Xiangdong Ji University of Maryland/SJTU Observing the proton Off the light-cone Xiangdong Ji University of Maryland/SJTU JLab Physics Colloquium, April 23, 2013

Collaborators and papers Feng Yuan, LBL Jianhui Zhang, Shanghai Jiao Tong University Xiaonu Xiong, Peking University Yong Zhao, U. Maryland 1. Phys. Rev. Lett. 109, 152005 (2012) 2. Phys.Lett. B717 (2012) 214-218 3. Parton picture for longitudinal polarization, Phys. Rev. D 4. Physics of gluon helicity 5. parton distribution from Euclidean lattice

Outline Frame dependence in relativistic quantum theories Two special frames for understanding the proton structure: the lab frame and infinite momentum frame (IMF) The proton spin structure in IMF Lattice QCD calculation of parton physics in IMF Conclusions

Important mission 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 Strong interactions, relativistic Test the fundamental theory: QCD

Constraints of Relativity The proton wave function is a frame dependent concept Boost operators, Ki, are interaction-dependent |P˃ = U(Λ(p)) |p=0> U is not just kinematical, it is dynamical!

Wave functions in Relativistic QM Wave function is not a Lorentz invariant concept as it is defined by observations of different space points in a fixed time (simultaneously) at a particular frame. Simultaneity for two events in one frame does not mean simultaneity in a different frame. In general, WF has not been very a popular concept in field theory. However, it is an important and intuitive concept.

Two special frames Lab frame: where the proton is at rest, or low momentum, relatively speaking Probe frame: in which the electron or virtual photon has the smallest momentum. In high-energy limit, the probe frame reaches the infinite momentum frame (IMF), i.e., the proton is probed as it travels at the speed of light!

Electron scattering four 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.

Reconcile physics in two frames Physics shall be frame-independent! In the rest frame, one probes the light-cone, time-dependent correlations (light-front) In the IMF, one probes the static correlations. The two different correlations are related by Lorentz boost. If one uses the light-front quantization in the rest frame (an effective theory), one gets the same physics as IMF! Proton WF in the IMF is what high-energy scattering study.

IMF and parton physics In the IMF, the interactions between particles are Lorentz-dilated, and thus the systems appear as if interaction free: the proton is made of free partons. 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 parton correlations. Partons provide a useful language to understand the proton structure.

Spin of the proton in QCD The spin of the proton is generated from the angular momentum of the internal quarks and gluons From general consideration, one can write down, The quark orbital contribution can be probed through GPDs These are frame-independent results.

The spin structure in IMF How is the spin of the proton generated from the AM of the underlying partons? Longitudinal polarization Transverse polarization

Longitudinal polarization  

Quark orbital AM: intrinsically twist-three Orbital AM is nominally a leading twist, however, it is actually a high-twist contribution because it involves the parton transverse momentum. The total contribution can be probed through the GPD sum rule (L=J-ΔΣ. However, the individual parton contributions do not yet have a simple parton picture. OAM parton distribution can be probed through twist-three GPDs. (Ji, xiong, & yuan)

Gluon polarization In IMF, however, the gluon partons have well-defined helicity ± 1 and densities g±(x) Gluon helicity distribution is g(x) = g+(x) – g-(x) and G = ʃdx g(x) +1 or -1 1/2

Total gluon helicity The total gluon helicity ΔG is gauge invariant, and has a complicated expression on the rest frame In light-cone gauge A+=0, it reduces to a simple expression , which is the spin of the gluon, but has no gauge-symmetric notion (Laudau, Jackson).

ALL from RHIC 2009

Perspective on Delta G

Total gluon helicity in the IMF A gauge potential can be decomposed into longitudinal and transverse parts (R.P. Treat,1972), The transverse part is gauge covariant, One can define the gauge-symmetric gluon spin as ExA┴ (X.Chen et al, 2009) It be shown that ΔG is the matrix element of about operator in the IMF. (Ji, Zhang, Zhao)

Reproducing the light-cone gauge result by boost In fact, transforming to the rest frame Which in the light-cone gauge, reduces to the standard expression ExA.

Transverse polarization Transverse polarization in term of the polarization vector S is of subleading, thus does not seem to have a simple parton picture However, this is incorrect. There are many misconceptions and pitfalls about transverse pol. The best language is not the transverse AM, rather, it shall be transverse Pauli-Lubanski spin. [Transverse AM does not commute with the hamiltonian]

Transverse polarization There is a leading twist-contribution, the subleading contributions either are cancelled or related to the leading one by symmetry. The leading contribution has a partonic interpretation. First pointed out by Burkardt (2005), but the connection to the spin sum rule was not rigourous. works for transverse spin!

A plane-wave derivation of transverse-spin parton sum rule Consider parton momentum density It has a “distribution” term depending on coordinate ξ, Calculating its contribution to the transverse spin density

Parton distributions The Feynman momentum is easier to understand in the infinite momentum frame: fraction of the longitudinal momentum carried by quarks x = kz/Pz, 0<x<1

Lattice QCD calculation Current approach is based on the rest frame formalism. To compute light-cone correlation difficult because the lattice cannot handle real time correlations One can calculate the moments (2-3) of parton distributions: local operators.

Lattice QCD calculations Parton distributions are calculated using the rest-frame formalism, the corresponding operators are light-cone correlations Since lattice cannot handle real-time dependence, only moments are calculated on lattice. Usually only a few moments (2-3) are calculated.

A new proposal Using the IMF formalism. Start with static correlation in the z-direction. X. Ji, to be published

Comments This can be proved using the local matrix elements formulation Approaching the light-cone through boost One-loop calculation demonstrates how this work.

One-loop calculation

The extension of the approach GPDs TMDs

The extension of the approach Wigner distribution Light-cone amplitudes Light-cone wave functions, higher-twists….

Conclusions In high-energy scattering, one probes parton physics, which can be understood both in the rest frame and IMF formulation Understanding the spin structure of the proton in term of partons needs more theoretical and experimental efforts Parton physics is best calculated in lattice QCD using IMF formalism.