# New quark distributions and semi-inclusive electroproduction on polarized nucleons

**ABSTRACT** The quark-parton model calculation including the effects of intrinsic transverse momentum and of all six twist-two distribution functions of quarks in polarized nucleons is performed. It is demonstrated that new twist-two quark distribution functions and polarized quark fragmentation functions can be investigated in semi-inclusive DIS at leading order in Q2. The general expression for the cross-section of semi-inclusive DIS of polarized leptons on polarized nucleons in terms of structure functions is also discussed.

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**ABSTRACT:**One of the important objectives of the COMPASS experiment (SPS, CERN) \cite{Abbon:2007pq} is the exploration of the transverse spin structure of the nucleon via spin dependent azimuthal asymmetries in single-hadron production in deep inelastic scattering of polarized leptons off transversely polarized targets. For this purpose a series of measurements were made in COMPASS, using 160 GeV/c longitudinally polarized muon beam and transversely polarized $^6LiD$ (in 2002, 2003 and 2004) and $NH_3$ (in 2007 and 2010) targets. In the past few years considerable theoretical interest and experimental efforts were focused on the study of Collins and Sivers transverse spin asymmetries. The experimental results obtained so far play an important role in the general understanding of the three-dimensional nature of the nucleon in terms of transverse momentum dependent parton distribution functions. In addition to these two measured leading-twist effects, the SIDIS cross-section includes six more target transverse spin dependent azimuthal asymmetries, which have their own well defined leading or higher-twist interpretation in terms of QCD parton model. COMPASS preliminary results for these six "beyond Collins and Sivers" asymmetries, obtained from transversely polarized deuteron and proton data have been presented at the previous conferences \cite{Parsamyan:2007ju} - \cite{Parsamyan:2013ug}. In this review we focus on the results obtained with the last "proton-2010" data sample.06/2013; -
##### Article: Studies of TMDs with CLAS

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##### Article: Evolution of the helicity and transversity Transverse-Momentum-Dependent parton distributions

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**ABSTRACT:**We examine the QCD evolution of the helicity and transversity parton distribution functions when including also their dependence on transverse momentum. Using an appropriate definition of these polarized transverse momentum distributions (TMDs), we describe their dependence on the factorization scale and rapidity cutoff, which is essential for phenomenological applications.Nuclear Physics B 03/2013; · 4.33 Impact Factor

Page 1

arXiv:hep-ph/9412283v3 10 Jan 1995

hep-ph/9412283

December 1994.

New Quark Distributions and Semi-Inclusive

Electroproduction on Polarized Nucleons

Aram KOTZINIAN1

Yerevan Physics Institute,

Alikhanian Brothers St. 2; AM-375036 Yerevan,

Armenia.

Abstract.

The quark-parton model calculation including the effects of intrinsic transverse momentum

and of all six twist-two distribution functions of quarks in polarized nucleons is performed. It is

demonstrated that new twist-two quark distribution functions and polarized quark fragmentation

functions can be investigated in semi-inclusive DIS at leading order in Q2. The general expression

for the cross-section of semi-inclusive DIS of polarized leptons on polarized nucleons in terms of

structure functions is also discussed.

1Now a visitor at CERN, PPE-Division, CH-1211, Geneva 23, Switzerland.

E-mail:ARAM@CERNVM.CERN.CH

1

Page 2

1 Introduction

Deep-inelastic scattering (DIS) of leptons on nucleons provides an excellent tool for prob-

ing the structure of nuclear matter. The leading-twist momentum and helicity distribu-

tion functions (DF) of quarks in nucleons, f1(x) and g1(x), have been intensively stud-

ied. There exists another independent leading-twist DF, h1(x) [1], [2], describing the

transverse-spin distribution of quarks in a transversely-polarized nucleon. These DF’s

depend on longitudinal momentum fraction (x) carried by quark and are integrated over

its intrinsic transverse momentum (?kT). In contrast to chirally even DF’s f1(x) and g1(x),

the chirally odd DF, h1(x), cannot be measured in simple DIS because in this case trans-

verse spin asymmetries are suppressed at high energies. It can be measured in the lepton

pair production process in nucleon-nucleon collisions with both nucleons polarized trans-

versely. To be sensitive to the transversity distribution, it is necessary that either the sea

quarks are highly polarized or that polarized antinucleons are used. Thus, at present this

experiment seems very difficult.

In semi-inclusive DIS (SIDIS) the transverse-spin DF can be probed if the transverse

polarization of the struck quark is measured in some way. This “quark polarimeter”

may be provided by an azimuthal dependence of the fragmentation function (FF) for

transversely-polarized quarks [3] (Collins effect). Another possibility of quark polarimetry

is based on the observed correlation of flavor and electric charge of the fragmenting quark

and leading hadron. In Ref. [4] it was proposed to measure the transverse polarization of

quarks by measuring the polarization of self-analyzing baryons from fragmentation. An

investigation of the transversity distribution, h1(x), in SIDIS on transversely-polarized

nucleons has been proposed by the HELP collaboration [5].

Semi-inclusive DIS on a longitudinally-polarized target has been considered in Ref.

