[PPT] - Process dependence of the gluon Sivers function in inclusive pp PowerPoint Presentation

SLIDE 1

Process dependence of the gluon Sivers function in inclusive pp collisions: theory

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

SLIDE 2

TMD factorization and process dependence

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SLIDE 3

TMD factorization

Two scale processes Q2 ≫ p2

T

Factorization proven

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SLIDE 4

TMD factorization

Factorization and correlators

Hard partonic interactions can be separated from nonperturbative correlators SIDIS Drell-Yan

P

h

P

h

k p p k ∆ Φ q P P q P

B

PA PB PA Φ p k Φ q q p k

γ∗p → h X p p → γ∗X Parton correlators Φ and ∆ describe the soft hadron ↔ parton transitions

i j p p P P Φ(p;P,S) P

h

P

h

(k;P ,S )

h h

∆ k k

Parametrized in terms of distribution and fragmentation functions

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SLIDE 5

TMD factorization

Gauge invariance of the correlators

Resummation of all gluon exchanges leads to gauge links in the correlators Φ, ∆

P

h

P

h

k p p k ∆ Φ q P P q P

h

P

h

p

1 1

p−p p k P P k q q P

h

P

h

p

1 1

p−p p P P k q k P

h

P

h

p

1 1

p−p−p

2

k P P q p p

2

q P

h

P

h 1

k−k p k p q P P k1 P

h

P

h 1

k

1

k−k p k q p P P

Boer, Mulders, Pijlman, NPB 667 (2003)

UC

[0,ξ] = Pexp

−ig
C[0,ξ]

dsµAµ(s)

The path C depends on the color interactions, i.e. on the specific process

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SLIDE 6

TMD factorization

Gauge invariance of the correlators

Gauge invariant definition of Φ (not unique) Φ[U] ∝

P, S
ψ(0) U C

[0,ξ] ψ(ξ)

P, S
SIDIS

P

h

P

h

k p p k ∆ Φ q P P q P

h

P

h

p

1 1

p−p p k P P k q q P

h

P

h

p

1 1

p−p p P P k q k P

h

P

h

p

1 1

p−p−p

2

k P P q p p

2

q

Belitsky, Ji, Yuan, NPB 656 (2003) Boer, Mulders, Pijlman, NPB 667 (2003)

ξ

−

ξ T ξ

−

ξ T

≡ U[+]

[0,ξ]

Possible effects in transverse momentum observables (ξT is conjugate to kT)

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SLIDE 7

TMD factorization

Process dependence of gauge links

SIDIS Drell-Yan

P

h

P

h

k p p k ∆ Φ q P P q P

B

PA PB PA Φ p k Φ q q p k ξ

−

ξ T ξ

−

ξ T

U[+]

[0,ξ]

U[−]

[0,ξ] Belitsky, Ji, Yuan, NPB 656 (2003) Boer, Mulders, Pijlman, NPB 667 (2003) Boer, talk at RBRC Synergies workshop (2017)

ξ

−

ξ T

dkT −

→ ξT = 0 − → the same in both cases

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SLIDE 8

TMD factorization

The quark Sivers function Fundamental test of TMD theory

f ⊥ [DY ]

1T

(x, k2

⊥) = −f ⊥ [SIDIS] 1T

(x, k2

⊥)

h⊥ [DY ]

1

(x, k2

⊥) = −h⊥ [SIDIS] 1

(x, k2

⊥) Collins, PLB 536 (2002)

FSI in SIDIS ISI in DY

ξ

−

ξ T ξ

−

ξ T

= − ISI/FSI lead to process dependence of TMDs, could even break factorization

Collins, Qiu, PRD 75 (2007) Collins, PRD 77 (2007) Rogers, Mulders, PRD 81 (2010) 8/24

SLIDE 9

Process dependence of gluon TMDs

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SLIDE 10

Gluon TMDs

The gluon correlator

Γ

αβ(p;P,S)

P P p p

Gauge invariant definition of Γµν

Γ[U,U′]µν ∝ P, S| Trc

F +ν(0) UC

[0,ξ] F +µ(ξ) UC′ [ξ,0]

|P, S

Mulders, Rodrigues, PRD 63 (2001) Buffing, Mukherjee, Mulders, PRD 88 (2013) Boer, Cotogno, Van Daal, Mulders, Signori, Zhou, JHEP 1610 (2016)

