productions in B decays Pyungwon Ko 1; Seoul National University, Seoul , Korea (October 3, 1995) Abstract

Transcription

1 SNUTP 95{06 Inclusive S wave charmonium productions in B decays Pyungwon Ko ; Department of Physics, Hong-Ik University, Seoul -79, Korea Jungil Lee ;y and H.S. Song ;z Department of physics and Center for Theoretical Physics Seoul National University, Seoul 5-7, Korea (October 3, 995) Abstract The inclusive S wave charmonium production rates in B decays are considered using the Bodwin-Braaten-Lepage (BBL) approach, including the relativistic corrections and the color-octet mechanism suggested as a possible solution to the 0 puzzle at the Tevatron. We rst consider relativistic and radiative corrections to J= e + e and J= Light Hadrons (LH), in order to determine two nonperturbative parameters, hj= jo ( 3 S )jj= i, hj= jp ( 3 S )jj= i, in the factorization formulae for these decays. Using these two matrix elements and including the color-octet cc( 3 S ) state contribution, we get a moderate increase in the decay rates for B decays into J= (or 0 ) + X. Our results, B(B J= (or 0 )+X)=0:5 (0:3)% for M b =5:3 GeV, get closer to the recent CLEO data. As a byproduct, we prefer a larger decay rate for c LH compared to the present data. Typeset using REVTEX pko@phyb.snu.ac.kr y jungil@phyy.snu.ac.kr z hssong@phyy.snu.ac.kr

2 I. INTRODUCTION It has been commonly believed that inclusive production rates of a heavy quarkonium state in various high energy processes can be adequately described by perturbative QCD (PQCD) within the color-singlet model []. However, recent observations of inclusive J= and 0 productions at high p T at the Tevatron suggest that a new mechanism is called for beyond the color-singlet model []. For the J= production, the lowest order subprocess comes from parton fusions, which are smaller than the data by more than an order of magnitude. If one includes the gluon fragmentations into J= and into cj (P ) states followed by cj (P ) J= + (which is the next-to-leading order in s ), one gets higher theoretical estimates that still underestimates the experimental yield by factor of 3 5. For the 0 production, the situation is even worse. Even with the gluon fragmentation included, the theoretical production rate falls below the data by factor of 30 or so. In order to resolve this puzzle, basically two scenarios have been suggested up to now : (i) existence of new charmonium states above the DD threshold, which can decay into J= and 0 with appreciable branching ratios [3]{ [5], and (ii) importance of the gluon fragmentation into a pointlike color-octet S wave cc ( 3 S ) state, and its subsequent evolution into 0 [6]. Both scenarios are quite intriguing in a sense that they call for new elements of physics within the standard model, new spectroscopy or new production mechanism for charmonium states. It would be useful to explore and test these suggestions in places other than pp colliders such as the Tevatron. In Ref. [7], one of us has explored the consequences of the rst scenario, nding that these hypothetical cj (P ) states should be observed at the level of (0:3 0:5)% (in branching ratio) in the decay channels of B c;j=; (P )+X (followed by c;j=; 0 + with 0% in branching ratio, if the rst scenario is to work). In this paper, we explore the consequences of the second scenario in the inclusive decays of B mesons into a S wavecharmonium. It is well known that the lowest order results in the heavy quark velocity and the strong coupling constant for (B J= + X) direct is smaller than the CLEO data by factor of 3 []. The situation does not get better even if the next-to-leading order corrections in s are included in the nonleptonic eective weak hamiltonian for B decays [9]. In Ref. [], only the color-singlet contribution has been included, since the color-octet contributions are higher order in v, hence suppressed relative to the singlet contributions. However, the Wilson coecient for the color-singlet contribution is much suppressed compared to that for the color-octet contribution. Therefore, the coloroctet contribution, being suppressed by v,maybenumerically important, because of the larger Wilson coecient. This is similar to the case of the gluon fragmentation into cj (P ) states [0], for which the color-octet contribution is lower order in s compared to the colorsinglet contribution, whereas both of them are of the same order in v. Also, in the case of 0 production through the gluon fragmentation into a color-octet cc state, the octet contribution is higher by v compared to the color-singlet contribution, but this is compensated by the large short distance factor = s compared to the color-singlet contribution [6]. In Sec. II, the S wavecharmonium production rates in B decays are calculated in the framework of Nonrelativistic QCD (NRQCD) [] including the relativistic corrections and a color-octet [cc( 3 S )] contribution which isofo(v ) compared to the nonrelativistic limit. The results contain three nonperturbative parameters, h0jo H ( 3 S )j0i; h0jp H ( 3 S )j0i and h0jo H ( 3 S )j0i. Among these three parameters appearing in the heavy quarkonium

