VOLUME 21, NUMBER 4 P H Y S I C A L R E V I E W L E T T E R S 22 JULY 1968

uum. From either the commutation relations [Eq. (2)] or the odd parity of X, we see that the multiple must be zero; i.e., Xj annihilates the vacuum.

We now obtain a contradiction by taking the vacuum expectation value of Eq. (3)10:

(01 (Y3773-773X3) 10) = 0 - 0

= (01(1+773773)10)

= 1 + S ! (0 IT7J«) ! 2 >0. n 3

Reviewing the arguments which led to this con­tradiction, we see that any attempt to formulate a relativistic quantum field theory with chiral transformations given by Eq. (1) must encounter difficulty. In particular, the "usual mass-spec­trum condition" cannot be imposed—that is, the pion must have zero mass.11

This investigation was prompted by lectures of Professor S. Weinberg and Professor B. Zumino at a symposium held at Northwestern University in January 1968. We acknowledge conversations with many of our colleagues at Case Western Re­serve University.

*Work supported in part by the U. S. Atomic Energy Commission.

1S. L. Adler, Phys. Rev. Letters 14, 1051 (1965); W. I. Weisberger, Phys. Rev. Letters 14, 1047 (1965).

2J. Schwinger, Ann. Phys. (N.Y.) 2, 407 (1957); M. Gell-Mann and M. Levy, Nuovo Cimento _16, 705 (1960); M. Gell-Mann, Phys. Rev. 125, 1067 (1962); Y. Nambu and D. Lurie, Phys. Rev. 125, 1429 (1962); S. Weinberg, Phys. Rev. Letters 1^, 507 (1967); J. Schwinger, Phys. Letters 24B, 473 (1967); J. Wess and B. Zumino, Phys. Rev. 163, 1727 (1967). This

It has been suggested1'2 that the idea of vector dominance in particle interactions can be formu­lated in terms of (approximate) identities be­tween the vector currents taking part in the in­teractions and vector fields. An attempt at a concrete realization of these identities in a con­sistent Lagrangian formalism leads one to a La-

list is not meant to be exhaustive. 3S. Weinberg, Phys. Rev. 166, 1568 (1968). 4J. C. Taylor, Phys. Rev. 110, 1216 (1958); M. L.

Goldberger and S. B. Treiman, Phys. Rev. 110, 1478 (1958).

5R. Haag, Phys. Rev. 112, 669 (1958); K. Nishijima, Phys. Rev. I l l , 995 (1958); M. Zimmerman, Nuovo Cimento 10, 597 (1958).

6This is not precisely true, since it can be shown that there are two distinguishable classes of nonlinear realizations. This point, which we shall discuss else­where, does not alter our conclusions.

7R. F. Streater and A. S. Wightman, PCT, Spin and Statistics, and All That (W. A. Benjamin, Inc., New York, 1964). Note that by interpreting the fields ^(x) as operators, we are already going far beyond their "phenomenological interpretation" as a collection of greek and latin letters.

8D. Ruelle, Helv. Phys. Acta 35, 147 (1962), Appen­dix. The proof applies only to bounded operators. Our argument can be recast in a form involving the chiral group operators eia'x rather than the generators; Ruelle's theorem then applies rigorously, and we are led to the same final result.

9We can see directly that if UXXT~lX is c-number it must vanish, by taking its vacuum expectation value and noting that U\o)=\0). We thank Professor K. Ko-walski for pointing out this simpler proof.

10A smearing function/ should be applied in order to make the left-hand side nonsingular. TT-MO) is normal-izable and is assumed to belong to the domain of X. If / is chosen to be non-negative, the right-hand side r e ­mains positive definite.

nThis does not answer the converse question of whether permitting the pion to have zero mass does make chiral symmetry possible. [Some complications associated with zero mass are discussed by J. Gold-stone, A. Salam, and S. Weinberg, Phys. Rev. 127, 965 (1962), especially Sec. III.] We regard massless fields as ter ra incognita and dare not guess the de­tailed geography.

grangian which is similar2 to the one in Yang-Mills-type theories3 except for the mass term of the vector particles. Local gauge-invariance is, therefore, lost and does not play any role in the theory. On the other hand, it has been shown4'5

that it is possible to obtain massive vector me­sons in a Yang-Mills-type theory if the local

GAUGE INVARIANCE, BROKEN SYMMETRIES, AND FIELD-CURRENT IDENTITIES

Tulsi Dass Department of Physics, Indian Institute of Technology, Kanpur, India

(Received 28 May 1968)

It is shown that (approximate) field-current identities appear in a natural way in a Yang-Mills—type theory if the local symmetry is spontaneously broken.