[6] and [7]. It was proposed to measure asymmetries for production of different types of

hadrons to get information about the flavor dependence of quark helicity DF in nucleons.

This kind of measurements was already performed in the SMC experiment [8] and is

planned by the HERMES collaboration [9].

In theoretical calculations of polarized SIDIS the intrinsic transverse momentum of the

quarks in the nucleon is usually ignored. For example in [10] the polarized SIDIS cross

section integrated over final hadron transverse momentum (?Ph

higher twist DF and FF. Because the intrinsic kT was neglected and the integration over

?Ph

three (∼ 1/Q). But, as was shown in Ref. [3], the target transverse-spin asymmetry may

exist at twist-two level in the azimuthal distribution of produced hadron. This asymmetry

arises from the azimuthal dependence of transversely-polarized quark fragmentation and

is sensitive to h1(x). The parton model picture of Ref. [3] is not symmetric in the sense

that the transverse momentum of final hadron with respect to the scattered quark was

taken into account but the intrinsic transverse momentum of the initial quark in nucleon

was neglected.

It is known that even in unpolarized SIDIS the effect of intrinsic momentum can

be significant [11],[12]. For a polarized nucleon the situation is more complicated. As

was shown in Ref. [1] and [13] for the nonzero kT case, the quark distribution in a

polarized nucleon is described by six DF’s already at twist-two (instead of three DF’s

when intrinsic kT effects are neglected). The three “new” DF’s relate the transverse

T) was considered, keeping

Twas assumed, the target transverse-spin asymmetry in this case appears only at twist-

2

Page 3

(longitudinal) polarization of the quark to the longitudinal (transverse) polarization of

nucleon. Measurement of these DF’s was proposed in the doubly-polarized Drell-Yan

process. Their contribution to polarized DIS appears only at twist-three (∼ 1/Q) [14].

The main subject of this article is the calculation of the polarized SIDIS cross section in

the quark-parton model with nonzero intrinsic kT. The lay-out of this paper is as follows:

section 2 contains the derivation and discussion of a general expression for the polarized

SIDIS cross section in terms of structure functions, section 3 contains a description of the

quark-parton model with intrinsic kTfor polarized SIDIS, in the section 4 a final expression

for the SIDIS cross section is derived assuming exponential dependence on transverse

momentum of DF’s and FF’s, section 5 contains a short discussion and illustrates possible

ways of experimental investigation of the effects of different DF’s and FF’s in polarized

SIDIS and finally section 6 contains some concluding remarks.

2 Structure functions for polarized SIDIS

Inelastic scattering of polarized leptons on polarized nucleons l+N → l′+N′+π was con-

sidered a long time ago by Dombey [15]. For polarized SIDIS, Gourdin [16] has counted

the number of structure functions and derived some constraints on them from the require-

ment of positivity. To derive a general formula for the cross section these authors used

the decomposition of the hadronic tensor into scalar structure functions corresponding to

different polarizations of the virtual photon and target.2

In this section I follow the method used in [15] and [16] and present explicit formulae

for polarized SIDIS cross-sections.

The Feynman diagram describing this process in the one-photon exchange approxima-

tion is depicted in Fig. 1.

Figure 1: Lowest order diagram for SIDIS.

The standard notation for DIS variables is used: l(E) and l′(E′) are the momenta

(energies) of the initial and final lepton; θland φl

azimuthal angles in the laboratory frame; q = l − l′is the exchanged virtual photon

momentum; θγis the virtual photon emission angle; P is the target nucleon momentum;

Ph(Eh) is the final hadron momentum (energy); Q2= −q2= 4EE′/sin2(θl/2); ν =

P · q/M; x = Q2/2P · q; y = ν/E; z = P · Ph

2A recent review on the subject of polarized lepton-nucleon scattering is given by S. Boffi, C.Giusti

and F.D. Pacati [17]. The author thanks D. von Harrach for mentioning this article.

labare the scattered electron polar and

T/P · q.

3

Page 4

For the cross-section one has

d6σl+N→l′+h+X=

1

4P · l

?4πα

Q2

?2

lµνWµν(2π)4

d3l

′

(2π)32E

′

d3Ph

(2π)32Eh. (1)

Here the leptonic tensor is given by QED:

lµν= 2

?

lµl

′

ν+ lνl

′

ν− l · l

′gµν+ iλǫµναβlαqβ?

, (2)

where λ is the initial lepton helicity (| λ |≤ 1).

The hadronic tensor in (1) is defined by

Wµν=

?

X

?

δ(4)(P +q−Ph−PX)

?

i∈X

d3Pi

(2π)32Ei?Ph,X | Jµ| P,S??P,S | Jν| Ph,X?, (3)

where S is the target nucleon polarization.

The hadronic tensor can be decomposed into a spin-independent (W(0)

dependent (W(S)

Wµν= W(0)

µν) and a spin-

µν) part:

µν+ W(S)

µν. (4)

For a spin-1

2target, W(S)

µνhas to be linear in S:

W(S)

µν= SρW(S)

µνρ. (5)

Note that, since S is a pseudovector, Wµνρis a pseudotensor.

