Background-field and Schwinger-DeWitt proper-time algorithm for the low-energy effective-field-theory action

Choonkyu Lee; Taehoon Lee; Hyunsoo Min

doi:10.1103/PhysRevD.39.1681

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Background-field and Schwinger-DeWitt proper-time algorithm for the low-energy effective-field-theory action

Choonkyu Lee and Taehoon Lee

Hyunsoo Min

Department of Physics, Seoul National University, Seoul, 151-742, Korea

Division of Liberal Arts, Seoul City University, Seoul, 130-743, Korea

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Phys. Rev. D 39, 1681 – Published 15 March, 1989

DOI: https://doi.org/10.1103/PhysRevD.39.1681

Abstract

For a quantum field theory consisting of fields with vastly different mass scales, we develop, within the one-loop approximation, a simple algorithm which allows the construction of the low-energy effective Lagrangian for light particles to any desired order in inverse powers of heavy masses. Nontrivial generalizations of the background-field method and Schwinger-DeWitt proper-time technique are given in connection with finding effective local vertices induced from ‘‘mixed’’ loop diagrams, i.e., from one-loop diagrams involving heavy- and light-particle internal lines simultaneously. The factorization theory concerning virtual heavy-particle effects finds a simple realization in our procedure. We illustrate our algorithm in the context of the most general renormalizable four-dimensional scalar field theory in flat space.

References (28)

T. Appelquist and J. Carazzone, Phys. Rev. D 11, 2856 (1975).
C. Lee, Nucl. Phys. B161, 171 (1979); Y. Kazama and Y. P. Yao, Phys. Rev. D 21, 1116 (1980); ibid. 21, 1138 (1980).
B. Ovrut and H. Schnitzer, Phys. Rev. D 21, 3369 (1980); S. Weinberg, Phys. Lett. 91B, 51 (1980).
W. Zimmermann, in Lectures on Elementary Particles and Quantum Field Theory, edited by S. Deser, M. Grisaru, and H. Pendleton (MIT Press, Cambridge, Massachusetts, 1971).
H. Georgi, Weak Interactions and Modern Particle Theory (Benjamin/Cummings, Menlo Park, California, 1984), p. 124.
Here note that the full S matrix of a given quantum field theory can be obtained directly from the effective action. See B. S. DeWitt, in Relativity, Groups and Topology II, proceedings of the Les Houches Summer School, Les Houches, France, 1983, edited by B. S. DeWitt and R. Stora (Les Houches Summer School Proceedings, Vol. 40) (North-Holland, Amsterdam, 1984), pp. 696–701; A. Jevicki and C. Lee, Phys. Rev. D 37, 1485 (1988).
B. S. DeWitt, Dynamical Theory of Groups and Fields (Gordon and Breach, New York, 1965); J. Honerkamp, Nucl. Phys. B36, 130 (1971); ibid. B48, 269 (1972); G. 't Hooft, in Acta Universitatis Wratislavensis No. 38, 12th Winter School of Theoretical Physics in Karpacz, Functional and Probabilistic Methods in Quantum Field Theory, Vol. 1 (1975); Vol. 1; D. G. Boulware, Phys. Rev. D 23, 389 (1981); L. F. Abbott, Nucl. Phys. B185, 189 (1981); B. S. DeWitt, in Quantum Gravity II, edited by C. Isham, R. Penrose, and D. Sciama (Oxford University Press, New York, 1981).
J. Schwinger, Phys. Rev. 82, 664 (1951); DeWitt (Ref. 7); J. Schwinger, Phys. Rep. 19, 295 (1975).
G. 't Hooft, Nucl. Phys. B62, 444 (1973); S. Ichinose and M. Omote, ibid. B203, 221 (1982); C. Lee and C. Rim, ibid. B255, 439 (1985).
B. W. Lee, Chiral Dynamics (Gordon and Breach, New York, 1972).
M. K. Gaillard and B. W. Lee, Phys. Rev. Lett. 33, 108 (1974); G. Altarelli and L. Maiani, Phys. Lett. 52B, 351 (1974).
B. S. DeWitt, Phys. Rep. 19, 295 (1975).
W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936).
C. Lee, H. Min and P. Y. Pac, Nucl. Phys. B202, 336 (1982); R. Ball and H. Osborn ibid. B263, 245 (1986).
T. Appelquist and C. Bernard, Phys. Rev. D 23, 425 (1981); R. Akhoury and Y. P. Yao, ibid. 25, 3361 (1982); L.-H. Chan, ibid. 36, 3755 (1987).
If one wishes to study the r-loop (r >= 1) contribution $Γ_{φ}^{(r)}$ ( φ ) through Eq. (2.5), one only needs to solve the effective field equation (2.6) to (r-1)-loop order.
W. A. Bardeen, A. J. Buras, D. W. Duke and T. Muta, Phys. Rev. D 18, 3998 (1978).
Stated differently, it will be sufficient to consider a certain finite number of insertions involving such higher-derivative vertices, given an effective field theory valid to some orders in $M^{- 1}$ . For a detailed theoretical justification of this, consult Ref. 2.
C. Lee, T. Lee and H. Min, following paper, Phys. Rev. D 39, 1701 (1989).
Mass terms for light-particle fields are included in B( φ ).
One might suspect that this infrared difficulty could be due to the fact that, in writing the $Λ^{- 1}$ expansions, we elected to expand the exponential factors $e^{- i τ m_{l}^{2} / Λ^{2}}$ ( $m_{l}$ : light-field masses) in powers of $Λ^{- 1}$ . Surely, if all light fields have nonvanishing masses, all term-by-term tau integrals could be made finite by maintaining those exponential factors explicitly. But this is no solution to the problem: the right-hand side of Eq. (2.25) will now receive increasingly large contributions from terms corresponding to larger k values in the $Λ^{- 1}$ expansion, and as a result the $Λ^{- 1}$ expansion becomes useless.
In the context of the linear sigma model with large $M_{sigma}$ , an analogous connection has been given by Appelquist and Bernard (Ref. 15). Here the low-energy effective field theory is taken by the nonliner sigma model.
But we may remind readers of the fact that interaction terms involving one heavy and two light fields correspond to one of the most common types in spontaneous broken gauge theories (which make the backbone of all more realistic models of elementary-particle interactions).
Chaiho P. Rim, Phys. Lett. B. 208, 381 (1988).
Choice of the finite renormalization terms needed to make decoupling manifest is not unique, although requiring Eq. (3.38) certainly appears to be one of the most natural. Especially, there exists considerable arbitrariness with respect to finite renormalizations of the parameters involving (partially or wholly) heavy fields.
In the given case this renormalization prescription is of course identical to that used in Eq. (2.11), except for the change in the normalization mass, μ -> $\tilde{μ}$ .
Lee and Rim (Ref. 9); I. Jack and H. Osborn, Nucl. Phys. B249, 472 (1985).
C. Lee, Nucl. Phys. B207, 157 (1982).

