- Email: [email protected]
On the peratization theory of Feinberg and Pais
.lSNALS
OF
On
PHYSICS:
the
27, 183-192 (1964)
Peratization
Theory
of Feinberg
and
Pais”
MIRZA A. BAQI Bti~ Institute
for
Advanced
Study,
Princeton,
New
Jersey
We present an alternate formulation of the peratization program of Feinberg and Pais and show that t.he principal results of their theory can be reproduced in a simple manner. Essential use is made of the self-damping of the Fermi interaction in t,he chain approximation, and of a circumstance t,hat follows therefrom; to wit, t,he existence of a continuously infinite set of theories equivalent in the ladder approximation. While some of the results are more general, the discussion is geared to the case of distinguishable and massless Fermions scattering at zero energy. iY0 pretensions are made to mathematical rigor. I. IKTRODUCTIOS
Recently, P‘einberg and Pais have drawn att,ention to the possibility t,hat in a nonrenormalizable theory a cognizance of higher order effects may lead to a modification of matrix element,sin leading order of the coupling constant (1, 2). In parbicular, they show that for leptons interacting via massive, charged vector bosons, the term in the leading order matrix element
B,,(y)
= y”--l,l’ - y2
(SPY - YrYd)
(1.1)
get.smodified to
(1.2) of the uncrossedladder graphs. Here p and v are t,o be contracted with the (V-s) type lepton currents, y is the 4-momentum transfer, 171 the mass of the boson, and g the sem-weak boson lepton coupling constant.’ The very simple structure of Eq. (1.2) is in marked contrast to the highly involved nature of the derivations hitherto advanced (1-3). The purpose of this on summation
* Work supported, in part, by t)he U. S. Atomic Energy Commission during the author’s stay at Brookhaven National Laboratory, Summer 1963, and in part by the National Science Foundation. 1 The notations used in this paper, in particular the metric, the representations of the T-matrices, and the normalization factors in the Feynmann Rules etc. are in accord with S. S. Schweber, “An Introduction to Relat,ivistic Quantum Field Theory.” Row, Peterson, New IYork, 1961.
183
184
BtiG
note is to present a simple deduction of equation (1.2) based on the self-damping of a Fermi interaction in the chain approximation. 2 Our motivation stems from the belief that Ecf. (1.2) is of sufficient importance to warrant examination from more than one angle; even if such scrutiny has no more than a didactic value. The extension of our procedure to problenrs involving an arbitrary number of coupled channels (such as semileptonic processes) is accomplished in a trivial manner, and leads to the now known result that selection rules are preserved in leading order.3 Xo pretensions will be made to mathematical rigor. While many of the results are more general, the discussion is geared to the case of distinguishable, massless particles scattering at zero energy. II.
NEUTRAL
VECTOR
MESON
THEORY
Consider first the hypothetical case of leptons interacting via neutral massive bosons. We are aware of the fact that this theory is renormalizable, nevertheless if one restricts oneself to uncrossed ladder graphs, the divergences which pile up in successive orders are not cancelled. (Cancellation comes about from the crossed graphs.) We write the amplitude for lepton-lepton scattering in the form
T = T,v WY1 - ~rs)lu~h”(l - hdh
(2.1)
B = B,, Ml
- ,idlm h”(l - in&) .
(2.2)
Here (1) and (2) refer to the two particles being scattered. convenient to split B into two parts BR and Bs where
We shall find it
and define
Vb)w = A2
gw
QPP” (Bs)w = *G2 2 lll?.
(2.3)
2 This statement can be proved in a variety of ways. For the sake of completeness, and for clarification of the meaning to be ascribed to highly divergent series occurring in this paper, we supply a proof in Appendix I. 3 See ref. 1, note added in proof. Independently of Feinberg and Pais, this result has also been derived by C. Bouchiat, B. D’Espagnat and J. Prentki (CERN preprint) and T. T. Wu (unpublished). The proof supplied in Section III, worked out independently by the present author, appears to be essentially identical to t.hat of Bouchiat et aE. and T. T. Wu. Since the work of Bouchiat et al. is due to be published soon, our inclusion of this proof is justified mainly for reasons of completeness. We thank Professor A. Pais for informing us of the work of Bouchiat et al. and Professor T. T. Wu for informing us of his own work.
PERATIZATION
The Bethe-Salpeter
equation
185
THEORY
may then be written,
in a matrix
notation,
T = B + BGT
as
(2.4)
where G stands for the product of lepton propagators (within a numerical factor of ---in; see Appendix II). The part of B which we label Bs plays havoc in the iterational solution of (2.4). The possibility, however, exists that the amplitude generat’ed by B, , satisfying the integral equation
Ts = Bs + BsGT,
(2.5)
may well be finite and well behaved. This type of situation where the difficulties presumably lie with the perturbation expansion, rather than any intrinsic defect of the theory, is by no means unprecedented. One may refer, for esample, to perturbative calculations of the correlation energy of an electron gas as providing an early precedent (4). Of course the problem at hand is quite different and the analogy with an electron gas can not be pushed further. We proceed to examine Eq. (2.5). The formal iterational solution is
T, = B, + BsGB, + B,GB,GB,
+ . ‘. .
