Outline Quadpack Computation of IR-divergent integrals E. de Doncker 1 J. Fujimoto 2 N. Hamaguchi 3 T. Ishikawa 2 Y. Kurihara 2 Y. Shimizu 2 . Yuasa 2 F 1 Department of Computer Science, Western Michigan University, Kalamazoo MI 49008, U. S. 2 High Energy Accelerator Research Organization (KEK), Oho 1-1, Tsukuba, Ibaraki, 305-0801, Japan tu-logo 3 Hitachi, Ltd., Japan ur-logo E. de Doncker, J. Fujimoto, N. Hamaguchi, T. Ishikawa, Y. Kurihara, Y. Shimizu, F. Yuasa
Outline Outline Introduction 1 Infrared singularity Digression on extrapolation Application to massless one-loop vertex 2 Asymptotics one off-shell, two on-shell particles Hypergeometric function and threshold singularity 3 tu-logo Vertex with one on-shell and two off-shell particles 4 ur-logo E. de Doncker, J. Fujimoto, N. Hamaguchi, T. Ishikawa, Y. Kurihara, Y. Shimizu, F. Yuasa
Outline Outline Introduction 1 Infrared singularity Digression on extrapolation Application to massless one-loop vertex 2 Asymptotics one off-shell, two on-shell particles Hypergeometric function and threshold singularity 3 tu-logo Vertex with one on-shell and two off-shell particles 4 ur-logo E. de Doncker, J. Fujimoto, N. Hamaguchi, T. Ishikawa, Y. Kurihara, Y. Shimizu, F. Yuasa
Outline Outline Introduction 1 Infrared singularity Digression on extrapolation Application to massless one-loop vertex 2 Asymptotics one off-shell, two on-shell particles Hypergeometric function and threshold singularity 3 tu-logo Vertex with one on-shell and two off-shell particles 4 ur-logo E. de Doncker, J. Fujimoto, N. Hamaguchi, T. Ishikawa, Y. Kurihara, Y. Shimizu, F. Yuasa
Outline Outline Introduction 1 Infrared singularity Digression on extrapolation Application to massless one-loop vertex 2 Asymptotics one off-shell, two on-shell particles Hypergeometric function and threshold singularity 3 tu-logo Vertex with one on-shell and two off-shell particles 4 ur-logo E. de Doncker, J. Fujimoto, N. Hamaguchi, T. Ishikawa, Y. Kurihara, Y. Shimizu, F. Yuasa
Introduction Massless one-loop vertex Infrared singularity Hypergeometric function Digression on extrapolation Vertex with one on-shell and two off-shell particles Outline Introduction 1 Infrared singularity Digression on extrapolation Application to massless one-loop vertex 2 Asymptotics one off-shell, two on-shell particles Hypergeometric function and threshold singularity 3 tu-logo Vertex with one on-shell and two off-shell particles 4 ur-logo E. de Doncker, J. Fujimoto, N. Hamaguchi, T. Ishikawa, Y. Kurihara, Y. Shimizu, F. Yuasa
Introduction Massless one-loop vertex Infrared singularity Hypergeometric function Digression on extrapolation Vertex with one on-shell and two off-shell particles Sample problem: Characteristics Integral Z 1 Z 1 2 ε − 2 1 ( ε − 1 ) ε = 2 ϕ ( ε ) I ( ε ) = dx dy ( x + y ) 2 − ε = (1) ε 0 0 where ϕ ( ε ) = ( 2 ε − 1 − 1 ) / ( ε − 1 ) converges for ε > 0 and has a non-integrable singularity when ε ≤ 0 . Taylor expansion of ϕ ( ε ) around ε = 0 results in I ( ε ) ∼ C − 1 + C 0 + C 1 ε + . . . . (2) ε tu-logo 2 log 2 2 . with C − 1 = 1 , C 0 = 1 − log 2 , C 1 = 1 − log 2 − 1 According to the asymptotic behavior, we can explore linear or nonlinear extrapolation for the computation of the coefficients of the leading terms. ur-logo E. de Doncker, J. Fujimoto, N. Hamaguchi, T. Ishikawa, Y. Kurihara, Y. Shimizu, F. Yuasa
Introduction Massless one-loop vertex Infrared singularity Hypergeometric function Digression on extrapolation Vertex with one on-shell and two off-shell particles Sample problem: Characteristics Integral Z 1 Z 1 2 ε − 2 1 ( ε − 1 ) ε = 2 ϕ ( ε ) I ( ε ) = dx dy ( x + y ) 2 − ε = (1) ε 0 0 where ϕ ( ε ) = ( 2 ε − 1 − 1 ) / ( ε − 1 ) converges for ε > 0 and has a non-integrable singularity when ε ≤ 0 . Taylor expansion of ϕ ( ε ) around ε = 0 results in I ( ε ) ∼ C − 1 + C 0 + C 1 ε + . . . . (2) ε tu-logo 2 log 2 2 . with C − 1 = 1 , C 0 = 1 − log 2 , C 1 = 1 − log 2 − 1 According to the asymptotic behavior, we can explore linear or nonlinear extrapolation for the computation of the coefficients of the leading terms. ur-logo E. de Doncker, J. Fujimoto, N. Hamaguchi, T. Ishikawa, Y. Kurihara, Y. Shimizu, F. Yuasa
