中国科技大学粒子物理与核物理讨论班

12011 年 8 月　中国科技大学粒子物理与核物理讨论班

中国科技大学粒子物理与核物理讨论班

核子结构研究与新进展

梁作堂山东大学物理学院

2011 年 8 月 25 日，合肥，中国科技大学


ContentsContents

引言单举轻子核子深度非弹性散射 (DIS) 实验，部分

子模型与部分子分布函数

--------------- A short overview

半单举轻子核子深度非弹性散射 (SIDIS) 与部分子关联函数

总结与展望


ContentsContents

引言

核子结构研究的核心问题、特点、意义

( ) | (0) ( ) |2 2

ixp zq

dzf x e p ψ ψ z p

pzψzψpedz

xf zixpq |)(),0(

2)0(|

2)( L

Why parton distribution function is defined as


引言：物质结构图象

物质结构

物质

原子(atom)

电子(electron)

原子核(nucleus, A)

质子 (p) 、中子 (n) ：统称为核子 (Nucleon, N)

夸克 (quark)

电子、夸克： <10-3fm

原子 : 10-10 m=1

核子： 1fm=10-15m

o

A


强子结构的夸克模型

强子由夸克构成

夸克有三种：

flavor （味道） charge （电荷） spin （自旋）

u (up) 2/3 1/2

d (down) -1/3 1/2

s (strange) -1/3 1/2

1964 年， Gell-Mann, Zweig

)(uudp

)(uddn

)( du),( du

)(2

10 dduu

Nobel prize in 1969.


高速运动的核子的结构？

量子场论的基本性质

真空涨落与真空激发vacuum excitation

A dynamical picture for a high energy nucleon that participates in high energy reaction.

核子结构研究的基本问题


Introduction

核子结构是粒子物理重要前沿

高能反应实验强相互作用性质、核子结构、强子化机制、 … …

粒子物理标准模型

微扰部分

电

弱

强量子色动力学 (QCD) 非微扰部分

弱电统一理论实验充分检验尚未解决的重大难题

Quark has color degree of freedom, strong interaction is color interaction


Introduction

关于核子结构的研究

核子结构的研究与原子分子结构的研究有很大的差别。

目前通常的做法

非微扰 QCD 求解困难与色禁闭

对高能反应理论描述的需要

合适的物理参量

核子结构定量描述与普适性

高能反应的预言


Introduction: DIS and Nucleon Structure

Inclusive deep-inelastic lepton-nucleon scattering (DIS) XeNe

)()2(|)0(||)0(|),( 44X

X

pqppJXXJppqW

),()2)(2(4

1),()(),( 2

222

12 QxFxpqxpqqpx

QxFq

qqgpqW

核子结构理想的探针

电磁相互作用点结构电子：完全确定

)',( llL

)(le )'(le

)( pN X

)(* q 22 Qq

'2

'),(

3

4

2

E

ldpqW

sQd em

)',( llL

leptonic tensor

的定义式：

一般形式：

质子波函数

W

structure function of nucleon

),()()( xxexJ 电磁流： :| p

The differential cross section ：

hadronic tensor

222 ,2/ qQpqQx


Introduction: DIS and Nucleon Structure

几十年实验与理论研究，取得了巨大成就：质子由更小的点粒子构成Bjorken Scaling

Gottfried 求和规则

Callen-Gross 关系

构成质子的点粒子带分数电荷构成质子的点粒子自旋为 1/2

)(),( 2 xFQxF ii

)(2)( 12 xxFxF

)]()([ 22 xFxFx

dx np 3/122 du ee

Nobel Prize 1990

Jerome I. Friedman Henry W. Kendall Richard E. Taylor

“We see quarks through deeply inelastic lepton-nucleon scattering.”

部分子模型：目前核子结构最成功的描述。


ContentsContents

单举轻子核子深度非弹性散射 (DIS) 实验，部分子模型与部分子分布函数


单举 DIS 过程与 ( 传统的 ) 部分子模型、

部分子分布函数

Higher twist 贡献


ee

q q

DIS and Parton Model （部分子模型）

Parton model:

qfA fast moving proton A beam of free partons

Lots of virtual processes such as

In a fast moving frame, because of time dilatation, they exist for quite long time.

