LETTERE AL NUOVO CIMENTO VOL. 14, N. 17 27 Dicembre 1975

Production of + and Charmed Vector Mesons.

C. S. KAI~MAN

L o y o l a C a m p u s , Concordia Univers i t y - Mon t rea l , C anada H 4 B I R 6

(ricevuto il 27 0t tobre 1975)

In a problem described by the formalism of quantum field theory, there is typically a set of states corresponding, say, to the spin-�89 + baryons and a set of operators cor- responding, say, to the spin 0- and 1 + mesons. Suppose that both the baryon states and the meson operators correspond to representations of the group S U 4. I t is assumed that the space of meson operators is a tensor representation buil t from tho direct product of (~ quark ~ and (~ ant iquark 7) operators; B4k, k ~ l , 2, 3, B,~ 4, r e = l , 2, 3 (0- mesons), and Bs~, n = 1 , 2 , 3 , 4 , B~5, p = 1 , 2 , 3 , 4 (1 + mesons). (For the 0- me- sons, the charmed quark operator is omitted.) The following shows some resulting identification of meson operators:

(1) •+ . - . C(r:+) B~4B41 ,

(2) ~ - . - . C(r:-) B14B4~ ,

(3) K + ~-~C(K +) B~,B43 ,

(4) K - .-~ C(K-) Ba4B4~,

(5) cp +-* C(~?) B35Bb3 ,

(6) ~ +-, C(+) B , sB54 ,

(7) D * - *-~ C(D*-)B~sB54 ,

(8) D *~ ~ C(D *~ B25B~, ,

(9) .F*- ',~ C(F*-)B35B54,

(10) 17*+ *-~ C(F *+) B , sBal ,

(11) D *~ *-* C(D *~ B4aB52,

(12) D *+ +~ C(D*+)B45Bsa ,

where C(7: +) etc. are all constants. The notat ion of GAILLARD, Lv.v. and I~OSNER (1) is used for the charm C = 1 states.

(1) ~ . GAILLARD, B. W. LEE and J. ROSNER: Rev. Mod. Phys., 47, 277 (1975).

605

606 c . s . KALMA~

The generators Bi t , i , j = 1, 2,3, of the group SU3 must be defined on bo th the baryons and meson operator is given by

(13)

(14)

(15)

(16)

operators. Transformation of the * quark ~ and ~ a n t i q u a r k ,

[Bo, B ~ ] = ~ B ~ t , i, j , m = 1, 2, 3 ,

[Bo, Bk4] = - - ~ k t B , 4 , i , j , k = 1 , 2 , 3 ,

[Bit, Bs~] = ($i,~Bst, ~, j , m = 1, 2, 3, 4 ,

[B~t, Bks] = - - ~kt B i s , k, j , m = 1, 2, 3, 4 .

I t is hypothesized tha t the closure of the operators under addi t ion and commu- ta t ion forms the Lie algebra of some Lie group. The smallest algebra occurs wi th the choice of

(17) [B4~, Bj4] = ~p:B44-- Bsi,

(18) [Bsi, B~5] = O(daiB55--Bji),

so tha t wi th

0 = + 1 the Lie group is SU 5,

0 - = - 1 the Lie group is SU4. ~.

BANDER and ITZYKS0N (3) have shown tha t the hydrogen a tom problem for both bound and scat ter ing states can be solved by use of an analogous dynamical group. In the bound-s ta te problem a representat ion of this group is labelled b y discrete parameters and, for the scattering states, a continuous parameter representat ion is used. Since the group S U 5 is compact, i t has only finite-dimensional representat ions characterized b y discrete parameters . The group S U4,1, on the other hand, has both discrete and con- t inuous parameter representations. A continuous parameter representat ion of this group will be used in this paper.

The representat ions of the group corresponding to the �89 baryons arc constructed b y means of a Gelfand basis (a):

(19) re(a) = t ~bl5 m25 ~35 145 9/b55 1

ml4 m2a ms4 m44

~r~13 m2a ross

m12 m22

/11

All the parameters are integers subject to certain conditions. The representat ion is chosen so tha t all the spin-�89 + baryons tha t can be constructed by a simple quark model approach (4) are contained in a single SU4.1 representation. The s ta te vectors corre- sponding to the 20-dimensional SU4 representat ion containing the S U 3 octet represen- ta t ion are shown in tables I and I I . The nota t ion for the charmed baryons is essential ly

(') ]Yi. BA2CDER and C. ITZYKSO~: Rev . Mod . P h y s . , 38, 330, 346 (1966). (') I. M. GEI~AND and lVl. I. GR~EV: A m e r . Math . Soc. Trans l . , Ser. 2, 64, 116 (1967). (') D. B. LIC~rENBER0: Indiana University Report COO-2009-92 (1974).

