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Miguel G. Echevarría

QCD Evolution of Sivers Asymmetry

May 12-16, 2014 Santa Fe (NM, US)

QCD Evolution Workshop

1

- QCD Evolution of the Sivers Asymmetry MGE, Ahmad Idilbi, Zhong-Bo Kang, Ivan Vitev. Phys.Rev. D89 (2014) 074013 [arXiv: 1401.5078] !- A Unified Treatment of the QCD Evolution of All (Un-)Polarized TMD Functions: Collins Function as a Study Case MGE, Ahmad Idilbi, Ignazio Scimemi. [arXiv: 1402.0869]

Siver Asymmetry: The Goal

2

• Distribution of unpolarized quarks inside a transversely polarized hadron:

• Improve our understanding of QCD (test theoretical tools: factorization, TMD framework,…) • How? By testing one of the main predictions of the TMD formalism: sign change of Sivers function between SIDIS and DY processes. • We want to extract it from SIDIS and make reliable predictions for DY.

fq/A"(x,k?, ~S) = f

q1 (x, k

2T )� f

?q1T (x, k2T )

(P ⇥ k?) · S?MA

QCD invariant under P and T

f

SIDISq/A" (x,k?, ~S) = f

DYq/A"(x,k?,�~

S)

f

?q1T (x, k2T )

SIDIS = �f

?q1T (x, k2T )

DY

• Different Wilson lines in SIDIS and DY processes • Soft function is universal

QCD Evolution Workshop 2014 (Santa Fe, US)

They include the soft function!

Miguel G. Echevarría - Nikhef/VU - May 12th 2014

The Object

3

• SIDIS cross-section with a transversely polarized hadron:

d�

dxBdydzhd2Ph?

= �0(xB , y,Q2)

hFUU + sin(�h � �s)F

sin(�h��s)UT

i


Asin(�h��s)UT =

F sin(�h��s)UT

FUU

e(`) +A"(P ) ! e(`0) + h(Ph) +X

• Sivers asymmetry is defined as:


The Object

3


d�

dxBdydzhd2Ph?

= �0(xB , y,Q2)


sin(�h��s)UT

i




FUU

e(`) +A"(P ) ! e(`0) + h(Ph) +X



SIDIS factorization…• Factorization? • Relation with Sivers function? • Evolution? • …

SIDIS Factorization (1/3)

4

• Applying the SCET machinery:

QCD Evolution Workshop 2014 (Santa Fe, US) Miguel G. Echevarría - Nikhef/VU - May 12th 2014

l(k) +N(P, S) ! l0(k0) + h(Ph, Sh) +X(PX)

JµQCD =

X

q

eq �µ

More details: MGE, Idilbi, Scimemi 1402.0869

JµSCET =

X

q

eq ⇠nWTn ST†

n �µSTn W

T†n ⇠n

T: MGE, Idilbi, Scimemi PRD'11


4

• Applying the SCET machinery:


l(k) +N(P, S) ! l0(k0) + h(Ph, Sh) +X(PX)

JµQCD =

X

q

eq �µ

Wµ⌫ = H(Q2/µ2)2

Nc

X

q

eq

Zd2kn?d

2kn?d2ks?�

(2)(q? + kn? � kn? + ks?)

⇥ Tr⇥�(0)(x,kn?, S)�

µ �(0)(z,kn?, Sh)�⌫⇤S(ks?)

More details: MGE, Idilbi, Scimemi 1402.0869

�(0)ij (x,kn?, S) =

1

2

Zdy

�d

2y?(2⇡)3

e

�i( 12y

�k+n�y?·kn?) hPS|

h⇠njW

Tn

i(0+, y�,y?)

hW

T†n ⇠ni

i(0) |PSi

�(0)ij (z, P h?, Sh) =

1

2

Zdy

+d

2y?(2⇡)3

e

i( 12y

+k�n �y?·kn?)

