Gauss hypergeometric function: reduction, epsilon-expansion for integer/half-integer parameters and Feynman diagrams

Abstract

The Gauss hypergeometric functions 2F1 with arbitrary values of parameters are reduced to two functions with fixed values of parameters, which differ from the original ones by integers. It is shown that in the case of integer and/or half-integer values of parameters there are only three types of algebraically independent Gauss hypergeometric functions. The epsilon-expansion of functions of one of this type (type F in our classification) demands the introduction of new functions related to generalizations of elliptic functions. For the five other types of functions the higher-order epsilon-expansion up to functions of weight 4 are constructed. The result of the expansion is expressible in terms of Nielsen polylogarithms only. The reductions and epsilon-expansion of q-loop off-shell propagator diagrams with one massive line and q massless lines and q-loop bubble with two-massive lines and q-1 massless lines are considered. The code (Mathematica/FORM) is available via the www at this URL http://theor.jinr.ru/~kalmykov/hypergeom/hyper.html

19 pages, LaTeX, 1-eps figure; v5: The code (Mathematica/FORM) is available via the www http://theor.jinr.ru/~kalmykov/hypergeom/hyper.html

References in corpus (4)

• Hypergeometric representation of the two-loop equal mass sunrise diagram
• Series and epsilon-expansion of the hypergeometric functions
• Representation of the Gauss hypergeometric function by multiple polylogarithms and relations of multiple zeta values
• Steps towards full two-loop calculations for 2 fermion to 2 fermion processes: running versus pole masses schemes

Cited by in corpus (32)

• HypExp 2, Expanding Hypergeometric Functions about Half-Integer Parameters
• Running coupling and power corrections in nonlinear evolution at the high--energy limit
• 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
• Hypergeometric representation of the two-loop equal mass sunrise diagram
• Effective potential at three loops
• On the all-order epsilon-expansion of generalized hypergeometric functions with integer values of parameters
• Optimized Negative Dimensional Integration Method (NDIM) and multiloop Feynman diagram calculation
• Counting master integrals: integration by parts vs. differential reduction
• Two-Loop Vertices in Quantum Field Theory: Infrared and Collinear Divergent Configurations
• New ideas for handling of loop and angular integrals in D-dimensions in QCD
• First order radiative corrections to Bhabha scattering in $d$ dimensions
• Functional equations for one-loop master integrals for heavy-quark production and Bhabha scattering
• Standard and epsilon-finite Master Integrals for the rho-Parameter
• On one master integral for three-loop on-shell HQET propagator diagrams with mass
• Functional reduction of one-loop Feynman integrals with arbitrary masses
• "Sixth root of unity" and Feynman diagrams: hypergeometric function approach point of view
• $\mathcal{O}(\alpha {\alpha }_s)$ corrections to the $\gamma t\bar{t}$ vertex at the top quark threshold
• Running-coupling effects in the triple-differential charmless semileptonic decay width
• q-derivatives of multivariable q-hypergeometric function with respect to their parameters
• Analytic Periods via Twisted Symmetric Squares
• Differential Reduction Algorithms for the All-Order Epsilon Expansion of Hypergeometric Functions
• When epsilon-expansion of hypergeometric functions is expressible in terms of multiple polylogarithms: the two-variables examples
• Expansion of hypergeometric functions in terms of polylogarithms with nontrivial variable change
• Renormalized $\epsilon$-finite master integrals and their virtues: the three-loop self energy case
• GKZ hypergeometric systems of the three-loop vacuum Feynman integrals
• Hypergeometric functions with rational arguments
• Evaluation of three-loop self-energy master integrals with four or five propagators
• Two-loop Feynman integrals for ${\varphi }^4$ theory with long-range correlated disorder
• Scalar 1-loop Feynman integrals as meromorphic functions in space-time dimension d, II: Special kinematics
• Threshold corrections to the gamma t anti-t vertex at O(alpha alpha(s))
• GKZ hypergeometric systems of the four-loop vacuum Feynman integrals