Volume 5, n u m b e r 4 P H Y S I C S L E T T E R S 15 Ju ly 1963

But ~o + K is a s e i f - c ro s sed react ion, hende d(t)=0. Thus both D~/A½ and D32_/A ~ approach zero for S --* vo.

The foregoing argument a lso applies to n + p scat ter ing, except that our analys is neglects the proton spin. This is s t r ic t ly valid for t = 0 because of helici ty conservat ion; d(t) = 0 then reduces to the sum rule of Goldberger et al. 31. For t not too large our neglection of spin will still be justified if helici ty flip is r a r e at high energies .

1) I. Ia. Pomeranchuk, J. Exptl. Theoret. Phys. (USSR) 34 (1958) 725; translation: Soviet Phys. JETP 7 (1958) 499.

2) I. Ia. Pomeranchuk, J. Exptl. Theoret. Phys. (USSR) 30 (1956) 423; translation: Soviet Phys. JETP 3 (1956) 306. L. B. Okun and I. Ia. Pomeranchuk, J. Exptl. Theoret. Phys. (USSR) 30 (1956) 424; translation: Soviet Phys. JETP 3 (1956) 307.

3) M.L.Goldberger, H.Miyazawa and R.Oehme, Phys. Rev. 99 (1955) 986. T.D. Spearman, Nuclear Phys. 16 (1960) 402.

75 I N V A R I A N C E *

K. JOHNSON Department of Physics and Laboratory for Nuclear Science,

Massachusetts Institute of Technology, Cambridge 39, Massachusetts

Received 24 June 1963

It has long been argued with the use of formal 75 invar tance, that a Lagrangian which contains no ba re mass for F e r m i par t i c les implies that no m a s s can be generated by an interact ion which " p o s s e s s e s " the s y m m e t r y , if the vacuum pos - s e s se s the symmet ry . If the vacuum does not pos ses s the s y m m e t r y then other objectionable fea tures a r i se (zero m a s s pseudosca la r states) 11. Recently it has been shown that another exception can occur if the Fermion m a s s opera tor has c e r - tain anomalies in its spec t rum 2). However, the s y m m e t r y is still p resen t in that two F e r m i pa r t i - c les appear (with m a s s e s ± m). This would then mean that the zero m a s s pseudosca la r par t ic les could be avoided but only with the cost of a doublet of Fe rmions with opposite par i t ies .

It is the purpose of this note to point out a m a - thematical (but nonetheless interest ing) way out of the di lemma. The fo rma l use of an invariance in the Lagrangian is ve ry dangerous because of the loosely defined cha rac te r of the opera to r s it con- tains. In fact , charge conservat ion holds for the example of a f r ee Fe rmion interact ing with an ex- ternal e lec t romagnet ic field only if the cur ren t is defined by ve ry careful l imiting p rocedures 3,41 which allow one to c i rcumvent the so-cal led photon m a s s te rm in the vacuum polar isa t ion, which gives a t e rm in j~(x) propor t ional to Al~ext(x) and hence a ~N(x) which is not conserved. Thus, the fo rmal

* Th i s work i s suppor t ed in p a r t t h r o u g h funds p rov ided by the Atomic E n e r g y C o m m i s s i o n u n d e r c o n t r a c t AT(30-1) -2098 .

use of the equations of motion together with the fo rmal express ion for the cur ren t

ju(x) = ½ [i~(x)v~, ¢(x)]

does not enable one safely to conclude that 3/jjU=O. In this problem the current must be defined by a limiting procedure applied to the product

x+ f ~(x+~) v~ ~(x) (I - i f d~ A~xt(~1),

x

where Ap ext is the external potential. This form is locally gauge invariant and this insures conserva- tion in the limit. Without the factor involving Ap ext the resulting operator is not conserved.

Since such great care is needed in this example to get an invariant theory, one might worry also about the "proof 'w of the conservat ion law for ~/~r5~b. In the e lec t rodynamic example one can c i r - cumvent the difficulty by using the t e r m s in the in- teraction (Ap) to generate a compensat ion for the singular te rm. One need not always be able to do this. In fact , a s imple, exactly soluable model in one space-one time dimension 51 i l lus t ra tes this ve ry nicely and provides a counter example to the fo rma l proof. The model is quantum e lec t rodynam- ics with m o = 0 for the charged F e r m i par t ic les . The Lagrangian is formal ly ~5 invariant. However, if one solves the model , one obtains for the cu r ren t - cur ren t vacuum expectation value

e2 e 2 (JP(q) jr(q)) = (gpu ~ + qP qU I . 5(q2 + --~-1 ,

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Volume 5, number4 P H Y S I C S L E T T E R S 15 July 1963

whe re e i s the coupl ing constant . One eas i ly s ees that this is cons i s ten t with cha rge c q n s e r v a t i o n , q# j# (q ) = O. However , in one s p a c e - o n e t ime di - mens ion s ince

~,5 = ~,o,/I , ~ 5 ~'/~ = ¢ / ~ ' u

t h e ' p s e u d o v e c t o r c u r r e n t is

J5 =

SO

<j~(q) j~(_q)) (_g#U e 2 e2 = __ + ~/ZXqxcU~q~ ) 5(q2 + ~-)

and thus , e 2 e 2

q# ( j~(q) j~(q)> = _ _~_qtJ 5(q2 + __~_) ¢ 0 .