⇥ 1

z

X

X

Zh0|

hW

T†n ⇠ni

i(y+, 0�,y?) |X;PhShi hX;PhSh|

h⇠nW

Tnj

i(0) |0i

S(ks?) =

Zd

2y?(2⇡)2

e

iy?·ks?1

Nch0|Tr

hS

T†n S

Tn

i(0+, 0�,y?)

hS

T†n S

Tn

i(0) |0i

JµSCET =

X

q

eq ⇠nWTn ST†

n �µSTn W

T†n ⇠n

T: MGE, Idilbi, Scimemi PRD'11

• Pure vs. naive collinear (double counting) • Matrix elements suffer from rapidity divergences


5QCD Evolution Workshop 2014 (Santa Fe, US) Miguel G. Echevarría - Nikhef/VU - May 12th 2014

lnS = Rs(bT ,↵s) + 2D(bT ,↵s) ln

✓�+��

Q2µ2

◆ lnS� =1

2Rs(bT ,↵s) +D(bT ,↵s)ln

✓(��)2

⇣Fµ2

◆

lnS+ =1


✓(�+)2

⇣Dµ2

◆

• The soft function can be split (in rapidity space) to all orders as:

⇣F ⇣D = Q4

S = S+ S�dD

dlnµ= �cusp

D ⌘ K See John’s talk



Fij(x,kn?, S; ⇣F , µ2;��) =

Zd

2b? e

ib?·kn? �(0)ij (x, b?, S;µ2;��) S�(bT ; ⇣F , µ

2;��)

Dij(z, P h?, Sh; ⇣D, µ

2;�+) =

Zd

2b? e

�ib?·kn? �(0)ij (z, b?, Sh;µ

2;�+) S+(bT ; ⇣D, µ

2;�+)


✓�+��

Q2µ2

◆ lnS� =1


✓(��)2

⇣Fµ2

◆

lnS+ =1


✓(�+)2

⇣Dµ2

◆


⇣F ⇣D = Q4

• To cancel rapidity divergences, TMDPDFs and TMDFFs are defined as:

S = S+ S�dD

dlnµ= �cusp




Fij(x,kn?, S; ⇣F , µ2;��) =

Zd

2b? e

ib?·kn? �(0)ij (x, b?, S;µ2;��) S�(bT ; ⇣F , µ

2;��)

Dij(z, P h?, Sh; ⇣D, µ

2;�+) =

Zd

2b? e

�ib?·kn? �(0)ij (z, b?, Sh;µ

2;�+) S+(bT ; ⇣D, µ

2;�+)


✓�+��

Q2µ2

◆ lnS� =1


✓(��)2

⇣Fµ2

◆

lnS+ =1


✓(�+)2

⇣Dµ2

◆


⇣F ⇣D = Q4

• To cancel rapidity divergences, TMDPDFs and TMDFFs are defined as:

This story is regulator IN-dependent!!

⇣F,D 6= ⇣ =(2v · P )2

v2• Straightforward to update the decomposition of TMDs given by [Boer, Mulders PRD’98] to include the soft function.

S = S+ S�dD

dlnµ= �cusp




Wµ⌫ = H(Q2/µ2)2

Nc

X

f

ef

Zd2b

(2⇡)2e�ib·P h?/z

⇥h⇣

F[�+]f/N (x, b?, S; ⇣F , µ

2) D[��]h/f (z, b?, Sh; ⇣D, µ2)

+F[�+�5]f/N (x, b?, S; ⇣F , µ

2) D[��5]h/f (z, b?, Sh; ⇣D, µ2)

⌘gµ⌫?

+F[i�i+�5]f/N (x, b?, S; ⇣F , µ

2) D[i�j��5]h/f (z, b?, Sh; ⇣D, µ2)

⇣g{µ?i g

⌫}j � g?ij g

µ⌫?

⌘i

• Fierzing, going to impact parameter space,…

F [�] ⌘ 1

2Tr (F�)

Soft function included in F/D

T (bT ; ⇣, µ2) =

✓⇣b2T

4e�2�E

◆�D(bT ;µ)

CQ/T (bT ;µ

2)⌦ t(µ2) +O (bT⇤QCD)



Wµ⌫ = H(Q2/µ2)2

Nc

X

f

ef

Zd2b

(2⇡)2e�ib·P h?/z

⇥h⇣

F[�+]f/N (x, b?, S; ⇣F , µ

2) D[��]h/f (z, b?, Sh; ⇣D, µ2)

+F[�+�5]f/N (x, b?, S; ⇣F , µ

2) D[��5]h/f (z, b?, Sh; ⇣D, µ2)

⌘gµ⌫?

