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 <x< 1 ) S (Q2 = 10 GeV2) 0.15 0.20 q ⇡ ÷ • The proton spin carried by the gluons is (for 0 . 05 <x< 1 ) 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 protons 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 γ⇤A 2 γ⇤ qq¯ 2 ~ σtot = ? 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 γ A 2 γ qq¯ 2 σtot⇤ (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 δ ab @Y WY [↵]=↵s d x d y a b ⌘~x ~y WY [↵] 2 ? ? ↵ (x−, ~x ) ↵ (y−, ~y ) ? ? 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