cristian pisano - institute for nuclear theory · manuale dÕuso del marchio universitË ... 0.4...

Gluon TMDs and Heavy QuarkProduction at an EIC

Cristian Pisano

INT-17-3 WorkshopHadron imaging at Jefferson Laband at a future EIC

September 25 - 29 2017Seattle (USA)

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GDR Juljan Rushaj

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Colore 01 Colore 03Colore 02

MANUALE D’USO DEL MARCHIO

UNIVERSITÀ DI PAVIA

Quark TMDs

Angeles-Martinez et al., Acta Phys, Pol. B46 (2015)

I h⊥ q1 : T -odd distribution of transversely polarized quarks inside an unp. hadron

I hq1T , h⊥ q1T : helicity flip distributions: T -even and chiral odd

I Transversity hq1 ≡ hq1T +p2T

2M2ph⊥ q

1T survives under pT integration

They are known and can all be accessed in semi-inclusive DIS (SIDIS)



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2/27

Gluon TMDs

Angeles-Martinez et al., Acta Phys, Pol. B46 (2015)Mulders, Rodrigues, PRD 63 (2001)

Meissner, Metz, Goeke, PRD 76 (2007)

I h⊥ g1 : T -even distribution of linearly polarized gluons inside an unp. hadron

I hg1T , h⊥ g1T : helicity flip distributions like hq1T , h⊥ q

1T , but T -odd, chiral even!

I hg1 ≡ hg1T +p2T

2M2ph⊥ g

1T does not survive under pT integration, unlike transversity

In contrast to quark TMDs, gluon TMDs are almost unknown



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3/27

The distribution of linearly polarized gluons

inside an unpolarized proton: h⊥ g1



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4/27

Linear polarization of gluons

Gluons inside an unpolarized hadron can be linearly polarized

It requires nonzero transverse momentum

±1

±1

∓1

±1

h⊥ g1

f g1

±1 ∓1

±1 ∓1

h⊥ g1

Interference between ±1 gluon helicity states

It does not need ISI/FSI to be nonzero, unlike the Sivers function. However it isaffected by them =⇒ process dependence



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5/27

Gluon polarization and the Higgs bosonp p → H X at the LHC

Higgs boson production happens mainly via gg → H

Pol. gluons affect the Higgs transverse spectrum at NNLO pQCD

Catani, Grazzini, NPB 845 (2011)

The nonperturbative distribution can be present at tree level and wouldcontribute to Higgs production at low qT

Boer, den Dunnen, CP, Schlegel, Vogelsang, PRL 108 (2012)



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6/27

Gluon polarization and the Higgs bosonp p → H X at the LHC

qT -distribution of the Higgs boson

1

σ

dσ

dq2T

∝ 1 + R(q2T ) R =

h⊥ g1 ⊗ h⊥ g

1

f g1 ⊗ f g1|h⊥ g

1 (x , p2T )| ≤

2M2p

p2T

f g1 (x , p2T )

Gaussian Model TMD evolution

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 2 4 6 8 10

qT

σ-1dσ/dqT2

[GeV]

[GeV-2]

lin. pol. > 0 r = 2/3

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 2 4 6 8 10

qT

σ-1dσ/dqT2

[GeV]

[GeV-2]

lin. pol. > 0 r = 1/3

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 2 4 6 8 10

qT

σ-1dσ/dqT2

[GeV]

[GeV-2]

lin. pol. > 0lin. pol. = 0

-0.2

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14

qT

R0

[GeV]

r = 2/3

-0.2

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14

qT

R0

[GeV]

r=1/3

-0.2

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14

qT

R0

[GeV]

0 5 10 15 200.00

0.01

0.02

0.03

0.04

0.05

0.06

QT [GeV]

R(QT )

mH = 126 GeV

Echevarria, Kasemets, Mulders, CP, JHEP 1507 (2015) 158

Study of H → γγ and interference with gg → γγ

Boer, den Dunnen, CP, Schlegel, PRL 111 (2013)



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7/27

C = +1 quarkonium production

qT -distribution of ηQ and χQJ (Q = c, b) in the kinematic region qT 2MQ

1

σ(ηQ)

dσ(ηQ)

dq2T

∝ f g1 ⊗ f g1 [1− R(q2T )] [pseudoscalar]

