Measurement of Generalized Form Factors near the Pion Threshold

• First time extracting preliminary E0+/GD multipole near the pion threshold through nπ+ channel

• Real part contribution is dominated• In experiment side : - Asymmetry data is important to determine L0+

- Cross check is needed by multipoles analysis - Direct use the F.F. of Gm

n from CLAS instead parameterization • In theory side : - Second order correction of pion mass - More serious study for error estimation (one-loop correction) - Improved radiative correction in sum rule - DA’s parameters from LQCD become available





CLAS Hadron Spectroscopy meeting Sep. 26, 2009 K. Park









Long term plan with high stat. data



Low-Energy Theorem (LET) for Q2=0Restriction to the charged pionChiral symmetry + current algebra for electroproduction

1954

1960s

Kroll-Ruderman

Nambu, Laurie, Schrauner

Re-derived LETsCurrent algebra + PCACChiral perturbation theory

1970sVainshtein, Zakharov

1990sScherer, Koch

pQCD factorization methodsBrodsky, Lepage, Efremov, Radyunshkin, Pobylitsa, Polyakov, Strikman, et al

• Constructed relating the amplitude for the radiative decay of Σ+(pγ) to properties of the QCD vacuum in alternating magnetic field.

• An advantage of study because soft contribution to hadron form factor can be calculated in terms of DA’s that enter pQCD calculation without other nonperturbative parameters.

• New technique : the expansion of the standard QCD sum rule approach to hadron properties in alternating external fields.


2* 2 2

| |8

fem

N

k dd

W W m

2

* 2 2| |

8fem

N

k dd

W W m

2

'

| | cos(2 ) 2 (1 ) cos( )

2 (1 ) sin( )

T L TT LT

LT

2

'

| | cos(2 ) 2 (1 ) cos( )

2 (1 ) sin( )

T L TT LT

LT

2

1 ,NT MGG 2

1 ,NT MGG

22 ,N

L EGG 22 ,N

L EGG

1 2Re ,Re , ,LN N

T E MG G G G 1 2Re ,Re , ,LN N

T E MG G G G

1 2/ Im , Im , ,L

N NT E MG G G G 1 2

/ Im , Im , ,LN N

T E MG G G G

0TT 0TT


1NG

1NG

2NG

2NG

MGMG EGEG

2 2 2 2 2 22 22 22 2 2 2 4 2

0 0 1 22 2 2 2 2 2

441 A f A fT L N NiN M L i N E

N N N

c g k c g kk QA D G Q m G k G m G

f m W m W m

2 2 2 2 2 22 2

2 22 2 2 2 4 20 0 1 22 2 2 2 2 2


N N N


f m W m W m

2 21 1 1 22 2 2

41Re Re

A i fT L N NM L N E

N

c g k kA D Q G G m G G

f W m

2 21 1 1 22 2 2

41Re Re

A i fT L N NM L N E

N


f W m

0 0 2 12 2 2

1Re 4 Re

A i fLT N NN M E

N

c g k kD D Qm G G G G

f W m

0 0 2 12 2 2

1Re 4 Re

A i fLT N NN M E

N


f W m

0 0 0TTC C 0 0 0TTC C


2 2 2 20( ) 1/(1 / )DG Q Q

2 20 0.71GeV

2 2[ ]Q GeV2 2[ ]Q GeV

2 2[ ]Q GeV 2 2[ ]Q GeV

V. M. Braun et al., Phys. Rev. D 77:034016, 2008.

Dashed Lines : pure LCSR

symbol index

Solid Lines : LCSR using experimental EM form factor as input


Q2 dependence of the Normalized E0+ Multipole by dipole F. F.

Q2 dependence of the Normalized E0+ Multipole by dipole F. F.



F1 = E0+ + 3*cos(θ)*(E1+ + M1+)

F2 = 2*M1+ + M1-

F3 = 3*(E1+ - M1+)

F4 = 0

F5 = S0+ + 6*cos(θ)*S1+

F6 = S1- - 2*S1+

H1 = (-1/sqrt(2))*cos(θ/2)*sin(θ)*(F3 + F4)

H2 = -1*sqrt(2)*cos(θ/2)*(F1 - F2 - sin(θ)*(F3 - F4))

H3 = (1/sqrt(2))*sin(θ/2)*sin(θ)*(F3 - F4)

H4 = sqrt(2)*sin(θ/2)*(F1 + F2 +(cos(θ/2))**2*(F3 + F4))

H5 = -1*(sqrt(Q2)/abs(k_cm))*cos(θ/2)*(F5 + F6)

H6 = (sqrt(Q2)/abs(k_cm))*sin(θ/2)*(F5 - F6)

Helicity amplitudes ( Hi ) :

Using six amplitudes ( Fi ) : ** if lπ = 1


σT+L = (1/2)*(H1² + (H2²)*(H3²) + H4²) + ε*(H5² + H6²)

σTT = H3*H2 - H4*H1

σLT = (-1/sqrt(2))*( H5*(H1 - H4) + H6*(H2 + H3) )

Structure functios vs. Helicity amplitudes ( Hi ) :


* E0+, S0+ are dominated in this regime.

