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The gluon Sivers function and its process dependence from RHIC data Cristian Pisano In collaboration with: U. DAlesio, C. Flore, F. Murgia, P. Taels The TMD Generalized Parton Model 2/17 TMD factorization Two scale processes Q 2 p 2 T

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SLIDE 1

The gluon Sivers function and its process dependence from RHIC data

Cristian Pisano

In collaboration with: U. D’Alesio, C. Flore, F. Murgia, P. Taels

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SLIDE 2

The TMD Generalized Parton Model

2/17

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SLIDE 3

TMD factorization

Two scale processes Q2 ≫ p2

T

Factorization proven

3/17

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SLIDE 4

TMD factorization

Phenomenological extension of the TMD formalism to processes like

X

pp → πX pp → jet πX and more (pp → jet X, pp → γX) Single scale processes

Anselmino, Boglione, Murgia, PLB 362 (1995), ... Aschenauer, D’Alesio, Murgia, EPJA52 (2016)

Transverse Momentum Dependent – Generalized Parton Model (GPM)

◮ Spin & k⊥-dependent distribution and fragmentation functions as in TMD scheme ◮ k⊥-dependence included in the hard scattering, unlike in the TMD formalism ◮ Universality and TMD factorization: assumption to be tested

4/17

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SLIDE 5

Color Gauge Invariant (CGI) GPM

The quark Sivers function

The CGI-GPM takes into account the effects of initial and final state interactions

Gamberg, Kang, PLB 696 (2011)

One-gluon exchange approx.: LO term of of the αS expansion of the gauge link SIDIS

p

a

p P ,

c

q S

A

k

T A

p

T a a c

S p P , p

− →

f ⊥q [SIDIS]

1T

x Hard part

qq′ → qq′

P ,

A ST

p

c

p

b a

p k

d

p

A

P ,

T

S k

b

p p

a

p

d

p

c A

p C

b F T c a

  • r

p

a c d

p S p C

I

P , p

− →

f ⊥q [SIDIS]

1T

x (CF x Hard part)

f ⊥q[SIDIS]

1T

is universal, process dependence absorbed in modified hard functions

5/17

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SLIDE 6

Color Gauge Invariant (CGI) GPM

The quark Sivers function

The CGI-GPM recovers the relation f ⊥ [DY ]

1T

= −f ⊥ [SIDIS]

1T

In the CGI-GPM TMDs are process dependent, different predictions w.r.t. GPM

Gamberg, Kang, PLB 696 (2011) D’Alesio, Gamberg, Kang, Murgia, CP, PLB 704 (2011)

  • 0.04
  • 0.02

0.02 0.04 0.06 0.1 0.2 0.3 0.4 0.5 0.6

x

F

AN

sinφS

ηj = 3.25

  • 0.04
  • 0.02

0.02 0.04 0.06 0.1 0.2 0.3 0.4 0.5 0.6

x

F

AN

sinφS

ηj = 3.25

  • 0.04
  • 0.02

0.02 0.04 0.06 0.1 0.2 0.3 0.4 0.5 0.6

x

F

AN

sinφS

ηj = 3.25

  • 0.04
  • 0.02

0.02 0.04 0.06 0.1 0.2 0.3 0.4 0.5 0.6

x

F

AN

sinφS

ηj = 3.25

  • 0.04
  • 0.02

0.02 0.04 0.06 0.1 0.2 0.3 0.4 0.5 0.6

x

F

AN

sinφS

ηj = 3.25 GPM

  • 0.04
  • 0.02

0.02 0.04 0.06 0.1 0.2 0.3 0.4 0.5 0.6

x

F

AN

sinφS

ηj = 3.25 CGI

  • 0.04
  • 0.02

0.02 0.04 0.06 0.1 0.2 0.3 0.4 0.5 0.6

x

F

AN

sinφS

ηj = 3.25 ANDY data

  • 0.04
  • 0.02

0.02 0.04 0.06 0.1 0.2 0.3 0.4 0.5 0.6

x

F

AN

sinφS

ηj = 3.25

p↑p → jet X (√s = 500 GeV)

Extension of the CGI-GPM to the gluon Sivers function is now completed

D’Alesio, Murgia, CP, Taels, PRD 96 (2017) D’Alesio, Flore, Murgia, CP, PRD 99 (2019)

Gluon Sivers function constrained from available data on p↑p → π0 X and p↑p → DX; predictions for p↑p → J/ψ X and p↑p → γX at RHIC Talk by F. Murgia, WG7

6/17

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SLIDE 7

Color Gauge Invariant (CGI) GPM

Two independent gluon Sivers functions with first transverse moments f ⊥(1)g (f /d)

1T

(x) =

  • d2kT

k2

T

2M2

p

f ⊥ g (f /d)

1T

(x, k2

T)

related to two different trigluon Qiu-Sterman functions T (f /d)

G

, involving the antisymmetric fabc and symmetric dabc color structures, respectively

