Small x Asymptotics of the Quark and Gluon Helicity Distributions Yuri Kovchegov The Ohio State University work with Dan Pitonyak and Matt Sievert, arXiv:1706.04236 [nucl-th] and 5 other papers
Outline • Goal: understanding the proton spin coming from small x partons • 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 • Main results (at large N c ): ✓ 1 ◆ α q r r 4 α s N c α s N c h α q ∆ q ( x, Q 2 ) ∼ with h = ≈ 2 . 31 √ 2 π 2 π x 3 ✓ 1 ◆ α G r r h α s N c α s N c α G ∆ G ( x, Q 2 ) ∼ 13 with h = ≈ 1 . 88 √ 2 π 2 π x 4 3 2
Our Goal: Proton Spin at Small x
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 one • 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 • ∆ f ( x, Q 2 ) ≡ f + ( x, Q 2 ) − f − ( x, Q 2 ) with the net quark helicity distribution u + ∆ d + ∆ ¯ ∆Σ ≡ ∆ u + ∆ ¯ d + ∆ s + ∆ ¯ s and ∆𝐻(𝑦, 𝑅 ' ) the gluon helicity distribution. 5
Proton Helicity Sum Rule Helicity sum rule: • 1 2 = S q + L q + S g + L g with the net quark and gluon spin 1 1 Z S q ( Q 2 ) = 1 Z S g ( Q 2 ) = dx ∆ G ( x, Q 2 ) dx ∆Σ ( x, Q 2 ) 2 0 0 L q and L g are the quark and gluon orbital angular momenta • 6
Proton Spin Puzzle 1 S q ( Q 2 ) = 1 Z dx ∆Σ ( x, Q 2 ) 2 0 • The spin puzzle began when the EMC collaboration measured the proton g 1 structure function ca 1988. Their data resulted in ∆Σ ≈ 0 . 1 ÷ 0 . 2 It appeared quarks do not carry all of the proton spin • (which would have corresponded to ). ∆Σ = 1 1 Missing spin can be 2 = S q + L q + S g + L g • Carried by gluons – In the orbital angular momenta of quarks and gluons – – At small x: 1 1 S q ( Q 2 ) = 1 Z Z dx ∆Σ ( x, Q 2 ) S g ( Q 2 ) = dx ∆ G ( x, Q 2 ) 2 0 0 Can’t integrate down to zero, use x min instead! Or all of the above! –
Proton Spin Pie Chart The proton spin carried by the • quarks is estimated to be (for ) 0 . 001 < x < 1 S q ( Q 2 = 10 GeV 2 ) ≈ 0 . 15 ÷ 0 . 20 The proton spin carried by the • gluons is (for ) 0 . 05 < x < 1 S G ( Q 2 = 10 GeV 2 ) ≈ 0 . 13 ÷ 0 . 26 • Unfortunately the uncertainties are large. Note also that the x-ranges are limited, with more spin (positive or negative) possible at small x. 8
How much spin is at small x? • E. Aschenaur et al, arXiv:1509.06489 [hep-ph] • Uncertainties are very large at small x! 9
Spin at small x • The goal of this project is to provide theoretical understanding of helicity PDF’s at very small x. • Our work would provide guidance for future hPDF’s 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
Quark Helicity Evolution 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]
Quark Helicity Observables at Small x k ⊥ k ⊥ γ ∗ γ ∗ γ ∗ γ ∗ y ⊥ x ⊥ σ σ x ⊥ y ⊥ q z q z σ ′ w ⊥ w ⊥ σ ′ Σ Σ • One can show that the g 1 structure function and quark helicity PDF ( D q) and TMD at small-x can be expressed in terms of the polarized dipole amplitude (flavor singlet case): 1 " # 1 N c N f Z dz Z X X g S 1 ( x, Q 2 ) = dx 2 | ψ T λσσ 0 | 2 | ψ L σσ 0 | 2 G ( x 2 01 ,z ) + 01 , z ) , 01 ( x 2 ( x 2 01 ,z ) 2 π 2 α EM z 2 (1 − z ) 2 λσσ 0 σσ 0 z i 1 1 zQ 2 dx 2 ∆ q S ( x, Q 2 ) = N c N f Z dz Z 01 G ( x 2 01 , z ) , 2 π 3 x 2 z 01 z i 1 zs 1 T ) = 8 N c N f Z Z d 2 x 01 d 2 x 0 0 1 e − ik · ( x 01 − x 0 0 1 ) x 01 · x 0 0 1 dz g S 1 L ( x, k 2 G ( x 2 01 , z ) x 2 01 x 2 (2 π ) 6 z 0 0 1 z i • Here s is cms energy squared, z i = L 2 / s, Z G ( x 2 d 2 b G 10 ( z ) 01 , z ) ≡ 12
