IL NUOV0 CIMENTO VOL. 46 A, N. 2 21 Luglio 1978

Radiative Decays of Heavy Particles in the Geometrodynamical Approach (*).

F. CSIKOR (**)

C E R N - Geneva, Switzerland

G. PREPARATA

C E R N - Geneva, Switzerland Istituto di Pisica dell' Universit~ - Bari, Italia

(rieevuto il 15 Marzo 1978)

Summary. - - We calculate the radiative decays of the new particles in the geometrodynamieal theory of hadrons. Very good results are obtained for E1 transitions, while the M1 transitions fail in describing the J/~ and ~' transitions to the X(2830) and the X(3455) interpreted as pseudoscalar states. On this basis we speculate that the X(2830) might be interpreted as nonrelativistie bound states (~ molecules ~) of (DD) and the X(3455) might be called ~r

I n a recen t pape r (1) we have discussed in the f r a m e w o r k of the geomet ro -

d y n a m i c a l a p p r o a c h (2-4) the possibi l i ty of defining a (( d i rect ~) cu r ren t -pa r t i c l e

coupl ing which obeys the necessary conse rva t ion or pa r t i a l - conse rva t ion con-

(*) To speed up publication, the authors of this paper have agreed to not receive the proofs for correction. (**) On leave from E6tvSs University, Budapest, Hungary. (1) F. CSlKOR and G. PR]~PARATA: Geometrodynamics ]or quarks and hadrons: the de]ini- tiou o] currents, CERN preprint TH. 2396 (1977). (2) G. PREPARATA and N. CRAIGIE: Nuel. Phys., 102 B, 478 (1976); N. CRAIGI]~ and G. PREPARATA: Nuct. Phys., 102B, 497 (1976); G. Pn]~PARAT.~: Proceedings of the 1975 Erice Summer School, edited by A. ZICHICm. (3) G. PREPARATA: Nucl. Phys., 122 B, 29 (1977). (4) G. PREPA~ATA: Geometrodynamics ]or quarks and hadrons: the mesonic states, CERN preprint TH. 2271 (1977).

230

RADIATIVE DECAYS OF HEAVY PARTICLES ETC. 2 ~ 1

ditions. As stressed in ref. (1), the definition of such currents is necessarily not unique, nevertheless, one can invoke the principle of simplicity to obtain an essentially unique choice.

In this paper we intend to take this problem up again and to show that , in the case of the radiat ive decays of the newly found heavy particles, not only our definition works remarkably well, bu t we can get interest ing clues as to the na ture of these particles.

The radiat ive transit ions of the heavy mesons made out of c~ pairs are the

best possibility to tes t the s t ructure of the mat r ix elements of the (, direct ~ contr ibut ion to the electromagnetic current. In fact the normalization of the wave functions of the mesons should be determined (a,4) by imposing tha t the charge of a given meson, as calculated from the (c direct )) and vector-domi- na ted terms, should come out correctly. I t has been shown (4) that , while for low-m~ss mesons the vector-dominated te rm dominates over the direct con- tr ibution, in the heavy-meson case the si tuat ion is reversed. Thus the cal- culated rates for radiat ive transit ions of heavy mesons will depend only on the part icular form assumed for the direct piece of the current.

The calculation of the rates is straightforward, even though it becomes quite tedious in practice. The relevant meson wave functions are given in ref.(4). We identify the J /~ as the L = 0 , n = l , S----1 state, the ~' as a m i x t u r e of the L ~- 0~ n --~ 2, S--~ 1 and L---- 2, n ~ 1, S = 1 states (with the mixing angle ~o taken as a parameter) and the Z states with the usual spin assignment as the L ~ 1, n ---- 1 states. The X(2.83) and the Z(3.455) shall be ten ta t ive ly identified with the L- -~0 , S----0, n = l and n ~ 2 states, respect ively ('). Our input parameters are

the charmed-quark mass mo ~ 1.86 GeV, the slope of Regge trajectories (which determines the spatial radii R~

of the bags as in ref. (4)),

the experimental meson masses (**).

The radii of the t ime bags R t are then determined by eq. (4.10) of ref. (4). The mat r ix element of the electromagnetic current is given by (1)

(1) (pl~rjg, q(0)lp27} = T r ( ~ Q [ ~ , ~ ] ) 1 ~ d4xexp i ~ ( p l - - p ~ ) �9

9~

(*) Note that our description is fully relativistic, thus orbital angular momentum and spin have only a limited sense. (**) The fine structure of the spectrum is, however, calculable by the methods of ref. (5). (5) G. FOGLI and G. PREPARATA: Unitarity and hadron sell-energies, CERN preprint TH. 2323 (1977).