3 productions, the rst two color-singlet matrix elements can be related another parameters, hhjo ( 3 S )jhi and hhjp ( 3 S )jhi, which enters in the heavy quarkonium decays in the vacuum saturation approximation [], via h0jo H j0i(j +)hhjo jhi +O(v ) : () In Sec. III A, the two parameters hhjo ( 3 S )jhi and hhjp ( 3 S )jhi (with H = J= ; 0 ) are determined by analyzing the decays of J= and 0 into light hadrons (LH) and e + e. Implications of this analysis on the decays of c into light hadrons and are discussed. Our result prefers larger decay rate for c LH. In Sec. III B, we present numerical estimates for S wavecharmonium production rates in B decays, using the factorization formulae obtained in Sec. II and h0jo H ( 3 S )j0i; h0jp H ( 3 S )j0i obtained in Sec. III A. We also discuss the polarization of the J= in B decays. In Sec. IV, the results are summarized, and possible improvements of the present work are speculated. II. HIGHER ORDER CORRECTIONS The eective Hamiltonian for b ccq ( with q = d; s) is written as [] H ef f = G F p C V cb V + C cq c ( 5 )c q ( 5 )b 3 +(C + +C )c ( 5 )T a c q ( 5 )T a b] ; () where C 's are the Wilson coecients at the scale M b. We have neglected penguin operators, since their Wilson coecients are small and thus they are irrelevant to our case. To leading order in s (M b ) and to all orders in s (M b ) ln(m W =M b ), the above Wilson coecients are C + (M b ) 0:7; C (M b ) :3: (3) According to the factorization theorem for the S decays, one has [] wave charmonium productions in B (b J= + X) = h0joj= ( 3 S )j0i 3Mc ( S 0 )j0i (b c + X) = h0joc M c ^(b (cc) ( 3 S )+X); () ^(b(cc) ( S 0 )+X); (5) in the nonrelativistic limit, where ^ are rates for hard subprocesses of b quark decaying into a cc pair with suitable angular momentum and vanishing relative momentum in the color-singlet : ^(b (cc) ( 3 S )+s; d) =(C + C ) + M c ^0; (6) Mb ^(b (cc) ( S 0 )+s; d) =(C + C ) ^0; (7) 3