1

σ(χQ0)

dσ(χQ0)

dq2T

∝ f g1 ⊗ f g1 [1 + R(q2T )] [scalar]

1

σ(χQ2)

dσ(χQ2)

dq2T

∝ f g1 ⊗ f g1

Boer, CP, PRD 86 (2012) 094007

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.5 1 1.5 2 2.5 3

qT

σ-1dσ / d (GeV-2)

⟨ pT2 ⟩ = 1 GeV2

qT2

(GeV)

h1⊥ g = 0)(

χ0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.5 1 1.5 2 2.5 3

qT

σ-1dσ / d (GeV-2)

⟨ pT2 ⟩ = 1 GeV2

qT2

(GeV)

h1⊥ g = 0)( χ2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.5 1 1.5 2 2.5 3

qT

σ-1dσ / d (GeV-2)

⟨ pT2 ⟩ = 1 GeV2

qT2

(GeV)

h1⊥ g = 0)(

ηQ

0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1 1.2 1.4

qT

σ-1dσ / d (GeV-2)

⟨ pT2 ⟩ = 0.25 GeV2

qT2

(GeV)

h1⊥ g = 0)(

χ0

0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1 1.2 1.4

qT

σ-1dσ / d (GeV-2)

⟨ pT2 ⟩ = 0.25 GeV2

qT2

(GeV)

h1⊥ g = 0)( χ2

0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1 1.2 1.4

qT

σ-1dσ / d (GeV-2)

⟨ pT2 ⟩ = 0.25 GeV2

qT2

(GeV)

h1⊥ g = 0)(

ηQ

Proof of factorization at NLO for p p → ηQ X in the Color Singlet Model (CSM)Ma, Wang, Zhao, PRD 88 (2013), 014027; PLB 737 (2014) 103



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8/27

Heavy quark pair production at an EIC



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9/27

Heavy quark pair production in DISProposal for the EIC

Gluon TMDs probed directly in e(`) + p(P,S) → e(`′) +Q(K1) +Q(K2) +X

Boer, Mulders, CP, Zhou, JHEP 1608 (2016)

I the QQ pair is almost back to back in the plane ⊥ to q and P

I q ≡ `− `′: four-momentum of the exchanged virtual photon γ∗

q

P

K1

K2

K1⊥

K2⊥

φ1φ2

δφ

qT ≡ K1⊥ + K2⊥

K⊥ ≡ (K1⊥ −K2⊥)/2

P

S

=⇒ Correlation limit: |qT | |K⊥|, |K⊥| ≈ |K1⊥| ≈ |K2⊥|

♠q



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10/27

Heavy quark pair production in DISAngular structure of the cross section

±1

±1

∓1

±1

h⊥ g1

f g1

±1 ∓1

±1 ∓1

h⊥ g1

φT , φ⊥, φS azimuthal angles of qT ,K⊥, ST

At LO in pQCD: only γ∗g → QQ contributes

dσ(φS , φT , φ⊥) = dσU(φT , φ⊥) + dσT (φS , φT , φ⊥)

Angular structure of the unpolarized cross section for ep → e′QQX , |qT | |K⊥|

dσU

d2qT d2K⊥∝AU

0 + AU1 cosφ⊥ + AU

2 cos 2φ⊥

fg

1 (x, q2T ) +

q2T

M2p

h⊥ g1 (x, q2

T )

×BU

0 cos 2φT + BU1 cos(2φT − φ⊥) + BU

2 cos 2(φT − φ⊥) + BU3 cos(2φT − 3φ⊥) + BU

4 cos 2(φT − 2φ⊥)

The different contributions can be isolated by defining

〈W (φ⊥, φT )〉 =

∫dφ⊥dφT W (φ⊥, φT ) dσ∫

dφ⊥dφT dσ, W = cos 2φT , cos 2(φ⊥ − φT ) , ...



