Other types of meson theory result from the replacement of γµ, by other expressions (for example by with a subsequent sum over all µ and ν for virtual mesons). Scalar mesons with vector coupling result from the replacement of γµ by µ−1 q where q is the final momentum of the nucleon minus its initial momentum, that is, it is the momentum of the meson if absorbed, or the negative of the momentum of a meson emitted. As is well known, this theory with neutral mesons gives zero for all processes, as is proved by our discussion on longitudinal waves in electrodynamics. Pseudoscalar mesons with pseudo-vector coupling corresponds to γµ being replaced by µ−1γ5q white vector mesons with tensor coupling correspond to using (2µ)−1 (γµq − qγµ). These extra gradients involve the danger of producing higher divergencies for real processes. For example, γ5q gives a logarithmically divergent interaction of neutron and electron.np Although these divergencies can be held by strong enough convergence factors, the results then are sensitive to the method used for convergence and the size of the cut-off values of λ. For low order processes µ−1γ5q is equivalent to the pseudoscalar interaction because if taken between free particle wave functions of the nucleon of momenta p1 and p2 = p1 + q, we have
since γ5 anticommutes with p2 and p2 operating on the state 2 equivalent to M as is 1 on the state 1. This shows that the γ5 interaction is unusually weak in the non-relativistic limit (for example the ex pected value of γ5 for a free nucleon is zero), but since is not small, pseudoscalar theory gives a more important interaction in second order than it does in first. Thus the pseudoscalar coupling constant should be chosen to fit nuclear forces including these important second order processes.nq The equivalence of pseudoscalar and pseudo-vector coupling which holds for low order processes therefore does not hold when the pseudoscalar theory is giving its most important effects. These theories will therefore give quite different results in the majority of practical problems.
In calculating the corrections to scattering of a nucleon by a neutral vector meson field (γµ) due to the effects of virtual mesons, the situation is just as in electrodynamics, in that the result converges without need for a cut-off and depends only on gradients of the meson potential. With scalar (1) or pseudoscalar (γµ) neutral mesons the result diverges logarithmically and so must be cut off. The part sensitive to the cut-off, however, is directly proportional to the meson potential. It may thereby be removed by a renormalization of mesonic charge g. After this renormalization the results depend only on gradients of the meson potential and are essentially independent of cut-off. This is in addition to the mesonic charge renormalization coming from the production of virtual nucleon pairs by a meson, analogous to the vacuum polarization in electrodynamics. But here there is a further difference from electrodynamics for scalar or pseudoscalar mesons in that the polarization also gives a term in the induced current proportional to the meson potential representing therefore an additional renormalization of the mass of the meson which usually depends quadratically on the cut-off.
Next consider charged mesons in the absence of an electromagnetic field. One can introduce isotopic spin operators in an obvious way. (Specifically replace the neutral γ5, say, by τiγ − 5 and sum over i = 1, 2, where τ1 = τ+ + τ−, τ2 = i(τ+ − τ−) and τ+ changes neutron to proton (τ+ on proton = 0) and τ− changes proton to neutron.) It is just as easy for practical problems simply to keep track of whether the particle is a proton or a neutron on a diagram drawn to help write down the matrix element. This excludes certain processes. For example in the scattering of a negative meson from q1 to q2 by a neutron, the meson q2 must be emitted first (in order of operators, not time) for the neutron cannot absorb the negative meson q1 until it becomes a proton. That is, in comparison to the Klein Nishina formula (15), only the analogue of second term (see Fig. 5(b)) would appear in the scattering of negative mesons by neutrons, and only the first term (Fig. 5 (a)) in the neutron scattering of positive mesons.
The source of mesons of a given charge is not conserved, for a neutron capable of emitting negative mesons may (on emitting one, say) become a proton no longer able to do so. The proof that a perturbation q gives zero, discussed for longitudinal electromagnetic waves, fails. This has the consequence that vector mesons, if represented by the interaction γµ, would not satisfy the condition that the divergence of the potential is zero. The interaction is to be takennr as γµ − µ−2qµq in emission and as γµ in absorption if the real emission of mesons with a non-zero divergence of potential is to be avoided. (The correction term µ−2qµq gives zero in the neutral case.) The asymmetry in emission and absorption is only apparent, as this is clearly the same thing as subtracting from the original γµ . . . γµ, a term µ−2q . . . q. That is, if the term −µ−2qµq is omitted the resulting theory describes a combination of mesons of spin one and spin zero. The spin zero mesons, coupled by vector coupling q, are removed by subtracting the term µ−2q . . . q.
