Proton Spin at Small x
Yuri Kovchegov The Ohio State University Outilne
• Goal: understanding the proton spin coming from helicities and OAM of small-x quarks and gluons. • Brief introduction to small-x physics and evolution equations. • Quark Helicity: • Quark helicity distribution at small x • Small-x evolution equations for quark helicity • Small-x asymptotics of quark helicity • Gluon Helicity: • Gluon helicity distribution at small x • Small-x evolution equations for gluon helicity • Small-x asymptotics of quark helicity TMDs • Quark and Gluon OAM at small x • Preliminary phenomenology • Conclusions Goal: Proton Spin at Small x
3 Proton Spin
Our understanding of nucleon spin structure has evolved:
• In the 1980’s the proton spin was thought of as a sum of constituent quark spins (left panel)
• Currently we believe that the proton spin is a sum of the spins of valence and sea quarks and of gluons, along with the orbital angular momenta of quarks and gluons (right panel) 4 Helicity Distributions • To quantify the contributions of quarks and gluons to the proton spin on defines helicity distribution functions: number of quarks/gluons with spin parallel to the proton momentum minus the number of quarks/gluons with the spin opposite to the proton momentum:
• The helicity parton distributions are 2 + 2 2 f(x, Q ) f (x, Q ) f (x, Q ) ⌘ with the net quark helicity distribution ⌃ u + u¯ + d + d¯+ s + s¯ ⌘ and ∆"($, &') the gluon helicity distribution.
5 Proton Helicity Sum Rule
• Helicity sum rule (Jaffe-Manohar form): 1 = S + L + S + L 2 q q g g
with the net quark and gluon spin 1 1 2 1 2 2 2 S (Q )= dx ⌃(x, Q ) Sg(Q )= dx G(x, Q ) q 2 Z0 Z0
• Lq and Lg are the quark and gluon orbital angular momenta
6 1 1 S (Q2)= dx ⌃(x, Q2) Proton Spin Puzzle q 2 Z0
• The spin puzzle began when the EMC collaboration measured the proton
g1 structure function ca 1988. Their data resulted in ⌃ 0.1 0.2 ⇡ ÷ • It appeared quarks do not carry all of the proton spin (which would have corresponded to ⌃ =1 ). 1 • = Sq + Lq + Sg + Lg Missing spin can be 2 – Carried by gluons – In the orbital angular momenta of quarks and gluons – At small x: 1 1 1 S (Q2)= dx ⌃(x, Q2) S (Q2)= dx G(x, Q2) q 2 g Z0 Z0
Can’t integrate down to zero, use xmin instead! – Or all of the above! Current Knowledge of Proton Spin
• The proton spin carried by the quarks is estimated to be (for 0 . 001 S (Q2 = 10 GeV2) 0.13 0.26 G ⇡ ÷ • Unfortunately the uncertainties are large. Note also that the x-ranges are limited, with more helicity (positive or negative) possible at small x. How much spin is at small x? • E. Aschenaur et al, arXiv:1509.06489 [hep-ph], (DSSV = de Florian, Sassot, Stratmann, Vogelsang, helicity PDF parametrization) • Uncertainties are very large at small x! (EIC may reduce them.) Spin at small x • The goal of this work is to provide theoretical understanding of helicity PDFs at very small x. • Our work would provide guidance for future hPDFs parametrizations of the existing and new data (e.g., the data to be collected at EIC). • Alternatively the data can be analyzed using our small-x evolution formalism. 10 Introduction: small-x physics Quasi-classical approximation Kinematics of DIS Ø Photon carries 4-momentum q µ , its virtuality is Q2 = q qµ µ Ø Photon hits a quark in the proton carrying momentum x Bj p with p being the proton s momentum. Parameter x Bj is the Bjorken x variable. 13 Dipole picture of DIS • In the dipole picture of DIS the virtual photon splits into a quark-antiquark pair, which then interacts with the target. • The total DIS cross section and structure functions are calculated via: q x x ⊥ ⊥ γ∗ 14 Dipole Amplitude • The total DIS cross section is expressed in terms of the (Im part of the) forward quark dipole amplitude N: 1 d2x dz ⇤A 2 ⇤ qq¯ 2 ~ tot = ? d b ! (~x ,z) N(~x , b ,Y) 2 ⇡ ? z (1 z) | ? | ? ? Z Z0 q z x x ⊥ ⊥ γ∗ 1 z b ,Y ? with rapidity Y=ln(1/x) 15 DIS in the Classical Approximation The DIS process in the rest frame of the target is shown below. It factorizes into A 2 qq¯ 2 tot⇤ (xBj,Q )= ⇤! N(x ,Y =ln1/xBj) | | ⌦ ? with rapidity Y=ln(1/x) 16 Dipole Amplitude • The quark dipole amplitude is defined by 1 N(x1,x2)=1 tr V (x1) V †(x2) Nc h i • Here we use the Wilson lines along⇥ the light-cone direction⇤ 1 + + V (x)=Pexp ig dx A (x ,x =0,x) 2 3 Z • In the classical Glauber4 -Mueller/ 1 McLerran-Venugopalan5 approach the dipole amplitude resums multiple rescatterings: x1 x ⊥ x2 17 Quasi-classical dipole amplitude x ⊥ A.H. Mueller, 90 Lowest-order interaction with each nucleon – two gluon exchange – the same resummation parameter as in the MV model: 2 1/3 ↵s A 18 DIS in the Classical Approximation The dipole-nucleus amplitude in the classical approximation is 2 2 x Qs 1 N(x ,Y)=1 exp ? ln ? 4 x ⇤ ? A.H. Mueller, 90 Black disk limit, 2 sptot < 2 R Color transparency 1/QS 19 Small-x evolution equations Quantum Evolution • The energy (x) dependence comes in through nonlinear small-x BK/JIMWLK evolution, which resums the long-lived s-channel gluon corrections: 1 ↵ ln ↵ Y 1 s x ⇠ s ⇠ These extra gluons bring in powers of aS ln 1/x, such that when aS << 1 and ln 1/x >>1 this parameter is aS ln 1/x ~ 1 (leading logarithmic approximation, LLA). 21 Resumming Gluonic Cascade In the large-NC limit of QCD the gluon corrections become color dipoles. Gluon cascade becomes a dipole cascade. A. H. Mueller, 93- 94 We need to resum dipole cascade, with each final state dipole interacting with the target. Yu. K. 99 22 Notation (Large-NC) ⃗x1 ⃗x1 ⊥ ⊥ ⃗x2 ⊥ ⃗x2 ⊥ Real emissions in the ⃗x0 ⃗x0 ⊥ ⊥ ⃗x1 ⃗x1 amplitude squared ⊥ ⊥ ⃗x2 ⊥ ⃗x2 (dashed line – all ⊥ ⃗x0 ⃗x0 Glauber-Mueller exchanges ⊥ ⊥ ⃗x1 at light-cone time =0) ⊥ ⃗x2 ⊥ ⃗x0 ⊥ Virtual corrections in the amplitude (wave function) 23 Nonlinear Evolution To sum up the gluon cascade at large-NC we write the following equation for the dipole S-matrix: 1 1 S ∂ ∂Y S 2 dashed line = S 0 0 all interactions 1 1 with the target 2 2 S S 0 0 ↵ N x2 @ S (Y )= s c d2x 01 [S (Y ) S (Y ) S (Y )] Y x0,x1 2⇡2 2 x2 x2 x0,x2 x2,x1 x0,x1 Z 02 21 Remembering that S= 1-N we can rewrite this equation in terms of the dipole scattering amplitude N. 24 Nonlinear evolution at large Nc As N=1-S we write 1 1 N ∂ ∂Y N 2 0 0 1 1 1 2 2 N N 0 0 1 dashed line = N 2 all interactions N with the target 0 BFKL ↵ N x2 @ N (Y )= s c d2x 01 [N (Y )+N (Y ) N (Y ) N (Y ) N (Y )] Y x0,x1 2 ⇡2 2 x2 x2 x0,x2 x2,x1 x0,x1 x0,x2 x2,x1 Z 02 21 Balitsky ‘96, Yu.K. ‘99 25 Beyond large-Nc: JIMWLK Equation • JIMWLK evolution equation (1997-2002): 2 1 2 2 ab @Y WY [↵]=↵s d x d y a b ⌘~x ~y WY [↵] 2 ? ? ↵ (x , ~x ) ↵ (y , ~y ) ? ? ( Z ? ? ⇥ ⇤ 2 a d x a ⌫~x WY [↵] ? ↵ (x , ~x ) ? ) with ab ab 4 2 ~x21 ~x20 ⌘ = d x · 1 U U † U U † + U U † ~x1 ~x0 2 2 2 2 2 ~x1 ~x2 ~x2 ~x0 ~x1 ~x0 ? ? g ⇡ x x ? ? ? ? ? ? 1 + + U~x =Pexp ig dx (x =0,x , ~x ) ? 8 A ? 9 26 Resummation parameter • BK equation resums powers of 1 ↵s Nc ln 27 Solution of BK equation 1 N(x⊥,Y) numerical solution 0.8 α SYY=0, = 0, 3, 1.2, 6, 9,2.4, 12 3.6, 4.8 by J. Albacete ‘03 (earlier solutions were 0.6 found numerically by Golec-Biernat, Motyka, Stasto, 0.4 by Braun and by Lublinsky et al in ‘01) 0.2 0 0.00001 0.0001 0.001 0.01 0.1 1 10 -1 x⊥ (GeV ) 1/Qs BK solution preserves the black disk limit, N<1 always (unlike the linear BFKL equation) qqA¯ =2 d2bN(x ,b ,Y) ? ? 28 Z Saturation scale Q (Y) (GeV) 10000 s 1000 100 10 1 0 1 2 3 4 5 6 αsY numerical solution by J. Albacete 29 Small-x Asymptotics ↵ 1 1 P • BFKL solution gives (x<<1) N ⇠ x with ✓ ◆ 4 ↵ N ↵ N ↵ 1= s c ln 2 2.77 s c P ⇡ ⇡ ⇡ • NLO corrections are known (Fadin, Lipatov ’98; Ciafaloni, Camici ’98; Balitsky, Chirilli ‘07 for BK). • Full BK equation solution also leads to saturation at very small x (x<<<1): N const ⇠ • Below we will refer to the BFKL-like linear regime as the “small-x asymptotics” of TMDs. It should be understood that at even smaller x saturation is expected to come in and stop the small x evolution. 