4 with ^0 jv cb j G F M 3 b M c M c M b : () The operator O H ( S+ S J ) is dened in terms of heavy quark eld operators in NRQCD. Its matrix element h0jo H ( S+ S J )j0i contains the nonperturbative eects in the heavy quarkonium production processes, and is proportional to the probability that a cc in a color-singlet S wave state fragments into a color-singlet S wave cc bound state such as aphysical J=, c or 0. It is also related to the matrix element hhjo ( S+ S J )jhi as in Eq. (), and also with the nonrelativistic quarkonium wavefunction as follows : h0jo J= ( 3 S )j0i3hj= jo ( 3 S )jj= i 9 jr (0)j ; (9) in the nonrelativistic limit. Similar expressions hold for the case of nonperturbative matrix elements appearing in the c productions and its decays : h0jo c ( S 0 )j0ih c jo ( S 0 )j c i 3 jrc(0)j : (0) Note that dependence on the radial quantum numbers n enters through the nonperturbative parameters, h0jo H ( 3 S )j0i. Using the leptonic decay width of J= and 0, one can determine hj= jo ( 3 S )jj= i:0 GeV 3 ; () h 0 jo ( 3 S )j 0i9:70 GeV 3 ; () in the nonrelativistic limit with s (M c )=0:7. Also, to the lowest order in v, one has h c jo ( S 0 )j c i = hj= jo ( 3 S )jj= i because of heavy quark spin symmetry []. From these expressions with M b 5:3 GeV, one can estimate the branching ratios for B decays into J= + X and 0 + X : B(B J= + X) =0:3%; (0:0 0:0)%; (3) B(B c + X) =0:%; (< 0:9% (90%C:L:)); () B(B 0 + X) =0:0%: (0:3 0:0 0:03)%: (5) We follow the notations in Ref. [], and will not give explicit forms for these dimension-six operators in this paper. The radiative corrections in s has not been included here for consistency. To be consistent with the velocity counting rules in the NRQCD in the Coulomb gauge for the heavy quarkonia [], one has to include the relativistic corrections as well, since v s (Mv)inheavy quarkonium system. If one includes the O( s ) radiative corrections to J= l + l without relativistic corrections, one gets a larger h0jo J= ( 3 S )j0i compared to the lowest order result, Eq. () : h0jo J= ( 3 S )j0i : 0 GeV 3. Relativistic corrections gives a further enhancement. See Eq. () below.

5 The recent data from CLEO [] are shown in the parentheses, where the cascades from B cj (P )+X followed by cj J= + have been subtracted in the data shown. In view of these results, we may conclude there are some important pieces missing in the calculations of decay rates for B (cc) ( 3 S )+X using the color-singlet model in the nonrelativistic limit. In Ref. [], it was noticed that the inclusion of color-octet piece, which is often neglected in the previous studies of the charmonium production in B decays, is mandatory in order to factorize the amplitude consistently without any infrared divergence in case of B decays into the P wavecharmonium. This also leads to nonvanishing decay rates for B (h c ; c0 ; c )+X, all of which vanish in the color-singlet model. In view of this, we rst estimate the color-octet contributions to B J= +X, motivated by the suggestion that the color-octet mechanism might be the solution to the 0 puzzle at the Tevatron. Although it is of higher order in v ( O(v )), it can be important in the case of the inclusive B decays into J= + X, since the Wilson coecient of the color-singlet part is suppressed compared to that of the color-octet part by a factor of s. ( In Eq. (), (C + C ) 0:, and (C + + C ) :0. ) Now, it is straightforward to calculate the contribution of (cc) ( 3 S )tobj= + X : (B (cc) ( 3 S )+X J= + X) = h0joj= ( 3 S )j0i Mc ( S 0 )j0i (C + + C ) + M c M b ^0; (6) (B (cc) ( S 0 )+X c +X)= 3h0jOc (C Mc + +C ) ^0: (7) (Similar expression holds for the B 0 (or 0 )+X except that h0jo H ( 3 S )j0i and M c =M b should change appropriately in order to account for the phase space eects.) Here, a new nonperturbative parameter h0jo ( 3 S )j0i comes in, which creates a (cc) pair in the color-octet state, projects into the subspace of states which contain J= in the asymptotic future, and then annihilates the (cc) pair at the creation point. The matrix element of this operator is proportional to the probability that the color-octet (cc) ( 3 S )to fragment into the physical J= state in the long distance scale. This type of a color-octet operator was rst considered in the gluon fragmentation function into P wavecharmonia in Ref. [0], and then in the gluon fragmentation into 0 to solve the 0 puzzle at the Tevatron. Braaten and Fleming xed h0jo 0 ( 3 S )j0i to be :0 3 GeV 3 (for M c :5 GeV), in order to t the total cross section for the inclusive 0 production cross section at the Tevatron, and found that this value of h0jo 0 ( 3 S )j0i yields the p T spectrum for the 0 production which nicely agrees with the measured shape. Then, Cho and Leibovich have performed a complete analysis for the color-octet contribution to Upsilon and Psi productions at the Tevatron for both low and high p T regions [3]. Their results are h0jo J= ( 3 S )j0i =:0 GeV 3 ; () h0jo 0 ( 3 S )j0i =7:30 3 GeV 3 : (9) Taking the ratio between the color-octet and the color-singlet contributions, one gets (B (cc) ( 3 S )+XH+X) (B (cc) ( 3 S )+XH+X) = 3h0jOH ( 3 S )j0i h0jo H ( 3 S )j0i (C + +C ) (C + C ) (0) =0:76 (:) for H = J= ( 0 ); () 5