** M1-, S1- were used from MAID2007 model prediction.

*** for I=3/2I=3/2 case, following correlation functions are acceptable.

→ GD′ =(1+Q2/mu_02)²

→ GM = 3.*exp(-0.21*Q2)/(1. +0.0273*Q2 -0.0086*Q2²)/GD′

→ M1+ = (Y_O/52.437)* GM * sqrt(((2.3933+Q2)/2.46)**2-0.88)* 6.786

→ E1+ = -0.02 * M1+

→ R_sm =-6.066 -8.5639*Q2 +2.3706*Q2² +5.807* sqrt(Q2) -0.75445* Q2²* sqrt(Q2)

→ S1+ = R_sm*M1+/100.

where, mu_02=0.71, Y_O is the interpolation value from SAID model.

Constraints :


Q2 dependence of the Normalized E0+ , L0+ and E1+ Multipole by dipole F. F.

Q2 dependence of the Normalized E0+ , L0+ and E1+ Multipole by dipole F. F.

1 1N nG G

1 1N nG G

2 2N nG G

2 2N nG G

2( )nM M n DG G G Q 2( )nM M n DG G G Q

P.E. Bosted Phys. Rev. C 51 (1995)


0 /nDE G

0 /nDE G

0nE EG G 0nE EG G

J.J. KellyPhys. Rev. C 70 (2004)

S. PlatchekovNucl.Phys. A 70 (1990)0n

E EG G 0nE EG G

J. J. Kelly et al., PRC 70:068202 (2004)


J. Lachniet et al., PRL 102:192001 (2009)


1 1N nG G

1 1N nG G

2 2N nG G

2 2N nG G



0 /nDE G

0 /nDE G

0nE EG G 0nE EG G J.J. Kelly

Phys. Rev. C 70 (2004)

J. Lachniet (2009)Phys. Rev. Lett. 102CLAS DATA


• As first time, E0+ multipole comparison near pion threshold between two methods (LCSR, multipole fit) was performed.

• Multipole analysis gives us same answer for extracting E0+ multipole with LCSR method.

however, the fitting chi2 should be improved ...(future plan)

• Direct use of neutron magnetic form factor from CLAS publication gives consistent result with F.F. parametrization.

• As first time, E0+ multipole comparison near pion threshold between two methods (LCSR, multipole fit) was performed.

• Multipole analysis gives us same answer for extracting E0+ multipole with LCSR method.

however, the fitting chi2 should be improved ...(future plan)

• Direct use of neutron magnetic form factor from CLAS publication gives consistent result with F.F. parametrization.

• Enrgy dependent generalized form factors generated by FSI

• Adding D-wave contributio model

• Tune calculation with low Q2 and high W experimental data

• Systematic approach in the global PWA analysis framework in N and *N scattering under QCD S-, P- and D partial waves.

• Enrgy dependent generalized form factors generated by FSI

• Adding D-wave contributio model

• Tune calculation with low Q2 and high W experimental data

• Systematic approach in the global PWA analysis framework in N and *N scattering under QCD S-, P- and D partial waves.


1 1N nG G

1 1N nG G

2 2N nG G

2 2N nG G


0nE EG G 0nE EG G

Due to low-energy theorem(LET) relates the S-wave multipoles or equivalently, the form factor G1, G2 @ threshold Due to low-energy theorem(LET) relates the S-wave multipoles or equivalently, the form factor G1, G2 @ threshold 0m 0m


2 2

12 2 2

1

2 22n nA

M AN N

gQ QG G G

m Q m

2 2

12 2 2

1

2 22n nA

M AN N

gQ QG G G

m Q m

2

2 2 2

2 20

2n nA N

EN

g mG G

Q m

2

2 2 2

2 20

2n nA N

EN

g mG G

Q m

Scherer, Koch, NPA534(1991)Vainshtein, Zakharov NPB36(1972)

Assumption in LCSRV.Braun PRD77(2008)


