LETTERE AL NUOVO CIMENT0 VOL. 16, N. 16 14 Agosto 1976

Effective Coupling Strengths of Quarks and Octet-Coloured Gluons.

S. IWAO

Depar tmen t o] Phys ics , College o/ L iberal A r t s - K a n a z a w a Univers i ty - K a n a z a w a

(ricevuto il 25 Maggio 1976)

The family of ~-particles (the so-called charmoniums)(1) showed us a distinct character of them compared to the ordinary hadrons classified by the S U a symmetry. The apparently large mass difference between ~b and ordinary hadrons gave us the evidence of the large symmetry breaking of S U 4 (2,3). In spite of this, the pseudoscalar meson and free-quark mass relation derived in the lowest order in chiral S U 4 breaking

(1) m ~ 2 2 2 mK m D m F

m u-~- m d m u ~ - m s m u-~ m c m~-~- mr

seems to be consistent with those estimated from the constituent group U 6 incorporated in the null-plane quark model of mesons (4). No definitive information on D- and F-mesons is available at this time, but we shall assume the above relation as working hypothesis in our discussion. The numerical values of the free-quark masses may be chosen to be mu ~ m a = 2.52, m~ = 62.4 and mc = 1228MeV, by defining the last value to that utilized in one of the charmonium models (5). This quanti ty is not much different from tha t estimated from the upper limit of the K L ~ ~+~- decay and K ~ - - K s mass dif- ference (6), me = 1.5 to a few GeV, or roughly to �89 The observed mass of the hadron is considered to be the sum of the effective masses of the constituent (valence) quarks, arising from the exchange of octet-coloured massless gluons obeying non- abelian gauge fields. The effective masses of the u- and d-quarks in the nucleon are, e.g., mu* ~_ m d-N_ ~m'~c. These observations suggest that the asymptotic freedom (7) is more or less well satisfied for the bound states of co and possibly for the gluon- charmed-quark couplings in the decay and production of the ~-particle, but not for ordinary hadrons.

(1) G . J . FELDMAN: The ~ resonances and searches lor new part icles , in t he F i l t h ln ternat ional Con]erence on Neu t r ino Science, Balatoli~red, Hungary , J u n e 1975, SLAC-PUB-1647 (September 1975); N e w particle searches and discoveries at S P E A R , in the 1975 Internat ional S y m p o s i u m on Lepton and Photon Interact ions at High Energies, S tanlord, C a l . , . A u g u s t 1975, SLAC-PUB-1662 (Octobcr 1975). (3) • . K . GAILLARD, ]3. W. LEE and J. L. ROSNER: Rev. Mod. P h y s . , 47, 277 (1975). (a) V. S. MATHUR, S. OKUBO and S. ]3ORCHARDT: P h y s . Rev. 1), l l , 2572 (1975). (~) H. LEUTWYLER: P h y s . Le f t . , 48 ]3, 431 (1974). (5) J . F. GUNION and R. S. ~VILLEY: P h y s . Rev. D , 12, 174 (1975). (6) ]3. VY'. LEE: Review of gauge theories, prep r in t , FERMILAB-Conf-75172-THY (September 1975). (~) D. J . GROSS and F. WILCZEK: P h y s . Rev . Le f t . , 30, 1343 (1973); P h y s . Rev. D, 8, 3633 (1973); ~I. D. POLITZER: P h y s . Rev. Le t t . , 30, 1346 (1973).

486

EFFECTIVE COUPLING STRENGTHS OF QUARKS AND OCTET-COLOURED GLUONS ~ 8 7

Last April, OKCN (s) gave an interesting lecture on the role of coloured-octet gluons in relation to new particles and quarks up to that time in a great detail.

We want to pursue, first of all, the effective coupling constants of quark-gluon couplings by taking into account the weakness (an approximate asymptotic freedom) of the c-quark-gluon coupling, then to apply the, results to the relative, production rate of neutral vector mesons in the inclusive production, to some of the hadronic decays of the if-particle and to other relevant problems. We hope that an elaboration of the method will admit the dynamical t reatment of various problems at finite energies.

Let us define the strong current associated with the octet-colour gluons transforming as Zi (a member of S U 3 generators) by

(2) jstro,g (X ~ - , t~i , , = g , u y # z i u q- g d d y ~ z i d q- g, gypZiS q- gcF:ytzZie

where u, d, s, c are the quark fields, gk (k = u, d, s, e) represent the effective coupling constants and yzt (It = l, 2, 3, 4) are the Dirae matrices. The g~:'s may be the tune- tions of the momentum transfer square between initial and final quarks. However, for finite-energy processes we assume that their t- (or s-) dependence may be neglected and assumed to be contained effectively in their definition. An approximate equality g, ~--ga is assumed throughout the paper.

