Xiangdong Ji University of Maryland Shanghai Jiao Tong University

1. Xiangdong Ji University of Maryland Shanghai Jiao Tong University Parton Physics on a Bjorken-frame lattice July 1, 2013

2. Knowledge of parton distributions is data-driven ─── Paul Reimer from the prevous talk of this workshop

3. Outline • Review of Bjorken frame and parton physics • Why parton physics is hard to calculate? • A new proposal, resource requirement, and applicability • Gluon polarization: its physics and calculation • Outlooks

4. High-energy scattering and Bjorken frame • In high-energy scattering, the nucleon has a large momentum relative to the probes. • In the Bjorken frame, the probes (electron or virtual photon) may have the smallest momentum, but the proton has a large momentum (infinite momentum frame, IMF) relative to the observer and travels at near the speed of light. • This frame has been used frequently in the old literature, but giving away to rest-frame light-front quantization in recent years.

5. Electron scattering in Bjorken frame 4-momentum transfer qµ = (v, q) is a space like vector v2-q2 < 0 and fixed. Smallest momentum happens when v=0, Q2=q2 Pq = P3Q = Q2/2x, thus P3 = Q/2x. In the scaling limit, P3 -> infinity.

6. Bjorken frame and Parton physics • The interactions between particles are Lorentz-dilated, and thus the system appears as if interaction-free: the proton is probed as free partons. • In QCD, parton physics emerges when working in light-cone gauge. A+=0. • In field theory, parton physics is cut-off dependent. • This is only true to a certain degree: leading twist. The so-called higher-twist contributions are sensitive to parton off-shellness, transverse momentum and correlations.

7. Quark and gluon parton distributions The Feynman momentum is, in the Bjorken frame, fraction of the longitudinal momentum carried by quarks: x = kz/Pz, 0<x<1

8. Parton Physics • Light-cone wave function, ψn(xi, k⊥i) • Distributions amplitudes, ψn(xi) • Parton distributions, f(x) • Transverse momentum dependent (TMD) parton distributions, f(x, k⊥) • Generalized parton distributions, F(x,𝜉,r⊥) • Wigner distributions, W(x, k⊥, r⊥) • Fragmentation functions…

9. Frame-independent formulation of parton physics • Over the years, the parton physics have been formulated in a boost-invariant way. In particular it can be described as the physics in the rest frame. • In this frame, the probe appears a light-front (light-like) correlation. • Thus light-cone quantization is the essential tool (S. Brodsky)!

10. Light-front quantization

11. Unique role of lattice QCD (1974) • Lattice is the only non-perturbative approach to solve QCD • Light-front quantization: many years of efforts but hard for 3+1 physics • AdS/CFT: no exact correspondence can be established, a model. • An intrinsically Euclidean approach • “time” is Eucliean 𝜏=i t, no real time • A4 = iA0 is real (as oppose to A0 is real) • No direct implementation of physical time.

12. Ken Wilson (1936-2013)

13. Don’t know how to calculate! • Parton physics? Light-like correlations • For parton distributions & distribution amplitudes: moments are ME of local operators, 2-3 moments. Very difficult beyond that… • For parton physics that cannot be reduced to local operators, there is no way to calculate! 𝜉0 𝜉+ 𝜉- 𝜉3

15. A Euclidean distribution • Consider space correlation in a large momentum P in the z-direction. • Quark fields separated along the z-direction • The gauge-link along the z-direction • The matrix element depends on the momentum P. • This distribution can be calculated using standard lattice method. 𝜉0 0 Z 𝜉3

16. Taking the limit P-> ∞ first • After renormalizing all the UV divergences, one has the standard quark distribution! • One can prove this using the standard OPE • One can also see this by writing |P˃ = U(Λ(p)) |p=0> and applying the boost operator on the gauge link • The Altarelli-Parisi evolution was derived this way! 𝜉- 𝜉0 𝜉+ 𝜉3

17. Finite but large P • The distribution at a finite but large P is the most interesting because it is potentially calculable in lattice QCD. • Since it differs from the standard PDF by simply an infinite P limit, it shall have the same infrared (collinear) physics. • It shall be related to the standard PDF by a matching condition in the sense that the latter is an effective theory of the former.

18. Relationship: factorization theorem • The matching condition is perturbative • The correction is power-suppressed.

19. Pictorial factorization ZZ(P, μ) q(x, P, μ) q(x, μ)

20. One-loop example • Pz dependence is mostly isolated in the large logs of the loop integral.

21. Practical considerations • For a fixed x, large Pz means large kz, thus, as Pz gets larger, the valence quark distribution in the z-direction get Lorentz contracted, z~1/kz. • Thus one needs increasing resolution in the z-direction for a large-momentum nucleon. Roughly speaking: aL/aT ~ γ

22. One needs special kinds of lattices γ=2 x,y z

23. γ=4 x,y z

24. Small x partons • The smallest x partons that one access for a nucleon momentum P is roughly, xmin = ΛQCD/P~ 1/3γ small x physics needs large γ as well. • Consider x ~ 0.01, one needs a γ factor about 10~30. This means 100 lattice points along the z-direction. • A large momentum nucleon costs considerable resources!