6 Thus, we nd that the color-octet (cc) ( 3 S ) contributions to B J= (or 0 )+X are about 76 % ( %) of the color-singlet contributions in the nonrelativistic limit. There is another subprocess in the lower order in v compared to the color-octet contribution considered above : the relativistic corrections to the color-singlet component which is an order of O(v ) compared to (),(5). Extending the Feynman rule in the presence of a heavy quarkonium [], we can derive that the relativistic corrections to B J= (or c )+X can be written as the following factorized form : (b J= + X) = h0jpj= ( 3 S )j0i 9Mc ( S 0 )j0i (b c + X) = h0jpc M c ^(b (cc) ( 3 S )+X); () ^(b(cc) ( S 0 )+X): (3) Here again, one can use the vacuum saturation approximation, Eq. () : h0jp H ( S+ S J )j0i(j +)hhjp ( S+ S J )jhi: () wave heavy quarkonium wave- The latter is related with the spin-weighted average of the S functions in the following way : hnsjp jnsi = 3Re (R ns r R ns ) +O(v ) ; (5) R ns is the spin-weighted average of the S wave wavefunctions [] : R ns (3R + R c ) : (6) In order to estimate the relativistic corrections, we need one more nonperturbative matrix element, h0jp ns ( 3 S )j0i or Re (R ns r R ns ). This is not available in the current literature now, and we will determine this parameter as well as h0jo ( 3 S )j0i in the following section. However, we note that the relativistic corrections make the decay rates for the S wave charmonium productions in B meson decays decrease because h0jp ns III. NUMERICAL ANALYSIS A. Analysis of the S wave charmonium decays ( 3 S )j0i > 0. In the previous section, we derived the S wave charmonium production rates in B decays to O(v ) for the color-singlet contributions, including one of the color-octet contributions at O(v ). The results depends on three nonperturbative parameters, h0jo ( 3 S )j0i; h0jp ( 3 S )j0i; h0jo ( 3 S )j0i or equivalently, jr (0)j and Re(R Sr R S ) for the rst two color-singlet matrix elements. Since the third parameter h0jo ( 3 S )j0i is xed from the t to the 0 production at the Tevatron [6], we consider the other two parameters in this subsection. In order to determine these two parameters, one has to invoke the lattice calculations, some potential models. Or, one can simply determine these parameters from the well measured decay rates of J= ; c which depend on the same parameters. We choose 6

