# Current Algebras And Universal Divergent

## Transcript Of Current Algebras And Universal Divergent

CURRENT ALGEBRAS AND UNIVERSAL DIVERGENT RADIATIVE CORRECTIONS*f G. W. Gaffney

Stanford Linear Accelerator Center Stanford University, Stanford, California 94305

(Submitted to Annals of Physics )

*Work supported by the U, S. Atomic Energy Commission.

'A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics.

ABSTRACT

We examine, using current algebras, the ultraviolet divergences occurring in the calculation of electromagnetic radiative corrections to any lowest order weak process a t arbitrary momentum transfer. We consider all orders in perturbation theory in the fine structure constant a. The divergent parts of the radiative corrections are expressed in terms of matrix elements of equal-time current commutators by using the Bjorken expansion of time-ordered products of currents at large momenta. We assume the validity of this expansion an&of the use of "naive" current commutation relations in discussing various current algebra models. We impose the condition that the divergences contribute only to an unobservable, universal weak coupling constant renormalization. It is shown that, in models with operator Schwinger terms in the current commutators, this condition cannot be satisfied for non-zero momentum transfer. Also, i t is not satisfied for a weak interaction theory mediated by a vector boson. Two current algebra models are exhibited which a r e satisfactory if the weak Hamiltonian has a local current-current form. For these models, the weak and electromagnetic currents of both the hadrons and the leptons obey the same commutation relations, and the Schwinger terms a r e c-numbers. One, a quark model of hadrons with integrally charged quarks together with the conventional lepton currents, gives finite radiative corrections. The second, the algebra of fields model for the total electromagnetic and weak currents, including leptons, contains only a universal divergent factor. These two results a r e shown to hold to all orders in a. In obtaining these results, divergent contributions to electromagnetic mass shifts and to electromagnetic renormalization effects in strong interaction processes a r e isolated and removed by adding a counter term to the interaction Hamiltonian. These divergences may thus be treated as a separate problem, which w e do not discuss in detail.

- 1-

I. INTRODUCTION

Our current experimental knowledge of leptonic and semileptonic weak interactions is well described by the familiar universal current-current form192 of the phenomenological interaction Lagrangian. One of the most remarkable features of this Lagrangian is that i t predicts, for p-decay and neutron &decay, the approximate equality of the respective vector coupling constants, which is consistent with experimental observations. Specifically, by using the conserved vector current hypothesis, one may infer from the equality of €he appropriate bare coupling constants that the renormalized coupling constants a r e equal even after the inclusion of strong interaction effects. The fact that the bare coupling constants may be chosen to be equal leads us to believe that the effective Lagrangian may have a more fundamental significance.

T o test this %n.iversality" of the weak interactions we must also include

--- the electromagnetic radiative corrections. We expect these to be small corrections e 2 at least of the o r d e r of a, where a = 47~ 137 is the fine structure constant. However, for semileptonic processes a serious problem arises in their calculation since divergent momentum integrals occur. It is these divergences we wish to study, to all orders in perturbation theory in the fine structure constant.

The most straightforward resolution of this problem would be to construct a theory in which all these divergences cancelled out so that the amplitude for any weak process would be finite to all orders in a. A less stringent condition would nevertheless be satisfactory since all that is required of a consistent theory is that measurable quantities, such as ratios of coupling constants, masses, o r form factors, be finite. Hence, i t is enough to impose the condition that any divergences arising in the calculation of electromagnetic radiative corrections to weak processes contribute only to an overall (possibly cut-off dependent)

-2-

constant factor times a finite matrix element. Of course, this factor must be universal, Le., it must be the same for all w e a k processes so that ratios of any measurable parameters will be finite. The divergent factor can then be absorbed into the definition of the weak coupling constant, since i t s overall scale is undetermined. Note that it is not sufficient merely to require that ratios of the coupling constants defined at zero momentum transfer be finite. The ratios of various form factors occurring for non-zero momentum transfers are measurable and therefore must also be finite and calculable. ..

