Suppression of maximal linear gluon polarization in angular - - PowerPoint PPT Presentation

Suppression of maximal linear gluon polarization in angular asymmetries

Daniël Boer REF 2017, Madrid, November 14, 2017

Sudakov suppression of Sivers and Collins effects Gluon TMDs & processes to probe them Linearly polarized gluons in unpolarized protons 𝛿*-jet production in pp and pA collisions at small x Conclusions about the pattern of suppression

Outline

TMD evolution

• f azimuthal asymmetries

TMD = transverse momentum dependent parton distribution

Transverse Momentum Dependence

Including transverse momentum of quarks involves much more than replacing f1(x) → f1(x,kT2) in collinear factorization expressions One deals with less inclusive processes and with TMD factorization & TMD evolution

TMD = transverse momentum dependent parton distribution

Transverse Momentum Dependence

The transverse momentum dependence can be correlated with the spin, e.g.

P

T

k

T

k

sT

q

sT

q

= /

• D. Sivers (’90):

kT × ST

Including transverse momentum of quarks involves much more than replacing f1(x) → f1(x,kT2) in collinear factorization expressions One deals with less inclusive processes and with TMD factorization & TMD evolution

TMD = transverse momentum dependent parton distribution

Transverse Momentum Dependence

The transverse momentum dependence can be correlated with the spin, e.g.

P

T

k

T

k

sT

q

sT

q

= /

• D. Sivers (’90):

kT × ST

Including transverse momentum of quarks involves much more than replacing f1(x) → f1(x,kT2) in collinear factorization expressions One deals with less inclusive processes and with TMD factorization & TMD evolution

• T

sT s k k

π

T

π

H⊥

1 =

T

≠

Similar effects can arise in the final state, such as the Collins effect, which is described by a TMD fragmentation function:

• J. Collins (’93):

Studies of TMD evolution of azimuthal asymmetries

TMD evolution studies initially focused on Collins and Sivers effect asymmetries

[D.B., 2001, 2009, 2013; Idilbi, Ji, Ma & Yuan, 2004; Aybat & Rogers, 2011; Aybat, Collins, Qiu, Rogers, 2012; Aybat, Prokudin & Rogers, 2012; Anselmino, Boglione, Melis, 2012; Godbole, Misra, Mukherjee, Rawoot, 2013; Sun & Yuan, 2013; …]

Generic features:

• decrease and broadening of

TMDs with increasing energy

• Gaussian develops a power-

law tail

Aybat & Rogers, 2011

Azimuthal asymmetries will decrease with Q Sivers function

TMD factorization dσ dΩd4q = Z d2b e−ib·qT ˜ W(b, Q; x, y, z) + O

• Q2

T /Q2

˜ W(b, Q; x, y, z) = X

a

˜ f a

1 (x, b2; ζF , µ) ˜

Da

1(z, b2; ζD, µ)H (y, Q; µ)

TMD factorization dσ dΩd4q = Z d2b e−ib·qT ˜ W(b, Q; x, y, z) + O

• Q2

T /Q2

˜ W(b, Q; x, y, z) = X

a

˜ f a

1 (x, b2; ζF , µ) ˜

Da

1(z, b2; ζD, µ)H (y, Q; µ)

µ = Q

H (Q; αs(Q)) ∝ e2

a

• 1 + αs(Q2)F1 + O(α2

s)

• This choice avoids large logarithms in H, but now they will appear in the TMDs

Take

TMD factorization dσ dΩd4q = Z d2b e−ib·qT ˜ W(b, Q; x, y, z) + O

• Q2

T /Q2

˜ W(b, Q; x, y, z) = X

a

˜ f a

1 (x, b2; ζF , µ) ˜

Da

1(z, b2; ζD, µ)H (y, Q; µ)

µ = Q

H (Q; αs(Q)) ∝ e2

a

• 1 + αs(Q2)F1 + O(α2

s)

• This choice avoids large logarithms in H, but now they will appear in the TMDs

Take Use renormalization group equations to evolve the TMDs to the scale: where S is the so-called Sudakov factor

