TMD and collinear functions for transverse and ... · h with ↵q h = 4 p 3 r ↵ s N c 2⇡ ... (Hatta and Yang (2018)) Siversand Collins Effects & A Nin ppCollisions D. PitonyakD

TMD and collinear functions for transverse and longitudinal spin observables at RHIC

Daniel PitonyakOld Dominion University, Norfolk, VA

supported byTMD Topical Collaboration

RHIC & AGS Users MeetingBrookhaven National Lab, Upton, NY

June 12, 2018

OutlineØ TMD and collinear twist-3 (CT3) functions

Ø The proton spin at small x

Ø Sivers and Collins effects & AN in pp collisions

Ø Toward a global analysis of transverse spin observables

Ø Summary

D. PitonyakD. Pitonyak

TMD and Collinear Twist-3 Functions


H pol.q pol.

f1

U

U

L

T

L T H pol. U

U

L

T

L Tq pol.

D1

D. Pitonyak

f?1T g1T

g1L

h?1

h?1L

h?1T

h1T D?1T G1T

G1L

H?1

H1T

H?1T

H?1L

TMD FFs (z, )p?

(Boer, Jakob, Mulders (1997))

TMD PDFs (x, kT)

D. Pitonyak

(Mulders, Tangerman (1996); Goeke, Metz, Schlegel (2005))

D. Pitonyak

H pol.q pol.

f1

g1

h1

U

U

L

T

L T

Collinear PDFs (x)

H pol. U

U

L

T

L T

Collinear FFs (z)q pol.

D1

G1

H1

unpolarized

helicity

transversity

Integrate TMDs over kT (or ) è collinear PDFs and FFsp?

D. Pitonyak

D. Pitonyak

Hadron Pol.

U

L

T

intrinsic kinematical intrinsic kinematicaldynamical dynamical

CT3 PDF (x) CT3 PDF (x, x1) CT3 FF (z) CT3 FF (z, z1)

e

hL

gT

E, H

HL, EL

DT , GT

h?(1)1

h?(1)1L

f?(1)1T ,

g?(1)1T

D?(1)1T ,

G?(1)1T

H?(1)1L

H?(1)1HF U

HF L

FF T , GF T

H<,=F U

D<,=F T , G<,=

F T

H<,=F L

D. Pitonyak

The Proton Spin at Small x



COMPASS (2016)

STAR (2014) & PHENIX (2018)

STAR (2018)

PHENIX, Talk by R. Seidl (DIS 2016)

valence & sea quarks gluons

D. Pitonyak

How much of the proton’s spin comes from small-x quarks and gluons?

D. Pitonyak

only data down to x ~ 0.05(now x ~ 0.001 from new STAR data)

only data down to x ~ 0.001

Aschenauer, Sassot, Stratmann (2016)

D. Pitonyak

Which trajectory will the quark and gluon spin follow?

D. Pitonyak


D. Pitonyak


No data yet to tell us (need EIC!)

D. Pitonyak


D. Pitonyak


No data yet to tell us (need EIC!)

Use small-x helicity evolution!

D. Pitonyak


D. Pitonyak

Derived and solved small-x helicity evolution equations in the “large-Nc” approximation (Kovchegov, DP, Sievert: JHEP 1601 (2016), PRL 118 (2017), PRD 95 (2017), PLB 772 (2017), JHEP 1710 (2017))

D. Pitonyak

These are direct inputs from theory!

