Radiative corrections and electroweak observables

Jonathan L. Rosner

doi:10.1103/PhysRevD.42.3107

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Radiative corrections and electroweak observables

Jonathan L. Rosner

Enrico Fermi Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637

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Phys. Rev. D 42, 3107 – Published 1 November, 1990

DOI: https://doi.org/10.1103/PhysRevD.42.3107

Abstract

Different electroweak observables probe different combinations of the parameters θ¯ (the weak mixing angle as defined in terms of couplings at the Z pole) and ρ== $M_{W}^{2}$ /( $M_{Z}^{2}$ $\cos^{2}$ θ¯). A simple, approximate method is given for relating these parameters to the angle θ defined in terms of gauge-boson masses by $\cos^{2}$ θ ${ieqM}_{W}^{2}$ / $M_{Z}^{2}$ , and it is shown how these various observables shed light on θ and quantities such as the top-quark mass that directly affect ρ.

References (67)

CDF Collaboration, F. Abe et al., Phys. Rev. Lett. 63, 720 (1989).
Mark II Collaboration, G. S. Abrams et al., Phys. Rev. Lett. 63, 724 (1989).
ALEPH Collaboration, D. Decamp et al., Phys. Lett. B 231, 519 (1989); , ibid. 235, 399 (1990); DELPHI Collaboration, P. Aarnio et al., ibid 231, 539 (1989); DELPHI Collaboration, P. Abreu et al., ibid 241, 435 (1990); L3 Collaboration, B. Adeva et al., ibid 231, 509 (1989); , ibid. 237, 136 (1990); OPAL Collaboration, M. Z. Akrawy et al., ibid 231, 530 (1989).
Preliminary results from CDF Collaboration, F. Abe et al., were presented by P. Schlabach and Hans Jensen, Bull. Am. Phys. Soc. 35, 999 (1990). The most recent value is based on F. Abe et al., CDF reports, 1990 (unpublished).
UA2 Collaboration, J. Alitti et al., Phys. Lett. B 241, 150 (1990).
Ugo Amaldi, Phys. Rev. D 36, 1385 (1987).
W. Marciano, in Physics at the 100 GeV Mass Scale, proceedings of the 17th SLAC Summer Institute, Stanford, California, 1989, edited by Eileen C. Brennan (SLAC, Stanford, California, 1990); in Proceedings of the TASI-90, Boulder, Colorado, 1990, edited by M. Cvetič and P. Langacker (World Scientific, Singapore, in press).
M. Peskin, in Physics at the 100 GeV Mass Scale (Ref. 7).
G. Altarelli, in Proceedings of the XIVth International Symposium on Lepton and Photon Interactions, Stanford, California, 1989, edited by M. Riordan (World Scientific, Singapore, 1990), p. 286.
Paul Langacker, presented at Symposium on ``Thinking about the Top Quark,'' Santa Barbara, California, 1990 (unpublished).
D. A. Ross and M. Veltman, Nucl. Phys. B95, 135 (1975); M. Veltman, ibid. B123, 89 (1977); M. S. Chanowitz, M. A. Furman and I. Hinchliffe, Phys. Lett. 78B, 285 (1978).
W. J. Marciano, Phys. Rev. D 20, 274 (1979); A. Sirlin, ibid. 22, 971 (1980); W. J. Marciano and A. Sirlin, ibid. 22, 2695 (1980); ibid. 29, 945 (1984); A. Sirlin, Comments Nucl. Part. Phys. 17, 279 (1987); Phys. Lett. B 232, 123 (1989); Nucl. Phys. B332, 20 (1990); Proceedings of Rencontres de Moriond, Les Arcs, France, 1990 (unpublished); S. Fanchiotti and A. Sirlin, Phys. Rev. D 41, 319 (1990); G. Degrassi, S. Fanchiotti, and A. Sirlin, New York University report, 1990 (unpublished).
