Integration by parts: The algorithm to calculate β-functions in 4 loops

doi:10.1016/0550-3213(81)90199-1

Nuclear Physics B

Volume 192, Issue 1, 23 November 1981, Pages 159-204

https://doi.org/10.1016/0550-3213(81)90199-1 Get rights and content

Abstract

The following statement is proved: the counterterm for an arbitrary 4-loop Feynman diagram in an arbitrary model is calculable within the minimal subtraction scheme in terms of rational numbers and the Riemann ζ-function in a finite number of steps via a systematic “algebraic” procedure involving neither integration of elementary, special, or any other functions, nor expansions in and summation of infinite series of any kind. The number of steps is a rapidly increasing function of the complexity of the diagram. To demonstrate further possibilities offered by the technique we compute the 5-loop diagram contributing to the anomalous field dimension γ₂(g) in the ϕ⁴ model, that defied, heretofore, analytical calculation by other methods.

References (30)

K.G. Chetyrkin et al.
Nucl. Phys.
(1980)
P. Binétruy et al.
Nucl. Phys.
(1981)
P. Binétruy et al.
Nucl. Phys.
(1981)
N.-P. Chang et al.
Phys. Rev.
(1980)
S. Weinberg
Phys. Lett.
(1980)
A.E. Terrano
Phys. Lett.
(1980)
A.P. Isaev
Diploma work
(1978)
L.J. Slater
Generalized hypergeometric functions
(1966)
K.G. Chetyrkin et al.
Phys. Lett.
(1981)
erratum, to be...
C.M. Bender et al.
Phys. Rev.
(1977)
I.S. Gradsteyn et al.
Tables of integrals, series and products
(1961)
S. Weinberg
Phys. Rev.
(1961)

L.N. Lipatov

ZhETF (USSR)

(1977)

D.I. Kazakov et al.

Fortsch. Phys.

(1980)

K.G. Chetyrkin et al.

INR P-0118

(1979)

E.C.G. Stueckelberg et al.

Helv. Phys. Acta

(1953)

M. Gell-Mann et al.

Phys. Rev.

(1954)

N.N. Bogoliubov et al.

Introduction to the theory of quantized fields

(1959)

G.'t Hooft

Nucl. Phys.

(1973)

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Integration by parts: The algorithm to calculate β-functions in 4 loops

Abstract

Nucl. Phys.

Nucl. Phys.

Nucl. Phys.

Phys. Rev.

Phys. Lett.

Phys. Lett.

Phys. Lett.

Phys. Rev.

Phys. Rev.

ZhETF (USSR)

Fortsch. Phys.

INR P-0118

Helv. Phys. Acta

Phys. Rev.

Introduction to the theory of quantized fields

Nucl. Phys.