Bomhof, Mulders, JHEP 0702 (2007) Buffing, Mukherjee, Mulders, PRD 88 (2013)

The two distributions have a different behavior under charge conjugation The Burkardt sum rule constraints only the f -type gluon Sivers function

  • a=q,¯

q,g

  • dx f ⊥(1)a

1T

(x) = 0

Boer, Lorc´ e, CP, Zhou, AHEP 2015 (2015) 7/17

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SLIDE 8

Gluon Sivers function in p↑ p → J/ψ X

AN in the GPM

In the Color Singlet Model, the dominant production channel is gg → J/ψ g

b b c a a p↑(PA) p (PB)

AN ≡ dσ↑ − dσ↓ dσ↑ + dσ↓ ≡ d∆σ 2dσ d∆σGPM = 2α3

s

s dxa xa dxb xb d2k⊥a d2k⊥b δ(ˆ s + ˆ t + ˆ u − M2) ×

  • −k⊥ a

Mp

  • f ⊥ g

1T (xa, k⊥a) cos φa fg/p(xb, k⊥b) HU gg→J/ψg(ˆ

s, ˆ t, ˆ u) f ⊥ g

1T : Gluon Sivers function (one and process independent) 8/17

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SLIDE 9

Gluon Sivers function in p↑ p → J/ψ X

AN in the CGI-GPM

GPM CGI-GPM

b b c a a p↑(PA) p (PB) b b c e a a′ d p↑(PA) p (PB) b b c a a′ d p↑(PA) p (PB)

− → CU C (f /d)

I

C (f /d)

Fc [Color Factors]

[GPM] f ⊥ g

1T

HU

gg→J/ψg −

→ f ⊥ g (f )

1T

HInc (f )

gg→J/ψg + f ⊥ g (d) 1T

HInc (d)

gg→J/ψg

[CGI − GPM] Two independent, universal f ⊥

1T’s, process dependence shifted into new hard parts

HInc (f /d)

gg→J/ψg ≡ C (f /d) I

+ C (f /d)

Fc

CU HU

gg→J/ψg 9/17

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SLIDE 10

Gluon Sivers function in p↑ p → J/ψ X

AN in the CGI-GPM

b b c a a′ d p↑(PA) p (PB)

c ¯ c pair in a color singlet state, no FSIs: C (f )

Fc = C (d) Fd = 0

  • F. Yuan, PRD 78 (2003)

b b c e a a′ d p↑(PA) p (PB)

C (f )

I

= − 1

2 CU

C (d)

I

= 0 Only f ⊥ g (f )

1T

contributes to AN

d∆σCGI = 2α3

s

s dxa xa dxb xb d2k⊥a d2k⊥b δ(ˆ s + ˆ t + ˆ u − M2) ×

  • − k⊥ a

Mp

  • f ⊥ g (f )

1T

(xa, k⊥a) cos φa fg/p(xb, k⊥b) HInc (f )

gg→J/ψg(ˆ

s, ˆ t, ˆ u) HInc (f )

gg→J/ψg = − 1

2 HU

gg→J/ψg 10/17

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SLIDE 11

Gluon Sivers function in p↑ p → D X

AN in the CGI-GPM

LO channels are gg → c ¯ c and q¯ q → c ¯

  • c. Color factors for gg → c ¯

c:

b a b0 a0 b e a c b0 a0 c b a b0 a0 c b a b0 a0

CU C (f /d)

I

C (f /d)

Fc

C (f /d)

Fd

Agreement with gluonic pole strenghts calculated for p↑ p → h h X C (f /d)

G

≡ C (f /d)

I

+ C (f /d)

Fc

+ C (f /d)

Fd

CU

Bomhof, Mulders, JHEP 0702 (2007)

Agreement with twist-three results for p↑ p → D X

Kang, Qiu, Vogelsang, Yuan, PRD 78 (2008)

Both f ⊥ g (f )

1T

and f ⊥ g (d)

1T

contribute to AN(p↑ p → D X)

11/17

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SLIDE 12

Gluon Sivers function in p↑ p → D X

Color factors for gg → c ¯ c C Inc (f /d)

I

≡ C (f /d)

I

+ C (f /d)

Fc D CU C(f)

I

C(f)

F c

C(f)

Fd

CInc (f) C(d)

I

C(d)

Fc

C(d)

Fd

CInc (d)

1 4Nc

−

Nc 8(N 2

c −1)

1 8Nc

−

1 8Nc(N 2

c −1) −

1 8Nc(N 2

c −1)

Nc 8(N 2

c −1)

1 8Nc 1 8Nc(N 2

c −1)

2N 2

c −1

8Nc(N 2

c −1)

1 4Nc

−

Nc 8(N 2

c −1) −

1 8Nc(N 2

c −1)

1 8Nc

−

N 2

c +1

8Nc(N 2

c −1) −

Nc 8(N 2

c −1) −

1 8Nc(N 2

c −1)

−

1 8Nc

−

N 2

c +1

8Nc(N 2

c −1)

Nc 2(N 2

c −1)

