Exclusive QCD predictions for proton-antiproton decay rates of 3D1-states charmonium states

Physics Letters B 27t (1991) 231-236 North-Holland

PHYSICS LETTERS 13

Exclusive QCD predictions for proton-antiproton decay rates of 3D -states charmonium states R.W. R o b i n e t t a n d L. W e i n k a u f Department (?fPhysics, Pennsylvania State University, University Park, PA 16802. USA

Received 20 August 1991

Using familiar methods of exclusive QCD, we derive the amplitude for the coupling of J= 1 quarkonium D-states to protonantiproton pairs. Including the 3D 1 contribution, we then evaluate the ~' -,pl5 decay width and find that the 12 3S~) - I 13DI ) mixing angle solution ~ = - ( 28 + 2 ) ° is favored by the observed ~¢' decay rate. Using this solution, we then find that F(W' ~ pp) / F ' ( ~ p p ) is in the range 0.07-0.31 depending on the assumed nucleon wavefunction.

There has been much progress in the last decade in the development of calculations for exclusive processes in Q C D [ 1 - 3 ] . Examples which have been considered in detail include baryon and meson electromagnetic form factors [4,5] and transition moments (ANy couplings ), large angle exclusive scattering processes [ 6 - 9 ] , weak decays of heavy flavored mesons, and quarkonium two-body decays. In the last case, the decay width for ~ (3S ~) -,pf~ [ 10 ] was among the first predictions made using the formalism and calculations for 3Pl, 3p2--+p13 have also appeared [ 1113]. (We note, however, that there have been occasional strong objections made to the application of perturbative Q C D to such exclusive processes [ 14 ]. ) Exclusive Q C D predictions for quarkonium decay widths have, so far, been confronted only with data for S- and P-wave c h a r m o n i u m states with somewhat mixed results. (See e.g. refs. [ 13,15 ] for comparison with data o n 3Sl, 3p2, 3P l widths. For experimental measurements of 3Pi. 2 data see ref. [16]. The pp width of the rio is more problematic and is discussed in this context in refs. [ 17,18 ]. ) The first data on exclusive charmonium production in resonant pl5 collisions from the next generation experiment (Fermilab E760) [ 19 ] is beginning to appear [ 20 ], so there is hope that more detailed and extensive information will soon be available. Besides the search for the 'P~ state (to verify an earlier measurement [21 ] ), the

exploration o f charmonium D-states may also eventually be possible. While exclusive Q C D predictions for plb decays have been confronted mostly with the N = 1 t~(3097) width, the ~' (3685) also has a well-measured prot o n - a n t i p r o t o n decay rate. Consideration of the leptonic width of the ~" ( 3770 ) suggests that the ~ ' and ~" are admixtures of the 12 3Si ) and I 1 3D l ) states [22,1 ] so that a complete comparison o f F ( ~ ' - ' P t ~ ) with existing data would necessitate the inclusion of its 3D l component while any prediction for ~"--,ptb would (presumably) be dominated by the D-wave contribution. Motivated by this, we have used standard methods of exclusive Q C D to derive the amplitude for 3Dj ~ pp and have applied our results to an analysis of ~¢' and ~" two-body decays and that is what we report in this letter. We note that familiar helicity conservation arguments from exclusive Q C D imply that the coupling of the ~D2 state to pl5 pairs will be suppressed (just as predicted for the ~S0 and ~PI, 3po) while the 3DI,2, 3 are all allowed at full strength. Both the 3D 2 and ID 2 states are predicted to be relatively narrow [22,1] because parity and angular m o m e n t u m conservation forbid their decays into DI) pairs and they are below DIS)* threshold. Thus a calculation o f the 3D 2 pO decay width would also be appropriate in the context. Because of the computational complexity of that case, however, it will be discussed elsewhere [23].

0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.