B. W. Lynn and R. G. Stuart, Nucl. Phys. B253, 216 (1985); B. W. Lynn, M. Peskin, and R. G. Stuart, in Physics at LEP, LEP Jamboree, Geneva, Switzerland, 1985, edited by J. Ellis and R. D. Peccei (CERN Report No. 86-02, Geneva, 1986); in Tests of Electroweak Theories: Polarized Processes and Other Phenomena, proceedings of 2nd Conference on Tests of Electroweak Theories, Trieste, Italy, 1985, edited by Bryan W. Lynn and Claudio Verzegnassi (World Scientific, Singapore, 1986), p. 213; D. C. Kennedy, B. W. Lynn, C. J.-C. Im and R. G. Stuart, Nucl. Phys. B321, 83 (1989).
Elizabeth Jenkins and Aneesh V. Manohar, Phys. Lett. B 237, 259 (1999); R. D. Peccei, Mod. Phys. Lett. A 5, 1001 (1990); Alain Blondel, CERN Report No. CERN-EP/90-10, 1990 (unpublished).
G. Passarino and M. Veltman, Phys. Lett. B 237, 537 (1990).
F. Jegerlehner, Z. Phys. C 32, 195 (1986); ibid. 32, 425 (1986); ibid. 38, 519(E) (1988); M. Consoli, W. Hollik and F. Jegerlehner, Phys. Lett. B 227, 167 (1989).
Paul Langacker, William J. Marciano and Alberto Sirlin, Phys. Rev. D 36, 2191 (1987).
Wolfgang F. L. Hollik, Fortschr. Phys. 38, 165 (1990). For a recent review, see Wolfgang Hollik, presented at the 1989 CERN-JINR School of Physics, Egmond-aan-Zee, The Netherlands (unpublished).
William A. Bardeen, Christopher T. Hill and Manfred Lindner, Phys. Rev. D 41, 1647 (1990).
Jonathan L. Rosner, in Physics at Fermilab in the 1990's, proceedings of a workshop at Breckenridge, Colorado, 1989, edited by D. Green and H. J. Lubatti (World Scientific, Singapore, 1990), p. 15.
William A. Bardeen and Christopher T. Hill (unpublished).
Ray Renken and Michael E. Peskin, Nucl. Phys. B211, 93 (1983); Michael E. Peskin and Tatsu Takeuchi, Phys. Rev. Lett. 65, 964 (1990).
Mitchell Golden and Lisa Randall, Fermilab Report No. Fermilab-Pub-90/83-T, 1990 (unpublished).
Paul Langacker, Commun. Nucl. Part. Phys. 19, 1 (1989) Phys. Rev. Lett. 63, 1920 (1989); Particle Data Group, Phys. Lett. B 239, 1 (1990).
G. Costa et al., Nucl. Phys. B297, 244 (1988); G. L. Fogli and J. Ellis, Phys. Lett. B 213, 526 (1988); G. L. Fogli and D. Haidt, Z. Phys. C 40, 379 (1988); D. Haidt, in Weak Interactions and Neutrinos, proceedings of the International Workshop, Ginosar, Israel, 1989, edited by P. Singer and Gad Eilam [Nucl. Phys. B (Proc. Suppl.) 13 (1989)].
Francis Halzen and Duncan A. Morris, Phys. Lett. B 237, 107 (1990).
K.-I. Aoki et al., Prog. Theor. Phys. Suppl. 73, 1 (1982); Z. Hioki, Nucl. Phys. B229, 284 (1983); Prog. Theor. Phys. 82, 875 (1989) Phys. Lett. B 224, 417 (1989); ibid. 228, 560(E) (1989).
H. Burkhardt, F. Jegerlehner, G. Penso, and C. Verzegnassi, in Polarization at LEP, edited by G. Alexander et al. (CERN Report No. 88-06, Geneva, Switzerland, 1989), p. 145; M. Consoli, W. Hollik, and F. Jegerlehner, in Z Physics at LEP I, proceedings of the Workshop, Geneva, Switzerland, 1989, edited by G. Altarelli, R. Kleiss, and C. Verzegnassi (CERN Report No. 89-08, Geneva, Switzerland, 1989), Vol. 1, p. 7.
R. Peccei, summary talk at the Z Phenomenology Symposium, Madison, Wisconsin, 1990 (unpublished); E. Fernandez, presented at Neutrino 90 Conference, Geneva, Switzerland, 1990 (unpublished); More recent Z properties have been presented by F. Dydak, presented at XXV International Conference on High Energy Physics, Singapore, 1990 (unpublished).
