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

Outline 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

TMD evolution of azimuthal asymmetries

Transverse Momentum Dependence Including transverse momentum of quarks involves much more than replacing f 1 (x) → f 1 (x,k T2 ) in collinear factorization expressions One deals with less inclusive processes and with TMD factorization & TMD evolution TMD = t ransverse momentum dependent parton distribution

Transverse Momentum Dependence Including transverse momentum of quarks involves much more than replacing f 1 (x) → f 1 (x,k T2 ) in collinear factorization expressions One deals with less inclusive processes and with TMD factorization & TMD evolution TMD = t ransverse momentum dependent parton distribution The transverse momentum dependence can be correlated with the spin, e.g. q q k k s T T T P = / k T × S T D. Sivers (’90): s T

Transverse Momentum Dependence Including transverse momentum of quarks involves much more than replacing f 1 (x) → f 1 (x,k T2 ) in collinear factorization expressions One deals with less inclusive processes and with TMD factorization & TMD evolution TMD = t ransverse momentum dependent parton distribution The transverse momentum dependence can be correlated with the spin, e.g. q q k k s T T T P = / k T × S T D. Sivers (’90): s T Similar effects can arise in the final state, such as the Collins effect, which is described by a TMD fragmentation function: k k π π s T T T - H ⊥ ≠ 1 = J. Collins (’93): s T

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; …] Sivers function Generic features: - decrease and broadening of TMDs with increasing energy - Gaussian develops a power- law tail Azimuthal asymmetries will decrease with Q Aybat & Rogers, 2011

TMD factorization d σ Z d 2 b e − i b · q T ˜ Q 2 T /Q 2 � � d Ω d 4 q = W ( b , Q ; x, y, z ) + O ˜ ˜ 1 ( x, b 2 ; ζ F , µ ) ˜ X 1 ( z, b 2 ; ζ D , µ ) H ( y, Q ; µ ) f a D a W ( b , Q ; x, y, z ) = a

TMD factorization d σ Z d 2 b e − i b · q T ˜ Q 2 T /Q 2 � � d Ω d 4 q = W ( b , Q ; x, y, z ) + O ˜ ˜ 1 ( x, b 2 ; ζ F , µ ) ˜ X 1 ( z, b 2 ; ζ D , µ ) H ( y, Q ; µ ) f a D a W ( b , Q ; x, y, z ) = a Take µ = Q H ( Q ; α s ( Q )) ∝ e 2 1 + α s ( Q 2 ) F 1 + O ( α 2 � � s ) a This choice avoids large logarithms in H, but now they will appear in the TMDs

TMD factorization d σ Z d 2 b e − i b · q T ˜ Q 2 T /Q 2 � � d Ω d 4 q = W ( b , Q ; x, y, z ) + O ˜ ˜ 1 ( x, b 2 ; ζ F , µ ) ˜ X 1 ( z, b 2 ; ζ D , µ ) H ( y, Q ; µ ) f a D a W ( b , Q ; x, y, z ) = a Take µ = Q H ( Q ; α s ( Q )) ∝ e 2 1 + α s ( Q 2 ) F 1 + O ( α 2 � � s ) a This choice avoids large logarithms in H, but now they will appear in the TMDs µ b ≈ 1 /b Use renormalization group equations to evolve the TMDs to the scale: 1 ( z, b 2 ; Q 2 , Q ) = e − S ( b,Q ) ˜ ˜ 1 ( x, b 2 ; Q 2 , Q ) ˜ b , µ b ) ˜ 1 ( x, b 2 ; µ 2 1 ( z, b 2 ; µ 2 f a D b f a D b b , µ b ) where S is the so-called Sudakov factor [Collins & Soper, 1981; Collins, Soper, Sterman, 1985; Ji, Ma, Yuan, 2004/5; Collins, 2011; Echevarria, Idilbi & Scimemi 2012/14; …]

Sudakov factors Z Q 2 ✓ Q 2 dµ 2 ✓ Q 2 ◆ ◆ γ F ( g ( µ ); 1) − 1 ˜ ⇥ ⇤ S ( b, Q ) = − ln K ( b, µ b ) − 2 ln γ K ( g ( µ )) µ 2 µ 2 µ 2 µ 2 b b At leading order in α s the perturbative expression for S is: Z Q 2 dµ 2 ln Q 2 ✓ µ 2 − 3 ◆ S p ( b, Q ) = C F + O ( α 2 µ 2 α s ( µ ) s ) 2 π µ 2 b

Sudakov factors Z Q 2 ✓ Q 2 dµ 2 ✓ Q 2 ◆ ◆ γ F ( g ( µ ); 1) − 1 ˜ ⇥ ⇤ S ( b, Q ) = − ln K ( b, µ b ) − 2 ln γ K ( g ( µ )) µ 2 µ 2 µ 2 µ 2 b b At leading order in α s the perturbative expression for S is: Z Q 2 dµ 2 ln Q 2 ✓ µ 2 − 3 ◆ S p ( b, Q ) = C F + O ( α 2 µ 2 α s ( µ ) s ) 2 π µ 2 b It can be used whenever the restriction b 2 ≪ 1/ Λ 2 is justified (e.g. at very large Q 2 ) If also larger b contributions are important, e.g. at moderate Q and small Q T = |q T |, then one needs to include a nonperturbative Sudakov factor W ( b ) ≡ ˜ ˜ p W ( b ∗ ) e − S NP ( b ) b ∗ = b/ 1 + b 2 /b 2 max ≤ b max such that W(b * ) can be calculated within perturbation theory In general S NP is Q dependent and often taken to be Gaussian to be fitted to data