The two extra gradients q . . . q make the problem of diverging integrals still more serious (for example the interaction between two protons corresponding to the exchange of two charged vector mesons depends quadratically on the cut-off if calculated in a straightforward way). One is tempted in this formulation to choose simply γµ . . . γµ and accept the admixture of spin zero mesons. But it appears that this leads in the conventional formalism to negative energies for the spin zero component. This shows one of the advantages of the method of second quantization of meson fields over the present formulation. There such errors of sign are obvious while here we seem to be able to write seemingly innocent expressions which can give absurd results. Pseudovector mesons with pseudovector coupling correspond to using γ5(γµ − µ−2qµq) for absorption and γ5γµ for emission for both charged and neutral mesons.
In the presence of an electromagnetic field, whenever the nucleon is a proton it interacts with the field in the way described for electrons. The meson interacts in the scalar or pseudoscalar case as a particle obeying the Klein-Gordon equation. It is important here to use the method of calculation of Bethe and Pauli, that is, a virtual meson is assumed to have the same “mass” during all its interactions with the electromagnetic field. The result for mass µ and for (µ2 + λ2)1/2 are subtracted and the difference integrated over the function G(λ)dλ. A separate convergence factor is not provided for each meson propagation between electromagnetic interactions, otherwise gauge invariance is not insured. When the coupling involves a gradient, such as γ − 5q where q is the final minus the initial momentum of the nucleon, the vector potential A must be subtracted from the momentum of the proton. That is, there is an additional coupling ±γ5A (plus when going from proton to neutron, minus for the reverse) representing the new possibility of a simultaneous emission (or absorption) of meson and photon.
Emission of positive or absorption of negative virtual mesons are represented in the same term, the sign of the charge being determined by temporal relations as for electrons and positrons.
Calculations are very easily carried out in this way to lowest order in g2 for the various theories for nucleon interaction, scattering of mesons by nucleons, meson production by nuclear collisions and by gamma-rays, nuclear magnetic moments, neutron electron scattering, etc., However, no good agreement with experiment results, when these are available, is obtained. Probably all of the formulations are incorrect. An uncertainty arises since the calculations are only to first order in g2, and are not valid if g2 /ħc is large.
The author is particularly indebted to Professor H. A. Bethe for his explanation of a method of obtaining finite and gauge invariant results for the problem of vacuum polarization. He is also grateful for Professor Bethe’s criticisms of the manuscript, and for innumerable discussions during the development of this work. He wishes to thank Professor J. Ashkin for his careful reading of the manuscript.
APPENDIX
In this appendix a method will be illustrated by which the simpler integrals appearing in problems in electrodynamics can be directly evaluated
. The integrals arising in more complex processes lead to rather complicated functions, but the study of the relations of one integral to another and their expression in terms of simpler integrals may be facilitated by the methods given here.
As a typical problem consider the integral (12) appearing in the first order radiationless scattering problem:
(1a)
where we shall take C(k2) to be typically −λ2(k2 − λ2)−1 and d4 k means (2π)−2dk1dk2dk3dk4. We first rationalize the factors (p − k − m)−1 = (p − k + m) ((p − k)2 − m2)−1 obtaining,
(2a)
The matrix expression may be simplified. It appears to be best to do so after the integrations are performed. Since AB = 2A · B − BA where A · B = AµBµ is a number commuting with all matrices, find, if R is any expression, and A a vector, since γµA = −Aγµ + 2Aµ,
(3a)
Expressions between two γµ’s can be thereby reduced by induction. Particularly useful are
(4a)
where A, B, C are any three vector-matrices (i.e., linear combinations of the four γµs),
In order to calculate the integral in (2a) the integral may be written as the sum of three terms (since k = kσγσ),
(5a)
where
(6a)
That is for J1 the (1; kσ ; kσkτ) is replaced by 1, for J2 by kσ and for J3 by kσ kτ.
More complex processes of the first order involve more factors like ((p3 − k)2 − m2)−1 and a corresponding increase in the number of k’s which may appear in the numerator, as kσ kr kν .... Higher order processes involving two or more virtual quanta involve similar integrals but with factors possibly involving k + k′ instead of just k, and the integral extending on k−2d4kC(k2)k′−2d4kC(′2). They can be simplified by methods analogous to those used on the first order integrals.
The factors (p − k)2 − m2 may be written
(7a)
where Δ = m2 where , etc., and we can consider dealing with cases of greater generality in that the different denominators need not have the same value of the mass m. In our specific problem (6a) = m2 that Δ1 = 0, but we desire to work with greater generality.
Now for the factor C(k2)/k2 we shall use −λ2(k2 − λ2)−1k−2. This can be written as
(8a)
Thus we can replace k−2 C(k2) by (k2 − L)−2 and at the end integrate the result with respect to L from zero to λ2. We can for many practical purposes consider λ2 very large relative to m2 or p2. When the original integrai converges even without the convergence factor, it will be obvious since the L integration will then be convergent to infinity. If an infra-red catastrophe exists in the integral one can simply assume quanta have a small λmin and extend the integral on L from to λ2, rather than from zero to λ2.
