On-shell gauge-parameter independence of <i>〈q¯q〉</i> contributions to electroweak quark self-energies

M. R. Ahmady; V. Elias; R. R. Mendel; M. D. Scadron; T. Steele

doi:10.1103/PhysRevD.39.2764

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On-shell gauge-parameter independence of 〈q¯q〉 contributions to electroweak quark self-energies

M. R. Ahmady, V. Elias, and R. R. Mendel

M. D. Scadron

T. Steele

Department of Applied Mathematics, University of Western Ontario, London, Ontario, Canada N6A 5B9

Department of Physics, University of Arizona, Tucson, Arizona 85721

Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, United Kingdom

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Phys. Rev. D 39, 2764 – Published 1 May, 1989

DOI: https://doi.org/10.1103/PhysRevD.39.2764

Abstract

We allow an external 〈q¯q〉 condensate to enter standard SU(2)×U(1) electroweak theory via the vacuum expectation value 〈0‖:q¯(x)q(y):‖0〉, as in QCD sum-rule applications. For a given flavor, we then find that any gauge-parameter dependence of quark self-energies on the ‘‘mass shell’’ is eliminated provided that the mass shell is made to coincide with both the expansion-parameter mass occurring in the operator-product expansion of 〈0‖:q¯(x)q(y):‖0〉 and the standard electroweak mass acquired via the Yukawa coupling to the usual scalar vacuum expectation value of spontaneous symmetry breaking. This result indicates that if the QCD-generated order parameter 〈q¯q〉 and associated dynamical mass(es) $m_{q}^{dyn}$ are utilized as external input parameters in electroweak calculations involving hadrons, then new corrections must be introduced into the q¯qW and q¯qZ vertices in order to preserve SU(2)×U(1) Ward identities.

References (20)

H. D. Politzer, Nucl. Phys. B117, 397 (1976).
V. Elias and M. D. Scadron, Phys. Rev. D 30, 647 (1984).
V. Elias, M. D. Scadron and R. Tarrach, Phys. Lett. 162B, 176 (1985).
L. J. Reinders and K. Stam, Phys. Lett. B 180, 125 (1986).
P. Pascual and R. de Rafael, Z. Phys. C 12, 127 (1982).
T. Larsson, Phys. Rev. D 32, 956 (1985).
V. Elias, T. G. Steele and M. D. Scadron, Phys. Rev. D 38, 1584 (1988).
V. Elias and M. D. Scadron, J. Phys. G 14, 1175 (1988).
This identification of the QCD propagator pole with the constituent-quark mass was first suggested in R. Tarrach, Nucl. Phys. B183, 384 (1981).
V. Elias and T. Steele, Phys. Lett. B 212, 88 (1988).
P. Pascual and R. Tarrach, in QCD: Renormalization for the Practitioner, edited by H. Araki, J. Ehlers, K. Hepp, J. Kippenhahn, H. A. Weidenmuller, and J. Zittartz (Lecture Notes in Physics, Vol. 194) (Springer, Berlin, 1984), pp. 168–184.
M. Shifman, A. Vainshtein and V. Zakharov, Nucl. Phys. B147, 385 (1979); ibid. B147, 448 (1979).
S. L. Glashow, Nucl. Phys. 22, 579 (1961); S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967); Abdus Salam, in Elementary Particle Theory: Relativistic Groups and Analyticity (Nobel Symposium No. 8), edited by N. Svartholm (Wiley, New York, 1969), p. 367.
See K. Aoki, Z. Hioki, R. Kawabe, M. Konuma and T. Muta, Suppl. Prog. Theor. Phys. 73, 1 (1982) for Feynman rules of spontaneously broken SU(2)timesU(1) expressed entirely in terms of particle masses, KM mixing parameters, and the electromagnetic coupling constant e.
R. Delbourgo and M. D. Scadron, Nuovo Cimento Lett. 44, 193 (1985).
Such condensates represent additional symmetry-breaking order parameters whose nonzero values would imply the occurrence of additional nonperturbative vacuum expectation values [e.g., < 0|: $ψ_{n}^{α}$ (y) ψ ${bar}_{r}^{β}$ (z) $A_{μ}^{b}$ (w):|0 > , as discussed in Ref. 7 and in L. J. Reinders and K. Stam, Phys. Lett. B 195, 465 (1987); V. Elias and T. G. Steele, ibid. 199, 547 (1987)].
Truncation of j >= 2 terms in (2.1) is discussed in V. Elias, T. Steele, M. D. Scadron and R. Tarrach, Phys. Rev. D 34, 3537 (1986).
See Eq. (7.4.15) of E. Hansen, A Table of Series and Products (Prentice-Hall, Englewood Cliffs, NJ, 1975). These series evaluations were also performed by S. Sherebrin using the symbolic manipulation program MAPLE_, which was brought to our attention by R. Corless.
The on-shell gauge-parameter independence of W-mediated self-energies analogous to Figs. 1 occurs through GIM-driven cancellations, in which the sum of u- and c-mediated analogs to Fig. 1(a) conspires to absorb the $χ^{+ -}$ -tadpole analogs to Fig. 1(c).
The SU(2)timesU(1)-symmetry breaking due to langleq¯q > ${}_{QCD}$ is ``explicit'' as long as we treat the two theories separately, i.e., treat SU(3) $_{c}$ interactions as external with respect to SU(2)timesU(1) and its vacuum. Clearly, in the context of the full SU(2)timesU(1)timesSU(3) $_{c}$ theory with its unique physical vacuum, the langleq¯q > ${}_{QCD}$ condensates should also be regarded as spontaneous-symmetry-breaking order parameters. Indeed, it is the latter statement that makes it possible for physical processes to be independent of $ξ_{Z}$ and $ξ_{W}$ allowing for renormalizability of the full theory.