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VOLUME 21, N U M B E R 4 P H Y S I C A L R E V I E W L E T T E R S 22 JULY 1968

symmetry is spontaneously broken. It is, there­fore, interesting to enquire if the proposed field-current identities also appear in theories of the latter type in some approximation. In this note, we propose to answer this question in the affir­mative.

In the case of constant-parameter symmetry groups, it is well known that when the symmetry is spontaneously broken, the current operators have terms linear in the fields. These terms can be identified6 as gradients of the Goldstone fields. We assume7 that these Goldstone fields are "ele­mentary" fields present in the Lagrangian. When the symmetry is extended to a local one and then spontaneously broken, the contribution of the scalar fields to the current operator contains a term linear in the gauge fields in addition to the Goldstone term. The two linear terms have just the right factors to combine into a single mas­sive vector field, the Goldstone fields now a s ­suming the role of longitudinal modes of the massive vector fields. For Abelian gauge groups, this result appears in Refs. 4 and 5 although the significance of the linear vector-field term in the current operator was not emphasized there. In the following paragraphs the same will be dem­onstrated in the case of non-Abelian gauge groups.

The current operators for a multiplet of (Her-mitian) scalar fields (fa(a = 1, • • «,iV) are given by

5 * = ~ih y*?„*<?& « = 1> ix a a|8 ^ (1)

where Ta are the imaginary antisymmetric gen­erators. Writing (Pa=Va + Xa where rja = (<Pa)o, we get

symmetry is spontaneously broken, we have

. a ^ <a A b h ^a j =d G -gA rjT T T]

jut M / i

- / (V v) *T a\ .

The matrix

"te-*V6)a(A>a

(5)

(6)

is the mass matrix of the vector mesons.5 The first two terms in Eq. (5) can be combined to give

M ba

where

V b~A b-gd GC(M~X) t

M jLl ]Lt Cb

(7)

(8)

are the massive vector fields.10 This identifica­tion can be easily confirmed from the Lagrang­ian:

£ = - i ( V cp) \v cp) -\F aF ° fi a ii ct \xv \xv

= -i(V x) V X) +£* GaA a IJL a (i a fi fi

-\M A A + • • •. ab ix ii

(9)

In the representation in which the mass matrix M is diagonal, the term (7) will take the form

-(ra /g)V a (i

(10)

J\i \i [iKa ap A£

where

°a-vj\

(2)

(3)

are easily identified as the Goldstone fields. This can be confirmed, for example, by applying Gilbert's arguments8 to the quantity (0\[jQ

a(x), Ga(0)] 10) which is obviously nonzero.9

When the symmetry is extended to a local one, the current operator is modified by

which gives the same proportionality factors be­tween currents and fields as appear in Ref. 1.

When other fields are present, they will add their own bilinear and higher terms to the cur­rent operator. It is clear that in a matrix ele­ment of the current operator, the linear term (10) will give a vector-meson pole, while the bi­linear and higher terms will give the cut s truc­ture. In the author's opinion, this is the most natural explanation of vector dominance. It also makes clear the nature of the approximation in­volved in writing a field-current identity.

8 <p-v <ps(8 +igA aTa)cp, (4) Ii ix ix H 9

where A^ are the gauge fields. Again, if the

1N. M. Kroll , Te D. Lee, and B . Zumino, Phys . Reve

157, 1376 (1967). 2 T. D. Lee and B. Zumino, Phys . Rev. 163, 1667

243

V O L U M E 21, N U M B E R 4 P H Y S I C A L R E V I E W L E T T E R S 22 JULY 1968

L. Mil ls , Phys . Rev. 96, 191 Phys . Rev. 10JL, 1597 (1956); Glashow, Ann. Phys . (N.Y.) 15,

(1967). 3C. N. Yang and R.