We are interested here in the implications of current algebra3 for the

problem of divergences in radiative corrections to weak interactions. It is our purpose in this paper to develop a technique for discussing these divergences which does not depend upon the particular weak process considered and which is valid for arbitrary momentum transfers. Furthermore, we wish to examine the

corrections to all orders in e2 , not merely to second order, as in all previous

. investigations

Since this topic has received considerable attention in the past by various authors, 4-12 let u s first briefly review the previous work before detailing our contribution to the subject. The status of the radiative corrections problem before the advent of current algebra w a s summarized by Berman and Sirlin. 13 They observed that the radiative corrections top-decay could be shown to be finite to all orders in Q! by performing a Fierz transf~rmation'~on the weak Hamiltonian. However, for decays involving bare hadrons, the corrections were in general logarithmically divergent. It was generally conjectured that when strong interactions were taken into account, they would provide the convergence factors necessary to make the semileptonic amplitudes finite, although Berman and Sirlin provided some qualitative arguments to the contrary.

-3-

Bjorken4 pointed out that the assumption of Gell-Mann's current algebra postulate3 implied that matrix elements of the exact hadronic currents behaved at large momenta like those of point particles, thus nullifying the above conjecture. Specifically, he showed that in the simple quark model with fractionally charged quarks, the second order (in e) radiative corrections to the vector part of the neutron @-decay amplitude at zero momentum transfer are logarithmically divergent, treating the strong interactions exactly. Abers, Norton, Dicus, and

5 Quinn generalized Bjorken's result and emphasized that certain contributions to the divergence depended only on the relatively model-independent commutators of the time components of the electromagnetic and weak currents. However, the

divergent corrections as a whole were model-dependent, and several models for the commutators of the space components of the hadronic currents were constructed6' so that the radiative corrections to neutron @-decaywould be finite in second order.

Using basically the same techniques as Bjorken and A b e r s - et2 a1 9 Callan9 and Preparata and Weisberger" generalized their work to include any semileptonic process. These authors did not restrict the discussion to only zero momentum transfer. Both papers considered only models constructed from renormalizable theories of strong interactions. They concluded that the models mentioned above, involving hadronic currents constructed from integrally charged quarks, gave finite second order radiative corrections to a general semileptonic process. Preparata and Weisberger further noted that currents containing bilinear products of spin zero fields yielded additional divergent corrections for non-zero momentum transfer.

Sirlin8 and Abers et al., 5 studied the second order radiative corrections to the vector part of p-decay and neutron @- decay in a weak interaction theory

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mediated by a vector boson. They showed that, at zero momentum transfer, only a universal divergence occurred. That is, the divergent part was merely a constant factor times the uncorrected matrix element, and this factor was the same for both p-decay and neutron &decay. Furthermore, this result depended

only on model independent current commutators involving time components of currents. For the weak boson theory, the order -e2 corrections to GA/GV, the ratio of the axial vector and vector coupling constants in neutron B-decay, were

shown by Mrlin" to be finite in the algebra of fields model, although this was not true in general. He used a technique which is very similar to ours although there m e some differences in detail. In particular, our interpretation of electromagnetic mass shift contributions is somewhat different from his.

The toolwhich we shall use to discuss divergent radiative corrections is the expansion of time-ordered products of operators at large momenta in terms

of equal-time commutators. The relevance of this technique to current algebra was first pointed out by Bjorken4 and Johnson and Low.15 A l l of the above mentioned papers a t some point used this device. Many also employed Ward identities to handle external line wave function renormalization and to exhibit cancellations of certain divergent contributions. Since the end result is that the divergences involve highly model-dependent current commutators, we shall make this explicit by applying the Bjorken expansion in a straightforward manner. We shall assume the expansion is justified for time-ordered products containing an arbitrary number of currents. (1) F o r point particles there is no problem, but for the exact hadronic currents it is by no means obvious that this is a valid assumption. Nevertheless, w e shall take it a s our starting point without further ado, since i t is certainly impossible to justify i t rigorously.

-5-

A related assumption we shall make is that we may use the "naiver1current commutation relations obtained from the canonical commutators and the equations of motion for any particular model of the hadronic currents. Our point of view here is that these models should not necessarily be taken seriously, but that perhaps the current algebra should be. For, if w e assume that Bjorken expansion is valid, then the requirement that no divergences occur in the calculation of physically measurable quantities restricts the form that the current commutators

may take. Several recent investigation^'^-^^ have shown that, when simple strong

interaction models are treated in perturbation theory, the naive commutators no longer hold. We shall comment on this point in the conclusion.