William Marciano and Jonathan L. Rosner, Brookhaven National Laboratory and Enrico Fermi Institute report, 1990 (unpublished).
G. Burgers, F. Jegerlehner, B. Kniehl, and J. Kühn, in Z Physics at LEP 1 (Ref. 28), Vol. 1, p. 55. This result is consistent with the one quoted by William J. Marciano, in Proceedings of the Salt Lake City Meeting, Annual Meeting of the Division of Particles and Fields of the APS, Salt Lake City, Utah, 1987, edited by Carleton DeTar and James Ball (World Scientific, Singapore, 1987), p. 319: 127.76+-0.3; and by Fanchiotti and Sirlin (Ref. 12): 127.9+-0.3. Different values are quoted by Altarelli (Ref. 9), 128.80+-0.13, and by Peskin (Ref. 8), 128.77+-0.12, both based on the result by Burkhardt et al.. (Ref. 28). We thank M. Peskin for noting that these last values are what one actually obtains when performing a careful estimate of photon vacuum-polarization effects alone, whereas the value quoted by Burgers et al. includes nonelectromagnetic radiative corrections as well, and that of Fanchiotti and Sirlin is in the MS bar_ scheme. For our purposes is the values of Burgers et al. which are most relevant, though our approach does not distinguish clearly among the different quantities.
We have added the result of Ref. 1 ( $M_{Z}$ =90.9 +- 0.3 +- 0.2 GeV/ $c^{2}$ ) to the compilation of results from SLC ( $M_{Z}$ =91.14 +-0.12 GeV/ $c^{2}$ ) and LEP ( $M_{Z}$ =91.169 +-0.032 GeV/ $c^{2}$ ) presented in Ref. 29. The corresponding average obtained by the Particle Data Group is 91.161+-0.031 GeV/ $c^{2}$ (see Ref. 24).
M. Veltman, Acta Phys. Pol. B 8, 475 (1977); M. Veltman and J. van der Bij, Nucl. Phys. B231, 205 (1984); Marciano and Sirlin (Ref. 12); Hollik (Ref. 18).
Jonathan L. Rosner, in The Santa Fe TASI-87, proceedings of the 1987 Theoretical Advanced Study Institute, Santa Fe, New Mexico, 1987, edited by Richard Slansky and Geoffrey West (World Scientific, Singapore, 1988), p. 3.
D. A. Ross and M. Veltman, Nucl. Phys. B95, 135 (1975); Michael S. Chanowitz and Mitchell Golden, Phys. Lett. 165B, 105 (1985); Mitchell Golden, 169B, 248 (1986) Paul Langacker, lecture presented at International School of Theoretical Physics, Szczyrk, Poland, 1987 (unpublished); G. Passarino, Phys. Lett. B 231, 458 (1989); D. Morris, as quoted by M. Golden, in Physics at Fermilab in the 1990's (Ref. 20), p. 103.
CDF Collaboration, F. Abe et al., presented at a seminar by S. Geer at Fermilab, 1990 (unpublished); presented by K. Sliwa, High Energy Hadronic Interactions, proceedings of the 25th Rencontres de Moriond, Les Arcs, France, 1990 (unpublished); and (unpublished); For earlier results see F. Abe et al., Phys. Rev. Lett. 64, 142 (1990); , ibid. 64, 147 (1990).
H. Abramowicz et al., Phys. Rev. Lett. 57, 298 (1986); CDHSW Collaboration, A. Blondel et al., Z. Phys. C 45, 361 (1990).
CHARM II Collaboration, J. V. Allaby et al., Phys. Lett. B 177, 446 (1986); , Z. Phys. C 36, 611 (1987).
CCFRR Collaboration, P. G. Reutens et al., Phys. Lett. 152B, 404 (1985); P. G. Reutens, Ph.D. thesis, University of Chicago, 1986.
