1242 J O H N W. : U R S O 149

wave NN-wir contribution is also suspect due to the strong cutoff dependence.

Now that there are other known wN resonances, these could be incorporated into the model. This would require a reparametrization of the irN model which could lead to significant effects in the nucleon-nucleon channel. This could be one way in which to test whether the agreement with experiment is largely a matter of chance or whether the model's success demonstrates the consistency of all low-energy phenomena involving pions and nucleons.

DURING the past year, there has been a great deal of research activity centered upon the algebras

generated by equal-time commutators of currents.1 At the beginning of these studies, it was pointed out that the proposed commutation relations between the cur­rents or their associated charges was the most natural extension of the concept of universality2 between the vector and axial-vector parts of the weak interaction. In this paper we propose to extend this universality to include the commutators of chiral charges and un-observables such as the baryon field.

The weak Lagrangian is assumed to be of the form of a current times a current. The current responsible for the weak decays is considered to be the sum of a vector and an axial-vector part. The charges associated with these currents are

Qi-(t)~-i d*xV,(x,t) (1)

and

X+(t)=-i d*xAi(x,t). (2)

Imposing the conserved-vector-current hypothesis,3 we can identify Q+(t) as one of the generators of isospin

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

i M . Gell-Mann and Y. Ne'eman, Ann. Phys. (N. Y.) 30, 360 (1964); M. Gell-Mann, Phys. Rev. 125, 1067 (1962).

2 N. Cabibbo, Phys. Rev. Letters 10, 351 (1963). 3 R. P. Feynman and M. Gell-Mann, Phys. Rev. 109, 193 (1958).

ACKNOWLEDGMENTS

The author wishes to express his thanks to Professor B. Vitale for many discussions and his hospitality while the author was at the Istituto di Fisica Teorica in Naples, and to Professor D. Amati for making the intermediate results of the ALV calculation available for comparison with the present calculation. The calcu­lations reported here were carried out at the Istituto Superiore di Sanita at the University of Rome and at the Computing Center of Michigan State University.

rotations. This identification allows us to write down the equal-time commutator of Q+(t) with the baryon field as

[ ^ ( / ) ^ n ( x , 0 ] = 0 , (3)

ce+(o,^(x,o]=-^(x,o, (4) where ^P(»)0M) is the renormalized proton (neutron) field. We restrict ourselves to equal times in the presence of possible symmetry breaking. Invoking universality, we assume that

[ X + ( 0 , * n ( x , 0 > 0 , (5)

and

[ X + ( 0 , ^ , ( x , 0 ] = - T 6 ^ » ( x , 0 . (6)

This appears as the simplest form consistent with space-time symmetries. We also point out that this form is implied by the free-field form of these operators.

For the future evaluation of the matrix elements which arise, we require the partially conserved axial-vector-current hypothesis (PCAC). We use PCAC in the form

d r -X+(t)=»*fr <Px4>+(x9t), (7) dt J

where fv is determined from the pion decay rate to be / ,=0 .935 .

Taking the matrix elements of (6) between a 7r+ and

P H Y S I C A L R E V I E W V O L U M E 1 4 9 , N U M B E R 4 3 0 S E P T E M B E R 1 9 6 6

Weak Axial-Vector Currents and the Baryon Field*

AUSTIN M. GLEESON

Physics Department, Syracuse University, Syracuse, New York (Received 9 May 1966)

We study a postulated commutation relation between the chiral charge and the nucleon field. This rela­tionship is used to generate a sum rule between gp7r7r, g^N*, and gr. We find that the p meson and N* baryon are insufficient to saturate the sum rule. The addition of strong S-wave dipion scattering is found to be adequate to explain the discrepancies of the above sum rule.