(2.6)
Each term in Eq. (2.6) is, of course, highly singular. The most sTnguZar parts can be summed by noticing that the degree of divergence encountered in each of the above terms is identical to that, encountered in the chain iteration of the Fermi interaction. More precisely, as we show in Appendix III, the most singular part of the (7~ + 1)th iteration of B,y is identical to the (n + 1)th iteration of the Fermi coupling -f~gp.(g2/~~~‘) in the chain approximation. Hence if we define
B, = -.1;; g,, (g’/d
tr’(l - i~s)lm h”(l - i-dloj
(3.7‘)
and refer to the amplitude generated by smnming the most singular parts of the Seumann Series for Ts as the “peratized” amplitude TsP, we get
TsP = Bs + ,$ BdGBF)n = B, - B, + 5 Bp(GBF)n. ,t=” The sum in Eq. (2.9) is the amplitude thus damps to zero. Hence
T,’
generated by the Fermi interaction
= B,s - B F.
(2.8:)
(2.9) and (2.10’)
The sum of the most singular parts, on the right hand side of (2.6) is therefore finite. It would seem now that we hare a way of extracting the “solution” of (2.5) by respective summation of the most singular t#erms, the next to most
singular terms, the nest to the left with a series of finite terms. unwieldy. We therefore take a different dix IV, we rewrite Eq. (2.5) in
nest most singular terms etc. etc. until we are This procedure, however, is bound to be rather approach. Using the argument the form
given in
Appcn-
(2.11)
Ts = Bs + XB, -I- (Bs + XB,)GTs
where X is any continuous parameter. (A must be chosen as real in order to preserve time reversal invariance.) In particular, therefore, we can write
17s = Tsp + TaPGT, Similarly
Eq. (2.4) may be written
(2.1’)
as (2.13’)
T = Tap + BR + (TsP + B,)GT.
The iterational development of (2.12) or (2.13), remarkably enough, yields The amplitude may therefore be written as finite answers on the ,mass she11.475
T,, = (l’s’ + B,!,v + Oh”!
(2.14)
= B;y + O(g”). Equation (2.14) reproduces the result of Feinberg and Pais for a single channel. It should be emphasized that the summation of most singular parts of the Seumann series has played a rather auxiliary role in the above deduction. We used this summation only to enable us to exercise intelligently our freedom to rewrite Eq. (2.5) in the form (2.11). III.
CHARGED VECTOR MULTICHANNEL
MESOS THEORY PERATIZATION
AND
We sketch in this section, mostly for the sake of completeness, the straight,forward generalization of t#he above procedure to the physically interesting case of charged vector bosons. There are now two coupled chamiels in the purely leptonic processes and 4 More precisely, let us write T in the form given in Eq. (AII.l). The explicit integral equation for T, then admits of an it,erational solution, that for Z’b does not. However the uss,cnaption that the latter integral equation has a “solution” (in some sense, which excludes a straightforward use of either the Neumann Series or Fredholm Theory) suffices to rule out the possibility of a pole in Ta(y2) at q? = 0. Thus the on-shell amplitude is determined entirely in terms of T,(O). In this connection, it should be pointed out that if a solution for !Z’b is defined by the use of a cutoff, this cutoff should be allowed to approach infinity afte? the limit q* + 0 has been taken. The limit is not very well defined in the reverse order. 5 The finiteness of the iterated amplitude on the mass shell, after incorporation of a suitably chosen Fermi counter term, has been noted independently by W. R. Frazer (private communication).
PERATIZrlTIOA-
187
THEORY
several coupled channels in semileptonic processes. Equations must now be read also as matrix equations in channel space. We shall find it convenient to rewrite the Born term as
(2.4),
(3.5) etc-
(3.1)’ where i and j are channel indices and t,he matrix b contains the relevant coupling constants. n’ow the matrix b is real and symmetric, owing to time reversal invariance. Hence it can be diagonalized by an orthogonal matrix 0. Since G is a c-number in channel space, 0 will also diagonalize T. Hence if
Xi being the eigenvalues
(0-l TO)ij
= TiGij
(3.3)
(0-l RO)ij
= XiS,jCY
13.3)
T, = XX + XiCVT, .
