Brodsky and de Teramond
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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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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


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


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)


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!


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)!


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

LF nonvalence

LF valence


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


H0=M0

1/4 for 1—

-3/4 for 0-+

Normalization:

Model DescriptionPRD59, 074015(99); PLB460, 461(99) by Choi and Ji


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


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


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*


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]


Comparison of LFQM respecting conformal symmetry with the Ads/CFT prediction

F(Q2)~exp(-m(Q2)2/4x(1-x)b2)


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)


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!


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


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)]


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


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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