30 Map of High Energy QCD Q2 (Y) saturation s 1 region 2 Qs Y = ln 1/x ⇠ x Geometric ✓ ◆ Scaling Saturation Scale BK/JIMWLK grows with energy energy BFKL, DGLAP – linear equations BK/JIMWLK – nonlinear BFKL DGLAP non-perturbative region 2 2 ΛQCD ln Q αs ~ 1 αs << 1 size of gluons 31 References Published in September 2012 by Cambridge U Press 32 Quark Helicity at Small x (flavor-singlet case) Yu.K., M. Sievert, arXiv:1505.01176 [hep-ph] Yu.K., D. Pitonyak, M. Sievert, arXiv:1511.06737 [hep-ph], arXiv:1610.06197 [hep-ph], arXiv:1610.06188 [hep-ph], arXiv:1703.05809 [hep-ph], arXiv:1808.09010 [hep-ph] 33 Observables • We want to calculate quark helicity PDF and TMD at small x. Leading Twist TMDs Nucleon Spin Quark Spin Quark Polarization Un-Polarized Longitudinally Polarized Transversely Polarized (U) (L) (T) ┴ — h1 = U ƒ1 = Boer-Mulders — L g1L = ┴ — h1L = Helicity h1 = — T ƒ ┴ = — ┴ Transversity Nucleon Polarization — 1T g1T = Sivers ┴ h1T = — 34 Quark Helicity TMD • We start with the definition of the quark helicity TMD with a future- pointing Wilson line staple. + 5 q 2 1 1 2 ik r g (x, k )= S d rdr e · p, S ¯(0) [0,r] (r) p, S + 1L T (2⇡)3 2 L h L| U 2 | Lir =0 b + Vx[b ,a ]=Pexp8ig dx A (x ,x)9 > > < aZ = :> ;> 35 Quark Helicity TMD at Small x • At high energy/small-x the proton is a shock wave, and we have the following contributions to the SIDIS quark helicity TMD: ABC ζξζξζξ x− x− 0 0 DEF ζξ ζξζξ + q 2 2p 2 2 ik (⇣ ⇠) 1 + 5 g (x, k )= d ⇣ d⇣ d ⇠ d⇠ e · ¯ (⇠) V [⇠ , ] X X V [ , ⇣ ] (⇣) 1L T (2⇡)3 2 ↵ ↵ ⇠ 1 | ih | ⇣ 1 36 Quark Helicity TMD at Small x ABC ζξ ζξζξ DEF ζξζ ξ ζξ • Diagram D does not transfer spin information from the target. Diagram C is canceled as we move t-channel quarks across the cut. • Diagram F is energy-suppressed, since the gluon should have no time to be emitted and absorbed inside the shock wave. • Diagrams of the types A and E++ can be shown to cancel each other at the leading (DLA) order (Ward identity). • We are left with the diagram B. 37 Quark Helicity TMD at Small x • Evaluating diagram B we arrive at 1 q 2 4Nc 2 2 2 ik (⇣ y) dz ⇣ w y w g (x, k )= d ⇣ d wd ye · G (zs) 1L T (2⇡)6 z ⇣ w 2 · y w 2 w,⇣ Z ⇤2Z/s | | | | where Gwz is the polarized dipole amplitude (defined on the next slide). • Here s is the cms energy squared, L is some IR cutoff, underlining denotes transverse vectors, z = smallest longitudinal momentum fraction of the dipole momentum out of those carried by the quark and the antiquark ζ ξ k1 w k2 38 Polarized Dipole • All flavor-singlet small-x helicity observables depend on one object, “polarized dipole amplitude”: 1 pol pol G10(z) Re Ttr V0V1 † +Ttr V1 V0† (z) unpolarized quark polarized quark: eikonal propagation, 1 non-eikonal spin-dependent interaction + + V exp ig dx A (x , 0 ,x) x ⌘ P 2 3 Z 1 • Double 4brackets denote an object5 with energy suppression scaled out: (z) zs (z) O ⌘ O 39 DD EE D E Polarized fundamental “Wilson line” • To complete the definition of the polarized dipole amplitude, we need to construct the definition of the polarized “Wilson line” Vpol, which is the leading helicity-dependent contribution for the quark scattering amplitude on a longitudinally-polarized target proton. p2 p2 + k p2 p2 + k σ σ′ σ σ′ a λ λ b k k − − p1 x1 x2 • At the leading order we can either exchange one non-eikonal t-channel gluon (with quark-gluon vertices denoted by blobs above) to transfer polarization between the projectile and the target, or two t-channel quarks, as shown above. 40 Polarized fundamental “Wilson line” p2 p2 + k p2 p2 + k σ σ′ σ σ′ a λ λ b k k − − p1 x1 x2 • In the end one arrives at (KPS ‘17; YK, Sievert, ‘18; cf. Chirilli ‘18) + 1 pol igp1 12 V = dx V [+ ,x ] F (x ,x) V [x , ] x s x 1 x 1 Z 1 2 + 1 1 g p1 b ba 1 + 5 a dx dx V [+ ,x ] t (x ,x) U [x ,x ] ¯ (x ,x) t V [x , ]. s 1 2 x 1 2 2 x 2 1 2 ↵ 1 x 1 1 ↵ Z xZ • The polarized dipole amplitude is then defined by 1 1 G (z) dx tr V [ , ]V [ ,x ]( ig) A˜(x ,x) V [x , ] +c.c. (z) 10 ⌘ 4N 0 1 1 1 1 r ⇥ 1 1 c Z 1 D h i E with the standard light-cone t Wilson line b + Vx[b ,a ]=Pexp8ig dx A (x ,x)9 > aZ > < = z :> ;> 1 0 proton 42 Polarized adjoint “Wilson line” • Quarks mix with gluons. Therefore, we need to construct the adjoint polarized Wilson line --- the