In extending the results on second order radiative corrections to all orders

in e2, w e shall see that our method allows us to isolate only those divergences due

to momentum loops containing virtual photons. Thus, w e ignore any divergences arising from momentum loop integrations in the hadron o r lepton "blobs" which

the photon lines enter. In fact, we - must neglect any such divergences to be con-

sjstent with our use of naive commutation relations, as we discuss in the conclusion. The underlying physical assumption is that the basic hadronic and leptonic theory of matter, whatever i t is, must be sufficiently convergent at high momenta that such divergences, if they occur at all, do not affect the current commutators.

Lest it be misunderstood, we should state that throughout this paper we shall use the manipulations of %aive quantum field theoryft. Thus, w e ignore any singularities of local products of quantum field operators except those which a r e explicit in the use of equations of motion and canonical commutation relations. These ambiguities in local products of operators provide a possible escape from the divergence difficulties, but we are interested in the more conventional solutions to the problem.

-6-

In Section 11we begin our discussion by considering second order electromagnetic radiative corrections. We first illustrate the Bjorken expansion for the time-ordered product of three operators which occurs there. We treat the hadron and lepton currents on the same footing so that no special weak process is singled out. An important part of the discussion of the divergent corrections to the weak coupling constant is the removal of divergent contributions to electromagnetic mass shifts and to radiative corrections to strong interaction parameters. We identify these terms and argue that they are removed by an appropriate counter t e r m in the interaction Hamiltonian. Because of this, these divergences may be considered as a separate problem, which we shall not study here since it has been examined considerably by others. 2o In removing these contributions some care is required in making the covariant generalization of the Bjorken limit, an explicitly non-covariant procedure. This is illustrated for several models.

Next we discuss the possibility that the remaining divergences contribute only to a universal constant factor times the uncorrected matrix element. We show that this is not possible if there are operator (q-number) Schwinger terms in the current commutators by considering, as an example, currents constructed from a bilinear product of spin-zero fields. These terms produce, for non-zero momentum transfer, contributions which are manifestly not proportional to the lowest order matrix element. lo If q-number Schwinger terms are absent, the current commutators involving time components of currents are the same for hadrons and leptons, independent of specific models. We point out that the divergent radiative corrections will then be universal if the commutators of the space components of the currents are also the same for hadrons as for leptons. Two models where this condition is satisfied are exhibited. These are the integrally charged quark models6’ mentioned above for hadrons together with

-7-

the conventional lepton currents, and the algebra of fields mode121 for total electro-

magnetic and weak currents, including leptons,as proposed by To Do Lee.22 In

the former model the radiative corrections are finite and in the latter a nonvanishing, but universal, divergence is found. 11,23

We conclude Section I1 with a discussion of second order radiative corrections in a weak interaction theory mediated by a vector boson. We show that

the positive result of Sirlin8 , l l and Abers et al. ,5 is not maintained for non-zero

momentum transfer. Non-universal divergent terms are found, -and a counter term having a local current-current form would have to be added to the interaction Hamiltonian to make the radiative corrections finite.

We consider the generalization to these second order results to all orders in e2 in Section III. After discussing the additional assumptions, we take up the two models which were satisfactory in second order with respect to universality. For the algebra of fields model we consider the fourth order calculation in some detail as an illustrative example, Here an important point to be mentioned is that in order to avoid ambiguities in making the covariant generalization of the Bjorken limit, we must let only one photon loop momentum go to infinity at a time holding all others fixed. This offers no problem since we are dealing with only logarithmically divergent integrals.

Then w e show that,for the algebra of fields mode1,to any given order in e2 the amplitude for any weak process may be expressed, once the divergent mass renormalization terms are removed, as a divergent constant factor times the finite part of the matrix element to the next lower order. This is precisely the condition for the divergences not to have any observable effects. It contributes simply to an overall rescaling of the weak coupling constant. In summing the series f o r all orders in e2 the divergent coefficient in second order exponentiates.

- 8-

Next w e consider the quark model which led to finite radiative corrections in second order and show, examining the fourth order case in detail, that the corrections are finite to all orders. It is pointed out that this result could have been anticipated, knowing the same is true for p-decay, since our technique is independent of any particular weak process.

In conclusion we present a critical discussion of our assumptions, in particular the use of naive commutation relations, We also point out the difficulties encountered in attempting to apply the Bjorken expansion to discuss divergences in non-renormalizable field theories. In the light of our results, w e summarize the current status of the problem of radiative corrections to weak interactions.

