Built on

K. G. Chetyrkin and F. V. Tkachov, Integration by Parts: The Algorithm to Calculate beta Functions in 4 Loops , Nucl. Phys. B192
1981
Earlier work this paper cites.
F. V. Tkachov, A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions , Phys. Lett. 100B
1981
Earlier work this paper cites.
D. J. Broadhurst, J. Fleischer and O. Tarasov, Two loop two point functions with masses: Asymptotic expansions and Taylor series, in any dimension , Z.Phys. C60
1993
Earlier work this paper cites.
J. Fleischer, A. V. Kotikov and O. L. Veretin, Analytic two loop results for selfenergy type and vertex type diagrams with one nonzero mass , Nucl. Phys. B547
1999
Earlier work this paper cites.
Wiley, 1999
G. Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer · 1999
Earlier work this paper cites.
S. Moch, P. Uwer and S. Weinzierl, Nested sums, expansion of transcendental functions and multi-scale multi-loop integrals , J. Math. Phys. 43
2002
Earlier work this paper cites.
U. Aglietti and R. Bonciani, Master integrals with 2 and 3 massive propagators for the 2 loop electroweak form-factor - planar case , Nucl. Phys. B698
2004
Earlier work this paper cites.
A. I. Davydychev and M. Y. Kalmykov, Massive Feynman diagrams and inverse binomial sums , Nucl. Phys. B699
2004
Earlier work this paper cites.
S. Weinzierl, Expansion around half-integer values, binomial sums and inverse binomial sums , J. Math. Phys. 45
2004
Earlier work this paper cites.
Springer-Verlag, Berlin, second ed., 2004, 10.1007/978-3-642-57739-0
W. P. Barth, K. Hulek, C. A. M. Peters and A. Van de Ven, Compact complex surfaces , vol. 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] · 2004
Earlier work this paper cites.
S. Pérez-Díaz, J. Sendra and J. R. Sendra, Parametrization of approximate algebraic surfaces by lines , Comput. Aided Geom. Design 22
2005
Earlier work this paper cites.
S. Laporta and E. Remiddi, Analytic treatment of the two loop equal mass sunrise graph , Nucl. Phys. B704
2005
Earlier work this paper cites.
Princeton University Press, Princeton, NJ, 2006
E. Bézout, General theory of algebraic equations · 2006
Earlier work this paper cites.
M. Y. Kalmykov, B. F. L. Ward and S. Yost, All order epsilon-expansion of gauss hypergeometric functions with integer and half/integer values of parameters , JHEP 02
2007
Earlier work this paper cites.
Springer, Berlin, 2008, 10.1007/978-3-540-73725-4
J. R. Sendra, F. Winkler and S. Pérez-Díaz, Rational algebraic curves , vol. 22 of Algorithms and Computation in Mathematics · 2008
Earlier work this paper cites.
Müller-Stach, Stefan and Weinzierl, Stefan and Zayadeh, Raphael, A Second-Order Differential Equation for the Two-Loop Sunrise Graph with Arbitrary Masses , Commun. Num. Theor. Phys. 6
Original
2012
Earlier work this paper cites.
J. M. Henn and V. A. Smirnov, Analytic results for two-loop master integrals for Bhabha scattering I , JHEP 11
Original
2013
Earlier work this paper cites.

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Then

A. von Manteuffel and L. Tancredi, A non-planar two-loop three-point function beyond multiple polylogarithms , JHEP 06
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2017
Later among the works it cites.
C. Bogner, A. Schweitzer and S. Weinzierl, Analytic continuation and numerical evaluation of the kite integral and the equal mass sunrise integral , Nucl. Phys. B922
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2017
Later among the works it cites.
E. Remiddi and L. Tancredi, An Elliptic Generalization of Multiple Polylogarithms , Nucl. Phys. B925
Original
2017
Later among the works it cites.
G.-M. Greuel, S. Laplagne and G. Pfister, normal.lib
2017
Later among the works it cites.
G. Pfister, W. Decker, H. Schoenemann and S. Laplagne, primdec.lib
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Later among the works it cites.