According to the quark model the relative magnitude of the decay amplitudes for an electron-positron pair of neutral vector mesons becomes

(3) M(p~ ~ e+e-) : M(co ~ e ~e-): M(~ -+ e+e-): M( 9 -~- e+e -) =

1 t (Q~ - Q,~) : ~ (Qu + Qa):Q, :Q~,

V Z

where Qk's are the electric charges of the composite quarks. In order to become con- sistent with the experimental values, the last amplitude should be reduced by ~ 500/0 . IIowever, it is difficult to find out the reason of the reduction due to the abelian character of the photon field. We shall follow the conventional choice in this paper.

In order to determine the quark-gluon coupling constants we shall use the relative decay rate of a leptonic pair and direct-hadronie decays of neutral vector mesons. APPELQUIST and POLITZER (0) discussed, first, this problem in analogy with ortho- positronium decays, According to them the relevant quanti ty for the ~b-meson becomes

F(9-~ e+e - ) lSz a 2 (4)

3 ~ F(~-+ direct hadrons) 5(.~- -- 9) ao

2 where c~o--gd4~r. A similar relation may immediately be written down for F(o-->e+e-) /F(o--+ 3~) and F(~-~ e+e-)/F(~-+ 37:)in referring to eqs. (3) and (4), accord- ing to OKUN. Here, we have interpreted the ~-> 3r: in terms of three-gluon exchange mechanism, although this process is not forbidden by the so-called Okubo-Zweig-Iizuka (0ZI) rule. Our approach, however, may admit ~ unified treatment of strong interac- tions and possibly including other interactions.

(s) L . B . OK[Yx: New particles anal quarks (in Russ ian) , problems o] elementary particle physics, Lec tures de l ive red a t The Eighth Session o] Spring School o] Experimenlal and Theoretical Physics, Irerevan, (April 1975), p. 9. (~) T. APP~,H~QUI~T and H. D. POLITZE[~: Phys. [~ev. LetL, 34, 43 (/975).

4 8 8 s. i w A o

Before making the numerical estimate it is important to separate out the three- gluon contributing widths from the total hadronic decays of the ~-meson. We should, especially, separate out the direct photonic decays, photon associated decays etc. which should not be counted as the thrce-gluon contributions. According to the previous analyses(1~ we estimate them as: ~ - ~ , - q ' , E § 20%; ~-->lepton pairs, 14%; ~-~ yv--> hadrons, 20% and ~-~ O~(2.8)q-y, 10%. The remaining direct hadronic decays are then estimated to be ~45~ . Making use of the above result and information on other vector-meson decays we find

(5) a~ --~ ud ~ 1.0I, a~ ~ 0.441 and ~ ~ 0.165.

We think that the last value is about a factor 1.5 higher than the true one, but shall use the above result in the following discussions.

The relative magnitude of these three constants becomes ~:g~:% = 16:7.0:2.6. I t is interesting to notice that the ratio obtained here has a common feature to that obtained before from a completely different approach (~2), where the corresponding ratio was 16:8.6:1.1. There is no direct relevance between the two approaches, but information on the relative weakness of the c-quark-gluon coupling is essential in t w o c a s e s .

a)

~TI c

Fig. 1. - a) Single ne u t r a l vec to r -meson product ion in associat ion wi th hadrons in a § b collision. Do t t ed l ines ind ica te gluons a nd t h e l i n e w i th a r rows me a n s q u a r k l i n e . Sol id l incs wi thou t a r row arc for hadrons . S imi la r definition appl ies to t he r e m a i n i n g d iag rams , b) Single n eu t r a l sca la r -meson produc- t i on in associat ion wi th hadrons in a q-b collision.

Let us discuss, first, the production ratio of the neutral vector mesons and pseudo- scalar mesons through the diagrams indicated in fig. la) and b), respectively. The former processes will proceed through three gluons, while the latter through two gluons. The relative production rate of (~, ~, ~ and "q, ~r are predicted, respectively, by

(6)

a n d

(7)

a(co):a(~):a(+)-----~=.~.~3" 3. 3 = 2 3 3 : 2 0 : 1

~(~) : . (~o) = ~i :~o = 3s :

at an appropriate set of finite energies.

(lo) G. J . FELDMAN a nd M. L. PERt : Phys . Rep. , ]9 C, 233 (1975). (11) F. J . GIL.~[AN : N e w particle spectroscopy and decays, i nv i t ed t a lk p resen ted a t the Orbis Scientiac 1976, Unive r s i t y of Miami , Coral Gables , F la . , SLAC-PUB-1720 ( J a n u a r y 1976). (12) S. IwAo: Left . Nuovo Cimen~o, 15, 331 (1976).

EFFECTIVE COUPLING STICENGTHS OF QUARKS AND OCTET-COLOURED GLUONS ~ 9

Experimentally, we know the photoproduction of neutral vector mesons at various energies (13-16). I t is questionable to compare the production cross-sections of particles with large mass difference at some finite energies. We shall, however, assume that the following sets of data will be appropriate in our comparison: co-production for E v -- (5.0--6.3) GeV, ~-production at E v -- 9.3 GeV (this is an average energy) and ~-production at Eu -- 13.5 GeV. These energies are not high enough compared to the masses of the produced mesons but somewhat above the production threshold for each of them. The experimental ratio becomes 692:86:1 corresponding to the cross-sections 3.6- 10 -3, 0.445- 10 -3 and 0.52. l0 5 rob, respectively. These choices, of course, do not justify our prediction. Notice, however, that the reduction of a~ by a factor of 3 will givc a better explanation. In any case the numerical values should not be taken too seriously.