7 the last method to x two nonperturbative parameters, h0jo ( 3 S )j0i; h0jp ( 3 S )j0i, in this work. For this purpose, we list the decay rates for ggg + gg LH(light hadrons) and l + l [] : ( LH) =Imf ( 3 S )hj= jo ( 3 S )jj= i +Img ( 3 S )hj= jp ( 3 S )jj= i + O(v ); (7) ( e + e )=Imf ee ( 3 S ) hj= jo ( 3 S )jj= i +Img ee ( 3 S ) hj= jp ( 3 S )jj= i + O(v ): () We need to know the short distance coecients Imf's to O( s ), and Img's to the leading order only in s, because of the velocity counting rule in the NRQCD []. From the results in the earlier literatures [] [], one can extract Imf Imf 3g (J= ggg)+imf (J= gg); (9) Imf 3g ( 3 S )= ( 9)(Nc )C F 5N c 3 s (M) +( 9:6()C F +:3(7)C A :6()n f ) s Img 3g ( 3 S )= :33 ( 9)(Nc )C F 5N c 3 s(m); (3) Imf ( 3 S )= ( 9)C F Q 3N s(m) +( 9:6C F +:75C A 0:77n f ) s c (3) Img ( 3 S )= :33 ( 9)C F Q 3N s(m); c (33) Imf ee ( 3 S )= Q s C F ; 3 (3) Img ee ( 3 S )= Q ; (35) Imf qq ( 3 S )=Q i Q i C s F ; (36) Img qq ( 3 S )= 3 Q i Q i : (37) In the above expressions, N c = C A = n f = 3 and C F ==3, and Q is the electric charge of a heavy quark in the unit of the proton charge. The strong coupling constant s (M) is dened in the modied minimal subtraction scheme (MS) for QCD with n f light quarks, renormalized at the scale = M. The last two are relevant toj= hadrons. Similar expressions for the c decays are ( c LH) =Imf ( S 0 )h c jo ( S 0 )j c i+img ( S 0 )h c jp ( S 0 )j c i+o(v ); (3) ( c )=Imf ( S 0 ) h c jo ( S 0 )j c i +Img ( S 0 ) h c jp ( S 0 )j c i + O(v ); (39) where (30) Imf ( S 0 )= C F N c " + s(m) ( 5 C F ) C A 9 n s f ; (0) 7

8 Img ( S 0 )= C F 3 N s(m); () c Imf ( S 0 )=Q " + 5 C F s ; () Img ( S 0 )= 3 Q : (3) The PDG lists the measured data for these decays as follows [5] : ( LH) = (60:7 :7) kev; () ( e + e )=(5:6 0:37) kev; (5) ( c LH) =0:3 +3: 3: MeV; (6) ( c )=7:0 +:0 :7 kev: (7) Since the data on c decays are not precise enough yet, we use the data on J= decays only in order to determine two parameters hj= jo ( 3 S )jj= i and hj= jp ( 3 S )jj= i. Since we don't have enough inputs available at this level, we use s (M) tobe0:5 0: instead of treating it as a free parameter. Although this choice is not fully systematic from the view point of perturbative QCD, our numerical results presented below should not strongly depend on the exact value of s (M). From LH and e + e,we determine hj= jo ( 3 S )jj= i0:0 (0:90) GeV 3 ; () hj= jp ( 3 S )jj= i0:05 (0:03) GeV 5 ; (9) for s =0:5 (0:), respectively. For 0,we get h 0 jo ( 3 S )j 0i0:77 (0:9) GeV 3 ; (50) h 0 jp ( 3 S )j 0i0:00 (0:0) GeV 5 : (5) Note that the radiative corrections to the J= and 0 decays are fairly large with or without the h0jp ( 3 S )j0i term. Radiative corrections increase hj= jo ( 3 S )jj= i in Eq. (9) by 0%. The relativistic correction term, hj= jp ( 3 S )jj= i, isabout9% of Eq. (9), hence decreases the B J= + X rate (Eq. ()) by %. Next, let us determine h c jo ( S 0 )j c i which is expected to be h c jo ( S 0 )j c i = hj= jo ( 3 S )jj= i ( + O(v )); (5) due to the heavy quark spin symmetry. Since h0jp ( 3 S )j0i is independent of the total spin S, one can determine h c jo ( S 0 )j c i from one of the decays, Eq. (3) or Eq. (39), and then predict the other and compare with the measured rate. Also, the relation (5) should be respected in order to be consistent with the NRQCD and the heavy quark spin symmetry. If we use (6) as an input, we get (with s =0:5) h c jo ( S 0 )j c i(0:9 +0:053 0:06) GeV 3 ; (53) ( c )=(: +: :0) kev; (5)

