1

Twist 4 Matrix elements

Su Houng Lee 1. S. Choi et al, PLB 312 (1993) 3512. Su Houng Lee, PRD 49 (1994) 2242 3. Su Houng Lee, PRC 57 (1998) 927

2S H Lee

DIS

Relation to Polarization Tensors

Some basics on matrix elements and moments

MQWQWdd

dEdd

M

2/,2

tan,2'

22

221

MWQxF

WQWQxFL2/,

/1,

22

2

222

12

24 de |0|

21 FFpjyjpyedW L

iqy

n

nn2,nL,

4 1/ Md Me |]0T[| xpjyjpyediT iqy

1

0

22-nn ),(M Im QxFdxxTW

1

02

24222-n

2

4n2

n),(),(MM

QQxFQxFdxx

Q

e (E,k)

X

e (E’,k’)

P

q 2

2

22

,/1or

,or

,2/

Qx

Q

QqpQx

x/1

3S H Lee

Diagrammatic rep of Structure function

Diagrammatic rep of OPE

X

P

24 de |0|

21 FFpjxjpxedW L

iqx

n

nn2,nL,

OPE4 Md Me |]00,T[| pQxQxpxediT iqx

P

P

x 0Q Q

P

n

P P

0 0

Q Q

OPE n

1

0

22

2-n1

0

22-n ),( ),( QxFdxxQxFdxx L

4S H Lee

Twist-2 Operators (LO)

|]00,T[|4 pQxQxpxediT iqx

P P

00

Q Q pDQpM |0| 22

2

02 LM

Twist-4 Operators

P P

00

Q Q

P P

00

Q Q

2

5, QFDigg aa

aaA

AAg

iQiQg

2

552

aaaa

V

VVgQg

FDQg

222

22

],[

Politzer (80), Shuryak, Vainshtein (81) , Jaffe, Soldate (81)

5S H Lee

Twist-4 Operators

OPE

gVgVA MM

QxMMM

QxT

83

411e

161

851d 2222

Operators

kP

k Mgmpp

2

41

25

2

2

, QFDig

VVg

AAg

g

aaV

aaA

mass Operators

2QDDmm

mm M

QxM

QxT

4181e

3141d 2222

gVL

gVA

MMdxF

MMMdxF

83

41

21

161

85

21

1

0

4

1

0

42

Politzer (80), Shuryak, Vainshtein (81) , Jaffe, Soldate (81)

Lee (94)

6S H Lee

Parameterizing F2 (=4)

For Cp: BCDMSdata and SLAC data +Virchauz,Milsztajin, PLB274 (92) 221

For Cp-Cn: NMC (combining NMC,SLAC, BCDMSdata)

75.007.0 ,, 222

42 xQxFxCF p

87.18 ,11.33 ,88.16 ,33.3 ,27.0 ,4.0 43210 aaaaab

0 ,1

44

33

2210

b

xxaxaxaxaaxC bp

• We fit to

75.007.0 1 2

2

2

2

2

2

xQ

xCxCFF

FF np

p

n

p

n

• We fit to 96.10 ,44.22 ,32.12 ,14.3 ,27.0 ,4.0 43210 aaaaab

2

23

211)1( :SoldateQxxxF

2

2

23

2111)1( :al.et Gunion Qx

QxxxF

7S H Lee

Parameterizing FL (=4)

Parameterization using transverse basis (Ellis, Furmanski, Petronzio 82)

2224 ,4 TTTL kxfkkdF

P P

00

Q Q

SLAC data analyzed by Sanchex Guillen etal. (91)

22222

24 GeV 01.003.0 ,8 QxFFL

34 1 1 Soldate xxFL

6.02.0 x

8S H Lee

Constraints for matrix elements from experiments

(neutron) GeV 004.0011.0(proton) GeV 004.0005.0

161

85

21

2

21

0

42

gVA MMMdxF

(neutron) GeV 008.0023.0(proton) GeV 012.0035.0

83

41

21

2

21

0

4 gVL MMdxF

Note that the matrix elements A’s for the proton and neutron data are independent.