TMD evolution of the Sivers asymmetry 1.5 1.5 Q � GeV � A � Q T ,max � 3.33 10 1 � Q 0.68 30 1 1 60 90 A � Q T � 0.5 0.5 0 0 0 1 2 3 4 5 0 20 40 60 80 100 Q T Q  � 0 . 184 ln Q The peak of the Sivers asymmetry S AR b 2 NP ( b, Q, Q 0 ) = + 0 . 332 2 Q 0 decreases as 1/Q 0.7±0.1 [Aybat & Rogers, 2011] [D.B., NPB 2013]

TMD evolution of the Sivers asymmetry 1.5 1.5 Q � GeV � A � Q T ,max � 3.33 10 1 � Q 0.68 30 1 1 60 90 A � Q T � 0.5 0.5 0 0 0 1 2 3 4 5 0 20 40 60 80 100 Q T Q  � 0 . 184 ln Q The peak of the Sivers asymmetry S AR b 2 NP ( b, Q, Q 0 ) = + 0 . 332 2 Q 0 decreases as 1/Q 0.7±0.1 [Aybat & Rogers, 2011] [D.B., NPB 2013] Very similar to the fall-off with Q, obtained  � 0 . 58 ln Q S LY b 2 before with CS81 factorization and LY NP ( b, Q, Q 0 ) = + 0 . 11 2 Q 0 [D.B., NPB 2001] [Ladinsky & Yuan, 1994] The power of the fall-off is a robust feature

TMD evolution of the Sivers asymmetry At low Q 2 (up to ~20 GeV 2 ), the Q 2 evolution is dominated by S NP [Anselmino, Boglione, Melis, PRD 86 (2012) 014028] Precise low Q 2 data can help to determine the form and size of S NP Uncertainty in S NP determines the ±0.1 in 1/Q 0.7±0.1

Double Collins Effect Double Collins effect gives rise to an azimuthal asymmetry cos 2 φ in e + e - → h 1 h 2 X DB, Jakob Mulders, NPB 504 (1997) 345 d σ ( e + e − → h 1 h 2 X ) ∝ { 1 + cos 2 φ 1 A ( q T ) } dz 1 dz 2 d Ω d 2 q T 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)

Double Collins Effect Double Collins effect gives rise to an azimuthal asymmetry cos 2 φ in e + e - → h 1 h 2 X DB, Jakob Mulders, NPB 504 (1997) 345 d σ ( e + e − → h 1 h 2 X ) ∝ { 1 + cos 2 φ 1 A ( q T ) } dz 1 dz 2 d Ω d 2 q T 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) 3. 3 Q � GeV � A � Q T ,max � 2.5 3.33 2.5 10 1 � Q 1.1 30 2. 2 60 90 A � Q T � 1.5 1.5 1 1 0.5 0.5 0 0 0 20 40 60 80 100 0 1 2 3 4 5 Q Q T Considerable Sudakov suppression ~1/Q (effectively twist-3) DB, NPB 603 (2001) 195 & 806 (2009) 23 & QCD evolution 2013 proceedings

TMD evolution of the double Collins asymmetry Does it work? BESIII (left) and BaBar data shown as function of P h ⊥ not Q T √ s = 10 . 58 GeV √ s = 3 . 65 GeV

TMD evolution of the double Collins asymmetry Does it work? BESIII (left) and BaBar data shown as function of P h ⊥ not Q T √ s = 10 . 58 GeV √ s = 3 . 65 GeV 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…

TMD evolution of the double Collins asymmetry Does it work? BESIII (left) and BaBar data shown as function of P h ⊥ not Q T √ s = 10 . 58 GeV √ s = 3 . 65 GeV 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… 0.15 ) S TMD evolution φ − Evolution from HERMES (<Q 2 > ~ 2.4 GeV 2 ) HERMES h φ sin ( to COMPASS (<Q 2 > ~ 3.8 GeV 2 ) seems to COMPASS UT 0.1 A work well, but very small energy range and can be quite S NP dependent 0.05 Aybat, Prokudin & Rogers, PRL 2012 0 0 0.2 0.4 0.6 0.8 1 1.2 P (GeV) h

Gluons TMDs

Gluons TMDs The gluon correlator: h i Γ µ ν [ U , U 0 ] F + ν (0) U [0 , ξ ] F + µ ( ξ ) U 0 ( x, k T ) ⌘ F . T . h P | Tr c | P i [ ξ , 0] g For unpolarized protons: ✓ k µ k 2 ⇢ ◆ � U ( x, k T ) = 1 T k ν Γ µ ν − g µ ν T f g + g µ ν h ⊥ g 1 ( x, k 2 T ( x, k 2 T T ) + T ) 1 T 2 x M 2 2 M 2 p p

Gluons TMDs The gluon correlator: h i Γ µ ν [ U , U 0 ] F + ν (0) U [0 , ξ ] F + µ ( ξ ) U 0 ( x, k T ) ⌘ F . T . h P | Tr c | P i [ ξ , 0] g For unpolarized protons: ✓ k µ k 2 ⇢ ◆ � U ( x, k T ) = 1 T k ν Γ µ ν − g µ ν T f g + g µ ν h ⊥ g 1 ( x, k 2 T ( x, k 2 T T ) + T ) 1 T 2 x M 2 2 M 2 p p unpolarized gluon TMD