Letters in Mathematical Physics

, Volume 97, Issue 1, pp 37–44 | Cite as

The Number of Master Integrals is Finite

  • Alexander V. Smirnov
  • Alexey V. Petukhov


For a fixed Feynman graph one can consider Feynman integrals with all possible powers of propagators and try to reduce them, by linear relations, to a finite subset of integrals, the so-called master integrals. Up to now, there are numerous examples of reduction procedures resulting in a finite number of master integrals for various families of Feynman integrals. However, up to now it was just an empirical fact that the reduction procedure results in a finite number of irreducible integrals. It this paper we prove that the number of master integrals is always finite.

Mathematics Subject Classification (2000)

81Q30 14F10 


Feynman integrals algebraic groups D-modules 


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  1. 1.
    ’t Hooft G., Veltman M.: Regularization and renormalization of Gauge fields. Nucl. Phys. B 44, 189 (1972)MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    Bollini C.G., Giambiagi J.J.: On spectral-function sum rules. Nuovo Cim. 12B, 20 (1972)Google Scholar
  3. 3.
    Smirnov V.A.: Feynman Integral Calculus, pp. 283. Springer, Berlin (2006)Google Scholar
  4. 4.
    Chetyrkin K.G., Tkachov F.V.: Integration by parts: The algorithm to calculate beta functions in 4 loops. Nucl. Phys. B 192, 159 (1981)ADSCrossRefGoogle Scholar
  5. 5.
    Laporta S.: High-precision calculation of multi-loop Feynman integrals by difference equations. Int. J. Mod. Phys. A 15, 5087 (2000) [hep-ph/0102033]MathSciNetADSGoogle Scholar
  6. 6.
    Gehrmann T., Remiddi E.: Two-loop master integrals for γ* → 3 jets: the planar topologies. Nucl. Phys. B 601, 248 (2001) [hep-ph/0008287]ADSCrossRefGoogle Scholar
  7. 7.
    Gehrmann T., Remiddi E.: Two-loop master integrals for γ* → 3 jets: the non-planar topologies. Nucl. Phys. B 601, 287 (2001) [hep-ph/0101124]ADSCrossRefGoogle Scholar
  8. 8.
    Anastasiou C., Lazopoulos A.: Automatic integral reduction for higher order perturbative calculations. JHEP 0407, 046 (2004) [hep-ph/0404258]ADSCrossRefGoogle Scholar
  9. 9.
    Smirnov A.V.: Algorithm FIRE - Feynman integral reduction. JHEP 0810, 107 (2008) [0807.3243]ADSCrossRefGoogle Scholar
  10. 10.
    Studerus, C.: Reduze - Feynman integral reduction in C+. [0912.2546]Google Scholar
  11. 11.
    Smirnov, A.V., Smirnov, V.A.: On the reduction of Feynman integrals to master integrals. PoS ACAT 2007:085 [0707.3993]Google Scholar
  12. 12.
    Lee R.N.: Group structure of the integration-by-part identities and its application to the reduction of multiloop integrals. JHEP 0807, 031 (2008) [0804.3008]ADSCrossRefGoogle Scholar
  13. 13.
    Coutinho, S.C.: A primer of algebraic D-modules. In: London Mathematical Society Student Texts, vol. 33. Cambridge University Press, Cambridge (1995)Google Scholar
  14. 14.
    Kapusta J.I., Gale C.: Finite-Temperature Field Theory Principles and Applications, 2nd edn. Cambridge University Press, Cambridge (2006)CrossRefGoogle Scholar
  15. 15.
    Schröder Y.: Loops for hot QCD. Nucl. Phys. Proc. Suppl. B 183, 296 (2008) [arXiv:0807.0500 [hep-ph]]ADSCrossRefGoogle Scholar

Copyright information

© Springer 2010

Authors and Affiliations

  1. 1.Scientific Research Computing CenterMoscow State UniversityMoscowRussia
  2. 2.Department of Higher Algebra of Moscow State UniversityMoscowRussia
  3. 3.Jacobs UniversityBremenGermany

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