�G(x) ⇠✓1

x

◆↵Gh

with ↵Gh =

13

4p3

r↵sNc

2⇡⇡ 1.88

r↵sNc

2⇡

↵s=0.3⇡ 0.712

<latexit sha1_base64="VSKakyRVrwJEk8hvGZsjXWY+DMc=">AAAC1nichVJdaxNBFJ1dv2r8aNRHX64GIX1ZdtNCA6VQVKhPUtG0gUy6TCaz2aGzu9OZu5qwrA+K+Opv880f4X9wkqxfreCFgcO55547c+9MtJIWw/Cb51+5eu36jY2brVu379zdbN+7f2yL0nAx4IUqzHDCrFAyFwOUqMRQG8GyiRInk7Nny/zJW2GsLPI3uNBinLFZLhPJGToqbn+nz4VCBofd+RZQKzOgSiTYpYlhvIpqqOY1NXKW4tZpRR9RpnTK4vT0sAZ6XrIpVNRk8E5i+pP4LYF9aGy262qH2nOD1XZdr8E604gtvIy569WjWtbOiGltijlEQb8P/5XTPadBxs+MUL8U+2HgmjZGznEvDHajXtzuhEG4CrgMogZ0SBNHcfsrnRa8zESOXDFrR1GocVwxg5IrUbdoaYV2rdlMjBzMWSbsuFqtpYYnjplCUhh3coQV+2dFxTJrF9nEKTOGqb2YW5L/yo1KTPrjSua6RJHzdaOkVIAFLHcMU2kER7VwgHEj3V2Bp8wNEN1PaLkhRBeffBkc94IoDKJXO52Dp804NshD8ph0SUR2yQF5QY7IgHDvtbfwPngf/aH/3v/kf15Lfa+peUD+Cv/LD+7I4zg=</latexit><latexit sha1_base64="VSKakyRVrwJEk8hvGZsjXWY+DMc=">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</latexit><latexit sha1_base64="VSKakyRVrwJEk8hvGZsjXWY+DMc=">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</latexit><latexit sha1_base64="VSKakyRVrwJEk8hvGZsjXWY+DMc=">AAAC1nichVJdaxNBFJ1dv2r8aNRHX64GIX1ZdtNCA6VQVKhPUtG0gUy6TCaz2aGzu9OZu5qwrA+K+Opv880f4X9wkqxfreCFgcO55547c+9MtJIWw/Cb51+5eu36jY2brVu379zdbN+7f2yL0nAx4IUqzHDCrFAyFwOUqMRQG8GyiRInk7Nny/zJW2GsLPI3uNBinLFZLhPJGToqbn+nz4VCBofd+RZQKzOgSiTYpYlhvIpqqOY1NXKW4tZpRR9RpnTK4vT0sAZ6XrIpVNRk8E5i+pP4LYF9aGy262qH2nOD1XZdr8E604gtvIy569WjWtbOiGltijlEQb8P/5XTPadBxs+MUL8U+2HgmjZGznEvDHajXtzuhEG4CrgMogZ0SBNHcfsrnRa8zESOXDFrR1GocVwxg5IrUbdoaYV2rdlMjBzMWSbsuFqtpYYnjplCUhh3coQV+2dFxTJrF9nEKTOGqb2YW5L/yo1KTPrjSua6RJHzdaOkVIAFLHcMU2kER7VwgHEj3V2Bp8wNEN1PaLkhRBeffBkc94IoDKJXO52Dp804NshD8ph0SUR2yQF5QY7IgHDvtbfwPngf/aH/3v/kf15Lfa+peUD+Cv/LD+7I4zg=</latexit>

�q(x) ⇠✓1

x

◆↵qh

with ↵qh =

4p3

r↵sNc

2⇡⇡ 2.31

r↵sNc

2⇡

↵s=0.3⇡ 0.874

=) �⌃(x) ⇠✓1

x

◆↵qh

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D. Pitonyak

Derived and solved small-x helicity evolution equations in the “large-Nc” approximation (Kovchegov, DP, Sievert: JHEP 1601 (2016), PRL 118 (2017), PRD 95 (2017), PLB 772 (2017), JHEP 1710 (2017))

D. Pitonyak

These are direct inputs from theory!

�G(x) ⇠✓1

x

◆↵Gh

with ↵Gh =

13

4p3

r↵sNc

2⇡⇡ 1.88

r↵sNc

2⇡

↵s=0.3⇡ 0.712

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�q(x) ⇠✓1

x

◆↵qh

with ↵qh =

4p3

r↵sNc

2⇡⇡ 2.31

r↵sNc

2⇡

↵s=0.3⇡ 0.874

=) �⌃(x) ⇠✓1

x

◆↵qh

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Universal (flavor-independent) intercept! è all quark/antiquark helicity PDFs should have the same small-x behavior


D. Pitonyak

More work to do…

-Need to go beyond the large-Nc approximation

-Must include these results in the next fits of helicity PDFs

D. Pitonyak

Could these pieces be even greater due to spin at small x?

What about OAM?OAM and helicity could cancel at small x (Hatta and Yang (2018))

Sivers and Collins Effects & AN in pp Collisions



Drell-Yan Sivers effect

STAR (2016) COMPASS (2017)


SIDIS Sivers effect ( sin(ϕh – ϕs ) )

HERMES (2009) JLab, Hall A (2011)

COMPASS (2015)

F sin(�h��S)UT = C

"� h · ~kT

Mf?1T D1

#


TMDs in Collins-Soper-Sterman (CSS) evolution formalism

Echevarria, et al. (2014)Anselmino, et al. (2017)

Belle (2008)

Also data from BaBar (2014) and BESIII (2016)


e+e- Collins effect ( cos(2ϕ0 ) )COMPASS (2015)

Also data from JLab Hall A (2011, 2014) and HERMES

SIDIS Collins effect ( sin(ϕh + ϕs ) )

F sin(�h+�S)UT = C

"� h · ~p?

Mhh1H

?1

#

F cos(2�0)UU = C

"2h · ~pa? h · ~pb? � ~pa? · ~pb?

