Conformal symmetry and pion form factor: Soft and hard contributions Ho-Meoyng Choi(Kyungpook Nat’l Univ.) 2007 APCTP Workshop on Frontiers in Nuclear and Neutrino Physics anti-de Sitter space geometry /conformal field theory(AdS/CFT) Correspondence [Maldacena,1998] QCD (with massless quark) Brodsky and de Teramond [PRL 96, (06), PRL 94, (05)] Light-front holographic wavefunction  LF display confinement at large inter-quark separation(z large) and conformal symmetry at short distances(z small). String amplitude  z   LF holographic mapping QCD: LFQM + PQCD Refs: PRD 74, (06); PRD74, xx(07, Feb.)[hep-ph/ ][Choi and Ji] z=0

Outline 1. Introduction on Light-Front(LF) formulation 2. Light-Front Quark Model(LFQM) description 3. LFQM prediction of pion form factor: (I) Quark distribution amplitude(DA) (II) Soft(LFQM) and hard(PQCD) contributions to pion form factor (III) Comparison of LFQM and Ads/CFT correspondence results on DA and form factor (IV)  transition form factor (V)  and Gegenbauer moments of pion 4. Conclusion

Comparison of equal-t and equal LF-  = t+z/c(=x + =x 0 +x 3 ) coordinates Equal t Equal  p·q=p 0 q 0 -p·q p·q=(p + q - +p - q + )/2-p  ·q  (p ± =p 0 ±p 3 ) p + : longitudinal mom. p - =LF energy p 2 = m 2 p 0 =  p 2 +m 2 p 2 =m 2 p - =(p 2  +m 2 )/p + (P + : positive) z=x 3 ct=x 0 ct+z=x 0 +x 3 =x + =c  x +  Poincare’ group(translations P , rotations L and boost K) Kinematic generators: P and L for ET(6) P +,P , L 3 and K for LF(7) Light front(LF)

t t’  = t+z/c z t v ct’=  (ct+  z) z’=  (z+  ct)  =v/c and  =1/(1-  2 ) 1/2  ’ = e    = cosh   = sinh  t=0 is not invariant under boost!  =0 is invariant under boost!  LF  x  k   x(=k + /P + ) 1-x Advantages of LF: (1) Boost invariance

Advantages of LF: (2) Vacuum structure k1k1 k2k2 k3k3 k 1 +k 2 +k 3 =0 k1+k1+ k2+k2+ k3+k3+ k k k 3 + =0 Equal t Equal  Not allowed ! since k + >0 t  Physical LF vacuum(ground state) in interacting theory is trivial(except zero mode k + =0)! k 0 =  k 2 +m 2 k - =(k 2  +m 2 )/k +

Advantages of LF: (3) Covariant vs. time-ordered diagram LF valence LF nonvalence

Electromagnetic Form factor of a pseudoscalar meson (q 2 =q + q - -q 2  <0 region) in LF  F(Q 2 ) =  [dx][d 2 k  ]   n (x,k’   n  x,k  ) in q + =0 frame e e’  P P+q = q 2 =-Q 2 x,k  +(1-x)q  x,k  nn nn  n+2 nn + q+q+ P=(P +,M 2 /P +,0  ), q=(0,2P.q/P +,q  ) in q + =0

Model Description PRD59, (99); PLB460, 461(99) by Choi and Ji 1/4 for 1 — -3/4 for 0 -+ H 0 =M 0 Normalization:

Fixing Model Parameters by variational principle Input for Linear potential: m u =m d =220 MeV, b=0.18 GeV 2 +  splitting fix a= GeV,  qq = GeV and  =0.313 Central potential V 0 (r) vs. r Phys. Rev. D 59, (99) by Choi and Ji