In the infinite momentum frame (IMF), they exist forever.

electron scatters with free parton

number density of parton q in nucleon.

known as parton distribution functions (PDF)

（部分子分布函数）

:)(xfq

2 2ˆ| ) | ( ) | ) |qq

(eN eX dx f x (eq eq M M

R.P. Feynman,Bjorken and Paschos

XeNe E.g.: for

qeqe

the elementary scattering process

scattering amplitude

qf

ee

NX



Parton model:

Impulse Approximation( 冲量近似 ):

During the interaction of the lepton with the nucleon

(1) interaction between the partons can be neglected;

( 自由的夸克 )

(2) the lepton interacts only with one single parton;

( 无多重散射 )

(3) interaction of the lepton with different partons can

be added incoherently. ( 传统、经典 )

qfA fast moving proton A beam of free partons

Lots of Virtual processes such as

In a fast moving frame, because of time dilatation, they exist for quite long time.

In the infinite momentum frame, they exist forever.

qf



Parton model: direct consequences

质子由更小的自由的点粒子构成

Bjorken Scaling:

Gottfried 求和规则

Callen-Gross 关系

构成质子的自由的点粒子带分数电荷

构成质子的自由的点粒子自旋为 1/2

)(),( 2 xFQxF ii

)(2)( 12 xxFxF

)]()([ 22 xFxFx

dx np 3/122 du ee

“We see quarks through deeply inelastic lepton-nucleon scattering.”

Nobel prize in 1990 for Friedman, Kendall and Taylor.

Q/1

不依赖 Q

22 1( ) 2 ( ) [ ( ) ( )]q q q

q

F x xF x e xf x xf x


Parton Model and Other High Energy Reactions

Parton model: A fast moving proton A beam of free partons

p

p

l

l

Xllpp Drell-Yan process: High transverse momentum jet production: Xjetpp

ˆ )( dxfd q

parton distributionfunctions

cross sections for elementary scattering process

The factorized form for the differential cross section:

jet

jet

The parton distribution functions have to rely on parameterizations.


Global QCD analysis and PDFLIB

Experimental data for the cross sections of, e.g.:

parameterizations of the parton distribution functions

Empirical forms of the PDF’s ),,( ),,( ),,( QxsQxdQxu

),( ),,( ),,( ),,( QxgQxsQxdQxu ;XeNe ;Xllpp ;Xjetpp

ˆ )( dxfd q

parton distributionfunctions

cross sections for elementary scattering process

... ...

Parton model



Parton distribution function library: many different sets

E.g.: CTEQ collaboration and CTEQ parameterizations

Home page at: http://www.phys.psu.edu/~cteq

The CTEQ Collaboration is founded by Wu-Ki Tung in Michgan State University

and colleagues in the early 1990s and now consists of 31 theorists and

experimentalists at 18 universities.

Early parameterizations:

Duke-Owens: D. Duke and J. Owens, Phys. Rev. D30, 49 (1984);

Morfin-Tung: J.G. Morfin and W.K. Tung, Z. Phys. C52, 13(1991).

Different versions of parameterizations: from CTEQ1 to CTEQ10.





Parton distribution functions: Some of the salient features:

Bjorken scaling violation:

Gluon contribution:

Symmetry breaking in the sea distributions:

QCD evolutions

5.0)( xxqdx ii

)()( xdxu

)()( xsxs

Quarks and anti-quarks carry only half of the momentum of proton

Gottfried sum rule violation

Brodsky, Ma; NuTev, ...

a

aaa xfxfxexxFxF )()()(2)( 212


qf

Polarized DIS and spin structure of nuleon

with polarized beams and targets

)]()([1

0

xqxqdxQ iii SDU

)]()([2

1)(

1

21 xqxqexg jj

N

jj

f

Integrated over x:

contribution of spins of quarks to that of proton

)()()( xqxqxq jjj ± means: helicity of qj = ± helicity of p

polarized photon make selection of polarization of quark/anti-quark

1)( * zs2/1),( inqsz

2/1),( finalqsz

Spin dependent structure function

c.m. frame of p**

p

ee

X

z


Extracting from Inclusive DIS data

DIS & Parton Model:

,3

22)(

6

1)( 01

1

0

1 83QQQxgdx pppp

),(2

13 DUQ ).2(

6

38 SDUQ ,

3

2

12

1)(

3

2

12

10 SDUQ

,||2 pApQSM apap )(

2)()( 5 zzzA

aa Operator expression:

SUf(3) Symmetry in Hyperon Decay

Wigner-Eckart Theorem: )(2|| DdFifMSA abcabcca

b

:)8,...2,1}({ aa the 8-dimensional representation of the SUf(3) group.

),(2

1| 54 ip E.g.: ),(

2

12||2 3

3 DFSMpApQSM pp

p

),3(32

12||2 8

8 DFSMpApQSM pp

p

... ...3

2

12

10 Q

)],()([2

1)(

1

21 xqxqexg p

jpj

N

jj

pf

Integrated over x:



Earliest experiments:

138.0094.0120.0 1989: EMC+E130+E80: “proton’s spin crisis”

SLAC E80, E130, in 1970s-80s.

Since then, intensive studies have been performed.



Collection of data available



Results:

138.0094.0120.0 1989: EMC+E130+E80: “proton’s spin crisis”

1990s: SLAC, SMC (CERN), HERMES (DESY)

2000s: COMPASS (CERN), HERMES, RHIC-Spin, JLab, … :

large contribution from gluon ?)%30~20(

G

orbital angular momentum important?%30~ G probably small?

Still puzzling situation for the origin of proton spin1

( ) ???2p q gJ L G L


The “QCD Analysis” and the x-Dependence

Parton Model with QCD corrections

)],(),(),(),(),(),(1

[2

1),(

1

1

21

f

Gf

NSqf

NSj

f

Sqf

fx

N

jj

p QzC

z

xgzC

z

xq

QzC

z

x

Nz

dzeQxg

f

)],,(),([),(1

fifi

N

if xqxqx

f

),(1

),(),(),( ff

fififNSi x

Nxqxqxq

:),( ),,( ),,(f

G

f

NSq

f

Sq

QzC

QzC

QzC

Wilson coefficients, calculable in

pQCD

Inclusive DIS data on g1 Empirical forms of q and g

Spin-dependent parton distribution functions ),( ),,( ),,( ),,( ),,( QxGQxsQxqQxdQxu


Early example of the results of “QCD Analysis”

NLO Results at Q2=4GeV2

Blümlein & Böttcher, NPB636(2002)225.


Other Measurements

Flavor tagging in semi-inclusive DIS

To the leading order:

)()(

)()(1 )()(

)()(

)(xdxd

xdxdehXepehXpeehXpe

ehXpeehXpe

xA

�

�

)()(

)()(

2

2

zDdzxqe

zDdzxqe

hfff f

hfff f

)(

)()(

)(

)()()(

)()(

'''

2'

2

xq

xqxP

xq

xq

f

f

f

hf

f

f

fzDdzxqe

zDdzxqe

hfff f

hfff

:)(xP hf purity, spin independent, calculable using e.g. LEPTO.

),,()( 111XeeDXeKepXeep AAAxA

),,,,()(d

d

u

u

s

s

d

d

u

uxQ

)()]([)( xQxPxA h

f

XtagheNe )(


Other Measurements

RHIC Spin program

Xpp 0

polarized pp collisions at 200, 500GeV


Global QCD analysis for polarized parton distributions

DSSV parameterization

well constraineddu ,

sdu ,, non-zero, with large uncertainties.

s with a node for sign changeg very large uncertainties.

de Florian, Sassot, Stratmann, Vogelsang, Phys.Rev.D(2009).


