Small x Asymptotics of the Quark and Gluon Helicity Distributions - - PowerPoint PPT Presentation

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Small x Asymptotics of the Quark and Gluon Helicity Distributions - - PowerPoint PPT Presentation

Oct 14, 2022 268 likes •947 views

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

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SLIDE 1

Small x Asymptotics

  • f 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

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SLIDE 2

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 Nc):

2

∆G(x, Q2) ∼ ✓ 1 x ◆αG

h

with αG

h = 13 4 √ 3

r αs Nc 2π ≈ 1.88 r αs Nc 2π ∆q(x, Q2) ∼ ✓ 1 x ◆αq

h

with αq

h =

4 √ 3 r αs Nc 2π ≈ 2.31 r αs Nc 2π

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SLIDE 3

Our Goal: Proton Spin at Small x

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SLIDE 4

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

  • f quarks and gluons (right panel)

4

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SLIDE 5

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

with the net quark helicity distribution and ∆𝐻(𝑦, 𝑅') the gluon helicity distribution.

5

∆f(x, Q2) ≡ f +(x, Q2) − f −(x, Q2)

∆Σ ≡ ∆u + ∆¯ u + ∆d + ∆ ¯ d + ∆s + ∆¯ s

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SLIDE 6

Proton Helicity Sum Rule

  • Helicity sum rule:

with the net quark and gluon spin

  • Lq and Lg are the quark and gluon orbital angular momenta

1 2 = Sq + Lq + Sg + Lg

Sq(Q2) = 1 2

1

Z dx ∆Σ(x, Q2)

Sg(Q2) =

1

Z dx ∆G(x, Q2)

6

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SLIDE 7

Proton Spin Puzzle

  • The spin puzzle began when the EMC collaboration measured the proton

g1 structure function ca 1988. Their data resulted in

  • It appeared quarks do not carry all of the proton spin

(which would have corresponded to ).

  • Missing spin can be

– Carried by gluons – In the orbital angular momenta of quarks and gluons – At small x: – Or all of the above!

∆Σ ≈ 0.1 ÷ 0.2

∆Σ = 1

Sq(Q2) = 1 2

1

Z dx ∆Σ(x, Q2)

1 2 = Sq + Lq + Sg + Lg

Sq(Q2) = 1 2

1

Z dx ∆Σ(x, Q2)

Can’t integrate down to zero, use xmin instead!

Sg(Q2) =

1

Z dx ∆G(x, Q2)

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SLIDE 8

Proton Spin Pie Chart

  • The proton spin carried by the

quarks is estimated to be (for )

  • The proton spin carried by the

gluons is (for )

  • Unfortunately the uncertainties are
  • large. Note also that the x-ranges

are limited, with more spin (positive or negative) possible at small x.

Sq(Q2 = 10 GeV2) ≈ 0.15 ÷ 0.20 SG(Q2 = 10 GeV2) ≈ 0.13 ÷ 0.26

0.001 < x < 1 0.05 < x < 1

8

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SLIDE 9

How much spin is at small x?

  • E. Aschenaur et al, arXiv:1509.06489 [hep-ph]
  • Uncertainties are very large at small x!

9

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SLIDE 10

Spin at small x

  • The goal of this project is to provide theoretical understanding
  • f 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

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SLIDE 11

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]

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SLIDE 12

Quark Helicity Observables at Small x

  • One can show that the g1 structure function and quark helicity PDF (Dq)

and TMD at small-x can be expressed in terms of the polarized dipole amplitude (flavor singlet case):

  • Here s is cms energy squared, zi=L2/s,

x⊥ y⊥ w⊥ x⊥ y⊥ w⊥ k⊥ k⊥ σ σ γ∗ γ∗ γ∗ γ∗ z z σ′ σ′ q q Σ Σ

gS

1 (x, Q2) =

Nc Nf 2 π2αEM

1

Z

zi

dz z2(1 − z) Z dx2

01

" 1 2 X

λσσ0

|ψT

λσσ0|2 (x2

01,z) +

X

σσ0

|ψL

σσ0|2 (x2

01,z)

# G(x2

01, z),

∆qS(x, Q2) = Nc Nf 2π3

1

Z

zi

dz z

1 zQ2

Z

1 zs

dx2

01

x2

01

G(x2

01, z),

gS

1L(x, k2 T ) = 8 Nc Nf

(2π)6

1

Z

zi

dz z Z d2x01 d2x001 e−ik·(x01−x001) x01 · x001 x2

01x2 001

G(x2

01, z)

G(x2

01, z) ≡

Z d2b G10(z)

12

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SLIDE 13

Polarized Dipole

  • All flavor singlet small-x helicity observables depend on one object,

“polarized dipole amplitude”:

  • Double brackets denote an object with energy suppression scaled out:

D D O E E (z) ≡ zs D O E (z)

G10(z) ≡ 1 2Nc D D tr h V0V pol †

1

i + tr h V pol

1

V † i E E (z)

polarized quark (“polarized Wilson line”): eikonal propagation, non-eikonal spin-dependent interaction unpolarized quark

Vx ≡ P exp 2 4ig

∞

Z

−∞

dx+ A−(x+, 0−, x) 3 5

13

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SLIDE 14

“Polarized Wilson line”

k p2 p2 − k p1 σ σ′

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: 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)

+

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SLIDE 15

“Polarized Wilson line”

  • A simple calculation in A-=0 gauge yields the “polarized

Wilson line”: where is the spin-dependent sub-eikonal gluon field of the plus- direction moving target with helicity S. (𝐵* is the unpolarized eikonal field.)

15

V pol

x

= 1 2 s

1

Z

1

dx P exp 8 < :ig

1

Z

x−

dx0 A+(x0, x) 9 = ; ig r ⇥ ˜ A(x, x) P exp 8 > < > : ig

x−

Z

1

dx0 A+(x0, x) 9 > = > ;

AΣ(x−, x) = Σ 2p+

1

˜ A(x−, x)

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SLIDE 16

Polarized Dipole Amplitude

  • The polarized dipole amplitude is then defined by

with the standard light-cone Wilson line

16

G10(z) ⌘ 1 4Nc

∞

Z

−∞

dx− D tr h V0[1, 1]V1[1, x−] (ig) r ⇥ ˜ A(x−, x) V1[x−, 1] i + c.c. E (z)

Vx[b−, a−] = P exp 8 > < > : ig

b−

Z

a−

dx− A+(x−, x) 9 > = > ;

1 t z proton

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SLIDE 17

Quark Helicity TMDs: Small-x Evolution

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SLIDE 18

Evolution for Polarized Quark Dipole

  • We can evolve the polarized dipole operator and obtain its small-x

evolution equation:

  • From the first two graphs on the right we get

“c.c.”

  • ther LLA diagrams

1 2 1 k1 k2 x−

1

x−

2

G10(z) = G(0)

10 (z) + αs

π2

z

Z dz0 z0 Z d2x2 x2

21

1 Nc D D tr h tbV0taV †

1

i U pol ba

2

E E + . . .

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SLIDE 19

Evolution for Polarized Quark Dipole

∂Y

1 1 1 1 1 1 2 2 2 2 2

One can construct an evolution equation for the polarized dipole:

Spin-dependent (non-eikonal) vertex polarized particle box = target shock wave similar to unpolarized BK evolution

19

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SLIDE 20

Resummation Parameter

  • For helicity evolution the resummation parameter is different from BFKL,

BK or JIMWLK, which resum powers of leading logarithms (LLA)

  • Helicity evolution resummation parameter is double-logarithmic (DLA):
  • 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.

αs ln2 1 x

αs ln(1/x)

20

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SLIDE 21

Evolution for Polarized Quark Dipole

1 Nc D D tr h V unp V pol †

1

iE E (z) = 1 Nc D D tr h V unp V pol †

1

iE E

0 (z) + αs

2π2

z

Z

zi

dz0 z0 Z

ρ02

d2x2 x2

21

× ⇢ θ(x10 − x21) 2 Nc D D tr h tb V unp ta V unp †

1

i U pol ba

2

E E (z0) + θ(x2

10z − x2 21z0) 1

Nc D D tr h tb V unp ta V pol †

2

i U unp ba

1

E E (z0) + θ(x10 − x21) 1 Nc hD D tr h V unp V unp †

2

i tr h V unp

2

V pol †

1

iE E (z0) − Nc D D tr h V unp V pol †

1

iE E (z0) i

∂Y

1 1 1 1 1 1 2 2 2 2 2

ρ0 2 = 1 z0 s

  • hh. . .ii = 1

z s h. . .i Equation does not close!

21

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SLIDE 22

Polarized Dipole Evolution in the Large-Nc Limit

In the large-Nc limit the equations close, leading to a system of 2 equations:

∂ ∂ ln z

=

1

G10(z)

∂ ∂ ln z′

1 2

Γ02,21(z′) =

z z z′

+ + − + + −

Γ02,21(z) S21(z) Γ03,32(z′) S23(z′) 3 2 1 z 1 2 z′ z S03(z′) G32(z′) S02(z) G21(z) S03(z′) G23(z′) S02(z) G12(z) Γ01,21(z) Γ02,32(z′)

G10(z) = G(0)