It is convenient to consider the hadronic tensor decomposition in terms of structure

functions in the target laboratory frame, with the z-axis chosen in the virtual photon

momentum direction and the x-axis along the final hadron transverse momentum?Ph

(see Fig. 2). This reference system is referred to as the laboratory gamma-hadron frame

(LGHF).

T

Figure 2: Lepton and produced hadron momenta in the LGHF.

To pass to this reference frame one has to perform three rotations:

1) Rotation around the initial lepton momentum by an angle φl

scattering plane.

lab, to pass to the lepton

4

Page 5

2) Rotation around the normal to the lepton scattering plane by an angle −θγ, defining

a new z-axis coinciding with the virtual photon momentum direction. This often-used

reference frame will be called the laboratory gamma-lepton frame (LGLF).

3) Rotation around the new z-axis by the azimuthal angle of the produced hadron

(φh

direction.

For the following it is important to note that, by definition,3 ˆ? z =ˆ? q,ˆ? x =ˆ?Ph

vectors andˆ? y = [ˆ? z ×ˆ? x] is an axial vector.

One of the advantages of the LGHF is that the hadronic tensor in this frame is inde-

pendent of the relative azimuthal angle between the lepton and hadron planes (φl

To decompose the hadronic tensor into scalar structure functions the following complete

basis for polarization vectors of the virtual photon (ǫµ) and nucleon (eµ) is chosen in the

laboratory frame:4

l), defining a new x-axis coinciding with the produced hadron transverse momentum

Tare polar

h=−φh

l).

ǫµ

0=1

Q(q3,0,0,q0),

1= (0,1,0,0),

ǫµ

3=qµ

Q,

eµ

0= (1,0,0,0),

ǫµ

eµ

eµ

1= (0,1,0,0),

2= (0,0,1,0),

2= (0,0,1,0),

(6)

ǫµ

eµ

3= (0,0,0,1).

By construction, ǫµ

closure properties hold for the basis vectors:

2and eµ

2are equal toˆ? y and thus are axial vectors. The following

gµνǫµ

gµνeµ

gabǫµ

gijeµ

aǫν

ieν

aǫν

ieν

b= gab,

j= gij,

b= gµν,

j= gµν.

(7)

Now, expanding both parts of the hadronic tensor over the complete basis of polar-

ization vectors,

W(0)

µν= ǫa

W(S)

µǫb

νH(0)

ρH(S)

ab,

µνρ= ǫa

µǫb

νei

abi. (8)

one has

lµνWµν= Lab?

H(0)

ab+ Sρei

ρH(S)

abi

?

, (9)

where

Lab= ǫa

µǫb

νlµν

(10)

can be treated as a virtual-photon polarization density matrix.

Using (7) one can express the scalar structure functions as

H(0)

H(S)

ab= ǫµ

aǫν

bW(0)

µν,

abi= ǫµ

aǫν

beρ

iW(S)

µνρ. (11)

3Everywhere in this article unit vectors are denoted by a hat:ˆ? a = ? a/|? a|.

4Of course, this decomposition can be performed in a Lorentz invariant form.

5

Page 6

Let us consider the restrictions imposed by the invariance properties of Wµν[16] :

- current conservation ⇒ structure functions with a = 3 or b = 3 are equal to zero,

- parity conservation ⇒ H(0)

if it contains an even number of indices 2,

- hermiticity of Wµν⇒ H(0)

- contribution of H(S)

- time reversal invariance does not give new constraints on SIDIS structure functions.

Taking into account these properties one can choose the following set of indepen-

dent real structure functions: five spin-independent H(0)

thirteen spin-dependent H(S)

ReH(S)

In the general case these structure functions depend on four variables Q2, ν , z and

Ph2

The lepton momenta in the LGHF have the form:

ab= 0 if it contains an odd number of indices 2 and H(S)

abi= 0

ab= H(0)∗

ba

and H(S)

abi= H(S)∗

0= Pρ/M and S · P = 0,

bai,

ab0to Wµνis zero since eρ

00,H(0)

021, ImH(S)

11, H(0)

021, ReH(S)

22, ReH(0)

01, ImH(0)

023, ImH(S)

01and

023, H(S)

002, ReH(S)

123, ImH(S)

012, ImH(S)

123, H(S)

012, eH(S)

112,

121, ImH(S)

121, ReH(S)

222.

T.

lµ

γh= E

?

?

1,sinθγcosφh

l,sinθγsinφh

l,cosθγ

?

,

l

′µ

γh= E(1 − y),sinθγcosφh

l,sinθγsinφh

l,(1 − y)cos(θγ+ θl)

?

,(12)

with

cosθγ=

?

Q2

Q2+ 4M2x2

?

1 +2M2x2y

Q2

?

,

sinθγ=

?

?

?

?

4M2x2

Q2+ 4M2x2

?

1 − y −M2x2y2

Q2

?

. (13)

By our choice of reference system the dependence on azimuthal angle φl

contained in the leptonic tensor. For Lab= (1 − ǫ)Lab/2Q2, one also has the hermiticity

property Lab= Lba∗and from (2), (10) and (12) it follows that

L00= ǫ,

L01=

2

?

