M. Anselmino Torino University & INFN

An intuitive approach to factorization

based on work with M. Boglione, U. D’Alesio, E. Leader, S. Melis, F. Murgia, A. Prokudin

INT program on “Gluons and the quark sea at high energySeptember 21, 2010, INT-Seattle

lepton plane

Zc.m. − axis

Xc.m. − axis

φh

φS

hadron plane

S

"

PT

"′

Ph

γ∗, q p

SIDIS processes with Trento kinematical conventions

d6σ ≡ d6σ!p↑→!hX

dxB dQ2 dzh d2P T dφS

! p→ ! hX

xB =Q2

2p · q

Q2 = −q2

zh =p · Ph

p · q

dσ

dφ= FUU + cos(2φ) F cos(2φ)

UU+

1Q

cos φ F cos φUU

+ λ1Q

sinφ F sin φLU

+ SL

{sin(2φ) F sin(2φ)

UL+

1Q

sinφ F sin φUL

+ λ

[FLL +

1Q

cos φ F cos φLL

]}

+ ST

{sin(φ− φS)F sin(φ−φS)

UT+ sin(φ + φS) F sin(φ+φS)

UT+ sin(3φ− φS) F sin(3φ−φS)

UT

+1Q

[sin(2φ− φS) F sin(2φ−φS)

UT+ sinφS F sin φS

UT

]

+ λ

[cos(φ− φS) F cos(φ−φS)

LT+

1Q

(cos φS F cos φS

LT+ cos(2φ− φS)F cos(2φ−φS)

LT

)]}

Gourdin, NP B49 (1972) 501Kotzinian, NP B441 (1995) 234

Mulders and Tangermann, NP B461 (1996) 197Boer and Mulders, PR D57 (1998) 5780Bacchetta et al., PL B595 (2004) 309

Bacchetta et al., JHEP 0702 (2007) 093

most general azimuthal dependence at O(1/Q)

SIDIS in the QCD parton model (with intrinsic motion)

dσ ! p→! hX =∑

q

fq(x,k⊥;Q2)⊗ dσ ! q→! q(y, k⊥;Q2)⊗Dhq (z,p⊥;Q2)

factorization holds at large Q2, and PT ≈ k⊥ ≈ ΛQCD

PT ! Q2Two scales:

Q2

P T ! p⊥ + zh k⊥

λq λ′q

p, Sp, S

Q2Q2

h h

recover the general structure within the factorized scheme

three “physical” steps:

p→ q + X ! q → ! q q → h + X

TMD-PDF hard scattering TMD-FF

dσ!(S!)+p(S)→!′+h+X

dxB dQ2 dzh d2P T dφS

= ρ!,S!

λ!,λ′!⊗ ρq/p,S

λq,λ′qfq/p,S(x,k⊥) ⊗ Mλ!,λq ;λ!,λq

M∗λ′

!,λ′q ;λ′

!,λ′q⊗ Dh

λq,λ′q(z,p⊥)

compute polarized cross section with helicity formalism

more precisely

dσ!(S!) p(S)→!′hX

dxB dQ2 dzh d2P T dφS! 1

2π

∑

q

∑

{λ}

116 π (xBs)2

∫d2k⊥ d2p⊥ δ(P T − zhk⊥ − p⊥)

× ρ!(S!)λ!λ′

!ρq/p,S

λqλ′q

fq/p,S(x,k⊥) Mλ!λq ;λ!λqM∗

λ!λ′q ;λ!λ′

qDh

λqλ′q(z,p⊥)

valid at O(PT /Q)

= helicity density matrixρλ λ′

in general, for a spin 1/2 Dirac particle

components of particle polarization vector in its helicity rest framePx, Py, Pz =

ρλ λ′ =12

(1 + Pz Px − iPy

Px + iPy 1− Pz

)

=12

(1 + PL PT e−iϕ

PT eiϕ 1− PL

)

first step and the quark correlator

ρa/A,SA

λa,λ′a

fa/A,SA(xa,k⊥a) =

∑

λA,λ′A

ρA,SA

λA,λ′A

∑

XA,λXA

∫Fλa,λXA

;λAF∗λ′

a,λXA;λ′

A

≡∑

λA,λ′A

ρA,SA

λA,λ′A

Fλa,λ′

a

λA,λ′A

Fλa,λXA;λA

(xa,k⊥a) helicity amplitude for the “process”: A→ a + X

from general properties of helicity amplitudes:

Fλa,λXA;λA

(xa,k⊥a) = Fλa,λXA;λA

(xa, k⊥a) eiλAφa

Fλa,λ′

a

λA,λ′A(xa,k⊥a) = F

λa,λ′a

λA,λ′A(xa, k⊥a) e(λA−λ′

A)φa

fa/A(xa,k⊥a) = fa/A,SL=

(F++

++ + F++−−

)

fa/A,ST(xa,k⊥a) =

(F++

++ + F++−−

)+ 2 ImF++

+− sin(φSA − φa)

P ax fa/A,SL

(xa,k⊥a) = 2 ReF+−++

P ax fa/A,ST

(xa,k⊥a) =(F+−

+− + F−++−

)cos(φSA − φa)

P ay fa/A,SL

(xa,k⊥a) = P ay fa/A = −2 ImF+−

++

P ay fa/A,ST

(xa,k⊥a) = −2 ImF+−++ +

(F+−

+− − F−++−

)sin(φSA − φa)

P az fa/A,SL

(xa,k⊥a) =(F++

++ − F++−−

)

P az fa/A,ST

(xa,k⊥a) = 2 ReF+++− cos(φSA − φa)

and there are eight independent Fλa,λ′a

λA,λ′A

F++++ , F++

−− , F+−+− , F−+

+− , F+++− , F+−

++

real real for quarks complex

examples

Sivers function: needs interference between two amplitudes with nucleon helicity flip (chiral-even)

Boer-Mulders function: needs interference between two amplitudes with quark helicity flip (chiral-odd)

∆Nfa↑/A = −2 ImF+−++ = −2 Im

∑

XA,λXA

∫F+,λXA

;+ F∗−,λXA;+

∆Nfa/A↑ = 4 ImF+++− = 4 Im

∑

XA,λXA

∫F+,λXA

;+ F∗+,λXA;−

(∆N fa/A↑ ≡ fa/ST− fa/−ST

∆N fa↑/A ≡ Py fa/A)