10 (z) + αs Nc

2π

z

Z

zi

dz0 z0

x2

10

Z

ρ02

dx2

21

x2

21

[2 Γ02, 21(z0) S21(z0) + 2 G21(z0) S02(z0) + G12(z0) S02(z0) − Γ01, 21(z0)]

Γ02, 21(z0) = Γ(0)

02, 21(z0) + αs Nc

2π

z0

Z

zi

dz00 z00 min{x2

02,x2 21 z0/z00}

Z

ρ002

dx2

32

x2

32

[2 Γ03, 32(z00) S23(z00) + 2 G32(z00) S03(z00) + G23(z00) S03(z00) − Γ02, 32(z00)]

S = found from BK/JIMWLK, it is LLA

22

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SLIDE 23

Your friendly “neighborhood” 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:

  • We denote the evolution in the neighbor dipole 02 by

1 1 2 2, z′ 3, z′′

x2

21 z0 x2 32 z00

Γ02, 21(z0)

23

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SLIDE 24

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’)

  • The initial conditions are given by the Born-level graphs

G(x2

10, z)= G(0)(x2 10, z) + αs Nc

2π

z

Z

1 x2

10s

dz0 z0

x2

10

Z

1 z0s

dx2

21

x2

21

⇥ Γ(x2

10, x2 21, z0) + 3 G(x2 21, z0)

⇤ Γ(x2

10, x2 21, z0) = Γ(0)(x2 10, x2 21, z0) + αs Nc

2π

z0

Z

1 x2

10s

dz00 z00 min

n x2

10,x2 21 z0 z00

  • Z

1 z00s

dx2

32

x2

32

⇥ Γ(x2

10, x2 32, z00) + 3 G(x2 32, z00)

⇤

Γ(0)(x2

10, x2 21, z) = G(0)(x2 10, z) 24

G(0)(x2

10, z) = α2 sCF

Nc π h CF ln zs Λ2 − 2 ln(zs x2

10)

i

x0 x1 z 1 − z x0 x1 z 1 − z x0 x1 z 1 − z x0 x1 z 1 − z x0 x1 z 1 − z

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SLIDE 25

Quark Helicity TMDs: Small-x Asymptotics

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SLIDE 26

Prior Results

  • Small-x DLA evolution for the g1 structure function was first considered by

Bartels, Ermolaev and Ryskin (BER) in ‘96.

  • Including the mixing of quark and gluon ladders, they obtained

with zs = 3.45 for 4 quark flavors and zS=3.66 for pure glue.

  • The power is large: it becomes larger than 1 for realistic strong coupling of

the order of αs = 0.2 − 0.3, resulting in polarized PDFs which actually grow with decreasing x fast enough for the integral of the PDFs over the low-x region to be (potentially) large (infinite).

∆Σ ∼ g1 ∼ ✓ 1 x ◆zs √ αs Nc

2 π

Sq(Q2) = 1 2

1

Z dx ∆Σ(x, Q2)

26

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SLIDE 27

Numerical Solution

  • We discretize the equations

and solve them by progressively populating each fixed-h row in s.

  • The solution for G looks like this:

27

Gij =G(0)

ij + ∆η ∆s j−1

X

j0=i j0

X

i0=i

[Γii0j0 + 3 Gi0j0] , Γikj =Γ(0)

ikj + ∆η ∆s j−1

X

j0=i j0

X

i0=max{i,k+j0−j}

[Γii0j0 + 3 Gi0j0]

η = r αsNc 2π ln zs Λ2

s10 = r αsNc 2π ln 1 x2

10 Λ2

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SLIDE 28

Solution of the large-NC Equations

  • The resulting small-x asymptotics is
  • Our result, 2.31, is about 35% smaller than BER’s 3.66 any-NC pure glue.

28

gS

1 (x, Q2) ∼ ∆qS(x, Q2) ∼ gS 1L(x, k2 T ) ∼

✓ 1 x ◆αh ≈ ✓ 1 x ◆2.31

r αsNc 2π

αh ≈ 2.31 r αsNc 2π

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SLIDE 29

Scaling

  • Our numerical solution has

a scaling property!