ContentsContents

单举轻子核子深度非弹性散射 (DIS) 实验，部分子模型与部分子分布函数


单举 DIS 过程与部分子模型、

部分子分布函数

部分子分布函数的定义与 Higher twist 贡献


Collinear approximation & operator expressions of PDF’s

Need for a proper theoretical framework:

Approximations:

What is neglected?

Are they indeed small so that we can neglect them?

Can we improve the accuracy (order by order)?

Parton distribution functions:

A proper definition (in terms of operators and wave functions) ?

Physical interpretations? Universality?

Calculable using models, QCD theory?



Towards an accurate theoretical frameworkXeNe

)()2(|)0(||)0(|),( 44X

X

pqppJXXJppqW

The differential cross section ： )(le )'(le

)( pN X

)(* q 22 Qq

'2

'),(

3

4

2

E

ldpqW

sQd em

)',( llL

leptonic tensor hadronic tensor

2| | 2| |

The hadronic tensor:



Parton model:

The hadronic tensor:

)()2(|)0(||)0(|),( 44X

X

pqppJXXJppqW

)'()2(|)0()0(|','','|)0()0(|2)2(

''

44

''3

3

kpqppkXkXpE

kdX

X k

,'|'|| kXX

'

)'('3

34 |)0(|'''|)0(|

2)2(

' '

X

zkpqpi

k

pXkXpeE

kdzd X

'

)'('3

34 |)(|'''|)0(|

2)2(

' X

zkqi

k

pzXkXpeE

kdzd

24

44 )()2(|)()()0(|

)2( qkpzkqpekd

zd zik

'|)'('|'|)( ' XekukXx xik



Parton model: )],(ˆ)(ˆTr[)2(

),( 4

4

qkHk,pφkd

pqW

pzψψpezdk,pφ ikz |)()0(|)(ˆ 4

2)()2()(),(ˆ qkqkqkH The hard part:

The matrix element:

Collinear approximation: )(ˆ)|,(ˆ),(ˆ xHqkHqkH xpk

)],(ˆ);(ˆTr[),( xHpxφdxpqW

)()2()2(2

1)()(])(Tr[

4

1),( 2 xfxpqxpq

pxqq

qqgxfpxqp

pqpqW

Leading twist (neglecting 1/Q suppressed terms):

);(ˆ )()2(

);(ˆ4

4

pkφp

kx

kdpxφ

...)(2

1);(ˆ xfppxφ

pzψψpedz

xf zixp |)(2

)0(|2

)(

parton distribution function



Parton distribution function:

pzψψpedz

xf zixp |)(2

)0(|2

)(

])()()()([)( ikzikz

k

ekvkbekukaz

However, the expression is not (local color) gauge invariant!

pxpbxpbxpaxpapxf |)()()()(|)(

the number density of quark and anti-quark.


Inclusive DIS with QCD interaction

SpzψψSpezdk,p,S ikz ,|)()0(|,)(ˆ 4)0(

),(ˆ)(ˆTr)2(

),,( )0()0(4

4)0( qkHk,p,S

kdSpqW

SpzψygAψSpyedzdp,S,kk ρyzikyik

ρ ,|)()()0(|,),(ˆ )(4421

)1( 21

),,(ˆ),(ˆTr)2()2(

),,( 21)1(

21)1(

42

4

41

4)1( qkkHp,S,kk

kdkdSpqW ρ

μνρμν

Not gauge invariant!

Parton density matrix:

…

We take the “multiple gluon scattering” into account

; )()2()(),(ˆ 2)0( qkqkqkH 212

2

21);,1( )()2()(

)()(),(ˆ qk

iqk

qkqkqkH siL

( , , )W q p S

(0) (1) (2)( , , ) ( , , ) ( , , ) ( , , )W q p S W q p S W q p S W q p S



Collinear approximation:

(1) (1) (1)1 2 1 1 2 2 1 2

ˆ ˆ ˆ( , , ) ( , , ) ( , )ρ ρ ρμν μν μνH k k q H k x p k x p q H x x

(0) (0) (0)ˆ ˆ ˆ( , ) ( , ) ( )μν μν μνH k q H k xp q H x

Keep only the longitudinal component of the gluon field:( ) ( ) ρρ

pA y n A y

n p

pkx /

Approximate k in the hard parts as k =xp:

Using the Ward identities such as, ixx

xHxxHp μνρ

μν

12

1(0)

21L)(1, )(ˆ

),(ˆ

)(2

130 kkk

)0,1,0(

n

)0,0,1(

n

4(0 ))

4(0ˆˆ( , , ) Tr ( )

(2 )( )μν

d kW q p S k, p, HS x

(0) 4ˆ ( ) , | (0) ( ) |) ,(0,ikzk, p,S d ze p S ψ ψ zz p S L

0

(0, ,0 )2

0 0 0

(0, ) 1 (0, ,0 ) ( ) (0, ,0 ) ' (0, ' , 0 ) ...

z

ig dy A y yz z

z Pe ig dy A y ig dy A y dy A y

L

the leading twist contributions can be obtained

gauge invariant!gauge link



Collinear expansion:

......),(ˆ

),(ˆ),(ˆ'1

'

1

21(1)

21(1)(1)

kk

xxHxxHqkH σ

ρμνρ

μνρ

μν

......)(ˆ

)(ˆ),(ˆ'

')0(

)0()0(

ρρ

μνμνμν k

k

xHxHqkH

Decomposition of the gluon field:

)()()( yAωpn

pyAnyA ρ'

ρ'ρ

ρρ

ρ'ρ

ρ'ρ

ρ'ρ nngω

pkx /

Ellis, Furmanski, Petronzio, (82) Qiu, Sterman (90,91)

Expanding the hard parts around k =xp:

Using the Ward identities such as,

to replace the derivatives etc.

(0)1(1,L)

1 22 1

ˆ ( )ˆ ( , ) μνρμν

H xp H x x

x x i

)(' xpkkω ρ'ρ

)(2

130 kkk

)0,1,0(

n

)0,0,1(

n

(0)(1)

ˆ ( ) ˆ ( , ),μν ρμνρ

H xH x x

k



SpzψzψSpezdk,p,S ikz ,|)(),0()0(|,)(ˆ 4)0( L

)(ˆTr)2(

),,(~ )0(

4

4)0( k,p,S

kdSpqWμν

...),,(~

),,(~

),,(~

),,( )2()1()0( SpqWSpqWSpqWSpqW μνμνμνμν

SpzψzyyDyψSpyedzdp,S,kk ρyzikyik

ρ ,|)(),()(),0()0(|,),(ˆ )(4421

)1( 21 LL

),(ˆTr)2()2(

),,(~

21)1(

42

4

41

4)1( p,S,kk

kdkdSpqW ρ'μν

Contain QCD interactions.(Color) gauge invariant !

...)0,',0(')0,,0()()0,,0(1),0(00

2

0

)0,,0(0

yzzyAdyig

yAdyyAdyigyAdyigPez

z

L

)()( ygAiyD ρρρ

)(ˆ )0( xH μν

),(ˆ21

)1( xxH ρμν

ρ'ρω

A consistent theoretical framework for inclusive DIS.

Inclusive DIS: Summary

…

collinear approximation

pzψψpedz

xf zixp |)(2

)0(|2

)(

collinear approximation

( ) | (0) (0, ) ( ) |2 2

ixp zdzf x e p ψ z ψ z p

L

collinear expansion

1( , ) Tr[ ( ) ] ( )

4W q p p q xp f x

q p

1( , ) Tr[ ( ) ] ( )

4W q p p q xp f x

q p

(0) 4ˆ ( ) | (0) (0, ) ( ) |ikzk, p d ze p ψ z ψ z p L

4(0) (0)

4ˆ( , ) Tr ( )

(2 )μν

d kW q p k, p

(0) (1) (2)( , ) ( , ) ( , ) ( , ) ...μν μν μν μνW q p W q p W q p W q p

4 4(1) (1)1 2

1 24 4ˆ( , ) Tr ( , )

(2 ) (2 )μν ρ'

d k d kW q p k ,k p

)(ˆ )0( xH μν

),(ˆ21

)1( xxH ρμν

ρ'ρω


1 2 ( )(1) 4 41 2

ˆ ( , ) | (0) (0, ) ( ) ( , ) ( ) |ik y ik z yρ ρk ,k p d zd ye p ψ y D y y z ψ z p L L

leading twist contribution, no gauge invariance.

leading twist contribution, with gauge invariance.

consistent framework for leading and higher twist, with gauge invariance.