+F[i�i+�5]f/N (x, b?, S; ⇣F , µ

2) D[i�j��5]h/f (z, b?, Sh; ⇣D, µ2)

⇣g{µ?i g

⌫}j � g?ij g

µ⌫?

⌘i

• Fierzing, going to impact parameter space,…

Known at 3 loops! Depends just on Q!

F [�] ⌘ 1

2Tr (F�)

• TMDs contain perturbative information at large kT (refactorization):

Collinear distribution (PDF, FF, ETQS,…)Perturbative

No divergencesAny TMD PDF/FF

Soft function included in F/D

Universal Evolution

7

• All the 8 TMDPDFs and 8 TMDFFs (leading twist) evolve with the same kernel:


F

[�]f/N (x, b?, S; ⇣F,f , µ

2f ) = F

[�]f/N (x, b?, S; ⇣F,i, µ

2i ) R

�bT ; ⇣F,i, µ

2i , ⇣F,f , µ

2f

�

D

[�]h/f (z, b?, Sh; ⇣D,f , µ

2f ) = D

[�]h/f (z, b?, Sh; ⇣D,i, µ

2i ) R

�bT ; ⇣D,i, µ

2i , ⇣D,f , µ

2f

�

˜R�b; ⇣i, µ

2i , ⇣f , µ

2f

�= exp

⇢Z µf

µi

dµ

µ�

✓↵s(µ), ln

⇣fµ2

◆�✓⇣f⇣i

◆�D(bT ;µi)

Universal Evolution

7

• All the 8 TMDPDFs and 8 TMDFFs (leading twist) evolve with the same kernel:


F

[�]f/N (x, b?, S; ⇣F,f , µ

2f ) = F

[�]f/N (x, b?, S; ⇣F,i, µ

2i ) R

�bT ; ⇣F,i, µ

2i , ⇣F,f , µ

2f

�

D

[�]h/f (z, b?, Sh; ⇣D,f , µ

2f ) = D

[�]h/f (z, b?, Sh; ⇣D,i, µ

2i ) R

�bT ; ⇣D,i, µ

2i , ⇣D,f , µ

2f

�

˜R�b; ⇣i, µ

2i , ⇣f , µ

2f

�= exp

⇢Z µf

µi

dµ

µ�

✓↵s(µ), ln

⇣fµ2

◆�✓⇣f⇣i

◆�D(bT ;µi)


✓�+��

Q2µ2

◆

The Soft function is universal!!

D(bT ;µi) = DR(bT ;µi) ✓(bTc � bT ) +DNP (bT ) ✓(bT � bTc)

MGE, Idilbi, Scimemi 1402.0869

Universal!!

Known at 3-loops!!

dD

dlnµ= �cusp

See John’s talk

D ⌘ K

The Object (again)

8


d�

dxBdydzhd2Ph?

= �0(xB , y,Q2)


sin(�h��s)UT

i




FUU

FUU =1

2⇡HDIS(Q)

Z 1

0db bJ0(Ph?b/zh)

X

q

e2qfq/A(xB , b;Q)Dh/q(zh, b;Q)

Fsin(�h��s)UT = �HDIS(Q)

Zd2b

(2⇡)2e�iPh?·b/zh P↵

h?X

q

e2qf?q(↵)1T,SIDIS(xB , b;Q)Dh/q(zh, b;Q)

e(`) +A"(P ) ! e(`0) + h(Ph) +X



fq/A(x, b;Q) =

Zd

2k? e

�ik?·bfq/A(x, k

2?;Q)

Dh/q(z, b;Q) =1

z

2

Zd

2pT e

�ipT ·b/zDh/q(z, p

2T ;Q)

f

?q(↵)1T (x, b;Q) =

1

M

Zd

2k? e

�ik?·bk

↵?f

?q1T (x, k2?;Q)

We have set: µ2 = ⇣A = ⇣h = Q2

Sivers Asymmetry: The Strategy

9

Describe unpolarized DY+SIDIS with proper QCD Evolution and a universal non-perturbative model