  • The solution is well

approximated by

  • This motivated us to look for the solution in the following scaling form:

29

G(s10, η) ∝ e2.31 (η−s10)

G(s10, η) = G(η − s10) Γ(s10, s21, η0) = Γ(η0 − s10, η0 − s21)

η = r αsNc 2π ln zs Λ2

s10 = r αsNc 2π ln 1 x2

10 Λ2

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SLIDE 30

Scaling Equations

  • The large-Nc evolution equations can be rewritten in terms of

the scaling variables (not a trivial property, does not work for the large-Nc&Nf equations):

  • For simplicity, pick the following initial conditions:

30

G(ζ) = 1 +

ζ

Z dξ

ξ

Z dξ0 [Γ(ξ, ξ0) + 3 G(ξ0)] , Γ(ζ, ζ0) = 1 +

ζ0

Z dξ

ξ

Z dξ0 [Γ(ξ, ξ0) + 3 G(ξ0)] +

ζ

Z

ζ0

dξ

ζ0

Z dξ0 [Γ(ξ, ξ0) + 3 G(ξ0)]

G(0) = 1, Γ(ζ0, ζ0) = G(ζ0)

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SLIDE 31

Analytic Solution

  • These scaling equations can be solved exactly via Laplace

transform + a few clever tricks, yielding

  • As usual, the high-energy asymptotics is given by the right-

most pole in the complex w-plane: the pole is at 𝜕 = + 3

  • .

31

G(ζ) = Z dω 2πi eω ζ+ ζ

ω

ω2 − 1 ω (ω2 − 3), Γ(ζ, ζ0) = 4 Z dω 2πi eω ζ0+ ζ

ω

ω2 − 1 ω (ω2 − 3) − 3 Z dω 2πi eω ζ0+ ζ0

ω

ω2 − 1 ω (ω2 − 3).

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SLIDE 32

Analytic Solution and Intercept

  • The (dominant part of the) scaling solution is
  • The corresponding helicity intercept is
  • This is in complete agreement with the numerical solution!

32

G(ζ) ≈ 1 3e

4 p 3 ζ

Γ(ζ, ζ0) ≈ 1 3e

4 p 3 ζ0 ✓

4e

ζζ0 p 3 − 3

◆ = G(ζ0) ✓ 4e

ζζ0 p 3 − 3

◆

αq

h =

4 √ 3 r αs Nc 2π ≈ 2.3094 r αs Nc 2π

αq

h ≈ 2.31

r αs Nc 2π

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SLIDE 33

Quark Helicity at Small x

  • The small-x asymptotics of quark helicity is (at large Nc)

∆q(x, Q2) ∼ ✓ 1 x ◆αq

h

with αq

h =

4 √ 3 r αs Nc 2π ≈ 2.31 r αs Nc 2π

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SLIDE 34

Impact of our DS on the proton spin

  • We have attached a curve to the existing hPDF’s fits

at some ad hoc small value of x labeled x0 :

34

∆˜ Σ(x, Q2) = N x−αh

10-9 10-7 10-5 10-3 x 0.02 0.04 0.06 0.08 xΔu(x,Q2)

This work (x0=0.001) This work (x0=0.01) This work (x0=0.03) DSSV14

10-7 10-4 x

  • 0.008
  • 0.006
  • 0.004
  • 0.002

xΔs(x,Q2)

This work (x0=0.001) This work (x0=0.01) This work (x0=0.03) DSSV14

“ballpark” phenomenology

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SLIDE 35

Impact of our DS on the proton spin

  • Defining we plot it for x0=0.03, 0.01,

0.001:

  • We observe a moderate to significant enhancement of quark spin.
  • More detailed phenomenology is needed in the future.

35 10-8 10-5 10-2 xmin 0.1 0.2 0.3 0.4 0.5 ΔΣ [xmin](Q2)

This work (x0=0.001) This work (x0=0.01) This work (x0=0.03) DSSV14

∆Σ[xmin](Q2) ≡ Z 1

xmin

dx ∆Σ(x, Q2)

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SLIDE 36

Impact on proton spin

36

  • Here we compare our results with DSSV,

now including the 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.

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SLIDE 37

Gluon Helicity TMDs

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SLIDE 38

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:
  • Here U[+] and U[-] are future and

past Wilson line staples (hence the name `dipole’ TMD,

  • F. Dominguez et al ’11 – looks

like a dipole scattering on a proton):

gG

1 (x, k2 T ) = −2i SL

x P + Z d⇠− d2⇠ (2⇡)3 eixP + ξ−−ik·ξ D P, SL|✏ij

T tr

h F +i(0) U[+]†[0, ⇠] F +j(⇠) U[−][⇠, 0] i |P, SL E

ξ+=0

t z j i U^[+] U^[−] proton

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SLIDE 39

Dipole Gluon Helicity TMD

  • At small x, the definition of dipole gluon helicity TMD can be massaged

into

  • Here we obtain a new operator, which is a transverse vector (written

here in A-=0 gauge):

  • Note that can be thought of

as a transverse curl acting on and not just on -- different from the polarized dipole amplitude!

gG dip

1

(x, k2

T ) = 8i Nc SL

g2(2⇡)3 Z d2x10 eik·x10 ki

⊥✏ij T

Z d2b10 Gj

10(zs = Q2

x )