[c1] T. Appelquist and J. Carazzone, Phys. Rev. D 11, 2856 (1975).

[c2] C. Lee, Nucl. Phys. B161, 171 (1979); Y. Kazama and Y. P. Yao, Phys. Rev. D 21, 1116 (1980); ibid. 21, 1138 (1980).

[c3] B. Ovrut and H. Schnitzer, Phys. Rev. D 21, 3369 (1980); S. Weinberg, Phys. Lett. 91B, 51 (1980).

[c4] W. Zimmermann, in Lectures on Elementary Particles and Quantum Field Theory, edited by S. Deser, M. Grisaru, and H. Pendleton (MIT Press, Cambridge, Massachusetts, 1971).

[c5] H. Georgi, Weak Interactions and Modern Particle Theory (Benjamin/Cummings, Menlo Park, California, 1984), p. 124.

[c6] Here note that the full S matrix of a given quantum field theory can be obtained directly from the effective action. See B. S. DeWitt, in Relativity, Groups and Topology II, proceedings of the Les Houches Summer School, Les Houches, France, 1983, edited by B. S. DeWitt and R. Stora (Les Houches Summer School Proceedings, Vol. 40) (North-Holland, Amsterdam, 1984), pp. 696–701; A. Jevicki and C. Lee, Phys. Rev. D 37, 1485 (1988).

