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Radiative corrections to weak interactions
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Nuclear Physics 37 (1962) 689--690; ( ~ ) N o r t h - H o l l a n d Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
RADIATIVE CORRECTIONS TO WEAK INTERACTIONS D. B. P E A R S O N and J. C. T A Y L O R
Dept. of Applied Mathematics and Theoretical Physics, University of Cambridge Received 21 May 1962 Abstract: It is proposed to use the freedom allowed by gauge invariance to clear up certain ambiguities, connected with divergent integrals and nucleon-structure-dependent integrals, encountered in the electrodynamics o f weak interactions.
When calculating radiative corrections to weak processes by perturbation theory one encounters two sorts of divergent integrals. The first sort occurs when strong interaction vertices or propagators are present. It is usually assumed and is quite possible, though not proved, that these divergent integrals occur only because of our failure to insert the strong interaction form factors properly. As a provisional measure, it is thoroughly reasonable to make such integrals finite by inserting a cut-off at energies somewhere between the pion and nucleon masses.
The second sort of divergent integrals involves only leptons and photons. No known physical effect can influence these integrals, unless one goes to energies for which weak interactions become strong (101. MeV), or electromagnetic interactions become strong 1) (e137 MeV), or gravitation becomes strong (also about e 137 MeV 1)). To cut-off these integrals around the nucleon mass implies the assumption that conventional electrodynamics fails already at these energies. It is certainly more economical if one can do without this hypothesis. Radiative corrections to muon decay 2)t contain only the second type of integral. However, the integrals are only logarithmically divergent, and, by what appears to be a happy accident, the divergent parts cancel in the complete expression. Radiative corrections to neutron /%decay 3) include both types of divergent integral. One might assume, as a condition of obtaining a meaningful result, that, when all the terms are combined, the truly divergent integrals of the second type are cancelled by contributions from integrals of the first type. In other words, the first class of integrals is not made completely convergent by the strong-interaction form factors, but includes just sufficient truly divergent terms to cancel the first-type divergent integrals exactly. The purpose of this note is to point to a method of putting this imprecise verbal prescription into practice. In the case of neutron fl-decay, there is only one second* Previous references are contained in ref. 2). 689
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type divergent integral, namely the electron wave-function renormalization integral. We can choose a photon gauge which makes this integral finite. In the appropriate gauge, the p h o t o n propagator 4) is
The remaining, first-type, integrals have to be calculated in this gauge also. But, by the working assumption we are making, they should now contain n o truly divergent terms. It is therefore correct to calculate the remaining integrals with form-factors approximated as well as may be by a cut-off of the order of the nucleon mass, provided the above photon propagator is used. We have applied this prescription to the calculation of fl-decay corrections, making in all other respects the same assumption and approximations as Berman 3). We find that Berman's numerical results are unchanged. One may therefore say that the argument of this note provides a justification for Berman's 3) procedure of cutting- off the second-type integrals at the same energy as those of first type; which it at first sight a very strange thing to do. The technique of using gauge freedom to remove ambiguities may also be applied to m u o n decay; for, although second-type divergences cancel here, it has been contended that the cancellation is not unambiguous t. The use of the above mentioned photon gauge renders all the integrals involved convergent, and yields unambiguously the usual result. The same technique might be used for computing corrections to electronic and muonic pion decay 6). But since it is only the ratio of these two decay rates which is interesting, the second-type divergent integrals cancel out anyway. t See footnote (8) of ref. ~). References 1) L. D. Landau, Niels Bohr and the Development of Physics edited by W. Pauli (Pergamon Press London, 1955) 2) T. Kinoshita and A. Sirlin, Phys. Rev. 113 (1959) 1652 3) S. M. Berman, Phys. Rev. 112 (1958) 267 4) L. D. Landau, A. A. Abrikosov and I. M. Khalatnikov, Doklady Akad. Nauk. S.S.S.R 95 (1954) 773; L. D. Landau and I. M. Khalatnikov, JETP 29 (1955) 89 [transl.: Soviet Physics JETP 2 (1956) 69] 5) Loyal Durand III, Leon F. Landovitz and R. B. Marr, Phys. Rev. Letters 4 (1960) 620 6) S. M. Berman, Phys. Rev. Letters 1 (1958) 468; T. Kinoshita, Phys. Rev. Letters 2 (1959) 477
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Nuclear Physics 37 (1962) 689--690; ( ~ ) N o r t h - H o l l a n d Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
RADIATIVE CORRECTIONS TO WEAK INTERACTIONS D. B. P E A R S O N and J. C. T A Y L O R
Dept. of Applied Mathematics and Theoretical Physics, University of Cambridge Received 21 May 1962 Abstract: It is proposed to use the freedom allowed by gauge invariance to clear up certain ambiguities, connected with divergent integrals and nucleon-structure-dependent integrals, encountered in the electrodynamics o f weak interactions.