  • Gi

10(z) ≡

1 4Nc

∞

Z

−∞

dx− D tr h V0[∞, −∞]V1[−∞, x−] (−ig) ˜ Ai(x−, x) V1[x−, ∞] i + c.c. E (z)

1 t z i proton

Gi

10(z)

ki

⊥ ✏ij T

˜ Ai(x−, x)

slide-40
SLIDE 40

Dictionary

  • We seem to have two operators:
  • Quark helicity
  • Gluon helicity

G10(z) ⌘ 1 4Nc

∞

Z

−∞

dx− D tr h V0[1, 1]V1[1, x−] (ig) r ⇥ ˜ A(x−, x) V1[x−, 1] i + c.c. E (z) Gi

10(z) ≡

1 4Nc

∞

Z

−∞

dx− D tr h V0[∞, −∞]V1[−∞, x−] (−ig) ˜ Ai(x−, x) V1[x−, ∞] i + c.c. E (z)

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SLIDE 41

Gluon Helicity TMDs: Small-x Evolution

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SLIDE 42

Evolution Equation

  • To construct evolution equation for the operator 𝐻1

governing the gluon helicity TMD we resum similar (to the quark case) diagrams:

“c.c.”

  • ther LLA diagrams

1 2 1 k1 k2 x−

1

x−

2

i i i i

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SLIDE 43

Large-Nc Evolution: Diagrams

  • At large-Nc the equations are

=

z

∂ ∂Y

Gi

10(zs)

1 i

+

1 z 2 z′ Γgen

21 , 20(z′s)

1 z 2 z′ Γgen

20 , 21(z′s)

+

1 z 2 z′ Γgen

20 , 21(z′s)

+

“c.c.”

+

“c.c.”

+

1 z 2 z′

+

“c.c.” i

+

1 z 2 z′ i G21(z′s) i Gi

12(z′s)

+

1 z 2 z′ Γi

10 , 21(z′s)

i i i

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SLIDE 44

Large-Nc Evolution: Diagrams

  • and

∂ ∂Y

1 Γi

10 , 21(z′s)

1 Γ30 , 31(z′′s) 1

=

1 Γ30 , 31(z′′s)

+ +

“c.c.”

+

“c.c.”

+

“c.c.”

+

1 1 2 z′ 2 z′ 2 z′ 2 z′ 2 z′ 2 z′ 1 Γi

10 , 31(z′′s)

i G30(z′′s) 3 z′′ 3 z′′ Γ31 , 30(z′′s) Gi

30(z′′s)

i 3 z′′ 3 z′′ i 3 z′′ 3 z′′ i i i i

+ + +

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SLIDE 45

Large-Nc Evolution: Equations

  • This results in the following evolution equations:

Gi

10(zs) = Gi (0) 10 (zs) + ↵sNc

2⇡2

z

Z

Λ2 s

dz0 z0 Z d2x2 ln 1 x21Λ ✏ij

T (x21)j ?

x2

21

h Γgen

20 , 21(z0s) + G21(z0s)

i − ↵sNc 2⇡2

z

Z

Λ2 s

dz0 z0 Z d2x2 ln 1 x21Λ ✏ij

T (x20)j ?

x2

20

h Γgen

20 , 21(z0s) + Γgen 21 , 20(z0s)

i + ↵sNc 2⇡

z

Z

1 x2 10s

dz0 z0

x2

10

Z

1 z0s

dx2

21

x2

21

h Gi

12(z0s) − Γi 10 , 21(z0s)

i Γi

10 21(z0s) = Gi (0) 10 (z0s) + ↵sNc

2⇡2

z0

Z

Λ2 s

dz00 z00 Z d2x3 ln 1 x31Λ ✏ij

T (x31)j ?

x2

31

h Γgen

30 , 31(z00s) + G31(z00s)

i − ↵sNc 2⇡2

z0

Z

Λ2 s

dz00 z00 Z d2x3 ln 1 x31Λ ✏ij

T (x30)j ?

x2

30

h Γgen

30 , 31(z00s) + Γgen 31 , 30(z00s)

i + ↵sNc 2⇡2

z0

Z

1 x2 10s

dz00 z00

min  x2

10 , x2 21

z0 z00

  • Z

1 z00s

dx2

31

x2

31

h Gi

13(z00s) − Γi 10 , 31(z00s)

i .

slide-46
SLIDE 46

Large-Nc Evolution: Equations

  • Here

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:

Gi

10(zs) = Gi (0) 10 (zs) + ↵sNc

2⇡2

z

Z

Λ2 s

dz0 z0 Z d2x2 ln 1 x21Λ ✏ij

T (x21)j ?