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11/27

h⊥ g1 in ep → e′QQX

Maximal asymmetries

Positivity bound for h⊥ g1 : |h⊥ g

1 (x , p2T )| ≤ 2M2

p

p2T

f g1 (x , p2T )

It can be used to estimate maximal values of the asymmetries

Asymmetries usually larger when Q and Q have same rapidities

Upper bounds on R ≡ |〈cos 2(φT − φ⊥)〉| and R ′ ≡ |〈cos 2φT 〉| at y = 0.01

0

0.05

0.1

0.15

0.2

0.25

0.3

5 10 15 20 25

|K⊥ |

RQ2

Q2

Q2

MQ = Mc

(GeV)

= 100 GeV2

0

0.05

0.1

0.15

0.2

0.25

0.3

5 10 15 20 25

|K⊥ |

RQ2

Q2

Q2

MQ = Mc

(GeV)

= 10 GeV2

0

0.05

0.1

0.15

0.2

0.25

0.3

5 10 15 20 25

|K⊥ |

RQ2

Q2

Q2

MQ = Mc

(GeV)

= 1 GeV2

0

0.05

0.1

0.15

0.2

0.25

0.3

5 10 15 20 25

|K⊥ |

RQ2

Q2

Q2

MQ = Mb

(GeV)

= 100 GeV2

0

0.05

0.1

0.15

0.2

0.25

0.3

5 10 15 20 25

|K⊥ |

RQ2

Q2

Q2

MQ = Mb

(GeV)

= 10 GeV2

0

0.05

0.1

0.15

0.2

0.25

0.3

5 10 15 20 25

|K⊥ |

RQ2

Q2

Q2

MQ = Mb

(GeV)

= 1 GeV2

0

0.1

0.2

0.3

0.4

0.5

5 10 15 20 25

|K⊥ |

R’Q2

Q2

Q2

MQ = Mc

(GeV)

= 100 GeV2

0

0.1

0.2

0.3

0.4

0.5

5 10 15 20 25

|K⊥ |

R’Q2

Q2

Q2

MQ = Mc

(GeV)

= 10 GeV2

0

0.1

0.2

0.3

0.4

0.5

5 10 15 20 25

|K⊥ |

R’Q2

Q2

Q2

MQ = Mc

(GeV)

= 1 GeV2

0

0.1

0.2

0.3

0.4

0.5

5 10 15 20 25

|K⊥ |

R’Q2

Q2

Q2

MQ = Mb

(GeV)

= 100 GeV2

0

0.1

0.2

0.3

0.4

0.5

5 10 15 20 25

|K⊥ |

R’Q2

Q2

Q2

MQ = Mb

(GeV)

= 10 GeV2

0

0.1

0.2

0.3

0.4

0.5

5 10 15 20 25

|K⊥ |

R’Q2

Q2

Q2

MQ = Mb

(GeV)

= 1 GeV2

CP, Boer, Brodsky, Buffing, Mulders, JHEP 1310 (2013)Boer, Brodsky, Mulders, CP, PRL 106 (2011)



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12/27

Spin asymmetries in ep↑ → e′QQX

Angular structure of the single polarized cross section for ep↑ → e′QQX , |qT | |K⊥|

dσT∝ sin(φS − φT )

[AT

0 +AT1 cosφ⊥ + AT

2 cos 2φ⊥

]f⊥ g

1T+ cos(φS − φT )

[BT

0 sin 2φT

+ BT1 sin(2φT − φ⊥) + BT

2 sin 2(φT − φ⊥) + BT3 sin(2φT − 3φ⊥) + BT

4 sin(2φT − 4φ⊥)

]h⊥ g1T

+

[B′ T

0 sin(φS + φT ) + B′ T1 sin(φS + φT − φ⊥) + B′ T

2 sin(φS + φT − 2φ⊥)

+B′ T3 sin(φS + φT − 3φ⊥) + B′ T

4 sin(φS + φT − 4φ⊥)

]hg1T

The φS dependent terms can be singled out by means of azimuthal moments AWN

AW (φS ,φT )N ≡ 2

∫dφT dφ⊥W (φS , φT ) dσT (φS , φT , φ⊥)∫

dφT dφ⊥ dσU(φT , φ⊥)

Asin(φS−φT )N ∝

f ⊥ g1T

f g1A

sin(φS+φT )N ∝ hg

1

f g1A

sin(φS−3φT )N ∝

h⊥ g1T

f g1

Same modulations as in SIDIS for quark TMDs (φT → φh)