Introduction Massless one-loop vertex Infrared singularity Hypergeometric function Digression on extrapolation Vertex with one on-shell and two off-shell particles Sample problem: Methods Linear system n − 1 X C k − 1 ε ℓ k , ε ℓ I ( ε ℓ ) = 1 ≤ ℓ ≤ n (3) k = 0 S ℓ = P n − 1 special case of k = 0 C k − 1 ϕ k ( ε ℓ ) , 1 ≤ ℓ ≤ n Numerical integration ˆ I ( ε ) ≈ I ( ε ) for ε > 0 is affected by the integrand singularity of (1) at x = y = 0 on the boundary of the integration domain, when 2 > 2 − ε > 0 . tu-logo Recursive (repeated) integration with the 1D integration code D QAGSE from Q UADPACK [8] is equipped to handle algebraic end-point singularities in each coordinate direction. ur-logo E. de Doncker, J. Fujimoto, N. Hamaguchi, T. Ishikawa, Y. Kurihara, Y. Shimizu, F. Yuasa
Introduction Massless one-loop vertex Infrared singularity Hypergeometric function Digression on extrapolation Vertex with one on-shell and two off-shell particles Sample problem: Methods Linear system n − 1 X C k − 1 ε ℓ k , ε ℓ I ( ε ℓ ) = 1 ≤ ℓ ≤ n (3) k = 0 S ℓ = P n − 1 special case of k = 0 C k − 1 ϕ k ( ε ℓ ) , 1 ≤ ℓ ≤ n Numerical integration ˆ I ( ε ) ≈ I ( ε ) for ε > 0 is affected by the integrand singularity of (1) at x = y = 0 on the boundary of the integration domain, when 2 > 2 − ε > 0 . tu-logo Recursive (repeated) integration with the 1D integration code D QAGSE from Q UADPACK [8] is equipped to handle algebraic end-point singularities in each coordinate direction. ur-logo E. de Doncker, J. Fujimoto, N. Hamaguchi, T. Ishikawa, Y. Kurihara, Y. Shimizu, F. Yuasa
Introduction Massless one-loop vertex Infrared singularity Hypergeometric function Digression on extrapolation Vertex with one on-shell and two off-shell particles Sample problem: Methods Linear system n − 1 X C k − 1 ε ℓ k , ε ℓ I ( ε ℓ ) = 1 ≤ ℓ ≤ n (3) k = 0 S ℓ = P n − 1 special case of k = 0 C k − 1 ϕ k ( ε ℓ ) , 1 ≤ ℓ ≤ n Numerical integration ˆ I ( ε ) ≈ I ( ε ) for ε > 0 is affected by the integrand singularity of (1) at x = y = 0 on the boundary of the integration domain, when 2 > 2 − ε > 0 . tu-logo Recursive (repeated) integration with the 1D integration code D QAGSE from Q UADPACK [8] is equipped to handle algebraic end-point singularities in each coordinate direction. ur-logo E. de Doncker, J. Fujimoto, N. Hamaguchi, T. Ishikawa, Y. Kurihara, Y. Shimizu, F. Yuasa
Introduction Massless one-loop vertex Infrared singularity Hypergeometric function Digression on extrapolation Vertex with one on-shell and two off-shell particles Sample problem: Methods Linear system n − 1 X C k − 1 ε ℓ k , ε ℓ I ( ε ℓ ) = 1 ≤ ℓ ≤ n (3) k = 0 S ℓ = P n − 1 special case of k = 0 C k − 1 ϕ k ( ε ℓ ) , 1 ≤ ℓ ≤ n Numerical integration ˆ I ( ε ) ≈ I ( ε ) for ε > 0 is affected by the integrand singularity of (1) at x = y = 0 on the boundary of the integration domain, when 2 > 2 − ε > 0 . tu-logo Recursive (repeated) integration with the 1D integration code D QAGSE from Q UADPACK [8] is equipped to handle algebraic end-point singularities in each coordinate direction. ur-logo E. de Doncker, J. Fujimoto, N. Hamaguchi, T. Ishikawa, Y. Kurihara, Y. Shimizu, F. Yuasa
Introduction Massless one-loop vertex Infrared singularity Hypergeometric function Digression on extrapolation Vertex with one on-shell and two off-shell particles Outline Introduction 1 Infrared singularity Digression on extrapolation Application to massless one-loop vertex 2 Asymptotics one off-shell, two on-shell particles Hypergeometric function and threshold singularity 3 tu-logo Vertex with one on-shell and two off-shell particles 4 ur-logo E. de Doncker, J. Fujimoto, N. Hamaguchi, T. Ishikawa, Y. Kurihara, Y. Shimizu, F. Yuasa
Introduction Massless one-loop vertex Infrared singularity Hypergeometric function Digression on extrapolation Vertex with one on-shell and two off-shell particles Digression on extrapolation For sequence { S ( ε ℓ ) } , an extrapolation is performed to create sequences that convergence to the limit S = lim ε ℓ → 0 S ( ε ℓ ) faster than the given sequence, based on Asymptotic expansion S ( ε ) ∼ S + a 0 ϕ 0 ( ε ) + a 2 ϕ 2 ( ε ) + . . . ε may be a parameter of the problem or of the method. For example in Romberg integration, ε ℓ = h ℓ is the step size. A linear extrapolation method solves (implicitly or explicitly [2]) linear systems of the form S ( ε ℓ ) = a 0 + a 1 ϕ 1 ( ε ℓ ) + . . . a ν ϕ ν ( ε ℓ ) , ℓ = 0 , . . . , ν ; tu-logo of order ( ν + 1 ) × ( ν + 1 ) in unknowns a 0 , . . . a ν are solved for increasing values of ν. ur-logo E. de Doncker, J. Fujimoto, N. Hamaguchi, T. Ishikawa, Y. Kurihara, Y. Shimizu, F. Yuasa
Recommend
More recommend