PRODUCTION OF ~ AND CHARMED VECTOR MESONS 6 0 7

TABLE I. - Identi/ication o/ the elements o/ the baryon �89 octet with a continuous repre- sentation o/ SU4.1.

l 2 1 0 0 2 1 0 0 N O = 2 1 0 .N "+ = 2 1 0

2 1 2 1 2 1

/ - - 2 + a 1 0 0 2-f-~ t i 2 1 0 0 Z ' - = 2 1 0 Z ' + =

2 0 2 1

- - 2 § 1 0 0 2A-~ t 2 1 0 0

2 1 0 2 0

0

i 2 1 0 0 27~ 2 1 0 ) A 2 0

1 / l - - 2 + a 1 0 0 2 + a 1 2 1 0 0

2 1 0 1 1

1

1 - - 2 + . 1 0 0 2 + ~ 1 2 1 0 0

2 - = 2 1 0 2 ~ 1 0

1 t - - 2 + ~ 1 0 0 2 + ~ 1 2 1 0 0 2 1 0

1 1 1

that of LICHTENBERG (s). Baryons are named according to their isospin I and strange- ness minus charm S--C. A subscript is used to denote the charm if it is not zero. Thus 2: + denotes a baryon with I = 1, S = - - I , C = 0. Z~ denotes I = 1, S-~ 0, C = + 1. Summing up the plus and minus signs on the symbol gives the charge of the particle. Finally 2+s corresponds to a 2 which is symmetric in the quark indices and 2+a to one which is antisymmetrie in the quark indices.

The cross-section for a reaction of the type

MI+B 1 +-4 M2-~-B 2 ,

where M I is a pseudosealar meson and M 2 an isoveetor meson and B1 and B e are baryon states, has the form

(20) a(M1B 1 ~ M~B 2) = ,l.t~te<B2.~, t M21int. states I M1BI> 2 .

(6) D. B. LICItTElqBERG: Left. Nuovo Cimento, 13, 346 (1975).

608 c.s. XAT.MA~

TABLr II. - Identi]icatio~t o] the �89 charmed baryons with a continstous representatio~t ol SU,,~.

/--2q- ~ 1 0 0 2-4-~ 1 i 2 1 0 0

2 0 2

- - 2 + a 1 0 0 2+~ t 2 1 0 0 2 0 0

2 0 0

t - - 2 + ~ 1 0 0 2 + a 1 2 1 0 0

Z'~_= 2 0 0 A~_= 2 0

2

--2 + ct 1 0 0 2+~ 1 2 1 0 0 1 1 0

1 1

1

i �9 1 o o 2 )2 1 11 11 ~ o o < 1 o o 2 )1 11 ol o~ o o ~ - ~ =

--2 + cr 1 0 0 2 + 8 1 2 1 0 0 ~7 = 2 o o Qu

0 0 0

- - 2 + a 1 0 0 2 + $ 1 2 l 0 0 1 0 0

0 0 0

t --2+m 1 0 0 2+~ 1 2 1 0 0

1 0 0 3~+ = 1 0

1 1 - -2+ ot 1 0 0 2 + $ 1 2 ] 0 0

1 0 0 1 0

0

PRODUCTION OF ~ AND CHARM]~D V~CTOR MESONS 609

Application of a vector meson of the form given in eqs. (5)-(12) to a baryon state described in tables I and I I yields

(m) :~IB~) = ~, C(M~)(A + BN~)IB/> , i

and that of a pseudosealar meson given in eqs. (1)-(4) yields

(22) M~[B,) = ~_~ C(M1)AIlB~}, i

where A, B, A 1 are constants. Applying eqs. (21), (22) to eq. {20) implies that

(23) a(M1B 1 r C2(M1)C*(M2)(a q- bills + cN*),

where a, b, c are all constants. Now MORRISON (6) showed that the parametrization