231

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14 November 1991

The formalism for extracting the nucleon and antinucleon spinors from the basic quark diagrams is described in refs. [ 7,13 ] and we follow their methods here. The quark (and antiquark) spinors arising from the fundamental diagrams are projected onto the nucleon spinors by using

presence of large numerical factors in several D-state processes (see also ref. [28] ) which can make their effects larger than otherwise expected. We begin by briefly reviewing the methods of ref. [ 24 ]. We write the amplitude for the annihilation interaction of a free fermion-antifermion pair as

< 01 e OkU~(Zl )U~(Z2 )d~(z3) IP)

d = ~(f, g) (5u(f, s),

= -~fN[(/~C)~e(TsN)7 V(zip) + (g75 C)o~p(N,)A (zgp)

- (au, p,,C)o~¢(yuysN)~T(z~p ) ] ,

( 1)

and a similar expression for the neutron. The functions V, A, and T are all related to a single, leadingtwist nucleon wavefunction, ¢(z~, z> z3), via 0(1, 2, 3 ) = V(1, 2, 3 ) - A ( 1 , 2, 3 ) ,

(2)

(5)

where f f a n d s, gare the particle and antiparticle momenta and spins and 6 is the relevant Dirac operator. The amplitude for a bound pair can then be described in a non-relativistic approximation by A=

fd, ~

~%(k)lT(f, g) (gu(f s) ,

(6)

f-f=2k=(O,

(7)

where

and 2T(1,2, 3 ) = 0 ( 1 , 3, 2 ) + 0 ( 2 , 3, 1 ) ,

(3)

2k),

and

and we have used

o',,,, = ½ [7u, 7,,.]

f+f=P=(M,O),

(4)

-

The diagrams we require (fig. 1 ) are identical to those for 3S~--,pp (including color factors) and the only new feature is the need to project the heavy quark wavefunctions into a D-wave quarkonium state. Recently, a covariant formalism for describing D-state annihilation decays has been developed [24 ] which has already been used for calculations of D-wave quarkonium decays into ggg and ggy [25] and large Px production of D-states in hadronic collisions [26 ]. (The formalism generalizes the much used methods for S- and P-states of Kuhn, Kaplan, and Safiani [ 27 ]. ) A further motivation from these results is the

J" d k l ~ g l 2 = 1.

(8)

M a n d m are the masses of the bound state and quark respectively, P is the quarkonium momentum, and we work in the rest frame of the bound state described by the momentum space wavefunction q&e(k). The necessary S-, P-, and D-state projections are obtained by expanding (5 to zeroth, first, and second order in k respectively and applying suitable spin projection operators. The resulting amplitudes [ 24 ] for the S = 1 D-states (3D j, J = 1, 2, 3) are then given by

/ 15 01;(0)

u(xlp) u(xZP) d(x3P)

I 3~

u(l

P) u(y2P)d(Y3P)

Fig. 1. Typical diagram for 3Dj coupling to pla states.

232

A = N/8g M 2M 2 { T r [ 7 " ~ y ~ ( ~ + M ) 7 ;]

+MYr[{ C?,/} + (~'+M)Tq + 1M2 Tr[ C~P(?+M)rV] }H, ap,

(9)

where ~o, (57, 6~ p are the zeroth, first, and second derivatives of (5 with respect to k s respectively (with k = 0 ) and 0~(0) is the second derivative of the radial wavefunction evaluated at the origin. The H,~pT(J) are the appropriate projection operators for the S = 1 states which for J = 1 is given by

Volume 271, number 1,2

PHYSICS LETTERSB

Hc@(J= 1 ) = --x/~oo [ (g~p-PaPp/M2)e~ + (g.@-Pi~Pp/M 2)e. _ 2 (g.a_p.pa/M 2)ep] ,

J/s (10)

where e. is the usual spin-one polarization vector satisfying e.P '~= O. For completeness, we note that the results for J = 2, 3 are

....

