L~TTF~R:E AL NUOVO CIMEI~TO VOL. 17, N. 6 9 0t tobre 1976

Baryon-Loop Model for Radiative Decays of Mesons.

A. BRAMON and J. A. I~A]~ROC~IA

Departamento de .Fisica Tedrica, _Facultad de Ciencias - Valencia

(ricevuto 1'11 Maggio 1976)

The experimental results concerning the radiative decays of pseudoscalar and vector mesons ( P ~ V 7 and V--~ P•) have been considered, during the last years, to be in good agreement with the predictions of the simple quark model (1,2). However, recent measurements(3,4) of the radiative transitions p-~u~ and K * - ~ K u performed by means of the Primakoff effect, strongly contradict the present validity of this simple scheme. This is clearly shown in the central columns of table I, where the available experimental information for some radiative decays can be compared with the typical predictions of the quark model (2).

In order to account for the new data, several models have already been proposed. Some of them are based on modifications of the original quark scheme (5,e), whereas other authors (7) have preferred to reconsider a more conventional SU3 approach. Par- tial success has certainly been achieved but the general situation seems to be still rather unsatisfactory.

The aim of this paper is to a present a model, based on the original octet model of S U 3 and without any mention to quarks, which is consistent with the presently available data. The main idea of our model was originally proposed by ST~INB~RG~R (8) in 1949 and has been reconsidered, more recently, by several authors (9,~o). Essentially, it consists in the substi tution of the tr iangular loop of quark lines generally considered to describe the Pu165 PV~, and PVV vertices by analogous baryon loops. In this way, the baryon-boson coupling constants and the baryon masses act as the essential para- meters of the model. Since they are known or can be evaluated in independent processes, the model results, at least in principle, free of intrinsic arbitrary parameters.

(1) R . H . DALITZ: Lectures given at Les Houehes Summer School, e d i t e d b y C. DE W n ' T a n d M. JACOB (1965); G. MORPURGO: Lectures given at Eriee School, e d i t e d b y A. ZICHIC~ (1971). (2) A . BRAMON a n 4 ]VI. GRECO: Phys. Left. , 48 B, 137 (197~)~ (a) B. GOBBI, J . L . ROSEN, H . h . SCOTT, S. L . SHAPIRO, L . STI~AWCZYNSKI alXd C. 1Vii. M~ELTZER: Phys. Rev. Left., 33, 1450 (1974). (4) 1NT. C. CARRITHERS, P . MOIt'LEMA_~N, D. UNDERWOOD a n d D. G. RYAN: Phys. Rev. Lett., 3 5 , 3 4 9 (1975). (5) S. ONO: Lelt. Nuovo Cimento, 14, 569 (1975). (6) G. GRUNBERG a n d F. M. RENARD: 1Vfontpellier l a repr in t P/~I/76/02 (1976). (~) B . J . EDWARDS a ~ d A. :N'. JAMAL: Phys. Rev. Left. , 36 , 241 (1976); B. H . BOAL, R . I t . GRAHAM a n d J . W . MOFFAT: Phys. Rev. Lelt. , 36, 714 (1976). (s) J . STEI~ERGER: Phys. Rev., 76, 1180 (1949). (~) B. LAUTRUP a n d P . OLESEN: Phys. Left., 22, 342 (1966). (10) R . R O C K , ORE: Phys. Rev. D, 11, 620 (1975) a n d r e f e r ences t h e r e i n .

205

2 0 6

T A B L E I .

A. BRAMON and J . A. PENARROCHA

Decay Exper imen ta l rate Quark model Presen t Model in KeY (ref.) (ref. (2))

=0 ~.fy (8.0 7}- 0.4)-10 3 (n) 8.4 cV

-~ y,( 0.324 _ 0.046 (I2) 0.34 eV

~--~=-y 35 • 10 (3) 83 ~ 6 KeV 42 KeV

K*~ K~ 75 -- 35 (~) 190 ~ 14 KeV 89 KeV

(o--> =ou 870 • 70 (~) i npu t 850 KcV

9 ---> r.~ 5.9 ~ 2.2 (i~) ~ 0 ~ 6 . 5 KeV

I~adiative decays are described by the mat r ix elements

G E:~ra kl EI k.z E2 ,

where ki, k 2 (and E l , Eo) are the four -momenta (and polarizat ion vectors) of the two spin-one particles involved in the vertex. Par t ia l -decay widths are then given by the well-known expressions

G"- 1 G 2 (1) F(V-~Py) = ~ ~ Ipy[ ~ , F(P->u165 = ~ [pyl a .