Hence only in the absence of coupl ing is the t h e o r y ~5 invar ian t . Th i s m e a n s s i m p l y that the c a r e r e - qu i red to make j #AI j a mean ingfu l and gauge inva- r i an t coupl ing a l so y ie lds the r e s u l t tha t the coup- l ing is not ac tua l ly ~5 invar ian t . The f o r m a l " p r o o f ' of tha t i nva r i ance is jus t i n c o r r e c t .

R e f e r e n c e s

1) Goldstone, Salam and Weinberg, Phys. Rev. 127 (1962) 965.

2) W. Thirring, Physics Letters 4 (1963) 167. 3) J. Schwinger, Phys. Rev. Letters 3 (1959) 296. 4) K.Johnson, Nuclear Phys. 25 (1961) 431; Nuovo

Cimento 20 (1961) 773. 5) J.Schwinger, Phys. Rev. 128 (1962) 2425.

$ $ * * $

1 ON THE VALIDITY OF THE IAII - 2

SELECTION RULE IN Z DECAY

P. FRANZINI and D. ZANELLO Istituto Nazionale di Fis ica Nuclea*'e, Sezione di Pisa

Istituto di Fis ica della UniversitY, Pisa Scuola No~'male Superiore, Pisa

Received 19 June 1963

Many ques t ions have been r a i s e d on the e x p e r i - men ta l va l id i ty of the ]AIJ = ½ se lec t ion ru le in s t r ange p a r t i c l e leptonic d e c a y s 1-3).

The s i tuat ion fo r non leptonic d e c a y s is at the m o m e n t not quite c lea r . On the one hand A o and K~ d e c a y s appea r to be in ~ood a g r e e m e n t with the p red ic t ions of the ru l e 4 , 5 L

In ref . 3) a l so the p r o c e s s K~ -. 3~ is d i s c u s s e d but the conc lus ion is s t i l l r a t h e r unce r t a in because of the poor s t a t i s t i c s due to the di f f icul ty of co l l ec t - ing enough even ts of th i s type.

The va l id i ty of the I All = ½ se lec t ion ru le has been r e c e n t l y ques t ioned 6) in the case of Z d e c a y into p ions and nuc leons on the bas i s of new e x p e r i - men ta l data.

Having r e e x a m i n e d the expe r imen ta l s i tuat ion fo r the p r o c e s s e s

E - -" ~- + n , ( la)

~+ -. y+ + n , ( lb)

~+ -, ~o + p , (Ic)

in a m o r e cons i s ten t and comple te way we have been led to the conclus ion that the data a r e not in d i s - a g r e e m e n t with the va l id i ty of the I All = ½ ru le .

F r o m the [A11 = ½ ru le it fo l lows that it is p o s - s ible to d e s c r i b e the t h r e e r e a c t i o n s (1) 7) with four

254

S and P wave ampl i t udes S1, P1, $3, P3, w h e r e 1 and 3 s tand fo r /final s ta te = ½ and ~ (I is the i s o - topic spin).

We can e x p r e s s the d e c a y r a t e s and a s y m m e t r y p a r a m e t e r s as t

r-:p-(Is312+ Ip3l 2) 2 Re S 3 P3*

Of- =

ls3l 2 + IP3I 2

r + = - ~ P + ( I S 3 + 2 S I 12+ IP3 + 2 P 1 1 2 )

2 Re (S 3 + 2S1) (P3 + 2P1)* (2)

Is3 + 2s l l 2 + le3 + 2 e l ] 2

r ° = ~ ° ( I s 3 - s112 + ]P3 - P l ]2)

2 R e (S 3 - S 1 ) ( P 3 - P l ) * ~ O =

IS3 - S 1 ] 2 + IP3 - P I ] 2

where the s u p e r s c r i p t s - , +, o r e f e r to the p r o - c e s s e s ( la ) , ( lb) , ( lc) r e s p e c t i v e l y and p is the phase space f ac to r .

Using a l so t ime r e v e r s a l i n v a r i a n c e , the four

We have written the decay rate in the form r = ]MI 2p with p = (E1E2/M) p.