Fλa,λ′

a

λA,λ′A(xa,k⊥a) is the quark correlator

Fλa,λ′

a

λA,λ′A

=∑

XA,λXA

∫Fλa,λXA

;λAF∗

λ′a,λXA

;λ′A

λA λ′A

λa λ′a

Fλa,λXA;λA

F∗λ′

a,λXA;λ′

A

XA, λXA

Φ(x,k⊥) =12

[f1/n+ + f⊥1T

εµνρσγµnν+kρ⊥Sσ

T

M+

(SL g1L +

k⊥ · ST

Mg⊥1T

)γ5/n+

+ h1T iσµνγ5nµ+Sν

T +(

SL h⊥1L +k⊥ · ST

Mh⊥1T

)iσµνγ5nµ

+kν⊥

M

+ h⊥1σµνkµ

⊥nν+

M

]

4 M. ANSELMINO

P, S

q q

k

k′

P, S

Fig. 1. – The handbag diagram for DIS. At leading QED order, the interaction between thelepton (not shown) and the nucleon is mediated by the exchange of a virtual photon. Thus, theDIS cross section is just the total cross section for the γ∗N → X process, which, by the opticaltheorem, is related to the forward scattering amplitude. In the parton model, at leading QCDorder, the virtual photon scatters off a single quark in the nucleon, as represented in the figure.The lower blob is thus the matrix element between the nucleon initial and final states of twoquark fields, one ”extracted from” and the other ”replaced into” the nucleon. It is a matrix inthe Dirac spinor space.

and it shows the chiral-odd nature of transversity, as it relates quarks with oppositehelicities. It is then clear why h1 cannot be measured in DIS: the bottom blob of fig. 2cannot be inserted in the handbag diagram of fig. 1, as the QED (and QCD) interactionsconserve helicity and there is no way, by photon or gluon couplings, of flipping the helicityof massles quarks.

A measurement of transversity requires a process in which h1 couples to anotherchiral-odd function. Several suggestions have been discussed in the literature. At themoment the most practicable way appears via SIDIS processes [7], in which h1 couplesto a chiral-odd fragmentation function, the Collins fragmentation function, as depictedin fig. 3. In principle, the cleanest and most direct way should be via the measurementof the double transverse spin asymmetry ATT in Drell-Yan processes, which couples twotransversity distributions (see fig. 4), as discussed in Section 5.

So far we have only considered collinear partonic configurations, in which the rele-vant degrees of freedom, describing the nucleon structure, are the parton longitudinalmomentum fraction x and the helicities. Yet, it is already clear that the spin transversedegree of freedom is at least as interesting, but much less known. It will be much moreso when also the intrinsic transverse motions of partons, k⊥, in addition to x, will beconsidered. Which requires a detour into the issue of SSA.

3. – The (problem of) transverse Single Spin Asymmetries

Let us consider a 2 into 2 physical process, like AB → C D, in the center of massreference frame, A(p) + B(−p) → C(p′) + D(−p′), like in fig. 5. We wonder whetheror not the cross section for such a process can depend on the spin polarization S of oneparticle only, say A; particle B is not polarized and the polarization of the final particles

Φij(k;P, S) =∑

X

∫d3P X

(2π)3 2EX(2π)4 δ4(P − k − PX)〈PS|Ψj(0)|X〉〈X|Ψi(0)|PS〉

=∫

d4 ξ eik·ξ〈PS|Ψj(0)Ψi(ξ)|PS〉

f1(xa, k⊥a) = F++++ + F++

−− = fa/A

k⊥a

Mf⊥1T (xa, k⊥a) = −2 ImF++

+−

g1L(xa, k⊥a) = F++++ − F++

−−k⊥a

Mg⊥1T (xa, k⊥a) = 2 ReF++

+−

k⊥a

Mh⊥1L(xa, k⊥a) = 2 ReF+−

++

k⊥a

Mh⊥1 (xa, k⊥a) = 2 ImF+−

++

h1(xa, k⊥a) = F+−+−

(k⊥a

M

)2

h⊥1T (xa, k⊥a) = 2F−++−

relations between different notations

XpA

P AP a

k⊥a

F. Murgia

Φ(xa, pA, k⊥a;P A,P a) =12

{fa/A(xa, k⊥a) · 1

+ ∆Nfa↑/A(xa, k⊥a) (pA × k⊥a) · P a

+12

∆Nfa/A↑(xa, k⊥a) (pA × k⊥a) · P A

+ ∆−fsy/ST(xa, k⊥a)

[P A · P a

− (pA · P A)(pA · P a)− (k⊥a · P A)(k⊥a · P a)]

+ ∆fsz/SZ(xa, k⊥a) (pA · P A)(pA · P a)

+ ∆fsx/SZ(xa, k⊥a) (pA · P A)(k⊥a · P a)

+ ∆fsz/ST(xa, k⊥a) (k⊥a · P A)(pA · P a)

+ ∆fsx/ST(xa, k⊥a) (k⊥a · P A)(k⊥a · P a)

}

P qj fq/p,SJ

(x,k⊥) = fqsj/SJ

(x,k⊥)− fq−sj/SJ

(x,k⊥) ≡ ∆fqsj/SJ

(x,k⊥)

u(pi, λi) =√

p0i

(1λi

)χλi

(pi) pi = (sin θi cos φi, sin θi sin φi, cos θi)

χ+(pi) =

cos(θi/2) e−iφi/2

sin(θi/2) eiφi/2

χ−(pi) =

− sin(θi/2) e−iφi/2

cos(θi/2) eiφi/2

!(p1) + q(p2)→ !(p3) + q(p4)

second step: elementary interaction and phases in (non-planar) helicity amplitudes

if scattering is not planar all phases are different and remain in the amplitudes

Dirac-Pauli helicity spinors, pi = (p0i , pi)

M ∝ u(p3, λ3) γµ u(p1, λ1) u(p4, λ4) γµ u(p2, λ2)

= M0 eiϕ

neglecting masses there are two independent helicity amplitudes

M1 = M++;++ ! 2 eqe2

[1y

e+iφ⊥ − 2√

1− y

y

k⊥Q

]

M2 = M+−;+− ! 2 eqe2

[(1− y)

ye−iφ⊥ − 2

√1− y

y

k⊥Q

]

Mλ!λq ;λ!λq

φ⊥ is azimuthal angle of k⊥

O(k⊥/Q)

dσ!q→!q

dt=

116πs2

14

∑

λq,λ!

|Mλ!,λq ;λ!,λq |2

unpolarized cross section

third step: fragmentation functions

= helicity amplitude for the “process”:

Dλh,λ′

hλq,λ′

q(z,p⊥) =

∑

X,λX

∫Dλh,λX ;λq

(z,p⊥) D∗λ′h,λX ;λ′

q(z,p⊥)

Dλh,λX ;λq

q → h + X

Dh/q(z) =12

∑

λq,λh

∫d2p⊥ D

λh,λhλq,λq

(z,p⊥)

usual unpolarized fragmentation function:

λq λ′q

p, Sp, S

Q2Q2

h h

from general properties of helicity amplitudes:

Collins function (spinless final particles)

Dλh,λ′

hλq,λ′

q(z,p⊥) = D

λh,λ′h

λq,λ′q(z, p⊥) ei(λq−λ′

q)φHh

Dλh,λX ;λq(z,p⊥) = Dλh,λX ;λq

(z, p⊥) eiλqφHh

φHh can be written in terms of external or integration variables

∆NDh/q↑(z, p⊥) ≡ −2i Dh/q+−(z, p⊥) = 2 ImDh/q

+−(z, p⊥) =2p⊥zMh

H⊥1 (z, p⊥)

∆NDh/a↑ = 2 ImDh/q+− = 2 Im

∑

X,λX

∫DλX ;+D∗λX ;−

X

spin-p┴ correlations in fragmentation process (case of final spinless hadron)

H⊥q1 (x,p2

⊥)

XDq

1(x,p2⊥)

“Collins effect”sq · (pq × p⊥)

Dh/q,sq(z,p⊥) = Dh/q(z, p⊥) +

12

∆NDh/q↑(z, p⊥) sq · (pq × p⊥)

= Dq1(z, p⊥) +

p⊥zMh

H⊥q1 (z, p⊥) sq · (pq × p⊥)

FIG. 2. One-loop corrections to the fragmentation of aquark into a pion.

Σ(k) = A /k + B m , (9)

Γ(k, p) = C + D /p + E /k + F /p /k. (10)

The real parts of the functions A, B, C etc. are UV-divergent and require in principle a proper renormaliza-tion. Though our model is renormalizable, we do nothave to deal with this question at all, since only theimaginary parts of the loop diagrams will turn out tobe important.

The contributions to the correlation function generatedby the diagrams (a) and (c) are given by:

∆(a)(1)(k, p) = −

g2

(2π)4(/k + m)

k2 − m2γ5 (/k − /p + m) γ5

×(/k + m)

k2 − m2Σ(k)

(/k + m)

k2 − m22π δ((k − p)2 − m2) , (11)

∆(c)(1)(k, p) = −

g2

(2π)4(/k + m)

k2 − m2γ5 (/k − /p + m) γ5

× Γ(k, p)(/k + m)

k2 − m22π δ((k − p)2 − m2). (12)

The contributions from diagrams (b) and (d) follow from

the hermiticity condition: ∆(b)(1)(k, p) = γ0∆(a)†

(1) (k, p)γ0,

∆(d)(1)(k, p) = γ0∆(c)†

(1) (k, p)γ0.

kk

p

! ( )( )i- " k,pg #5k

FIG. 3. One-loop self-energy and vertex corrections.

Summing the contributions of the four diagrams andinserting the resulting correlation function in Eq. (1), weobtain the result

H⊥1 (z, k2

T)

=g2 mπ

4π3

p−

z

∫

dk+δ((k − p)2 − m2)

×(

m Im (A + B)

(k2 − m2)2+

Im (D + E + mF )

(k2 − m2)

)∣

∣

∣

∣

k−= p−

z

=g2 mπ

8π3

1

1 − z

(

m Im (A + B)

(k2 − m2)2

+Im (D + E + mF )

(k2 − m2)

)∣

∣

∣

∣

k2= z1−z

k2

T+ m2

1−z+

m2π

z

. (13)

Thus the actual value of the Collins function in thismodel depends only on the imaginary parts of the co-efficients defined in Eqs. (9–10). The lack of an imag-inary component in these coefficients would inevitablyresult in a vanishing Collins function. We can computethe imaginary parts by applying the Cutkosky rule tothe self-energy and vertex diagram of Fig. 3. In this way,as mentioned before, we can avoid the issues related torenormalization, which affect only the real parts of thediagrams. Explicit calculation leads to

Im (A + B) =g2

16π2

(

1 −m2 − m2

π

k2

)

I1 , (14)

Im (D + E + m F ) = −g2

8π2m

k2 − m2 + m2π

λ(k2, m2, m2π)

×[

I1 + (k2 − m2 − 2m2π)I2

]

, (15)

where we have introduced the so-called Kallen function,λ(k2, m2, m2

π) = [k2 − (m + mπ)2][k2 − (m − mπ)2], andthe factors

I1 =

∫

d4l δ(l2 − m2π) δ((k − l)2 − m2)

=π

2k2

√

λ(k2, m2, m2π) θ(k2 − (m + mπ)2) , (16)

I2 =

∫

d4lδ(l2 − m2

π) δ((k − l)2 − m2)

(k − p − l)2 − m2

= −π

2√

λ(k2, m2, m2π)

ln

(

1 +λ(k2, m2, m2

π)

k2m2 − (m2 − m2π)2

)

× θ(k2 − (m + mπ)2). (17)

These integrals are finite and vanish below the thresh-old of quark-pion production, where the self-energy andvertex diagrams do not possess any imaginary part.

3

FIG. 2. One-loop corrections to the fragmentation of aquark into a pion.

Σ(k) = A /k + B m , (9)

Γ(k, p) = C + D /p + E /k + F /p /k. (10)

The real parts of the functions A, B, C etc. are UV-divergent and require in principle a proper renormaliza-tion. Though our model is renormalizable, we do nothave to deal with this question at all, since only theimaginary parts of the loop diagrams will turn out tobe important.

The contributions to the correlation function generatedby the diagrams (a) and (c) are given by:

∆(a)(1)(k, p) = −

g2

(2π)4(/k + m)

k2 − m2γ5 (/k − /p + m) γ5

×(/k + m)

k2 − m2Σ(k)

(/k + m)

k2 − m22π δ((k − p)2 − m2) , (11)

∆(c)(1)(k, p) = −

g2

(2π)4(/k + m)

k2 − m2γ5 (/k − /p + m) γ5

× Γ(k, p)(/k + m)

k2 − m22π δ((k − p)2 − m2). (12)

The contributions from diagrams (b) and (d) follow from

the hermiticity condition: ∆(b)(1)(k, p) = γ0∆(a)†

(1) (k, p)γ0,

∆(d)(1)(k, p) = γ0∆(c)†

(1) (k, p)γ0.

kk

p

! ( )( )i- " k,pg #5k

FIG. 3. One-loop self-energy and vertex corrections.

