Quark orbital angular momentum in the proton

evaluated using a direct derivative method

Michael Engelhardt

LHPC

Acknowledgments:M. Burkardt, S. Liuti

Gauge ensembles provided by:R. Edwards, B. Joo and K. Orginos

Direct evaluation of quark orbital angular momentum

LU3 = dx d2kT d2rT (rT × kT )3 WU(x, kT , rT ) Wigner distribution

∫ ∫ ∫

LU3n

=ǫij

∂∂zT,i

∂∂∆T,j

〈p′, S | ψ(−z/2)γ+U [−z/2, z/2]ψ(z/2) | p, S〉∣

∣

∣

∣

∣ z+=z−=0 , ∆T=0 , zT→0

〈p′, S | ψ(−z/2)γ+U [−z/2, z/2]ψ(z/2) | p, S〉| z+=z−=0 , ∆T=0 , zT→0

n : Number of valence quarks

p′ = P + ∆T/2, p = P −∆T/2, P, S in 3-direction, P → ∞

This is the same type of operator as used in TMD studies – generalization to off-forward matrix elementadds transverse position information

Direct evaluation of quark orbital angular momentum

LU3n

=ǫij

∂∂zT,i

∂∂∆T,j

〈p′, S | ψ(−z/2)γ+U [−z/2, z/2]ψ(z/2) | p, S〉∣

∣

∣

∣

∣ z+=z−=0 , ∆T=0 , zT→0

〈p′, S | ψ(−z/2)γ+U [−z/2, z/2]ψ(z/2) | p, S〉| z+=z−=0 , ∆T=0 , zT→0

Role of the gauge link U :

�

η→∞

z

2ηv +

z

2

−z

2ηv −

z

2

v

Y. Hatta, M. Burkardt:

• Straight U [−z/2, z/2] −→ Ji OAM

• Staple-shaped U [−z/2, z/2] −→ Jaffe-Manohar OAM

• Difference is torque accumulated due to final state interaction

Direct evaluation of quark orbital angular momentum

LU3n

=ǫij

∂∂zT,i

∂∂∆T,j

〈p′, S | ψ(−z/2)γ+U [−z/2, z/2]ψ(z/2) | p, S〉∣

∣

∣

∣

∣ z+=z−=0 , ∆T=0 , zT→0

〈p′, S | ψ(−z/2)γ+U [−z/2, z/2]ψ(z/2) | p, S〉| z+=z−=0 , ∆T=0 , zT→0

Role of the gauge link U :

�

η→∞

z

2ηv +

z

2

−z

2ηv −

z

2

v

Direction of staple taken off light cone (rapidity divergences)

Characterized by Collins-Soper parameter ζ ≡P · v

|P ||v|

Are interested in ζ −→ ∞; synonymous with P −→ ∞ in theframe of the lattice calculation (v = e3)

Ensemble details

Clover ensemble provided by R. Edwards, B. Joo and K. Orginos (JLab / W&M Collaboration)

L3 × T a(fm) mπ (MeV) mN (GeV) #conf. #meas.

323 × 96 0.11403(77) 317(2)(2) 1.077(8) 967 23208

Direct evaluation of quark orbital angular momentum

LU3n

=ǫij

∂∂zT,i

∂∂∆T,j

〈p′, S | ψ(−z/2)γ+U [−z/2, z/2]ψ(z/2) | p, S〉∣

∣

∣

∣

∣ z+=z−=0 , ∆T=0 , zT→0

〈p′, S | ψ(−z/2)γ+U [−z/2, z/2]ψ(z/2) | p, S〉| z+=z−=0 , ∆T=0 , zT→0

�

η→∞

z

2ηv +

z

2

−z

2ηv −

z

2

v

Parameters to consider: ∆, z, ζ , η

Direct evaluation of quark orbital angular momentum

LU3n

=ǫij

∂∂zT,i

∂∂∆T,j

〈p′, S | ψ(−z/2)γ+U [−z/2, z/2]ψ(z/2) | p, S〉∣

∣

∣

∣

∣ z+=z−=0 , ∆T=0 , zT→0

〈p′, S | ψ(−z/2)γ+U [−z/2, z/2]ψ(z/2) | p, S〉| z+=z−=0 , ∆T=0 , zT→0

Perform ∆T -derivative using Rome method (Phys. Lett. B718 (2012) 589)

G(x, y; ~p) = e−i~p(~x−~y)G(x, y)

∂

∂pjG(x, y; ~p)

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣~p=0

= −i G(x, z)ΓjVG(z, y)

Clover fermions: ΓjVG(z, y) = U

†j (z − )

1 + γj

2G(z − , y)− Uj(z)

1− γj

2G(z + , y)

(further contributions from derivatives of source/sink smearings)

∑

z

Direct evaluation of quark orbital angular momentum

0 1 2 3 4-0.5

-0.4

-0.3

-0.2

-0.1

0.0

d

L3/n

u-d quarks

m = 317 MeV

= 0.315

|v| = 0|v| = 3a|v| = 7a

∂f

∂zT,i

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣zT,i=0

=1

2da(f (daei)− f (−daei))

Direct evaluation of quark orbital angular momentum

LU3n

=ǫij

∂∂zT,i

∂∂∆T,j

〈p′, S | ψ(−z/2)γ+U [−z/2, z/2]ψ(z/2) | p, S〉∣

∣

∣

∣

∣ z+=z−=0 , ∆T=0 , zT→0

〈p′, S | ψ(−z/2)γ+U [−z/2, z/2]ψ(z/2) | p, S〉| z+=z−=0 , ∆T=0 , zT→0

�

η→∞

z

2ηv +

z

2

−z

2ηv −

z

2

v

Remaining parameters to consider: ζ , η

Ji quark orbital angular momentum: η = 0

� �

�

��

��

0.2 0.4 0.6 0.8�0.30

�0.25

�0.20

�0.15

�0.10

�0.05

0.00

Ζ�

L3�Η�0� �n�Η�0�

u�d quarks

mΠ � 518 MeV

�0.0

Biased evaluation of ∂f/∂∆T Unbiased derivative method

From Ji to Jaffe-Manohar quark orbital angular momentum

From Ji to Jaffe-Manohar quark orbital angular momentum

From Ji to Jaffe-Manohar quark orbital angular momentum

Burkardt’s torque – extrapolation in ζ

τ3 = (L(η=∞)3 /n(η=∞))− (L

(η=0)3 /n(η=0))

Integrated torque accumulated by struck quark leaving proton

Flavor separation – from Ji to Jaffe-Manohar quark orbital angular momentum

Flavor separation – Burkardt’s torque

τ3 = (L(η=∞)3 /n(η=∞))− (L

(η=0)3 /n(η=0))

Integrated torque accumulated by struck quark leaving proton

Conclusions

• Quark orbital angular momentum can be accessed directly in Lattice QCD, continuously interpolating be-tween the Ji and Jaffe-Manohar definitions.

• Direct evaluation of derivative w.r.t. momentum transfer (Rome method) rectifies bias affecting earlier ex-ploration. Result obtained for Ji orbital angular momentum agrees with standard Ji sum rule result.

• Difference between the Ji and Jaffe-Manohar definitions (torque accumulated by struck quark leaving aproton) is clearly resolvable, sizeable (∼ 1/3 of the original Ji OAM, at mπ = 317 MeV), and leads to anenhancement of Jaffe-Manohar OAM relative to Ji OAM.

• Im future studies, explore lighter pion masses, higher longitudinal momenta.

And, as an outlook ...

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