1nG

1nG

2nG

2nG

2 2 2

0 13

44

8pemn n

p

Q Q mE G

m f

2 2 2

0 13

44

8pemn n

p

Q Q mE G

m f

2 2 2

0 23

44

32pemn n

p

Q Q mL G

m f

2 2 2

0 23

44

32pemn n

p

Q Q mL G

m f

1/ DG1/ DG

1/ DG1/ DG


W-dependence of L0+ from MAID2007


Lower Q2, Smaller W dependence

Bold –Real part Thin – Imaginary

part


* E=5.754GeV(pol.), LH2 target (unpol.)* IB=3375/6000A, Oct. 2001-Jan. 2002


Cross Section (φ*)

Cross Section (φ*)

Blue shade : systematic uncertainty estimation

Red data : CRS with new analysis


T L T L

TTTT0E 0E insensitive !insensitive !

0E 0E sensitive !sensitive !


0E 0E sensitive !sensitive !LTLT


*0 1 1(cos )T L T L

T L L D D P

0TT

TT D

1 1N nG G

1 1N nG G

2 2N nG G

2 2N nG G


0nE EG G 0nE EG G



2 2

12 2 2

1

2 22n nA

M AN N

gQ QG G G

m Q m

2 2

12 2 2

1

2 22n nA

M AN N

gQ QG G G

m Q m

2

2 2 2

2 20

2n nA N

EN

g mG G

Q m

2

2 2 2

2 20

2n nA N

EN

g mG G

Q m




2 2 22 2 22 2 2

0 0 12 2 2 2

41 A fT L n nip M

p p

c g kk QA D G Q m G

f m W m

2 2 22 2 22 2 2

0 0 12 2 2 2

41 A fT L n nip M

p p

c g kk QA D G Q m G

f m W m

21 1 12 2 2

41Re

A i fT L n nM

p

c g k kA D Q G G

f W m

2

1 1 12 2 2

41Re

A i fT L n nM

p

c g k kA D Q G G

f W m

0 0 0LTD D 0 0 0LTD D

1.267Ag 1.267Ag

2c 2c

93f MeV 93f MeV 0 0 0TTC C 0 0 0TTC C

1 1 1nG x iy

1 1 1nG x iy

2 2 2nG x iy

2 2 2nG x iy

1nG

1nG

2nG

2nG


1 1N nG G

1 1N nG G

2 2N nG G

2 2N nG G


0nE EG G 0nE EG G

2 2

12 2 2

1

2 22n nA

M AN N

gQ QG G G

m Q m

2 2

12 2 2

1

2 22n nA

M AN N

gQ QG G G

m Q m

2

2 2 2

2 2

2n nA N

EN

g mG G

Q m

2

2 2 2

2 2

2n nA N

EN

g mG G

Q m




2 2 2 2 2 22 22 22 2 2 2 4 2

0 0 1 22 2 2 2 2 2


N N N


f m W m W m

2 2 2 2 2 22 2

2 22 2 2 2 4 20 0 1 22 2 2 2 2 2


N N N


f m W m W m

2 21 1 1 22 2 2

41Re Re

A i fT L N NM L N E

N


f W m

2 21 1 1 22 2 2

41Re Re

A i fT L N NM L N E

N


f W m

0 0 2 12 2 2

1Re 4 Re

A i fLT N NN M E

N


f W m

0 0 2 12 2 2

1Re 4 Re

A i fLT N NN M E

N


f W m

0 0 0TTC C 0 0 0TTC C

1 1 1nG x iy

1 1 1nG x iy

2 2 2nG x iy

2 2 2nG x iy

1nG

1nG

2nG

2nG

1.267Ag 1.267Ag

2c 2c

93f MeV 93f MeV


1 1 1nG x iy

1 1 1nG x iy

2 2 2nG x iy

2 2 2nG x iy

* Real parts x1, x2 can be determined by A1, D0 legendre coeff.

* 3 Eqs. 4 parameter should be determined

* Imaginary parts y1, y2 can be determined in 2cases

* Asymmetry helps to determine complete form factor

/ /0 0 2 12 2 2

1Im 4 Im

A i fLT N NN M E

N


f W m

/ /0 0 2 12 2 2

1Im 4 Im

A i fLT N NN M E

N


f W m


1nG

1nG

2nG

2nG

2 2 2

0 13

44

8pemn n

p

Q Q mE G

m f

2 2 2

0 13

44

8pemn n

p

Q Q mE G

m f

2 2 2

0 23

44

32pemn n

p

Q Q mL G

m f

2 2 2

0 23

44

32pemn n

p

Q Q mL G

m f

1/ DG1/ DG

1/ DG1/ DG


• Historically, threshold pion in the photo- and electroproduction is the very old subject that has been receiving continuous attention from both experiment and theory sides for many years.