Summing the contributions of the four diagrams andinserting the resulting correlation function in Eq. (1), weobtain the result

H⊥1 (z, k2

T)

=g2 mπ

4π3

p−

z

∫

dk+δ((k − p)2 − m2)

×(

m Im (A + B)

(k2 − m2)2+

Im (D + E + mF )

(k2 − m2)

)∣

∣

∣

∣

k−= p−

z

=g2 mπ

8π3

1

1 − z

(

m Im (A + B)

(k2 − m2)2

+Im (D + E + mF )

(k2 − m2)

)∣

∣

∣

∣

k2= z1−z

k2

T+ m2

1−z+

m2π

z

. (13)

Thus the actual value of the Collins function in thismodel depends only on the imaginary parts of the co-efficients defined in Eqs. (9–10). The lack of an imag-inary component in these coefficients would inevitablyresult in a vanishing Collins function. We can computethe imaginary parts by applying the Cutkosky rule tothe self-energy and vertex diagram of Fig. 3. In this way,as mentioned before, we can avoid the issues related torenormalization, which affect only the real parts of thediagrams. Explicit calculation leads to

Im (A + B) =g2

16π2

(

1 −m2 − m2

π

k2

)

I1 , (14)

Im (D + E + m F ) = −g2

8π2m

k2 − m2 + m2π

λ(k2, m2, m2π)

×[

I1 + (k2 − m2 − 2m2π)I2

]

, (15)

where we have introduced the so-called Kallen function,λ(k2, m2, m2

π) = [k2 − (m + mπ)2][k2 − (m − mπ)2], andthe factors

I1 =

∫

d4l δ(l2 − m2π) δ((k − l)2 − m2)

=π

2k2

√

λ(k2, m2, m2π) θ(k2 − (m + mπ)2) , (16)

I2 =

∫

d4lδ(l2 − m2

π) δ((k − l)2 − m2)

(k − p − l)2 − m2

= −π

2√

λ(k2, m2, m2π)

ln

(

1 +λ(k2, m2, m2

π)

k2m2 − (m2 − m2π)2

)

× θ(k2 − (m + mπ)2). (17)

These integrals are finite and vanish below the thresh-old of quark-pion production, where the self-energy andvertex diagrams do not possess any imaginary part.

3

2

q

p

q

p

p ! k ! l

k + q + l

k k + l

p ! k

k + q

l

p ! k

k + q

!

(a) (b)

FIG. 1: Tree-level and one-loop diagrams for the specator-model calculation of the Sivers function. The dashed line indicatesboth the scalar and axial-vector diquarks.

where S is the spin of the target. The correlator Φ(x,!kT ) can be written as [17]

Φ(x,!kT ; S) =

∫

dξ− d2ξT

(2π)3e+ik·ξ〈P, S|ψ(0)L[0−,∞−]L[0T ,∞T ]L[∞T ,ξT ]L[∞−,ξ−]ψ(ξ)|P, S〉

∣

∣

∣

∣

ξ+=0

, (2)

where the notation L[a,b] indicates a straight gauge link running from a to b. In Drell-Yan processes the link runsin the opposite direction, to −∞ [17]. For the calculation of the unpolarized function f1 the transverse part of thegauge link does not play a role and the entire gauge link can be reduced to unity. Therefore, for this first part of thecalculation it is sufficient to consider only the handbag diagram.

At tree level, we follow almost exactly the spectator model of Jakob, Mulders and Rodrigues [21]. In this model,the proton (with mass M) can couple to a constituent quark of mass m and a diquark. The diquark can be both ascalar particle, with mass Ms, or an axial-vector particle, with mass Mv. The relevant diagram at tree level (identicalfor the scalar and axial-vector case) is depicted in Fig. 1 (a). In our model, the nucleon-quark-diquark vertices are

Υs = gs(k2), Υµ

v =gv(k2)√

2γ5γ

µ. (3)

We make use of the dipole form factor

gs/v(k2) = Ns/v

(k2 − m2) (1 − x)2(

!k2T + L2

s/v

)2 , (4)

where

!k2T = −(1 − x) k2 − xM2

s/v + x (1 − x)M2, (5)

L2s/v = (1 − x)Λ2 + xM2

s/v − x (1 − x)M2. (6)

The only difference with respect to Ref. [21] is the form of Υv – the vertex involving nucleon, quark, and axial-vectordiquark. This change modifies the original results only slightly. Note that our choice of the form factor, definedin Eq. (4), is very different from the Gaussian form factor employed in Ref. [19]. Both choices have the effect ofeliminating the logarithmic divergences arising from kT integration and suppress the influence of the high kT region,where anyway perturbative corrections should be taken into account [20].

The final results for the unpolarized distribution function f1 are

fs1 (x,!k2

T ) =g2

s

[

(xM + m)2 + !k2T

]

2 (2π)3 (1 − x) (k2 − m2)2=

N2s (1 − x)3

[

(xM + m)2 + !k2T

]

16π3(

!k2T + L2

s

)4 , (7)

fv1 (x,!k2

T ) =g2

v

[

(xM + m)2 + !k2T + 2xmM

]

2 (2π)3 (1 − x) (k2 − m2)2=

N2v (1 − x)3

[

(xM + m)2 + !k2T + 2xmM

]

16π3(

!k2T + L2

v

)4 . (8)

Bacchetta, Conti, Radici, Guagnelli, Gamberg, Goldstein, Mukherjee, Metz, Amrath, Schaefer, Yang, Brodsky, Schmidt, Hwang, Pasquini, Xiao,

Yuan, Scopetta, Courtoy, Frattini, Vento ....

models for Collins function

and for Sivers function

(a) (b)

y1, !1⊥

y2, !2⊥ x2, k2⊥

x1, k1⊥

yn, !n⊥ xn, kn⊥

FIG. 1: Light-front time-order perturbation Feynman diagrams for the phase contribution from

one-gluon exchange between two constituent quarks.

where∑

k− represents the sum of all partons energy k−i , d[i]

′ represents the integral of

(yi, !i⊥). The interaction kernel K can be calculated from the light-front time-order pertur-

bation theory [2]. The wave functions ψn and ψ′n may differ. From the above expression,

we find that the phase of ψn may come from the wave function in the right hand side ψ′n

or the interaction kernel K. In the following, we assume that the wave function ψ′n is real,

for example, from model calculation such as constituent quark model [18]. We will focus on

the contribution from the interaction kernel. We will calculate, in particular, the one-gluon

exchange contribution to the interaction kernel.