MaMbH

?1 H

?1

#

Kang, et al. (2016)

TMDs in CSS formalism


Anselmino, et al. (2015)


Hadron in a jet Collins effect

Theory curves from D’Alesio, et al. (2017) & Kang, et al. (2017)

Transversity from dihadron FF

STAR (2017)

STAR (2015)

Bacchetta and Radici (2017)


AN in pp -> γ X

(Kanazawa, Koike, Metz, DP – PRD 91 (2015))(See also Gamberg, Kang, Prokudin (2013))

Qiu-Sterman term is the main cause of AN in pp -> γ X

Qiu-Sterman function

d��⇡ ⇠ H ⌦ f1 ⌦ FF T (x, x)

Test of the process dependence of the Sivers function


AN in pp -> γ X

Anselmino, et al. (2013)

GPM

Twist-3

GPM predicts a positive asymmetry while twist-3 predicts a negative one

Kanazawa, et al. (2015)


AN in pp -> π X – PUZZLE FOR 40+ YEARS!

1976


AN in pp -> π X – PUZZLE FOR 40+ YEARS!PHENIX (2014)

STAR (2012)

PHENIX, Talk by J. Bok (DIS 2018)


(Qiu and Sterman (1999), Kouvaris, et al. (2006))FFT ⇠ TF

For many years the Qiu-Sterman/Sivers-type contribution was thought to be the dominant source of TSSAs in p"p ! ⇡ X

d��⇡ ⇠ H ⌦ f1 ⌦ FF T (x, x)


d��⇡ ⇠ H ⌦ f1 ⌦ FF T (x, x)

(Kang, Qiu, Vogelsang, Yuan (2011); Kang and Prokudin (2012); Metz, DP, Schäfer, Schlegel, Vogelsang, Zhou (2012))


(Metz and DP - PLB 723 (2013))

Eh

d��Frag(ST )d3 ~Ph

=� 4↵2sMh

S✏P0PPhST

X

i

X

a,b,c

Z 1

0

dz

z3

Z 1

0dx0Z 1

0dx �(s + t + u)

1s (�x0t� xu)

⇥ ha

1(x) fb

1(x0)

("H?(1),c1 (z)� z

dH?(1),c1 (z)dz

#S

i

H?1

+1zH

c(z) Si

H

+2z

Z 1

z

dz1

z21

1⇣

1z� 1

z1

⌘2 Hc,=FU

(z, z1)Si

HF U

)

d��⇡ ⇠ H ⌦ f1 ⌦ FF T (x, x)

d��⇡ ⇠ h1 ⌦ S ⌦

H?(1)1 ,H,

Zdz1

z21

H=F U

(1/z � 1/z1)2

!

We now believe the TSSAs in are due fragmentation effects as the partons

form pions in the final state

p"p ! ⇡ X


QCD e.o.m. relation (EOMR)

Hq(z) = �2z H

?(1),q1 (z) + 2z

Z 1

z

dz1

z21

11z �

1z1

Hq,=FU (z, z1)

⌘ Hq(z)

d��⇡ ⇠ h1 ⌦ S ⌦

H?(1)1 ,H,

Zdz1

z21

H=F U

(1/z � 1/z1)2

!


Fragmentation term is the main cause of AN in pp -> π X

(Kanazawa, Koike, Metz, DP, PRD 89(RC) (2014))

Also included the Qiu-Sterman term

d��⇡ ⇠ h1 ⌦ S ⌦

H?(1)1 ,H,

Zdz1

z21

H=F U

(1/z � 1/z1)2

!

⇡ FF T (x, x) = f?(1)1T (x)

d��⇡ ⇠ h1 ⌦ S ⌦

H?(1)1 , H,

Zdz1

z21

H=F U

(1/z � 1/z1)2

!


(Kanazawa, Koike, Metz, DP, Schlegel, PRD 93 (2016))

Hq(z)z

= �✓

1� zd

dz

◆H?(1),q1 (z)� 2

z

Z 1

z

dz1

z21

Hq,=FU (z, z1)

(1/z � 1/z1)2Lorentz

invariance relation (LIR)

d��⇡ ⇠ h1 ⌦ S ⌦

H?(1)1 , H,

Zdz1

z21

H=F U

(1/z � 1/z1)2

!


Si

H?1⌘

Si

H?1� Si

HF U

�x0t� xuand Si

H⌘

Si

H� Si

HF U

�x0t� xuwhere

Eh


=� 4↵2sMh

S✏P0PPhST

X

i

X

a,b,c

Z 1

0

dz

z3

Z 1

0dx

0Z 1

0dx �(s + t + u)

1s

⇥ ha

1(x) fb

1(x0)

("H

?(1),c1 (z)� z

dH?(1),c1 (z)dz

#S

i

H?1

+�2H

?(1),c1 (z) +

1zH

c(z)�

Si

H

)

(Gamberg, Kang, DP, Prokudin, PLB 770 (2017))

d��⇡ ⇠ h1 ⌦ S ⌦⇣H

?(1)1 , H

⌘

d��⇡ ⇠ h1 ⌦ S ⌦

H?(1)1 , H,

Zdz1

z21

H=F U

(1/z � 1/z1)2

!



d��⇡ ⇠ h1 ⌦ S ⌦⇣H

?(1)1 , H

⌘

d��⇡ ⇠ h1 ⌦ S ⌦

H?(1)1 , H,

Zdz1

z21

H=F U

(1/z � 1/z1)2

!