• Pion mass vanishing approximation in Chiral Symmetry allows us to make an exact prediction for threshold cross section known as LET

• The LET established the connection between charged pion electroproduction and axial form factor in nucleon.

• Therefore, It is very interesting to extracting Axial Form Factor which is dominated by S- wave transverse multipole E0+ in LCSR


• Constructed relating the amplitude for the radiative decay of Σ+(pγ) to properties of the QCD vacuum in alternating magnetic field.

• An advantage of study because soft contribution to hadron form factor can be calculated in terms of DA’s that enter pQCD calculation without other nonperturbative parameters.

• New technique : the expansion of the standard QCD sum rule approach to hadron properties in alternating external fields.


1 1N nG G

1 1N nG G

2 2N nG G

2 2N nG G


0nE EG G 0nE EG G



2 2

12 2 2

1

2 22n nA

M AN N

gQ QG G G

m Q m

2 2

12 2 2

1

2 22n nA

M AN N

gQ QG G G

m Q m

2

2 2 2

2 20

2n nA N

EN

g mG G

Q m

2

2 2 2

2 20

2n nA N

EN

g mG G

Q m




2 2 22 2 22 2 2

0 0 12 2 2 2

41 A fT L n nip M

p p

c g kk QA D G Q m G

f m W m

2 2 22 2 22 2 2

0 0 12 2 2 2

41 A fT L n nip M

p p

c g kk QA D G Q m G

f m W m

21 1 12 2 2

41Re

A i fT L n nM

p

c g k kA D Q G G

f W m

2

1 1 12 2 2

41Re

A i fT L n nM

p

c g k kA D Q G G

f W m

0 0 0LTD D 0 0 0LTD D

1.267Ag 1.267Ag

2c 2c

93f MeV 93f MeV 0 0 0TTC C 0 0 0TTC C

1 1 1nG x iy

1 1 1nG x iy

2 2 2nG x iy

2 2 2nG x iy

1nG

1nG

2nG

2nG


1 1N nG G

1 1N nG G

2 2N nG G

2 2N nG G


0nE EG G 0nE EG G

2 2

12 2 2

1

2 22n nA

M AN N

gQ QG G G

m Q m

2 2

12 2 2

1

2 22n nA

M AN N

gQ QG G G

m Q m

2

2 2 2

2 2

2n nA N

EN

g mG G

Q m

2

2 2 2

2 2

2n nA N

EN

g mG G

Q m




2 2 2 2 2 22 22 22 2 2 2 4 2

0 0 1 22 2 2 2 2 2


N N N


f m W m W m

2 2 2 2 2 22 2

2 22 2 2 2 4 20 0 1 22 2 2 2 2 2


N N N


f m W m W m

2 21 1 1 22 2 2

41Re Re

A i fT L N NM L N E

N


f W m

2 21 1 1 22 2 2

41Re Re

A i fT L N NM L N E

N


f W m

0 0 2 12 2 2

1Re 4 Re

A i fLT N NN M E

N


f W m

0 0 2 12 2 2

1Re 4 Re

A i fLT N NN M E

N


f W m

0 0 0TTC C 0 0 0TTC C

1 1 1nG x iy

1 1 1nG x iy

2 2 2nG x iy

2 2 2nG x iy

1nG

1nG

2nG

2nG

1.267Ag 1.267Ag

2c 2c

93f MeV 93f MeV


1 1 1nG x iy

1 1 1nG x iy

2 2 2nG x iy

2 2 2nG x iy

* Real parts x1, x2 can be determined by A1, D0 legendre coeff.

* 3 Eqs. 4 parameter should be determined

* Imaginary parts y1, y2 can be determined in 2cases

* Asymmetry helps to determine complete form factor

/ /0 0 2 12 2 2

1Im 4 Im

A i fLT N NN M E

N


f W m

/ /0 0 2 12 2 2

1Im 4 Im

A i fLT N NN M E

N


f W m


1nG

1nG

2nG

2nG

2 2 2

0 13

44

8pemn n

p

Q Q mE G

m f

2 2 2

0 13

44

8pemn n

p

Q Q mE G

m f

2 2 2

0 23

44

32pemn n

p

Q Q mL G

m f

2 2 2

0 23

44

32pemn n

p

Q Q mL G

m f

1/ DG1/ DG

1/ DG1/ DG


Q2 dependence of the scaled cross section for three different W bins

dashsolid


Measurement of Generalized Form Factors near the Pion Threshold

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