At the lowest order of the light-front time-order perturbation theory, we have one gluon

exchange contribution to the interaction kernel. This can be expressed as a sum of all

diagrams with gluon connection between all possible pair of constituents in the light-front

wave function. For example, the contribution from the gluon exchange between the ith and

jth quark can be written as,

K[k; !]ij =uλi

(xi, ki⊥)√xi

γµuλ′

i(yi; !i⊥)√yi

dµνuλj

(xj, kj⊥)√xj

γνuλ′

j(yj; !j⊥)√yi

×

1

P− − q− − k−i − !−j −

∑

α$={i,j}k−α + iε

θ(q+)

q+

+1

P− − q′− − k−j − !−i −

∑

α$={i,j}k−α + iε

θ(q′+)

q′+

, (3)

where λ represents the helicity for the associated quarks, q+ = k+j − !+j and q′+ = k+

i − !+i ,

and the color factors are implicit in the above equation. Similar expression shall hold for the

5

Brodsky, Pasquini, Xiao, Yuan, arXiv:1001.1163 Pasquini, Yuan, arXiv:1001.5398

Sivers function from light-front wave function

see also Hwang, arXiv:1003.0867 - incorporation of final state interactions into the light-cone wave function

convoluting the three steps gives

dσ!(S!)p(S)→!′hX

dxBdQ2dzh d2P T dφS

=12π

∑

q

116 π (xBs)2

∫d2k⊥

12

×{

fq/p,S(x,k⊥)(

|M1|2 + |M2|2)

Dh/q(z, p⊥)

+ P !z P q

z fq/p,S(x,k⊥)(

|M1|2 − |M2|2)

Dh/q(z, p⊥)

+[P q

y fq/p,S(x,k⊥)(Re(M1M

∗2 ) cos φh

q − Im(M1M∗2 ) sinφh

q

)

− P qx fq/p,S(x,k⊥)

(Im(M1M

∗2 ) cos φh

q + Re(M1M∗2 ) sinφh

q

)]∆NDh/q↑(z, p⊥)

}

cos φhq =

PT

|p⊥|

[cos(φh − φ⊥)− zh

k⊥PT

]

sin φhq =

PT

|p⊥| sin(φh − φ⊥)

with

with the final expression dσ!(S!)p(S)→!′hX

dxB dQ2 dzh d2P T dφS=

2 α2

Q4

×{

1 + (1− y)2

2FUU + (2− y)

√1− y cos φh F cos φh

UU + (1− y) cos 2φh F cos 2φh

UU

+ SL

[(1− y) sin 2φh F sin 2φh

UL + (2− y)√

1− y sinφh F sin φh

UL

]

+ SL P lz

[1− (1− y)2

2FLL + y

√1− y cos φh F cos φh

LL

]

+ ST

[1 + (1− y)2

2sin(φh − φS) F sin(φh−φS)

UT

+ (1− y)(

sin(φh + φS) F sin(φh+φS)UT + sin(3φh − φS) F sin(3φh−φS)

UT

)

+ (2− y)√

1− y(

sinφS F sin φS

UT + sin(2φh − φS) F sin(2φh−φS)UT

)]

+ ST P lz

[1− (1− y)2

2cos(φh − φS)F cos(φh−φS)

LT

+ y√

1− y(

cos φS F cos φS

LT + cos(2φh − φS) F cos(2φh−φS)LT

)]}

in agreement with the general decomposition and the expression of the FXY in terms of TMDs given in the literature

Bacchetta et al., JHEP 0702 (2007) 093

dσD−Y =∑

a

fq(x1,k⊥1;Q2)⊗ fq(x2,k⊥2;Q2) dσqq→!+!−

same approach for Drell-Yan processes

factorization holds, two scales, M2, and qT << M

p p

Q2 = M2

qT

qL

l+

l–

direct product of TMDs no fragmentation process

Extension to hadronic processes?

p p→ π0 XCross section for in pQCD, only one scale, PT

factorization theorem holds and works well in collinear configuration

dσ =∑

a,b,c,d=q,q,g

fa/p(xa)⊗ fb/p(xb)⊗ dσab→cd ⊗Dπ/c(z)

PDF FF pQCD elementary

interactions

ab

cX

X

σ

but no Single Spin Asymmetries...

good pQCD description of data at 200 GeV, at all rapidities, down to pT of 1-2 GeV/c

Polarization-averaged cross sections at √s=200 GeV

rather good agreement even at at √s=62.4 GeV

11% normalization uncertainty not included

mid-rapidity pions

Comparison of NLO pQCD calculations with BRAHMS π data at high rapidity. The

calculations are for a scale factor of µ=pT, KKP (solid) and DSS (dashed) with CTEQ5

and CTEQ6.5.

STAR-RHIC √s = 200 GeV 1.2 < pT < 2.8

Good description of unpolarized cross-section, with collinear factorization. And AN ... ?

patterns of polarization signs. The unfilled 9 bunches aresequential and correspond to the abort gap needed to ejectthe stored beams. Pb was measured every 3 h during RHICstores by a polarimeter that detected recoil carbon ionsproduced in elastic scattering of protons from carbon rib-bon targets inserted into the beams. The effective AN of thispolarimeter was determined from p" þ p" elastic scatteringfrom a polarized gas jet target [24] thereby determiningPb ¼ 55:0# 2:6% (56:0# 2:6%) for the Blue (Yellow)beam in the 2006 run [25].

The FPD comprises four modules, each containing amatrix of lead glass (PbGl) cells of dimension 3:8 cm$3:8 cm$ 18 radiation lengths. Pairs of modules werepositioned symmetrically left (L) and right (R) of thebeam line in both directions, at a distance of %750 cmfrom the interaction point [21]. The modules facing theYellow (Blue) beam are square matrices of 7$ 7 (6$ 6)PbGl cells. Data from all FPD cells were encoded for eachbunch crossing, but only recorded when the summed en-ergy from any module crossed a preset threshold.