The AN data from RHIC can be used to constrain transversity at large x!



The AN data from RHIC can be used to constrain transversity at large x!


d��⇡ ⇠ h1 ⌦ S ⌦

H?(1)1 , H,

Zdz1

z21

H=F U

(1/z � 1/z1)2

!

STAR had the first data EVER on transversity in 2004!

d��⇡ ⇠ h1 ⌦ S ⌦⇣H

?(1)1 , H

⌘


STAR shows no suppression with A (neutral pions, larger xF region)

AN in pA -> π X

PHENIX, Talk by J. Bok (DIS 2018) STAR, Talk by C. Dilks (DIS 2016)

PHENIX shows A1/3 suppression (charged hadrons, smaller xF region)


Eh


=� 4↵2sMh

S✏P0PPhST

X

i

X

a,b,c

Z 1

0

dz

z3

Z 1

0dx0Z 1

0dx �(s + t + u)

1s (�x0t� xu)

⇥ ha

1(x) fb

1(x0)

("H?(1),c1 (z)� z

dH?(1),c1 (z)dz

#S

i

H?1

+1zH

c(z) Si

H

+2z

Z 1

z

dz1

z21

1⇣

1z� 1

z1

⌘2 Hc,=FU

(z, z1)Si

HF U

)

2013 expression from Metz and DP


Eh


=� 4↵2sMh

S✏P0PPhST

X

i

X

a,b,c

Z 1

0

dz

z3

Z 1

0dx0Z 1

0dx �(s + t + u)

1s (�x0t� xu)

⇥ ha

1(x) fb

1(x0)

("H?(1),c1 (z)� z

dH?(1),c1 (z)dz

#S

i

H?1

+1zH

c(z) Si

H

+2z

Z 1

z

dz1

z21

1⇣

1z� 1

z1

⌘2 Hc,=FU

(z, z1)Si

HF U

)


~ A-1/3

~ A-1/3

Include small x multiple-rescatteringsto calculate pA TSSA (Hatta, Xiao, Yoshida, Yuan (2017))

~ A0


Eh


=� 4↵2sMh

S✏P0PPhST

X

i

X

a,b,c

Z 1

0

dz

z3

Z 1

0dx0Z 1

0dx �(s + t + u)

1s (�x0t� xu)

⇥ ha

1(x) fb

1(x0)

("H?(1),c1 (z)� z

dH?(1),c1 (z)dz

#S

i

H?1

+1zH

c(z) Si

H

+2z

Z 1

z

dz1

z21

1⇣

1z� 1

z1

⌘2 Hc,=FU

(z, z1)Si

HF U

)


EOMR + LIR è

~ A0

2z

Z 1

z

dz1

z21

1⇣

1z �

1z1

⌘2 H⇡/c,=FU (z, z1) = H

?(1),c1 (z) + z

dH?(1),c1 (z)dz

� 1zH

c(z)


Eh


=� 4↵2sMh

S✏P0PPhST

X

i

X

a,b,c

Z 1

0

dz

z3

Z 1

0dx0Z 1

0dx �(s + t + u)

1s (�x0t� xu)

⇥ ha

1(x) fb

1(x0)

("H?(1),c1 (z)� z

dH?(1),c1 (z)dz

#S

i

H?1

+1zH

c(z) Si

H

+2z

Z 1

z

dz1

z21

1⇣

1z� 1

z1

⌘2 Hc,=FU

(z, z1)Si

HF U

)


Calculate pieces involving the (first kT-moment of the) Collins function to get an updated estimate for the term in blue

~ A0

2z

Z 1

z

dz1

z21

1⇣

1z �

1z1

⌘2 H⇡/c,=FU (z, z1) = H

?(1),c1 (z) + z

dH?(1),c1 (z)dz

� 1zH

c(z)

EOMR + LIR è


Fragmentation term as the cause of AN in pp collisionsis not inconsistent with the RHIC pA AN data.

Reduced theoretical uncertainties is needed.



Sivers ~ sin(ϕh – ϕs ), Collins ~ sin(ϕh + ϕs ), ...Collins ~ cos(ϕa + ϕb ), ...

AN ~ dσL – dσRSivers ~ sin(ϕs ) (lepton pair) / Sivers ~ cos(ϕW/Z ) (boson)


H?1h1, f

?1T , H

?1

f?1T h1, FF T , H

?(1)1 , H


H?1h1, f

?1T , H

?1

f?1T h1, FF T , H

?(1)1 , H


Figure from EIC Whitepaper

One naively expects that we can obtain collinear functions by integrating TMDs over kT


“Original CSS” (Collins, Soper, Sterman (1985); Ji, Ma, Yuan (2005); Collins (2011); ...)

Zd2kT f1(x, kT ;Q2, µQ) = f1(x, bT ! 0; Q2, µQ) = 0 !