leading helicity-dependent part of the gluon scattering amplitude on the longitudinally polarized target. p2 µ p2 + k p2 − k ′ ′ ν b b p2 a a c ′ σ σ ′ λ λ b λ µ λ a ρ k k − − p1 x1 x2 • The calculation is similar to the quark scattering case. It yields (cf. Chirilli ‘18) + + 1 pol ab 2igp1 12 + ab (U •) = dx U [+ ,x ] (x =0,x ,x) U [x , ] x s x 1 F x 1 Z 1 2 + 1 1 g p1 aa0 a0 1 + b0 b0b dx dx U [+ ,x ] ¯(x ,x) t V [x ,x ] t (x ,x) U [x , ] c.c. s 1 2 x 1 2 2 x 2 1 2 5 1 x 1 1 Z xZ 43 Small-x Evolution at large Nc • At large Nc the evolution is gluon-driven. We will evolve a gluon dipole, remembering that at large Nc the relation between the adjoint and fundamental longitudinally-polarized gluon dipoles is Gadj(z)=4G (z) (Note that the factor is 4, not 2 like in the unpolarized dipole case.) 44 Evolution for Polarized Quark Dipole One can construct an evolution equation for the polarized dipole: 00 2 ∂Y 1 1 0 2 Spin-dependent (non-eikonal) vertex polarized 1 particle 0 2 1 similar to 0 unpolarized 2 1 BK evolution box = 0 target shock 2 wave 1 45 Evolution for Polarized Quark Dipole 00 2 ∂Y 1 1 0 2 1 0 1 2 ... = ... 1 hh ii zsh i 0 2 1 2 1 0 ⇢0 = z0 s 2 1 z 2 1 unp pol 1 unp pol ↵s dz0 d x2 † † tr V0 V1 (z)= tr V0 V1 (z)+ 2 2 Nc Nc 0 2⇡ z0 x21 2 DD h iEE DD h iEE zZi ⇢Z0 2 b unp a unp pol ba ✓(x10 x21) tr t V0 t V1 † U2 (z0) ⇥ Nc Equation does not close! ⇢ DD h i EE 2 2 1 b unp a pol unp ba + ✓(x10z x21z0) tr t V0 t V2 † U1 (z0) Nc DD h i EE 1 unp unp unp pol unp pol + ✓(x10 x21) tr V0 V2 † tr V2 V1 † (z0) Nc tr V0 V1 † (z0) Nc 46 hDD h i h iEE DD h iEE i Polarized Dipole Evolution in the Large-Nc Limit In the large-Nc limit the equations close, leading to a system of 2 equations: 0 0 Γ02,21(z) S02(z) S02(z) ∂ G10(z) = 2 + + Γ01,21(z) ∂ ln z − z z S21(z) G21(z) G12(z) 1 1 0 0 Γ03,32(z′) S03(z′) S03(z′) ∂ Γ02,21(z′) = 3 + + Γ02,32(z′) ∂ ln z′ − S23(z′) G32(z′) G23(z′) z′ z′ 2 2 1 1 z z 2 z x10 2 (0) ↵s Nc dz0 dx21 G10(z)=G10 (z)+ 2 [2 02, 21(z0) S21(z0)+2G21(z0) S02(z0) 2⇡ z0 x21 2 zZi ⇢Z0 + G12(z0) S02(z0) 01, 21(z0)] 2 2 z0 min x02,x21 z0/z00 { } 2 (0) ↵s Nc dz00 dx32 02, 21(z0)= 02, 21(z0)+ 2 [2 03, 32(z00) S23(z00)+2G32(z00) S03(z00) 2⇡ z00 x32 Z Z2 zi ⇢00 + G23(z00) S03(z00) 02, 32(z00)] S = found from BK/JIMWLK, it is LLA 47 “Neighbor” dipole • There is a new object in the evolution equation – the neighbor dipole. • This is specific for the DLA evolution. Gluon emission may happen in one dipole, but, due to transverse distance ordering, may ’know’ about another dipole: 0 0 3,z′′ 2 2,z′ 1 1 2 2 x z0 x z00 21 32 • We denote the evolution in the neighbor dipole 02 by 02, 21(z0) 48 Large-Nc Evolution • In the strict DLA limit (S=1) and at large Nc we get (here G is an auxiliary function we call the ‘neighbour dipole amplitude’) (KPS ‘15) 2 z x10 2 2 (0) 2 ↵s Nc dz0 dx21 2 2 2 G(x10,z)= G (x10,z)+ 2 (x10,x21,z0)+3G(x21,z0) 2⇡ z0 x21 Z1 Z1 2 z0s ⇥ ⇤ x10s 2 2 z0 min x10,x21 z0 z00 n o 2 2 2 (0) 2 2 ↵s Nc dz00 dx32 2 2 2 (x10,x21,z0)= (x10,x21,z0)+ 2 (x10,x32,z00)+3G(x32,z00) 2⇡ z00 x32 Z1 Z1 2 z00s ⇥ ⇤ x10s • The initial conditions are given by the Born-level graphs 1 z 1 z 1 z x0 − x0 − x0 − x1 x1 x1 z z z 0 0 0 1 z 1 z (0) 2 2 (0) 2 x0 − x0 − (x10,x21,z)=G (x10,z) x1 x1 z z 0 0 2 (0) 2 ↵sCF zs 2 G (x10,z)= ⇡ CF ln 2 2ln(zsx10) Nc ⇤ h i 49 Resummation Parameter • For helicity evolution the resummation parameter is different from BFKL, BK or JIMWLK, which resum powers of leading logarithms (LLA) ↵s ln(1/x) • Helicity evolution resummation parameter is double-logarithmic (DLA): 1 ↵ ln2 s x • The second logarithm of x arises due to transverse momentum (or transverse coordinate) integration being logarithmic both in the UV and IR. • This was known before: Kirschner and Lipatov ’83; Kirschner ’84; Bartels, Ermolaev, Ryskin ‘95, ‘96; Griffiths and Ross ’99; Itakura et al ’03; Bartels and Lublinsky ‘03. 