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

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

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

Jlab 12 GeV & EIC An important mission of Jlab 12 GeV upgrade and EIC is to study the internal structure of proton and neutron at a new level. An important mission of Jlab 12 GeV upgrade and EIC is to study the internal structure of proton and neutron at a new level. A complex system of the quarks and gluons A complex system of the quarks and gluons Strong interactions, relativistic Test the fundamental theory: QCD

Wave Function and Relativity The proton wave function is a frame-dependent concept: The proton wave function is a frame-dependent concept: Simultaneity for two events in one frame does not mean simultaneity in a different frame. Boost operators, K i, are interaction-dependent |P ˃ = U(Λ(p)) |p=0> U is not just kinematical, it is dynamical!

High-energy scattering and Bjorken frame In high-energy scattering, the nucleon has a large momentum relative to the probes. 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. 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. This frame has been used frequently in the old literature, but giving away to rest-frame light- front quantization in recent years.

Electron scattering in Bjorken frame 4-momentum transfer q µ = (v, q) is a space like vector v 2 -q 2 < 0 and fixed. Smallest momentum happens when v=0, Q 2 =q 2 Smallest momentum happens when v=0, Q 2 =q 2 Pq = P 3 Q = Q 2 /2x, thus P 3 = Q/2x. Pq = P 3 Q = Q 2 /2x, thus P 3 = Q/2x. In the scaling limit, P 3 -> infinity.

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. 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.

Quark and gluon parton distributions The Feynman momentum is, in the Bjorken frame, fraction of the longitudinal momentum carried by quarks: x = k z /P z, 0<x<1

Parton Physics Light-cone wave function, ψ n (x i, k ⊥ i ) Light-cone wave function, ψ n (x i, k ⊥ i ) Distributions amplitudes, ψ n (x i ) Distributions amplitudes, ψ n (x i ) Parton distributions, f(x) Parton distributions, f(x) Transverse momentum dependent (TMD) parton distributions, f(x, k ⊥ ) Transverse momentum dependent (TMD) parton distributions, f(x, k ⊥ ) Generalized parton distributions, F(x,, r ⊥ ) Generalized parton distributions, F(x,, r ⊥ ) Wigner distributions, W(x, k ⊥, r ⊥ ) Wigner distributions, W(x, k ⊥, r ⊥ ) Fragmentation functions… Fragmentation functions…

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. 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. In this frame, the probe appears a light- front (light-like) correlation. Thus light-cone quantization is the essential tool (S. Brodsky)! Thus light-cone quantization is the essential tool (S. Brodsky)!

Light-front quantization

Unique role of lattice QCD (1974) Lattice is the only non-perturbative approach to solve QCD 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 An intrinsically Euclidean approach “time” is Eucliean =i t, no real time A 4 = iA 0 is real (as oppose to A 0 is real) No direct implementation of physical time.

Ken Wilson ( )

Don’t know how to calculate! Parton physics? Light-like correlations 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 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! For parton physics that cannot be reduced to local operators, there is no way to calculate!

A Euclidean distribution Consider space correlation in a large momentum P in the z-direction. 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. 3 0 Z 0

Taking the limit P-> ∞ first After renormalizing all the UV divergences, one has the standard quark distribution! 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!

Finite but large P The distribution at a finite but large P is the most interesting because it is potentially calculable in lattice QCD. 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. 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. 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.

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

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

One-loop example P z dependence is mostly isolated in the large logs of the loop integral. P z dependence is mostly isolated in the large logs of the loop integral.

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

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

z x,y γ=4

Small x partons The smallest x partons that one access for a nucleon momentum P is roughly, The smallest x partons that one access for a nucleon momentum P is roughly, x min = Λ QCD /P~ 1/3γ x min = Λ QCD /P~ 1/3γ small x physics needs large γ as well. 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. 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! A large momentum nucleon costs considerable resources!

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

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

1/P 2 correction Two types (to be published) Two types (to be published) The nucleon mass corrections in the traces of the twsit-2 matrix elements can easily be calculated. 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. 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. Higher-order corrections can similarly be handled.

Other applications This approach is applicable for all parton physics This approach is applicable for all parton physics Recipe: 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.

GPDs and TMDs GPDs GPDs TMDs TMDs

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

Gluon helicity distribution

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

A LL from RHIC

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

Light-cone gauge In light-cone gauge A + =0, the above expression reduces to a simple form In light-cone gauge A + =0, the above expression reduces to a simple form which is the spin of the photon (gluon) ! 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).

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

Electric field of a charge

A moving charge

Gauge potential

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) 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)

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

Large momentum limit As the charge velocity approaches the speed of light, E ┴ >>E ║, B ~ E ┴, thus 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) Weizsacker-William equivalent photon approximation (J. D. Jackson) Thus gauge-invariant A ┴ appears to be now physical which generates the E ┴ & B. Thus gauge-invariant A ┴ appears to be now physical which generates the E ┴ & B.

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

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

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

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

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

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. 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. It will be a while before that “data driving” era is over. However, we know how to get there.

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