T is well known that many people found difficulty in reconciling the SU(4) or SU(3) symmetry(l-5', vector meson dominance model, the Okubo-Zweig-Iizuka (OZI) ...
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VOL. 18, NO. 1
CHINESE JOURNAL OF PHYSICS
SPRING, 1980
Radiative Decays of Vector Mesons C H I E N - ER L EE (+gz) and Y u - H U A LEE (+-H$) Department of Physics, National Cheng Kung Universiiy Tainan, Taiwan, Republic of China
(Received: November 24, 1979) A simple scheme with single parameter for the description of the radiative decays of all vector mesons is proposed. The prediction is found to be reasonably good, except possibly for (ll--r~r. _
I. INTRODUCTION T is well known that many people found difficulty in reconciling the SU(4) or SU(3) symmetry(l-5’,
II vector meson dominance model, the Okubo-Zweig-Iizuka (OZI) rule and the quadratic mass mixing scheme with the radiative decay data of vector mesons. In struggling out of this dilemma, some people introduce more parameters@*6), or break unitary symmetry”) in the VPV coupling constants, or relax the OZI rule(‘) and discard the SU(4) quadratic mass mixing scheme. In our present paper, we try to find a way to recoile the radiative decay data of vector mesons with all the above mentioned simple schemes. That.is, in analyzing the radiative decays of vector mesons, we assume the following: (1) SU(4) symmetry which is beoken by physical mass and mixing only. (2) SU(4) quadratic mass mixing scheme. (3) The OZI rule which is broken by mixing only. (4) The vector meson dominance model. The basic question is this: what is the most suitable invariant coupling constants which would best reconcile the data with symmetry ? In the radiative decays of vector mesons, there are six types of invariant coupling constants: The universal photon-vector-meson coupling constant, and five invariant VPV coupling constants which are g 151515, g11515, 91511% Sl5151 and gill where gy2y3y1 denotes the SU(4) invariant coupling constants with Yi’s representing the multiplicity of the SU(4) irreducible representation. Owing to charge conjugation invariance, g151sl, should be of the symmetric type, denoted by gS. Then through the OZI Iule, we can relate (*I all other invariant VPV coupling constants to the symmetric-type coupling constants gs. Therefore we have only two invariant coupling (1) B. J. Edwards and A. N. Kamal, Phys. Rev. Letters 36, 241 (1976). (2) D. H. Boal, R. H. Graham and J. W. Moffat, Phys. Rev. Letters 36, 714 (1976). (3) P. J. O’Donnell, Phys. Rev. Letters 36, 177 (1976). (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (1%
G. Gounaris, Phys. Letters 63B. 307 (1976). D.P. Roy, Phys. Letter 63B, 324 (1976). Seiji, Okubo, “Radiative Decays of Vector Mensons and VMD with Corrected Photon-Vector Coupling ”. Hokkaido University Report. No. 002 (1978). J. Randa and A. Donnachie, Nucl. Phys. B125, 303 (1977). Chien-Er Lee and Jen-Fa Min, Chinese J. Phys. 14, 189 (1978). V. Chaloupka et al., Phys. Letters 50B, 109 (1974). B. Gobbi et al., Phys. Rev. Letters 33, 1450 (1974). W. C. Carithers et al., Phys. Rev. Letters 35, 345 (1975). D. E. Andrews er al., Phys. Rev. Letters 38, 198 (1977). C. Bemporad, Proc. SLAG Lepton Photon Conf. (1975). B. H. Wiik, Invited talk, Tbilisi Conf. DESY 76/52 (1976). B. II. Wiik and G. Wolf, DESY Report No. DESY 77/01. 1
2
RADIATIVE DECAYS OF VECTOR MESONS
constants, the universal photon-vector coupling constant and g,, left. We find that via quark model, quadratic mass mixing’ formala and current field identy, the universal photon-vector-meson coupling constant can be naturally defined. The SU(4) invariant gs is extracted from the VPV coupling constants which although differ from the usual ones only slightly in definition, but has important effect on the decay rates of the vector mesons. Therefore, via vector meson dominance, we have only one adjustable parameter in our analysis. This parameter is just the product of the universal photonvector meson coupling constant and the SU(4) invariant symmetric type coupling constant. The fitting of all V-G7 data, including old and charmed particles, with this single parameter is found to be reasonably good. The only serious difficulty in our scheme is the decay #-+vr which is about one order of magnitude larger. But we think our scheme is still encouraging since so far no other scheme can accommodate all assumptions used here without catastrophic contradiction with data, and the data for 4-t~~ is still uncertain. In section 2, we present the universal photon-vector meson coupling constant. In section 3, we prestent the vector meson dominance model with our choice of the SU(4) invariant coupling constants In section 4, we present our calculations. We conclude our work and make some discussions in the last section. II. UNIVERSAL PHOTON-VECTOR MESON COUPLING CONSTANT The photon-vector meson coupling constant is defined by the following expression (2#‘* -4~/fv(~))~,(q, 4,
(1)
where V denotes the vector meson, and d is the helicity. Under the quark model of four flavours, the electromagnetic current can be writen as jp=e(f ar,u-fd;,d-f
sr,s+-~-- Cyc > .