(24) a = constant X (Pl~b) '7

fits two-body reactions. Moreover in actual fits to experimental data (7) for B 1, B 2 elements of t h e , nucleon octet ~, P~b ~> 2 GeV/c, ~ ------ 2. Consistence between eqs. (24) and (23) is required. Suppose that

(25) I~l = i.,=k,

as P~,,b increases the term c[~[4= cP~-~4b tends to zero and at some point the equation is dominated by the term bla[ ~ = bP~, b.-* According to the experimental data this occurs for Plab ~> 2 GeV/e. I t should be pointed out that ul t imately this term also decays and a~C*(M1)CS(M2)a = constant. Such a region of asymptotia presumably occurs for larger values of P~b than are possible in present accelerators. For consistency with current experimental data, the constant term is neglected and the cross-sections are given by

(26)

Now consider

(27)

a(M1B1 ~-*M~B2) = C~(M1) C2(M2)bP'{~.

x = a(K-p ~ Ao)/a(K-p-~E~162

According to eq. (26)

(28) x = bl/b 2 ,

where b 1 and b~ are constants that are completely determined in the model {parameter free). Thus the calculation of x is a good text of the model.

(6) D . n . O . ~ORRId~ON-." Proceedings of the S t o n y Brook ~on]erence on Two-Body Interact ions (Stony Brook, N . Y . ) . (7) E. BRACCI, J . t). DROULEZ, E . FLAI~IINO, J . D. HANSEN and D. R. O. MORRISO~: C E R N Repor t , C E R N / H e r a 72-1, C E R N - H e r a 72-2 (1972).

610 r s. ~LMA~

NoW

a(K-p--> A~) = 1.5C(K-)C(?)[(0.6(9 + P ~ ) - -0 .0292(4 + p~)]2__~ 15.85C(K-)C(~)P[~,

-3 -2 0.044(4 + .P~)]~.-~ a (K-p~X~ = 0.5[0.4(9 + P~,,b) + 0.111(1 + Pl,b) +

-+ 3.89 C(K-) C(~)P~.

Hence a ( K - p ~ A ) ) / a ( K - p - + Z 0 ~ ) = 4 . 1 . This is to be compared with Lindsey ' s r e su l t ( s ) of 4 . 4 • a t P l~b=2 .58GeV/c and London 's result(9) of 5 . 2 ~ 2 . 1 a t -PI,b : 2.24 GeV/c.

As above (eqs. (20), (23), (26)) the cross-sections for product ion of the ~ and charmed vector mesons have the form

(29) a(M1B~ ,--+M2B2) = ~2(M1) C2(M2)k[ (4 + [~le) /6]~ 1.33kCe(M~)C2(M2)PT~,

TABL~ I I I . - Production cross-sections.

React ion P~ba/[C2(M1) C~(M~)]

~r+~l --* ~p 5.33

rc-p --> ~/~ 5.33

K-p--* ~A 2.00

K - p - * ~E- 1.33

K-~ -~ ~/Z ~ 0.67

~+~ --~ D*-Z~. 0.89

~-p --> D*~165 0.89 K - p -->- D*-~:~_, 0.44

~-p --~ D*-Z~_ 0.44

r:+~ --> D*~ 0.44

K - p ~ D*~ 0.44

~ -p -+ D*-A+ 0.40

rr --, ~*OA+ 0.40

K-p--> F*-A+ 0.30

K-~ -~ F*-E+ 0.22

K-~-+ D*-E+, 0.11

K - p - ~ F*-E~ 0.10

K-~ -+ D*-E+, 0.10

K-p--> D * ~ 0.10

where k is a react ion-dependent constant. In table I I I , the values of 1.33 k are given for reactions tha t produce the ~ or charmed vector mesons. To a first approximat ion i t is probably val id to take C2(M1) to have the same order of magni tude for Ml=r : , K and s imilar ly with C~(M2) for M s = F * or D*. Thus table I I I shows at a glance the relat ive sizes of the various product ion cross-sections.

(s) J. S. LINDSEY alld G. A. SMITH: Phy8. Rev., 147, 913 (1966). (*) G . W . LONDON: Phgs. Rev., 143, 1034 (1966).