…



Lesson (I): Quark distribution is not merely

…i.e., it contains “intrinsic motion” and “multiple gluon scattering”.

but

Intrinsic transverse momentum and multiple gluon scattering effects always mix up to give us the finally observed effects.

Multiple gluon scattering effects are contained in the gauge link.Collinear expansion is the necessary procedure to obtain the correct form of gauge invariant parton distributions.

Lesson (II):


ContentsContents

半单举轻子核子深度非弹性散射 (SIDIS) 与部分子关联函数



pzψzψpedz

xf zixpq |)(),0(

2)0(|

2)( L

What can we learn through inclusive DIS about the structure of nucleon?

Quark distribution function:

pzψzψpedz

xf zixpq |)(),0(

2)0(|

2)( 5 L

Quark helicity distribution:

描述核子结构的矩阵元： pzψψp |)()0(|

有 8 个独立的分量 (“ 广义结构函数” ) ，研究了其中的两个。

目前有大量实验，如 HERMES, COMPASS, JLab, FAIR, ........., 未来 eRHIC?

半单举深度非弹性散射过程

pzψyDψp |)()()0(|

的其它分量

更复杂的矩阵元，如)()( zigAzD

pzψψp |)()0(|

All concentrate on

(semi-inclusive DIS)


Semi-inclusive DIS and Parton Correlation Functions

Semi-inclusive DIS: XheNe e

e

N

DMuch more to learn!

In inclusive DIS, only longitudinal distributions are studied, e.g.

pzψzψpedz

xf zixpq |)(),0(

2)0(|

2)( L

pzψzψpedz

xf zixpq |)(),0(

2)0(|

2)( 5 L

In semi-inclusive DIS, longitudinal and transverse (transverse momentum dependent, TMD) distributions are studied, e.g.

pzψzψpezddz

kxf zkizixpq |)(),0(

2)0(|

)2(),( 3

2

L

pzψzψpezddz

kxf zkizixpq |)(),0(

2)0(|

)2(),( 5

3

2

L

Furthermore


Quark distribution/correlation functions at leading twist

The 8 independent TMD quark distribution/correlation functions at leading twist

Quark

NucleonUn-polarized Longitudinally

polarizedTransversely polarized

momentum distribution

helicity distribution

transversity

Un-polarized

Longitudinal polarization

Transverse polarization

Boer-Mulders function

Sivers function),( kxq

),( kxq

),( kxq

Pretzelocity


x

y

zk

k

s

proton spin, up

proton momentum forward

right

left

xp

S


Sivers function2

3( , ) , | (0) (0, ) ( ) | ,(2 ) 2

ixp z ik zdz d zf x k ,S e p S ψ z ψ z p S

L

represents the correlation between the transverse momentum of quark and the spin of proton.

1 1

1 1

1 ˆ( , ) ( ) ( , )

1( , ) | | | | sin ( , )

T

s T

f x k S k p f x kM

f x k S k f x kM

single-spin left-right asymmetry.

xeSS

||

:s azimuthal angle



Sivers function

Single-spin left-right asymmetry AN has been clearly observed in pp collision experiments, e.g. by E704 at FNAL in 1990s.

Xπpp )(RL

RLN NN

NNA

up

forward

left

right

spin



Sivers function

1991, Sivers function was first proposed by D. Sivers to account for AN in pp;

1993, Boros, ZTL, Meng: AN can be obtained from quark orbital angular momentum & “surface effect” in pp collisions;

1993, proof by J. Collins that Sivers function = 0;

2001, HERMES: first measurement of single spin asymmetry in Semi-inclusive DIS which shows that Sivers effect is non-zero;

2002, Brodsky, Hwang, Schmidt: explicit calculations show that, if quark orbital angular momentum & “final state interaction” are taken into account, Sivers effect is non-zero;

2002, Collins: 1993’s proof is wrong because forgot gauge link which represents the “final state QCD interactions”.