An appendix examines certain details concerning divergent contributions to external line wave function renormalization.

-9-

Stanford Linear Accelerator Center Stanford University, Stanford, California 94305

(Submitted to Annals of Physics )

*Work supported by the U, S. Atomic Energy Commission.

'A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics.

ABSTRACT

We examine, using current algebras, the ultraviolet divergences occurring in the calculation of electromagnetic radiative corrections to any lowest order weak process a t arbitrary momentum transfer. We consider all orders in perturbation theory in the fine structure constant a. The divergent parts of the radiative corrections are expressed in terms of matrix elements of equal-time current commutators by using the Bjorken expansion of time-ordered products of currents at large momenta. We assume the validity of this expansion an&of the use of "naive" current commutation relations in discussing various current algebra models. We impose the condition that the divergences contribute only to an unobservable, universal weak coupling constant renormalization. It is shown that, in models with operator Schwinger terms in the current commutators, this condition cannot be satisfied for non-zero momentum transfer. Also, i t is not satisfied for a weak interaction theory mediated by a vector boson. Two current algebra models are exhibited which a r e satisfactory if the weak Hamiltonian has a local current-current form. For these models, the weak and electromagnetic currents of both the hadrons and the leptons obey the same commutation relations, and the Schwinger terms a r e c-numbers. One, a quark model of hadrons with integrally charged quarks together with the conventional lepton currents, gives finite radiative corrections. The second, the algebra of fields model for the total electromagnetic and weak currents, including leptons, contains only a universal divergent factor. These two results a r e shown to hold to all orders in a. In obtaining these results, divergent contributions to electromagnetic mass shifts and to electromagnetic renormalization effects in strong interaction processes a r e isolated and removed by adding a counter term to the interaction Hamiltonian. These divergences may thus be treated as a separate problem, which w e do not discuss in detail.

- 1-

I. INTRODUCTION

Our current experimental knowledge of leptonic and semileptonic weak interactions is well described by the familiar universal current-current form192 of the phenomenological interaction Lagrangian. One of the most remarkable features of this Lagrangian is that i t predicts, for p-decay and neutron &decay, the approximate equality of the respective vector coupling constants, which is consistent with experimental observations. Specifically, by using the conserved vector current hypothesis, one may infer from the equality of €he appropriate bare coupling constants that the renormalized coupling constants a r e equal even after the inclusion of strong interaction effects. The fact that the bare coupling constants may be chosen to be equal leads us to believe that the effective Lagrangian may have a more fundamental significance.

T o test this %n.iversality" of the weak interactions we must also include

--- the electromagnetic radiative corrections. We expect these to be small corrections e 2 at least of the o r d e r of a, where a = 47~ 137 is the fine structure constant. However, for semileptonic processes a serious problem arises in their calculation since divergent momentum integrals occur. It is these divergences we wish to study, to all orders in perturbation theory in the fine structure constant.

The most straightforward resolution of this problem would be to construct a theory in which all these divergences cancelled out so that the amplitude for any weak process would be finite to all orders in a. A less stringent condition would nevertheless be satisfactory since all that is required of a consistent theory is that measurable quantities, such as ratios of coupling constants, masses, o r form factors, be finite. Hence, i t is enough to impose the condition that any divergences arising in the calculation of electromagnetic radiative corrections to weak processes contribute only to an overall (possibly cut-off dependent)

-2-

constant factor times a finite matrix element. Of course, this factor must be universal, Le., it must be the same for all w e a k processes so that ratios of any measurable parameters will be finite. The divergent factor can then be absorbed into the definition of the weak coupling constant, since i t s overall scale is undetermined. Note that it is not sufficient merely to require that ratios of the coupling constants defined at zero momentum transfer be finite. The ratios of various form factors occurring for non-zero momentum transfers are measurable and therefore must also be finite and calculable. ..