µb ≈ 1/b

˜ f a

1 (x, b2; Q2, Q) ˜

Db

1(z, b2; Q2, Q) = e−S(b,Q) ˜

f a

1 (x, b2; µ2 b, µb) ˜

Db

1(z, b2; µ2 b, µb)

[Collins & Soper, 1981; Collins, Soper, Sterman, 1985; Ji, Ma, Yuan, 2004/5; Collins, 2011; Echevarria, Idilbi & Scimemi 2012/14; …]

Sp(b, Q) = CF π Z Q2

µ2

b

dµ2 µ2 αs(µ) ✓ ln Q2 µ2 − 3 2 ◆ + O(α2

s)

Sudakov factors

At leading order in αs the perturbative expression for S is:

S(b, Q) = − ln ✓Q2 µ2

b

◆ ˜ K(b, µb) − Z Q2

µ2

b

dµ2 µ2 ⇥ γF (g(µ); 1) − 1 2 ln ✓Q2 µ2 ◆ γK(g(µ)) ⇤

Sp(b, Q) = CF π Z Q2

µ2

b

dµ2 µ2 αs(µ) ✓ ln Q2 µ2 − 3 2 ◆ + O(α2

s)

Sudakov factors

At leading order in αs the perturbative expression for S is:

S(b, Q) = − ln ✓Q2 µ2

b

◆ ˜ K(b, µb) − Z Q2

µ2

b

dµ2 µ2 ⇥ γF (g(µ); 1) − 1 2 ln ✓Q2 µ2 ◆ γK(g(µ)) ⇤

It can be used whenever the restriction b2 ≪ 1/Λ2 is justified (e.g. at very large Q2) If also larger b contributions are important, e.g. at moderate Q and small QT = |qT|, then one needs to include a nonperturbative Sudakov factor

˜ W(b) ≡ ˜ W(b∗) e−SNP (b) b∗ = b/ p 1 + b2/b2

max ≤ bmax

such that W(b*) can be calculated within perturbation theory In general SNP is Q dependent and often taken to be Gaussian to be fitted to data

TMD evolution of the Sivers asymmetry

20 40 60 80 100 0.5 1 1.5 Q 1Q0.68 AQT,max 1 2 3 4 5 0.5 1 1.5 QT AQT

90 60 30 10 3.33

Q GeV

[D.B., NPB 2013]

The peak of the Sivers asymmetry decreases as 1/Q0.7±0.1

SAR

NP (b, Q, Q0) =

 0.184 ln Q 2Q0 + 0.332

• b2

[Aybat & Rogers, 2011]

TMD evolution of the Sivers asymmetry

20 40 60 80 100 0.5 1 1.5 Q 1Q0.68 AQT,max 1 2 3 4 5 0.5 1 1.5 QT AQT

90 60 30 10 3.33

Q GeV

[D.B., NPB 2013]

The peak of the Sivers asymmetry decreases as 1/Q0.7±0.1

SAR

NP (b, Q, Q0) =

 0.184 ln Q 2Q0 + 0.332

• b2

[Aybat & Rogers, 2011]

SLY

NP (b, Q, Q0) =

 0.58 ln Q 2Q0 + 0.11

• b2

[Ladinsky & Yuan, 1994]

Very similar to the fall-off with Q, obtained before with CS81 factorization and LY The power of the fall-off is a robust feature

[D.B., NPB 2001]

At low Q2 (up to ~20 GeV2), the Q2 evolution is dominated by SNP

[Anselmino, Boglione, Melis, PRD 86 (2012) 014028]

TMD evolution of the Sivers asymmetry

Precise low Q2 data can help to determine the form and size of SNP Uncertainty in SNP determines the ±0.1 in 1/Q0.7±0.1

Double Collins effect gives rise to an azimuthal asymmetry cos 2φ in e+e- → h1 h2 X

