conformal symmetry and pion form factor: soft and hard contributions ho-meoyng choi(kyungpook...
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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, 201601(06), PRL 94, 201601(05)]
Light-front holographic wavefunction LF display confinement at large inter-quark separation(z large) and conformal symmetry at short distances(z small).
2/
/
/1~
)1(8
)(
QF
xxx
CFTAds
CFTAds
String amplitude z LFholographic mapping
QCD:LFQM+PQCD
Refs: PRD 74, 093010(06); PRD74, xx(07, Feb.)[hep-ph/0701177][Choi and Ji]
z=0
Outline
1. Introduction on Light-Front(LF) formulation2. Light-Front Quark Model(LFQM) description3. 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 pion4. Conclusion
Comparison of equal-t and equal LF-=t+z/c(=x+=x0+x3) coordinates
Equal t Equal
p·q=p0q0-p·q
p·q=(p+q-+p-q+)/2-p·q
(p±=p0±p3)
p+: longitudinal mom. p-=LF energy
p2= m2
p0=p2+m2
p2=m2
p-=(p2+m2)/p+ (P+:
positive)
z=x3
ct=x0
ct+z=x0+x3=x+
=c
x+
Poincare’ group(translations P, rotations L and boost K)Kinematic generators: P and L for ET(6) P+,P, L3 and K for LF(7)
Light front(LF)
tt’
=t+z/cz
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!
LFxk
x(=k+/P+)
1-x
Advantages of LF: (1) Boost invariance
Advantages of LF: (2) Vacuum structure
k1
k2
k3
k1+k2+k3=0
k1+
k2+
k3+
k1+ + k2
+ + k3+=0
Equal t Equal Not allowed !since k+>0
t
Physical LF vacuum(ground state) in interacting theory is trivial(exceptzero mode k+=0)!
k0=k2+m2 k-=(k2+m2)/k+
Advantages of LF: (3) Covariant vs. time-ordered diagram
LF valence
LF nonvalence
Electromagnetic Form factor of a pseudoscalar meson (q2=q+q--q2
<0 region) in LF
<p+q‘J+(0)|p>F(Q2) =[dx][d2k]
n(x,k’nx,k)
in q+=0 frame
ee’
P P+q
=
q2=-Q2
x,k+(1-x)qx,k
n n
n+2
n
+
q+
P=(P+,M2/P+,0), q=(0,2P.q/P+,q) in q+=0
Model DescriptionPRD59, 074015(99); PLB460, 461(99) by Choi and Ji
coul
z
JJnlm
JJnlm
V
andkk
zz
2
QQ2
hyp0QQ
222
QQ22
Q22
QQQ
m3m
S2S
3r
4- ]br[br a
(r)V(r)VV
k where
]Vkmkm[H
1/4 for 1—
-3/4 for 0-+
H0=M0
x
mk
x
mkM
mmM
pvpu
x
i
iinlm
1
])([2
),(),(
)/2kexp(-k4
)k,(x
)k(x,)k,(x),k,(x
22
221
220
221
20
2251100
22z3/2
3/4
i
JJ,iii
JJ,
21
z
21
z
1),(16
2
1003
21
0
kxkd
dx zJJ
Normalization:
Fixing Model Parameters by variational principle
Input for Linear potential: mu=md=220 MeV, b=0.18 GeV2
+ splitting
fix
a=-0.724 GeV, qq=0.3659 GeVand =0.313
0|]V[H| 00
Central potential V0(r) vs. rPhys. Rev. D 59, 074015(99) by Choi and Ji
Ground state meson spectra[MeV]PLB 460, 461(99); PRD 59, 074015(99) by Choi and Ji
Model Parameters and Decay
constants PRD 74(07) (Choi and Ji)
159.80(1.4)(44)161[155]0.3886[0.3419]0.45 [0.48]
130.70(10)(36)130[131]0.3659[0.3194]0.22 [0.25]
fexp[MeV]fth[MeV]qQ[GeV]mQ[GeV]
246[215](fL)188[173](fT)
220(2)(fL)160(10)[SR:Ball]
0.22 [0.25] 0.3659[0.3194]
K* 0.45 [0.48] 0.3886[0.3419] 256[223](fL)210[191](fT)
217(5)(fL)170(10)[SR:Ball]
Linear[HO]
2/)(
2/)(
)()()(
)()()(
**
**
L
KKT
K
L
KKT
Kfff
Sum-rule[Leutwyler, Malik]:
K
*[For heavy meson sector: hep-ph/0701263(Choi)] important for LCSRpredictions for B to or K*
Quark DA and soft form factor for pion
PRD 59, 074015(99); PRD74,093010 [Choi and Ji]PRD74, 093010(06)[Choi and Ji]
22
),(16
),(3
2
kkx
kdx
F(Q2)~exp(-m2/4x(1-x)2)
Comparison of LFQM respecting conformal symmetry with the Ads/CFT prediction
F(Q2)~exp(-m(Q2)2/4x(1-x)2)
e- e-
M M
e- + M e- + M
q
x
1-x
y
1-y
TH
),(),,,,(),(]][[)( *332 kxlkqyxTlyldkdQF MHMPQCDM
Hard contribution to meson form factor
where
(x,k)=R(x,k)x (spin w.f.) (,) +(,)
PQCD analysis of pion form factor
D1 D2 D3=D1 D4 D5 D6=D4
kg
A1 A2 A3
B1 B2 B3
),( 1 kx
),( 2 kx
),( 11 lqyy
),( 22 lqyy
q
dx][dy](x,Q2)TH(x,y,Q2)(y,Q2)leading twist
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, 093010(06)[Choi and Ji]
HO Linear
(,) +(,)
(,)
(,)
AdS/CFT=(16/9) x PQCD
PQCD
Transition Form Factor
x,kT
1-x,-kT
0,qT
1,qT
2222
221
0
2
1
0
2
'
)1()(~)(
)1(6
),(
2
mk
mx
km
kkddxQF
xx
Qxdx
f
FQ
RNLO
LO
PQCD
Ads/CFT =(4/3) PQCD
Linear(LO)
HO(LO)
NLO
L. Del Debbio[Few-Body Sys. 36,77(05)]
(Lattice)
(CLEO Collab.)
(E791 Collab.)
(asymp)
(Transverse lat.)
(Chernyak and Zhitnitsky)
<2>
Second moment of pion
Our results[PRD75(07):Choi and Ji]
<2>= 0.24 for linear =0.22 for HO
Ours
1
0 21
1
2/3
1
0 )(
)12(),(
)1(6)(
)12()(1)(),(
1),(
xxxxdx
and
xxx
where
xCaxx
dxx
nn
as
nnnas
VP
Gegenbauer moments
1-error ellipse
twist-two
Gegenbauer moments a2 and a4 for pion
twist-four
asymp.
CZ
Ours:a2[a4]= 0.12[-0.003] for
linear=0.05[-0.03]for HO
424
22
77
8
35
8
35
35
1
35
12
aa
a
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/Q2 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]