The different baryonie .coupl ing constants (with explici~ const ruct ion of S U 3 invar ian t s ) are defined according to (10)

(2)

B B P :

BBV:

BBy :

- - V~] Tr ( [BiFs , B]P) + ~/2d Tr({Biy5, B } P ) ,

- - ~ /2F Tr ([BT~, B] V") § x /2D Tr ({Bye, B} V~),

1 - - ~ e Tr ( [ B y y , B] : l / ' ) ,

wherc B, P and V are the usual baryon, pseudoscalar and vector-meson traceless matrices, respectively, and A = ).3 § (1/~J-3))-s.

The re levant coupling cons tants G, appearing in eqs. (1), are eva lua ted using s t andard techniques (9) in terms of those defined by means of expressions (2). As i t is well known (s,9), hav ing neglected any tensor coupling for the B B ? and BBV vertices, the in tegra t ion over the loop fou r -momen tum generates no divergencies. For the =o_~ y,v t rans i t ion one obta ins

1 G_:%, v -- ~ ]i e:2d,

(1L) A. BROW3LtN, J, DE WIRE, B. GITTELSIAX, K . 3[. H ~-~SO~, D. LARSON an 4 E. L o g : Phys . Rev. Lef t . , 33, 1400 (1974). (12) A. BRO'*%%IAN ", J . DE %VIRE, B. GITTI~I,SIAN, K . ~1. HANSON ~Ild E. LOH: Phys . RO). Loll. 32, 1067 (1974). (la) PARTICLE DATA GROUP: Phys . Lel l . , 50 B, 1 (1974). (14) C. BE-~IPORAD: Proceedings o/ the 1975 International Syn~posium on Lepton and Photon Interactions at High Et~ergies (Stalfford, Cal . , 1975).

BA R Y O N -L 00P MODEL FOR RADIATIVE DECAYS OF MESONS 207

where M ~ 1.16 GeV stands for the average mass of charged baryons. The va lue of the pa rame te r d = %g is ra ther uncer ta in due to i ts dependence on % = d / (d -~ 1), whose va lue is a round 0.6 when deduced f rom weak-decay da ta , b u t can also decrease unt i l 0.5, as recent resul ts (from st rong interact ions) seem to indica te (~). W e take % ~ �89 and g:/4z~ ~ 14 (1~,~) or, equiva len t ly , d ~ f " 6.6 in such a w a y t h a t our model d i rec t ly predicts F(~:~ 7y) = 8.4 eV in good agreement w i th the exper imenta l deter- minat ions (n,~a).

F u r t h e r confidence on our choice of % ~ �89 can be obta ined f rom the -~ ~ 7y decay. Defining V ~ c o s 0 , ~ s - - s i n 0 , v ~ one can deduce

g ~ 1 s R , - - - - ~/~ cos 0~ - - ~-~ sin 0 , ,

gpp~

Rr Gnvy ~ cos 0, - - ~ - sin 0 , ,

where s s tands for the BBv~ unknown coupling constant . F r o m the exper imenta l (~:) resul t /R, ~ 0.67 =k 0.08 and approx imat ing c o s 0 , _ 1 as general ly accepted, even if the precise va lue of 0" is essential ly unknown, one obtains / ~ v _ ~ 0.79 and the excel- lent va lue for F(~--> yy) quo ted in table I.

We now tu rn to the V--> Py transit ions. W i t h the same techniques as before one arr ives at

1 1 1 Gp-~-y -- 2 GK*~ = 4 ~ 2 ed (F -t- D) = 4 z ~ 2 ed g, ,~, ,

where all the paramete rs have been prev ious ly fixed excep t t he coupl ing / v + D = g~pp, whose definite va lue is st i l l no t wel l established. F r o m ref. (18) we adop t gp,p ~ ~Gpo~-~v-- 2.0, obta in ing the sa t is factory resul ts / '(p--->~:-u = 42 KeV and F ( K * ~ K~ = 89 KeV.

In order to discuss the co, ~0-+ ~ decays, the mix ing angle 0 v of these vec tor mesons and the coupling cons tan t of the i r SU3 singlet pa r t to the baryons, BBco 1 = S, are also required. Wr i t ing ~0 = - - c O S 0 v W s + sin0vo) 1 and co ~ sin Ov(Os-~ cOS0v~ % our model leads to

G~n.~ ~ - - V'3 cos OvG~r~ r ~ sin Ov6e/s/4~ ~- M ,

G~,~y ~ ~/3 sin 0 v G~u -]- cos Ov6e]s/4~ ~ M .