DB, Jakob Mulders, NPB 504 (1997) 345

Double Collins Effect

Clearly observed in experiment by BELLE (Seidl et al., PRL 2006; PRD 2008), BaBar (I. Garzia at Transversity 2011 & Lees et al., PRD 2014) and BESIII (PRL 2016)

dσ(e+e− → h1h2X) dz1dz2dΩd2qT ∝ {1 + cos 2φ1A(qT )}

Double Collins effect gives rise to an azimuthal asymmetry cos 2φ in e+e- → h1 h2 X

DB, Jakob Mulders, NPB 504 (1997) 345

Double Collins Effect

Clearly observed in experiment by BELLE (Seidl et al., PRL 2006; PRD 2008), BaBar (I. Garzia at Transversity 2011 & Lees et al., PRD 2014) and BESIII (PRL 2016)

dσ(e+e− → h1h2X) dz1dz2dΩd2qT ∝ {1 + cos 2φ1A(qT )}

DB, NPB 603 (2001) 195 & 806 (2009) 23 & QCD evolution 2013 proceedings

Considerable Sudakov suppression ~1/Q (effectively twist-3)

20 40 60 80 100 0.5 1 1.5 2. 2.5 3. Q 1Q1.1 AQT,max

1 2 3 4 5 0.5 1 1.5 2 2.5 3 QT AQT

90 60 30 10 3.33

Q GeV

Does it work? BESIII (left) and BaBar data shown as function of Ph⊥ not QT

TMD evolution of the double Collins asymmetry

√s = 3.65 GeV √s = 10.58 GeV

Does it work? BESIII (left) and BaBar data shown as function of Ph⊥ not QT

TMD evolution of the double Collins asymmetry

Rough estimate: the peak of 2.5% at BaBar would be increased by factor (10.58/3.65)1.1 = 3.2, giving 8% at BESIII. The right ball-park…

√s = 3.65 GeV √s = 10.58 GeV

Does it work? BESIII (left) and BaBar data shown as function of Ph⊥ not QT

TMD evolution of the double Collins asymmetry

Rough estimate: the peak of 2.5% at BaBar would be increased by factor (10.58/3.65)1.1 = 3.2, giving 8% at BESIII. The right ball-park…

√s = 3.65 GeV √s = 10.58 GeV

)

S

φ −

h

φ sin ( UT

A

(GeV)

h

P

0.2 0.4 0.6 0.8 1 1.2 0.05 0.1 0.15

HERMES COMPASS

TMD evolution

Evolution from HERMES (<Q2> ~ 2.4 GeV2) to COMPASS (<Q2> ~ 3.8 GeV2) seems to work well, but very small energy range and can be quite SNP dependent

Aybat, Prokudin & Rogers, PRL 2012

Gluons TMDs

Gluons TMDs

The gluon correlator: For unpolarized protons:

Γµν[U,U0]

g

(x, kT ) ⌘ F.T.hP|Trc h F +ν(0) U[0,ξ] F +µ(ξ) U0

[ξ,0]

i |Pi

Γµν

U (x, kT ) = 1

2x ⇢ − gµν

T f g 1 (x, k2 T ) +

✓kµ

T kν T

M 2

p

+ gµν

T

k2

T

2M 2

p

◆ h⊥ g

1

(x, k2

T )

Gluons TMDs

unpolarized gluon TMD The gluon correlator: For unpolarized protons:

Γµν[U,U0]

g

(x, kT ) ⌘ F.T.hP|Trc h F +ν(0) U[0,ξ] F +µ(ξ) U0

[ξ,0]

i |Pi

Γµν

U (x, kT ) = 1

2x ⇢ − gµν

T f g 1 (x, k2 T ) +

✓kµ

T kν T

M 2

p

+ gµν

T

k2

T

2M 2

p

◆ h⊥ g

1

(x, k2

T )

Gluons TMDs

unpolarized gluon TMD The gluon correlator: For unpolarized protons:

Γµν[U,U0]

g

(x, kT ) ⌘ F.T.hP|Trc h F +ν(0) U[0,ξ] F +µ(ξ) U0

[ξ,0]

i |Pi

linearly polarized gluon TMD Gluons inside unpolarized protons can be polarized!

