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Perturbative QCD analysis of e+e- J/y+c process Ho-Meoyng Choi(KNU) (Phys.Rev.D 76, 094010(2007) with C.-R. Ji) Outline 1. Motivation 2. Model Description - Light-front(LF) quark model(LFQM) - Nonfactorized LF PQCD approach 3. Numerical Results - Quark distribution amplitudes(DAs), Decay constants and x moments for J/y and c - Form Factor and Cross section for e+e- J/y +c 4. Conclusion LC2008(Mulhouse,July7-11) 1 1. Motivation Exclusive heavy meson pair productions provide a unique opportunity to investigate asymptotic behaviors of meson form factor within PQCD: - If factorization theorem is applicable, the invariant amplitude is given by 1 M [dxi ][ dyi ] ( xi , q 2 )TH ( xi , yi , q 2 ) ( yi , q 2 ) 0 NRQCD factorization approach for heavy meson pair production assumes that (x,q2) ~ d(x– mh/(mh+ml)) (peaking approximation). However, the large discrepancy of (e+e- J/y c) at s1/2=10.6 GeV between t he NRQCD predictions and experiments. Braaten, Lee (03) e+ (e+e- J/y c) = 2 ~ 5 fb (NRQCD) Liu, He, Chao(03) * How to resolve the discrepancy? (e e J / y c ) * Br (n 2) 25.6 2.8 3.4 fb (Belle) e- (e e J / y c ) * Br (n 2) 17.6 2.8 2.1fb (Babar) 2 Several suggestions to resolve the problems Processes proceeding via two virtual photons may be important (J/y + J/y) (Bodwin and Lee , 03) The final state contains a charmonium state and a glueball (Brodsky, Goldhaber & Lee, 03; Dulat, Hagiwara 04) Belle’s updated analysis shows: (PRD 70 (2004) 071102) e+e-** J/y + J/y contamination is small no evidence for e+e- J/y + glueball Use broad shape of quark DA instead of d-shaped quark DA (Ma and Si, 04 ; Bondar and V.L. Chernyak, 05) Calculate the NLO QCD corrections with NRQCD factorization formalism. The ratio of NLO/LO (K-factor) can reach ~ 2, which reduces the large discrepancy. (Zhang, Gao & Chao, 06) But the discrepancy still exists in the O(as). Need to better understand peaking(i.e. nonrelativistic) and beyond peaking(i.e. relativistic) approximations in O(as) Going beyond peaking approximation: - In the LF framework, the quark DA is computed from the valence LF wave function ( xi , ki ) and is given by 2 k q 2 ( xi , q ) [d ki ]( xi , ki ) 2 2 -If (x) is not exactly d-type, factorization theorem is no longer valid! - Going beyond the peaking approximation, we use the nonfactorized PQCD and the invariant amplitude should be expressed in terms of LF wave function instead of quark DA, i.e. 2 M dxdyd k d l ( x, k )TH ( x, y, k , l , q )( y, l ) 2 Our aim is to investigate e+e- J/y c process using in O(as) 3 2. Model Description PRD59, 074015(99); PLB460, 461(99) by Choi and Ji (1) LFQM: Using the variational principle to the QCD-motivated effective Hamiltonian, we fix the model parameters! Variational Principle H Q Q m k m k VQ Q 2 2 2 2 Q Q |[H 0 V0 ]| where 0 VQ Q V0 (r) Vhy p(r) and 4a s 2SQ SQ 2 a br - VCoul M Q Q |[H 0 VQ Q ]| 3r 3m Q m Q cr and 2 QSzQ (x, k ) (x, k )S,QSzQ (x, k ) S, Input parameters for the linear confining potential (x, k ) ~ exp(-k2 /2 2 ) : mu=md=220 MeV b=0.18 GeV2 4 Fit of the ground state meson masses obtained from our LFQM Experiment Linear potential Harmonic oscillator (HO)potential Input masses 5 (2) Non-factorized LF PQCD Formalism for e+e- J/y(Pv) c(Pp) e+ Form factor is defined as * J /y ( P , h)c ( PP ) | J em | 0 * P PP F (q 2 ) V V and the cross section is given by 2 3/ 2 e- a 2 4M h (e e J / y c ) | F ( s ) | 2 1 6 s Crossing Symmetry Using crossing symmetry, we first calculate 2 c(Pp) *(q) J/y(Pv) in q+=0 frame( q 2 q Q 2) e- e- and then Q2 -q2 in timelike region! q TH Hard contribution to meson form factor y x FM (Q ) [d k ][d l ]M ( y, l )TH ( x, y, q , k , l )M ( x, k ) Hard 2 3 3 * c 1-x 1-y J/y (obtained from J+ with h=+) 6 Leading order(in as) LF time-ordered diagrams for TH q A1 A2 A3 ( x1 , k ) ( y1 , y1q l ) kg ( x2 ,k ) ( y 2 , y 2 q l ) D1 D2 D3=D1 D4 D5 D6=D4 B1 B2 B3 2 2 2 (k q ) m1 k m2 2 2 E denominators: D1 M q 2 etc. x1 x2 Ti Bi TAi ( x y, k l ) i 7 Sample Calculation of Feynman Diagram A1 q (k g ) TA1 (1) u (k1 q ) u (k1 ) k g D1 D2 k1 l1 u (l1 ) u (k1 q )d u (l2 ) u (k 2 ) A1: kg k2 l2 (k g ) N A ) N A ( D1 D2 1 kg D1 D2 singular where in LF gauge(A+=0): regular k k g d (k g , g ) * (k g , g ) g g g k g ( , , ) (0,2,0 ) where the regular part NA is common for all Ai(i=1,2,3)! (c.f. instantaneous part(~1/(k+g)2) is absorbed into kg by kg- kg-=P- + q- - l1- – k2-) 8 Effective treatment of singular parts [ Ji, Pang & Szczepaniak 95, Choi and Ji 06] In terms of