2

L11=1

L12= −ǫ

2

L22=1

his completely

?

ǫ(1 + ǫ)

cosφl

h+ iλ

?

ǫ(1 − ǫ)

2

?

sinφl

h,

L02= −

ǫ(1 + ǫ)

sinφl

h+ iλ

ǫ(1 − ǫ)

2

cosφl

h,

2+ǫ

2cos2φl

h, (14)

2sin2φh

2−ǫ

e+ iλ

√1 − ǫ2,

2cos2φl

h,

where

ǫ =

1

1 + 2ν2+Q2

Q2 tg2θl

2

=

2(1 − y) − Mxy/E

1 + (1 − y)2+ Mxy/E

(15)

describes the virtual-photon polarization.

6

Page 7

One can see that all “memory” of the scattered lepton azimuthal angle, φl

in the polarization density matrix of virtual photon when SIDIS is described in the LGHF.

Finally, for the product of the hadronic and leptonic tensors we get

h, is contained

1 − ǫ

2Q2lµνWµν

=1

2

+λ

2ǫ(1 − ǫ)ImH(0)

−S1

?1

2

−S3

+λS1

(γh)

?

H(0)

11+ H(0)

22

?

+ ǫH(0)

00+

?

2ǫ(1 + ǫ)ReH(0)

01cosφl

h+ǫ

2

?

H(0)

11− H(0)

22

?

cos2φl

h

?

01sinφl

h

(γh)

??

2ǫ(1 + ǫ)ReH(S)

021sinφl

h+ ǫReH(S)

121sin2φl

h

?

012cosφl

−S2

(γh)

?

2ǫ(1 + ǫ)ReH(S)

−√1 − ǫ2ImH(S)

??

?

(γh), S2

are related to the target nucleon longitudinal SL

components in the LGLF by

H(S)

112+ H(S)

222

?

023sinφl

+ ǫH(S)

002+

?

2ǫ(1 + ǫ)ReH(S)

h+ǫ

2

?

H(S)

112− H(S)

222

?

cos2φl

h

?

(γh)

??

?

h+ ǫReH(S)

123sin2φl

h

?

(16)

121+

?

2ǫ(1 − ǫ)ImH(S)

?

?

(γh)are the target nucleon spin components in the LGHF. They

γland transverse?ST

021cosφl

h

?

−λS2

+λS3

(γh)

2ǫ(1 − ǫ)ImH(S)

−√1 − ǫ2ImH(S)

012sinφl

h

(γh)

123+2ǫ(1 − ǫ)ImH(S)

023cosφl

h

?

,

where S1

(γh)and S3

γl= (S1

γl,S2

γl) spin

S1

S2

S3

γh= ST

γh= ST

γh= SL

γlcos(φS

γlsin(φS

l− φh

l− φh

l),

l),(17)

γl,

and?Sγlis related to the target polarization in the laboratory frame by

S1

S2

S3

γl= ST

γl= ST

γl= −ST

labcosθγcos(φS

labsin(φS

labsinθγcos(φS

lab− φl

lab),

lab− φl

lab) + SL

labsinθγ,

lab− φl

(18)

lab) + SL

labcosθγ,

where SL

respect to initial lepton momentum.

Note that in (16)-(18) the dependence of the polarized SIDIS cross section on target

spin components and produced hadron azimuthal angle is expressed in explicit form. In

principle, it is possible to separate the different structure functions contribution by using

a “Fourier analysis” on φh

unpolarized case in Ref. [12].

Let us now consider the cross section expression in the LGLF

laband?ST

labare the nucleon spin longitudinal and transverse components with

lfor different beam and target polarizations as was done for the

d6σl+N→l′+h+X

dxdydφl

labdzdPh2

Tdφh

l

=

ν

4Ph

?

α2y

2Q4lµνWµν,(19)

7

Page 8

and integrate it over φh

l. One gets

?2π

0

dφh

l

d6σl+N→l′+h+X

dxdydφl

labdzdPh2

Tdφh

l

=

ν

4Ph

?

2πα2y

Q2(1 − ǫ)

?1

2

?

H(0)

11+ H(0)

22

?

+ ǫH(0)

00

+S2

(γl)

?

ǫ(1 + ǫ)

2

?

?

ReH(S)

021− ReH(S)

012

?

(20)

+λS1

(γl)

ǫ(1 − ǫ)

2

?

ImH(S)

021− ImH(S)

012

?

− λS3

(γl)

√1 − ǫ2ImH(S)

123

?

.

It is interesting to note that, due to the presence of the third term in the rhs of the

last equation, when integrated over φh

asymmetry. Furthermore integration over Ph2

cross section times the mean hadron multiplicity (?nh(x,Q2)?).

One can show that by integrating over the produced hadron phase space, the first and

last two terms in (22) precisely reproduce the exact formula for the polarized DIS cross

section, given for example in Ref. [18], and that the following kinematical sum rules hold:

l, the cross section can still have a single target-spin

and z has to give us the polarized DIS

T

?d3Ph

2Eh

?d3Ph

2Eh

?d3Ph

2Eh

?d3Ph

2Eh

(x,Q2), FDIS

?1

?1

?2Mx

Q

Q

2Mx

2

?

