Is an effective Lagrangian a convergent series?

Ariel R. Zhitnitsky

doi:10.1103/PhysRevD.54.5148

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Is an effective Lagrangian a convergent series?

Ariel R. Zhitnitsky^*

Physics Department, University of British Columbia, 6224 Agricultural Road, Vancouver, British Columbia, Canada V6T 1Z1

^*Electronic address: arz@physics.ubc.ca

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Phys. Rev. D 54, 5148 – Published 15 October, 1996

DOI: https://doi.org/10.1103/PhysRevD.54.5148

Abstract

We present some generic arguments demonstrating that an effective Lagrangian $L_{eff}$ which, by definition, contains operators $O^{n}$ of arbitrary dimensionality in general is not convergent, but rather an asymptotic series. It means that the behavior of the far distant terms has a specific factorial dependence $L_{eff} \sim Σ_{n} (\frac{c_{n} O^{n}}{M^{n}})$ , $c_{n} \sim n!$ , $n ≫ 1$ . We explain the main ideas by using QED as a toy model. However we expect that the obtained results have a much more general origin. We speculate on possible applications of these results to various physical problems with typical energies from 1 GeV to the Planck scale.

References (14)

H. Georgi, Annu. Rev. Nucl. Part. Sci. 43, 355 (1994)
A. Manohar, Effective Field Theories, Report No. hep-ph/9508245 (unpublished)
D. Kaplan, Effective Field Theories, Report No. hep-ph/9506035 (unpublished)
V. Bhansli and H. Georgi, Running Nonlocal Lagrangians, Report No. hep-ph/9205242, 1992 (unpublished)
V. BhansliH. GeorgiLarge Order Behavior of Perturbative Theory, edited by J.C. Gillou and J. Zinn-Justin, Current Physics-Sources and Comments Vol. 7 (North-Holland, Amsterdam, 1990)
M.A. Shifman, in Proceedings of the Workshop "Coutinuous Advances in QCD," edited by A. Smilga (World Scientific, Singapore, 1994), pg. 249 talk at PASCOS, Johns Hopkins University, Baltimore, March 1995, Report No. TPI-MINN-95/15-T, hep-ph/9505289 (unpublished) talk at XVIII Kazimier Meeting, Ames, Iowa, May 1995, Report No. TPI-MINN-95/32-T, hep-ph/9511469 (unpublished)
A. Zhitnitsky, Phys. Rev. D 53, 5821 (1996)
C. Itzykson and J. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980)
P.H. Damgaard, H.B. Nielsen, and R. Sollacher, Nucl. Phys. B414, 541 (1994)
J.F. Donoghue, Report No. gr-qc/9512024, hep-ph/9512287 (unpublished)
T. Fugleberg and A. Zhitnitsky, Perturbative series at large order in quantum gravity (work in progress)
A. Linde, Particle Physics and Inflationary Cosmology (Harwood Academic, Chur, Switzerland, 1990)
R.H. Brandenberger, in $CP$ Violation and the Limits of the Standard Model, Proceedings of the Theoretical Advanced Study Institute, Boulder, Colorado, 1994, edited by J. Donoghue (World Scientific, Singapore, 1995)
R. Brandenberger and A. Zhitnitsky, Report No. hep-ph 9604407, 1996 (unpublished)

[c1] H. Georgi, Annu. Rev. Nucl. Part. Sci. 43, 355 (1994)

[c2] A. Manohar, Effective Field Theories, Report No. hep-ph/9508245 (unpublished)

[c3] D. Kaplan, Effective Field Theories, Report No. hep-ph/9506035 (unpublished)

[c4] V. Bhansli and H. Georgi, Running Nonlocal Lagrangians, Report No. hep-ph/9205242, 1992 (unpublished)

[c5] V. BhansliH. GeorgiLarge Order Behavior of Perturbative Theory, edited by J.C. Gillou and J. Zinn-Justin, Current Physics-Sources and Comments Vol. 7 (North-Holland, Amsterdam, 1990)

[c6] M.A. Shifman, in Proceedings of the Workshop "Coutinuous Advances in QCD," edited by A. Smilga (World Scientific, Singapore, 1994), pg. 249 talk at PASCOS, Johns Hopkins University, Baltimore, March 1995, Report No. TPI-MINN-95/15-T, hep-ph/9505289 (unpublished) talk at XVIII Kazimier Meeting, Ames, Iowa, May 1995, Report No. TPI-MINN-95/32-T, hep-ph/9511469 (unpublished)

[c7] A. Zhitnitsky, Phys. Rev. D 53, 5821 (1996)

[c8] C. Itzykson and J. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980)

[c9] P.H. Damgaard, H.B. Nielsen, and R. Sollacher, Nucl. Phys. B414, 541 (1994)

[c10] J.F. Donoghue, Report No. gr-qc/9512024, hep-ph/9512287 (unpublished)

[c11] T. Fugleberg and A. Zhitnitsky, Perturbative series at large order in quantum gravity (work in progress)

[c12] A. Linde, Particle Physics and Inflationary Cosmology (Harwood Academic, Chur, Switzerland, 1990)

[c13] R.H. Brandenberger, in $CP$ Violation and the Limits of the Standard Model, Proceedings of the Theoretical Advanced Study Institute, Boulder, Colorado, 1994, edited by J. Donoghue (World Scientific, Singapore, 1995)

[c14] R. Brandenberger and A. Zhitnitsky, Report No. hep-ph 9604407, 1996 (unpublished)

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