14 November 1991

/(s(°)= f [C~¢] f [d'~y]j6o),

and .is(O) =

¢(x)O(y)xjy3 + 2T(x)T(y)x2Y3

In

the

X (2y~-

H~aY(J=2) = ~

above we have defined and 1

1

~(1 0

0

--Xl--X2--X3) XIX2X3

0

(19)

(11)

where e.~ is the symmetric spin-two polarization tensor which satisfies

The corresponding decay rate is then

g"ae.a=O, P"e.~=O,

F(3Sl -,pp) = (~Ogs) 6 3892

(12)

and where the sum over polarizations is given by the familiar expression 2

Y. Gb(m)e*y(m)

m = --2

where

P.Pb

~ b = - - g ~ b + M~T ,

(14)

while II~ap(J= 3 ) = e.ap is the J = 3 polarization tensor. For comparison, the corresponding expression for the 3S~ state is given by A=-

~(0)

- -

~

lrtCo(~+M)¢] .

10¢s(0)f 2 m(3s1 -+P(ql )15(q2) ) = (47tO/s) 3 ~q/~M 81M 5

where

21252 ICs(0) 12f 4 rid2. M lo

(20)

This expression for the decay rate differs by a numerically insignificant (but strange) factor of s0 from the corresponding expression in ref. [ 13] (i.e., their eq. (23)). Then, using our D-wave formalism, we find (after extensive algebra) that the corresponding result for 3Dwstates is obtained by the substitution 0s(0)o:~/s-' 5,/2 m~ 2( 0 ) ~#D,

(21)

where

(15)

Eqs. (9) and (10) then generalize the results of the appendix of ref. [ 13 ] to D-states. The advantage of an explicitly covariant formalism is its ability to be used directly with popular algebraic manipulation packages and we have made extensive use of FORM in deriving the results below. The amplitude for 3S~ *-+ p13 has been derived in ref. [ 13 ] and in our language the result reads

X ~7(q~ )¢N(q2)~/gs,

D~-(2xi-1)

1) - 1

(e°"~E~PP°'P~g°°'

z fro-') , +ea~G.v~'Pg

(18)

D1 D2

Dj D3

1

i

(17)

(16)

corresponding to the contributions from the zeroth, first, and second derivative terms in eq. (9) respectively. The integrands are given by s~o) = ~ ( x ) ~ ( y ) (x,y3 +x2)

DI D3 2T(x)

+

T(y)(X2J23 "]-Xl ) DID2 ,

(23)

+ E2 E3 j~l) =O(x)O(Y)[ D, D31 \D,(E' ~3)+ 4~D2] +T(x)T(y)

[DI--~

+~

2

(F2

F3~],

D7 + D2}I

(24)

where 233

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El

PHYSICS LETTERS B

2x~Y3 --X2 + X l

= 2x2y3 -

,

(25)

+ x3Y3 ,

(26)

,

(27)

F~ = 1 + 2 x 3 ,

(28)

E3 = 2 - x 3

2x1 Y2 -- X 2 + Xl ,

(29)

F3 = 2 x I y 2 -- 2XI Y2 --X~ "{-X2 ,

(30)

F2 .= 2 x ~ y 2 -

and 2 O ( x ) O ( y ) (2G,

J#'-

H~

12T(x)T(y)(I, 5D, D3

+ 2x 2~y~~ - x 2 y 2 - x I X~ -J~X1X 2 "1-X 1Y2

--xIy 2 ,

(40)

J3 = 2Xl y3 - 2xl x 2 y ~ - - X l y ~ + Xl X 2 •

(41

)

To evaluate the hard-scattering amplitudes, we make use of several recent parameterizations of the nucleon wavefunction which satisfy sum rule constraints and lead to reasonable predictions for nucleon form factors and ~, ~,2 decays. Specifically we consider for O(x) ~---0(Xl, X2, X3)

-- 1.68X3 --2.94)

2H2

+ D2

(39)

OCZ = 0AS (X) ( 18.06X{ + 4.62x22 + 8.82x59

G3 )