(3.4)
of b,
The procedure used in the one channel case (to go from (2.4 be applied mutatis mutandis to Eq. i 3.4). One obtains (TpY)r = M’,, Illvcrting
hack, ho the original
+ o/m)
.;i gpv + Oh”).
representation,
T,, = B,, + (h/,~“)-?~
j to 13.14)) can (3.5)
we get
g,,y + O(g*,.
(.3&J
Thus the amplitudes for processes with a nonvanishing Born term are modified as in (2.14); the amplitudes for processes forbidden in leading order are still zero in that order. Hence the lack of emergence of neut)ral lepton currents, AS = -AQ or 1ASI > 1 transitions in leading order.3 I\‘.
coscLusIo~
We have presented an alternat’e formulation of the peratization program of Feinberg and Pais and shown &at the principal result’s of their theory can be underst,ood in a simple and elegant. nlatuler. In order to preserve continuity and facility of underst,anding we dropped all prehensions to nlathematical rigor. This feature of our presentation is based, in part, on our belief t#hat any attempt to inject fornlalistic rigor into the present cont8ext may well be tantamount to self deception. One should content oneself, rather, wit,h a set of rules sllficiently well defined to enable two different individuals to obtain t)he same answer to the same problem. As emphasized by E’einberg and Pais (1, 2,) t.he raison d’6trB for t,he theory would depend entirely on con>parison with experiment..
188
B&G
We carried through the peratization by invoking the self-damping of the Fermi interaction in the chain approximation. The reader may well ask whethel one is not at liberty to choose a suitable bare coupling such that the Icermi coupling is“renormalized” rather than damped out. Such an inquiry, however, is meaningful only within the context of a renormalizable theory. Since the theory is not renormalizable one also has the liberty of not making such a delicate choice for the bare coupling. APPENDIX SELF-DAMPING
OF CHAIN
IN
Let us write
the interaction
I
THE FERMI APPROXIMATION
INTERACTION
in the form
where 1 and 2 are the particle labels and we have omitted the spinor wave functions. The scattering amplitude generated by this interaction can be explicitly evaluated in the chain approximation (see Fig. 1). In momentum space the answer turns out to be T, = (T,),,[$‘(
1 - .&)]&“(l
- i~5)1M
where
with P*(P - n) I(Y2) = / d4P p2(p _ *)” . Here q is the sum of the two incoming (or outgoing) momenta, which need not be on their mass shells. The integral I is quadratically divergent; it may be given a finite value by introduction of a suitable cutoff A such that I --f A2 -+ m as h + m . We now ascribe a nlectning to the dive?-gent series occur&g in (AI.3) by dejining it to be the development of its formal sum, viz the function
(1 -s&-‘.
(AI.5)
With this definition of the divergent series, we find that T, + 0 as A’ --3 5. The above deduction should be compared with Appendix 3 of ref. 2.
x + )o( FIG. mation.
1. These
graphs
illustrate
+ joo( the iteration
of the Fermi
+ -----interaction
in the chain
approxi-
PERATIZATION
APPENDIX EXPLICIT
189
THEORY
REPRESENTATION8
II OF
THE
INTEGRAL
EQUATIONS
For the sake of completeness, we reproduce here the explicit representations of the integral equations that were handled symbolically in the text. These representations are specialized to the case of massless particles scattering at zero energy. We therefore take the two final momenta to be zero; the two initial momenta which we denote by p and -p respectively, can be put on their mass shells only after the equations have been solved. The (off-shell) scattering amplitude may be written as T,,(P)
= T,(p”j
Y,, + Tb(p’)
(AII.1)
(~~grYp2 - p,pyJ
where the tensor combinations have been chosen to be orthogonal to each other and the dependence of T, and Tt, on p” alone follows from Lorentx invariance. Equation (3.4) of the t#ext reduces to two decoupled equations
& BJ(9 - pmw) Tdp”) = Bdp2) + ; j 9”
(AII.2)
cl49Bb[(q- d'] Tb(p2)= Bb($) + & . f . j (12 (AII.3)
x @j&q - p)’ - [p-(9 - p)}21Tb(q2) where H, and & are the projections of B on the tensor basis defined in (AII.l). The explicit representations of Eqs. (2.5)) (2.11), and (2.12) can nom be immediately written down by substitution of the appropriate expressions corresponding to T,L , Tb and B, ad Bb . APPENDIX MOST
SINGULAR
PARTS
OF
THE
III NEUMANN
8E:RIJ-S
FOR
2’S
Let us denote by ill” a term of the form BsCGBs . . . CBS where Bs occurs n times. Then dl ii +1 = B.&W. (AIII.l) Choosing external momenta in the same way as in Appendix
II, we get
= [gurllf:+l (p” ) + (?Qi?wr p” - p, p<1x+’ ( p2j 1
.[Y”(l - iY5)ldYr(l - iYdl(2,
190
BtiG
where
It is convenient
to introduce
the notation 1 - ci) = (p - q)’ -
Ah
dp, q) = xp2hJ
- UY -
(AIII.5)
1122
{P. (P - Y) 12.