(2)
We may express the above expression in terms of the renormalized particle fields. Since @=(uz-da)/,/T, and the quadratic mass mixing@) implies /4 w 0 (I
= (
-0.082
-0.992
0.990
-0.093
0.118
0.090
ua+dd
0.100 7 -0.110 ss . 0.989 )( CC i
(3)
We obtain the following current-field identity j,=eF(0.707p,+0.378~,+0.191~,+0.657~,),
(4) where F has dimension of mass squared and is chosen as the universal photon-vector meson coupling constant in our analysis. Substitute the expression (4) into the expression (I), and compare it with the following expression for a renormalized vector field (2&Y “&/#(%A>.
(5)
We can express the photon-vector meson coupling constant l/fp(q”) in term of the universal F a s follows +I.183 F(qz), P hF=0.363 F(@),
(6) T;?T-0.312 F(qZ), *
-j;/2i -0.069 F(qa).
CHIEN-ER LEE AND YU-HUA LEE
3
Where F(@) has dimension GeV*. For real photon emission, we set 9-O. We note that this choice of F, defined by the expression (4), as the universal photon-vector meson coupling constant is reasonable as can be seen in the following: The expression (6) implies
=
LA : QL- : L&) : /r;fi3.79 : 1.16 : 1 : 0.22,
(7)
while the data from V+e’e+ give
= 2.96 : 1.15 : 1
‘b
:
1.24.
(8)
Since the theoretical ratio (7) is predicted at equal @, and the masses of p, 4 and o mesons are close to each other, we see that the fitting is reasonanly good. The large difference in @ meson, simply implies that the photon-vector meson coupling constant is indeed qz dependent. We note that our universal photon-vector meson coupling is different from the usual one by a physical mass squared and is also used by Okubo@) in a different context. In our scheme, this choice brings down the p decay width considerably.
III. VECTOR MESON DOMINANCE It is well known that under vector meson dominance model, the coupling constant for a vector meson decaying into a pseudoscalar meson and a photon is writen as glPr= 3 frto)
(1
LbPP’.
(9)
where $=O is used for real photon, and gvp+ is the coupling constant for the vector-pseudoscalarvector meson vertex, which represents the role played by the strong interaction in the radiative transition between vector and pseudoscalar mesons. This strong coupling constant used in our analysis is defined by the following expression (2qo 2~: 2~0)“’ /
. LJVPT =le -2 - Eppo T,(q, 1’) Ev(P, 1) Pp4.. w
C(
(11)
The above defined coupling constant gyp, is different from the usual one by a factor I/m;. Under this choice of coupling constant, the decay width of Y+ Pr is found to be T(V+P++
s4Pr&(l--gg
(12)
where a! is the fine structue constant and mp(my) is the mass of the pseudoscalar (vector) meson. We not that gYPr has a dimension of mass. .
IV. CALCULATIONS In our calculation of the decay width for a vector meson decays into a pseudoscalar meson and a photon, we first express the photon-vector meson coupling constant l/MO) in term of the universal
4
RADIATIVE DECAYS OF VECTOR MESONS
photon-vector meson coupling constant P(O), as given in the expression (6). We then extract the SU(4) invariant coupling constant by dividing the vector-pseudoscalar-vector coupling constant by the appropriate SU(4) Clebsch-Gordan coefficient. The SU(4) invariant coupling constant for JJ~ 4 J+,J+ is denoted by gvawsvl with Vi’s as the multiplicity of the irreducible representations. Owing to charge . . . conjugation Invariance, glsl,15 should be of the symmetric type. Hereafter we write glsl~ls-g,. By imposing the OZI rule on the SU(4) symmetry, the five SU(4) invariant coupling constants become related to each other. The relations@) are 1
g111-g11515-g15115 - - - g151sr
1
(13)
a--
/15
,/ 6 ‘I’
Therefore, we have only one independent SU(4) invariant coupling constant gS. Then through the
vector meson dominant model expressed by the equation (9), all vector-pseudoscalar-photon coupling constants are expressed in term of a single parameter which is the product of the universal photonvector coupling constant and the SU(4) invariant coupling constant, and are given in the Table 1. In expressing gVpr in term of P(O) g,, the quadratic mass mixing is used. For vector mesons, it is given by the expression@) 0 .,762 = 0.647 -0.006 .o i Q 9 w
0.470 -0.472 0.782
- 0 . 5 0 3 77; 0.598 r]:, , 0 . 6 2!i 3 r]: )
(14)
0 . 1 4 2 71~ 0.806 71s . 0 . 5 7 jt 5 71 )
(13
and for pseudoscalar mesons, the expression is 0.984 -0.178 0.006 ii! 71, 71 pt
=
-0.107 -0.565 0.818
We then calculate the single parameter P(O)g, by using the decay width expression (12) with the input r(o + n’r) = 870 Kev. It gives The decay widths of V-+P7
0.676
870 (Input)
-
870~1~80
I
Reference
-0.166
52.805
35-110
9 10
0.415
104.804
75*35
11
-0.268
43.751