x2

21

h Γgen

20 , 21(z0s) + G21(z0s)

i − ↵sNc 2⇡2

z

Z

Λ2 s

dz0 z0 Z d2x2 ln 1 x21Λ ✏ij

T (x20)j ?

x2

20

h Γgen

20 , 21(z0s) + Γgen 21 , 20(z0s)

i + ↵sNc 2⇡

z

Z

1 x2 10s

dz0 z0

x2

10

Z

1 z0s

dx2

21

x2

21

h Gi

12(z0s) − Γi 10 , 21(z0s)

i

Γgen

20,21(z0s) = θ(x20 − x21) Γ20,21(z0s) + θ(x21 − x20) G20(z0s)

slide-47
SLIDE 47

Initial Conditions

  • Initial conditions for this evolution are given by the lowest order t-channel

gluon exchanges:

  • Note that these initial conditions have no ln s, unlike the initial conditions

for the quark evolution:

b 1

i

1

i

b

Z d2b10 Gi (0)

10 (zs) =

Z d2b10 Γi (0)

10,21(zs) = −↵2 sCF

Nc ⇡ ✏ij xj

10 ln

1 x10 Λ Z d2b10 G(0)

10 (zs) =

Z d2b10 Γ(0)

10,21(zs) = −α2 sCF

Nc π ln(zs x2

10)

slide-48
SLIDE 48

Gluon Helicity TMDs: Small-x Asymptotics

slide-49
SLIDE 49

Large-Nc Evolution Equations: Scaling

  • Just like in the quark helicity evolution case, the equations

simplify once we recognize the following scaling property:

  • The equations become

G2(x2

10, zs) = G2

r αs Nc 2π ln(zsx2

10)

! Γ2(x2

10, x2 21, z0s) = Γ2

r αs Nc 2π ln(z0sx2

10),

r αs Nc 2π ln(z0sx2

21)

!

G2(ζ) = −1 2 r αs Nc 6π e

4 p 3 ζ −

ζ

Z dξ

ξ

Z dξ0 Γ2(ξ, ξ0), Γ2(ζ, ζ0) = −1 2 r αs Nc 6π e

4 p 3 ζ −

ζ0

Z dξ

ξ

Z dξ0 Γ2(ξ, ξ0) −

ζ

Z

ζ0

dξ

ζ0

Z dξ0 Γ2(ξ, ξ0)

slide-50
SLIDE 50

Large-NcEvolution Equations: Solution

  • These equations can be solved in the asymptotic high-energy region

using a combination of ODE solving and Laplace transform, yielding

  • The small-x gluon helicity intercept is
  • We obtain the small-x asymptotics of the gluon helicity distributions:

G2(ζ 1) = 1 3 r 2 αs Nc π 19 p 3 64 e

13 4 p 3 ζ,

Γ2(ζ 1, ζ0 1) = 1 3 r 2 αs Nc π "p 3 4 e

4 p 3 ζ p 3 4 ζ0

+ 3 p 3 64 e

4 p 3 ζ0 p 3 4 ζ

#

αG

h =

13 4 √ 3 r αs Nc 2π ≈ 1.88 r αs Nc 2π

∆G(x, Q2) ∼ gG dip

1L

(x, k2

T ) ∼

✓ 1 x ◆ 13

4 √ 3

√ αs Nc

2π

slide-51
SLIDE 51

Impact of our DG on the proton spin

  • We have attached a curve to the existing hPDF’s fits

at some ad hoc small value of x labeled x0 :

51

10-8 10-5 10-2 x 0.02 0.04 0.06 0.08 0.10 xΔg(x,Q2)

This work (x0=0.001) This work (x0=0.05) This work (x0=0.08) DSSV14

“ballpark” phenomenology ∆ ˜ G(x, Q2) = N x−αG

h

slide-52
SLIDE 52

Impact of our DG on the proton spin

  • Defining we plot it for x0=0.08, 0.05,

0.001:

  • We observe a moderate enhancement of gluon spin.
  • More detailed phenomenology is needed in the future.

52

10-8 10-5 10-2 xmin 0.1 0.2 0.3 0.4 SG

[xmin](Q2)

This work (x0=0.001) This work (x0=0.05) This work (x0=0.08) DSSV14

S[xmin]

G

(Q2) ≡ Z 1

xmin

dx ∆G(x, Q2)

slide-53
SLIDE 53

Conclusions

  • We conclude that the small-x asymptotics of gluon helicity (at large Nc)

is while the quark helicity asymptotics is

  • Preliminary results indicate a possible enhancement of quark and gluon

spin coming from small x as compared to DSSV.

  • Future work may involve including running coupling and saturation

corrections + solving the large-NC&Nf equations. We can use our method to determine the small-x asymptotics of quark and gluon OAMs.

