<i>μ</i> decay at T≠<b>0</b> and infrared singularities

K. Jagannathan

doi:10.1103/PhysRevD.41.1667

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μ decay at T≠0 and infrared singularities

K. Jagannathan

Department of Physics, Amherst College, Amherst, Massachusetts 01002

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Phys. Rev. D 41, 1667 – Published 1 March, 1990

DOI: https://doi.org/10.1103/PhysRevD.41.1667

Abstract

The thermal corrections to the decay rate of the muon are calculated to lowest order in α and lowest nonvanishing order in the temperature T. An analytical expression is derived for ΔΓ, the correction to the decay rate, under the assumption that T≪m, where m is the mass of the electron. A rather puzzling feature regarding cancellation of the infrared divergence is discussed at some length. Turning next to mass singularities, in the approximation of T≪m, it would be inconsistent to let m→0, and, indeed, there are correction terms of the form ln(m/M), where M is the mass of the muon, which do not cancel. In order to consider the m→0 limit, the proper scheme is m≪T≪M; the additional corrections arising from the thermal bath of $e^{+}$ $e^{-}$ pairs, under these conditions, remove the mass singularities.

References (8)

J. F. Donoghue and B. R. Holstein, Phys. Rev. D 28, 340 (1983); ibid. 29, 3004(E) (1984). Also see J. F. Donoghue, B. R. Holstein and R. W. Robinett, Ann. Phys. (N.Y.) 164, 233 (1985).
D. Dicus et al., Phys. Rev. D 26, 2694 (1982).
J. L. Cambier, J. Primack and M. Sher, Nucl. Phys. B209, 372 (1982).
K. Ahmed and S. Saleem, Phys. Rev. D 35, 1861 (1987), and references there.
The wedge BAC of Fig. 1 has a simple analytical form: It is bounded by the straight line $p_{0}$ =( $M^{2}$ + $m^{2}$ )/2M and the curve $p_{0}$ =( $t^{2}$ + $m^{2}$ )/2t, where t=M+ ${2 k}_{0}$ .
See, for example, R. E. Marshak, Riazzuddin, and C. P. Ryan, Theory of Weak Interactions in Particle Physics (Wiley-Interscience, New York, 1969).
The absence of mass singularities in zero-temperature field theory is the Kinoshita-Lee-Nauenberg theorem: T. Kinoshita, J. Math. Phys. 3, 650 (1962); T. D. Lee and M. Nauenberg, Phys. Rev. 133, B1549 (1963). See also G. Sterman, Phys. Rev. D 14, 2123 (1976); E. Poggio and H. Quinn, ibid. 14, 578 (1976). It is reasonable to expect that even when T != 0 physical quantities should be well behaved as the fermion masses go to zero, but to my knowledge, there is no general proof.
W. Keil, Phys. Rev. D 40, 1176 (1989); P. Aurenche, Fermilab Report No. 107, 1989 (unpublished).

[c1] J. F. Donoghue and B. R. Holstein, Phys. Rev. D 28, 340 (1983); ibid. 29, 3004(E) (1984). Also see J. F. Donoghue, B. R. Holstein and R. W. Robinett, Ann. Phys. (N.Y.) 164, 233 (1985).

[c2] D. Dicus et al., Phys. Rev. D 26, 2694 (1982).

[c3] J. L. Cambier, J. Primack and M. Sher, Nucl. Phys. B209, 372 (1982).

[c4] K. Ahmed and S. Saleem, Phys. Rev. D 35, 1861 (1987), and references there.

[c5] The wedge BAC of Fig. 1 has a simple analytical form: It is bounded by the straight line $p_{0}$ =( $M^{2}$ + $m^{2}$ )/2M and the curve $p_{0}$ =( $t^{2}$ + $m^{2}$ )/2t, where t=M+ ${2 k}_{0}$ .

[c6] See, for example, R. E. Marshak, Riazzuddin, and C. P. Ryan, Theory of Weak Interactions in Particle Physics (Wiley-Interscience, New York, 1969).

[c7] The absence of mass singularities in zero-temperature field theory is the Kinoshita-Lee-Nauenberg theorem: T. Kinoshita, J. Math. Phys. 3, 650 (1962); T. D. Lee and M. Nauenberg, Phys. Rev. 133, B1549 (1963). See also G. Sterman, Phys. Rev. D 14, 2123 (1976); E. Poggio and H. Quinn, ibid. 14, 578 (1976). It is reasonable to expect that even when T != 0 physical quantities should be well behaved as the fermion masses go to zero, but to my knowledge, there is no general proof.

[c8] W. Keil, Phys. Rev. D 40, 1176 (1989); P. Aurenche, Fermilab Report No. 107, 1989 (unpublished).

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μ decay at T≠0 and infrared singularities

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