The gluon correlator depends on two path-dependent gauge links ep → e′QQX, ep → e′ jet jet X probe gluon TMDs with [++] gauge links pp → γγX (and/or other CS final state) probes gluon TMDs with [−−] gauge links pp → γ jet X probes an entirely independent gluon TMD: [+−] links (dipole)

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SLIDE 11

The gluon Sivers functions

Sign change test

Related Processes ep↑ → e′QQX, ep↑ → e′ jet jet X probe GSF with [++] gauge links (WW) p↑p → γγX (and/or other CS final state) probe GSF with [−−] gauge links

Analogue of the sign change of f ⊥ q

1T

between SIDIS and DY (true also for hg

1 and h⊥ g 1T )

f ⊥ g [e p↑→e′ QQ X]

1T

= −f ⊥ g [p↑ p→γ γ X]

1T

= −

Boer, Mulders, CP, Zhou (2016)

Motivation to study gluon Sivers effects at both RHIC and the EIC

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SLIDE 12

The gluon Sivers functions

The dipole GSF

Complementary Processes ep↑ → e′QQX probes a GSF with [++] gauge links (WW) p↑p → γ jet X (gq → γq) probes a gluon TMD with : [+−] links (DP) = − At small-x the WW Sivers function appears to be suppressed by a factor of x compared to the unpolarized gluon function, unlike the dipole one The DP gluon Sivers function at small-x is the spin dependent odderon (single spin asymmetries from a single Wilson loop matrix element)

Boer, Echevarria, Mulders, Zhou, PRL 116 (2016) Boer, Cotogno, Van Daal, Mulders, Signori, Zhou, JHEP 1610 (2016) 12/24

SLIDE 13

The Generalized Parton Model

The first transverse moments of the WW and DP gluon Sivers functions 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) 13/24

SLIDE 14

The Generalized Parton Model

The TMD Generalized Parton Model

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SLIDE 15

The Generalized Parton Model

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

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SLIDE 16

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

16/24

SLIDE 17

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, in preparation

Gluon Sivers function constrained from available data on p↑p → π0 X, p↑p → J/ψ X, p↑p → DX, predictions for p↑p → γX at RHIC

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SLIDE 18

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) 18/24

SLIDE 19

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 19/24

SLIDE 20

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 20/24

SLIDE 21

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)

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SLIDE 22

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, in preparation 22/24

SLIDE 23

Gluon Sivers function in p↑ p → γ X

AN in the CGI-GPM

LO channels are qg → γq, q¯ q → γg and gq → γq.

Gamberg, Kang, PLB 696 (2011) D’Alesio, Flore, Murgia, Pisano, Taels, in preparation

Color factors for gq → γq:

D CU C(f)

I

C(f)

Fd CInc (f) C(d) I

C(d)

Fd CInc (d) 1 2Nc − 1 4Nc 1 4Nc

−

1 4Nc 1 4Nc 1 4Nc 1 4Nc

Simple color structure, both f ⊥ g (f )

1T

and f ⊥ g (d)

1T

contribute to AN(p↑ p → γ X) HInc (f )

gq→γq = HInc (f ) g ¯ q→γ ¯ q = −1

2 HU

gq→γq = − 1

2Nc

− ˆ

u ˆ s − ˆ s ˆ u

HInc (d)

gq→γq = −HInc (d) g ¯ q→γ ¯ q = 1

2 HU

gq→γq =

1 2Nc

− ˆ

u ˆ s − ˆ s ˆ u

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SLIDE 24

Conclusions

◮ Single spin asymmetries AN for p↑ p → h(γ) X can be described within the

GPM, which includes both spin and transverse momentum effects

◮ In the CGI-GPM, effects of ISI/FSI are taken into account in the one-gluon

exchange approximation

◮ As a consequence, TMDs become process dependent ◮ In this framework, the gluon Sivers effect is given by the convolution of two

independent gluon Sivers functions with modified hard functions

◮ These two distributions can be in principle singled out by looking at

available data on AN in p↑p → π0 X, p↑p → J/ψ X, p↑p → DX, p↑p → γX at RHIC

Talk by U. D’Alesio

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