(AIII.6)
Then using the fact that il@ = B, , we get
.[(p
-
x [A(p
*l)*(ql
-
-
q,)A(ql
p212
*a*
(cm-1
-
- cm) a.* A(qn-I
(AIII.7)
dl
-
dlA(d.
(AIII.8) .[&I,
addn1,
X [A(P - &(YI
q2)
. . . d&l,
Vn)l
- UZ) ... A(%-1 - un)b(qn).
By straightforward power counting, we see that M,“+l diverges as Azn where A is the cutoff momentum. The same technique naively applied to Mb”+’ would indicate that it too diverges as A”“. However since the angular average of the (D’Sis zero the degree of divergence in Mb”+’ is considerably reduced. Since the angular integration gets rid of at least two powers of momentum, Mb”+’ is always bounded by (log A) * A2(n-m)where 1~is a positive integer such that 1 < M 5 n. Hence the most singular part of M”+’ is given by [~“(l
- &)]u,[r’(
= [y”(l
1 - ~+~)]~~jgo{ [Most singular part of MZ+‘l
- iYS)l~l~V(l -
ir5hgusw7r4Y
c-x
g”l~~w+l II(O)l”
(AIII.9
)
where I is the integral defined in (AI.4). Comparing with (AI.3), we see that (AIII.9) is nothing but the amplitude resulting from ‘n + 1 iterations of the Fermi interaction B, defined in Eq. (2.7) of the test.
PERATIZATION
APPENDIX EQUIVALENCE
191
THEORY
OF A CONTINUOUSLY IN THE LADDER
IV INFINITE APPROXIMATION
SET
OF
THEORIES
We wish to show that Eq. (2.11) of the text has the same “solution” for all values of X. The proof sketched here is based on the explicit representations introduced in Appendix II. It is thus specialized to the case of zero energy; however there seems to be no difficulty of principle in extending it to finite energies. In the notation of Appendix II, only the equation for Ts” depends on X. We have T,“(p’,
X) = [Bs”(p”)
+ 11 + (i/n41 1 [B,“((p
- q12) + xl ~ql&‘Y~“,
A).
(AIV. I)
Since
we can write Bs” (p’) + X as f(p’) + C where f(p’) constant. Equation (AIV.l) can therefore be written Ts”(X)
= S + c + Cf + c)sTs”O)
where we have introduced a symbolic The iterational solution of (AIV.2) T:(X)
p-” as p’ + ~1 and C is a as (AIV.‘)
notation. is given by
= Tj + T, + TI ST, + T, ST, + . . .
(,4IV.3)
where Tf = c:=o
i.fs)“f
(AIV.4)
and T, = C’ + C’sT, .
(AIV.:)
Clearly T, is constant and may be identified with the TF of Appendix I through the substitution x Gw = C. Consider now any term on the right hand side of (AIV.3)) such that f and T, occur YII and n times respectively. S would then occur ‘n + m - 1 times. By straightforward power counting, we see that one can always choose a number N such that the multiple integral occurring in this term is bounded by A4(n+m-1) h--2(n+7n-1) A-“” (log A)” = Az7L-2 (log A)” [A - cutoff momentum]. The TLfold occurrence of T, however provides a damping proportional to A-‘%. Hence as h -+ m, this term goes to zero for any N, 7~or ‘~Fz.Hence TsW,)
= Tf = Tsa(0).
Yote added in proof: By explicit summation of the series, (AIV. 3), one can verify that there is no dangerous exponential wish to thank Messrs. J. Rosner and I). Bailin for emphasizing
on the right -hand side of Eq. pile-up of the logarithms. I the necessity of this check.
ACKNOWLEDGMENTS The author is deeply indebted to Professor A. Pais for many discussions aud helpful advice. He wishes to thank Professor J. R. Oppenheimer for hospitality at the Institute for Advanced Study and Professor C;. C. Wick for hospitality at Brookhaven National Lnboratory. It is a pleasure to thank Professors G. Feinberg, T. T. Wu, and W. R. Frazer for interesting conversations. RECEIVED:
October 7, 1963 REFERENCES
1. G. FEINBERG AND A. PAIS, A field theory of weak interactions I. Phys. Rev. 131,2724 (1963). 2. G. FEINBERG AND A. PAIS, A field theory of weak interactions II. Phys. Rev. 133, B477 (1964). 3. Y. Pwu AND T. T. WV, Phys. Rev. 133, B778 (1964). 4. M. GELL-MANN AND K. A. BRUECHNER, Phys. Rev. 106,364 (1957).