Miguel G. Echevarría - Nikhef/VU - May 12th 2014QCD Evolution Workshop 2014 (Santa Fe, US)

Model Sivers function and fit it with SIDIS data and using previously extracted parameters

Flip sign to make predictions for planned measurements in DY

1.-

2.-

3.-

• W/Z boson production (CDF, D0, CMS) • DY lepton-pair production at (E288, E605) • SIDIS process (COMPASS, HERMES)

• SIDIS process (JLab, COMPASS, HERMES)

• DY lepton-pair production (Fermilab, CERN) • W boson production (RHIC)

We perform the resummation of large logarithms at NLL accuracy

HDIS,DY (Q2, µ2) = 1 +↵sCF

2⇡

3ln

Q2

µ2� ln2

Q2

µ2� 8 +

⇡2

6+⇡2

�

Unpolarized DY+SIDIS: Perturbative

10

• The necessary perturbative ingredients at NLL are:

LO hard part



2⇡

3ln

Q2

µ2� ln2

Q2

µ2� 8 +

⇡2

6+⇡2

�


10


F

i/h

(x, b;µ) =X

a

Z 1

x

d⇠

⇠

C

i/a

✓x

⇠

, b;µ

◆f

a/h

(⇠, µ) +O(b⇤QCD)

LO hard part



2⇡

3ln

Q2

µ2� ln2

Q2

µ2� 8 +

⇡2

6+⇡2

�


10


F

i/h

(x, b;µ) =X

a

Z 1

x

d⇠

⇠

C

i/a

✓x

⇠

, b;µ

◆f

a/h

(⇠, µ) +O(b⇤QCD)

Natural scale: c/bfq/A(x, b; c/b) = fq/A(x, c/b) + · · ·

Dh/q(z, b; c/b) =1

z

2Dh/q(z, c/b) + · · ·LO coefficients

c = 2e��E

LO hard part



11

˜R(b; c/b,Q) = exp

(�Z Q

c/b

dµ

µ

✓�cusp ln

Q2

µ2+ �V

◆)✓Q2

(c/b)2

◆�D(b;c/b)

�(1) = CF

�(2) =CF

2

CA

✓67

18� ⇡2

6

◆� 10

9TRnf

�

�V (1) = �3

2CF

D(1) =CF

2lnµ2b2

c2�! D(1)(b;µ = c/b) = 0



11

˜R(b; c/b,Q) = exp

(�Z Q

c/b

dµ

µ

✓�cusp ln

Q2

µ2+ �V

◆)✓Q2

(c/b)2

◆�D(b;c/b)

�(1) = CF

�(2) =CF

2

CA

✓67

18� ⇡2

6

◆� 10

9TRnf

�

�V (1) = �3

2CF

D(1) =CF

2lnµ2b2

c2�! D(1)(b;µ = c/b) = 0

↵s(µ)

4⇡=

1

�0x� �1

�

30

lnx

x

2+

�

21

�

50

ln2x� lnx� 1

x

3+

�2

�

40

1

x

3+ · · ·

For NLL

x = lnµ

2

⇤2QCD

We take into account the thresholds and integrate analytically


Unpolarized DY+SIDIS: Non-Perturbative

12

• Thus, the perturbative part of the TMD PDF/FF is written at NLL as:

• We need to separate and parameterize the non-perturbative large b contributions:

Fpert(x, b;Q) = f(x, c/b) exp

(�Z Q

c/b

dµ

µ

✓�cusp ln

Q

2

µ

2+ �

V

◆)

1/b � ⇤QCD

We use simple gaussian NP input No x/z dependence No flavor-dependence g2 is universal

We use here the b* prescription proposed by CSS

fq/A(x, b;Q) = fq/A(x, c/b⇤) e�gpdf

1 b2exp

(�Z Q

c/b⇤

dµ

µ

✓�cusp ln

Q

2

µ

2+ �

V

◆)✓Q

2

Q

20

◆� 14 g2b

2

Dh/q(z, b;Q) =

1

z

2Dh/q(z, c/b⇤) e

�gff1 b2

exp

(�Z Q

c/b⇤

dµ

µ

✓�cusp ln

Q

2

µ

2+ �

V

◆)✓Q

2

Q

20

◆� 14 g2b

2


Q20 = 2.4GeV2

b⇤ =bp

1 + (b/bmax

)2


13

• From [Konychev, Nadolsky PLB'06]:


0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5

110

120

130

140

150

0.1

0.15

0.2

0.25

0.3

0

0.1

0.2

0.3

0.4

0.5

0.60

-0.05

-0.1

-0.15

-0.2

a 2!GeV

2 "

bmax !GeV−1"

Renormalonanalysis !Tafat"

a 2!GeV

2 "

a 3!GeV

2 "

bmax !GeV−1"

r2

C3=b0

C3=2 b0

a 1!GeV

2 "


g2 = 0.184± 0.018 GeV2

bmax

= 1.5 GeV�1

Extracted from DY and W/Z


13

• From [Konychev, Nadolsky PLB'06]:


0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5

110

120

130

140

150

0.1

0.15

0.2

0.25

0.3

0

0.1

0.2

0.3

0.4

0.5

0.60

-0.05

-0.1

-0.15

-0.2

a 2!GeV

2 "

bmax !GeV−1"

Renormalonanalysis !Tafat"

a 2!GeV

2 "

a 3!GeV

2 "

bmax !GeV−1"

r2

C3=b0

C3=2 b0

a 1!GeV

2 "


g2 = 0.184± 0.018 GeV2

bmax

= 1.5 GeV�1

Extracted from DY and W/Z

0 1 2 3 4 5 6 bTHGeV-1L0.00.20.40.60.81.01.21.4

HQf êQiL-2Dg2=0.16

DR NLL

Qi = 2.4 GeVQf = 2 GeV

✓Qf

Qi

◆�2hD(b⇤;µb)+

R Qiµb

dµµ �cusp+ 1

4 g2b2i

✓Qf

Qi

◆�2DR(b;Qi)

• We choose g2=0.16 for bmax=1.5:

MGE, Idilbi, Schafer, Scimemi EPJC'13


14

• Regarding the intrinsic widths:


hk2?iQ0 = 0.25� 0.44 GeV2

hp2T iQ0 = 0.16� 0.20 GeV2 gpdf1 =hk2?iQ0

4

g↵1 =hp2T iQ0

4z2

Schweitzer, Teckentrup, Metz PRD'10 Torino group

Good description of HERMES data


14



hk2?iQ0 = 0.25� 0.44 GeV2


4

g↵1 =hp2T iQ0

4z2

• The tuned parameters that give a good global description of DY+SIDIS are:



hk2?iQ0 = 0.38 GeV2

hp2T iQ0 = 0.19 GeV2

g2

= 0.16 GeV2

bmax

= 1.5 GeV�1


14



hk2?iQ0 = 0.25� 0.44 GeV2


4

g↵1 =hp2T iQ0

4z2




hk2?iQ0 = 0.38 GeV2


g2

= 0.16 GeV2

bmax

= 1.5 GeV�1

• TMD factorization works for qT<Q, so we consider data accordingly!!


14



hk2?iQ0 = 0.25� 0.44 GeV2


4

g↵1 =hp2T iQ0

4z2




hk2?iQ0 = 0.38 GeV2


g2

= 0.16 GeV2

bmax

= 1.5 GeV�1

• TMD factorization works for qT<Q, so we consider data accordingly!!

Plots…

15

0

0.02

0.04

0.06

0.08

0 5 10 15 20

CMS Zp+p 3s=7 TeV

pT (GeV)

1/m

dm

/dp T

(GeV

-1)

0

200

400

600

800

0 5 10 15 20

CDF Z run 1D0 Z

p+p– 3s=1.8 TeV

pT (GeV)

dm/d

p T (p

b/G

eV)

0

500

1000

1500

2000

2500

0 5 10 15 20

D0 Wp+p

– 3s=1.8 TeV

pT (GeV)

dm/d

p T (p

b/G

eV)

• W/Z boson production at Tevatron (D0 and CDF) and LHC (CMS)


Unpolarized DY+SIDIS: Plots (1/4)


16

• DY lepton-pair production at Fermilab (E288 and E605)

10-2

10-1

1

10

0 0.2 0.4 0.6 0.8 1 1.2 1.4

E288 3s=19.4 GeV

pT (GeV)