9 the former of which severely violates the heavy quark spin symmetry, (5). For s =0:, the numbers above change into 0:6 +0:039 0:035 GeV3 and : 0:7 +0: kev. On the other hand, if we use (7) as an input with with s =0:5, we get h c jo ( S 0 )j c i(0:36 +0:09) 0:00 GeV3 ; (55) ( c LH) = (3 +7 6) MeV; (56) For s =0:, the numbers become 0:3 +0:097 0:03 GeV3, and 3 +9 MeV, respectively. Now, the relation () is better obeyed, although the dierence between hj= jo ( 3 S )jj= i and h c jo ( S 0 )j c i is not that small, and the predicted rate for c LH is quite large compared to the data, (6). One may conclude that the factorization formulation by BBL in terms of the NRQCD predicts rather large value of c LH, compared to the current experimental values, considering the large uncertainties in the measurements. A better determination of ( c LH) would test the validity of the factorization approach too(v ). B. Results for B decays Since all the relevant nonperturbative parameters are in hand now, we are ready to estimate the branching ratio for B J= + X which includes O(v ) corrections and one of the O(v ) color-octet contribution. Adding up the change in h0jo H ( 3 S )j0i and the color-octet contributions, we get a moderate increase in the branching ratio by factor of (:0+0:0 0:0 + 0:76) = :5 of the lowest order prediction (3) for the B J= + X case, and (:0+0:0 0:0+:) = :9 for the 0 case : B(B J= + X) =0:5 %; (57) B(B 0 + X) =0:3 %; (5) compared to the data, (0:00:0)% and (0:30:00:03)%. We note that the agreements between theoretical estimates and the data get improved, after the radiative corrections and the color-octet mechanism have been included. There is a residual uncertainty related with the b quark mass M b. In this work, we havechosen M b 5:3 GeV, and normalized the decay rate to that of the semileptonic B decay in order to reduce the uncertainty from less known M b []. If we use M b =:5 GeV, for example, all the decay rates should be multiplied by a factor of 5:3=:5 :. For B c + X, our prediction is tampered by less known h0jo c ( S 0 )j0i as well as h0jo c ( S 0 )j0i. Still, we expect that the decay rate increases by a factor of or more over the lowest order result, (). Let us nally consider the polarization of J= 's produced in B decays. It is convenient to dene two parameters, and as follows : = T (B J= + X) T+L(B J= + X) ; (59) = 3 : (60) The quantity can be readily measured through the polar angle distribution of dileptons in J= l + l in the rest frame of J= : 9