50 Quark Helicity at Small x • These equations can be solved both numerically and analytically. (KPS ‘16-’17) ↵sNc 1 ↵sNc zs s10 = ln ⌘ = ln 2⇡ x2 ⇤2 2 r 10 r 2⇡ ⇤ • The small-x asymptotics of quark helicity is (at large Nc) ↵q 1 h 4 ↵ N ↵ N q(x, Q2) with ↵q = s c 2.31 s c ⇠ x h p3 2⇡ ⇡ 2⇡ ✓ ◆ r r Small-x Evolution at large Nc&Nf • At large Nc&Nf there is no simple relation between the adjoint and fundamental polarized dipole amplitudes. We need to construct coupled evolution equations mixing them with each other. • Here’s the adjoint dipole evolution: 0 inhomogeneous term 1 0− ′ x− x− ′ I I II 2 1 II 0 0 0 0 2 2 k 2 k1 k 2 k1 k2 k1 k2 k1 2 2 1 1 1 1 − − − − − − − − − x− x1 0 x2 x1 0 x2 x1 0 0 2 III III′ eikonal 0 0 0 2 2 2 k1 k2 k1 k2 k1 k2 other eikonal diagrams 1 1 1 − − − − − − − − − x1 0 x2 x1 0 x2 x1 0 x2 52 Small-x Evolution at large Nc&Nf • At large Nc&Nf there is no simple relation between the adjoint and fundamental polarized dipole amplitudes. We need to construct coupled evolution equations mixing them with each other. • Here’s the fundamental dipole evolution: 0 inhomogeneous term 1 0− ′ x− x− ′ I I II 2 1 II 0 0 0 0 2 2 k 2 k1 k 2 k1 k2 k1 k2 k1 2 2 1 1 1 1 − − − − − − − − − x− x1 0 x2 x1 0 x2 x1 0 0 2 III eikonal 0 0 2 2 k1 k2 k1 k2 other eikonal diagrams 1 1 − − − − − − x1 0 x2 x1 0 x2 53 Small-x Evolution at large Nc&Nf • The resulting equations are 2 z x10 2 (0) ↵sNc dz0 dx21 1 adj 1 adj ¯ Q10(zs)=Q10 (zs)+ 2 02,21(z0)+ G21 (z0)+Q12(z0) 01,21(z0) 2⇡ z0 x21 2 2 2 ( ) ⇤Z 1/(Zz0s) s 2 z x10z/z0 2 ↵s Nc dz0 dx21 + 2 Q21(z0), 4 ⇡ z0 x21 2 ⇤Z/s 1/(Zz0s) 2 z x10 2 adj adj (0) ↵s Nc dz0 dx21 adj adj G10 (z)=G10 (z)+ 2 10,21(z0)+3G21 (z0) 2⇡ z0 x21 2 2 max ⇤ ,Z1/x /s 1/(Zz0 s) h i { 10} 2 z x10z/z0 2 ↵s Nf dz0 dx21 ¯ These are yet to be solved. 2 02;21(z0), 2 ⇡ z0 x21 2 ⇤Z/s 1/(Zz0 s) 2 2 z0 min x10,x21z0/z00 { } 2 adj adj (0) ↵s Nc dz00 dx32 adj adj 10,21(z0)= 10,21 (z0)+ 2 10,32(z00)+3G32 (z00) 2⇡ z00 x32 2 2 max ⇤ ,Z1/x /s 1/(zZ00 s) h i { 10} 2 z0 x21z0/z00 2 ↵s Nf dz00 dx32 ¯ 2 03;32(z00), 2 ⇡ z00 x32 2 ⇤Z/s 1/(zZ00 s) 2 2 z0 min x10,x21z0/z00 { } 2 ¯ ¯(0) ↵sNc dz00 dx32 1 adj 1 adj ¯ 10,21(z0)= 10,21(z0)+ 2 03,32(z00)+ G32 (z00)+Q32(z00) 01,32(z00) 2⇡ z00 x32 2 2 2 ( ) ⇤Z 1/(Zz00s) s 2 z0 x21z/z0 2 ↵s Nc dz00 dx32 + 2 Q32(z00). 4 ⇡ z00 x32 ⇤2Z/s 1/(Zz s) Yu.K., D. Pitonyak, M. Sievert, arXiv:1706.04236 [nucl-th] Dipole Gluon Helicity TMD • Now let us repeat the calculation for gluon helicity TMDs. • We start with the definition of the gluon dipole helicity TMD: 2 + G 2 2iSL d⇠ d ⇠ ixP ⇠ ik ⇠ ij +i [+] +j [ ] g1 (x, kT )= e · P, SL ✏T tr F (0) †[0, ⇠] F (⇠) [⇠, 0] P, SL xP+ (2⇡)3 | U U | ⇠+=0 Z D h i E U^[+] t • Here U[+] and U[-] are future and j past Wilson line staples (hence the name `dipole’ TMD, F. Dominguez et al ’11 – looks like a dipole scattering on a z proton): i U^[−] proton Dipole Gluon Helicity TMD • At small x, the definition of dipole gluon helicity TMD can be massaged into 2 8iNc SL Q Gdip 2 2 ik x10 i ij 2 j g1 (x, kT )= d x10 e · k ✏T d b10 G10(zs = ) g2(2⇡)3 ? x Z Z • Here we obtain a new operator, which is a transverse vector (written here in A-=0 gauge): 1 i 1 i G (z) dx tr V [ , ]V [ ,x ]( ig) A˜ (x ,x) V [x , ] +c.c. (z) 10 ⌘ 4N 0 1 1 1 1 1 1 c Z 1 D h t i E • i ij Note that k ✏ T can be thought of ? i i as a transverse curl acting on G10(z) i and not just on A ˜ ( x ,x ) -- different z from the polarized dipole amplitude! 1 0 proton Dipole TMD vs dipole amplitude • Note that the operator for the dipole gluon helicity TMD 1 i 1 i G (z) dx tr V [ , ]V [ ,x ]( ig) A˜ (x ,x) V [x , ] +c.c. (z) 10 ⌘ 4N 0 1 1 1 1 1 1 c Z 1 D h i E is different from the polarized dipole amplitude 1 1 G (z) dx tr V [ , ]V [ ,x ]( ig) A˜(x ,x) V [x , ] +c.c. (z) 10 ⌘ 4N 0 1 1 1 1 r ⇥ 1 1 c Z 1 D h i E • We conclude that the dipole gluon helicity TMD does not depend on the polarized dipole amplitude! (Hence the ‘dipole’ name may not even be valid for such TMDs.) • This is different from the unpolarized gluon TMD case. Evolution Equation • To construct evolution equation for the operator !" governing the gluon helicity TMD we resum similar (to the quark case) diagrams: 0 0 2 k1 k2 “c.c.” 1 i i 1 i x1− x2− other LLA diagrams i Large-Nc Evolution: Diagrams • At large-Nc the equations