We then have to do integrals of the form
(9a)
where by (1;kσ ; kσkτ) we mean that in the place of this symbol either 1, or kσ or kσkτ may stand in different cases. In more complicated problems there may be more factors (k2 − 2pi · k − Δi)−1 or other powers of these factors (the (k2 − L)−2 may be considered as a special case of such a factor with pi = 0, Δi = L) and further factors like kσkτkρ ... in the numerator. The poles in all the factors are made definite by the assumption that L, and the Δ’s have infinitesimal negative imaginary parts.
We shall do the integrals of successive complexity by induction. We start with the simplest convergent one, and show
(10a)
For this integral is where the vector K, of magnitude K = (K · K)1/2 is k1, k2, k3. The integral on k4 shows third order poles at k4 = + (K2 + L)1/2 and k4 = −(K2 + L)1/2. Imagining, in accordance with our definitions, that L has a small negative imaginary part only the first is below the real axis. The contour can be closed by an infinite semi-circle below this axis, without change of the value of the integral since the contribution from the semi-circle vanishes in the limit. Thus the contour can be shrunk about the pole k4 = +(K2 + L)1/2 and the resulting k4, integral is − 2π i times the residue at this pole. Writing k4 = (k2 + L)1/2 + ε and expanding in powers of ε, the residue, being the coefficient of the term ε−1, is seen to be 6(2(K2 + L)1/2)−5 so our integral is
establishing (10a).
We also have ∫ kσd4k(k2 − L)−3 = 0 from the symmetry in the k space. We write these results as
(11a)
where in the brackets (1;kσ) and (1;0) corresponding entries are to he used.
Substituting k = k′ − p in (11a) and calling L − p2 = Δ shows that
(12a)
By differentiating both sides of (12a) with respect to Δ or with respect to pτ there follows directly
(13a)
Further differentiations give directly successive integrals including more k factors in the numerator and higher powers of (k2 − 2p · k − Δ) in the denominator.
The integrals so far only contain one factor in the denominator. To obtain results for two factors we make use of the identity
(14a)
(suggested by some work of Schwinger’s involving Gaussian integrals). This represents the product of two reciprocals as a parametric integral over one and will therefore permit integrals with two factors to be expressed in terms of one. For other powers of a, b we make use of all of the identities, such as
(15a)
deducible from (14a) by successive differentiations with respect to a or b. To perform an integral, such as
(16a)
write, using (15a),
where
(17a)
(note that Δx is not equal to so that the expression (16a) is (8i) which may now be evaluated by (12a) and is
(18a)
where, px, Δx are given in (17a). The integral in (18a) is elementary, being the integral of ratio of polynomials, the denominator of second degree in x. The general expression although readily obtained is a rather complicated combination of roots and logarithms.
Other integrals can be obtained again by parametric differentiation. For example differentiation of (16a), (18a) with respect to Δ2 or p2τ gives
(19a)
again leading to elementary integrals.
As an example, consider the case that the second factor is just (k2 − L)−2 and in the first put p1 = p, Δ1 = Δ. Then px = xp, Δx − xΔ + (1 − x)L. There results
(20a)
Integrals with three factors can be reduced to those involving two by using (14a) again. They, therefore, lead to integrals with two parameters (e.g., see application to radiative correction to scattering below).
The methods of calculation given in this paper are deceptively simple when applied to the lower order processes. For processes of increasingly higher orders the complexity and difficulty increases rapidly, and these methods soon become impractical in their present form.
A. Self-Energy
The self-energy integral (19) is
(19)
so that it requires that we find (using the principle of (8a)) the integral on L from 0 to λ2 of
since (p − k)2 − m2 = k2 − 2p · k, as p2 = m2. This is of the form (16a) with Δ1 = L, p1 = 0, Δ2 = 0, p2 = p so that (18a) gives, since px = (1 − x)p, Δx = xL,
or performing the integral on L, as in (8),
Assuming now that λ2 >> m2 we neglect (1 − x)2m2 relative to xλ2 in the argument of the logarithm, which then becomes (λ2/m2)(x/(1 − x)2). Then since and find
so that substitution into (19) (after the (p − k − m)−1 in (19) is replaced by (p − k + m)(k2 − 2p · k)−1) gives
(20)
using (4a) to remove the γµ’s. This agrees with Eq. (20) of the text, and gives the self-energy (21) when p is replaced by m.
B. Corrections to Scattering
The term (12) in the radiationless scattering, after rationalizing the matrix denominators and using requires the integrals (9a), as we have discussed. This is an integral with three denominators which we do in two stages. First the factors (k2 − 2p1 · k) and (K2 − 2p2 · k) are combined by a parameter y;
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