INCOMPATIBILITY OF THE QUARK-FIELD [U(6) 8 U(6)] COMMUTATION RELATIONS WITH VECTOR MESON DOMINANCE IN nodecay

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1 SLAC-PUB-382 INCOMPATIBILITY OF THE QUARK-FIELD [U(6) 8 U(6)] COMMUTATION RELATIONS WITH VECTOR MESON DOMINANCE IN nodecay George W. Barry The Enrico Fermi Institute and the Department of Physics University of Chicago, Chicago, Illinois J. J. Sakuraif Stanford Linear Accelerator Center Stanford University, Stanford, California ABSTRACT Using the Bjorken limit theorem, we demonstrate that the commutation relations of the free-field quark model [or U(6) 8 U(6) commutation relations] are incompatible with the vector meson dominance of the a0-2y amplitude. The gauge field algebra does not suffer from this difficulty. The present status of vector meson dominance in r decay is also reviewed. Submitted to Phys. Rev. Letters *Work supported in part by the U0 S, Atomic Energy Commission. ton leave from the Enrico Fermi Institute and the Department of Physics, University of Chicago, Chicago, Illinois

2 I If the commutation relations of the vector and axial-vector currents are really part of the basic laws of physics, it is disturbing that as yet there has been no unanimity of opinions on the nature of the commutator between two space components of various current densities. For instance, the free-field quark model [or U(6) X U(6)] leads to 1 (x, 0), jzrn (0) = -2ie2 d( )(z) eopv7 [ (2/3) 3/2.O 1 j3 J57- * 3 5T t- S.T. (1) whereas the same commutator is given in the gauge field algebra2 by (2, 0), jr (0) 3 = 0 + ST., (2) where S. T. stands for the Schwinger terms (which are irrelevant in the present discussion). * We wish to point out that if the success of vector meson dominance in?r decay is not accidental, the commutation relations of the gauge field algebra must be favored. According to Bjorkeny the high energy behavior of the matrix element for can be evaluated as follows: 7r (9)-y(k) + ~4% (q = k + W M =i d xe CLV s -ik*x < 0 1 T (j;i1 (x) j; (0)) 1 7r > I ko--a: I -- 1 ko s d3xe-ii;o;f < 0 0), j; (0) II To > (3) where the limit is to be taken in such a way that c, zl, < and q. are kept finite -2 -

3 as we let lkol cz 1 kb 1 go to infinity. The quark-field commutation relation (1) then gives EoplJTq7- (4) (where ct is the charged pion decay constant numerically sult already obtained by B. -L, Young. 4 equal to 94 MeV), a re- We now note that the behavior (4) is incompatible with the vector meson dom- inance model for r decay, If the r decay amplitude is dominated by p and w (in the sense of Gell-Mann, Sharp and Wagner), 536 we expect + (s-s ) (5) where F(s) s ) is defined by M PV =EpVhT k k AT F(s,s )/m, 6% with s = -k2, s1 = -kf2 and Ap =emz I fp, AU= - (emi/2fy)sin Oy. (7) As I 01 = I 01 1~ goes to infinity withif, %!, < and q. finite, M cly based on (5) goes like l/k%, in contradiction with the behavior (4). In contrast, the commutation relation of the gxuge field algebra is consistent with the high-energy behavior of the vector meson dominance model since Eq. (2) requires the absence of the l/k0 term in the Bjorken limit. Note also that any additional form factor effect one may care to consider in conjunction with the vector-meson dominance model tends to strengthen our argument against the quark-field commutation relations. -3-

4 Next we comment on the paper of Young, 4 who claims to have shown that the observed pion lifetime is in good agreement with the quark-field commutation relation (1). Being aware of the incompatibility between the no-subtraction assumption for F(a,s ) and the quark-field commutation relation, he proceeds to write down a once-subtracted dispersion relation in s and s : F(s,s )-F(O,O)= + higher mass contributions (8) where F(O,O) is directly related to the 7r Met.i me via ~(7 r -.yy) = (1/64n) (9) Comparing (8) with the Bjorken limit (3), he obtains 7r, Wa) g Pw= _ fpfw-yn e 2 + fpc7rmn 6rnE (lob) where m o=mpand A f PPV =A wwy-rr f have been used* and the higher mass contributions have been neglected. At this point, however, Young uses an obsolete value for AW I hp characteristic of unbroken SU(6), viz. AU I hp = - l/3; instead, one should take into account symmetry-breaking effects provided by spectral-function sum rules of the Weinberg type, which yield h w I hp = -0.81/3. Thus we get in the -4-

5 I free-field quark model 10 1 F(os )lquark = (3.9 * 0.4) x 1o-3 (2.6 * 0.4) X 1O-3, 0.25 f 0.06 Wb) if The predictions of the gauge field algebra can be obtained simply by setting % = 0 in Eqs, (loa) and (lob); we then recover the results of the vector meson dominance model (as we must). Numerically IF(0, O)jVMD = (3.3 f 0.4) x 1o-3 (=a) dip - = 0.4OrtO T (1W Eirperimentally, the observed pion lifetime = (0.89 f 0.18) X set 1 corresponds to IWWIexp = (3.3 f 0.4) x 1o-3 (13) which coincides exactly with (12a). Thus we see that the gauge field algebra (or the vector meson dominance model) is in even better agreement 7r lifetime. 11 with the observed One of us (J. J. Sakurai) wishes to thank Professors W. K. H, Panofsky and S. D. Drell for their hospitality at the Stanford Linear Accelerator Center, -5 -

6 I REFERENCES 1. R. P. Feynman, M. Gell-Mann and G. Zweig, Phys. Rev. Letters l.3-, 678 (1964). 2. T. D. Lee, S. Weinberg and B. Zumino, Phys. Rev. Letters l8-, 1029 (1967). 3. J. D. Bjorken, Phys. Rev. 148, 1467 (1966), 4. B.-L. Young, Phys. Rev. 161, 1615 (1967). 5. M. Gell-Mann, D. Sharp and W. G. Wagner, Phys. Rev. Letters 5, 216 (1962). See also M. Gell-Mann and F. Zachariasen, Phys. Rev. 1.24, 953 (1961). 6. We ignore the $ meson contribution because the p+7i coupling appears to be anomalously weak. 7. Our vector-meson coupling constants fp and fy are normalized as in the papers of J. J. Sakurai [Annals of Phys. 11, 1(1960)] and N. M. Kroll, T. D. Lee and B. Zumino [Phys. Rev. 157, 1376 (1967)]. The constants gpwr, fpynand fwyrr are all dimensionless since we have taken out the factor l/n+ 8. It may be mentioned that once we abandon vector meson dominance, the re- lation hpfp ~ =h f w wy7i is more difficult to justi@- 9. R. J. Oakesand 5, J. Sakurai, Phys, Rev. Letters 19, 1266 (1967); and references therein. 10. For numerical estimates we use p 47r = 2.6 and fiyn 4n = 0.15cY f f l I The major contribution to the errors in F(O,O) and gp we comes from T(w -n r). 11. Models based on integrally charged triplets [such as that of K. Johnson, F, E. Low and I-I. Suura, Phys. Rev. Letters l8-, 1224 (1967)] are in worse agreement than the quark model.