Agreement w i th the exper imenta l results, as shown in table I, is then achieved wi th s = 2 . 9 _ ~ V ~ ( F ~- D) and the mix ing angle 0 v _ ~ 24 ~ which tu rns out to be substan- t ia l ly smaller t h a n the general ly accepted ideal value, 0 v ~ 35 ~ Note, however , t h a t our va lue corresponds to tha t known as << current mix ing ~> (is) and has also been ob- t a ined by GRUNBERG and :RENARD (6) in a con tex t similar to the present one. W i t h this value, s imple S U 3 predicts / ' ( p - + ~ ) / F ( @ - - > K K ) = 50 in reasonable agreement w i th the exper imenta l (13) rat io 44 • 4. The re la t ive couplings of the pho ton to p0, r

(16) ~]~. M. N~kGELS, T. A. I=~IJKEN anf~ J . J . DE SWART; Phys. Rev. D, 12, 744 (1975). (le) I~. PILKL~H,~, ", ~V. SCIT~IDT, A. D. ~IzkRTIN, C. MIOH&EL, F. STEINER, B. R. MARTIN, ~ . ~r NAGELS a n d J . J . DE SW=~RT; Nucl. Phys. , 6 5 B , 460 (1973 (I?) j . p . BORIGHT, D. R. BOWEN, D. E. GROOM, J. OREAR, D. P. OWEN, A. P. PAWLICKI a n d D. H . WHITE; Phys. Lelt., 33 B, 615 (1970). (18) S. COLE~L~Ig and H . J , SCHNITZER: Phys. Rev., 134, B 863 (1964).

208 A. BRAMO~ and J. A. PENARROCHA

and 9, measured from storage rings, which could be simply explained (*) by 0 v_~ 35 ~ can be also justified by our value 0 v_~ 24 ~ provided one allows the photon to ac- quire u small SUa singlct component when it couples to a neutral vector meson. But the existence of such a component is naturally justified in our model where the c0~-*y transition proceeds through p]~ and ~ - ~ - pairs whose exact cancellation is not pos- sible due to the nonnegligible baryon mass differences. Finally, we also note that in our model, which in agreement with Zwcig's rule suppresses the 9 ~ and 9=p couplings, no longer forbids the ~\%~'9 vertex as recent results tend to indicate (~9).

A rough prediction for the r 9 - .=+= =0 decay widths is equally possible in terms of the previously fixed parameters plus the D/tz ratio for the BBV couplings. Taking D/F ~ - - 0.2, as suggested by a Regge analysis (~s), proceeding as before and making use of previous (2o) phase-space evaluations, one obtains (**)

/'(r 3=) _~ 7 MeV (9 MeV), F ( 9 ~ 3=) ~ 1 MeV (0.7 McV),

in reasonable agreement with the corresponding experimental (~3) rates quoted in parenthesis.

In conclusion, a reasonable agreement with the data can be achieved in terms of a few coupling constants whose values have been fixed to be those previously known from other experimental results and, besides S, only the 0)-9 mixing angle 0 v has been internally fixed in our model. I t turns out to be smaller than the generally accepted ideal value but in agreement with the conclusions of several authors. The model can be enlarged to account for other transitions. In particular, decays involving ~-mesons (other than ~-*~,() can be discussed in a similar way but since the relevant exper- imental information is not clear (14) and one new unknown mixing angle 0p is required we neglect, for the moment, such a discussion.

{*) Note , however , t h a t our va lue 0 v ~ 2~: ~ is to be p re fe r red to 0 v_~ 35 ~ in order to expla in t h e V~ coupl ing ob ta ined in V~ exper imen t s . See, A. SILVERMAN: Proceedings o] the 1975 In ternat ional S y m p o s i u m on Lep ton and Photon Interact ions at High Energies (Stanford, Cal . , 1975). (19) V. BLOBEL, J=[ FESEFELDT, I~. FRANZ, t . HOFSIANN, CIt. JETTER, I-I. NEUSIAN.N B. SCHRODER a n d P. SSDING: p rep r in t D E S Y 75/31 (1975). (so) A. BARACCX and A. BRA.~mN: N u o v o Cimento, 6 9 A , 613 (1970). (**) I n de r iv ing the co~-7~ coupl ing we have adop ted the m a s s of the nucleon ins tead of the a v e r a g e m a s s -rv/= 1.16 GeV since in th i s ve r t e x , the nucleon loop plays a d o m i n a n t role. Note , also, t h a t in these p u r e l y hadron ic VVP ve r t i c e s the m o d e l i s expec ted to be less rel iable t h a n w h e n work ing wi th r ad i a t i v e ve r t i ces .