Boer, Mulders, PRD D57 (1998)



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13/27

Spin asymmetries in ep↑ → e′QQXUpper bounds

Maximal values for |AWN |, W = sin(φS + φT ), sin(φS − 3φT ) (|K⊥| = 1 GeV)

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1

y

ANW

max

MQ = Mc

Q2 = 100 GeV2

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1

y

ANW

max

MQ = Mc

Q2 = 10 GeV2

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1

y

ANW

max

MQ = Mc

Q2 = 1 GeV2

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1

y

ANW

max

MQ = Mb

Q2 = 100 GeV2

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1

y

ANW

max

MQ = Mb

Q2 = 10 GeV2

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1

y

ANW

max

MQ = Mb

Q2 = 1 GeV2



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14/27

Asymmetries in ep↑ → e′jet jetXUpper bounds

Contribution to the denominator also from γ∗q → gq, negligible at small-x

Asymmetries much smaller than in cc case for Q2 ≤ 10 GeV2

Upper bounds for AWN for K⊥ ≥ 4 GeV

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 0.2 0.4 0.6 0.8 1

y

ANW Q2

Q2

Q2

max = 100 GeV2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 0.2 0.4 0.6 0.8 1

y

ANW Q2

Q2

Q2

max

= 50 GeV2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 0.2 0.4 0.6 0.8 1

y

ANW Q2

Q2

Q2

max

= 10 GeV2



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15/27

Process dependence of gluon TMDs



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16/27

The gluon Sivers functionsSign change test

Related Processes

ep↑ → e′QQX , ep↑ → e′ jet jetX 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 ⊥ q1T between SIDIS and DY (true also for hg1 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)

We expect to measure the same h⊥ g1 and f g1 in both processes

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



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17/27

The gluon Sivers functionsThe dipole GSF

Complementary Processes

ep↑ → e′QQX probes a GSF with [++] gauge links (WW)

p↑p → γ jetX (gq → γq) probes a gluon TMD with : [+−] links (DP)

= −

At small-x the WW Sivers function appears to be suppressed by a factor of xcompared to the unpolarized gluon function, unlike the dipole one

The DP gluon Sivers function at small-x is the spin dependent odderon (singlespin 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)



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18/27

J/ψ-pair production at the LHCLansberg, CP, Scarpa, Schlegel, in progress



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19/27

J/ψ-pair production at the LHC

J/ψ’s are relatively easy to detect. Accessible at the LHC: already studied byLHCb, CMS & ATLAS LHCb PLB 707 (2012)

CMS JHEP 1409 (2014)ATLAS EPJC 77 (2017)

gg fusion dominant, negligible qq contributions even at AFTER@LHC energiesLansberg, Shao, NPB 900 (2015)

µ ν

P2

PQ,2x1P1 + k1T

x2P2 + k2T

P1Φµνg (x1, k1T , ζ1, µ)

Φρσg (x2, k2T , ζ2, µ)

ρ σ

PQ,1

No final state gluon needed for the Born contribution in the Color Singlet Model.Pure colorless final state, hence simple color structure because one has only ISI

Lansberg, Shao, PRL 111 (2013)



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20/27

J/ψ-pair production at the LHCColor Singlet vs Color Octet

Negligible Color Octet contributions, in particular at low PΨΨT

[Black/dashed curves vs blue ones]Lansberg, Shao PLB 751 (2015)



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21/27

J/ψ-pair production at the LHCSingle vs double parton scattering

At low PΨΨT , small double parton scattering (DPS) contributions, otherwise

required by the CMS, ATLAS data at large rapidity separations ∆y of the J/ψ’sLansberg, Shao, PLB 751 (2015)



GDR Juljan Rushaj

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22/27

J/ψ-pair production at the LHCCalculation of the cross section

At LO pQCD in the Color Singlet Model, one needs to consider 36 diagrams

Qiao, Sun, Sun, JPG 37 (2010)



GDR Juljan Rushaj

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23/27

J/ψ-pair productionStructure of the cross section

dσ

dQdY d2qTdΩ≈ A f g1 ⊗ f g1 + B f g1 ⊗ h⊥g

1 cos(2φCS) + C h⊥g1 ⊗ h⊥g

1 cos(4φCS)

