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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,QSzQ (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      8a s C F        ( ) 2m( x1l  y1k  x1 y2 q)
                                                                                        L      8a 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  8a s C F          ( )   2m( x2l  y2 k  x2 y2 q) L      8a 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
fc    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         0.300..02   (Note that (x) ~ d(x-1/2) as v0
                                   0 04
                        cc                                 ~ 6x(1-x) as v1 )

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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