(Collins, Gamberg, Prokudin, Rogers, Sato, Wang (2016))

(Gamberg, Metz, DP, Prokudin, PLB 781 (2018))

Zd2kT

k2T

2M2f?1T (x, kT ;Q2, µQ) ⌘ f?(1)

1T (x;Q2, µQ) = f?(1)1T (x, bT ! 0; Q2, µQ) = 0 !

Takes into account “complications” of QCD (e.g., parton re-scattering and gluon radiation)





Zd2kT

k2T

2M2f?1T (x, kT ;Q2, µQ) ⌘ f?(1)

1T (x;Q2, µQ) = f?(1)1T (x, bT ! 0; Q2, µQ) = 0 !

TMDs lose their physical interpretation in the “Original CSS” formalism!







Zd2kT

k2T

2M2f?1T (x, kT ;Q2, µQ) ⌘ f?(1)

1T (x;Q2, µQ) = f?(1)1T (x, bT ! 0; Q2, µQ) = 0 !


hkiT (x)iUT =

Zd2kT ki

T

�

~kT ⇥ ~ST

Mf?1T (x, kT )

!

avg. TM of unpolarized

quarks in a transversely

polarized spin-1/2 target

A. Prokudin (2012)








Zd2kT

k2T

2M2f?1T (x, kT ;Q2, µQ) ⌘ f?(1)

1T (x;Q2, µQ) = f?(1)1T (x, bT ! 0; Q2, µQ) = 0 !


hkiT (x)iUT =

Zd2kT ki

T

�

~kT ⇥ ~ST

Mf?1T (x, kT )

!

avg. TM of unpolarizedquarks in a transversely

polarized spin-1/2 target


A. Prokudin (2012)


Zd2~kT f1(x, kT ; Q2, µQ; C5) = f1(x, bc(0); Q2, µQ) = f1(x; µc) + O(↵s(Q)) + O((m/Q)p)

Zd2~pT D1(z, pT ; Q2, µQ; C5) = D1(z, bc(0); Q2, µQ) = D1(z; µc) + O(↵s(Q)) + O((m/Q)p)

Zd2~kT

~k2T

2M2f?1T (x, kT ; Q2, µQ; C5) = f?(1)

1T (x, bc(0); Q2, µQ) = ⇡ FF T (x, x; µc) + O(↵s(Q)) + O((m/Q)p0)

Zd2~pT

~p2T

2z2M2h

H?1 (z, pT ; Q2

, µQ; C5)= H?(1)1 (z, bc(0); Q2

, µQ)= H?(1)1 (z; µc)+ O(↵s(Q))+ O((m/Q)p00

))

At LO in the “Improved CSS” formalism we recover the relations one expects from the “naïve” operator definitions of the functions


(Collins, Gamberg, Prokudin, Rogers, Sato, Wang (2016))“Improved CSS” (Unpolarized)


Zd2~kT f1(x, kT ; Q2, µQ; C5) = f1(x, bc(0); Q2, µQ) = f1(x; µc) + O(↵s(Q)) + O((m/Q)p)

Zd2~pT D1(z, pT ; Q2, µQ; C5) = D1(z, bc(0); Q2, µQ) = D1(z; µc) + O(↵s(Q)) + O((m/Q)p)

Zd2~kT

~k2T

2M2f?1T (x, kT ; Q2, µQ; C5) = f?(1)

1T (x, bc(0); Q2, µQ) = ⇡ FF T (x, x; µc) + O(↵s(Q)) + O((m/Q)p0)

Zd2~pT

~p2T

2z2M2h

H?1 (z, pT ; Q2

, µQ; C5)= H?(1)1 (z, bc(0); Q2

, µQ)= H?(1)1 (z; µc)+ O(↵s(Q))+ O((m/Q)p00

))

The “Improved CSS” formalism (approximately) restores the physical interpretation of TMDs!

At LO in the “Improved CSS” formalism we recover the relations one expects from the “naïve” operator definitions of the functions


(Collins, Gamberg, Prokudin, Rogers, Sato, Wang (2016))“Improved CSS” (Unpolarized)


Zd2~kT

~k2T

2M2f?1T (x, kT ; Q2, µQ; C5) = ⇡ FF T (x, x; µc) + O(↵s(Q)) + O((m/Q)p0

)

The Qiu-Sterman function can fundamentally be understood as an avg. TM, and the first kT-moment of the Sivers function (using “Improved CSS”) retains this interpretation at LO

avg. TM of unpolarized quarks in a transversely polarized spin-1/2 target

hkiT (x; µ)iUT

=12

Zd2kT ki

T

Zdb�

2⇡

Zd2bT

(2⇡)2eixP +b�

e�i~kT ·~bT hP, S| (0)�+WDIS(0; b) (b)|P, Si��b+=0

=12

Zdb�dy�

4⇡eixP +b�hP, S| (0)�+W(0; y�)gF +i(y�) W(y�; b�) (b�)|P, Si

= �⇡ M✏ijSjT FF T (x, x; µ)


Recall also the Burkardt sum ruleX

a=q,q,g

Z 1

0dxF a

FT (x, x) = 0<latexit sha1_base64="mETthIR5tAVnMkp8kJ5YY/Tikzw=">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</latexit><latexit sha1_base64="mETthIR5tAVnMkp8kJ5YY/Tikzw=">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</latexit><latexit sha1_base64="mETthIR5tAVnMkp8kJ5YY/Tikzw=">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</latexit>


H?1h1, f

?1T , H

?1

f?1T h1, FF T , H

?(1)1 , H

Toward a Global Analysis of Transverse Spin Observables



Recall the current phenomenology of TMD observables...