5D-gTN\-b-7-,+D75 -+N

+ 3 # ( x ) 0 ( y ) (H1 IODLD2 ~

-F

G2

JI = 2x3y2 - 2 x l x 2 y j 2 + x l 2x 2 - x l y 22, J2 = - x 2 x 2

E2 =- 2x~ X3Y3 "t'-2x~ y2 _ Xl x3 _ 3xt Y3 -- X2

14 November 1991

(42)

(see ref. [ 31 ),

D~2,]

0KS = 0As(X) [20.16X~ + 15.12X~ + 22.68X32

12)

-- 6.72X3 + 1.68 (Xl --X2) --5.04]

~-1 + D33

(43)

(see ref. [291 ), and + 8 T ( x ) TD2 (y)

(

J2 J3) ~212 + D - - ~ 2 +~222 ,

(31)

0coz = 0AS (X) (23.814X~ + 12.978X~ + 6.174X~ + 5.88X3 -- 7.098 )

where G~ = 2 x 3 y 3 - 2x~ X3y21 "k X~X3 --X21Y3 , G2 ~- x 2 x3 - 6 x 2 1 x 3 Y 3 +

(32)

4x2 y 2 + X2~y3

--2Xl x2 +6XlX3Y3 --XlX3

-2xly~ --xly3 ,

(33)

G3 = 4X~ y3 +X~ y2 _ 10X 1x3y 2 + 6X~ x2y3 --XlX 2 ,

(34)

H~ =Xl (X2 --Y2 ) ,

(35)

H 2 -~ 2 x 2 x 2 Y 2 - - x 2 x2 - x 2 y 2 +

2xl y~ x 2 - 2x~ Yl x2

--XtX2--2X~X2Y2+2X~X2--Xly2+2xly2,

(36)

11 = & (X3 --Yx ) ,

(37)

12 =X3( 1 --X3) ,

(38)

( 44 )

(see ref. [301). In these expressions, one defines (~As(X) = 120XIX2X3 which is the asymptotic wavefunction which should be reached at infinitelyAarge values of Q2. Using these wavefunctions, we can evaluate the various overlap integrals in eqs. (17) and (22) and we give the results in table 1. To make contact with experiment, we recall that the 2 381-1 3D 1 mixing can be expressed as [22,1 ] [ W' ) = c o s ( 0 ) 12 3Sl ) - - s i n ( 0 ) I 1 3D1 ) ,

(45)

and J IIF' ) = sin(~) ] 23Sl ) + c o s ( ~ )

lgDt) .

(46)

The observed leptonic widths of the ~ ' and ~" then are reproduced for the two solutions 0 = ( 13 + 2) ° and

Table 1 Values of the overlap integrals ,ks and L1¢D for various proposed nucleon wavefunctions, ~cz(X) [ 3 ], Ocoz(X) [ 13 ], and ¢Ks(X) [29 ].

¢cz

Ocoz 0KS 234

0.73× 104 0.86× 104

1.11 × 104

2.12× 104

6.11 × 104

7.08× 104

1.53× 105

4.58X 104 7.63X 104

9.53× 104 1.48× 105

9.86× 104 1.55X 105

2.40X 105 3.79× l0 s

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( 2 8 + 2 ) ° with the second solution slightly favored by data on V', V" ~ rtTtV decays. In order to minimize the strong dependence of the decay widths on factors like ce6, we prefer to examine ratios of pf) decay widths. So, using the mixing formulation above and our results for the 3 D l ~ plb amplitudes we find that O =

-

F(V'--+pp)

" 2 6---10 (a~(M,~,)](M(~)]

F(V-*PlO) - \ a s ( M ~ ) / / rCO s

\M(~u' )J

//02S ( 0 ) ~

--Sln(0)tMS-01s~ )t~TJ

"