(1954); R. Utiyama, M. Gell-Mann and S. 437 (1961).

4 P. W. ffiggs, Phys . Rev. 145; 1156 (1966). 5 T. W. B . Kibble, Phys . Rev. 155, 1554 (1967). 6 L. Leplae, R. N. Sen, and H. Umezawa, P r o g r .

Theore t . Phys . Suppl. ex t r a number 645 (1965). 7When the Goldstone pa r t i c l e s a r e composi te , the

condition of vanishing renormal iza t ion constants may be imposed on the " e l e m e n t a r y " f ields.

8W. Gilbert , Phys . Rev. Le t t e r s 12, 713 (1964).

9Note that the Goldstone fields Ga a r e nonzero only when Tar]^01 i . e . , when the s y m m e t r y corresponding to T° i s spontaneously broken . For s y m m e t r i e s which r e m a i n intact in dynamics , we have (Ref. 5) Tar] = 0, and t h e r e a r e no Goldstone f ields.

10When some components of the s y m m e t r y r e m a i n u n ­broken, the m a s s ma t r i x M is s ingular (Ref.5). In th is ca se only i ts nonsingular pa r t i s to be cons idered . F r o m h e r e it i s c l e a r that only c u r r e n t s of broken s y m m e t r i e s will give the cor responding vec to r -meson po les . This explains why we do not have, for example , the photon pole in the ma t r i x e lements of the e l e c t r o ­magnetic c u r r e n t .

FINITE-WIDTH CORRECTIONS TO THE VECTOR-ME SON-DOMINANCE PREDICTION FORp-~e+e~*

G J. Gounaris and J. J Sakurai Department of Physics, and The Enrico Fermi Institute, The University of Chicago, Chicago, Illinois

(Received 17 May 1968)

Finite-width corrections based on a generalized effective range formula for pion-pion scattering modify by a non-negligible amount the well-known relation between T{p-~e+e~) and Tip-*- 7r7r) derived on the basis of vector-meson dominance. We also present a new current-algebra prediction for the shape and magnitude of a(e+e~—* 7r+7r"~) and estimate the p-meson contribution to the Schwinger term.

It was pointed out more than six years ago1

that the hypothesis of vector-meson dominance can be used to compute the lepton-pair decay rate of a neutral vector meson in terms of the vector-meson coupling constant appearing in strong interactions, in much the same way as the hypothesis of partially conserved axial-vector current relates the pion decay constant to the pi-on-nucleon coupling constant. If we call the cou­pling constant at the y-p junction emp

2/fpy the complete p dominance of the electromagnetic form factor of IT implies

fp fpirir>

or equivalently,

r(p-e+e-) R-

a2 Im 2-4m 2 \ 3 / 2

I n ir \

T(P-TTTT) 36 \ - - £

(i)

(2)

lepton-pair branching ratio R as well as the p -meson width is to rely on the colliding beam r e ­action

+ -e e (3)

We therefore focus our attention on the electro­magnetic form factor of the pion E^is), which is related to the colliding-beam cross section via

a{e+e ^7r+7r )

7TQ?2 ( s - 4 r a 2 ) 3 / 2

7T Q c 5 / 2 \F (s)

IT (4)

Our starting assumption is that for a wide energy range (s <1 BeV2) the/>-wave pion-pion scatter­ing phase shift 61 satisfies a generalized effec­tive-range formula of the Chew-Mandelstam type2:

In deriving the above relation, however, we have assumed that the p meson is essentially stable. In this note we demonstrate how Eq. (2) must be modified when we take the finite p width into ac­count. We also discuss the current (field) alge­bra predictions on T(p — irir), a(e+e "" — 7r+7r~), and the magnitude of the Schwinger term.

Experimentally the cleanest way to obtain the

(k3/Ss) cot61 =k2h(s) + a + bk2, (5)

where

k = {\s-m 2 ) 1 / 2 , 7T

. , N 2 k . (Vs + 2k h{s) = --rln[ —

7r vs \ 2m

(6)

244