Next interesting examples on the similar line are the test of the hadronic-decay branching ratios of the ~-particlc. We shall consider two examples: one of them is the relative rate of the decays ~-+ 3~ and ~ KKr~ and the other is that discussed by 0KUBO (16), v i z . , ~ - > ~ and ~-> conn. All these are forbidden decays in the sense of the 0ZI rule. Typical graphs are shown in fig. 2a) and b). In the assumed pictures the ratios become

F(0-~ K*K---~ KK~) a~ - - 2 ~ 0 . 2 , (8) F(~--> pr:-> an) ~.

and

r (~ -~ ~ ) ~ ( 9 ) / ' ( + ~ - o ) n n ) - - ~z,.,3 - - 0 . 1 .

The experimental values for the first and the second rates (1,11,16) is about ~ 0.4 and 0.2 • 0.1, respectively. We consider that these may bc counted good evidences to carry out the gluon dynamics. We need many more examples in order to arrive at the solid basis of the model.

~ K* / ~

a) b)

Fig. 2. - a ) ~ d e c a y i n to K* a n d K . b) ~ d e c a y i n t o t~ a n d ~.

(in) W. STRUCZINSKI, P . DITTMANN, V. ECKARDT, 1 ). JOOS, A. LADAGE, ~-I. MEYER, D. NOTZ, G. HENTS- CHEL, J . KNOBLOCH, E. t~ABE, n . TAUREG, M. GRIMM, I . ])ERADO, P. SCttACIIT a n d R . ~[EINKE; S t u d y o] photoproduction on hydrogen in streamer chamber with tagged photons ]or 1.6 GeV < E y < 6.3 GeV, in Topological and Reaction Cross-Sections, D E S Y 75/36 (Sep t embe r 1975). (la) A. SILVERMAN : Review O/high-energy photoproduetion, a t a l k g i v e n a t t h e V l I Internath~nal S y m p o s i u m on Lepton and Photon hderact ions at High Energies, Augus t 1975, CLNS-320 (Oc tobe r 1975). (1~) T. NASII, A. BELOUSOV, B. (JOVORKOV, D. O. CALDWELL, J . 1 ). CU)LkLAT, A. M. EISNER, R . J . MOR- RISON, F. V. MURPHY, S. J . YELLIN, P. J . Ds R . ~r EGLOFF, G. LUSTE a n d J . D. PRENTICE: Measurement o] J /~(3100) photoproduetion in deuter ium at 5 5 G e V , F E R M I L A B - P u b - 7 6 / 2 5 - E X P (March 1976). (~6) S. OKUBO: P h y s . R~:V. Le f t . , 36, 117 (1976).

4 9 0 s. IWAO

As a final comment , the value ar = 0.165 (or a s l ightly smaller than this) can still be used as an explana t ion of the enhancement of A I = �89 ampl i tude in K-meson de- cays. The effective Hami l ton ian of this ampl i tude was in t roduced as a cross t e rm of left- and r igh t -handed weak currents by making use of the heaviness of the mass of the cha rmed quark (17). The necessary magn i tude of the enhancement is known in a cer- ta in uni t (18). In order to make the quan t i t a t i ve t r e a t m e n t we need a fur ther assump- t ion on quark-gluon dynamics so we shall pos tpone a detai led account in future .

In this paper we picked up only some favorable examples. The single b-product ion m a y be al lowed by the quark-gluon model wi thou t worry ing about the OZI rule. There is an exper imenta l tes t on this p rob lem in referring to the ~-meson product ions (~9). Thus, the comple te s tudy of the g luon-quark dynamics requires to clarify how the O Z I - and g luon-qua rk -mechan i sms are re la ted in the case t h a t the former is not forbidden.

We wan t to emphasize once more t h a t the proper account of a sympto t i c f reedom (weakness) for the b-part icle fami ly in re la t ion to o ther hadronic states is one of the keys for the quark-gluon dynamics .

(1~) H . FRITZSCtI a n d P . MINKOWSKI: Phys. Left., 61 B, 275 (1976). (1,) M. A. A ~ I E D a n d G. G. R o s s : Phys. Left., 6 1 B , 287 (1976). (~l) R . A . DONALD, D. N. EDWARDS, ~r I-IouLDEN, R . WAKEFORD a n 4 W.WILLIAMS: Phys. Left., 61 B, 210 (1976); V. BLOBEL, H . FESEFELDT, I-1. FRANZ, A. HOFMANN, CH. JETTER, I-I. NEUMANN, B. SCHRODER a n d P . SODING: Phys. Left., 59 B, 88 (1975).