14 November 1991

see that the negative values of 0 give results which agree, within errors, with experiment for all of the model wavefunctions we consider while the positive values are all too small (very roughly 2.5a different). Thus the mixing solution with ¢ = - (28_+2) ° is favored. (We note that if one uses the asymptotic wavefunction, ¢~,s(x), the absolute predictions for V and V' widths are 20-30 times too small but that the ratio of widths is almost identical to that for ¢coz. This is consistent with our hope that ratios are far less sensitive to the details of the nucleon wavefunction. ) Using this value of ¢, we can then predict values for the remaining linear combination, namely

(47)

The final ingredient is the values of the potential model wavefunctions which we take from previous fits [22,1,31,3] to charmonium and bottomonium data. Specifically we use

2

6

F ( v ~ p p ) - \ a , ( M ~ ) J \ M ( v " )J •

02s(O)

/ 5,,/ 2 0 7>( O) \ /~tG ~G2

¢ls(O)=l.O1 GeV 3/2,

0 2 s ( 0 ) = 0 . 7 3 4 G e V 3/2,

O'I'D(0) =0.121 GeV 7/2 .

10

+c°s(o)t

(50)

(48) For the three model wavefunctions we find the results

The values for the ratio, eq. (48), depending on the mixing angle 0 and on the model wavefunctions eqs. ( 4 2 ) - ( 4 4 ) are presented in table 2 and should be compared with the experimental values [33] which give F(V'-+015)

exo.=0"31 +O.11,

(49)

when the various errors are added in quadrature. We

Table 2 Values of F( W-+plS)/F(~s-+p0) for various 123S,) - I 13Dt ) mixing angles, O, and for various proposed nucleon wavefunctions, 0cz(X) [3], 0coz(X) [13], and 01~s(X) [29]. The experimental value is 0.31 _+0.11. 0 (deg.)

0cz

0coz

0KS

11 13 15

0.026 0.021 0.015

0.018 0.012 0.008

0.011 0.006 0.003

--26 -28 - 30

0.182 0.191 0.199

0.243 0.258 0.273

0.306 0.327 0.348

F(V" ) I F ( v ) = (0.07-0.09), = (0,16-0.19), =(0.26-0.31)

(51)

for CZ, COZ, and KS wavefunctions respectively. The relatively large spread in values, compared to those in the V' case (over a factor of 4 compared to a factor of 2 in that case), is due to the destructive interference between the two terms which is present in this case when 0 is negative. Predictions for the 3 D 2 c a s e will involve the D-state contributions only and so will not suffer from any additional variation due to such interference effects. In conclusion, we have calculated the exclusive QCD amplitudes for the coupling of3Dl quarkonium states to P0 pairs and have used these to estimate the previously unevaluated proton-antiproton decay rates for V' and V" states. Confrontation with experimental data on the V' width has lent additional support to one of the two mixing solutions, namely 0 = (28 + 2) °. Many previous studies using this formulation of exclusive QCD have focused on examining different processes to probe the extent to which 235

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t h e s e m e t h o d s c a n o f f e r a c o h e r e n t p i c t u r e o f exclusive h a d r o n i c p r o c e s s e s a n d s h e d l i g h t o n t h e n u c l e o n w a v e f u n c t i o n . W h i l e also e x t e n d i n g t h i s m e t h o d to p r e v i o u s l y u n e x a m i n e d c h a r m o n i u m s t a t e s as a furt h e r ( a n d s i m i l a r ) test, i n a d d i t i o n , we h a v e att e m p t e d t o u s e o u r r e s u l t s to e x t r a c t i n f o r m a t i o n o n d e t a i l s o f c h a r m o n i u m s p e c t r o s c o p y . T h e use o f ratios o f d e c a y r a t e s w h i c h are m u c h less s e n s i t i v e t o v a r i a t i o n s in t h e n u c l e o n w a v e f u n c t i o n m a k e s t h e use o f e x c l u s i v e Q C D m e t h o d s to e x t r a c t p h y s i c s f r o m o t h e r s y s t e m s m o r e b e l i e v a b l e t h a n i f we w e r e r e l y i n g on absolute predictions. W e t h a n k R. L e w i s for n u m e r o u s u s e f u l c o n v e r s a tions. This work was supported in part by the National Science Foundation under PHY-9001744 (R.R.) and by the Texas National Research Laborat o r y C o m m i s s i o n u n d e r a n SSC J u n i o r F a c u l t y Fellowship (R.R.).