  • One may use our approach to combine experiment and theory to

constrain the quark and gluon spin (and OAM) at small x (in progress, a long-term goal).

∆G(x, Q2) ∼ ✓ 1 x ◆αG

h

with αG

h = 13 4 √ 3

r αs Nc 2π ≈ 1.88 r αs Nc 2π ∆q(x, Q2) ∼ ✓ 1 x ◆αq

h

with αq

h =

4 √ 3 r αs Nc 2π ≈ 2.31 r αs Nc 2π

slide-54
SLIDE 54

INT Program on EIC Physics, Fall 2018

  • Probing Nucleons and Nuclei in High Energy Collisions (INT-18-3)

October 1 - November 16, 2018

  • Y. Hatta, Y. Kovchegov, C. Marquet, A. Prokudin
  • Institute for Nuclear Theory, Seattle, WA
  • Please mark

your calendars!

slide-55
SLIDE 55

Backup Slides

55

slide-56
SLIDE 56

Large-Nc&Nf Evolution

  • The evolution equations read (in the strict DLA limit, S=1):

56

slide-57
SLIDE 57

Comparison with BER

A

p1 p2 k1 k2 k1 − k2 k2 k1 σ1 σ2

B

p1 p2 k1 k2 k1 − k2 k2 σ1 σ2

C

p1 p2 k1 k2 k1 − k2 k2 σ1 σ2 p1 − k1 + k2 p1 − k2

D

p1 p2 k1 k1 − k2 k2 σ1 σ2 p1 − k1 + k2 p2 − k1 + k2

E

p1 p2 k1 k1 − k2 k2 σ1 σ2 p1 − k2 p2 − k1 + k2

F

p1 p2 k1 − k2 k2 σ1 σ2 p1 − k1 + k2 k2

57

H G I

p1 p2 k1 − k2 k2 σ1 σ2 k2 p1 p2 k1 − k2 k2 σ1 σ2 k2 p1 p2 k1 − k2 k2 σ1 σ2 k2

To better understand BER work, we tried calculating one (real) step of DLA helicity evolution for the qq->qq scattering. It appears that we have identified the k2>> k1 (or k1>> k2) regime in which diagrams A, B, C, D, E, I are DLA, which was not considered by BER for B, C, … I.

slide-58
SLIDE 58

Intercepts

58

Here we plot our (flavor-singlet) helicity intercept as a function of the coupling. We show BER result and LO BFKL (all twist and leading twist) for comparison.

0.1 0.2 0.3 0.4 0.5 αs 0.5 1.0 1.5 2.0 2.5 3.0 Intercept

αh (BER) αh (this work) LO BFKL (twist-2) LO BFKL (all twist)

slide-59
SLIDE 59

Helicity Evolution at Small x

flavor non-singlet case

Yu.K., D. Pitonyak, M. Sievert, arXiv:1610.06197 [hep-ph]

59

slide-60
SLIDE 60

Flavor Non-Singlet Observables

  • In the flavor non-singlet case, all helicity observables again depend on the

polarized dipole amplitude:

  • Polarized dipole amplitude is different (difference instead of sum):
  • This is related to the definition

60

gNS

1

(x, Q2) = Nc 2 π2αEM

1

Z

zi

dz z2(1 − z) Z dx2

01

" 1 2 X

λσσ0

|ψT

λσσ0|2 (x2

01,z) +

X

σσ0

|ψL

σσ0|2 (x2

01,z)

# GNS(x2

01, z),

∆qNS(x, Q2) = Nc 2π3

1

Z

zi

dz z

1 zQ2

Z

1 zs

dx2

01

x2

01

GNS(x2

01, z),

gNS

1L (x, k2 T ) = 8 Nc

(2π)6

1

Z

zi

dz z Z d2x01 d2x001 e−ik·(x01−x001) x01 · x001 x2

01x2 001

GNS(x2

01, z)

GNS

10 (z) ≡

1 2Nc D D tr h V0V pol †

1

i − tr h V pol

1

V † i E E (z)

∆qNS(x, Q2) ≡ ∆qf(x, Q2) − ∆¯ qf(x, Q2)

slide-61
SLIDE 61

Flavor Non-Singlet Evolution

  • Evolution equations end up being much simpler in the non-singlet case:
  • Analytical solution (in the DLA case, S=1) leads to (in agreement with

Bartels et al, ‘95)

  • The resulting intercept is smaller than the flavor-singlet intercept.

