Resolution of apparent SU(6) violation in vector meson magnetic dipole decays

Volume 63B, number 1

PHYSICS LETTERS

5 July 1976

R E S O L U T I O N O F A P P A R E N T SU(6) V I O L A T I O N IN VECTOR MESON MAGNETIC DIPOLE DECAYS T. BARNES Lauritsen-Downs Lab. of Physics, Pasadena, Ca. 91125, USA Received 18 May 1976 We correct the M1 vector meson electromagneticdecays for recoil effects of the final pseudoscalar and for several smaller effects, using results from the FKR and bag models of meson structure. The apparent SU(6) violations in these decays seen in naive SU(6) are then found to be artifacts of recoil neglect, with the single exception p- ~ nO'. Recently there has been a great deal of interest in the apparent breaking of SU(6) in the magnetic dipole electromagnetic decays of the S-wave vector mesons [ 1 - 3 ] . The reported violations are in the SU(6) predictions of the transition moments between the following vectors and pseudoscalars: 6o ~ nov, p - -~ 7r-7, ~0~ 7/7, and K *° -~ K°7. The problem is as follows; in nonrelativistic unbroken SU(6) one conventionally assumes that the non-vanishing part of the magnetic moment operator between quarks with no orbital angular momentum is of the form [4]

It = gt.t ~ elSq = la ~ eq•q q e q e

(1)

Where g is the quark gyromagnetic ratio (assumed = 2) and/1 is a universal scale which is fixed by/a(proton) = #. In the long photon wavelength approximation the magnetic dipole transition rate between a vector and a pseudoscalar meson is given by [4] 43 _ ,illtl0-, f)l 2 -=~k.rlPtl 43 2 (2) P(1- ~ 0 - 3 , ) =~k~l(1 where Pt is the transition moment between the vector and pseudoscalar. Implicit in these equations are two important assumptions: (A1) There is a universal scale for the magnetic moments of all hadrons (1), and (A2) the photon wavelength may be assumed large compared to the scale of the hadron, which leads to the form (2) for the M1 decay rate. To go further two additional assumptions are conventionally made; (A3) the amplitude to find the mesons in states containing gluons or other exotic configurations may be neglected, and (A4) recoil of the final state pseudoscalar may be neglected. Assumptions (A1), (A2), and (A4) have each been checked in at least one model, and all

have been found to be invalid in cases of experimental interest. The assumption (A2) which is k~ ~ a -1 is certainly invalid, as in our cases we have k.ra ~" 1.5-2. The effect of not making this assumption has been investigated in the bag model [5], where it is found that the rate predicted for magnetic dipole decays is suppressed by a factor of from 0.75 (K *° -* K°3', assuming k,ra ~ 1.5) to 0;6 (w ~ 7r°7 and p - -~ zr-7, assuming k.ra "~ 2). This suppression is important in calculating the overall scale of the rates correctly, although it will only correct the SU(6) predictions for the ratios of decay rates by <~ 25%. The reported violations of SU(6) in these ratios of rates is significantly larger than 25%, so we shall continue to assume k.ra "~ 1 and concern ourselves only with the ratios. Assumption (A3) has been tested in a simple model for one special case [5], F(p- ~ n - 7 ) / P ( ~ -~ 7r3'), and it has been shown that mixing of the Iqr:t) and lowest-lying Iq?tA) state to lowest order in the quarkquark gluon coupling does not affect this ratio. The validity of assumption (A3) has not been checked on other reactions. Assumption (A4) was investigated by Feynman, Kislinger and Ravndal [6] in their relativistic harmonic oscillator quark model, and was found to be especially bad for m(1 -)~re(O-) much larger than unity. In particular, they find with their harmonic oscillator quark wave functions that the inclusion of recoil in computing the transition moment matrix element modifies the SU(6) recoilless decay rate as follows [7] :

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Volume 63B, number 1

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P ( 1 - -~ 0-3' with recoil)

(3) =F" 2m(l_~ ] 3 17(1- ~ 0-~,neglecting recoil). Lm(1-) +m(0-)J We see that in the limit m(1 - ) >> m ( 0 - ) this enhances the naive recoilless SU(6) prediction by a factor of 8. Since there is a large spread of m(1-)/m(O-) in the measured decays it is obvious that recoil effects must be included and that assumption (A2) can result in errors for ratios of rates of an o~der of magnitude. Assumption (A1) is clearly invalid in that it predicts #(A)/#(P) = -0.33, which is experimentally known to be - 0 . 2 4 -+ 0.02. The origin of the violations of the universal magnetic moment scale is presumably the mass of the strange quark, which suppresses the magnitude of the strange quark moment relative to the u, d quark moments. This speculation is supported by a bag model calculation which predicts (/aA//ap) = -0.23 ± 0.02 [8] with no free parameters. (The uncertainty is due to an arbitrariness in the choice of measured baryon parameters used in obtaining values for the revelant bag model parameters, B o and ms). The effect of the strange quark mass in the magnetic dipole vector meson decays is treated by assuming a different moment for each quark, which modifies eq. (1); /a = e ~ / a q Oq.