10 d (J= l + l ) dcos / +cos ; (6) where is the angle between the ight direction of a lepton in the rest frame of J= and the ight direction of J= in the rest frame of the initial B meson. For the unpolarized J=, we would have = =3( = 0) corresponding to the at cos distribution of dileptons. Assuming the factorization for B J= + X in the color-singlet component in the eective Hamiltonian (), one nds that J= 's produced in B decays are polarized with [6] = M b M 0:9 ( 0:36) (6) Mb + M or = 0: (0:9) for M b = 5:3 (:5) GeV, which nicely compares with the CLEO measurement [7], exp = 0: 0:6. One may wonder if the color-octet mechanism considered in this work can change the polarization of J= substantially. However, the structure of the amplitude for b J= + X due to the color-octet 3 S state is the same as that due to the color-singlet mechanism (including both the lowest and the next-to-leading order terms in v ). Namely, the amplitude for b J= + X is proportional to M/ s ( 5 )b: (63) Thus, the prediction for remains the same even in the presence of the (cc) ( 3 S ) color-octet contribution. This is in sharp contrast with the case of 0 production through the gluon fragmentation into the color-octet cc state which in turn evolves into 0. In the case of gluon fragmentation (g (cc)( 3 S ) J= + X), the initial gluon (with q (M c ) and q 0 >> M c ) is almost on shell, being almost transverse up to q =q 0.Thus, the polarization of the color-octet cc is also almost transverse. Because of the spin symmetry of heavy quark system, the polarization of the daughter J= is the same as the parental color-octet cc state. S IV. CONCLUSIONS In this work, we considered the relativistic corrections and a color-octet contribution to wavecharmonium productions in B decays. Our results, (57) and (5), give moderate increase to the previous analyses based on the color-singlet mechanism in the nonrelativistic limit. Compared to the previous analyses of the lowest order in v and s,we get an 0% increase in hj= jo ( 3 S )jj= i from the radiative corrections to J= e + e and J= LH, % decrease from the relativistic correction through the hj= jp ( 3 S )jj= i term, and 76% increase in the decay rate from the color-octet contribution, h0jo J= ( 3 S )j0i. Thus, the color-octet mechanism, which has been proposed as a possible solution to the 0 puzzle at the Tevatron, could give an enhancement of the B J= + X decay rate by a moderate amount. It should be kept in mind that what we considered in this work is only one of the coloroctet operators that may contribute to B J= +X. Wehavechosen only h0jo J= ( 3 S )j0i, since we know the numerical value of this matrix element from the work of Braaten and Fleming [6]. This matrix element is rather special in the sense that it is the only color-octet operator which is relevant tog(cc) ( 3 S )J= + X in the leading order in v and s. However, for the B decays, other color-octet operators can contribute as well ; e.g., 0

11 (B (cc) ( S 0 )+XJ= + X) = 3h0jOJ= ( S 0 )j0i Mc ( 3 S )j0i (B (cc) ( 3 S )+X c +X)= h0joc M c (C + + C ) ^0; (6) (C + +C ) and similar expressions for the contributions of h0jo H (3 P J )j0i : (B (cc) ( 3 P )+X J= + X) = h0joj= ( 3 P )j0i Mc (B (cc) ( 3 P )+X c +X)= ( h0joc 3 P )j0i Mc (C + + C ) (C + +C ) + M c M b + M c + M c M b Mb ^0; (65) ^0; (66) ^0; (67) In order to estimate the eects of these color-octet matrix elements h0jo J= ( S 0 )j0i and h0jo J= ( 3 P )j0i, we need to consider other processes as well, such as+pj= + X and e + e J= +X []. For example, the height of the elastic peak for photoproduction of J= depends on these color-octet matrix elements [6]. Complete analysis of color-octet contributions to the S wavecharmonia in p collisions is called for as well [9]. Once these new color-octet matrix elements are determined from other processes, our results in this work will provide an independent test of the hypothesis of color-octet mechanism as a possible solution to 0 anomaly at the Tevatron. We have also analyzed the leptonic and the inclusive hadronic decays of J= and 0 to O(v ) in the framework of the BBL's factorization scheme, and did extract two nonperturbative parameters, hhjo ( 3 S )jhi and hhjp ( 3 S )jhi with H = J= and 0. These are important inputs in many other theoretical calculations of the S wave charmonia productions in various high energy processes and their subsequent decays. As a by-product, we have found that the inclusive hadronic decay rate for c may be larger than the current PDG value by factor of, if the BBL's factorization formulae to O(v )works with the charmonium system. The better measurements of ( c LH) would test our predictions based on the factorization approach for the heavy quarkonium decays in the framework of NRQCD to O(v ). ACKNOWLEDGMENTS We are grateful to Dr. Seyong Kim for discussions on this subject as well as the current status of the lattice calculations of the nonperturbative parameters, jr S (0)j and Re(R Sr R S ) in the system. This work is supported in part by KOSEF through CTP at Seoul National University. Also, P.K. was supported in part by NON DIRECTED RESEARCH FUND, Korea Research Foundations, and in part by the Basic Science Research Institute Program, Ministry of Education, 99, Project No. BSRI{9{5.

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