FMM Collaboration, D. Bogert et al., Phys. Rev. Lett. 55, 1969 (1985). For more recent results see T. S. Mattison et al., Phys. Rev. D 42, 1311 (1990).
C. H. Llewellyn Smith, Nucl. Phys. B228, 205 (1983).
Jon Pumplin, Phys. Rev. Lett. 64, 2751 (1990).
R. Brock, in Physics at Fermilab in the 1990's (Ref. 20), p. 334.
E. A. Paschos and L. Wolfenstein, Phys. Rev. D 7, 91 (1973).
P. Q. Hung and J. J. Sakurai, Phys. Lett. 63B, 295 (1976). The notation we use is that of J. E. Kim et al., Rev. Mod. Phys. 53, 211 (1981).
M. C. Noecker, B. P. Masterson and C. E. Wieman, Phys. Rev. Lett. 61, 310 (1988).
V. A. Dzuba, V. V. Flambaum and O. P. Sushkov, Phys. Lett. A 141, 147 (1989).
Very similar corrections have been calculated more recently by S. A. Blundell, W. R. Johnson, and J. Sapirstein, Notre Dame University report, 1990 (unpublished).
The corrections of Ref. 48 yield $Q_{W}$ (Cs) =-71.04 +- 1.58 +-0.88.
Paul Langacker, Richard W. Robinett and Jonathan L. Rosner, Phys. Rev. D 30, 1470 (1984).
David London and Jonathan L. Rosner, Phys. Rev. D 34, 1530 (1986).
L. Stanley Durkin and Paul Langacker, Phys. Lett. 166B, 436 (1986).
Signatures of extra Z's in $e^{+}$ $e^{-}$ annihilations, and references to further literature, are given by David London, G. Bélanger, and J. N. Ng, in Selected Topics in Electroweak Interactions, proceedings of the Second Lake Louise Winter Institute on New Frontiers in Particle Physics, Lake Louise, Canada, 1987, edited by J. M. Cameron et al. (World Scientific, Singapore, 1987), p. 520; Jonathan L. Rosner, ibid, p. 91; R. Gatto, R. Casalbuoni, D. Dominici, and F. Feruglio, in Z Physics at LEP I, (Ref. 28), Vol. 2, p. 147.
Sheldon L. Glashow and Uri Sarid, Phys. Rev. Lett. 64, 725 (1990).
Carl Wieman (private communication).
C. Y. Prescott et al., Phys. Lett. 77B, 347 (1978); , ibid. 84B, 524 (1979).
W. Heil et al., Nucl. Phys. B327, 1 (1989). We thank Paul Langacker for informing us of this reference.
CHARM II Collaboration, D. Geiregat et al., Phys. Lett. B 232, 539 (1989).
Earlier values (as quoted in Ref. 58) are 0.211+- 0.035 +-0.011: CHARM Collaboration, J. Dorenbosch et al., Z. Phys. C 41, 567 (1989), and 0.199+-0.020+-0.013: Brookhaven National Laboratory Experiment E-734, K. Abe et al., Phys. Rev. Lett. 58, 636 (1987).
R. C. Allen et al., Phys. Rev. Lett. 64, 1330 (1990).
CDF Collaboration, presented by Melissa E. B. Franklin at XXV International Conference on High Energy Physics, (Ref. 29). The value (4.38) supersedes an earlier one of 0.216+- 0.015 +-0.010 presented by Myron Campbell, in Proceedings of the XIVth Symposium on Lepton and Photon Interactions (Ref. 9), p. 274, and by Peter Hurst at the 8th Topical Workshop on Proton-Antiproton Collider Physics, Castiglione, Italy, 1989 (unpublished).
ALEPH Collaboration, D. Decamp et al., Phys. Lett. B 244, 551 (1990): Γ ${}_{b b bar}$ / $Γ_{had}$ =0.215 +- 0.017 +- 0.024.
L3 Collaboration, B. Adeva et al., Phys. Lett. B 241, 416 (1990).
J. H. Kühn and P. M. Zerwas, Phys. Rep. 167, 321 (1988); C. Ahn et al., SLAC Report No. SLAC-329, 1988 (unpublished).
S. Jadach et al., in Z Physics at LEP I (Ref. 28), Vol. 1, p. 235.
V. Barger, J. L. Hewett and T. G. Rizzo, Phys. Rev. Lett. 65, 1313 (1990).
C. S. Kim, Jonathan L. Rosner and C.-P. Yuan, Phys. Rev. D 42, 96 (1990).