[c7] B. S. DeWitt, Dynamical Theory of Groups and Fields (Gordon and Breach, New York, 1965); J. Honerkamp, Nucl. Phys. B36, 130 (1971); ibid. B48, 269 (1972); G. 't Hooft, in Acta Universitatis Wratislavensis No. 38, 12th Winter School of Theoretical Physics in Karpacz, Functional and Probabilistic Methods in Quantum Field Theory, Vol. 1 (1975); Vol. 1; D. G. Boulware, Phys. Rev. D 23, 389 (1981); L. F. Abbott, Nucl. Phys. B185, 189 (1981); B. S. DeWitt, in Quantum Gravity II, edited by C. Isham, R. Penrose, and D. Sciama (Oxford University Press, New York, 1981).

[c8] J. Schwinger, Phys. Rev. 82, 664 (1951); DeWitt (Ref. 7); J. Schwinger, Phys. Rep. 19, 295 (1975).

[c9] G. 't Hooft, Nucl. Phys. B62, 444 (1973); S. Ichinose and M. Omote, ibid. B203, 221 (1982); C. Lee and C. Rim, ibid. B255, 439 (1985).

[c10] B. W. Lee, Chiral Dynamics (Gordon and Breach, New York, 1972).

[c11] M. K. Gaillard and B. W. Lee, Phys. Rev. Lett. 33, 108 (1974); G. Altarelli and L. Maiani, Phys. Lett. 52B, 351 (1974).

[c12] B. S. DeWitt, Phys. Rep. 19, 295 (1975).

[c13] W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936).

[c14] C. Lee, H. Min and P. Y. Pac, Nucl. Phys. B202, 336 (1982); R. Ball and H. Osborn ibid. B263, 245 (1986).

[c15] T. Appelquist and C. Bernard, Phys. Rev. D 23, 425 (1981); R. Akhoury and Y. P. Yao, ibid. 25, 3361 (1982); L.-H. Chan, ibid. 36, 3755 (1987).

[c16] If one wishes to study the r-loop (r >= 1) contribution $Γ_{φ}^{(r)}$ ( φ ) through Eq. (2.5), one only needs to solve the effective field equation (2.6) to (r-1)-loop order.

[c17] W. A. Bardeen, A. J. Buras, D. W. Duke and T. Muta, Phys. Rev. D 18, 3998 (1978).

[c18] Stated differently, it will be sufficient to consider a certain finite number of insertions involving such higher-derivative vertices, given an effective field theory valid to some orders in $M^{- 1}$ . For a detailed theoretical justification of this, consult Ref. 2.

[c19] C. Lee, T. Lee and H. Min, following paper, Phys. Rev. D 39, 1701 (1989).

[c20] Mass terms for light-particle fields are included in B( φ ).

[c21] One might suspect that this infrared difficulty could be due to the fact that, in writing the $Λ^{- 1}$ expansions, we elected to expand the exponential factors $e^{- i τ m_{l}^{2} / Λ^{2}}$ ( $m_{l}$ : light-field masses) in powers of $Λ^{- 1}$ . Surely, if all light fields have nonvanishing masses, all term-by-term tau integrals could be made finite by maintaining those exponential factors explicitly. But this is no solution to the problem: the right-hand side of Eq. (2.25) will now receive increasingly large contributions from terms corresponding to larger k values in the $Λ^{- 1}$ expansion, and as a result the $Λ^{- 1}$ expansion becomes useless.

[c22] In the context of the linear sigma model with large $M_{sigma}$ , an analogous connection has been given by Appelquist and Bernard (Ref. 15). Here the low-energy effective field theory is taken by the nonliner sigma model.

[c23] But we may remind readers of the fact that interaction terms involving one heavy and two light fields correspond to one of the most common types in spontaneous broken gauge theories (which make the backbone of all more realistic models of elementary-particle interactions).

[c24] Chaiho P. Rim, Phys. Lett. B. 208, 381 (1988).

[c25] Choice of the finite renormalization terms needed to make decoupling manifest is not unique, although requiring Eq. (3.38) certainly appears to be one of the most natural. Especially, there exists considerable arbitrariness with respect to finite renormalizations of the parameters involving (partially or wholly) heavy fields.

[c26] In the given case this renormalization prescription is of course identical to that used in Eq. (2.11), except for the change in the normalization mass, μ -> $\tilde{μ}$ .

[c27] Lee and Rim (Ref. 9); I. Jack and H. Osborn, Nucl. Phys. B249, 472 (1985).

[c28] C. Lee, Nucl. Phys. B207, 157 (1982).

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Background-field and Schwinger-DeWitt proper-time algorithm for the low-energy effective-field-theory action

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