Neutral pions are reconstructed via the decay !0 ! "".The offline event analysis included conversion of the datato energy for each cell, formation of clusters and recon-struction of photons using a fit with the function thatparametrizes the average transverse profile of electromag-netic showers. Collision events were identified by requiringa coincidence between the east and west STAR beam-beamcounters, as used for cross section measurements [26].Events were selected when two reconstructed photonswere contained in a fiducial volume, whose boundaryexcludes a region of width 1=2 cell at the module edges.Detector calibration was determined from the !0 peakposition in diphoton invariant mass (M"") distributions.

The estimated calibration accuracy is 2%. The analysis wasvalidated by checking against full PYTHIA/GEANT simula-tions [27]. The reconstructed !0 energy resolution is givenby #E!=E! & 0:16=

ffiffiffiffiffiffiffiE!

p.

Because of the limited acceptance there is a strongcorrelation between xF and pT for reconstructed !0

(Fig. 1). Spin effects in the xF-pT plane are studied bypositioning the calorimeters at different transverse dis-tances from the beam, maintaining L=R symmetry for pairsof modules. Figure 1 shows loci from h$i ¼ 3:3, 3.7, and4.0. There is overlap between the loci, providing cross-checks between the measurements. Because the measure-ments were made at a colliding beam facility, both xF > 0and xF < 0 results are obtained concurrently.Events with 0:08<M"" < 0:19 GeV=c2 were counted

separately by spin state from one or the other beam, withno condition on the spin state of the second beam, in the xFbins shown in Fig. 1. For each run i, AN;i for each bin wasthen determined by forming a cross ratio

AN;i ¼1

Pb

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNL";iNR#;i

p ' ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNL#;iNR";i

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNL";iNR#;i

p þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNL#;iNR";i

p ; (1)

whereNLðRÞ"ð#Þ;i is the number of events in the L (R) modulewhen the beam polarization was up (down). Equation (1)cancels spin dependent luminosity differences throughsecond order. Statistical errors were approximated by!AN;i ¼ ½Pb

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNL";i þ NL#;i þ NR";i þ NR#;i

p +'1, valid forsmall asymmetries. All measurements of Pb for a storewere averaged and applied to get AN;i for each bin. Therun-averaged AN #!AN values are shown in Fig. 2.

FIG. 1 (color online). Correlation between pion longitudinalmomentum scaled by

ffiffiffis

p=2 (xF) and transverse momentum (pT)

for all events. Bins in xF used in Figs. 2 and 4 are indicated bythe vertical lines. There is a strong correlation between xF andpT at a single pseudorapidity (h$i).

FIG. 2 (color online). Analyzing powers in xF bins (see Fig. 1)at two different h$i. Statistical errors are indicated for eachpoint. Systematic errors are given by the shaded band, excludingnormalization uncertainty. The calculations are described in thetext. The inset shows examples of the spin-sorted invariant massdistributions. The vertical lines mark the !0 mass.

PRL 101, 222001 (2008) P HY S I CA L R EV I EW LE T T E R Sweek ending

28 NOVEMBER 2008

222001-4

PRL 101, 222001 (2008)

STAR data

interesting PT dependence

Systematic errors potentially arise from several sources.The bunch counter, used for the spin directions, identifiesevents in the abort gaps arising from single-beam back-grounds. They account for <5! 10"4 of the observedyield. Systematic effects from gain variations with timeare controlled by polarization reversals of the stored beambunches, as demonstrated by examples of spin-sorted M!!

for L;R modules in the inset of Fig. 2. Distributions of thesignificance, Si ¼ ðAN;i " ANÞ=!AN;i, are well describedby zero mean value Gaussian distributions with " equal tounity, as expected if the uncertainties are dominated bystatistics, except near the trigger threshold where larger "is observed. Systematic errors are estimated from "!!AN and differences in AN associated with #0 identifica-tion, with the largest value chosen. The upper limit on acorrelated systematic error, common to all points, arisingfrom instrumental effects is $AN & 4! 10"4.

The same pair of modules concurrently measure AN

values consistent with zero for xF < 0 and AN that in-creases with xF for xF > 0, depending on which beamspin is chosen. Null results at xF < 0 are natural since apossible gluon Sivers function is probed where the unpo-larized gluon distribution is large. For xF > 0, a calculation[13,28] using quark Sivers functions fit [29] to SIDIS data[7] best describes our results at h%i ¼ 3:3. Twist-3 calcu-lations [16] that fit p" þ p ! #þ X data at

ffiffiffis

p ¼ 20 GeV[4] and preliminary RHIC results from the 2003 and 2005runs at

ffiffiffis

p ¼ 200 GeV [21,22] best describe the data ath%i ¼ 3:7. Both calculations are in fair agreement with thevariation of AN with xF. Neither calculation describes dataat both h%i.

Events from modules at different h%i that overlap in thexF-pT plane (Fig. 1) provide consistent results. Hence, it ispossible to further bin the results not only by xF but also bypT . For this analysis, pT is determined from the measuredenergy, the fitted position of the #0 within an FPD module,and the measured position of the module relative to thebeam pipe and to the collision vertex. The z component ofthe event vertex uses a coarse time difference between theeast and west beam-beam counters, and is determined to(20 cm resulting in !pT=pT ¼ 0:04, where !pT is theuncertainty in pT . One method of determining the pT

dependence (Fig. 3) was to select events with jxFj> 0:4.AN is consistent with zero for xF <"0:4. For xF > 0:4,there is a hint of an initial decrease of AN with pT , althoughthe statistical errors are large, since h%i ¼ 4:0 data wereonly obtained in the 2003 and 2005 runs with limitedintegrated luminosity and polarization. For pT >1:7 GeV=c, AN tends to increase with pT for xF > 0:4.This is contrary to the theoretical expectation that AN

decreases with pT .The results in Fig. 3 may still reflect small correlations

between xF and pT for each point, rather than the depen-dence of AN on pT at fixed xF. To eliminate this correla-tion, event selection from Fig. 1 was made in bins of xF,

followed by bins in pT . The resulting variation of AN withpT is shown in Fig. 4, compared to calculations [13] usinga Sivers function fit to p" þ p ! #þ X data [4] and twist-3 calculations [16]. For each point, the variation of hxFi issmaller than 0.01. There is a clear tendency for AN toincrease with pT , and no significant evidence over themeasured range for AN to decrease with increasing pT , asexpected by the calculations. This discrepancy may arisefrom unexpected TMD fragmentation contributions, xF; pT

dependence of the requisite color-charge interactions, evo-lution of the Sivers functions, or from process dependencenot accounted for by the theory.In summary, we have measured the xF and pT depen-

dence of the analyzing power for forward #0 production inp" þ p collisions at

ffiffiffis

p ¼ 200 GeV in kinematics (0:3<xF < 0:6 and 1:2< pT < 4:0 GeV=c) that straddle theregion where cross sections are found in agreement withpQCD calculations. The xF dependence of the #0 AN is in

FIG. 3 (color online). Analyzing powers versus #0 transversemomentum (pT) for events with scaled #0 longitudinal momen-tum jxFj> 0:4. Errors are as described for Fig. 2.