Ground state meson spectra[MeV] PLB 460, 461(99); PRD 59, (99) by Choi and Ji

Model Parameters and Decay constants PRD 74(07) (Choi and Ji) (1.4)(44) 161[155] [0.3419] 0.45 [0.48] (10)(36) 130[131] [0.3194] 0.22 [0.25] f exp [MeV]f th [MeV]  qQ [GeV] m Q [GeV]  246[215](f L ) 188[173](f T ) 220(2)(f L ) 160(10)[SR:Ball] 0.22 [0.25] [0.3194] K*K* 0.45 [0.48] [0.3419] 256[223](f L ) 210[191](f T ) 217(5)(f L ) 170(10)[SR:Ball] Linear[HO] Sum-rule [Leutwyler, Malik]:  K *[For heavy meson sector: hep-ph/ (Choi)] important for LCSR predictions for B to  or K*

Quark DA and soft form factor for pion PRD 59, (99); PRD74, [Choi and Ji]PRD74, (06)[Choi and Ji] F(Q 2 )~exp(-m 2 /4x(1-x)  2 )

Comparison of LFQM respecting conformal symmetry with the Ads/CFT prediction F(Q 2 )~exp(-m(Q 2 ) 2 /4x(1-x)  2 )

e-e- e-e- MM e - + M q x 1-x y 1-y THTH Hard contribution to meson form factor where  (x,k  )=  R (x,k  )x (spin w.f.) ( ,  ) + ( ,  ) PQCD analysis of pion form factor D1D1 D2D2 D 3 =D 1 D4D4 D5D5 D 6 =D 4 kgkg A1A1 A2A2 A3A3 B1B1 B2B2 B3B3 q  dx][dy]  (x,Q 2 )T H (x,y,Q 2 )  (y,Q 2 ) leading twist

D1D1 D2D2 kgkg A1:A1: q k1k1 k2k2 l1l1 l2l2  where in LF gauge(A + =0): Calculation of Feynman Diagram regular singular: need higher Fock state! If then T H (1/k + g ) =0 finite contribution

Effective treatment of singular part In terms of LF energy differences defined by we obtain In zeroth order of  x and  y, i.e Therefore, total hard scattering amplitude is given by, the singular parts vanish via

Suppresion of DA at the end points leads to enhancement(suppression) of soft(hard) form factor! Soft(LFQM) and hard(PQCD) contribution to pion form factor PRD74, (06)[Choi and Ji] HOLinear ( ,  ) + ( ,  ) ( ,  ) ( ,  ) AdS/CFT=(16/9) x PQCD PQCD

 Transition Form Factor    x,k T 1-x,-k T 0,q T 1,q T PQCD Ads/CFT =(4/3) PQCD Linear(LO) HO(LO) NLO

Quark DA Transverse size of meson Decay constant Quark DA in Gegenbauer polynomials Quark distribution amplitudes(DAs) and decay constants

L. Del Debbio[Few-Body Sys. 36,77(05)] (Lattice) (CLEO Collab.) (E791 Collab.) (asymp) (Transverse lat.) (Chernyak and Zhitnitsky) <2><2> Second  moment of pion Our results [PRD75(07):Choi and Ji] = 0.24 for linear =0.22 for HO Ours Gegenbauer moments

1  -error ellipse twist-two Gegenbauer moments a 2 and a 4 for pion twist-four asymp. CZ Ours: a 2 [a 4 ]= 0.12[-0.003] for linear =0.05[-0.03]for HO LCSR-based CLEO-data analysis

Conclusions and Discussions 1. We investigated quark DA and electromagnetic form factor of pion using LFQM. 2. Our LFQM is constrained by the variational principle for QCD-motivated effective Hamiltonian establish the extent of applicability of our LFQM to wider ranging hadronic phenomena. (a)Our quark DA is somewhat broader than the asymptotic one and quite comparable with AdS/CFT prediction (b) In massless limit, our gaussian w.f. leads to the scaling behavior F~1/Q 2 consistent with the Ads/CFT prediction (c) We found correlation between the quark DA and (soft and hard) form factors (d) Our  and Gegenbauer moments of pion are quite comparable with other model predictions such as (1) Electromagnetic form factors of PS and V[PRD56,59,63,65,70 ] (2) Semileptonic and rare decays of (PS to PS) and (PS to V)[PRD58,59,65,67,72; PLB460,513] (3) Deeply Virtual Compton Scattering and Generalized Parton Distributions(GPDs)[PRD64,66] (4) PQCD analysis of meson pair production in e+e- annihilations[PRD 73]