25. Ideal lattice configurations • Time direction also needs longer evolution because the energy difference between excited states and the ground state goes like 1/ γ • Thus ideal configurations for parton physics calculations will be 242x(24γ)2 or 362x(36γ)2 There are not yet available!

26. Sea quarks • The parton picture is clearest in the axial gauge AZ =0. • In this gauge, see quarks correspond to backward moving quarks (Pz>0, kz<0) or forward moving antiquark, but otherwise having arbitrary transverse momentum (with cut-off μ) and energy (off-shellness). • In the limit of Pz->∞, the contribution does not vanish. • Flavor structure? (Hueywen Lin’s talk)

28. 1/P2 correction • Two types (to be published) • The nucleon mass corrections in the traces of the twsit-2 matrix elements can easily be calculated. • Corrections in twist-four contributions can also be directly calculated on lattice. The contribution is suspected to be smaller than the mass correction. • Higher-order corrections can similarly be handled.

29. Other applications • This approach is applicable for all parton physics • Recipe: • Replace the light-cone correlation by that in the z-direction. • Replace the gauge link in the light-cone direction by that in the z-direction. • Derive factorizations of the resulting distributions in terms of light-cone parton physics.

30. GPDs and TMDs • GPDs • TMDs

31. Wigner distributions and LC amplitudes • Wigner distribution • Light-cone amplitudes • Light-cone wave functions • Higher-twists….

32. Gluon helicity distribution

33. ∆g(x) • An important part of the nucleon spin structure • Much attention has been paid to this quantity experimentally • DIS semi-inclusive • RHIC spin • … • In principle, it can be calculated from the approach discussed previously. However, it is still difficult to get ∆G, the integral.

34. ALL from RHIC 2009

35. QCD expression • The total gluon helicity ΔG is gauge invariant quantity, and has a complicated expression in QCD factorization (Manohar, 1991) • It does not look anything like gluon spin or helicity! Not in any textbook!

36. Light-cone gauge • In light-cone gauge A+=0, the above expression reduces to a simple form which is the spin of the photon (gluon) ! (J. D. Jackson, CED), but is not gauge-symmetric: There is no gauge symmetry notion of the gluon spin! (J. D. Jackson, L. Landau & Lifshitz).

37. Two long-standing problems • ∆G does not have a gauge-invariant notion of the gluon spin. • There is no direct way to calculate ∆G, unlike ∆∑, and orbital angular momentum.

38. Electric field of a charge

39. A moving charge

40. Gauge potential

41. Suggestion by X. Chen et al • Although the transverse part of the vector potential is gauge invariant, the separately E┴ does not transform properly, under Loretez transformation, and is not a physical observable (X. Chen et al, x. Ji, PRL)

42. Gauge invariant photon helicity • X. Chen et al (PRL, 09’) proposed that a gauge invariant photon angular momentum can be defined as ExA┴ • This is not an observable when the system move at finite momentum because (X. Ji) • A ┴ generated from A║ from Lorentz boost. • A lorentz-transformed A has different decomposition A = A┴ + A║ in different frames. • There is no charge that separately responds to A┴ and A║

43. Large momentum limit • As the charge velocity approaches the speed of light, E┴ >>E║, B ~ E┴, thus • E┴ become physically meaningul • The E┴ & B fields appear to be that of the free radiation • Weizsacker-William equivalent photon approximation (J. D. Jackson) • Thus gauge-invariant A┴appears to be now physical which generates the E ┴ & B.

44. Theorem • The total gluon helicity ΔG shall be the matrix element of ExA┴ in a large momentum nucleon. • We proved in the following paper X. Ji, J. Zhang, and Y. Zhao (arXiv:1304.6708) is just the IMF limit of the matrix element of ExA┴

45. QCD case • A gauge potential can be decomposed into longitudinal and transverse parts (R.P. Treat,1972), • The transverse part is gauge covariant, • In the IMF, the gauge-invariant gluon spin becomes

46. One-loop example • The result is frame-dependent, with log dependences on the external momentum • Anomalous dimension coincides with X. Chen et al.

47. Taking large P limit • If one takes P-> ∞ first before the loop integral, one finds • This is exactly photon (gluon) helicity calculated in QCD factorization! Has the correct anomalous dimension.

48. Matching condition • Taking UV regularization before p-> ∞ (practical calculation, time-independent) • One can get one limit from the other by a perturbative matching condition, Z. • A┴ can be obtained from Coulomb gauge fixing on lattice.

49. Conclusions • Parton physics can be explored in lattice QCD calculations using the Bjorken frame. This opens the door for precision comparisons of high-energy scattering data and fundamental QCD calculations. • It will be a while before that “data driving” era is over. However, we know how to get there.

50. ” He (Wilson) was decades ahead of his time with respect to computing and networks.” ─── Paul Ginsparg, Cornell