[c1] CDF Collaboration, F. Abe et al., Phys. Rev. Lett. 63, 720 (1989).

[c2] Mark II Collaboration, G. S. Abrams et al., Phys. Rev. Lett. 63, 724 (1989).

[c3] ALEPH Collaboration, D. Decamp et al., Phys. Lett. B 231, 519 (1989); , ibid. 235, 399 (1990); DELPHI Collaboration, P. Aarnio et al., ibid 231, 539 (1989); DELPHI Collaboration, P. Abreu et al., ibid 241, 435 (1990); L3 Collaboration, B. Adeva et al., ibid 231, 509 (1989); , ibid. 237, 136 (1990); OPAL Collaboration, M. Z. Akrawy et al., ibid 231, 530 (1989).

[c4] Preliminary results from CDF Collaboration, F. Abe et al., were presented by P. Schlabach and Hans Jensen, Bull. Am. Phys. Soc. 35, 999 (1990). The most recent value is based on F. Abe et al., CDF reports, 1990 (unpublished).

[c5] UA2 Collaboration, J. Alitti et al., Phys. Lett. B 241, 150 (1990).

[c6] Ugo Amaldi, Phys. Rev. D 36, 1385 (1987).

[c7] W. Marciano, in Physics at the 100 GeV Mass Scale, proceedings of the 17th SLAC Summer Institute, Stanford, California, 1989, edited by Eileen C. Brennan (SLAC, Stanford, California, 1990); in Proceedings of the TASI-90, Boulder, Colorado, 1990, edited by M. Cvetič and P. Langacker (World Scientific, Singapore, in press).

[c8] M. Peskin, in Physics at the 100 GeV Mass Scale (Ref. 7).

[c9] G. Altarelli, in Proceedings of the XIVth International Symposium on Lepton and Photon Interactions, Stanford, California, 1989, edited by M. Riordan (World Scientific, Singapore, 1990), p. 286.

[c10] Paul Langacker, presented at Symposium on ``Thinking about the Top Quark,'' Santa Barbara, California, 1990 (unpublished).

[c11] D. A. Ross and M. Veltman, Nucl. Phys. B95, 135 (1975); M. Veltman, ibid. B123, 89 (1977); M. S. Chanowitz, M. A. Furman and I. Hinchliffe, Phys. Lett. 78B, 285 (1978).

[c12] W. J. Marciano, Phys. Rev. D 20, 274 (1979); A. Sirlin, ibid. 22, 971 (1980); W. J. Marciano and A. Sirlin, ibid. 22, 2695 (1980); ibid. 29, 945 (1984); A. Sirlin, Comments Nucl. Part. Phys. 17, 279 (1987); Phys. Lett. B 232, 123 (1989); Nucl. Phys. B332, 20 (1990); Proceedings of Rencontres de Moriond, Les Arcs, France, 1990 (unpublished); S. Fanchiotti and A. Sirlin, Phys. Rev. D 41, 319 (1990); G. Degrassi, S. Fanchiotti, and A. Sirlin, New York University report, 1990 (unpublished).

[c13] B. W. Lynn and R. G. Stuart, Nucl. Phys. B253, 216 (1985); B. W. Lynn, M. Peskin, and R. G. Stuart, in Physics at LEP, LEP Jamboree, Geneva, Switzerland, 1985, edited by J. Ellis and R. D. Peccei (CERN Report No. 86-02, Geneva, 1986); in Tests of Electroweak Theories: Polarized Processes and Other Phenomena, proceedings of 2nd Conference on Tests of Electroweak Theories, Trieste, Italy, 1985, edited by Bryan W. Lynn and Claudio Verzegnassi (World Scientific, Singapore, 1986), p. 213; D. C. Kennedy, B. W. Lynn, C. J.-C. Im and R. G. Stuart, Nucl. Phys. B321, 83 (1989).

[c14] Elizabeth Jenkins and Aneesh V. Manohar, Phys. Lett. B 237, 259 (1999); R. D. Peccei, Mod. Phys. Lett. A 5, 1001 (1990); Alain Blondel, CERN Report No. CERN-EP/90-10, 1990 (unpublished).