are 0 0 0 gen Γ20 , 21(z′s) ∂ 2 i 2 ∂Y G10(zs) + + “c.c.” = z′ z′ G21(z′s) z z z i i i 1 1 1 0 0 gen Γ20 , 21(z′s) 2 2 “c.c.” + z′ + z′ + gen Γ21 , 20(z′s) z z i i 1 1 0 0 i Γ10 , 21(z′s) 2 + + z′ 2 + “c.c.” z′ i G12(z′s) z z i i 1 1 Large-Nc Evolution: Diagrams 0 0 0 • and ′′ Γ30 , 31(z s) ∂ 3 i z′s ∂Y Γ10 , 21( ) = 3 + z′′ + “c.c.” z′′ 2 2 2 z′ z′ z′ ′′ G30(z s) i i i 1 1 1 0 0 ′′ Γ30 , 31(z s) + 3 + 3 + “c.c.” z′′ z′′ 2 2 z′ z′ ′′ Γ31 , 30(z s) i i 1 1 0 0 i z′′s Γ10 , 31( ) 3 3 + + + “c.c.” z′′ z′′ 2 z′ i ′′ G30(z s) i i 1 1 Large-Nc Evolution: Equations • This results in the following evolution equations: z ij j i i (0) ↵sNc dz0 2 1 ✏T (x21) gen G10(zs)=G10 (zs)+ 2 d x2 ln 2 ? 20 , 21(z0s)+G21(z0s) 2⇡ z0 x21⇤ x21 ⇤Z2 Z h i s z ij j ↵sNc dz0 2 1 ✏T (x20) gen gen 2 d x2 ln 2 ? 20 , 21(z0s)+ 21 , 20(z0s) 2⇡ z0 x21⇤ x20 ⇤Z2 Z h i s 2 z x10 2 ↵sNc dz0 dx21 i i + 2 G12(z0s) 10 , 21(z0s) 2⇡ z0 x21 Z1 Z1 2 z s h i x10s 0 z0 ij j i i (0) ↵sNc dz00 2 1 ✏T (x31) gen 10 21(z0s)=G10 (z0s)+ 2 d x3 ln 2 ? 30 , 31(z00s)+G31(z00s) 2⇡ z00 x31⇤ x31 ⇤Z2 Z h i s z0 ij j ↵sNc dz00 2 1 ✏T (x30) gen gen 2 d x3 ln 2 ? 30 , 31(z00s)+ 31 , 30(z00s) 2⇡ z00 x31⇤ x30 ⇤Z2 Z h i s 2 2 z0 min x10 ,x21 z0 z00 2 ↵sNc dz00 dx31 i i + 2 2 G13(z00s) 10 , 31(z00s) . 2⇡ z00 x31 Z1 Z1 2 z s h i x10s 00 Large-Nc Evolution: Equations • Here gen (z0s)=✓(x x ) (z0s)+✓(x x ) G (z0s) 20,21 20 21 20,21 21 20 20 is an object which we know from the quark helicity evolution, as the latter gives us G and G. • Note that our evolution equations mix the gluon (Gi) and quark (G) small-x helicity evolution operators: z ij j i i (0) ↵sNc dz0 2 1 ✏T (x21) gen G10(zs)=G10 (zs)+ 2 d x2 ln 2 ? 20 , 21(z0s)+G21(z0s) 2⇡ z0 x21⇤ x21 ⇤Z2 Z h i s z ij j ↵sNc dz0 2 1 ✏T (x20) gen gen 2 d x2 ln 2 ? 20 , 21(z0s)+ 21 , 20(z0s) 2⇡ z0 x21⇤ x20 ⇤Z2 Z h i s 2 z x10 2 ↵sNc dz0 dx21 i i + 2 G12(z0s) 10 , 21(z0s) 2⇡ z0 x21 Z1 Z1 2 z s h i x10s 0 Initial Conditions • Initial conditions for this evolution are given by the lowest order t-channel gluon exchanges: 0 0 1 i 1 i b b ↵2C 1 d2b Gi (0)(zs)= d2b i (0) (zs)= s F ⇡✏ij xj ln 10 10 10 10,21 N 10 x ⇤ Z Z c 10 • Note that these initial conditions have no ln s, unlike the initial conditions for the quark evolution: ↵2C d2b G(0)(zs)= d2b (0) (zs)= s F ⇡ ln(zsx2 ) 10 10 10 10,21 N 10 Z Z c Large-Nc Evolution: Power Counting • The kernel mixing Gi or Gi with G and G is LLA: z ij j i i (0) ↵sNc dz0 2 1 ✏T (x21) gen G10(zs)=G10 (zs)+ 2 d x2 ln 2 ? 20 , 21(z0s)+G21(z0s) 2⇡ z0 x21⇤ x21 ⇤Z2 Z h i s z ij j ↵sNc dz0 2 1 ✏T (x20) gen gen 2 d x2 ln 2 ? 20 , 21(z0s)+ 21 , 20(z0s) 2⇡ z0 x21⇤ x20 ⇤Z2 Z h i s 2 z x10 2 LLA ↵sNc dz0 dx21 i i + 2 G12(z0s) 10 , 21(z0s) 2⇡ z0 x21 Z1 Z1 2 z s h i x10s 0 DLA • But, the initial conditions for G and G have an extra ln # as compared to i i & & G and G , making the two terms comparable (order-$% in $% '( # ~1 DLA power counting). Large-Nc Evolution Equations: Solution • These equations can be solved in the asymptotic high-energy region yielding the small-x gluon helicity intercept G 13 ↵s Nc ↵s Nc ↵h = 1.88 4p3 r 2⇡ ⇡ r 2⇡ • We obtain the small-x asymptotics of the gluon helicity distributions: 13 p ↵s Nc 1 4p3 2⇡ G(x, Q2) gGdip(x, k2 ) ⇠ 1L T ⇠ x ✓ ◆ 66 Main Physics Results • At large Nc we get for helicity ↵q 1 h 4 ↵ N ↵ N q(x, Q2) with ↵q = s c 2.31 s c ⇠ x h p3 2⇡ ⇡ 2⇡ ✓ ◆ r r G 1 ↵h ↵ N ↵ N G(x, Q2) with ↵G = 13 s c 1.88 s c ⇠ x h 4p3 2⇡ ⇡ 2⇡ ✓ ◆ r r 67 Orbital Angular Momentum • The small-x asymptotics of the quark and gluon orbital angular momentum (OAM) in the Jaffe-Manohar decomposition shown above can be tackled similarly (YK, 2019). • In the large-NC limit we obtain 4 p ↵s Nc 1 p3 2⇡ L (x, Q2)= ⌃(x, Q2) , q+¯q ⇠ x ✓ ◆ 13 p ↵s Nc 1 4p3 2⇡ L (x, Q2) G(x, Q2) G ⇠ ⇠ x Yu.K., D. Pitonyak, M. Sievert, arXiv:1706.04236 [nucl-th] Impact of our DS on the proton spin 2 ↵ • We have attached a ⌃˜ ( x, Q )= Nx h curve to the existing hPDF’s fits at some ad hoc small value of x labeled x0 : xΔu(x,Q2) 0.08 DSSV14 This work (x0 =0.03) 0.06 This work (x0 =0.01) 0.04 This work (x0 =0.001) 0.02 x “ballpark” 10-9 10-7 10-5 10-3 phenomenology xΔs(x,Q2) DSSV14 x This work (x =0.03) 10-7 10-4 0 -0.002 This work (x0 =0.01) This work (x0 =0.001) -0.004 -0.006 -0.008 70 Impact of our DS on the proton spin 1 [x ] 2 2 • Defining ⌃ min ( Q ) dx ⌃ ( x, Q ) we plot it for x0=0.03, 0.01, ⌘ 0.001: Zxmin ΔΣ [xmin](Q 2) 0.5 DSSV14 0.4 This work (x0 =0.03) 0.3 This work (x0 =0.01) This work (x =0.001) 0.2 0 0.1 xmin 10-8 10-5 10-2 • We observe a moderate to significant enhancement of quark spin. • More detailed phenomenology is needed in the future. 