LF energy difference M 2 M 0 (M0 = invariant mass), 2 we expand TH in terms of as TH = [TH](0) + [TH](1) + 2[TH](2) + … We find that the sum of six diagrams for the singular part vanishes in the limit of 0 (i.e. zeroth order of binding energy limit) ! In zeroth order of , the net contribution to the hard scattering amplitude is NA TH [TH ] ( 0) lim ( x y, k l ) 0 ( y x ) D D 2 2 1 2 Some features of our method: (i) Our [TH ](0) includes the binding energy effect (i.e. k , l 0 ) that was neglected in the zero-binding(or peaking) approximation. (ii) We effectively include all higher orders of the relative quark velocity beyond <v2> 9 Table. Leading helicity contributions to the hard scattering amplitude NA STAi(=0) NB STBi(=0) 2m( x1l y1k x2 y1q) L 8a s C F ( ) 2m( x1l y1k x1 y2 q) L 8a s C F NA N B) ( x2 y1 y2 ( y2 x2 ) D1 D 2 x2 y1 y2 ( y2 x2 ) D2 D 8 2m( x2l y2 k x2 y2 q) L 8a s C F ( ) 2m( x2l y2 k x2 y2 q) L 8a s C F ( y2 x2 ) D1 D 2 NA N B) ( x1 y1 y2 x1 y1 y2 ( y2 x2 ) D2 D 8 Di = energy denominator in the limit of =x=y=0 (c.f. see subleading helicity contributions in PRD 76, 094010(07)) 3. Numerical Results Decay constants of c and J/y (in unit of MeV) Linear HO HO’ Exp. HO’ was used (=0.6509) (=0.6998) (=0.7278) for sensitivity check of our model fc 326 354 370 335 75 fJ/y 360 395 416 416 6 x(=x1-x2)-moments x xn for c and J/y n C J /y <xn> Ours BC NRQCD QCD sum [1] [2] rules[3] n=2 0.13 0.075 [1] Bondar and Chernyak, 0.084 0..004 0 007 0.070 0.007 PLB 612, 215(05) n=4 0.040 0.010 [2]Bodwin, Kang and Lee, 0.017 0.001 0.003 0.012 0.002 PRD74, 014014(06) n=6 0.066 0.0017 [3] Braguta, Likhoded, and 0.0047 0..0006 0 0010 0.0031 0.0008 Luchinsky, PLB 646, 80(07) 10 Quark DAs for Charmoninum Properties of our results: (i) Our model satisfies 2 k Braguta,Likhoded,Luchinsky (ii) Negligible in the regions Ours x<0.1 and x>0.9 where the motion of cc pair is expected to be highly relativistic. Bondar -Chernyak (iii) Far from d-function type (i.e. 0 limit) (iv) Quark DA becomes broader and more enhanced at the end points as increases. 11 Relative quark velocity and NRQCD v-scaling rules In our LFQM, we get [mrv2/2 ~ 2(mc2 + k2)1/2 – 2mc] 2 0.300..02 (Note that (x) ~ d(x-1/2) as v0 0 04 cc ~ 6x(1-x) as v1 ) According to the relations (1)<xn> ~<vn>/(n+1)(n=2,4,6) [Braguta,Likhoded, Luchinsky 07] and (2) (mcv2)2 << (mcv)2 << mc2 [Bodwin, Bratten, Lepage 95] Our result for x-moments satisfy the v-scaling rules but Bondar-Chernyak(BC) moments do not! Form factor s2F(s) for e+e- J/y + c Fd ~ fhcfJ/y as/q4 FHT(leading + subleading) =NRQCD (dotted line) FLT ( ) FLT = Leading twist factorized FHT(leading helicities) form factor taking into account of relative motion of valence quarks(short-dashed line) Fd FHT = Higher twist nonfactorized form factor -leading helicities(long-dashed line) -(leading + subleading) helicities (solid line) Subleading helicities 12 Cross section for e+e- J/y + c Cross section at s1/2 = 10.6 GeV d ( J /y c ) 2.34 0..50 [ fb ] 0 69 LT ( J /y c ) 10 .57 3..15 [ fb ] 4 02 HT ( J /y c ) 8.76 1..61 [ fb ] Full 2 84 where the central, upper and lower values correspond to HO, HO’ and linear potential parameters. 13 Sensitivity check of our model parameters for the cross section Parameter (mc,b) dependence of (e+e- J/y c) using the nonfactorized higher twist form factor with all helicity contributions: -Cross section increases as b(mc) increases(decreases) -Cross section is more sensitive to the variation of gaussian parameter than the the charm quark mass 14 4. Conclusions We investigated the transverse momentum effect on the exclusive J/y+ c pair production in e+e- annihilation using the nonfactorized PQCD and LFQM that goes beyond the peaking approximation. (1) Quark DAs for J/y and c take substantially broad shape far from the d-type DA. - Relative motion of valence quark is significant! - Factorization theorem is no longer applicable! (2) In going beyond the peaking approximation, we stressed a consistency by keeping the transverse momentum both in the wave function part and the hard scattering part. -Even if the used LF wave function leads to the similar shape of quark Das, predictions for the cross section are different between the factorized and nonfactorized analyses. (3) Our higher twist result including all helicity contributions enhances the NRQCD result by a factor of 3~4 while it redueces that of the leading twist result by 20% - Cross section increases as (mc) increases(decreases). 15

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