AM VM

gM gM

proton

neutrondata

proton

neutronMIT Bag

2GeV 1.0 ,12.0 ,6.0,, solution typicalOne gVA MMM

1.0

1.0

9S H Lee

MIT Bag model calculations (Jaffe-Soldate 81)

Definitions

ppm

M

MgMpp

kii

k

N

k

kP

k

|31|2

41

00

2

Calculations

B 0,E ,)(

urgr

rfx • operators

• Normalizations by Jaffe (75)

||21

|0|21

34

4

pyjxjpyexdd

pjxjpxedW

yxiq

iqx

pyypydm

M kii

k

N

k |31|2

003

Vpppp '2'| 33

10S H Lee

Calculations- cont

25

22

552

, QFDig

Qg

QiQig

g

aaV

aaA

• calculations involve spin and spatial parts

pyypydm

M kii

k

N

k |31|2

003

rgrfdrrrgrfdrr 2222222 ,

11S H Lee

MIT Bag model vs experimental constraint

F2: Q2=5 GeV2 s = 0.5

(neutron) GeV 015.0(proton) GeV 027.0

Bag MIT 2

2

s

s

(neutron) GeV 008.0023.0(proton) GeV 012.0035.0

Experiment2

2

gVA MMMdxF

161

85

211

0

42

(neutron) GeV 026.0(proton) GeV 022.0

Bag MIT 2

2

s

s

FL: Q2=5 GeV2 s = 0.5

(neutron) GeV 004.0011.0(proton) GeV 004.0005.0

Experiment2

2

075.0

161064.0

85018.0

21

gV

L MMdxF83

41

211

0

4

075.0

83064.0

41

21

1.0

16112.0

8506.0

21 Typical numbers

1.0

8312.0

41

21

Typical numbers

12S H Lee

a) Bag model calculations only measures correlations between valance quarks

c) Non-trivial test of low energy models of hadrons and QCD

b) Need much more correlation such as

Need more correlations

13S H Lee

A Parameterization based on flavor structure

gudd

gduu

gnp

Vudd

Vduu

Vnp

Auddu

Audd

Aduu

Anp

KQKQM

KQKQM

KQQKQKQM

)(2

)(2

)(

)(2

)(2

)(

2)(

2)(

2)( 2/

pqFDqpMigK

pdduuqiqpM

K

pdiduiupM

K

pdiduiuqiqpM

K

gq

Vq

Aud

Aq

|,|2

||2

|2|2

||2

52

52

552

5552

• Unknowns:: 7 • Constraints: F2 (Proton, Neutron),FL,(proton, neutron) 4

Flavor Structure

Flavor Assumptions

xxuxux

xxdxdx

KK

KK

KK

gu

gd

Vu

Vd

Au

Ad

d)()(

d)()(

• Unknowns:: 7-3 = 4 • Constraints: F2 (Proton, Neutron),FL,(proton) 3

Au

Aud

Ad KKK

14S H Lee

puFDupMigK

pdduuuupM

KpdiduiuuupM

K

gu

Au

Au

|,|2

||2 ,||2

52

2555

2

Typical Result

KuA Ku

V Kug

0.173 GeV2 0.203 -0.2380.112 0.110 -0.3000.083 0.066 -0.329

2552 GeV 083.0||2

pdiduupM

K Aud

2GeV 1.0 ,12.0 ,6.0,, solution typicalOne gVA MMM

15S H Lee

1. Twist-4 matrix elements are interesting itself because, a) First experimental measurements of multiparticle correlation inside

proton b) Old data seems to suggest need much more correlation than

such as

Summary - i

16S H Lee

2. Twist-4 matrix elements are relevant for Vector and Axial vector in medium

Summary - ii

.....0),(

.....0),(

44

44

AVAAVOPEiqxAA

AVVAVOPEiqxVV

KKQcKK

QcpertAxATdxe

KKQcKK

QcpertVxVTdxe

22

1,

2

0,

21,

0,,

)()0,(

)()0,(

0,0,, Re

qs

ss

sds

qq

TLTL

TLTLVVTL

99) (SHLmatter nuclear at ,0)( to ofon contributi Nontrivial 1TL, AVKK

17S H Lee

2. Analysis suggests Large cancellation in Twist-4 effect in F2

Summary - iii

(neutron) GeV 004.0011.0(proton) GeV 004.0005.0

2

2

(neutron) GeV 008.0023.0(proton) GeV 012.0035.0

,2

21

0

24 dxQxFL

1.0

16112.0

8506.0

21

gVA MMMdxF

161

85

211

0

42

3. OPE suggests Large Twist-4 effect in FL and proportional to Twist-2

,8 222

24 QxFFL