71 Impact on proton spin • Here we compare our results for the total quark helicity with DSSV, now including their error band. • We observe consistency of our lower two curves with DSSV. • Our upper curve disagrees with DSSV, but agrees with NNPDF (Nocera, Santopinto, ‘16). • Better phenomenology is needed. EIC would definitely play a role. 72 Impact of our DG on the proton spin 2 ↵G • We have attached a G ˜ ( x, Q )= Nx h curve to the existing hPDF’s fits at some ad hoc small value of x labeled x0 : xΔg(x,Q2) 0.10 DSSV14 0.08 This work (x0 =0.08) 0.06 This work (x0 =0.05) 0.04 This work (x0 =0.001) 0.02 x 10-8 10-5 10-2 “ballpark” phenomenology 73 Impact of our DG on the proton spin 1 • Defining S [ x min ]( Q 2 ) dx G ( x, Q 2 ) we plot it for x =0.08, 0.05, G ⌘ 0 0.001: Zxmin [xmin] 2 SG (Q ) 0.4 DSSV14 This work (x0 =0.08) 0.3 This work (x0 =0.05) 0.2 This work (x0 =0.001) 0.1 xmin 10-8 10-5 10-2 • We observe a moderate enhancement of gluon spin. • More detailed phenomenology is needed in the future. 74 EIC & Spin Puzzle • Parton helicity distributions are sensitive to low-x physics. • EIC would have an unprecedented low-x reach for a polarized DIS experiment, allowing to pinpoint the values of quark and gluon contributions to proton’s spin: 1 3 10 Current polarized DIS data: Q2 = 10 GeV2 CERN DESY JLab SLAC current Current polarized BNL-RHIC pp data: 0.5 data PHENIX π0 STAR 1-jet 2 ) 10 2 0.95 ≤ y 0.95 G ≤ ≤ y ∆ 0 (GeV ≤ 2 Q 10 √s= 140 GeV, 0.01 √s= 45 GeV, 0.01 DSSV+ EIC -0.5 EIC 5×100 EIC 5×250 EIC 20×250 2 1 all uncertainties for ∆χ =9 -4 -3 -2 -1 -1 10 10 10 10 1 0.3 0.35 0.4 0.45 x ∆Σ • DG and DS are integrated over x in the 0.001 < x < 1 interval. 75 EIC: Solving the Spin Puzzle ) ) 1 2 2 0.6 DSSV 2014 DSSV 2014 ) 2 2 2 0.6 eRHIC data: with 90% C.L. band (x,Q Q = 10 GeV 15 100 GeV g(x,Q × 0.8 (x,Q eRHIC data: ∆Σ ∆ + 15 × 250 GeV 15 × 100 GeV g] + 20 × 250 GeV min 1 ∆ + 15 250 GeV dx × x min ∫ 1 dx x 0.4 all bands 90% C.L. ∫ + 20 × 250 GeV 0.4 + 0.6 all bands 90% C.L. arXiv:1409.1633 ∆Σ 0.4 0.2 DSSV 2014 min 0.2 1 dx [1/2 x with 90% C.L. band ∫ eRHIC data: 0.2 15 × 100 GeV − 0 + 15 × 250 GeV 2 2 + 20 × 250 GeV 1/2 Q = 10 GeV all bands 90% C.L. 2 2 0 Q = 10 GeV 0 -0.2 -6 -5 -4 -3 -2 -1 -6 -5 -4 -3 -2 -1 -6 -5 -4 -3 -2 -1 10 10 10 10 10 10 1 10 10 10 10 10 10 1 10 10 10 10 10 10 1 xmin xmin xmin 1/2 - Gluon - Quarks = orbital angular • Above plot shows the • EIC will also reduce momentum running integral of the uncertainty on the Constraints on gluon and 2 ∆g(x,Q ) from xmin to 1 as a quark contribution to quark contributions will function of xmin the proton spin provide information on the orbital angular momentum • Large reduction in component of proton spin uncertainty on ∆G from EIC can be seen E. Aschenaur et al, arXiv:1409.1633 [hep-ph] 76 Conclusions • At large Nc we have obtained the following small-x asymptotics: q ↵h 2 1 q 4 ↵s Nc ↵s Nc q(x, Q ) with ↵h = 2.31 ⇠ x p3 r 2⇡ ⇡ r 2⇡ ✓ ◆ G 1 ↵h ↵ N ↵ N G(x, Q2) with ↵G = 13 s c 1.88 s c ⇠ x h 4p3 2⇡ ⇡ 2⇡ ✓ ◆ r r 4 p ↵s Nc 1 p3 2⇡ L (x, Q2)= ⌃(x, Q2) , q+¯q ⇠ x ✓ ◆ 13 p ↵s Nc 1 4p3 2⇡ L (x, Q2) G(x, Q2) G ⇠ ⇠ x 78 Quark Helicity TMD at Small x • Dominance of diagram B can also be obtained by applying crossing symmetry to the SIDIS process (KS ‘15): x y y ⊥ ⊥ x ⊥ ⊥ c.c. z z =0 ⊥ ⊥ y x ⊥ ⊥ z c.c. ⊥ = the answer • Compare the last line ζ ξ to the diagram B: reflecting the cc amplitude into the k1 w amplitude reduces the above k2 diagram to the one on the right. 