?

H(0)

11+ H(0)

22

??

?

= ?nh(x,Q2)?FDIS

?

1

(x,Q2),

2

H(0)

11+ H(0)

22

+ H(0)

00

2xQ2

Q2+ 4M2x2= ?nh(x,Q2)?FDIS

?

2

(x,Q2),

?

?

ImH(S)

021− ImH(S)

012

?

?

− 2ImH(S)

123

= ?nh(x,Q2)?gDIS

1

(x,Q2), (21)

?

ImH(S)

021− ImH(S)

012

+ 2ImH(S)

123

?

= ?nh(x,Q2)?gDIS

2

(x,Q2),

where FDIS

entering in the spin-independent and spin-dependent part of the polarized DIS cross sec-

tion.

As is well known, a single target-spin asymmetry is forbidden in simple DIS by time

reversal invariance. This means that the following “sum rule” holds:

12

(x,Q2), gDIS

1

(x,Q2) and gDIS

2

(x,Q2) are the structure functions

?d3Ph

2Eh

?

ReH(S)

021− ReH(S)

012

?

= 0.(22)

Thus, one can have a single target-spin asymmetry in a SIDIS cross section integrated

over φh

confirmed in the simple quark-parton model.

l, which disappears after integration over Ph2

T

and z. This observation will be

3 Parton model with intrinsic transverse momentum

Unpolarized SIDIS is described in a simple way by the factorized parton model.

this model the lepton knocks out a quark, which subsequently fragments into hadrons.

Sixteen years ago it was shown by Cahn [11] that nonperturbative effects of the intrinsic

transverse momentum (kT) of the quarks inside the nucleon may induce significant hadron

In

8

Page 9

asymmetries in the relative azimuthal angle φh

azimuthal asymmetry at the level of up to 15-20%, which arises mainly from the effects

of the intrinsic kT of the struck quark with ?k2

Here I generalize the calculation of Ref. [11] to the polarized SIDIS process. Calcula-

tions will be performed in the electron-quark scattering Breit-frame (BF) (Fig. 3), which

can be reached from the LGLF by a Lorentz boost along the z-axis.

l. The EMC experiment [12] found an

T? ≥ (0.44GeV )2.

Figure 3: The quark-parton model picture for SIDIS in the BF.

The azimuthal angle, φh

virtual photon four-momentum, we have qµ= (0,0,0,Q). In the following calculations

the terms ∼ 1/Q2are neglected. In this approximation the lepton and quark (k,k′)

four-momenta in the BF are given by:

e, remains unchanged under this transformation, and, for the

lµ

Br=Q

2

?2 − y

?2 − y

?

?

y

,2√1 − y

,2√1 − y

y

ˆ?lT,1

?

,

l′µ

Br=Q

2yy

ˆ?lT,−1

?

,

kµ

Br=Q

2

1,2?kT

Q,−1

1,2?kT

Q,1

?

, (23)

k′µ

Br=Q

2

?

,

whereˆ?lT = (1,0) is the unit transverse two-vector in the lepton transverse momentum

direction. Note that because of a nonzero?kT, the electron and the quark scattering planes

do not coincide in general (see Fig. 3, where the angle between these planes is denoted

by φq

In the quark-parton model the description of the SIDIS process is similar to that of

double scattering experiments, in which the polarization of the particle produced after

the first scattering is measured. To calculate the cross section one has to know:

1) the description of the initial quark state,

2) the noncoplanar polarized l + q → l′+ q′scattering cross section,

3) the description of the polarized quark fragmentation.

l).

9

Page 10

3.1

The initial quark state.

The polarization of free quarks is most clearly described by the density matrix of partially

polarized fermions, which in the ultrarelativistic case takes the form [19]

ρ =1

2(γ · k)

?

1 + γ5sL+ γ5? γT·? sT

?

,(24)

where sLand? sTare the longitudinal and transverse (with respect to quark three-momentum

?k) components of twice the quark polarization vector in its rest frame, s2

For the initial quark state in the nucleon one has to use, instead of (2), the following

expression:

L+ |? sT|2≤ 1.

ρq (in)

N

=1

2Pq

N(x,k2

T)(γ · k)

?

1 + γ5s(in)

L (x,?kT) + γ5? γT·? s(in)

T

(x,?kT)

?

, (25)

where Pq

verse polarization distributions. They can easily be found from the density matrix by

calculating the appropriate trace

N(x,k2

T), s(in)

L (x,?kT) and ? s(in)

T

(x,?kT) are the probability, longitudinal and trans-

Pq

N(x,k2

T) =

1

2p0tr

?

ρq (in)

N

γ0?

1

2p0tr

1

2p0tr

,

Pq

N(x,k2

T)s(in)

L (x,?kT) =

?

?

ρq (in)

N

γ5γ0?

γ5? γTγ0?

, (26)

Pq

N(x,k2

T)? s(in)

T

(x,?kT) =

ρq (in)

N

.