Edm

/d3 p

(pb/

GeV

2 )

10-2

10-1

1

10

0 0.2 0.4 0.6 0.8 1 1.2 1.4

E288 3s=23.8 GeV

pT (GeV)

Edm

/d3 p

(pb/

GeV

2 )

10-2

10-1

1

10

0 0.2 0.4 0.6 0.8 1 1.2 1.4

E288 3s=27.4 GeV

pT (GeV)

Edm

/d3 p

(pb/

GeV

2 )

10-2

10-1

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

E605 3s=38.8 GeV

pT (GeV)

Edm

/d3 p

(pb/

GeV

2 )


Data from top to bottom correspond to different invariant mass Q.



17

• SIDIS Multiplicities at CERN (COMPASS)

10-1

1

10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

COMPASSDeuteron h+

�xB� = 0.093�Q2�=7.57 GeV2

pT (GeV)

dN/d

z d2 p T

(GeV

-2)

10-2

10-1

1

10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

COMPASSDeuteron h-

�xB� = 0.093�Q2�=7.57 GeV2

pT (GeV)

dN/d

z d2 p T

(GeV

-2)

Data from top to bottom correspond to different z regions




18

• SIDIS Multiplicities at DESY (HERMES)

10-1

1

10

0 0.1 0.2 0.3 0.4 0.5 0.6

HERMESProton /+

�xB� = 0.117

�Q2�=2.45 GeV2

pT (GeV)

dN/d

z d2 p T

(GeV

-2)

10-2

10-1

1

10

0 0.1 0.2 0.3 0.4 0.5 0.6

HERMESProton /-

�xB� = 0.117

�Q2�=2.45 GeV2

pT (GeV)

dN/d

z d2 p T

(GeV

-2)





18

• SIDIS Multiplicities at DESY (HERMES)

10-1

1

10

0 0.1 0.2 0.3 0.4 0.5 0.6

HERMESProton /+

�xB� = 0.117

�Q2�=2.45 GeV2

pT (GeV)

dN/d

z d2 p T

(GeV

-2)

10-2

10-1

1

10

0 0.1 0.2 0.3 0.4 0.5 0.6

HERMESProton /-

�xB� = 0.117

�Q2�=2.45 GeV2

pT (GeV)

dN/d

z d2 p T

(GeV

-2)




We have a good description of SIDIS+DY data


Sivers Function: Parameterization

19

• Sivers function is OPEd onto the Qiu-Sterman matrix element:


f

↵SIDIS1T f/P (z, bT ; ⇣F , µ) =

X

j

✓ib

↵?2

◆Z 1

z

dx1 dx2

x1 x2Cf/j(z/x1, z/x2, bT ; ⇣F , µ)TF j/P (x1, x2;µ) +O(bT⇤QCD)

• With evolution and non-perturbative model:

µ2 = ⇣F

Q20 = 2.4GeV2

• We parameterize the Qiu-Sterman function as:

Tq,F (x, x, µ) = Nq(↵q + �q)(↵q+�q)

↵

↵qq �

�q

q

x

↵q (1� x)�qfq/A(x, µ)

f↵SIDIS1T f/P (z, bT ; c/bT ) =

✓ib↵?2

◆TF f/P (z, z; c/bT ) + · · ·

˜f↵SIDIS1T f/P (z, b;Q) =

✓ib↵

2

◆TF f/P (z, z; c/b⇤)e

�gsivers1 b2

⇥ exp

(�Z Q

c/b⇤

dµ

µ

✓�cusp ln

Q2

µ2+ �V

◆)✓Q2

Q20

◆� 14 g2b

2

Kouvaris, Qiu, Vogelsang, Yuan PRD’06

Sivers Function: Fit

20

• We use the parameters obtained from the analysis of the unpolarized DY+SIDIS:


�

2/d.o.f. = 1.3

↵u = 1.051+0.192�0.180 ↵d = 1.552+0.303

�0.275

↵sea = 0.851+0.307�0.305 � = 4.857+1.534

�1.395

Nu = 0.106+0.011�0.009 Nd = �0.163+0.039

�0.046

Nu = �0.012+0.018�0.020 Nd = �0.105+0.043

�0.060

Ns = 0.103+0.548�0.604 Ns = �1.000±1.757

hk2s?i = 0.282+0.073�0.066 GeV2


hk2?iQ0 = 0.38 GeV2


g2

= 0.16 GeV2

bmax

= 1.5 GeV�1

• We fit 11 parameters, chi2/dof=1.3:

gsivers1 =hk2s?i4

Sivers Function: Fit

20

• We use the parameters obtained from the analysis of the unpolarized DY+SIDIS:


�

2/d.o.f. = 1.3

↵u = 1.051+0.192�0.180 ↵d = 1.552+0.303

�0.275

↵sea = 0.851+0.307�0.305 � = 4.857+1.534

�1.395

Nu = 0.106+0.011�0.009 Nd = �0.163+0.039

�0.046

Nu = �0.012+0.018�0.020 Nd = �0.105+0.043

�0.060

Ns = 0.103+0.548�0.604 Ns = �1.000±1.757

hk2s?i = 0.282+0.073�0.066 GeV2


hk2?iQ0 = 0.38 GeV2


g2

= 0.16 GeV2

bmax

= 1.5 GeV�1

• We fit 11 parameters, chi2/dof=1.3:

gsivers1 =hk2s?i4

Plots…

Sivers Asymmetry: Fit

21

• COMPASS proton target

-0.1

-0.05

0

0.05

0.1

AU

T

Sive

rsh-

COMPASS Proton

-0.1

-0.05

0

0.05

0.1

10 -2 10 -1

xB

AU

T

Sive

rs

h+

0.25 0.5 0.75 1zh

0.2 0.4 0.6 0.8ph� (GeV)



22

• COMPASS deuteron target

-0.1

-0.05

0

0.05

0.1

AU

T

Sive

rs

/-

COMPASS Deuteron

-0.1

-0.05

0

0.05

0.1

10 -2 10 -1

xB

AU

T

Sive

rs

/+

0.25 0.5 0.75 1zh

0.2 0.4 0.6 0.8ph� (GeV)

-0.1

-0.05

0

0.05

0.1

AU

T

Sive

rs

K-

COMPASS Deuteron

-0.1

-0.05

0

0.05

0.1

10 -2 10 -1

xB

AU

T

Sive

rs

K+

0.25 0.5 0.75 1zh

0.2 0.4 0.6 0.8ph� (GeV)



23

• HERMES proton target

-0.1

-0.05

0

0.05

0.1AU

T

Sive

rs

/0

HERMES Proton

-0.1

-0.05

0

0.05

0.1AU

T

Sive

rs

/-

-0.1

-0.05

0

0.05

0.1

10 -1

xB

AU

T

Sive

rs

/+

0.2 0.4 0.6zh

0.2 0.4 0.6ph� (GeV)



24

-0.1

-0.05

0

0.05

0.1

0.15A

UT

Sive

rsK-

HERMES Proton

-0.1

-0.05

0

0.05

0.1

0.15

10 -1

xB

AU

T

Sive

rs

K+

0.2 0.4 0.6zh

0.2 0.4 0.6ph� (GeV)

• HERMES proton target



25

• JLAB neutron target

-0.2

0

0.2AU

T

Sive

rs

/-JLAB Neutron

-0.2

0

0.2

0.1 0.2 0.3 0.4xB

AU

T

Sive

rs/+



26

• Qiu-Sterman function:

-0.1

-0.05

0

0.05

0.1

10 -3 10 -2 10 -1

u

d

x

x T q,

F(x,

x)

10 -3 10 -2 10 -1

u–

d–

x10 -3 10 -2 10 -1 1

s

s–

x

• SIDIS data only constrain u and d quark flavours. Sea quarks are not constrained.