H?(1)1 (z, bT ; Q2

, µQ) ⇠ H?(1)1 (z; µb⇤) exp

h�Spert(b⇤(bT );µb⇤ , Q, µQ)� S

H?1

NP(bT , Q)

i

f?(1)1T (x, bT ; Q2, µQ) ⇠ FF T (x, x; µb⇤) exp

h�Spert(b⇤(bT );µb⇤ , Q, µQ)� S

f?1TNP (bT , Q)

i

The CT3 functions (along with the NP g-functions) are what get extracted in analyses of TSSAs in TMD processes that use CSS evolution!

(Echevarria, Idilbi, Kang, Vitev (2014); Kang, Prokudin, Sun, Yuan (2016))

gf?1T(x, bT ) + gK(bT ) ln(Q/Q0)

gH?1

(z, bT ) + gK(bT ) ln(Q/Q0)


H?(1)1

This is NOT a “new” function!

FF T h1, FF T , H?(1)1 , H

h1, FF T , H?(1)1


Asin �S

UT in SIDIS integrated over PT (Mulders, Tangerman (1996); Bacchetta, et al. (2007))

in e+e- è h1h2 X integrated over qT (Boer, Jakob, Mulders (1997))

F sin �S

UT /X

a

e2a

2Mh

Qha

1(x)H

a(z)z

Fsin �S

UT /X

a,a

e2a

2M2

QD

a1(z1)

DaT (z2)z2

+2M1

Q

H(z1)z1

Ha1 (z2)

!

And also the TMD version of these (and other) observables (but with many more terms)

Asin �S

UT


FF T

H?(1)1 , H

h1, FF T , H?(1)1 , H

h1, FF T , H?(1)1 , H

SummaryD. PitonyakD. Pitonyak

• The proton spin could receive a moderate enhancement from small-x quarks and gluons, although one must also study carefully OAM at small x to get the full picture. Longitudinal spin data from RHIC and a future EIC will help us better understand this region.

• TMD and collinear functions are highly interconnected, especially for observables involving transverse spin, and we should treat both observables on the same footing.

• The current TMD formalism using “Improved CSS” allows one to rigorously connect these two different types of functions, and at LO we can restore the physical interpretation of (integrated) TMDs.

• A global analysis of TMD (Sivers and Collins effects ) AND collinear twist-3 (AN in pp, in SIDIS) transverse-spin observables is possible.Asin �s

UT

Back-up Slides


Ø The “sign mismatch” issuep"p! h X

RHIC, STAR (2012) CERN, COMPASS (2013) `N" ! `0 h X

⇡ FFT (x, x) = f?(1)1T (x)

RHIC, PHENIX (2014)




⇡ FFT (x, x) = f?(1)1T (x)

RHIC, PHENIX (2014)




⇡ FFT (x, x) = f?(1)1T (x)

RHIC, PHENIX (2014) �sign mismatch� (Kang, Qiu, Vogelsang, Yuan (2011))


D. Pitonyak

Proton-proton data from BRAHMS for π+ (left) and π -(right)

Kang and Prokudin (2012)

D. Pitonyak

Nodes in Sivers cannot resolve issue

Sivers input agrees reasonably well with the JLab data FIRST INDICATION on the PROCESS DEPENDENENCE of the Sivers function (see also Gamberg, Kang, Prokudin (2013))

KQVY input gives the wrong sign Qiu-Sterman function cannot be the main cause of the large TSSAs seen in pion production from pp collisions

JLab E07-013, Hall A (2013)

Neutron TSSA in inclusive DIS

D. Pitonyak

Metz, DP, Schäfer, Schlegel, Vogelsang, Zhou - PRD 86 (2012)

Proton-proton data from BRAHMS for π+ (left) and π -(right)

Kang and Prokudin (2012)

Nodes in Sivers cannot resolve issue

D. Pitonyak



“b-space” correlator

�[�+](x,~bT ;Q2, µQ) = f1(x, bT ; Q2, µQ)� iM✏ijbiT Sj

T

�

1M2

1bT

@

@bTf?1T (x, bT ; Q2, µQ)

�

⌘ f?(1)1T (x, bT ; Q2, µQ)

Boer, Gamberg, Musch, Prokudin (2011)


f1(x, bT ; Q2, µQ) ⇠⇣Cf1(x/x, b⇤(bT );µ2

b⇤ , µb⇤ , ↵s(µb⇤))⌦ f1(x; µb⇤)⌘

⇥ exph�Spert(b⇤(bT ); µb⇤ , Q, µQ)� Sf1

NP (bT , Q)i




T

�

1M2

1bT

@


�


Collins (2011); ...