OF
On
PHYSICS:
the
27, 183-192 (1964)
Peratization
Theory
of Feinberg
and
Pais”
MIRZA A. BAQI Bti~ Institute
for
Advanced
Study,
Princeton,
New
Jersey
We present an alternate formulation of the peratization program of Feinberg and Pais and show that t.he principal results of their theory can be reproduced in a simple manner. Essential use is made of the self-damping of the Fermi interaction in t,he chain approximation, and of a circumstance t,hat follows therefrom; to wit, t,he existence of a continuously infinite set of theories equivalent in the ladder approximation. While some of the results are more general, the discussion is geared to the case of distinguishable and massless Fermions scattering at zero energy. iY0 pretensions are made to mathematical rigor. I. IKTRODUCTIOS
Recently, P‘einberg and Pais have drawn att,ention to the possibility t,hat in a nonrenormalizable theory a cognizance of higher order effects may lead to a modification of matrix element,sin leading order of the coupling constant (1, 2). In parbicular, they show that for leptons interacting via massive, charged vector bosons, the term in the leading order matrix element
B,,(y)
= y”--l,l’ - y2
(SPY - YrYd)
(1.1)
get.smodified to
(1.2) of the uncrossedladder graphs. Here p and v are t,o be contracted with the (V-s) type lepton currents, y is the 4-momentum transfer, 171 the mass of the boson, and g the sem-weak boson lepton coupling constant.’ The very simple structure of Eq. (1.2) is in marked contrast to the highly involved nature of the derivations hitherto advanced (1-3). The purpose of this on summation
* Work supported, in part, by t)he U. S. Atomic Energy Commission during the author’s stay at Brookhaven National Laboratory, Summer 1963, and in part by the National Science Foundation. 1 The notations used in this paper, in particular the metric, the representations of the T-matrices, and the normalization factors in the Feynmann Rules etc. are in accord with S. S. Schweber, “An Introduction to Relat,ivistic Quantum Field Theory.” Row, Peterson, New IYork, 1961.
183
184
BtiG
note is to present a simple deduction of equation (1.2) based on the self-damping of a Fermi interaction in the chain approximation. 2 Our motivation stems from the belief that Ecf. (1.2) is of sufficient importance to warrant examination from more than one angle; even if such scrutiny has no more than a didactic value. The extension of our procedure to problenrs involving an arbitrary number of coupled channels (such as semileptonic processes) is accomplished in a trivial manner, and leads to the now known result that selection rules are preserved in leading order.3 Xo pretensions will be made to mathematical rigor. While many of the results are more general, the discussion is geared to the case of distinguishable, massless particles scattering at zero energy. II.
NEUTRAL
VECTOR
MESON
THEORY
Consider first the hypothetical case of leptons interacting via neutral massive bosons. We are aware of the fact that this theory is renormalizable, nevertheless if one restricts oneself to uncrossed ladder graphs, the divergences which pile up in successive orders are not cancelled. (Cancellation comes about from the crossed graphs.) We write the amplitude for lepton-lepton scattering in the form
T = T,v WY1 - ~rs)lu~h”(l - hdh
(2.1)
B = B,, Ml
- ,idlm h”(l - in&) .
(2.2)
Here (1) and (2) refer to the two particles being scattered. convenient to split B into two parts BR and Bs where
We shall find it
and define
Vb)w = A2
gw
QPP” (Bs)w = *G2 2 lll?.
(2.3)
2 This statement can be proved in a variety of ways. For the sake of completeness, and for clarification of the meaning to be ascribed to highly divergent series occurring in this paper, we supply a proof in Appendix I. 3 See ref. 1, note added in proof. Independently of Feinberg and Pais, this result has also been derived by C. Bouchiat, B. D’Espagnat and J. Prentki (CERN preprint) and T. T. Wu (unpublished). The proof supplied in Section III, worked out independently by the present author, appears to be essentially identical to t.hat of Bouchiat et aE. and T. T. Wu. Since the work of Bouchiat et al. is due to be published soon, our inclusion of this proof is justified mainly for reasons of completeness. We thank Professor A. Pais for informing us of the work of Bouchiat et al. and Professor T. T. Wu for informing us of his own work.
PERATIZATION
The Bethe-Salpeter
equation
185
THEORY
may then be written,
in a matrix
notation,
T = B + BGT
as
(2.4)
where G stands for the product of lepton propagators (within a numerical factor of ---in; see Appendix II). The part of B which we label Bs plays havoc in the iterational solution of (2.4). The possibility, however, exists that the amplitude generat’ed by B, , satisfying the integral equation
Ts = Bs + BsGT,
(2.5)
may well be finite and well behaved. This type of situation where the difficulties presumably lie with the perturbation expansion, rather than any intrinsic defect of the theory, is by no means unprecedented. One may refer, for esample, to perturbative calculations of the correlation energy of an electron gas as providing an early precedent (4). Of course the problem at hand is quite different and the analogy with an electron gas can not be pushed further. We proceed to examine Eq. (2.5). The formal iterational solution is
T, = B, + BsGB, + B,GB,GB,
+ . ‘. .