149 W E A K A X I A L - V E C T O R C U R R E N T S A N D B A R Y O N F I E L D 1243

a proton, we have

<*+|X+(*)[0)<0|^(x,0|pA>

+ £ < * + | X + « | v > < v | ^ ( x , * ) | p A > V

-£<*W*,')ln><iil*+(0|pX>

= - Y 5 < * + | ^ ( x 5 / ) | p A > . (8)

The sum over v symbolically represents the contribution from all physical states of charge zero and baryon num­ber zero. We point out that the usual G-parity require­ments of the axial-vector current4 (contained in PCAC) imply that v contains only an even number of pions. The sum over rj is the contribution from all states of baryon number one and charge plus two. Instead of attempting to evaluate these matrix elements rigor­ously, we will evaluate them by means of simple but physically sound models. We also follow Adler5 and Fubini6 in taking the iz+= (0,0,q)= p and q—> oo. Be­cause the matrix elements of the charge contain the space integral of a local operator, the vacuum term is nonzero only in the rest frame of the pion. I t is also in this limit that we feel safe in substituting stable reso­nances for the continuum contributions of the above matrix elements.5,7 In this sense, we assume that the v term is dominated by the p resonance and the r\ con­tribution by the N*++. We further simplify this calcu­lation by using the technique of effective Lagrangians. Although not entirely satisfactory theoretically, this technique offers the advantage of giving a good measure of all the matrix elements directly. We justify the use of this expedient by noting that in the previous calcula­tions of this nature,5 the off-the-mass-shell effects were small and did not significantly affect the results. This point is discussed in greater detail later in this paper.

Proceeding with the calculation, we write the La-grangian as

£ = gpinrQ\' ((pXdx^ + igpNNQ^'Hxh^ , (9 )

where

gp7T7r2/47r=2.1 and gp7rir=gpNN.

With this Lagrangian, we calculate the p contribution

l gp^2 1 / M

2 + ™ A =i»*f„ 1

fjL2—mp2 fji2 W J V \ 2mp2 /

XmNux(v)e~^+-^\ (10)

We calculate the N*++ contribution from the

4 S. Weinberg, Phys. Rev. 112, 1375 (1958). 5 S. L. Adler, Phys. Rev. 140, B736 (1965); W. I. Weisberger,

Phys. Rev. Letters 14, 1047 (1965) and references listed therein. 6 S. Fubini and G. Furlan, Physics 1, 229 (1965). 7 1 . S. Gerstein and B . W. Lee, Phys. Rev. 144, 1142 (1966).

Lagrangian

«C= (g*NN*/»)(&$)' Atfyfh.c.,

where A* is the set of 2X4 isospin matrices which force \pA\f/» to transform as a Cartesian vector under isospin rotations.

We use gTj\riv*2/47r=0.43. Proceeding as before, the N* contribution is

E(^|^(x,0|*?)^|x+W|px) V

igwNN*2 r fM2+mN2 mN /mN

2—fx2\ = (mN*+mN)\ 1 [ J

MM2 L 3 3niN*\ 2 /

2 /}i2+mN*2\/mN*2+mN2Yl y?fT

3mN*2\ 2 A 2 JjmN*2-i m^* —mN

Xux(p)e .—i(ir+—p)x (U)

Finally, we evaluate the right-hand side of the com­mutator. For this case, we get

<^+|^(x , / ) |pX)

(ir+—p)2-\-niN2 ••yJ2igrym\(v)e~i{lv+~p)x'. (12)

Putting all these results together, we find the sum rule

- i /xgpir» 8(0.119BeV- 1K(p)^C T +- p ) a ?

~if.g*NN*2(5.66 BeV-1)«x(p)^< ( '+-p ) f l !

= - ^ r ( 1 . 5 1 BeV-^uxip)*-^*-*)*. (13)

Using the experimental values of the coupling, we can evaluate the two sides of this equation. The left side yields the value —£31.61 BeV""1. The right-hand side is — i20.4 BeV"1. It is obvious that even the assumption of large uncertainties in the coupling con­stants (i.e., 10 to 20% uncertainty) would not be sufficient to explain this large difference. There are several possible explanations for this disagreement. The simplest would be that the assumed commutation rule does not hold. There are several reasons for desiring that this relation remain valid; the simplicity of the assumption, universality of the weak currents, and multiplicative renormalization effects. We therefore seek another explanation for this discrepancy.