(4)

q

5 July 1976

We shall eventually assume/a u = --2/ad, since the prediction/a N//ap = - 2 / 3 is well satisfied and the isomultiplet splittings make it clear that m u - m d is small compared to m u - m s and m d - m s. Reworking the transition moments with (4) gives us a rate suppression factor which is a function of ~s//ad) for each reaction: r s = 1 for p - -+ n-~, and co ~ 7ro~/, r s = ~(1 +/.ts/#d) 2 for K *° ~ K°7, and r s = 0/s//ad) 2 for ~0 777. If'we take #s/#d = 0.76 for these mesons, which follows from the zeroth order bag model with Blo/4= 140 MeV and m s = 270 MeV. These factors become 1 (pO and co), 0.77 (K*°), and 0.58 (~o) respectively. Now that we have estimates of the corrections to the magnetic dipole decay rate ratios after the invalid assumptions are discarded we may compare our predictions with experiment. We choose to normalize the predicted rates to the rate F(~o ~ rr°7) = 870 -+ 80 keV [9] which is known with the greatest relative certainty. The experimental results and the SU(6) predictions with and without corrections for assumptions (A1) and (A4) are shown in table 1. We note that the large SU(6) breaking apparent in the ~0~ 7?7 and K *° ~ K°~' decays relative to the w n°7 decay is largely due to quark recoil effects in the model of Feynman, Kislinger, and Ravndal, and that the relative transition matrix elements are SU(6) invariant. The only experiment that shows significant SU(6) breaking in IOTI2 is the p - ~ n-~' measure-

Table 1 Effect of correcting invalid quark model assumptions on predicted M1 decay rates. (We take 0 s = - 1 0 o and r s = 0.76. S and C are sin 0 s and cos 0s).

reaction ~o ~ ,ro~ p-~r 7 K*°'-~K°'Y ~°'*rD'

[ I#TI2 ~ \l~rrocol 2 ]

F(m'~m"/) KeV

Naive SU(6)

(A1) corrected

(A4) corrected

(A1,4) corrected

experiment 870 ± 80 [9] 3 5 ± 10[10]

~- 1 1/9

870 ± 80 input for all cases 91±8.3 ~-

88±8.1

.-

49~c l +2r s ~J

200 + 19

160 + 15

87 ± 8.0

68±6.2

75+

35 [11 ]

160±15

98±9.0

73±6.7

43+4.0

65±

15[ 11 ]

~(I+x/2-SC-1S2)rB 1

s

co--+~

T~(1-2x/~SC+S

2)

5.9±0.55

+-

1.9 ± 0 . 1 8

~

< 50[9]

p~rr),

~-(1-2x/~SC+S

2)

45±4.2

~

15±1.3

~

<160112]

n ' ~ to7

2"~(1 + x/2SC - 1S2)

3.4 ± 0.32

~-

0.92 + 0.085

,,-

<

r/' ~ P3'

} ( 1 + x/~'SC - 1 S 2 )

38 + 3.5

,--

1 0 ± 0.96

~

<270[9]

9 ( 2 - rs) 2

52±4.8

22±2.0

34±3.1

<

K*+~ K+"/

66

80±7.3

80 [9]

8019]

Volume 63B, number 1

PHYSICS LETTERS

ment, which is a very difficult experiment involving p - production from a rr- in a nuclear Coulomb field. Since there appears to have been some uncertainty regarding the phase angle of the photo- and direct- p production through the nucleus, which critically affects this rate, we suggest that this experiment may well be in error. As the other apparent deviations from SU(6) symmetry may be due to recoil effects it is important that this experiment be repeated. The remaining decays we predict to be far below current experimental limits with the exception of K *+ ~ K+3, , which is not a particularly good test of the assumptions (A1) and (A4) in that removing them corrects the SU(6) prediction in opposite directions. An r?' total width experiment would be an unambiguous test o f our results, as we predict here Ftot(r/' ) 35 keV while uncorrected SU(6) predicts Ptot(r/') "" 125 keV. (We get these numbers by dividing the partial rate 77' ~ p~, by the observed branching ratio r(rT' -~ p3,)/r07' -~ all) = 0.304 - 0.017 [13] ). In lieu of such a difficult experiment the best test of our results is a repetition of the p - ~ rr-3, experiment, which is the only measured decay for which we do not obtain reasonable agreement.

5 July 1976

I would like to thank R.P. Feynman for several helpful discussions and in particular for pointing out the importance of recoil corrections in these decays.

References [1] D.H. Boal et al., Univ. of Toronto preprint, May 1975. [2] D.H. Boal et al., Univ. of Toronto preprint M5S 1A7, July 1975. [3] B.J. Edwards et al., Phys. Rev. Lett. 36 (1976) 241. [4] J.J. Kokkedee, The quark model (1969) pp 64-6. [5] T. Barnes, unpublished calculation (relevant work is in ref. [8]). [6] R.P. Feynmann et al., Phys. Rev. D3 (1971) 2706. [7] R.P. Feynmann, private communication. [8] T. Barnes, Ph.D. thesis, CalTech (1976) (unpublished). [9] V. Chaloupka et al., Phys. Lett. 50 B (1974) 1. [10] B. Gobbi et al., Phys. Rev. Lett. 33 (1974) 1450. [ 11] C. Bemporad, Proe. Conf. on Lepton and photon interactions at high energies, SLAC (Aug. 1975). [121 M.E. Nordberg et al., Phys. Lett. 51B (1974) 106. [13] Review of Particle Properties, Rev. Mod. Phys. 48 (April 1976).

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