Polarized Dipole • All flavor singlet small-x helicity observables depend on one object, “polarized dipole amplitude”: 1 D D h i h i E E V 0 V pol † V † V pol G 10 ( z ) ≡ tr + tr ( z ) 1 1 0 2 N c unpolarized quark polarized quark (“polarized Wilson line”): eikonal propagation, non-eikonal 2 3 ∞ Z dx + A − ( x + , 0 − , x ) V x ≡ P exp 4 ig spin-dependent interaction 5 −∞ • Double brackets denote an object with energy suppression scaled out: D D E E D E ( z ) ≡ zs ( z ) O O 13
“Polarized Wilson line” To obtain an explicit expression for the “polarized Wilson line” operator due to a sub-eikonal gluon exchange (as opposed to the sub-eikonal quarks exchange), consider multiple Coulomb gluon exchanges with the target: p 2 p 2 − k - σ ′ σ k + p 1 Most gluon exchanges are eikonal spin-independent interactions, while one of them is a spin-dependent sub-eikonal exchange. (cf. Mueller ‘90, McLerran, Venugopalan ‘93)
“Polarized Wilson line” • A simple calculation in A - =0 gauge yields the “polarized Wilson line”: 8 9 8 9 x − 1 1 = 1 > > Z Z Z dx � P exp < = < = dx 0� A + ( x 0� , x ) dx 0� A + ( x 0� , x ) ; ig r ⇥ ˜ V pol A ( x � , x ) P exp : ig ig x 2 s > > : ; �1 x − �1 Σ where ˜ A Σ ( x − , x ) = A ( x − , x ) 2 p + 1 is the spin-dependent sub-eikonal gluon field of the plus- direction moving target with helicity S . ( 𝐵 * is the unpolarized eikonal field.) 15
Polarized Dipole Amplitude • The polarized dipole amplitude is then defined by ∞ 1 Z dx − D h i E V 0 [ 1 , �1 ] V 1 [ �1 , x − ] ( � ig ) r ⇥ ˜ G 10 ( z ) ⌘ tr A ( x − , x ) V 1 [ x − , 1 ] + c.c. ( z ) 4 N c −∞ with the standard light-cone t Wilson line 8 9 b − > > Z < = dx − A + ( x − , x ) V x [ b − , a − ] = P exp ig > > : ; a − z 1 0 proton 16
Quark Helicity TMDs: Small-x Evolution
Evolution for Polarized Quark Dipole • We can evolve the polarized dipole operator and obtain its small-x evolution equation: 0 0 2 “c.c.” k 1 k 2 1 1 x − x − 1 2 other LLA diagrams • From the first two graphs on the right we get Z dz 0 Z d 2 x 2 z 1 10 ( z ) + α s D D h i E E G 10 ( z ) = G (0) t b V 0 t a V † U pol ba tr + . . . 1 2 x 2 π 2 z 0 N c 21
Evolution for Polarized Quark Dipole One can construct an evolution equation for the polarized dipole: 0 0 2 ∂ Y 1 1 0 Spin-dependent (non-eikonal) vertex 2 polarized 1 particle 0 2 1 similar to 0 unpolarized 2 BK evolution 1 box = target shock 0 wave 2 1 19
Resummation Parameter • For helicity evolution the resummation parameter is different from BFKL, BK or JIMWLK, which resum powers of leading logarithms (LLA) α s ln(1 /x ) • Helicity evolution resummation parameter is double-logarithmic (DLA): α s ln 2 1 x • The second logarithm of x arises due to transverse momentum (or transverse coordinate) integration being logarithmic both in 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. 20
Evolution for Polarized Quark Dipole 0 0 ∂ Y 2 1 1 0 2 1 hh . . . ii = 1 0 z s h . . . i 2 1 0 2 1 1 ρ 0 2 = z 0 s 0 2 1 z d 2 x 2 1 ( z ) = 1 0 ( z ) + α s Z dz 0 Z D D h iE E D D h iE E V unp V pol † V unp V pol † tr tr 0 1 0 1 x 2 2 π 2 N c N c z 0 21 z i ρ 0 2 ⇢ θ ( x 10 − x 21 ) 2 D D h i E E t b V unp t a V unp † U pol ba ( z 0 ) tr × Equation does not close! 0 1 2 N c 21 z 0 ) 1 D D h i E E t b V unp t a V pol † U unp ba + θ ( x 2 10 z − x 2 ( z 0 ) tr 0 2 1 N c i� + θ ( x 10 − x 21 ) 1 hD D h i h iE E D D h iE E V unp V unp † V unp V pol † V unp V pol † tr tr ( z 0 ) − N c tr ( z 0 ) 0 2 2 1 0 1 N c 21
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