[c15] G. Passarino and M. Veltman, Phys. Lett. B 237, 537 (1990).

[c16] F. Jegerlehner, Z. Phys. C 32, 195 (1986); ibid. 32, 425 (1986); ibid. 38, 519(E) (1988); M. Consoli, W. Hollik and F. Jegerlehner, Phys. Lett. B 227, 167 (1989).

[c17] Paul Langacker, William J. Marciano and Alberto Sirlin, Phys. Rev. D 36, 2191 (1987).

[c18] Wolfgang F. L. Hollik, Fortschr. Phys. 38, 165 (1990). For a recent review, see Wolfgang Hollik, presented at the 1989 CERN-JINR School of Physics, Egmond-aan-Zee, The Netherlands (unpublished).

[c19] William A. Bardeen, Christopher T. Hill and Manfred Lindner, Phys. Rev. D 41, 1647 (1990).

[c20] Jonathan L. Rosner, in Physics at Fermilab in the 1990's, proceedings of a workshop at Breckenridge, Colorado, 1989, edited by D. Green and H. J. Lubatti (World Scientific, Singapore, 1990), p. 15.

[c21] William A. Bardeen and Christopher T. Hill (unpublished).

[c22] Ray Renken and Michael E. Peskin, Nucl. Phys. B211, 93 (1983); Michael E. Peskin and Tatsu Takeuchi, Phys. Rev. Lett. 65, 964 (1990).

[c23] Mitchell Golden and Lisa Randall, Fermilab Report No. Fermilab-Pub-90/83-T, 1990 (unpublished).

[c24] Paul Langacker, Commun. Nucl. Part. Phys. 19, 1 (1989) Phys. Rev. Lett. 63, 1920 (1989); Particle Data Group, Phys. Lett. B 239, 1 (1990).

[c25] G. Costa et al., Nucl. Phys. B297, 244 (1988); G. L. Fogli and J. Ellis, Phys. Lett. B 213, 526 (1988); G. L. Fogli and D. Haidt, Z. Phys. C 40, 379 (1988); D. Haidt, in Weak Interactions and Neutrinos, proceedings of the International Workshop, Ginosar, Israel, 1989, edited by P. Singer and Gad Eilam [Nucl. Phys. B (Proc. Suppl.) 13 (1989)].

[c26] Francis Halzen and Duncan A. Morris, Phys. Lett. B 237, 107 (1990).

[c27] K.-I. Aoki et al., Prog. Theor. Phys. Suppl. 73, 1 (1982); Z. Hioki, Nucl. Phys. B229, 284 (1983); Prog. Theor. Phys. 82, 875 (1989) Phys. Lett. B 224, 417 (1989); ibid. 228, 560(E) (1989).

[c28] H. Burkhardt, F. Jegerlehner, G. Penso, and C. Verzegnassi, in Polarization at LEP, edited by G. Alexander et al. (CERN Report No. 88-06, Geneva, Switzerland, 1989), p. 145; M. Consoli, W. Hollik, and F. Jegerlehner, in Z Physics at LEP I, proceedings of the Workshop, Geneva, Switzerland, 1989, edited by G. Altarelli, R. Kleiss, and C. Verzegnassi (CERN Report No. 89-08, Geneva, Switzerland, 1989), Vol. 1, p. 7.

[c29] R. Peccei, summary talk at the Z Phenomenology Symposium, Madison, Wisconsin, 1990 (unpublished); E. Fernandez, presented at Neutrino 90 Conference, Geneva, Switzerland, 1990 (unpublished); More recent Z properties have been presented by F. Dydak, presented at XXV International Conference on High Energy Physics, Singapore, 1990 (unpublished).

[c30] William Marciano and Jonathan L. Rosner, Brookhaven National Laboratory and Enrico Fermi Institute report, 1990 (unpublished).