We are interested here in the implications of current algebra3 for the

problem of divergences in radiative corrections to weak interactions. It is our purpose in this paper to develop a technique for discussing these divergences which does not depend upon the particular weak process considered and which is valid for arbitrary momentum transfers. Furthermore, we wish to examine the

corrections to all orders in e2 , not merely to second order, as in all previous

. investigations

Since this topic has received considerable attention in the past by various authors, 4-12 let u s first briefly review the previous work before detailing our contribution to the subject. The status of the radiative corrections problem before the advent of current algebra w a s summarized by Berman and Sirlin. 13 They observed that the radiative corrections top-decay could be shown to be finite to all orders in Q! by performing a Fierz transf~rmation'~on the weak Hamiltonian. However, for decays involving bare hadrons, the corrections were in general logarithmically divergent. It was generally conjectured that when strong interactions were taken into account, they would provide the convergence factors necessary to make the semileptonic amplitudes finite, although Berman and Sirlin provided some qualitative arguments to the contrary.

-3-

Bjorken4 pointed out that the assumption of Gell-Mann's current algebra postulate3 implied that matrix elements of the exact hadronic currents behaved at large momenta like those of point particles, thus nullifying the above conjecture. Specifically, he showed that in the simple quark model with fractionally charged quarks, the second order (in e) radiative corrections to the vector part of the neutron @-decay amplitude at zero momentum transfer are logarithmically divergent, treating the strong interactions exactly. Abers, Norton, Dicus, and

5 Quinn generalized Bjorken's result and emphasized that certain contributions to the divergence depended only on the relatively model-independent commutators of the time components of the electromagnetic and weak currents. However, the

divergent corrections as a whole were model-dependent, and several models for the commutators of the space components of the hadronic currents were constructed6' so that the radiative corrections to neutron @-decaywould be finite in second order.

Using basically the same techniques as Bjorken and A b e r s - et2 a1 9 Callan9 and Preparata and Weisberger" generalized their work to include any semileptonic process. These authors did not restrict the discussion to only zero momentum transfer. Both papers considered only models constructed from renormalizable theories of strong interactions. They concluded that the models mentioned above, involving hadronic currents constructed from integrally charged quarks, gave finite second order radiative corrections to a general semileptonic process. Preparata and Weisberger further noted that currents containing bilinear products of spin zero fields yielded additional divergent corrections for non-zero momentum transfer.

Sirlin8 and Abers et al., 5 studied the second order radiative corrections to the vector part of p-decay and neutron @- decay in a weak interaction theory

-4-

mediated by a vector boson. They showed that, at zero momentum transfer, only a universal divergence occurred. That is, the divergent part was merely a constant factor times the uncorrected matrix element, and this factor was the same for both p-decay and neutron &decay. Furthermore, this result depended

only on model independent current commutators involving time components of currents. For the weak boson theory, the order -e2 corrections to GA/GV, the ratio of the axial vector and vector coupling constants in neutron B-decay, were

shown by Mrlin" to be finite in the algebra of fields model, although this was not true in general. He used a technique which is very similar to ours although there m e some differences in detail. In particular, our interpretation of electromagnetic mass shift contributions is somewhat different from his.

The toolwhich we shall use to discuss divergent radiative corrections is the expansion of time-ordered products of operators at large momenta in terms

of equal-time commutators. The relevance of this technique to current algebra was first pointed out by Bjorken4 and Johnson and Low.15 A l l of the above mentioned papers a t some point used this device. Many also employed Ward identities to handle external line wave function renormalization and to exhibit cancellations of certain divergent contributions. Since the end result is that the divergences involve highly model-dependent current commutators, we shall make this explicit by applying the Bjorken expansion in a straightforward manner. We shall assume the expansion is justified for time-ordered products containing an arbitrary number of currents. (1) F o r point particles there is no problem, but for the exact hadronic currents it is by no means obvious that this is a valid assumption. Nevertheless, w e shall take it a s our starting point without further ado, since i t is certainly impossible to justify i t rigorously.

-5-

A related assumption we shall make is that we may use the "naiver1current commutation relations obtained from the canonical commutators and the equations of motion for any particular model of the hadronic currents. Our point of view here is that these models should not necessarily be taken seriously, but that perhaps the current algebra should be. For, if w e assume that Bjorken expansion is valid, then the requirement that no divergences occur in the calculation of physically measurable quantities restricts the form that the current commutators

may take. Several recent investigation^'^-^^ have shown that, when simple strong

interaction models are treated in perturbation theory, the naive commutators no longer hold. We shall comment on this point in the conclusion.