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[ 13 ] V.L. Chernyak, A.A. Ogloblin and 1.R. Zhitnitsky, Z. Phys. C42 (1989) 583. [14]N. Isgur and C.H. Llewellyn Smith, Phys. Rev. Lett. 52 (1984) 1081; Phys. Lett. B 217 (1989) 535; N. Isgur, in: Proc. Conf. on Hadronic physics in the 1990's with multi-GeV electrons (Seillac, France, 1988), eds. B. Frois, D. Goutte, P.J. Mulders and P.K.A. De Witt Huberts, Nucl. Phys. A 497 (1989) 299c. [ 15 ] W. Buchmfiller, in: Physics with antiprotons at LEAR in the ACOL era, Proc. third LEAR Workshop (Tignes-Savoie, France, 1985 ), eds. U. Gastaldi, R. Klapisch, J.M. Richard and J. Tran Thanh Van (l~ditions Fronti6res, Dreux, 1986 ) p. 327. [ 16] R704 Collab., C. Baglin et al., Phys. Lett. B 172 (1986) 455; B 195 (1987) 86. [ 17 ] S.J. Brodsky, in: Particle physics: the factory era, Proc. 1991 Lake Louise Winter Institute (Canada, 1991), and references therein. [ 18 ] M. Anselmino, F. Caruso, S. Forte and B. Pire, Phys. Rev. D38 (1988) 3516. [19]See, e.g., E. Menichetti, in: Proc. First Workshop on Antimatter physics at low energy, eds. B.E. Bonner and L.S. Pinsky (Fermilab, Batavia, IL, 1986 ) p. 95. [20] A. Hasan (E760 Collab.), in: Proc. Lake Louise Winter Institute (Alberta, Canada, 1991 ), to appear. [21 ] R704 Collab., C. Baglin et al., Phys. Lett. B 171 (1986) 135. [22]J.L. Rosner, in: Proc. Banff Summer Institute (CAP) (1988), eds. A.N. Kamal and F.C. Khanna (World Scientific, Singapore, 1989 ) p. 395. [23] R.W. Robinen and L. Weinkauf, in progress. [24] L. Bergstr/Sm, H. Grotch and R.W. Robinett, Phys. Rev. D 43 (1991) 2157. [ 25 ] L. Bergstr6m and P. Ernstr6m, Stockholm preprint USITP91-06, unpublished. [ 26 ] L. Bergstr6m, P. Ernstr6m and R.W. Robinen, Pennsylvania State preprint PSU/TH/88, unpublished. [27 ] J.H. Ki.ihn, J. Kaplan and E.G.O. Safiani, Nucl. Phys. B 157 (1979) 125; see also B. Guberina, J.H. Kiihn, R.D. Peccei and R. Ri.ickl, Nucl. Phys. B 174 (1980) 317. [28] G. B61anger and P. Moxhay, Phys. Lett. B 199 (1987) 575. [29] I.D. King and C.T. Sachrajda, Nucl. Phys. B 279 (1987) 785. [30] I.R. Zhitnitsky, A.A. Ogloblin and V.L. Chernyak, Sov. J. Nucl. Phys. 48 (1988) 536. [ 31 ] P. Moxhay and J.L. Rosner, Phys. Rev. D 28 ( 1983 ) 1132; P. Moxhay, private communication. [32] W. Kwong and J.L. Rosner, Phys. Rev. D 38 (1988) 279. [33] Particle Data Group, G.P. Yost et al., Review of particle properties, Phys. Lett. B 204 (1988) 1.