79 Quark and Gluon OAM at Small x Quark OAM: Definition • We begin by writing the (Jaffe-Manohar) quark OAM in terms of the Wigner distribution as 2 2 + d b db d k dk L = ? ? (b k) W (k, b) z (2⇡)3 z q,SIDIS 2 ik r 1 1 1 + W (k, b)=2 d rdr e · ¯↵ b r V 1 b r , X 2 b r 2 1 | i 2 ↵ X Z ⌧ 2 X ⇥ ⇤ 1 1 X V 1 ,b + r b + r b+ r 2 2 2 + 1 d d ib ˆ(b, r) = e · P + ˆ(0,r) P O 2P + (2⇡)3 2 O 2 • The resulting quark OAM “PDF” is + 2 2P 2 2 2 ik (⇣ ⇠) ⇣ + ⇠ 1 + Lq(x, Q )= d k d ⇣ d⇣ d ⇠ d⇠ e · k ¯↵(⇠) V⇠[⇠ , ] X (2⇡)3 ? 2 ⇥ 1 | i 2 ↵ X Z ✓ ◆ X D X V⇣ [ , ⇣ ] (⇣) + q 2 2p 2 2 ik (⇣ ⇠) 1 + 5 g (x, k )= d ⇣ d⇣ d ⇠ d⇠ e · ¯ (⇠) V [⇠ , ] V [ , ⇣ ] (⇣) 1L T (2⇡)3 2 ↵ ⇠ 1 ⇣ 1 ζξζξζξ x− x− 0 0 DEF ζξ ζξζξ Quark OAM: small-x expression • After some algebra we arrive at the following small-x expression for quark OAM: 1 8Nc x k dz 2 2 2 2 ik x10 10 f 2 f 2 Lq+¯q(x, Q )= d k d x10 d x1 e · x1 k G10(zs) [ q (x, Q )+ q¯ (x, Q )] (2⇡)5 ? x2 ⇥ 2 ⇥ z 10 k f Z ⇤2Z/s • The result is written in terms of the polarized dipole amplitude G10 (z). It seems we are done, right? • This is almost correct. The remaining minor technicality is that the above quark OAM depends on the “first moment” of the polarized dipole amplitude k 2 k I (x10,zs)= d x1 x1 G10(zs) • It turns out that the “first moment” of the polarized amplitude is subleading. It grows with energy as a smaller power of energy 2p ↵sNc Ik(x ,zs) zsx2 2⇡ • The gluon OAM story is similar. We start with the Wigner distribution definition 2 2 + d b db d k dk L = ? ? (b k) W (k, b) z (2⇡)3 z + Gdip 4 2 ixP ⇠ ik ⇠ W (k, b)= d⇠ d ⇠ e · xP + ? Z +i 1 [+] 1 1 +i 1 [ ] 1 1 tr F (b 2 ⇠) [b 2 ⇠,b+ 2 ⇠] F (b + 2 ⇠) [b + 2 ⇠,b 2 ⇠] + 2 4 2 2 2 ixP ⇠ ik ⇠ LG(x, Q )= d b db d k d⇠ d ⇠ (b k) e · (2⇡)3 x ? ? ? ⇥ Z +i 1 [+] 1 1 +i 1 [ ] 1 1 tr F (b 2 ⇠) [b 2 ⇠,b+ 2 ⇠] F (b + 2 ⇠) [b + 2 ⇠,b 2 ⇠] • Gluon OAM at small x can (similarly to the quark OAM) be rewritten in i terms of the “moment” of the polarized dipole amplitude G10 for the gluon helicity TMD. This object is different from the polarized amplitude for the quark. • We get 2 8iNc Q 2 2 2 ik x10 2 LG(x, Q )= d x10 d k e · (k x10) G5 x10,zs= g2 (2⇡)3 ? x 2 j i i j 2 jk k 2 d x1 x1 10 G10(zs)=x10 G4(x10,zs)+✏ x10 G5(x10,zs) • We write down and solve the equations for G5. Gluon OAM: small-x asymptotics • We arrive at the following relation ↵q Q2 L (x, Q2)= h ln G(x, Q2) G 4 ⇤2 • We conclude that ↵G 13 p ↵s Nc 1.88 p ↵s Nc 1 h 1 4p3 2⇡ 1 2⇡ L (x, Q2) G(x, Q2) G x x x • We have constructed the small-x asymptotics of the quark and gluon OAM (in the Jaffe-Manohar decomposition). • In the large-NC limit we obtain 4 p ↵s Nc 1 p3 2⇡ L (x, Q2)= ⌃(x, Q2) , q+¯q ⇠ x ✓ ◆ 13 p ↵s Nc 1 4p3 2⇡ L (x, Q2) G(x, Q2) G ⇠ ⇠ x • Theoretical calculations by Bartels, Ermolaev and Ryskin (BER, 1996) and by YK, Pitonyak and Sievert (2015-17, KPS). • KPS results (large-NC, this talk): ↵q 1 h 4 ↵ N ↵ N q(x, Q2) with ↵q = s c 2.31 s c ⇠ x h p3 2⇡ ⇡ 2⇡ ✓ ◆ G r r 1 ↵h ↵ N ↵ N G(x, Q2) with ↵G = 13 s c 1.88 s c ⇠ x h 4p3 2⇡ ⇡ 2⇡ ✓ ◆ r r • BER results at large-NC (for both quark and gluon helicity distributions): ↵BER = 17+p97 ↵sNc 3.66 ↵sNc h 2 2⇡ ⇡ 2⇡ q q q For finite NC and Nf=4 BER have 3.66 à 3.45. Helicity PDFs at Small-x 1 ↵ PDF(x) ⇠ x • The summary of the existing powers of x is • Similar disagreement exists for OAM distributions at small x. • However, even the smaller KPS powers lead to potentially important contributions to the proton spin coming from small x. (next) Comparison with BER σ1 σ1 p1 − k1 + k2 σ1 p1 − k2 p1 p1 p1 k1 k1 k1 k1 To better understand BER work, k1 − k2 k1 − k2 k1 − k2 we tried calculating one (real) k2 k2 k2 k2 k2 k2 step of DLA helicity evolution σ2 σ2 σ2 p2 p2 p2 for the qq->qq scattering. A B C p1 − k1 + k2 p1 − k2 p1 − k1 + k2 σ1 σ1 σ1 p1 p1 p1 k1 k1 k1 − k2 It appears that we have identified k1 − k2 k1 − k2 k2 k2 k2 k2 the k2>> k1 (or k1>> k2) regime σ2 σ2 σ2 p2 p2 p2 in which diagrams A, B, C, D, E, I p2 − k1 + k2 p2 − k1 + k2 D E F are DLA, which was not σ1 σ1 σ1 considered by BER for B, C, … I. p1 p1 p1 k1 − k2 k1 − k2 k1 − k2 k2 k2 k2 k2 k2 k2 σ2 σ2 σ2 p2 p2 p2 G H I 91