FIG. 4 (color online). Analyzing powers versus #0 transversemomentum (pT) in fixed xF bins (see Fig. 1). Errors are asdescribed for Fig. 2. The calculations are described in the text.

PRL 101, 222001 (2008) P HY S I CA L R EV I EW LE T T E R Sweek ending

28 NOVEMBER 2008

222001-5

AN xF-dependence in pT slices, √s = 200 GeV(C. Aidala talk at Transversity 2008)

Extension to TMD case?

(first proposed by Field-Feynman in unpolarized case)M.A., M. Boglione, U. D’Alesio, E. Leader, S. Melis, F. Murgia, A. Prokudin, ...

ab

cX

X

σ

dσ↑ =∑

a,b,c=q,q,g

fa/p↑(xa,k⊥a)⊗ fb/p(xb,k⊥b)⊗ dσab→cd(k⊥a,k⊥b)⊗Dπ/c(z,p⊥π)

single spin effects in TMDs

see also talk by Gamberg

TMD factorization

factorization assumed

TMD - PDF

TMD - PDF

TMD - FF non planar pQCD

dynamics

non planar pQCD

dynamics

General formalism with helicity amplitudes (Cagliari-Torino group)

M.A., M. Boglione, U. D’Alesio, E. Leader, S. Melis, F. Murgia, PR D71, 014002 (2005), PR D73, 014020 (2006)

dσ(A,SA)+(B,SB)→C+X =∑

ρa/A,SA

λa,λ′a

fa/A,SA(xa,k⊥a)⊗ ρb/B,SB

λb,λ′b

fb/B,SB(xb,k⊥b)

⊗ Mλc,λd;λa,λbM∗

λ′c,λd;λ′

a,λ′b(k⊥a,k⊥b) D

λC ,λ C

λc,λ′c

(z,k⊥C)

polarized pQCD processes

PDFs for polarized partons inside polarized hadrons

polarized FFs

Take into account all intrinsic motions: in parton distributions, fragmentation and elementary interactions.

It brings dependence on plenty of phases...

Phenomenology - TMD factorization

dσ↑ − dσ↓ ≡ Eπ dσ p→π X

d3pπ

− Eπ dσ p→π X

d3pπ

= [dσ↑ − dσ↓]Sivers + [dσ↑ − dσ↓]Collins

AN =dσ↑ − dσ↓

dσ↑ + dσ↓main contribution from Sivers

and Collins effects

[dσ↑ − dσ↓]Sivers =∑

qa,b,qc,d

∫dxa dxb dz

16 π2 xa xb z2sd2k⊥a d2k⊥b d3p⊥ δ(p⊥ · pc) J(p⊥) δ(s + t + u)

× ∆Nfa/(xa, k⊥a) cos φa

× fb/p(xb, k⊥b)12

[|M0

1 |2 + |M02 |2 + |M0

3 |2]

ab→cdDπ/c(z, p⊥)

Sivers phase

negligible contributions from other TMDs

[dσ↑ − dσ↓]Collins =∑

qa,b,qc,d

∫dxa dxb dz

16 π2 xa xb z2sd2k⊥a d2k⊥b d3p⊥ δ(p⊥ · pc) J(p⊥) δ(s + t + u)

× ∆T qa(xa, k⊥a) cos(φa + ϕ1 − ϕ2 + φHπ )

× fb/p(xb, k⊥b)[M0

1 M02

]

qab→qcd∆NDπ/qc

(z, p⊥)

Collins + scattering phases

contributions to A_N of SIDIS extracted Sivers, Collins and transversity distributions

STAR

-0.1

-0.05

0

0.05

0.1

0.15

0.2 0.4 0.6

AN

xF

Sivers effect

!=3.3

0.2 0.4 0.6

xF

Sivers effect

!=3.7

-0.1

-0.05

0

0.05

0.1

0.15

0.2 0.4 0.6

AN

xF

Collins effect

!=3.3

0.2 0.4 0.6

xF

Collins effect

!=3.7

BRAHMS

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2 0.3 0.4

AN

xF

Sivers effect

!=2.3!

"+

"-

0.2 0.3 0.4

xF

Sivers effect

!=4.0!

"+

"-

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2 0.3 0.4

AN

xF

Collins effect

!=2.3!

"+

"-

0.2 0.3 0.4

xF

Collins effect

!=4.0!

"+

"-

a combination of Sivers and Collins effect might explain data

3

FIG. 1: Kinematical configuration and conventions for the p↑! → h X process.

in terms of the integration variables and the observed final hadron momentum. We consider allpartons as massless, neglecting heavy quark contributions. Full details can be found in Ref. [67]and useful expressions are given in Appendix B.

• With massless partons, the function J is given by [42]

J(p⊥) =

(

Eh +√

P 2h − p2

⊥

)2

4(P 2h − p2

⊥)· (6)

In the kinematical regions which we shall consider J is always very close to 1.

• ρq/p,Sλq,λ′

qis the helicity density matrix of parton q inside the polarized proton p, with spin state S.

fq/p,S(x,k⊥) is the distribution function of the unpolarized parton q inside the polarized proton

p. The products ρq/p,Sλq,λ′

qfq/p,S(x,k⊥) are directly related to the Transverse Momentum Dependent

distribution functions, with a dependence on φ, the azimuthal angle of k⊥ [67].

• The Mλq,λ ;λq,λ ’s are the helicity amplitudes for the elementary process q # → q #, normalized sothat the unpolarized cross section, for a collinear collision, is given by

dσq"→q"

dt=

1

16πs2

1

4

∑

λq,λ

|Mλq,λ ;λq,λ |2 . (7)

At lowest perturbative order q # → q # is the only elementary interaction which contributes; noticethat, in the presence of parton intrinsic motion, it is not a planar process and depends on the intrinsicmomenta, including their phases. Neglecting lepton and quark masses there are two independenthelicity amplitudes:

M++;++(s, t, u, k⊥) = M∗−−;−− = −8 π eq α

s

teiϕ1 ≡ M0

1 eiϕ1 (8)

M+−;+−(s, t, u, k⊥) = M∗−+;−+ = 8 π eq α

u

teiϕ2 ≡ M0

2 eiϕ2 , (9)

where ϕ1,2 are phases explicitly given in Appendix A, Eqs. (A8) and (A9).