⌘ f?(1)1T (x, bT ; Q2, µQ)



b⇤ , µb⇤ , ↵s(µb⇤))⌦ f1(x; µb⇤)⌘


NP (bT , Q)i




T

�

1M2

1bT

@


�


f?(1)1T (x, bT ; Q2, µQ) ⇠

⇣Cf?1T(x1, x2, b⇤(bT );µ2

b⇤ , µb⇤ , ↵s(µb⇤))⌦ FF T (x1, x2; µb⇤)⌘

⇥ exph�Spert(b⇤(bT );µb⇤ , Q, µQ)� S

f?1TNP (bT , Q)

i

Aybat, Collins, Qiu, Rogers (2012); Echevarria, Idilbi, Kang, Vitev (2014); ...

Collins (2011); ...

⌘ f?(1)1T (x, bT ; Q2, µQ)


perturbative Sudakov factor

different for each TMD

universal

non-perturbative Sudakov factor

same for unpol. and pol.

gf1(x, bT ) + gK(bT ) ln(Q/Q0)


b⇤ , µb⇤ , ↵s(µb⇤))⌦ f1(x; µb⇤)⌘


NP (bT , Q)i

“Original CSS”

� ln(Q/µb⇤)K(b⇤, µb⇤)�Z µQ

µb⇤

dµ0

µ0 [�(↵s(µ0); 1)� �K(↵s(µ0)) ln(Q/µ0)]

(Collins, Soper, Sterman (1985); Ji, Ma, Yuan (2005); Collins (2011); ...)




universal





b⇤ , µb⇤ , ↵s(µb⇤))⌦ f1(x; µb⇤)⌘


NP (bT , Q)i

“Original CSS”


µb⇤

dµ0

µ0 [�(↵s(µ0); 1)� �K(↵s(µ0)) ln(Q/µ0)]


b⇤(bT ) ⌘

sb2T

1 + b2T /b2

max

µb⇤ = C1/b⇤(bT )

Note: problematic large logarithms in Spert




universal





b⇤ , µb⇤ , ↵s(µb⇤))⌦ f1(x; µb⇤)⌘


NP (bT , Q)i

“Original CSS”


µb⇤

dµ0

µ0 [�(↵s(µ0); 1)� �K(↵s(µ0)) ln(Q/µ0)]


b⇤(bT ) ⌘

sb2T

1 + b2T /b2

max

µb⇤ = C1/b⇤(bT )

b⇤(0) = 0 and (µb⇤)b⇤!0 = 1(Bozzi, Catani, de Florian, Grazzini (2006); Collins, Gamberg, Prokudin, Rogers, Sato, Wang (2016))

Note: problematic large logarithms in Spert




universal





b⇤ , µb⇤ , ↵s(µb⇤))⌦ f1(x; µb⇤)⌘


NP (bT , Q)i

“Original CSS”


µb⇤

dµ0

µ0 [�(↵s(µ0); 1)� �K(↵s(µ0)) ln(Q/µ0)]


b⇤(bT ) ⌘

sb2T

1 + b2T /b2

max

µb⇤ = C1/b⇤(bT )

b⇤(0) = 0 and (µb⇤)b⇤!0 = 1(Bozzi, Catani, de Florian, Grazzini (2006); Collins, Gamberg, Prokudin, Rogers, Sato, Wang (2016))


(Collins, Gamberg, Prokudin, Rogers, Sato, Wang (2016))*

Place a lower cut-off on bT :

µb⇤ ! µ ⌘ C1

b⇤(bc(bT )) so µb⇤ is cut o↵ at µc ⇡C1C5Q

b0

*Other modifications are discussed in this reference that attempt to improve the agreement of the CSS W+Y formulation with the differential cross section over all transverse momentum regions.

bT ! bc(bT ) where bc(bT ) =q

b2T + b2

0/(C5Q)2

“Improved CSS” (Unpolarized)




µb⇤ ! µ ⌘ C1


b0


b2T + b2

0/(C5Q)2


f1(x, bc(bT ); Q2, µQ) ⇠⇣Cf1(x/x, b⇤(bc(bT )); µ2, µ, ↵s(µ))⌦ f1(x; µ)

⌘

⇥ exph�Spert(b⇤(bc(bT )); µ, Q, µQ)� Sf1

NP (bc(bT ), Q)i




µb⇤ ! µ ⌘ C1


b0


b2T + b2

0/(C5Q)2


(Gamberg, Metz, DP, Prokudin, PLB 781 (2018))“Improved CSS” (Polarized)


⌘


NP (bc(bT ), Q)i


T f?(1)1T (x, bT ; Q2, µQ)


