Advertisement

Note on the Intermediate Field Representation of \(\phi ^{2k}\) Theory in Zero Dimension

  • Luca Lionni
  • Vincent Rivasseau
Article
  • 8 Downloads

Abstract

This note is a sequel to Rivasseau and Wang (J. Math. Phys. 51, 092304, 2010). We correct the intermediate field representation for the stable \(\phi ^{2k}\) field theory in zero dimension introduced there and extend it to the case of complex conjugate fields. For \(k = 3\) in the complex case we also provide an improved representation which relies on ordinary convergent Gaussian integrals rather than oscillatory integrals.

Keywords

Constructive field theory Loop vertex expansion Borel summability 

Mathematics Subject Classification (2010)

81T08 

References

  1. 1.
    Rivasseau, V., Wang, Z.: Loop Vertex Expansion for Phi**2K Theory in Zero Dimension. J. Math. Phys. 51, 092304 (2010).  https://doi.org/10.1063/1.3460320 arXiv:1003.1037 [math-ph]ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Rivasseau, V.: Constructive Matrix Theory. JHEP 0709, 008 (2007). arXiv:0706.1224 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Magnen, J., Rivasseau, V.: Constructive \(\phi ^{4}\) field theory without tears. Ann. Henri Poincare 9, 403 (2008). arXiv:0706.2457 [math-ph]ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Rivasseau, V.: Constructive Field Theory in Zero Dimension, vol. 2010.  https://doi.org/10.1155/2009/180159 arXiv:0906.3524 [math-ph] (2010)
  5. 5.
    Rivasseau, V., Wang, Z.: How to Resum Feynman Graphs. Ann. Henri Poincare 15(11), 2069 (2014).  https://doi.org/10.1007/s00023-013-0299-8 arXiv:1304.5913 [math-ph]ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Magnen, J., Noui, K., Rivasseau, V., Smerlak, M.: Scaling behaviour of three-dimensional group field theory. Class. Quant. Grav. 26, 185012 (2009). arXiv:0906.5477 [hep-th]ADSCrossRefMATHGoogle Scholar
  7. 7.
    Gurau, R.: The 1/N Expansion of Tensor Models Beyond Perturbation Theory. Commun. Math. Phys. 330, 973 (2014).  https://doi.org/10.1007/s00220-014-1907-2 arXiv:1304.2666 [math-ph]ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gurau, R., Krajewski, T.: Analyticity results for the cumulants in a random matrix model, arXiv:1409.1705 [math-ph]
  9. 9.
    Rivasseau, V., Wang, Z.: Corrected loop vertex expansion for \({{\Phi }_{2}^{4}}\) theory. J. Math. Phys. 56 (6), 062301 (2015).  https://doi.org/10.1063/1.4922116 arXiv:1406.7428 [math-ph]ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Delepouve, T., Gurau, R., Rivasseau, V.: Universality and Borel Summability of Arbitrary Quartic Tensor Models, arXiv:1403.0170 [hep-th]
  11. 11.
    Delepouve, T., Rivasseau, V.: Constructive Tensor Field Theory: The \({T^{4}_{3}}\) Model, arXiv:1412.5091 [math-ph]
  12. 12.
    Lahoche, V.: Constructive Tensorial Group Field Theory I:The \(U(1)-{T^{4}_{3}}\) Model, arXiv:1510.05050 [hep-th]
  13. 13.
    Lahoche, V.: Constructive Tensorial Group Field Theory II: The \(U(1)-{T^{4}_{4}}\) Model, arXiv:1510.05051 [hep-th]
  14. 14.
    Gurau, R., Rivasseau, V.: The Multiscale Loop Vertex Expansion. Ann. Henri Poincare 16(8), 1869 (2015).  https://doi.org/10.1007/s00023-014-0370-0 arXiv:1312.7226 [math-ph]ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Rivasseau, V., Vignes-Tourneret, F.: Constructive tensor field theory: The \({T^{4}_{4}}\) model, arXiv:1703.06510 [math-ph]
  16. 16.
    Delepouve, T., Rivasseau, V.: Constructive Tensor Field Theory: The \({T^{4}_{3}}\) Model. Commun. Math. Phys. 345 (2), 477 (2016).  https://doi.org/10.1007/s00220-016-2680-1 arXiv:1412.5091 [math-ph]ADSCrossRefMATHGoogle Scholar
  17. 17.
    Lionni, L., Rivasseau, V.: Intermediate Field Representation for Positive Matrix and Tensor Interactions, arXiv:1609.05018 [math-ph]
  18. 18.
    Rivasseau, V.: Loop Vertex Expansion for Higher Order Interactions. Lett. Math. Phys. 108(5), 1147 (2018).  https://doi.org/10.1007/s11005-017-1037-9 arXiv:1702.07602 [math-ph]ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Krajewski, T., Rivasseau, V., Sazonov, V.: Constructive Matrix Theory for Higher Order Interaction, arXiv:1712.05670 [math-ph]
  20. 20.
    Sokal, A.D.: An improvement of watson’s theorem on borel summability. J. Math. Phys. 21, 261 (1980)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Bonzom, V., Gurau, R., Riello, A., Rivasseau, V.: Critical behavior of colored tensor models in the large N limit. Nucl.Phys.B 853, 174 (2011).  https://doi.org/10.1016/j.nuclphysb.2011.07.022 arXiv:1105.3122 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Laboratoire de Physique Théorique, CNRS UMR 8627Université Paris XIOrsay CedexFrance

Personalised recommendations