(2.6)
Each term in Eq. (2.6) is, of course, highly singular. The most sTnguZar parts can be summed by noticing that the degree of divergence encountered in each of the above terms is identical to that, encountered in the chain iteration of the Fermi interaction. More precisely, as we show in Appendix III, the most singular part of the (7~ + 1)th iteration of B,y is identical to the (n + 1)th iteration of the Fermi coupling -f~gp.(g2/~~~‘) in the chain approximation. Hence if we define
B, = -.1;; g,, (g’/d
tr’(l - i~s)lm h”(l - i-dloj
(3.7‘)
and refer to the amplitude generated by smnming the most singular parts of the Seumann Series for Ts as the “peratized” amplitude TsP, we get
TsP = Bs + ,$ BdGBF)n = B, - B, + 5 Bp(GBF)n. ,t=” The sum in Eq. (2.9) is the amplitude thus damps to zero. Hence
T,’
generated by the Fermi interaction
= B,s - B F.
(2.8:)
(2.9) and (2.10’)
The sum of the most singular parts, on the right hand side of (2.6) is therefore finite. It would seem now that we hare a way of extracting the “solution” of (2.5) by respective summation of the most singular t#erms, the next to most
singular terms, the nest to the left with a series of finite terms. unwieldy. We therefore take a different dix IV, we rewrite Eq. (2.5) in
nest most singular terms etc. etc. until we are This procedure, however, is bound to be rather approach. Using the argument the form
given in
Appcn-
(2.11)
Ts = Bs + XB, -I- (Bs + XB,)GTs
where X is any continuous parameter. (A must be chosen as real in order to preserve time reversal invariance.) In particular, therefore, we can write
17s = Tsp + TaPGT, Similarly
Eq. (2.4) may be written
(2.1’)
as (2.13’)
T = Tap + BR + (TsP + B,)GT.
The iterational development of (2.12) or (2.13), remarkably enough, yields The amplitude may therefore be written as finite answers on the ,mass she11.475
T,, = (l’s’ + B,!,v + Oh”!
(2.14)
= B;y + O(g”). Equation (2.14) reproduces the result of Feinberg and Pais for a single channel. It should be emphasized that the summation of most singular parts of the Seumann series has played a rather auxiliary role in the above deduction. We used this summation only to enable us to exercise intelligently our freedom to rewrite Eq. (2.5) in the form (2.11). III.
CHARGED VECTOR MULTICHANNEL
MESOS THEORY PERATIZATION
AND
We sketch in this section, mostly for the sake of completeness, the straight,forward generalization of t#he above procedure to the physically interesting case of charged vector bosons. There are now two coupled chamiels in the purely leptonic processes and 4 More precisely, let us write T in the form given in Eq. (AII.l). The explicit integral equation for T, then admits of an it,erational solution, that for Z’b does not. However the uss,cnaption that the latter integral equation has a “solution” (in some sense, which excludes a straightforward use of either the Neumann Series or Fredholm Theory) suffices to rule out the possibility of a pole in Ta(y2) at q? = 0. Thus the on-shell amplitude is determined entirely in terms of T,(O). In this connection, it should be pointed out that if a solution for !Z’b is defined by the use of a cutoff, this cutoff should be allowed to approach infinity afte? the limit q* + 0 has been taken. The limit is not very well defined in the reverse order. 5 The finiteness of the iterated amplitude on the mass shell, after incorporation of a suitably chosen Fermi counter term, has been noted independently by W. R. Frazer (private communication).
PERATIZrlTIOA-
187
THEORY
several coupled channels in semileptonic processes. Equations must now be read also as matrix equations in channel space. We shall find it convenient to rewrite the Born term as
(2.4),
(3.5) etc-
(3.1)’ where i and j are channel indices and t,he matrix b contains the relevant coupling constants. n’ow the matrix b is real and symmetric, owing to time reversal invariance. Hence it can be diagonalized by an orthogonal matrix 0. Since G is a c-number in channel space, 0 will also diagonalize T. Hence if
Xi being the eigenvalues
(0-l TO)ij
= TiGij
(3.3)
(0-l RO)ij
= XiS,jCY
13.3)
T, = XX + XiCVT, .
