativistic QCD with elliptics
• NumExp: Numerical epsilon expansion of hypergeometric functions
• Analytic results for planar three-loop integrals for massive form factors
• Exponential suppression with four legs and an infinity of loops
• Counting master integrals: integration by parts vs. differential reduction
• Three-loop massive tadpoles and polylogarithms through weight six
• Geometrical methods in loop calculations and the three-point function
• Series and epsilon-expansion of the hypergeometric functions
• On a two-loop crossed six-line master integral with two massive lines
• Next-to-leading order QCD corrections to paired $B_c$ production in $e^{+}{e}^{-}$ annihilation
• Maclaurin's series expansions for positive integer powers of inverse (hyperbolic) sine and related functions, specific values of partial Bell polynomials, and two applications
• A new approach to the epsilon expansion of generalized hypergeometric functions
• Laurent series expansion of a class of massive scalar one-loop integrals to ${\cal O}(\ep^2)
• Functional equations for one-loop master integrals for heavy-quark production and Bhabha scattering
• First order radiative corrections to Bhabha scattering in $d$ dimensions
• Four-point function in general kinematics through geometrical splitting and reduction
• Two-Loop $\mathcal{O}(\alpha {\alpha }_s)$ Corrections to the On-Shell Fermion Propagator in the Standard Model
• DGLAP and BFKL evolution equations in the N=4 supersymmetric gauge theory
• Full two-loop electroweak corrections to the pole masses of gauge bosons
• Several combinatorial identities derived from series expansions of powers of arcsine
• Taylor's series expansions for real powers of functions containing squares of inverse (hyperbolic) cosine functions, explicit formulas for special partial Bell polynomials, and series representations for powers of circular constant
• "Sixth root of unity" and Feynman diagrams: hypergeometric function approach point of view
• The Epsilon Expansion of Feynman Diagrams via Hypergeometric Functions and Differential Reduction
• Laurent series expansion of sunrise-type diagrams using configuration space techniques
• Bosonic corrections to the effective leptonic weak mixing angle at the two-loop level
• Non-planar Feynman integrals, Mellin-Barnes representations, multiple sums
• The inverse Mellin transform via analytic continuation
• The propagator seagull: general evaluation of a two loop diagram
• Algorithms to Evaluate Multiple Sums for Loop Computations
• Post Gaussian effective potential in the Ginzburg Landau theory of superconductivity
• Appell Functions and the Scalar One-Loop Three-point Integrals in Feynman Diagrams
• Laurent series expansion of a class of massive scalar one-loop integrals up to ${\cal O}(\ep^2)$ in terms of multiple polylogarithms
• Heavy quark form factors in the large ${\beta }_0$ limit
• The vacuum seagull: evaluating a 3-loop Feynman diagram with 3 mass scales
• Automatizing the application of Mellin-Barnes representations for Feynman integrals
• The difference between two Stieltjes constants
• Further results for the two-loop Lcc vertex in the Landau gauge
• Ginzburg Landau theory of superconductivity at fractal dimensions
• On conjectured local generalizations of anisotropic scale invariance and their implications
• Critical behavior of isotropic systems with strong dipole-dipole interaction: three-loop study
• Dimensional Regularization and Two-Loop Vacuum Polarization Operator: Master Integrals, Analytic Results and Energy Shifts
• Geometrical splitting and reduction of Feynman diagrams
• QCD cusp anomalous dimension: current status
• GKZ-system of the 2-loop self energy with 4 propagators
• Special Values of Generalized Log-sine Integrals
• Renormalized $\epsilon$-finite master integrals and their virtues: the three-loop self energy case
• Angular integrals with three denominators via IBP, mass reduction, dimensional shift, and differential equations
• Analytical evaluation of certain on-shell two-loop three-point diagrams
• Log-sine evaluations of Mahler measures, II
• A Feynman Integral and its Recurrences and Associators
• Sun's Series via Cyclotomic Multiple Zeta Values
• The four-propagator three-loop vacuum integral by the hypergeometry
• Effective action of beta-deformed N = 4 SYM theory: Farewell to two-loop BPS diagrams
• Hypergeometric functions with rational arguments
• Two-loop Feynman integrals for ${\varphi }^4$ theory with long-range correlated disorder
• Next-to-next-to-leading order $\mathcal{O}({\alpha }^2{\alpha }_s^2)$ results for top quark pair production in photon--photon collisions: The one-loop squared contributions
• Steps towards full two-loop calculations for 2 fermion to 2 fermion processes: running versus pole masses schemes
• Two-Loop QCD-EW Master Integrals for Z Plus Jet Production at Large Transverse Momentum
• Next-to-leading order contributions to the pole mass of gluino in minimal gauge mediation
• A new treatment of mixed virtual and real IR-singularities
• Counting master integrals: Integration by parts vs. functional equations
• One-loop Jet Functions by Geometric Subtraction
• HQET vertex diagram: $\epsilon$ expansion
• Note on relations among multiple zeta-star values
• New Evaluations of Inverse Binomial Series via Cyclotomic Multiple Zeta Values
• Numerator Seagull and Extended Symmetries of Feynman Integrals
• Two loop Kahler potential in beta-deformed N=4 SYM theory
• Three loop effective potential for $⟨\frac{1}{2}{A_{\mu }^a}^2⟩$ in the Landau gauge in QCD
• Log-sine evaluations of Mahler measures
• Hyper-Mahler measures via Goncharov-Deligne cyclotomy
• Explicit Evaluation of Euler-Apéry Type Multiple Zeta Star Values and Multiple $t$-Star Values
• Alternating Apéry-Type Series and Colored Multiple Zeta Values of Level Eight
• Recurrence Relations and Dispersive Techniques for Precision Multi-Loop Calculations