At first sight, it seems that presence of the intrinsic transverse momentum of the

quark can not give a sizable effect on the spin distribution. Consider, for example, a

transversely-polarized quark in the quark-parton model. In this case the projection of

quark polarization vector onto the nucleon momentum direction is proportional to kT/k?=

2kT/Q, and one might conclude that the contribution of the quark transverse polarization

to the nucleon longitudinal polarization is suppressed at high Q2. However, the nucleon

polarization has to be calculated in its rest frame, and the factor 1/Q will disappear after

a Lorentz transformation to nucleon rest frame.

Consider a simple example for spin transfer from a polarized nucleon to a quark. Let

Aµ and aµ be the polarization four-vectors of the nucleon and the quark in the frame

where the nucleon has large momentum (for example in the BF), and suppose that they

are related by a0 = αA0, a3 = αA3, a1 = βA1 and a2 = βA2, with A0 = PNSL/M,

A3= ENSL/M,?AT =?ST, where?S is the nucleon polarization in its rest frame and PN

(EN) is the momentum (energy) of the nucleon in the BF. Now one can calculate the

quark polarization in its rest frame (? s) assuming that mq= xM. After rotation of the

coordinate system of the BF and a Lorentz boost along the quark momentum one gets:

sL= αSL+β?kT·?ST/mqand ? sT= β?ST−α?kTSL/mq. Thus, in this “toy” parton model, the

longitudinal spin of the quark receives two contributions: from both the longitudinal and

transverse spin of the nucleon. It is important to note that neither of these contributions

is suppressed at high Q2. The same behavior is true for the transverse spin of the quark.

10

Page 11

General consideration of the quark DF in a polarized nucleon in the case of nonvanish-

ing kT has been done by Ralston and Soper [1] and recently by Tangerman and Mulders

[13]. They have found that at the leading twist one needs six independent DF’s depending

on x and k2

are given by [13]

T: f,g1L, g1T, h1T, h⊥

1L, and h⊥

1T. The distributions s(in)

L (x,?kT) and ? s(in)

T

(x,?kT)

Pq

Pq

N(x,k2

T) = f(x,k2

T),

N(x,k2

T)s(in)

L (x,?kT) = g1L(x,k2

T)SL+ g1T(x,k2

T)

?kT·?ST

mD

, (27)

Pq

N(x,k2

T)? s(in)

T

(x,?kT) = h1T(x,k2

T)?ST +

h⊥

1L(x,k2

T)SL+ h⊥

1T(x,k2

T)

?kT·?ST

mD

?kT

mD,

where mDis an unknown mass parameter, SLand?ST are the nucleon longitudinal and

transverse polarization with respect to its momentum. The “new” DF’s have clear phys-

ical interpretation: for example, g1T describes the quark longitudinal polarization in a

transversely-polarized nucleon. It is important to notice that due to this DF even the

initial quark longitudinal spin distribution in a polarized nucleon exhibits an azimuthal

asymmetry.

3.2

Noncoplanar polarized l + q → l′+ q′scattering

Using standard methods of QED [19], it is easy to calculate the cross section of a polarized

lepton scattering on a polarized quark in the one-photon approximation:

d2σl+q→l′+q′

i

dydφl

h

=

α2e2

2Q2f(x,k2

iy

T)

??

T) − 4u(s(in)

s2+ u2??

1 + s(in)

L s′

L

?

+

?

s2− u2?

λ

?

T· l)

s(in)

L

+ s′

L

?

+2su(s(in)

T

· s′

T

· l)(s′

T· l′) − 4s(s(in)

T

· l′)(s′

?

. (28)

Here, eiis the electric charge of the quark in positron charge units, s′

ter describing the longitudinal spin component of the final quark and s′

describing its transverse spin; s, t and u are the usual Mandelstam variables given by

Lis the parame-

Tis four-vector

s = 2MEx

?

1 − 2

?

1 − y

?kT·ˆ?lT

Q

?

,

t = −Q2= −2MExy,(29)

u = −2MEx(1 − y)

?

1 −

2

√1 − y

?kT·ˆ?lT

Q

?

.

Note that for noncoplanar l + q → l′+ q′scattering s and u depend on the relative

azimuthal angle between the quark and lepton scattering planes, φq

corrections of order 1/Q will arise due to this dependence.

l, and kinematical

11

Page 12

The transverse polarization four-vectors of the quarks in the BF in this approximation

are

s(in)µ

T

= (0,? s(in)

T

,2

Q

?kT·? s(in)

T

),

s′µ

T= (0,? s′T,−2

Q

?kT·? s′T). (30)

Substituting (29) and (30) into (28) the following expression for the cross section is

obtained:

d2σl+q→l′+q′

i

dydφl

h

where

=α2e2

2Q2yf(x,k2

i

T)

?

a + bLs′

L+?bT·? s′T

?

, (31)

a = 1 + (1 − y)2− 4(2 − y)

?kT·ˆ?lT

Q

+ λs(in)

L y

2 − y − 4

?

1 − y

?kT·ˆ?lT

Q

,

bL= λy

2 − y − 4

?