[c31] G. Burgers, F. Jegerlehner, B. Kniehl, and J. Kühn, in Z Physics at LEP 1 (Ref. 28), Vol. 1, p. 55. This result is consistent with the one quoted by William J. Marciano, in Proceedings of the Salt Lake City Meeting, Annual Meeting of the Division of Particles and Fields of the APS, Salt Lake City, Utah, 1987, edited by Carleton DeTar and James Ball (World Scientific, Singapore, 1987), p. 319: 127.76+-0.3; and by Fanchiotti and Sirlin (Ref. 12): 127.9+-0.3. Different values are quoted by Altarelli (Ref. 9), 128.80+-0.13, and by Peskin (Ref. 8), 128.77+-0.12, both based on the result by Burkhardt et al.. (Ref. 28). We thank M. Peskin for noting that these last values are what one actually obtains when performing a careful estimate of photon vacuum-polarization effects alone, whereas the value quoted by Burgers et al. includes nonelectromagnetic radiative corrections as well, and that of Fanchiotti and Sirlin is in the MS bar_ scheme. For our purposes is the values of Burgers et al. which are most relevant, though our approach does not distinguish clearly among the different quantities.

[c32] We have added the result of Ref. 1 ( $M_{Z}$ =90.9 +- 0.3 +- 0.2 GeV/ $c^{2}$ ) to the compilation of results from SLC ( $M_{Z}$ =91.14 +-0.12 GeV/ $c^{2}$ ) and LEP ( $M_{Z}$ =91.169 +-0.032 GeV/ $c^{2}$ ) presented in Ref. 29. The corresponding average obtained by the Particle Data Group is 91.161+-0.031 GeV/ $c^{2}$ (see Ref. 24).

[c33] M. Veltman, Acta Phys. Pol. B 8, 475 (1977); M. Veltman and J. van der Bij, Nucl. Phys. B231, 205 (1984); Marciano and Sirlin (Ref. 12); Hollik (Ref. 18).

[c34] Jonathan L. Rosner, in The Santa Fe TASI-87, proceedings of the 1987 Theoretical Advanced Study Institute, Santa Fe, New Mexico, 1987, edited by Richard Slansky and Geoffrey West (World Scientific, Singapore, 1988), p. 3.

[c35] D. A. Ross and M. Veltman, Nucl. Phys. B95, 135 (1975); Michael S. Chanowitz and Mitchell Golden, Phys. Lett. 165B, 105 (1985); Mitchell Golden, 169B, 248 (1986) Paul Langacker, lecture presented at International School of Theoretical Physics, Szczyrk, Poland, 1987 (unpublished); G. Passarino, Phys. Lett. B 231, 458 (1989); D. Morris, as quoted by M. Golden, in Physics at Fermilab in the 1990's (Ref. 20), p. 103.

[c36] CDF Collaboration, F. Abe et al., presented at a seminar by S. Geer at Fermilab, 1990 (unpublished); presented by K. Sliwa, High Energy Hadronic Interactions, proceedings of the 25th Rencontres de Moriond, Les Arcs, France, 1990 (unpublished); and (unpublished); For earlier results see F. Abe et al., Phys. Rev. Lett. 64, 142 (1990); , ibid. 64, 147 (1990).

[c37] H. Abramowicz et al., Phys. Rev. Lett. 57, 298 (1986); CDHSW Collaboration, A. Blondel et al., Z. Phys. C 45, 361 (1990).

[c38] CHARM II Collaboration, J. V. Allaby et al., Phys. Lett. B 177, 446 (1986); , Z. Phys. C 36, 611 (1987).

[c39] CCFRR Collaboration, P. G. Reutens et al., Phys. Lett. 152B, 404 (1985); P. G. Reutens, Ph.D. thesis, University of Chicago, 1986.

[c40] FMM Collaboration, D. Bogert et al., Phys. Rev. Lett. 55, 1969 (1985). For more recent results see T. S. Mattison et al., Phys. Rev. D 42, 1311 (1990).

[c41] C. H. Llewellyn Smith, Nucl. Phys. B228, 205 (1983).

[c42] Jon Pumplin, Phys. Rev. Lett. 64, 2751 (1990).

[c43] R. Brock, in Physics at Fermilab in the 1990's (Ref. 20), p. 334.

[c44] E. A. Paschos and L. Wolfenstein, Phys. Rev. D 7, 91 (1973).