232 F. CSIKOR and G. pREPARATA

where W~ and ~o 2 denote the particle wave functions, k~ and k2 are simple factors occurring natural ly in the current definition (4), and the integrat ion extends over the intersection ~' of the bag domains of the two particles. The first t e rm in (1) is the ~( naive ~ definition of the current, while the second te rm represents the surface charge contr ibution necessary to ensure current con-

servation. I t was shown in ref. (1) tha t j~ad is given by

~q~ q~ ~q~ S ,

where q, is the momentum of the current and S is the scalar product of q, b y the first te rm of (1). The arbitrariness in the definition of j~ad lies in the dif- ferent ways in which S may be expressed as a function of qu. We choose the (~ simplest ,> possibility (discussed in ref. (1)) which consists in keeping in the covariant tensors of the invar iant decomposition of S the smallest number of q, factors, and in considering the invariant functions to depend on q~ only. For particles of unequal mass an additional factor (Pl + P2)q/M2~ - M22 is a t tached to 2, whenever the :p~oduet of the i~trinsie p~rities of the particles

is ~-1 . I t is easy to see tha t the 1- -+ 0- q- y transitions (usually called M1) are

not affected by j~dd while the 1 - - + 0+ q - y , 1 + + y, 2 + + y (El) transitions D '

are modified f rom their naive expression. Our results for the E t transitions appear in the table. The mixing angle

of the L = 0 and Z = I states building up the ~' is determined with a small uncer ta in ty to be ~ = ( - -51 =t=3) ~ (very close to maximal mixing!). The reason why ~ is so well determined is t l lat the ra te of ~ '-+ X(3414) q- Y is quite sensitive to it. Taking ~ = -- 51 ~ we obtain F(~ ' -+ e+e-)/F(~ -> e+e -) ----- 0.42, to be compared with the exper imental value 0.44 i 0.12, (A striking con- firmation of the mixing scheme proposed in ref. (4),)

TABLE I . - E1 transitions o/ ~'-+Z's and X's-+J/~.

State jec ~' ~ X+Y X ~ J/~+Y

F ~~ F ~p (D F~ (keY) (keY) (keV)

X(3552) 2++ 18.9 184-7 61

X(3508) 1++ 10.4 20=t=7 184

X(3414) O++ 18.7 18 4- 6 187

Our results for the 21/1 transitions (which are independent of the ambigui ty Df the current definition} do not support the identification of X(2.830) with

RADIATIVE DECAYS OF [HEAVY PARTICLES E~[C, 233

the L = 0, S = O, n = 1 c~ state. We get F~176162 -+ ;(y) ---- 16.6 keV (*), while the exper imenta l upper limit is 1.3 keV (~). An identification of the X(3.455) with the lowest-lying (L = O, S = 0) c~ state (suggested in (7)) does not lead to diffic~itie~, in fzc t we get I '~(~'->Z(3.455} ~ y) ---- 0.98 keV, the e~- per imenta l upper limit, on the other hand, is 6 keV (~).

In the figure we plot the e~lculated radiat ive r~tes, ~ssuming tha t the pseudoseal~r (%) ~nd its r~dial exci tat ion lie below the J /~ and ~b', respectively, the mass difference between the vectors and the p~eudosealars being denoted by AM. W~ not~ that , whiZe the si tuation i~ z~fe for the ~7: in view of the ~maJ!

a)

c)

> a,r

'JO

10 ' ~ I 0 100 200 300

#M(MeV)

Fig. 1. - Calculated values of the rzd~ative widths for the decays of J/d/ and ~' to ~c(M) and ~(M)' as a function of the mass differences A M = Mjj+--_~I~ and

(*) Interpreting the X{3.455) as the ~'~. we get F~'~(~'~ X(3.455)+~)= 2.5keV (experimental value < 6 key (~)). (6) J, S, WHITAKER, W. TANI~.NBAUM, G, 8. ABRAMS, M. ~. ALAM, ~k, M, BOYARSKI, M. BREIDENBACII, W. CI{INOWSKY', R. G, D~EVoE, G. J. F:ELDMAN, C. E. FRIEDB:ERG, D. FRYhERGER, G. GOLDHABER, G. HANSON, J. fl~. JAROS, A. D. JOHNSON, J. A. KADVK,

234 F. CSIKOR and G. PIREPAIRATA

r a d i a t i v e r a t e s , one m i g h t be in t r o u b l e as to t h e F(d?'--> ~o -}- y) , t h e u p p e r

l i m i t of wh ich has been e s t i m a t e d (~) to be axound 4 keV.