1 − y

?kT·ˆ?lT

Q

+ s(in)

·ˆ?lT)ˆ?lT

·ˆ?lT)?kT+ (? s(in)

L )

1 + (1 − y)2− 4(2 − y)

?

1 − y

?kT·ˆ?lT

Q

(32)

,

?bT= 2(1 − y)

?

? s(in)

T

− 2(? s(in)

(? s(in)

T

T

+2

2 − y

Q√1 − y

?

T

·?kT)ˆ?lT− (ˆ?lT·?kT)? s(in)

T

? ?

.

The final quark polarization according to the general rules (see [19],§65) is given by

=bL

s(f)

L

a,

?bT

a.

? s(f)

T

=

(33)

From (32) and (33) one can see that the final quark can be transversely polarized only

if the transverse polarization of the initial quark is not equal to zero. The sideways (Dss)

and normal (Dnn) transverse spin transfer coefficients to leading order in 1/Q are given

by

Dnn= −Dss=

2(1 − y)

1 + (1 − y)2+ λs(in)

L y(2 − y).

′S

l) in this approximation is very

(34)

The azimuthal angle of the final quark transverse spin (φ

simply connected with that of the initial quark (φS

l):

φ

′S

l

= π − φS

l

(35)

For unpolarized leptons, expression (34) coincides with the depolarization factor in Ref.

[3].

Note also that, in contrast with transverse polarization, the longitudinal polarization

of the final quark is not equal to zero even if the initial quark is unpolarized but the initial

lepton is longitudinally polarized.

12

Page 13

For the final quark state before fragmentation, one can now write the density matrix

as

ρq (f)=α2e2

i

2Q2yf(x,k2

T)a

?

1 + γ5s(f)

L + γ5? γT·? s(f)

T

?

. (36)

One can see that the final quark state has an azimuthal asymmetry in the relative

angle between the lepton and quark scattering planes, φq

a kinematical origin, as the third term in the expression for a in (32). The azimuthal

asymmetry in unpolarized SIDIS arises from this term [11]. But there also exist terms

reflecting the azimuthal anglular dependence of the initial quark distributions, which are

not suppressed at high Q2.

l. Part of this asymmetry has

3.3

Polarized quark fragmentation.

In analogy with the quark probability distribution in the nucleon one can write the prob-

ability of producing a hadron, h, in the polarized quark fragmentation as

Ph

q(z,?Ph

q T) =

1

2p0tr

?

ρq (f)F(z,?Ph

q T)γ0?

, (37)

where F(z,?Ph

and the hadron transverse momentum with respect to the final quark momentum?Ph

?Ph

parts [3].

q T) is the polarized quark fragmentation function, depending on z = Eh/Eq′

q T=

T− z?kT. This FF can be presented as a sum of spin-independent and spin-dependent

F(z,?Ph

q T) = F(0)(z,Ph 2

q T) + γ5(? γT· [ˆ?k′×

?Ph

mF

q T

])F(S)(z,Ph 2

q T), (38)

where mF is another unknown mass parameter. The two terms on the rhs of equation

(40) are the only ones allowed by parity invariance.

Calculating the trace in (37) one gets

Ph

q(z,?Ph

q T) =

α2e2

2Q2yf(x,k2

α2e2

i

2Q2yf(x,k2

i

T)

?

aF(0)(z,Ph 2

q T) − (?bT· [ˆ?k′×

q T) + |?bT||?Ph

?Ph

mF])F(S)(z,Ph 2

q T

q T)

?

=

T)

?

aF(0)(z,Ph 2

q T|

mF

F(S)(z,Ph 2

q T)sinφcol

?

, (39)

where φcolis the angle between?Ph

In contrast to the ordinary FF, F(0)(z,P2

quark FF, F(S)(z,P2

has never been measured.

q Tand the final quark transverse polarization.

T), the spin-dependent part of the polarized

T), or, in other words, the analyzing power of polarized fragmentation,

4 Results

In this section the polarized SIDIS cross section is calculated taking into account all six

twist-two quark DF’s and the Collins effect in polarized quark fragmentation. Kinematical

corrections of order 1/Q also are kept.

13

Page 14

To calculate the cross section one has to integrate over?kTand sum over all quark and

antiquark types the probability of hadron production in final quark fragmentation (the

initial quark probability distribution and l+q → l′+q′scattering cross section is already

included in the final quark density matrix):

d6σl+N→l′+h+X

dxdydφl

labdzdPh2

Tdφh

l

=

?

q

?

d2kTPh

q(z,?Ph

q T), (40)

where d2kT= 1/2dk2

This integration can be performed analytically if one supposes that the transverse

momentum dependence in the DF’s and FF’s may be written in factorized exponential

form:

Tdφq

land φq

lis the quark azimuthal angle in the BF.

dJ(x,k2

T) =

1

πaJexp

?

−k2

T

aJ

?

?

dJ(x),

F(0,S)(z,p2

T) =

1

πa(0,S)

F

exp−

p2

T

a(0,S)

F

?