In extending the results on second order radiative corrections to all orders

in e2, w e shall see that our method allows us to isolate only those divergences due

to momentum loops containing virtual photons. Thus, w e ignore any divergences arising from momentum loop integrations in the hadron o r lepton "blobs" which

the photon lines enter. In fact, we - must neglect any such divergences to be con-

sjstent with our use of naive commutation relations, as we discuss in the conclusion. The underlying physical assumption is that the basic hadronic and leptonic theory of matter, whatever i t is, must be sufficiently convergent at high momenta that such divergences, if they occur at all, do not affect the current commutators.

Lest it be misunderstood, we should state that throughout this paper we shall use the manipulations of %aive quantum field theoryft. Thus, w e ignore any singularities of local products of quantum field operators except those which a r e explicit in the use of equations of motion and canonical commutation relations. These ambiguities in local products of operators provide a possible escape from the divergence difficulties, but we are interested in the more conventional solutions to the problem.

-6-

In Section 11we begin our discussion by considering second order electromagnetic radiative corrections. We first illustrate the Bjorken expansion for the time-ordered product of three operators which occurs there. We treat the hadron and lepton currents on the same footing so that no special weak process is singled out. An important part of the discussion of the divergent corrections to the weak coupling constant is the removal of divergent contributions to electromagnetic mass shifts and to radiative corrections to strong interaction parameters. We identify these terms and argue that they are removed by an appropriate counter t e r m in the interaction Hamiltonian. Because of this, these divergences may be considered as a separate problem, which we shall not study here since it has been examined considerably by others. 2o In removing these contributions some care is required in making the covariant generalization of the Bjorken limit, an explicitly non-covariant procedure. This is illustrated for several models.

Next we discuss the possibility that the remaining divergences contribute only to a universal constant factor times the uncorrected matrix element. We show that this is not possible if there are operator (q-number) Schwinger terms in the current commutators by considering, as an example, currents constructed from a bilinear product of spin-zero fields. These terms produce, for non-zero momentum transfer, contributions which are manifestly not proportional to the lowest order matrix element. lo If q-number Schwinger terms are absent, the current commutators involving time components of currents are the same for hadrons and leptons, independent of specific models. We point out that the divergent radiative corrections will then be universal if the commutators of the space components of the currents are also the same for hadrons as for leptons. Two models where this condition is satisfied are exhibited. These are the integrally charged quark models6’ mentioned above for hadrons together with

-7-

the conventional lepton currents, and the algebra of fields mode121 for total electro-

magnetic and weak currents, including leptons,as proposed by To Do Lee.22 In

the former model the radiative corrections are finite and in the latter a nonvanishing, but universal, divergence is found. 11,23

We conclude Section I1 with a discussion of second order radiative corrections in a weak interaction theory mediated by a vector boson. We show that

the positive result of Sirlin8 , l l and Abers et al. ,5 is not maintained for non-zero

momentum transfer. Non-universal divergent terms are found, -and a counter term having a local current-current form would have to be added to the interaction Hamiltonian to make the radiative corrections finite.

We consider the generalization to these second order results to all orders in e2 in Section III. After discussing the additional assumptions, we take up the two models which were satisfactory in second order with respect to universality. For the algebra of fields model we consider the fourth order calculation in some detail as an illustrative example, Here an important point to be mentioned is that in order to avoid ambiguities in making the covariant generalization of the Bjorken limit, we must let only one photon loop momentum go to infinity at a time holding all others fixed. This offers no problem since we are dealing with only logarithmically divergent integrals.

Then w e show that,for the algebra of fields mode1,to any given order in e2 the amplitude for any weak process may be expressed, once the divergent mass renormalization terms are removed, as a divergent constant factor times the finite part of the matrix element to the next lower order. This is precisely the condition for the divergences not to have any observable effects. It contributes simply to an overall rescaling of the weak coupling constant. In summing the series f o r all orders in e2 the divergent coefficient in second order exponentiates.

- 8-

Next w e consider the quark model which led to finite radiative corrections in second order and show, examining the fourth order case in detail, that the corrections are finite to all orders. It is pointed out that this result could have been anticipated, knowing the same is true for p-decay, since our technique is independent of any particular weak process.

In conclusion we present a critical discussion of our assumptions, in particular the use of naive commutation relations, We also point out the difficulties encountered in attempting to apply the Bjorken expansion to discuss divergences in non-renormalizable field theories. In the light of our results, w e summarize the current status of the problem of radiative corrections to weak interactions.

An appendix examines certain details concerning divergent contributions to external line wave function renormalization.

-9-