HERMES results on azimuthal asymmetry in SIDIS

Xp)(p

HERMES, PRD (2001)



Lessons:

(I) Quark distributions are not merely

but include

…

(II) Azimuthal asymmetries should be studied in Semi-inclusive DIS.

universality nuclear effects spin effects higher twist effects

are important issues.

不是孤立的核子内的部分子的动量分布，而是包含了“末态相互作用”的贡献


e

e

N

D


Sivers function in Drell-Yan process

);,();,( 11 DYkxfDISkxf TT

Xllpp )(

Collins, 2002

p

p

l

l

Xllpp Drell-Yan process:

v.s. that in DIS Xhepe

Xhepe DIS

“final state interaction” “initial state interaction”



Replace N by A, gluons can connect to different nucleons in A.

Nuclear dependence:

Nuclear enhancement; transverse momentum broadening

),(

),( 22 /)(2

2

lxfeldΔ

Akxf N

qlk

F

Aq

F

ZTL, Wang, Zhou, 2008.

AzψzψAezddz

kxf zkizixpAq |)(),0(

2)0(|

)2(),( 3

2

L

+ +…+ … +…


Azimuthal asymmetry in unpolarized SIDIS

In 1978: R. Cahn studied intrinsic momentum effects

Although no final state interaction is considered, the results already show clearly that higher twist effects should be taken into account.

2)1(1

1)2(2||cos

y

yy

Q

k

Generalize parton model to include an intrinsic transverse momentum :k

22

2

)1(1

)1(2||2cos

y

y

Q

k

(twist 3)

(twist 4)

hp

*

ee

lepton plane

hp

sNSz x

production plane

Higher twist contribution:


需要解决的关键问题核心问题：

（一）找出合适的可测量，并确定它们与描述核子结构的物理参量（即部分子分布函数）之间的关系

（二）部分子分布函数的物理意义及其唯像学研究。

从单举过程学到的

（三）利用已有的实验数据确定部分子分布函数的形式

)(),( 221 xfeQxF qq

部分子分布函数库 (PDFLIB) 中的各种参数化

部分子分布函数的几率解释、求和规则、核依赖性、模型计算，等等。

共线展开

pzψzψpedz

xf zixpq |)(),0(

2)0(|

2)( L

规范不变的

),,()2

1(4 2

1

2

4

2

2 QxFy

yQdxdQ

d em


需要解决的关键问题核心问题：半单举过程 XheNe

（一）找出合适的可测量，并确定它们与描述核子结构的物理量之间的关系灵敏的可测量：各种方位角不对称描述核子结构：各种关联函数

e

e

N

D

各种情形：非极化、极化纵向极化（沿运动方向）横向极化（垂直于运动方向）单极化（电子或质子极化）双极化（电子与质子都极化）

高扭度贡献规范联结

半单举过程中共线展开

方位角不对称与关联函数的关系？

方位角极化

半单举横向

单举纵向

hp

*

ee

lepton planhp

sNSz x


需要解决的关键问题核心问题：其它半单举反应过程

Xllpp 例如，半单举 Drell-Yan 过程 :

Xhhee 21

例如，半单举湮灭过程 :ee

p

p

l

l

e

e

1h

2h

D

D

特点：结构函数的二次方，无碎裂函数

特点：碎裂函数的二次方，无结构函数


需要解决的关键问题核心问题：

（一）找出合适的可测量，并确定它们与描述核子结构的物理参量（即部分子分布函数）之间的关系

（二）部分子分布函数的物理意义及其唯像学研究。

从单举过程学到的

（三）利用已有的实验数据确定部分子分布函数的形式

)(),( 221 xfeQxF qq

部分子分布函数库 (PDFLIB) 中的各种参数化

部分子分布函数的几率解释、求和规则、核依赖性、模型计算，等等。

共线展开

pzψzψpedz

xf zixpq |)(),0(

2)0(|

2)( L

规范不变的

),,()2

1(4 2

1

2

4

2

2 QxFy

yQdxdQ

d em


e

e

N

D

Semi-Inclusive DIS with QCD final state interaction

Leading twist, collinear approximation, better studied theoretically.