F(0,S)(z). (41)

Here the index J = f1,g1L,g1T,h1T,h⊥

of transverse momentum distribution for each distribution (fragmentation) function aJ∼

?k2

After some calculations the final result looks like:

1L,h⊥

1Tenumerates the different DF’s. The width

T? (a(0,S)

F

∼ ?Ph2

q T?) can in principle depend on x (z).

d6σl+N→l′+h+X

dxdydφl

labdzdPh2

Tdφh

l

=

?

q

α2e2

2Q2y

q

?

CUP+ CDP+ CSP

?

, (42)

where the contributions to unpolarized, double (beam and target) and single (target only)

polarized parts of the cross section are given in the following three formulae.

CUP= A(0)

f1

1 + (1 − y)2− 4y(2 − y)α(0)

f1Ph

Q

T

cosφh

l

, (43)

CDP

= λSL

γlA(0)

g1Ly

?

2 − y − 4

?

1 − yα(0)

1 − yα(0)

g1LPh

Q

T

cosφh

l

?

+ λST

γlA(0)

g1Ty

??

2 − y − 4

1 − yb(0)

?

g1TPh

Q

T

cosφh

l

?α(0)

g1TPh

mD

T

cos(φh

l− φS

l)(44)

−2

?

g1T

mDQcos(φS

l)

?

,

CSP

= −21 − y

mF

?

ST

γlA(S)

h1T(1 − zα(S)

A(S)

h⊥

1L

mD

h1T)Ph

Tsin(φh

l+ φS

l)

+SL

γl

α(S)

h⊥

1L(1 − zα(S)

h⊥

1L)Ph2

T sin2φh

l

14

Page 15

+ST

γl

A(S)

h⊥

m2

+(1

1T

D

2− zα(S)

?

A(S)

h⊥

1L

mD

A(S)

h⊥

1T

m2

D

Ph

T

?

α(S)2

h⊥

1T(1 − zα(S)

h⊥

1T)Ph2

T cos(φh

l− φS

l)sin2φh

l

h⊥

1T)b(S)

h⊥

1Tsin(φh

l+ φS

l)

?

− 2

2 − y

Q√1 − y

?

ST

γlA(S)

h1T

α(S)

h1T(1 − zα(S)

h1T)Ph2

T

− zb(S)

h1T

?

sinφS

l

(45)

+SL

γl

?

α(S)2

h⊥

1L(1 − zα(S)

h⊥

1L)Ph2

T + (1 − 2zα(S)

h⊥

1L)b(S)

h⊥

1L

?

Ph

Tsinφh

l

+ST

γl

?1

2z(α(S)2

h⊥

1TPh2

T

+ 2b(S)

h⊥

1T)b(S)

h⊥

1TsinφS

l

−α(S)

h⊥

1T[α(S)2

h⊥

1T(1 − zα(S)

h⊥

1T)Ph2

T + (2 − 3zα(S)

h⊥

1T)b(S)

h⊥

1T]Ph2

T cos(φh

l− φS

l)sinφh

l

???

.

Here the following notation has been adopted5

B(0,S)

J

= a(0,S)

F

+ z2aJ,α(0,S)

J

=

zaJ

B(0,S)

J

,b(0,S)

J

=a(0,S)

B(0,S)

F

aJ

J

,

A(0,S)

J

=

1

πB(0,S)

J

exp

?

−Ph2

B(0,S)

J

T

?

dJ(x)F(0,S)(z).(46)

One can check that both the φh

exactly coincide with that required by the general structure function analysis of section

2.

Integrating (43)-(45) over hadron azimuthal angle one gets

land y dependence of the parton model result (42)-(45)

?2π

?2π

?2π

0

dφh

lCUP

= 2πA(0)

f1

?

1 + (1 − y)2

?

,

0

dφh

lCDP

= 2πλy

= 4π(2 − y)√1 − y

?

SL

γlA(0)

g1L(2 − y) − 2ST

γl

A(0)

g1T

mDQ

?

1 − y

?

α(0)2

g1TPh2

T + b(0)

g1T

?

cosφS

l

?

,

0

dφh

lCSP

QmF

ST

γl

?

2A(S)

h1T

?

α(S)

h1T(1 − zα(S)

h1T)Ph2

T − zb(S)

h1T

?

(47)

+

A(S)

h⊥

m2

1T

D

?

z

?

1T(1 − zα(S)

α(S)2

h⊥

1TPh2

T + 2b(S)

h⊥

1T

?

b(S)

h⊥

1T

−α(S)

h⊥

1T

?

α(S)2

h⊥

h⊥

1T)Ph2

T + (2 − 3zα(S)

h⊥

1T)b(S)

h⊥

1T

?

Ph2

T

??

sinφS

l.

As is clear from the last equation, even when integrated over hadron azimuthal angle,

the SIDIS cross section can still have a single target spin asymmetry. Further integration

over hadron transverse momentum gives

?

d2Ph

TCUP

=

?

1 + (1 − y)2

?

f1(x)F(0)(z),

5To simplify notation the index q has been suppressed where possible. Obviously, all quantities related

with DF (FF) depend on quark flavor q (and final hadron type h).

15