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2018

Rationalizing roots: an algorithmic approach

Besier, Marco, van Straten, Duco, Weinzierl, Stefan

Understand

In the computation of Feynman integrals which evaluate to multiple polylogarithms one encounters quite often square roots.

To express the Feynman integral in terms of multiple polylogarithms, one seeks a transformation of variables, which rationalizes the square roots.
In this paper, we give an algorithm for rationalizing roots.
The algorithm is applicable whenever the algebraic hypersurface associated with the root has a point of multiplicity $(d-1)$, where $d$ is the degree of the algebraic hypersurface.

Built on

K. G. Chetyrkin and F. V. Tkachov, Integration by Parts: The Algorithm to Calculate beta Functions in 4 Loops , Nucl. Phys. B192
1981
Earlier work this paper cites.
F. V. Tkachov, A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions , Phys. Lett. 100B
1981
Earlier work this paper cites.
D. J. Broadhurst, J. Fleischer and O. Tarasov, Two loop two point functions with masses: Asymptotic expansions and Taylor series, in any dimension , Z.Phys. C60
1993
Earlier work this paper cites.
J. Fleischer, A. V. Kotikov and O. L. Veretin, Analytic two loop results for selfenergy type and vertex type diagrams with one nonzero mass , Nucl. Phys. B547
1999
Earlier work this paper cites.
Wiley, 1999
G. Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer · 1999
Earlier work this paper cites.
S. Moch, P. Uwer and S. Weinzierl, Nested sums, expansion of transcendental functions and multi-scale multi-loop integrals , J. Math. Phys. 43
2002
Earlier work this paper cites.
U. Aglietti and R. Bonciani, Master integrals with 2 and 3 massive propagators for the 2 loop electroweak form-factor - planar case , Nucl. Phys. B698
2004
Earlier work this paper cites.
A. I. Davydychev and M. Y. Kalmykov, Massive Feynman diagrams and inverse binomial sums , Nucl. Phys. B699
2004
Earlier work this paper cites.
S. Weinzierl, Expansion around half-integer values, binomial sums and inverse binomial sums , J. Math. Phys. 45
2004
Earlier work this paper cites.
Springer-Verlag, Berlin, second ed., 2004, 10.1007/978-3-642-57739-0
W. P. Barth, K. Hulek, C. A. M. Peters and A. Van de Ven, Compact complex surfaces , vol. 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] · 2004
Earlier work this paper cites.
S. Pérez-Díaz, J. Sendra and J. R. Sendra, Parametrization of approximate algebraic surfaces by lines , Comput. Aided Geom. Design 22
2005
Earlier work this paper cites.
S. Laporta and E. Remiddi, Analytic treatment of the two loop equal mass sunrise graph , Nucl. Phys. B704
2005
Earlier work this paper cites.
Princeton University Press, Princeton, NJ, 2006
E. Bézout, General theory of algebraic equations · 2006
Earlier work this paper cites.
M. Y. Kalmykov, B. F. L. Ward and S. Yost, All order epsilon-expansion of gauss hypergeometric functions with integer and half/integer values of parameters , JHEP 02
2007
Earlier work this paper cites.
Springer, Berlin, 2008, 10.1007/978-3-540-73725-4
J. R. Sendra, F. Winkler and S. Pérez-Díaz, Rational algebraic curves , vol. 22 of Algorithms and Computation in Mathematics · 2008
Earlier work this paper cites.
Müller-Stach, Stefan and Weinzierl, Stefan and Zayadeh, Raphael, A Second-Order Differential Equation for the Two-Loop Sunrise Graph with Arbitrary Masses , Commun. Num. Theor. Phys. 6
Original
2012
Earlier work this paper cites.
J. M. Henn and V. A. Smirnov, Analytic results for two-loop master integrals for Bhabha scattering I , JHEP 11
Original
2013
Earlier work this paper cites.