[c1] H. D. Politzer, Nucl. Phys. B117, 397 (1976).

[c2] V. Elias and M. D. Scadron, Phys. Rev. D 30, 647 (1984).

[c3] V. Elias, M. D. Scadron and R. Tarrach, Phys. Lett. 162B, 176 (1985).

[c4] L. J. Reinders and K. Stam, Phys. Lett. B 180, 125 (1986).

[c5] P. Pascual and R. de Rafael, Z. Phys. C 12, 127 (1982).

[c6] T. Larsson, Phys. Rev. D 32, 956 (1985).

[c7] V. Elias, T. G. Steele and M. D. Scadron, Phys. Rev. D 38, 1584 (1988).

[c8] V. Elias and M. D. Scadron, J. Phys. G 14, 1175 (1988).

[c9] This identification of the QCD propagator pole with the constituent-quark mass was first suggested in R. Tarrach, Nucl. Phys. B183, 384 (1981).

[c10] V. Elias and T. Steele, Phys. Lett. B 212, 88 (1988).

[c11] P. Pascual and R. Tarrach, in QCD: Renormalization for the Practitioner, edited by H. Araki, J. Ehlers, K. Hepp, J. Kippenhahn, H. A. Weidenmuller, and J. Zittartz (Lecture Notes in Physics, Vol. 194) (Springer, Berlin, 1984), pp. 168–184.

[c12] M. Shifman, A. Vainshtein and V. Zakharov, Nucl. Phys. B147, 385 (1979); ibid. B147, 448 (1979).

[c13] S. L. Glashow, Nucl. Phys. 22, 579 (1961); S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967); Abdus Salam, in Elementary Particle Theory: Relativistic Groups and Analyticity (Nobel Symposium No. 8), edited by N. Svartholm (Wiley, New York, 1969), p. 367.

[c14] See K. Aoki, Z. Hioki, R. Kawabe, M. Konuma and T. Muta, Suppl. Prog. Theor. Phys. 73, 1 (1982) for Feynman rules of spontaneously broken SU(2)timesU(1) expressed entirely in terms of particle masses, KM mixing parameters, and the electromagnetic coupling constant e.

[c15] R. Delbourgo and M. D. Scadron, Nuovo Cimento Lett. 44, 193 (1985).

[c16] Such condensates represent additional symmetry-breaking order parameters whose nonzero values would imply the occurrence of additional nonperturbative vacuum expectation values [e.g., < 0|: $ψ_{n}^{α}$ (y) ψ ${bar}_{r}^{β}$ (z) $A_{μ}^{b}$ (w):|0 > , as discussed in Ref. 7 and in L. J. Reinders and K. Stam, Phys. Lett. B 195, 465 (1987); V. Elias and T. G. Steele, ibid. 199, 547 (1987)].

[c17] Truncation of j >= 2 terms in (2.1) is discussed in V. Elias, T. Steele, M. D. Scadron and R. Tarrach, Phys. Rev. D 34, 3537 (1986).

[c18] See Eq. (7.4.15) of E. Hansen, A Table of Series and Products (Prentice-Hall, Englewood Cliffs, NJ, 1975). These series evaluations were also performed by S. Sherebrin using the symbolic manipulation program MAPLE_, which was brought to our attention by R. Corless.

[c19] The on-shell gauge-parameter independence of W-mediated self-energies analogous to Figs. 1 occurs through GIM-driven cancellations, in which the sum of u- and c-mediated analogs to Fig. 1(a) conspires to absorb the $χ^{+ -}$ -tadpole analogs to Fig. 1(c).

[c20] The SU(2)timesU(1)-symmetry breaking due to langleq¯q > ${}_{QCD}$ is ``explicit'' as long as we treat the two theories separately, i.e., treat SU(3) $_{c}$ interactions as external with respect to SU(2)timesU(1) and its vacuum. Clearly, in the context of the full SU(2)timesU(1)timesSU(3) $_{c}$ theory with its unique physical vacuum, the langleq¯q > ${}_{QCD}$ condensates should also be regarded as spontaneous-symmetry-breaking order parameters. Indeed, it is the latter statement that makes it possible for physical processes to be independent of $ξ_{Z}$ and $ξ_{W}$ allowing for renormalizability of the full theory.

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On-shell gauge-parameter independence of 〈q¯q〉 contributions to electroweak quark self-energies

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