[Mulders, Rodrigues, 2001]

Γµν

U (x, kT ) = 1

2x ⇢ − gµν

T f g 1 (x, k2 T ) +

✓kµ

T kν T

M 2

p

+ gµν

T

k2

T

2M 2

p

◆ h⊥ g

1

(x, k2

T )

Gluons TMDs

unpolarized gluon TMD The gluon correlator: For unpolarized protons:

Γµν[U,U0]

g

(x, kT ) ⌘ F.T.hP|Trc h F +ν(0) U[0,ξ] F +µ(ξ) U0

[ξ,0]

i |Pi

linearly polarized gluon TMD Gluons inside unpolarized protons can be polarized!

[Mulders, Rodrigues, 2001]

an interference between ±1 helicity gluon states

±1 ∓1

h⊥ g

1

f1 and h1⊥g (both kT-even and T

• even) are process

dependent through the two gauge links 𝒱 and 𝒱’

Γµν

U (x, kT ) = 1

2x ⇢ − gµν

T f g 1 (x, k2 T ) +

✓kµ

T kν T

M 2

p

+ gµν

T

k2

T

2M 2

p

◆ h⊥ g

1

(x, k2

T )

Sensitive processes

±1 ±1 ∓1 ∓1

h⊥ g

1

h⊥ g

1

±1 ±1 ±1

fg

1

±1

fg

1

Linearly polarized gluons can be probed in:

• pp→𝛿𝛿X [Nadolsky, Balazs, Berger, C.-P

. Yuan, 2007; Qiu, Schlegel, Vogelsang, 2011]

• pp→HX [Catani, Grazzini, 2010; Sun, Xiao,

Yuan, 2011; D.B., Den Dunnen, Pisano, Schlegel, Vogelsang, 2012]

• pp→[QQ]X with JPC=0±+ [D.B., Pisano, 2012]
• pp→ J/ψ 𝛿 X and Υ 𝛿 X [Den Dunnen, Lansberg, Pisano, Schlegel, 2014]
• pp→ J/ψ J/ψ X [Lansberg, Pisano, Scarpa, Schlegel, 2017]
• pp→ (π jet) X [D’Alesio, Murgia, Pisano, 2011]
• pp→ H jet X [D.B., Pisano, 2015]
• ep→e’ Q Q X and ep→e’ jet jet X [D.B., Brodsky, Mulders, Pisano, 2010]

_ _

RHIC RHIC EIC LHC LHC LHC LHC

Higgs production is angular independent, but generally

• ne needs angular asymmetries
• f almost back-to-back pairs

LHC

Insensitive processes

Linearly polarized gluons can also not be accessed (safely) in:

• pp→ 𝛿 jet X [D.B, Mulders, Pisano, 2008]
• pp→ J/ψ X or Υ X [D.B., Pisano, 2012]
• pp→ Q Q X [Akcakaya, Schäfer, Zhou, 2013]
• pp→ jet jet X
• pp→ 𝛿* X

_

Power suppressed Landau-Yang theorem Problem with TMD factorization idem Landau-Yang theorem

Insensitive processes

Linearly polarized gluons can also not be accessed (safely) in:

• pp→ 𝛿 jet X [D.B, Mulders, Pisano, 2008]
• pp→ J/ψ X or Υ X [D.B., Pisano, 2012]
• pp→ Q Q X [Akcakaya, Schäfer, Zhou, 2013]
• pp→ jet jet X
• pp→ 𝛿* X

_

Power suppressed Landau-Yang theorem Problem with TMD factorization idem Landau-Yang theorem

Generally when the color flow is in too many directions: factorization breaking

[Collins & J. Qiu '07; Collins '07; Rogers & Mulders '10]

Such processes may become effectively TMD factorizing at small x (small-x factorization or hybrid factorization)

[Mueller, 1990 & 1994; Kovchegov & Mueller, 1998; Mueller, Xiao, Yuan, 2013]

Insensitive processes

Linearly polarized gluons can also not be accessed (safely) in:

• pp→ 𝛿 jet X [D.B, Mulders, Pisano, 2008]
• pp→ J/ψ X or Υ X [D.B., Pisano, 2012]
• pp→ Q Q X [Akcakaya, Schäfer, Zhou, 2013]
• pp→ jet jet X
• pp→ 𝛿* X

_

Power suppressed Landau-Yang theorem Problem with TMD factorization idem Landau-Yang theorem

Generally when the color flow is in too many directions: factorization breaking

[Collins & J. Qiu '07; Collins '07; Rogers & Mulders '10]

Such processes may become effectively TMD factorizing at small x (small-x factorization or hybrid factorization)

[Mueller, 1990 & 1994; Kovchegov & Mueller, 1998; Mueller, Xiao, Yuan, 2013]

A process where the jury is still out: pp→ 𝛿* jet X It may be factorizing, at least at small x

[D.B., Mulders, Zhou & Zhou, 2017]

For most processes of interest there are 2 relevant unpolarized gluon distributions

Dominguez, Marquet, Xiao, Yuan, 2011

Process dependence

[+,+] [+,-] For unpolarized gluons [+,+] = [-,-] and [+,-] = [-,+]

For most processes of interest there are 2 relevant unpolarized gluon distributions

Dominguez, Marquet, Xiao, Yuan, 2011

Process dependence

[+,+] [+,-] At small x the two correspond to the Weizsäcker-Williams (WW) and dipole (DP) distributions, which are generally different in magnitude and width: WW DP For unpolarized gluons [+,+] = [-,-] and [+,-] = [-,+]

For most processes of interest there are 2 relevant unpolarized gluon distributions

Dominguez, Marquet, Xiao, Yuan, 2011

Process dependence

Different processes probe one or the other or a mixture, so this can be tested [+,+] [+,-] At small x the two correspond to the Weizsäcker-Williams (WW) and dipole (DP) distributions, which are generally different in magnitude and width: WW DP For unpolarized gluons [+,+] = [-,-] and [+,-] = [-,+]

Buffing, Mukherjee, Mulders, 2013

Process dependence

= = Also for the linear gluon polarization: [+,+] = [−,−] and [+,−] = [−,+]

Buffing, Mukherjee, Mulders, 2013

Process dependence

= = Also for the linear gluon polarization: [+,+] = [−,−] and [+,−] = [−,+] Γ[+,−] ij(x, kT )

x→0

− → ki

T kj T

2πL Γ[⇤]

0 (kT )

The small-x limit of the DP correlator in the TMD formalism:

D.B., Cotogno, van Daal, Mulders, Signori & Ya-Jin Zhou, JHEP 2016

U [⇤] = U [+]

[0,y]U [−] [y,0]

Probes of linear gluon polarization

pp → γ γ X pA → γ⇤ jet X e p → e0 Q Q X pp → ηc,b X pp → J/ψ γ X e p → e0 j1 j2 X pp → H X pp → Υ γ X h? g [+,+]

1

(WW) √ × √ √ √ h? g [+,]

1

(DP) × √ × × ×

Selection of processes that probe the WW or DP linearly polarized gluon TMD: h1⊥g is power suppressed in pp→ 𝛿 jet X [D.B, Mulders, Pisano, 2008] It is not power suppressed in pp→ 𝛿* jet X if Q2 ~ P⊥,jet2 [D.B, Mulders, Zhou & Zhou, 2017] Consider Q2 ~ P⊥,jet2 to avoid a three-scale problem

Probes of linear gluon polarization

pp → γ γ X pA → γ⇤ jet X e p → e0 Q Q X pp → ηc,b X pp → J/ψ γ X e p → e0 j1 j2 X pp → H X pp → Υ γ X h? g [+,+]

1

(WW) √ × √ √ √ h? g [+,]