• Dλh,λ′

h

λq,λ′q(z, p⊥) is the product of fragmentation amplitudes for the q → h + X process

Dλh,λ′

h

λq,λ′q

=∑

∫

X,λX

Dλh

, λX

;λqD∗

λ′h

, λX

;λ′q, (10)

left-right asymmetry

consider p↑l→ h X large PT processes (one current jet events - EIC)

AN =dσ↑(P T )− dσ↓(P T )dσ↑(P T ) + dσ↓(P T )

=dσ↑(P T )− dσ↑(−P T )

2 dσunp(PT )

TMD factorization with one large scale

|M++;++|2 ≡ |M01 |2 = 64π2α2e2

qs2

t2

|M+−;+−|2 ≡ |M02 |2 = 64π2α2e2

qu2

t2

M++;++ M∗−+;−+ = 64 π2α2e2

qs(−u)

t2e−i(φ−φ′)

Eh dσ(p,S)+!→h+X

d3P h=

∑

q,{λ}

∫dx dz

16 π2x z2sd2k⊥ d3p⊥ δ(p⊥ · p′q) δ(s + t + u)

× ρq/p,Sλq,λ′

qfq/p,S(x,k⊥)

12

Mλq,λ ;λq,λ M∗λ′

q,λ ;λ′q,λ D

λh,λhλq,λ′

q(z,p⊥)

TMD-PDFs TMD-FFs

elementary interaction (at lowest order); phases due to non collinear, non planar configuration

assume TMD factorization:

AN =

∑

q,{λ}

∫dx dz

16 π2x z2sd2k⊥ d3p⊥ δ(p⊥ · p′q) δ(s + t + u)× [Σ(↑)− Σ(↓)]q"→q"

∑

q,{λ}

∫dx dz

16 π2x z2sd2k⊥ d3p⊥ δ(p⊥ · p′q) δ(s + t + u)× [Σ(↑) + Σ(↓)]q"→q"

∑

{λ}

[Σ(↑)− Σ(↓)]q"→q" =12

∆N fq/(x, k⊥) cos φ[

|M01 |2 + |M0

2 |2]

Dh/q(z, p⊥)

+ h1(x, k⊥) M01 M0

2 ∆NDh/q(z, p⊥) cos(φ′ + φHh )

Sivers

Collins x phases

∑

{λ}

[Σ(↑) + Σ(↓)]q"→q" = fq/p(x, k⊥)[

|M01 |2 + |M0

2 |2]

Dh/q(z, p⊥)

Ejet dσ(p,S)+!→jet+X

d3P jet=

∑

q,{λ}

∫dx

16 π2x sd2k⊥ δ(s + t + u)

× ρq/p,Sλq,λ′

qfq/p,S(x,k⊥)

12

Mλq,λ ;λq,λ M∗λq,λ ;λ′

q,λ

∑

{λ}

[Σ(↑)− Σ(↓)]q"→q"jet =

12

∆N fq/(x, k⊥) cos φ[

|M01 |2 + |M0

2 |2]

AjetN =

∑

q,{λ}

∫dx

16 π2x sd2k⊥ δ(s + t + u)× [Σ(↑)− Σ(↓)]q"→q"

jet

∑

q,{λ}

∫dx

16 π2x sd2k⊥ δ(s + t + u)× [Σ(↑) + Σ(↓)]q"→q"

jet

∑

{λ}

[Σ(↑) + Σ(↓)]q"→q"jet = fq/p(x, k⊥)

[|M0

1 |2 + |M02 |2

]

Even simpler: AN for p↑l→ jet X (only Sivers effect)

9

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1

AN

xF

PT=1.5 GeV Sivers effect

!+

!-

!0

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-0.6 -0.4 -0.2 0 0.2 0.4 0.6A

NxF

PT=2.5 GeV Sivers effect

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

AN

xF

PT=2.5 GeV Collins effect

FIG. 6: Same as in Fig. 2 but for ENC kinematics at√

s = 50 GeV.

FIG. 7: Expected density of events for ENC kinematics in the xF -Q2 plane, at fixed PT = 1.5 GeV (left panel) and 2.5 GeV(right panel).

Appendix A: Kinematics

1. Hadron production

We work in the proton-lepton center of mass frame, with the incoming proton and lepton moving along the Zcm

axis and the outgoing hadron emitted in the (XZ)cm plane:

p =

√s

2(1, 0, 0, 1) (A1)

! =

√s

2(1, 0, 0,−1) (A2)

Ph = (Eh, PT , 0, PL) E2h = P 2

T + P 2L , (A3)

where s is the proton-lepton c.m. square energy and where we have assumed all particles to be massless. Thekinematical variables for the elementary underlying process result in

pq =

(

x√

s

2+

k2⊥

2x√

s, k⊥ , x

√s

2−

k2⊥

2x√

s

)

(A4)

! =

√s

2(1, 0, 0,−1) (A5)

p′q =Eh +

√

E2h − p2

⊥

2z

[

1,1

√

E2h − p2

⊥

(PT − px⊥,−py

⊥, PL − pz⊥)

]

(A6)

!′ = pq + ! − p′q , (A7)

AN for ENC kinematics, TMDs from SIDIS √s = 50 GeV

0

0.02

0.04

0.06

0.08

0.1

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1

AN

xF

ENC-JET

pT=1.5 GeV

0

0.02

0.04

0.06

0.08

0.1

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1

AN

xF

ENC-JET

pT=2.5 GeV

jet production

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1

AN

xF

JLAB 12

pT=1.5 GeV Collins effect

!+

!-

!0

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1

AN

xF

JLAB 12

pT=1.5 GeV Sivers effect

!+

!-

!0

AN p↑!→ πX

AN for Jlab 12 kinematics, TMDs from SIDIS

ConclusionsTMD factorization is an

approximation; it should be tested experimentally

same physical approach for SIDIS, D-Y and large PT pp inclusive processes ? Two scales

vs. only one large scale

SIDIS with one large scale only (final hadron with large PT, no final lepton detected) might help

thank you