Tq,F (x, x, µ) = Nq(↵q + �q)(↵q+�q)

↵

↵qq �

�q

q

x

↵q (1� x)�qfq/A(x, µ)

Q20 = 2.4GeV2

Sivers Asymmetry: Predictions for DY

27

• DY at RHIC: proton(polarized)-proton at √s=510GeV

-0.04

-0.03

-0.02

-0.01

0

0.01

-4 -3 -2 -1 0 1 2 3 4y

AN

Integrate: pT: [0,1]GeV Q: [4,9]GeV

Integrate: pT: [0,3]GeV

-0.03-0.02-0.01

00.010.020.030.04

-2 -1.5-1 -0.5 0 0.5 1 1.5 2

W+

y

AN

0

0.01

0.02

0.03

0.04

-2 -1.5-1 -0.5 0 0.5 1 1.5 2

W<

y

AN


Forward rapidity region Prediction: 2%-3%

Prediction: 2%-3%

Sivers Asymmetry: Predictions for DY

28

• DY at CERN (COMPASS): 190GeV Pi- beam to scatter on polarized proton target. √s=18.9GeV.

• DY at Fermilab: 120GeV proton beam to scatter on proton target. Either polarized beam or polarized target. √s=15.1GeV.

-0.04

-0.03

-0.02

-0.01

0

-0.6 -0.4 -0.2 -0 0.2 0.4 0.6xF

AN

-0.02

-0.01

0

0.01

0.02

-0.6 -0.4 -0.2 -0 0.2 0.4 0.6xF

AN


xF = xa � xb

Polarized hadronProjected measurement at xF=-0.2 Prediction: 3%-4%

Polarized target Polarized beam

Prediction: 1%-2%



Conclusions & Outlook

29

• TMDs should contain the soft function to be well-defined.

• The evolution of all the 8 TMDPDFs and 8 TMDFFs is driven by the same evolution kernel.

• We know the perturbative ingredients to get resummation up to NNLL.

!• We found a simple non-perturbative model that can describe reasonably well all SIDIS, DY and W/Z boson data.

• Using that model and proper QCD evolution, we fit the Sivers asymmetry in SIDIS and made reliable predictions for DY and W boson to be compared with coming measurements (RHIC, CERN (COMPASS), Fermilab).

!• Improvements: NLL to NNLL, different models, and different ways to avoid the Landau pole.


See Stefano's talk

Back up slides


Unpolarized DY+SIDIS

31

• SIDIS process:

• We measure the hadron multiplicity:

e(`) +A(P ) ! e(`0) + h(Ph) +X

dN

dzhd2Ph?

=d�

dxBdQ2dzhd

2Ph?

�d�

dxBdQ2

d�

dxBdQ2= �

DIS0

X

q

e

2qfq/A(xB , Q)

d�

dxBdQ2dzhd2Ph?=

�DIS0

2⇡HDIS(Q)

X

q

e2q

Z 1

0db bJ0(Ph?b/zh)fq/A(xB , b;Q)Dh/q(zh, b;Q)



31

• SIDIS process:

• We measure the hadron multiplicity:

e(`) +A(P ) ! e(`0) + h(Ph) +X

dN

dzhd2Ph?

=d�

dxBdQ2dzhd

2Ph?

�d�

dxBdQ2

d�

dxBdQ2= �

DIS0

X

q

e

2qfq/A(xB , Q)

d�

dxBdQ2dzhd2Ph?=

�DIS0

2⇡HDIS(Q)

X

q

e2q

Z 1

0db bJ0(Ph?b/zh)fq/A(xB , b;Q)Dh/q(zh, b;Q)

Collins 11 MGE, Idilbi, Scimemi JHEP’12, PLB'13

Well-defined TMDsDepends just on Q

See John’s talkWe have set: µ2 = ⇣A = ⇣h = Q2



32

• Drell-Yan process:

A(PA) +B(PB) ! [�⇤ !]`+`�(y,Q, p?) +X

• W/Z boson production:

A(PA) +B(PB) ! W/Z(y, p?) +X

d�

dQ2dyd2p?=

�DY0

2⇡HDY (Q)

X

q

e2q

Z 1

0db bJ0(p?b)fq/A(xa, b;Q)fq/B(xb, b;Q)

d�W

dyd2p?=

�W0

2⇡HDY (Q)

X

q,q0

|Vqq0 |2Z 1

0db bJ0(q?b)fq/A(xa, b;Q)fq0/B(xb, b;Q)

d�Z

dyd2p?=

�Z0

2⇡HDY (Q)

X

q

�V 2q +A2

q

� Z 1

0db bJ0(q?b)fq/A(xa, b;Q)fq0/B(xb, b;Q)


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