T f?(1)1T (x, bT ; Q2, µQ)




µb⇤ ! µ ⌘ C1


b0


b2T + b2

0/(C5Q)2



⌘


NP (bc(bT ), Q)i

NO bT -> bc(bT) replacement –kinematic factor NOT associated

with the scale evolution

bT -> bc(bT) bT -> bc(bT)

“Improved CSS” (Polarized) (Gamberg, Metz, DP, Prokudin, PLB 781 (2018))

�[�+](x,~bT , bc(bT );Q2, µQ) = f1(x, bc(bT ); Q2, µQ)� iM✏ijbiT Sj

T f?(1)1T (x, bc(bT ); Q2, µQ)




µb⇤ ! µ ⌘ C1


b0


b2T + b2

0/(C5Q)2



⌘


NP (bc(bT ), Q)i

f?(1)1T (x, bc(bT ); Q2, µQ) ⇠

⇣Cf?1T(x1, x2, b⇤(bc(bT )); µ2, µ, ↵s(µ))⌦ FF T (x1, x2; µ

⌘

⇥ exph�Spert(b⇤(bc(bT )); µ, Q, µQ)� S

f?1TNP (bc(bT ), Q)

i

“Improved CSS” (Polarized) (Gamberg, Metz, DP, Prokudin, PLB 781 (2018))


H?(1)1 (z, bc(bT ); Q2

, µQ) ⇠⇣CH

?1 (z/z, b⇤(bc(bT )); µ2, µ, ↵s(µ))⌦H

?(1)1 (z; µ)

⌘

⇥ exph�Spert(b⇤(bc(bT )); µ, Q, µQ)� S

H?1

NP(bc(bT ), Q)

i

Analogous modification for fragmentation functions...

D1(z, bc(bT ); Q2, µQ) ⇠⇣CD1(z/z, b⇤(bc(bT )); µ2, µ, ↵s(µ))⌦D1(z; µ)

⌘

⇥ exph�Spert(b⇤(bc(bT )); µ, Q, µQ)� SD1

NP (bc(bT ), Q)i


We then define the momentum-space functions...

f1(x, kT ; Q2, µQ; C5) ⌘Z

dbT

2⇡bT J0(kT bT )f1(x, bc(bT ); Q2, µQ)

~k2T

2M2f?1T (x, kT ; Q2, µQ; C5) ⌘ kT

ZdbT

4⇡b2T J1(kT bT ) f?(1)

1T (x, bc(bT ); Q2, µQ)

D1(z, pT ; Q2, µQ; C5) ⌘Z

dbT

2⇡bT J0(pT bT ) D1(z, bc(bT ); Q2, µQ)

~p 2T

2z2M2h

H?1 (z, pT ; Q2

, µQ; C5) ⌘ pT

ZdbT

4⇡b2T J1(pT bT ) H

?(1)1 (z, bc(bT ); Q2

, µQ)



Zd2~kT

~k2T


)



(Boer, Mulders, Teryaev (1998); Burkardt (2004); Meissner, Metz, Goeke (2007))

Zd2~kT

~k2T


)

hkiT (x; µ)iUT

=12

Zd2kT ki

T

Zdb�

2⇡

Zd2bT

(2⇡)2eixP +b�


=12

Zdb�dy�




hkiT (x; µ)iUT

=12

Zd2kT ki

T

Zdb�

2⇡

Zd2bT

(2⇡)2eixP +b�


=12

Zdb�dy�




Zd2~kT

~k2T


)

“Naïve” TMD operator – UV renormalization at LO and soft factor at LO, Wilson lines on the lightcone


hkiT (x; µ)iUT

=12

Zd2kT ki

T

Zdb�

2⇡

Zd2bT

(2⇡)2eixP +b�


=12

Zdb�dy�




Zd2~kT

~k2T


)

This is NOT the operator that defines TMDs in CSS “Naïve” TMD operator – UV renormalization at LO

and soft factor at LO, Wilson lines on the lightconeX


hkiT (x; µ)iUT

=12

Zd2kT ki

T

Zdb�

2⇡

Zd2bT

(2⇡)2eixP +b�


=12

Zdb�dy�




Zd2~kT

~k2T


)

“Naïve” collinear operator – LO term of the UV renormalized correlator, Wilson lines on the lightcone


hkiT (x; µ)iUT

=12

Zd2kT ki

T

Zdb�

2⇡

Zd2bT

(2⇡)2eixP +b�


=12

Zdb�dy�




This IS the operator for the Qiu-Stermanfunction that enters the OPE within CSS

Zd2~kT

~k2T


)

“Naïve” collinear operator – LO term of the UV renormalized correlator, Wilson lines on the lightcone


Documents

TMD and collinear functions for transverse and ... · h with ↵q h = 4 p 3 r ↵ s N c 2⇡ ... (Hatta and Yang (2018)) Siversand Collins Effects & A Nin ppCollisions D. PitonyakD