(3.4)
of b,
The procedure used in the one channel case (to go from (2.4 be applied mutatis mutandis to Eq. i 3.4). One obtains (TpY)r = M’,, Illvcrting
hack, ho the original
+ o/m)
.;i gpv + Oh”).
representation,
T,, = B,, + (h/,~“)-?~
j to 13.14)) can (3.5)
we get
g,,y + O(g*,.
(.3&J
Thus the amplitudes for processes with a nonvanishing Born term are modified as in (2.14); the amplitudes for processes forbidden in leading order are still zero in that order. Hence the lack of emergence of neut)ral lepton currents, AS = -AQ or 1ASI > 1 transitions in leading order.3 I\‘.
coscLusIo~
We have presented an alternat’e formulation of the peratization program of Feinberg and Pais and shown &at the principal result’s of their theory can be underst,ood in a simple and elegant. nlatuler. In order to preserve continuity and facility of underst,anding we dropped all prehensions to nlathematical rigor. This feature of our presentation is based, in part, on our belief t#hat any attempt to inject fornlalistic rigor into the present cont8ext may well be tantamount to self deception. One should content oneself, rather, wit,h a set of rules sllficiently well defined to enable two different individuals to obtain t)he same answer to the same problem. As emphasized by E’einberg and Pais (1, 2,) t.he raison d’6trB for t,he theory would depend entirely on con>parison with experiment..
188
B&G
We carried through the peratization by invoking the self-damping of the Fermi interaction in the chain approximation. The reader may well ask whethel one is not at liberty to choose a suitable bare coupling such that the Icermi coupling is“renormalized” rather than damped out. Such an inquiry, however, is meaningful only within the context of a renormalizable theory. Since the theory is not renormalizable one also has the liberty of not making such a delicate choice for the bare coupling. APPENDIX SELF-DAMPING
OF CHAIN
IN
Let us write
the interaction
I
THE FERMI APPROXIMATION
INTERACTION
in the form
where 1 and 2 are the particle labels and we have omitted the spinor wave functions. The scattering amplitude generated by this interaction can be explicitly evaluated in the chain approximation (see Fig. 1). In momentum space the answer turns out to be T, = (T,),,[$‘(
1 - .&)]&“(l
- i~5)1M
where
with P*(P - n) I(Y2) = / d4P p2(p _ *)” . Here q is the sum of the two incoming (or outgoing) momenta, which need not be on their mass shells. The integral I is quadratically divergent; it may be given a finite value by introduction of a suitable cutoff A such that I --f A2 -+ m as h + m . We now ascribe a nlectning to the dive?-gent series occur&g in (AI.3) by dejining it to be the development of its formal sum, viz the function
(1 -s&-‘.
(AI.5)
With this definition of the divergent series, we find that T, + 0 as A’ --3 5. The above deduction should be compared with Appendix 3 of ref. 2.
x + )o( FIG. mation.
1. These
graphs
illustrate
+ joo( the iteration
of the Fermi
+ -----interaction
in the chain
approxi-
PERATIZATION
APPENDIX EXPLICIT
189
THEORY
REPRESENTATION8
II OF
THE
INTEGRAL
EQUATIONS
For the sake of completeness, we reproduce here the explicit representations of the integral equations that were handled symbolically in the text. These representations are specialized to the case of massless particles scattering at zero energy. We therefore take the two final momenta to be zero; the two initial momenta which we denote by p and -p respectively, can be put on their mass shells only after the equations have been solved. The (off-shell) scattering amplitude may be written as T,,(P)
= T,(p”j
Y,, + Tb(p’)
(AII.1)
(~~grYp2 - p,pyJ
where the tensor combinations have been chosen to be orthogonal to each other and the dependence of T, and Tt, on p” alone follows from Lorentx invariance. Equation (3.4) of the t#ext reduces to two decoupled equations
& BJ(9 - pmw) Tdp”) = Bdp2) + ; j 9”
(AII.2)
cl49Bb[(q- d'] Tb(p2)= Bb($) + & . f . j (12 (AII.3)
x @j&q - p)’ - [p-(9 - p)}21Tb(q2) where H, and & are the projections of B on the tensor basis defined in (AII.l). The explicit representations of Eqs. (2.5)) (2.11), and (2.12) can nom be immediately written down by substitution of the appropriate expressions corresponding to T,L , Tb and B, ad Bb . APPENDIX MOST
SINGULAR
PARTS
OF
THE
III NEUMANN
8E:RIJ-S
FOR
2’S
Let us denote by ill” a term of the form BsCGBs . . . CBS where Bs occurs n times. Then dl ii +1 = B.&W. (AIII.l) Choosing external momenta in the same way as in Appendix
II, we get
= [gurllf:+l (p” ) + (?Qi?wr p” - p, p<1x+’ ( p2j 1
.[Y”(l - iY5)ldYr(l - iYdl(2,
190
BtiG
where
It is convenient
to introduce
the notation 1 - ci) = (p - q)’ -
Ah
dp, q) = xp2hJ
- UY -
(AIII.5)
1122
{P. (P - Y) 12.
