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2007 APCTP Workshop on Frontiers in Nuclear and Neutrino Physics

Brodsky and de Teramond [PRL 96, 201601(06), PRL 94, 201601(05)]. QCD (with massless quark). String amplitude F( z ) Y LF. holographic mapping. QCD: LFQM + PQCD. Conformal symmetry and pion form factor: Soft and hard contributions Ho-Meoyng Choi(Kyungpook Nat’l Univ.).

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2007 APCTP Workshop on Frontiers in Nuclear and Neutrino Physics

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  1. Brodsky and de Teramond [PRL 96, 201601(06), PRL 94, 201601(05)] QCD (with massless quark) String amplitude F(z) YLF holographic mapping QCD: LFQM + PQCD Conformal symmetry and pion form factor: Soft and hard contributionsHo-Meoyng Choi(Kyungpook Nat’l Univ.) Refs: PRD 74, 093010(06); PRD74, xx(07, Feb.)[hep-ph/0701177][Choi and Ji] anti-de Sitter space geometry /conformal field theory(AdS/CFT) Correspondence[Maldacena,1998] Light-front holographic wavefunction YLF display confinement at large inter-quark separation(z large) and conformal symmetry at short distances(z small). z=0 2007 APCTP Workshop on Frontiers in Nuclear and Neutrino Physics

  2. 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) p-g transition form factor (V) x and Gegenbauer moments of pion 4. Conclusion

  3. Comparison of equal-t and equalLF-t=t+z/c(=x+=x0+x3)coordinates ct=x0 ct+z=x0+x3=x+ =ct z=x3 Light front(LF) x+=0 Poincare’ group(translations Pm, rotations L and boost K) Kinematic generators: P and L for ET(6) P+,P^, L3 and K for LF(7)

  4. t=t+z/c t x(=k+/P+) t YLF(x,k^) v t’ 1-x Advantages of LF: (1) Boost invariance z t’= eft g= coshf bg= sinhf ct’=g(ct+bz) z’=g(z+bct) b=v/c and g=1/(1-b2)1/2 t=0 is not invariant under boost! t=0is invariant under boost!

  5. Advantages of LF: (2) Vacuum structure k-=(k2^+m2)/k+ k0=Ök2+m2 Not allowed ! since k+>0 Equal t Equal t k1+ k1 k2+ k2 t t k3+ k3 k1+ + k2+ + k3+=0 k1+k2+k3=0 Physical LF vacuum(ground state) in interacting theory is trivial(except zero mode k+=0)!

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

  7. Electromagnetic Form factor of a pseudoscalar meson (q2=q+q--q2^<0 region) in LF q+ q2=-Q2 e’ e x,k^ + x,k^+(1-x)q^ = g* yn yn yn yn+2 P P+q P=(P+,M2/P+,0), q=(0,2P.q/P+,q) in q+=0 <p+q,l‘|J+(0)|p,l>=F(Q2) =Sò[dx][d2k^] y*n(x,k’^)yn(x,k^) in q+=0 frame

  8. H0=M0 1/4 for 1— -3/4 for 0-+ Normalization: Model DescriptionPRD59, 074015(99); PLB460, 461(99) by Choi and Ji

  9. Central potential V0(r) vs. rPhys. Rev. D 59, 074015(99) by Choi and Ji Fixing Model Parameters by variational principle Input for Linear potential: mu=md=220 MeV, b=0.18 GeV2 + r-p splitting fix a=-0.724 GeV, bqq=0.3659 GeV and k=0.313

  10. Ground state meson spectra[MeV]PLB 460, 461(99); PRD 59, 074015(99) by Choi and Ji

  11. Sum-rule[Leutwyler, Malik]: Model Parameters and Decay constantsPRD 74(07) (Choi and Ji) Linear[HO] mQ[GeV] bqQ[GeV] fth[MeV] fexp[MeV] qQ p 0.22 [0.25] 130[131] 130.70(10)(36) 0.3659[0.3194] 220(2)(fL) 160(10)[SR:Ball] r 246[215](fL) 188[173](fT) 0.22 [0.25] 0.3659[0.3194] K 0.45 [0.48] 0.3886[0.3419] 161[155] 159.80(1.4)(44) 217(5)(fL) 170(10)[SR:Ball] 256[223](fL) 210[191](fT) 0.45 [0.48] 0.3886[0.3419] K* *[For heavy meson sector: hep-ph/0701263(Choi)] important for LCSR predictions for B to r or K*

  12. Quark DA and soft form factor for pion F(Q2)~exp(-m2/4x(1-x)b2) PRD74, 093010(06)[Choi and Ji] PRD 59, 074015(99); PRD74,093010 [Choi and Ji]

  13. Comparison of LFQM respecting conformal symmetry with the Ads/CFT prediction F(Q2)~exp(-m(Q2)2/4x(1-x)b2)

  14. q e- e- A1 A2 A3 q TH kg y e- + M e- + M x D1 D2 D3=D1 D4 D5 D6=D4 M M 1-x 1-y B1 B2 B3 PQCD analysis of pion form factor Hard contribution to meson form factor leading twist ò[dx][dy]f(x,Q2)TH(x,y,Q2)f(y,Q2) where Y(x,k^)=fR(x,k^)x (spinw.f.) (hi,ih) +(ii,hh)

  15. Soft(LFQM) and hard(PQCD) contribution to pion form factor PRD74, 093010(06)[Choi and Ji] HO Linear AdS/CFT=(16/9) x PQCD (hi,ih) PQCD (hi,ih) +(ii,hh) (ii,hh) Suppresion of DA at the end points leads to enhancement(suppression) of soft(hard) form factor!

  16. g* 0,qT x,kT 1-x,-kT g 1,qT g* g p-gTransition Form Factor Ads/CFT =(4/3) PQCD Linear(LO) PQCD HO(LO) NLO

  17. Ours Second x moment of pion Gegenbauer moments <x2> (Lattice) (E791 Collab.) (CLEO Collab.) (Transverse lat.) (Chernyak and Zhitnitsky) (asymp) Our results[PRD75(07):Choiand Ji] <x2>= 0.24 for linear =0.22 for HO L. Del Debbio[Few-Body Sys. 36,77(05)]

  18. Ours: a2[a4]= 0.12[-0.003] for linear =0.05[-0.03]for HO Gegenbauer moments a2 and a4 for pion asymp. twist-two CZ 1s-error ellipse twist-four LCSR-based CLEO-data analysis

  19. 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. Conclusions and Discussions 1. We investigated quark DA and electromagnetic form factor of pion using LFQM. • 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/Q2 consistent with the Ads/CFT prediction • (c) We found correlation between the quark DA and (soft and hard) form factors • (d) Our x 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]

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