61

∂ ∂ ln z

=

1

GNS

10 (z)

z GNS

21 (z)

S10(z) 1 z 2 GNS

10 (z) = GNS (0) 10

(z) + αsNc 4π

z

Z

Λ2 s

dz0 z0

x2

10

z z0

Z

1 z0s

dx2

21

x2

21

S10(z0) GNS

21 (z0)

gNS

1

(x, Q2) ∼ ∆qNS(x, Q2) ∼ gNS

1L (x, k2 T ) ∼

✓ 1 x ◆αNS

h

≈ ✓ 1 x ◆

r αsNc π

slide-62
SLIDE 62

Dipole TMD vs dipole amplitude

  • Note that the operator for the dipole gluon helicity TMD

is different from the polarized dipole amplitude

  • 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 TMD.)

  • This is different from the unpolarized gluon TMD case.

Gi

10(z) ≡

1 4Nc

∞

Z

−∞

dx− D tr h V0[∞, −∞]V1[−∞, x−] (−ig) ˜ Ai(x−, x) V1[x−, ∞] i + c.c. E (z) G10(z) ⌘ 1 4Nc

∞

Z

−∞

dx− D tr h V0[1, 1]V1[1, x−] (ig) r ⇥ ˜ A(x−, x) V1[x−, 1] i + c.c. E (z)

slide-63
SLIDE 63

Large-Nc Evolution: Power Counting

  • The kernel mixing Gi or Gi with G and G is LLA:
  • But, the initial conditions for G and G have an extra ln 𝑡 as compared to

Gi and Gi, making the two terms comparable (order-𝛽6

' in 𝛽6 𝑚𝑜'𝑡 ~1

DLA power counting).

Gi

10(zs) = Gi (0) 10 (zs) + ↵sNc

2⇡2

z

Z

Λ2 s

dz0 z0 Z d2x2 ln 1 x21Λ ✏ij

T (x21)j ?

x2

21

h Γgen

20 , 21(z0s) + G21(z0s)

i − ↵sNc 2⇡2

z

Z

Λ2 s

dz0 z0 Z d2x2 ln 1 x21Λ ✏ij

T (x20)j ?

x2

20

h Γgen

20 , 21(z0s) + Γgen 21 , 20(z0s)

i + ↵sNc 2⇡

z

Z

1 x2 10s

dz0 z0

x2

10

Z

1 z0s

dx2

21

x2

21

h Gi

12(z0s) − Γi 10 , 21(z0s)

i

LLA DLA

slide-64
SLIDE 64

Large-NcGluon Helicity Evolution Equations: Solution

  • To solve the equations, first decompose the relevant object as follows:
  • It turns out that only G2 and G2 contribute to evolution and to the gluon

helicity TMD.

Z d2b Gi

10(z) = xi 10 G1(x2 10, z) + ✏ij xj 10 G2(x2 10, z)

Z d2b Γi

10(z) = xi 10 Γ1(x2 10, z) + ✏ij xj 10 Γ2(x2 10, z)

slide-65
SLIDE 65

Large-NcEvolution Equations: Solution

  • Plugging in the analytic solution for the quark helicity operator, we get

G2(x2

10, zs) = G(0) 2 (x2 10, zs) − αsNc

3π 1

4 p 3

q

αs Nc 2π

  • zsx2

10

4

p 3

√ αs Nc

2π

ln 1 x10Λ − αsNc 2π

z

Z

1 x2 10s

dz0 z0

x2

10

Z

1 z0s

dx2

21

x2

21

Γ2(x2

10, x2 21, z0s),

Γ2(x2

10, x2 21, z0s) = G(0) 2 (x2 10, z0s) − αsNc

3π 1

4 p 3

q

αs Nc 2π

  • z0sx2

10

4

p 3

√ αs Nc

2π

ln 1 x10Λ − αsNc 2π

z0

Z

1 x2 10s

dz00 z00

min  x2

10 , x2 21

z0 z00

  • Z

1 z00s

dx2

31

x2

31

Γ2(x2

10, x2 31, z00s)

slide-66
SLIDE 66

Scaling Solution Cross-Check

  • One can check the scaling property of our analytic

solution in the numerical solution of our equations:

  • ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■

◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ 7 8 9 10 η 1.0 1.1 1.2 1.3 Γ2(s10,s21,η)/G2(s10,η) ◆ s21-s10=0.8 ■ s21-s10=0.5

  • s21-s10=0.1
  • ■

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ 0.05 0.10 0.15 0.20 0.25 0.30 0.35 (s21-s10) 1.00 1.01 1.02 1.03 1.04 1.05 Γ2(s10,s21,η)/G2(s10,η) ◆ η=9.5 ■ η=8.5

  • η=7.5

η = r αsNc 2π ln zs Λ2

s10 = r αsNc 2π ln 1 x2

10 Λ2

G2(s10, η) = G2(η − s10) Γ2(s10, s21, η0) = Γ2(η0 − s10, η0 − s21) Γ2 G2 = f(s21 − s10)