Brodsky, Schimdt, Collins, Ji, Yuan, J. Ma, Boer, Mulders, Anselmino, B. Ma, ... ...

Higher twist, collinear expansion? higher twist calculations?

Relations between parton distributions and the azimuthal asymmetries established;

Physical significances of the parton distributions are studied;

Preliminary parameterizations obtained.

Collinear expansion is shown to be applicable to SIDIS e+Ne+q(jet)+X (Liang & Wang, 2007);

Calculations up to twist-4 have been done for unpolarized case (Boer, Mulders; Mentz; Gao, ZTL, Song, Wang);

Much more to do, ..................


SIDIS with : differential cross-section to 1/Q2k

A complete twist-4 result for

cos),(||

1)2(4),()1(12 2

2

22

2 kxxfQ

kyykxfy

yQ

e

kdxdyd

dqq

qem

2cos),(~),(||

)1(4 )1(2

)1(22

2

kxkxxQ

ky

),(2

),(~),(||

)1(8 )(2

22)1(

2)1(

22

2

kxfQ

Mxkxkxx

Q

ky q

),(

||)1(12 ),2(

22

22 kxxQ

ky L

XqeNe

pzψzψpezddz

kxf zkizixpq |)(

2),0()0(|

)2(),( 3

2

L

pzψk

kzψpe

zddzkxf zkizixp

q |)(2

),0()0(|)2(

),( 23

2

L

TMD quark distribution:

Higher twist quark correlations:

pzψzDzψpezddz

dkkkxk zkizixp |)()(2

),0()0(|)2(

)ˆˆ2(),( 3

2)1(

22

L


SIDIS with : Azimuthal asymmetriesk

Up to twist-4),(

),(||

)1(1

1)2(2cos 2

kxf

kxxf

Q

k

y

yy

q

q

),(

)],(~),([||

)1(1

)1(22cos

)1(2

)1(2

2

2

2

kxf

kxkxx

Q

k

y

y

q

If g=0, i.e., no “final state interaction” (results by Cahn):

,||

)1(1

)1(2|2cos 2

2

20 Q

k

y

yg

,||

)1(1

1)2(2|cos 20 Q

k

y

yyg

proportional to ,

||

Q

k

proportional to ,||

2

2

Q

k

A good place to study such correlation functions and the effects of “final state interaction”.


Higher twist parton correlation functions

At leading twist (twist-2), there are 8 independent quark distribution functions, they are all from

pzψψp |)()0(|

Twist-3 quark correlation functions are from

pzψψp |)()0(| pzψyDψp |)()()0(|

Questions: How many needed to describe semi-inclusive DIS?

their physical interpretations?

parameterizations?

...........

and all have clear physical interpretations.

and


Summary and Outlook

对单举 DIS 过程的研究，使我们对核子结构有了很深的认识，对部分子分布函数有了系统细致的研究，对 QCD提供了很好的检验，如：

也揭示出许多问题，如：

高速运动的核子由部分子构成，部分子包括夸克和胶子，部分子分布函数的形式由 PDFlib给出， Bjorken scaling 的破坏与部分子分布函数 QCD演化，

海夸克分布的不对称，质子自旋的来源，胶子的极化、夸克轨道角动量， ............


Summary and Outlook

目前对半单举 DIS 过程的研究，使我们对核子结构的研究进入了一个新的时代，由部分子分布函数深入到关联函数，将对一系列重要问题做出回答，并对 QCD 做进一步检验，如：

谢谢！

质子自旋、部分子自旋、部分子横向运动的关联？ “ 末态 QCD 相互作用”的效应与普适性？核子内夸克的轨道角动量？部分子关联函数的 QCD演化？核效应？ .....................

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中国科技大学粒子物理与核物理讨论班