I valid up to corrections O (qT /Q)

I Y : rapidity of the J/ψ-pair, along the beam in the hadronic c.m. frame

I dΩ = d cos θCS dφCS : solid angle for J/ψ-pair in the Collins-Soper frame

Analysis similar to the one for pp → γγX , pp → Jψ γ(∗) X , pp → H jetXQiu, Schlegel, Vogelsang, PRL 107 (2011)

den Dunnen, Lansberg, CP, Schlegel, PRL 112 (2014)Lansberg, CP, Schlegel, NPB 920 (2017)

Boer, CP, PRD 91 (2015)

The three contributions can be disentangled by defining the transverse moments

〈cos nφCS 〉 ≡

∫ 2π0 dφCS cos(n φCS ) dσ

dQdYd2qT dΩ∫ 2π0 dφCS

dσdQdYd2qT dΩ

(n = 2, 4)

∫dφCS dσ =⇒ f g1 ⊗ f g1

〈cos 2φCS〉 =⇒ f g1 ⊗ h⊥g1

〈cos 4φCS〉 =⇒ h⊥g1 ⊗ h⊥g

1



GDR Juljan Rushaj

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24/27

J/ψ-pair productionExtraction of f g1 at

√s = 13 TeV

We consider qT = PΨΨT ≤ MΨΨ/2 in order to have two different scales

dσ

/dP

ψψ

T /

∫ 0M-ψ

ψ /2

dσ

/dP

ψψ

T

[G

eV

-1]

PψψT [GeV]

Gaussian f1g model, <kT

2> fitted over [0 ; M

-ψψ/2]

LHCb 13 TeV data

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 2 4 6 8 10 12 14

<kT2> = 4.90 ± 0.78 GeV

2

M-

ψψ = 8 GeV

LHCb Coll., JHEP 06 (2017)

Gaussian model: f g1 (x , k2T ) =

f g1 (x)

π〈k2T 〉

exp

(−

k2T

〈k2T 〉

)



GDR Juljan Rushaj

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25/27

J/ψ-pair production〈cos nφCS 〉 at

√s = 13 TeV

At ∆y ≈ 0 the modulation C of cos4φ saturates its upper bound |C | = A

The cos 2φ modulation can fix the sign of h⊥ g1

MΨΨ = 8, 12, 21 GeV relevant for LHCb, CMS, ATLAS

〈cos 2φCS 〉 (%) 〈cos 4φCS 〉 (%)

-8

-7

-6

-5

-4

-3

-2

-1

0

0 2 4 6 8 10

S(2

;co

s θ

CS)

[in

%]

PψψT [GeV]

<kT2> = 4.90 GeV

2

|cos θCS| < 0.25

Mψψ = 8 GeV

Mψψ = 12 GeV

Mψψ = 21 GeVModel 1Model 2

0

10

20

30

40

50

0 2 4 6 8 10S

(4;c

os θ

CS)

[in

/%

]PψψT

[GeV]

<kT2> = 4.90 GeV

2

|cos θCS| < 0.25

Mψψ = 8 GeV

Mψψ = 12 GeV

Mψψ = 21 GeV

Model 1Model 2

Models for h⊥ g1

(1) h⊥g1 (x, k2

T ) =2M2

k2T

fg

1 (x, k2T ) ; (2) h

⊥g1 (x, k2

T ) =M f

g1 (x)

π〈k2T〉3/2

√2e(1− r)

rexp

(−

1

r

k2T

〈k2T〉

)r = 2/3



GDR Juljan Rushaj

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26/27

Conclusions

I Azimuthal asymmetries in heavy quark pair and dijet production in DIScould probe WW-type gluon TMDs (similar to SIDIS for quark TMDs)

I Asymmetries maximally allowed by positivity bounds of gluon TMDs can besizeable in specific kinematic region

I Different behavior of WW and dipole gluon TMDs accessible at RHIC,AFTER@LHC and at EIC, overlap of both spin and small-x programs

I First extraction of unpolarized gluon TMD from LHC data on di-J/Ψ



GDR Juljan Rushaj

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