(AIII.6)
Then using the fact that il@ = B, , we get
.[(p
-
x [A(p
*l)*(ql
-
-
q,)A(ql
p212
*a*
(cm-1
-
- cm) a.* A(qn-I
(AIII.7)
dl
-
dlA(d.
(AIII.8) .[&I,
addn1,
X [A(P - &(YI
q2)
. . . d&l,
Vn)l
- UZ) ... A(%-1 - un)b(qn).
By straightforward power counting, we see that M,“+l diverges as Azn where A is the cutoff momentum. The same technique naively applied to Mb”+’ would indicate that it too diverges as A”“. However since the angular average of the (D’Sis zero the degree of divergence in Mb”+’ is considerably reduced. Since the angular integration gets rid of at least two powers of momentum, Mb”+’ is always bounded by (log A) * A2(n-m)where 1~is a positive integer such that 1 < M 5 n. Hence the most singular part of M”+’ is given by [~“(l
- &)]u,[r’(
= [y”(l
1 - ~+~)]~~jgo{ [Most singular part of MZ+‘l
- iYS)l~l~V(l -
ir5hgusw7r4Y
c-x
g”l~~w+l II(O)l”
(AIII.9
)
where I is the integral defined in (AI.4). Comparing with (AI.3), we see that (AIII.9) is nothing but the amplitude resulting from ‘n + 1 iterations of the Fermi interaction B, defined in Eq. (2.7) of the test.
PERATIZATION
APPENDIX EQUIVALENCE
191
THEORY
OF A CONTINUOUSLY IN THE LADDER
IV INFINITE APPROXIMATION
SET
OF
THEORIES
We wish to show that Eq. (2.11) of the text has the same “solution” for all values of X. The proof sketched here is based on the explicit representations introduced in Appendix II. It is thus specialized to the case of zero energy; however there seems to be no difficulty of principle in extending it to finite energies. In the notation of Appendix II, only the equation for Ts” depends on X. We have T,“(p’,
X) = [Bs”(p”)
+ 11 + (i/n41 1 [B,“((p
- q12) + xl ~ql&‘Y~“,
A).
(AIV. I)
Since
we can write Bs” (p’) + X as f(p’) + C where f(p’) constant. Equation (AIV.l) can therefore be written Ts”(X)
= S + c + Cf + c)sTs”O)
where we have introduced a symbolic The iterational solution of (AIV.2) T:(X)
p-” as p’ + ~1 and C is a as (AIV.‘)
notation. is given by
= Tj + T, + TI ST, + T, ST, + . . .
(,4IV.3)
where Tf = c:=o
i.fs)“f
(AIV.4)
and T, = C’ + C’sT, .
(AIV.:)
Clearly T, is constant and may be identified with the TF of Appendix I through the substitution x Gw = C. Consider now any term on the right hand side of (AIV.3)) such that f and T, occur YII and n times respectively. S would then occur ‘n + m - 1 times. By straightforward power counting, we see that one can always choose a number N such that the multiple integral occurring in this term is bounded by A4(n+m-1) h--2(n+7n-1) A-“” (log A)” = Az7L-2 (log A)” [A - cutoff momentum]. The TLfold occurrence of T, however provides a damping proportional to A-‘%. Hence as h -+ m, this term goes to zero for any N, 7~or ‘~Fz.Hence TsW,)
= Tf = Tsa(0).
Yote added in proof: By explicit summation of the series, (AIV. 3), one can verify that there is no dangerous exponential wish to thank Messrs. J. Rosner and I). Bailin for emphasizing
on the right -hand side of Eq. pile-up of the logarithms. I the necessity of this check.
ACKNOWLEDGMENTS The author is deeply indebted to Professor A. Pais for many discussions aud helpful advice. He wishes to thank Professor J. R. Oppenheimer for hospitality at the Institute for Advanced Study and Professor C;. C. Wick for hospitality at Brookhaven National Lnboratory. It is a pleasure to thank Professors G. Feinberg, T. T. Wu, and W. R. Frazer for interesting conversations. RECEIVED:
October 7, 1963 REFERENCES
1. G. FEINBERG AND A. PAIS, A field theory of weak interactions I. Phys. Rev. 131,2724 (1963). 2. G. FEINBERG AND A. PAIS, A field theory of weak interactions II. Phys. Rev. 133, B477 (1964). 3. Y. Pwu AND T. T. WV, Phys. Rev. 133, B778 (1964). 4. M. GELL-MANN AND K. A. BRUECHNER, Phys. Rev. 106,364 (1957).