When calculating radiative corrections to weak processes by perturbation theory one encounters two sorts of divergent integrals. The first sort occurs when strong interaction vertices or propagators are present. It is usually assumed and is quite possible, though not proved, that these divergent integrals occur only because of our failure to insert the strong interaction form factors properly. As a provisional measure, it is thoroughly reasonable to make such integrals finite by inserting a cut-off at energies somewhere between the pion and nucleon masses.
The second sort of divergent integrals involves only leptons and photons. No known physical effect can influence these integrals, unless one goes to energies for which weak interactions become strong (101. MeV), or electromagnetic interactions become strong 1) (e137 MeV), or gravitation becomes strong (also about e 137 MeV 1)). To cut-off these integrals around the nucleon mass implies the assumption that conventional electrodynamics fails already at these energies. It is certainly more economical if one can do without this hypothesis. Radiative corrections to muon decay 2)t contain only the second type of integral. However, the integrals are only logarithmically divergent, and, by what appears to be a happy accident, the divergent parts cancel in the complete expression. Radiative corrections to neutron /%decay 3) include both types of divergent integral. One might assume, as a condition of obtaining a meaningful result, that, when all the terms are combined, the truly divergent integrals of the second type are cancelled by contributions from integrals of the first type. In other words, the first class of integrals is not made completely convergent by the strong-interaction form factors, but includes just sufficient truly divergent terms to cancel the first-type divergent integrals exactly. The purpose of this note is to point to a method of putting this imprecise verbal prescription into practice. In the case of neutron fl-decay, there is only one second* Previous references are contained in ref. 2). 689
600
D.B.
PEARSON AND J. C. TAYLOR
type divergent integral, namely the electron wave-function renormalization integral. We can choose a photon gauge which makes this integral finite. In the appropriate gauge, the p h o t o n propagator 4) is
The remaining, first-type, integrals have to be calculated in this gauge also. But, by the working assumption we are making, they should now contain n o truly divergent terms. It is therefore correct to calculate the remaining integrals with form-factors approximated as well as may be by a cut-off of the order of the nucleon mass, provided the above photon propagator is used. We have applied this prescription to the calculation of fl-decay corrections, making in all other respects the same assumption and approximations as Berman 3). We find that Berman's numerical results are unchanged. One may therefore say that the argument of this note provides a justification for Berman's 3) procedure of cutting- off the second-type integrals at the same energy as those of first type; which it at first sight a very strange thing to do. The technique of using gauge freedom to remove ambiguities may also be applied to m u o n decay; for, although second-type divergences cancel here, it has been contended that the cancellation is not unambiguous t. The use of the above mentioned photon gauge renders all the integrals involved convergent, and yields unambiguously the usual result. The same technique might be used for computing corrections to electronic and muonic pion decay 6). But since it is only the ratio of these two decay rates which is interesting, the second-type divergent integrals cancel out anyway. t See footnote (8) of ref. ~). References 1) L. D. Landau, Niels Bohr and the Development of Physics edited by W. Pauli (Pergamon Press London, 1955) 2) T. Kinoshita and A. Sirlin, Phys. Rev. 113 (1959) 1652 3) S. M. Berman, Phys. Rev. 112 (1958) 267 4) L. D. Landau, A. A. Abrikosov and I. M. Khalatnikov, Doklady Akad. Nauk. S.S.S.R 95 (1954) 773; L. D. Landau and I. M. Khalatnikov, JETP 29 (1955) 89 [transl.: Soviet Physics JETP 2 (1956) 69] 5) Loyal Durand III, Leon F. Landovitz and R. B. Marr, Phys. Rev. Letters 4 (1960) 620 6) S. M. Berman, Phys. Rev. Letters 1 (1958) 468; T. Kinoshita, Phys. Rev. Letters 2 (1959) 477