1

(DP) × √ × × ×

Selection of processes that probe the WW or DP linearly polarized gluon TMD: h1⊥g is power suppressed in pp→ 𝛿 jet X [D.B, Mulders, Pisano, 2008] It is not power suppressed in pp→ 𝛿* jet X if Q2 ~ P⊥,jet2 [D.B, Mulders, Zhou & Zhou, 2017] Consider Q2 ~ P⊥,jet2 to avoid a three-scale problem pp→ 𝛿* jet X offers a unique opportunity to study the Wilson loop matrix element for unpolarized protons, if factorization is okay at small x

Magnitude of linear gluon polarization effects

Linear gluon polarization at small x

There is no theoretical reason why h1⊥g should be small, especially at small x

Linear gluon polarization at small x

There is no theoretical reason why h1⊥g should be small, especially at small x The perturbative tail of h1⊥g has the same 1/x growth as f1

˜ h⊥g

1 (x, b2; µ2 b, µb) = αs(µb)CA

2π Z 1

x

dˆ x ˆ x ✓ ˆ x x − 1 ◆ fg/P (ˆ x; µb) + O(α2

s)

Linear gluon polarization at small x

There is no theoretical reason why h1⊥g should be small, especially at small x The perturbative tail of h1⊥g has the same 1/x growth as f1

˜ h⊥g

1 (x, b2; µ2 b, µb) = αs(µb)CA

2π Z 1

x

dˆ x ˆ x ✓ ˆ x x − 1 ◆ fg/P (ˆ x; µb) + O(α2

s)

Γ[+,−] ij(x, kT )

x→0

− → ki

T kj T

2πL Γ[⇤]

0 (kT )

Γij

U (x, kT ) = x

2 " − gij

T f1(x, k2 T ) + kij T

M 2 h⊥

1 (x, k2 T )

#

x→0

− → ki

T kj T

2M 2 e(k2

T )

lim

x→0 xf1(x, k2 T ) = k2 T

2M 2 lim

x→0 xh⊥ 1 (x, k2 T ) = k2 T

2M 2 e(k2

T )

In the TMD formalism the DP h1⊥g becomes maximal when x → 0 The small-x limit of the DP correlator in the TMD formalism:

D.B., Cotogno, van Daal, Mulders, Signori & Ya-Jin Zhou, JHEP 2016

U [⇤] = U [+]

[0,y]U [−] [y,0]

MV model calculations show that the CGC gluons are linearly polarized

h⊥g

1,W W ⌧ f ⊥g 1,W W

for k⊥ ⌧ Qs, h⊥g

1,W W = 2f ⊥g 1,W W

for k⊥ Qs

Metz, Zhou '11

Polarization of the CGC

MV model calculations show that the CGC gluons are linearly polarized

h⊥g

1,W W ⌧ f ⊥g 1,W W

for k⊥ ⌧ Qs, h⊥g

1,W W = 2f ⊥g 1,W W

for k⊥ Qs

Metz, Zhou '11

Polarization of the CGC

h⊥ g

1 W W

f1 W W ∝ 1 ln Q2

s/k2 ⊥

The WW h1⊥g is (moderately) suppressed for small transverse momenta:

MV model calculations show that the CGC gluons are linearly polarized

h⊥g

1,W W ⌧ f ⊥g 1,W W

for k⊥ ⌧ Qs, h⊥g

1,W W = 2f ⊥g 1,W W

for k⊥ Qs

Metz, Zhou '11

Polarization of the CGC

h⊥ g

1 W W

f1 W W ∝ 1 ln Q2

s/k2 ⊥

The WW h1⊥g is (moderately) suppressed for small transverse momenta:

The CGC can be 100% polarized, but its observable effects depend on the process and (as will be discussed) on the energy scale in the process

MV model calculations show that the CGC gluons are linearly polarized

h⊥g

1,W W ⌧ f ⊥g 1,W W

for k⊥ ⌧ Qs, h⊥g

1,W W = 2f ⊥g 1,W W

for k⊥ Qs

Metz, Zhou '11

Polarization of the CGC

h⊥ g

1 W W

f1 W W ∝ 1 ln Q2

s/k2 ⊥

The WW h1⊥g is (moderately) suppressed for small transverse momenta:

The CGC can be 100% polarized, but its observable effects depend on the process and (as will be discussed) on the energy scale in the process The “kT-factorization" approach (CCFM) yields maximum polarization too:

Catani, Ciafaloni, Hautmann, 1991

Γµν

g (x, pT )max pol = pµ T pν T

p2

T

x f g

1

𝛿*-jet production

Linear gluon polarization not power suppressed in pp→𝛿* jet X for Q2 ~ P⊥,jet2 leading to a cos(2φ) asymmetry, where φ=φT-φ⊥

Azimuthal asymmetry in 𝛿*-jet production

This process probes the [+,−] link structure At high gluon density (large A and/or small x) the DP linear gluon polarization is expected to become maximal, as was first shown in the MV model for the CGC

[Metz & Jian Zhou, 2011]

Linear gluon polarization not power suppressed in pp→𝛿* jet X for Q2 ~ P⊥,jet2 leading to a cos(2φ) asymmetry, where φ=φT-φ⊥

Azimuthal asymmetry in 𝛿*-jet production

This process probes the [+,−] link structure At high gluon density (large A and/or small x) the DP linear gluon polarization is expected to become maximal, as was first shown in the MV model for the CGC

[Metz & Jian Zhou, 2011]

In a hybrid factorization approach (assumed to be applicable at small x) at LO:

Linear gluon polarization not power suppressed in pp→𝛿* jet X for Q2 ~ P⊥,jet2 leading to a cos(2φ) asymmetry, where φ=φT-φ⊥

Azimuthal asymmetry in 𝛿*-jet production

This process probes the [+,−] link structure At high gluon density (large A and/or small x) the DP linear gluon polarization is expected to become maximal, as was first shown in the MV model for the CGC

[Metz & Jian Zhou, 2011]

In a hybrid factorization approach (assumed to be applicable at small x) at LO: This leads to a sizeable asymmetry: Hcos(2φ)

Born

HBorn ≈ −0.1 for z = 0.5 & Q = P⊥ = 6 GeV

[D.B, Mulders, Zhou & Zhou, 2017]

Azimuthal asymmetry in 𝛿*-jet production

Including resummation of large logarithmic corrections:

Azimuthal asymmetry in 𝛿*-jet production

Including resummation of large logarithmic corrections: Start evolution from Qs where we use the MV model expressions:

[D.B, Mulders, Zhou & Zhou, 2017]

Azimuthal asymmetry in 𝛿*-jet production

Including the resummation in the TMDs: Substantial Sudakov suppression Fall-off from μ=6 to 90 GeV is approximately 1/μ0.95

[D.B, Mulders, Zhou & Zhou, 2017]

Sudakov suppression of linear gluon polarization

D.B., Mulders, Zhou & Zhou, 2017

Despite the maximal DP linear gluon polarization at small x, there is Sudakov suppression of the cos(2φ) asymmetry in pA→𝛿* jet X: ~5% asymmetry at RHIC ≈ −0.1·0.4 It becomes effectively power suppressed as Q~P⊥ increases from 6 to 90 GeV

10 20 30 40 50 60 0.00 0.05 0.10 0.15 0.20 QT @GeVD RHQTL

126 63 25 9.9 3.4

Q @GeVD

mχc0

mχb0

TMD evolution in pp → scalar X

D.B. & den Dunnen, 2014

xA = xB = Q/(8TeV)

MSTW08 LO gluon distribution

Q−0.85

Fall-off at QT=0:

wH = (pT · kT )2 − 1

2p2 T k2 T

2M 4

R(QT ) ≡ C[wH h⊥g

1

h⊥g

1 ]

C[f g

1 f g 1 ]

The relative effect of linearly polarized gluons:

[Echevarria, Kasemets, Mulders, Pisano, 2015]