W e conc lude w i t h a few o b s e r v a t i o n s .

i) The g e o m e t r o d y n a m i c a l a p p r o a c h w o r k s v e r y wel l for t h e E1 t r ans -

i t i ons ~ m o n g c~ s t a t e s ; t h e r e su l t s of t h e t a b l e l e n d s u p p o r t to t h e co r rec tness

of t h e de f in i t ions of t h e (~ d i r ec t ~) c u r r e n t coup l ing of ref . (~).

ii) T h e f~i lure of ~he c a l c u l a t e d M1 t r a n s i t i o n s sugges t s t h a t t h e X(3455)

a n d t h e X(2830) h a v e been m i s i n t e r p r e t e d as t h e ~o a n d i ts r a d i a l e x c i t a t i o n .

W e p r o p o s e t h a t X(2830) be i n t e r p r e t e d as a D D (~ molecu le ,~ (s).

iii) One can e x p l a i n t h e fa i lu re to obse rve t h e D: b y n o t i c i n g t h a t , be-

cause of t he m i x e d n a t u r e of t h e r such a s t a t e is m o s t l i ke ly to l ie a b o v e t h e ~ ' .

iv) T h e puzz le of t h e ~~ r e m a i n s ; we c a n n o t e xc lude t h a t t h e ~]o l ies also

a b o v e t h e J / ~ a n d t h a t i t m u s t be i den t i f i ed w i th t he s t a t e a t 3340 (3455) MeV (e).

R. R. LARSEN, D. L~KE, V. Li~TH, H. L. LVNCa, R. J. MADARAS, C. C. MOREIIOUSE, H. K. ~GUYEN, J. M. PETERSON, M. L. PERL, I. PERUZZI, M. PICCOLO, F. ~ . PIERRE, T. P. PUN, P. R~PIDIS, B. RICtfTER, B. SADOULET, ~ . H. SCI~[INDLE]~, R. F. SCttWITT]5~S, J. SI~R~ST, G. H. TR~LL]N~, F. VANNUCC~ and J. E. W~ss: Phys. Rev. Lett., 37, 1596 (1976); D. M. BADTKE et al.: Contribution to the X V I I I International Con]erence on High-Energy Physics (Tbilisi, 1976). (7) J. _h~RAFUNE, M. FUKUGITA and Y. 0YANAGI: Phys. Rev. D, 16, 772 (1977). (s) The existence of these states has been conjectured in : L. B. OKUN and M. VOLOSnIN : Zur~a. Eksp. Teor. Fiz., 23, 369 (1976); A. DE RUJULA, HO~VAI~D GEORG[ and S. L. GLASltOW: Phys. Rev. Lett., 38, 317 (1977).

�9 R I A S S U N T O

Si ealcolano i dee~dimeati radi~tivi delle nuove particelle helle teori~ geometrodinamica degli adroni. Si ottengono ot t imi r isul tat i nel caso delle $ransizioni E l , mentre le tran- sizioni M1 non descrivono corret tamente le transizioni delle J /~ e ~' in X(2830) e X(3455), quando si interpret ino come s ta t i pseudoscalari. Su questa base si speeula che 1'X(2830) sia uno st~to legato non relativistieo DD e che il )~(3455) si debba invece chiamare ~)r

Pa~IHaIIHOHHMe pacHa~lbl TSl~:eJ'lblX qaCTHU B reOMeTpo2iaHaMaqeCgOM HO~lXO~le.

PeamMe (*). - - Mb~ BblqHCgI,CleM panHat~onnbm pacnanbx nOBb|x qacTntt B renMeTpo- J!HHaMH~IeCKO~ Teopnr~ a~IpOnOB. HoJIyqaloTC~ xopomHe pe3yJIbTaTbI ~JI~ E l nepexo- ~OB, xor~a raK M1 nepexo~hi He B COCTOaHnn onncaTb J/~ H ~' nepexo~lbl B X(2830)

X(3455), rorop~Ie n~repnper~pymrc~ Ka~ ~ce~x~ocran~p~e c o c x o ~ . Ha OC~OBe 3TOFO, Mbl OTMe~iaeM, ~lro X(2830) MO~eT 6/~ITb 14HTeprlpeTl~pOBalt KaK ttepefl~lTi~- BnCXcKoe CBn3annoe cocxonnne (<< Monexyna >)) (DD) n X(3455) Momex 6blXt, na3Bano n,(3340).

(*) 11epe6eJeuo pec)anque~.