Another source of error in the evaluation of this com­mutator may be the use of the effective Lagrangians. As was implied in the earlier parts of this paper, the momentum limits required to allow for the rapid saturation of commutation relation also requires the mass-shell extrapolations to go to zero mass. There are two cases where this extrapolation takes place. The pion field which enters through PCAC is used in the zero-mass limit and the nucleon field occurring in the original commutator is extrapolated to zero mass. The extension of the pion mass to zero has been treated

1244 A U S T I N M . G L E E S O N 149

before5 and the effects are seen to be quite small. The case of the nucleon field is not expected to be so simple since in units of energy the extrapolation is over a much greater range. We point out that the use of effective Lagrangians gives us a model which is very reasonable and includes the dominant mass-shell effects. This can be seen by noting that the matrix elements we require are evaluated via the equation

, NI , <Y|/*,n(x,*)|a) <YI*P,»(X,0I«>=-7 r> (14)

(7—a)2+mN2

where y is any one of the zero-baryon states required in Eq. (8), a is one of the one-baryon states, and fP,n(x,t) is the designated baryon source. The point to note is that by using effective Lagrangians we are forced to use nucleon-pole dominance of the original matrix ele­ment. We also see that the numerator of Eq. (14) is the usual vertex function for p, n—> 7 + a and realize that the lowest lying structure for this vertex is the pion-plus-nucleon channel in a P1/21/2 partial wave. This channel is notoriously weak and the lowest lying strong effects are at the Roper resonance with a mass slightly below that of two nucleons. Therefore, owing to a fortuitous set of circumstances any structure in the nucleon vertex is reasonably far away. Studied in light of these facts, it seems that this application of the effective Lagrangians is a fairly good first-order ap­proximation to the zero-nucleon-mass vertex functions. Because this mass extrapolation is not too well known and appears to be small we will look elsewhere for a possible explanation for the discrepancy of the sum rule.

We feel that another possible explanation for the error of the sum rule lies in the neglect of strong 5-wave dipion scattering. This effect has appeared in some ex­perimental results8 and been required for some theo­retical calculations.5 In order to determine the effects of the 5-wave pion-pion scattering, we once again resort to effective Lagrangians. Following the case of the p meson as closely as possible we assume

where

£ = MgS7r7rSO' (p) + gSNN^Ti\l/, (15)

8 L. M. Brown and P. Singer, Phys. Rev. Letters 8, 460 (1962); Phys. Rev. 133, B812 (1964); C. Kacser, P. Singer, and T. Truong, ibid. 137, B1605 (1965).

< : ) •

and S is the 5-wave field, and we choose gsirir^gsNN-This will add a factor to the v sum in the form

fx2~-ms2 (n2m) -^x(p)e~*(7r+~p)x. (16)

Selecting nts=4:Q0 MeV, we find that the sum rule predicts gsT7r

2/47r=6.4. This coupling can be used to predict the width of the 5-wave resonance, 1/T0=VS = 120 MeV. These values are not in disagreement with the parameters of the elusive a meson. A more usual labeling of the strong 5-wave scattering is by means of a scattering length. The above parameters predict a scattering length of #0=0.625 F. The previous values available for the scattering length indicate that this value is too low. The accepted values of a0 are approxi­mately 1 F. We point out that the value of a0 is very sensitive to the mass used in the 5-wave field amplitude. This prediction of the 5-wave dipion scattering pa­rameters is sufficiently close to the previously en­countered values that this appears as the strongest possible explanation for the discrepancies of the sum rule. We therefore conclude from this calculation that the assumed commutation rules for the baryon field are probably exact and that the deviations of the sum rule can be corrected by including two effects. The primary correction would be the addition of strong dipion 5-wave scattering. In addition, the effects at zero-mass nucleons would be expected to improve the predictions.

The author would like to acknowledge the con­structive criticism of Professor A. P. Balachandran.

After this work had been completed, the author dis­covered an independent analysis of this same com­mutator under somewhat different circumstances and treated in a different manner.9

9 H. Suura and L. 598 (1966).

M. Simmons, Jr., Phys. Rev. Letters 16,