[c45] P. Q. Hung and J. J. Sakurai, Phys. Lett. 63B, 295 (1976). The notation we use is that of J. E. Kim et al., Rev. Mod. Phys. 53, 211 (1981).

[c46] M. C. Noecker, B. P. Masterson and C. E. Wieman, Phys. Rev. Lett. 61, 310 (1988).

[c47] V. A. Dzuba, V. V. Flambaum and O. P. Sushkov, Phys. Lett. A 141, 147 (1989).

[c48] Very similar corrections have been calculated more recently by S. A. Blundell, W. R. Johnson, and J. Sapirstein, Notre Dame University report, 1990 (unpublished).

[c49] The corrections of Ref. 48 yield $Q_{W}$ (Cs) =-71.04 +- 1.58 +-0.88.

[c50] Paul Langacker, Richard W. Robinett and Jonathan L. Rosner, Phys. Rev. D 30, 1470 (1984).

[c51] David London and Jonathan L. Rosner, Phys. Rev. D 34, 1530 (1986).

[c52] L. Stanley Durkin and Paul Langacker, Phys. Lett. 166B, 436 (1986).

[c53] Signatures of extra Z's in $e^{+}$ $e^{-}$ annihilations, and references to further literature, are given by David London, G. Bélanger, and J. N. Ng, in Selected Topics in Electroweak Interactions, proceedings of the Second Lake Louise Winter Institute on New Frontiers in Particle Physics, Lake Louise, Canada, 1987, edited by J. M. Cameron et al. (World Scientific, Singapore, 1987), p. 520; Jonathan L. Rosner, ibid, p. 91; R. Gatto, R. Casalbuoni, D. Dominici, and F. Feruglio, in Z Physics at LEP I, (Ref. 28), Vol. 2, p. 147.

[c54] Sheldon L. Glashow and Uri Sarid, Phys. Rev. Lett. 64, 725 (1990).

[c55] Carl Wieman (private communication).

[c56] C. Y. Prescott et al., Phys. Lett. 77B, 347 (1978); , ibid. 84B, 524 (1979).

[c57] W. Heil et al., Nucl. Phys. B327, 1 (1989). We thank Paul Langacker for informing us of this reference.

[c58] CHARM II Collaboration, D. Geiregat et al., Phys. Lett. B 232, 539 (1989).

[c59] Earlier values (as quoted in Ref. 58) are 0.211+- 0.035 +-0.011: CHARM Collaboration, J. Dorenbosch et al., Z. Phys. C 41, 567 (1989), and 0.199+-0.020+-0.013: Brookhaven National Laboratory Experiment E-734, K. Abe et al., Phys. Rev. Lett. 58, 636 (1987).

[c60] R. C. Allen et al., Phys. Rev. Lett. 64, 1330 (1990).

[c61] CDF Collaboration, presented by Melissa E. B. Franklin at XXV International Conference on High Energy Physics, (Ref. 29). The value (4.38) supersedes an earlier one of 0.216+- 0.015 +-0.010 presented by Myron Campbell, in Proceedings of the XIVth Symposium on Lepton and Photon Interactions (Ref. 9), p. 274, and by Peter Hurst at the 8th Topical Workshop on Proton-Antiproton Collider Physics, Castiglione, Italy, 1989 (unpublished).

[c62] ALEPH Collaboration, D. Decamp et al., Phys. Lett. B 244, 551 (1990): Γ ${}_{b b bar}$ / $Γ_{had}$ =0.215 +- 0.017 +- 0.024.

[c63] L3 Collaboration, B. Adeva et al., Phys. Lett. B 241, 416 (1990).

[c64] J. H. Kühn and P. M. Zerwas, Phys. Rep. 167, 321 (1988); C. Ahn et al., SLAC Report No. SLAC-329, 1988 (unpublished).

[c65] S. Jadach et al., in Z Physics at LEP I (Ref. 28), Vol. 1, p. 235.

[c66] V. Barger, J. L. Hewett and T. G. Rizzo, Phys. Rev. Lett. 65, 1313 (1990).

[c67] C. S. Kim, Jonathan L. Rosner and C.-P. Yuan, Phys. Rev. D 42, 96 (1990).

Physical Review D

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Radiative corrections and electroweak observables

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