−

Nc 4(N 2

c −1)

Nc 8(N 2

c −1)

Nc 8(N 2

c −1)

−

Nc 8(N 2

c −1)

Nc 8(N 2

c −1)

−

Nc 8(N 2

c −1)

Nc 8(N 2

c −1)

Nc 4(N 2

c −1)

−

Nc 8(N 2

c −1)

Nc 8(N 2

c −1)

Nc 8(N 2

c −1)

Nc 8(N 2

c −1)

Nc 4(N 2

c −1)

Nc 4(N 2

c −1)

−

Nc 8(N 2

c −1)

Nc 8(N 2

c −1)

Nc 8(N 2

c −1)

Nc 8(N 2

c −1)

Nc 4(N 2

c −1)

−

Nc 4(N 2

c −1)

Nc 8(N 2

c −1)

−

Nc 8(N 2

c −1)

Nc 8(N 2

c −1)

Nc 8(N 2

c −1)

Nc 8(N 2

c −1)

Nc 8(N 2

c −1)

−

Nc 4(N 2

c −1)

Nc 8(N 2

c −1)

−

Nc 8(N 2

c −1)

Nc 8(N 2

c −1)

Nc 8(N 2

c −1)

Nc 8(N 2

c −1)

Nc 8(N 2

c −1)

−

1 4Nc(N 2

c −1)

−

1 8Nc(N 2

c −1) −

1 8Nc(N 2

c −1) −

1 8Nc(N 2

c −1)

−

1 8Nc(N 2

c −1)

1 8Nc(N 2

c −1) −

1 8Nc(N 2

c −1)

−

1 4Nc(N 2

c −1)

−

1 8Nc(N 2

c −1) −

1 8Nc(N 2

c −1) −

1 8Nc(N 2

c −1)

−

1 8Nc(N 2

c −1)

1 8Nc(N 2

c −1) −

1 8Nc(N 2

c −1)

D’Alesio, Murgia, Pisano, Taels, PRD 96 (2017)

Modified hard functions HInc (f /d) are not simply proportional to HU Similar tables and results for all the channels in p↑ p → π X

D’Alesio, Flore, Murgia, Pisano, Taels, PRD 99 (2019) 12/17

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SLIDE 13

Comparison with data and results

13/17

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SLIDE 14

Gluon Sivers function in p↑p → π0X

Upper bounds

Assumption: the GSFs have a factorized form in x-k⊥, Gaussian k⊥-dependence

  • 0.04
  • 0.02

0.02 0.04 0.06 0.08 0.1 1 2 3 4 5 6 7 8 9 10 xF = 0 √s = 200 GeV GPM CGI f-type CGI d-type CGI quark (fit) Ng(x) = + 1 AN (max) pT (GeV) p↑p → π0 X

Maximized GSFs (Ng = +1)

PHENIX Collaboration, PRD 90 (2014)

The f -type GSF is dominant in the CGI-GPM approach

14/17

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SLIDE 15

Gluon Sivers function in p↑p → π0X

Conservative scenario

Reduced f -type GSF (N(f )

g

= 0.1), negative saturated d-type GSF (N(d)

g

= −1)

  • 0.01
  • 0.005

0.005 0.01 1 2 3 4 5 6 xF = 0 √s = 200 GeV CGI AN pT (GeV) p↑p → π0 X full [Ng

(f)=0.1, Ng (d)=-1]

Shaded area represents a ±20% uncertainty on N(f )

g 15/17

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SLIDE 16

Gluon Sivers function in p↑p → D0X

Conservative scenario

f ⊥ (d)

1T

dominant, data imply |N(d)

g | ≤ 0.15; choice: N(d) g

= −0.15 ⇒ N(f )

g

= +0.05

  • 0.3
  • 0.2
  • 0.1

0.1 0.2 0.3

  • 0.15
  • 0.1
  • 0.05

0.05 0.1 0.15 CGI d-type √s = 200 GeV AN (D → µ+) xF p↑p → µ+ X Ng

(d)(x) = 0.15

Ng

(d)(x) = -0.15

GPM

  • 0.3
  • 0.2
  • 0.1

0.1 0.2 0.3

  • 0.15
  • 0.1
  • 0.05

0.05 0.1 0.15 CGI d-type √s = 200 GeV AN (D → µ−) xF p↑p → µ− X Ng

(d)(x) = 0.15

Ng

(d)(x) = -0.15

GPM PHENIX Collaboration, PRD 95 (2017)

Muon SSAs obtained from our D-meson estimates by Jeongsu Bok (PHENIX)

16/17

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SLIDE 17

First k⊥-moment of the GSFs

10-3 10-2 10-1 100 101 102 103 0.01 0.1 Q2 = 2 GeV2 |f1T

⊥(1)g(x)|

x d-type f-type GPM

  • pos. bound

◮ First attempt towards an extraction of the (process dependent) GSFs ◮ Data are not sufficient to discriminate between the GPM and the CGI-GPM

17/17