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 5 10 15 20 R qT [GeV] bc = 3 GeV−1 λQ = 0.01 λf = 0.01 λh = 1 xA = xB = Q/√s 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 5 10 15 20 R qT [GeV] bc = 3 GeV−1 λQ = 0.01 λf = 0.01 λh = 1 xA = xB = Q/√s 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 5 10 15 20 R qT [GeV] bc = 3 GeV−1 λQ = 0.01 λf = 0.01 λh = 1 xA = xB = Q/√s 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 5 10 15 20 R qT [GeV] bc = 3 GeV−1 λQ = 0.01 λf = 0.01 λh = 1 xA = xB = Q/√s 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 5 10 15 20 R qT [GeV] bc = 3 GeV−1 λQ = 0.01 λf = 0.01 λh = 1 xA = xB = Q/√s 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 5 10 15 20 R qT [GeV] bc = 3 GeV−1 λQ = 0.01 λf = 0.01 λh = 1 xA = xB = Q/√s 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 5 10 15 20 R qT [GeV] bc = 3 GeV−1 λQ = 0.01 λf = 0.01 λh = 1 xA = xB = Q/√s 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 5 10 15 20 R qT [GeV] bc = 3 GeV−1 λQ = 0.01 λf = 0.01 λh = 1 xA = xB = Q/√s Q = 9.39 GeV Q = 126 GeV

mηb

NP parameters [Echevarria, Idilbi, Schäfer, Scimemi, 2013; D'Alesio, Echevarria, Melis, Scimemi, 2014]

Higher order resummation

TMD approach including higher order resummation and quark contributions

Fall-off at QT=0 varies from Q-0.84 to Q-1.1 depending on SNP

Pattern of suppression

Linear gluon polarization in Higgs production: Unpolarized: b J0 Sivers: kT sin(ϕ) b2 J1 1/Q0.7±0.1 Double Collins: (kT sin(ϕ))2 b3 J2 1/Q1.1

• Lin. pol. in Higgs prod: (kT)4 b5 J0 1/Q0.9-1.1
• Lin. pol. in 𝛿*-jet prod: kT2 cos(2ϕ) b3 J2 1/Q0.95

For e p→e’ Q Q X one can expect the same 1/Q suppression (testable at EIC) _

Pattern of suppression

Linear gluon polarization in Higgs production: Unpolarized: b J0 Sivers: kT sin(ϕ) b2 J1 1/Q0.7±0.1 Double Collins: (kT sin(ϕ))2 b3 J2 1/Q1.1

• Lin. pol. in Higgs prod: (kT)4 b5 J0 1/Q0.9-1.1
• Lin. pol. in 𝛿*-jet prod: kT2 cos(2ϕ) b3 J2 1/Q0.95

For e p→e’ Q Q X one can expect the same 1/Q suppression (testable at EIC) _ Conclusion: kT2 cos(2ϕ) and kT4 have similar 1/Q suppression Penalty for increasing power of b is similar to that of increasing harmonic

Pattern of suppression

Linear gluon polarization in Higgs production: Unpolarized: b J0 Sivers: kT sin(ϕ) b2 J1 1/Q0.7±0.1 Double Collins: (kT sin(ϕ))2 b3 J2 1/Q1.1

• Lin. pol. in Higgs prod: (kT)4 b5 J0 1/Q0.9-1.1
• Lin. pol. in 𝛿*-jet prod: kT2 cos(2ϕ) b3 J2 1/Q0.95

For e p→e’ Q Q X one can expect the same 1/Q suppression (testable at EIC) _ Conclusion: kT2 cos(2ϕ) and kT4 have similar 1/Q suppression Penalty for increasing power of b is similar to that of increasing harmonic Expectation: for kT4 cos(4ϕ) there may be a suppression as much as 1/Q2

Conclusions

• Linear polarization of gluons in unpolarized hadrons can affect many processes

differently depending on the link structure and the kinematics

• In pp collisions a promising channel is 𝛿*-jet production, as it probes the [+,−] link

structure, which at small x becomes a single Wilson loop, which in turn implies maximal linear gluon polarization

• Although the azimuthal asymmetry that probes this gluon polarization suffers from

Sudakov suppression (like any other TMD azimuthal asymmetry), it may be measurable at RHIC and LHC and allow for a comparison to the [+,+] link cases

• TMD or small-x hybrid factorization needs to be established for this process still

Conclusions