Now probe the internal structure of nucleon by scattering processes, which is a task at future EIC. It is a bound state problem, but not many-body or few body, instead parametrize non-perturbative physics with factorization. It is different from pure internal structure of nucleon which is a many body problem. Also different from chiPT which views nucleon as point particle.

The basic ingredient is the bilinear quark current with nucleon states. Separating the states with hard fields, collinear fields, and soft fields. Depending on the formalism, one can describe the states in covariant Lagrangian formalism, or light-front Hamiltonian formalism.

There is no many-body dynamics, just parametrize non-perturbative dynamics into factorized distribution amplitude or parton distribution. So the key point here is the factorization.

Twist form factor and its expansion: spin orbital momentum decomposition

Jaffe and Kazuhiro Tanaka, Xiangdong Ji with higher d


  • iro Kodaira and Kazuhiro Tanaka, Polarized Structure Functions in QCD, review

  • Novel twist-three transverse-spin sum rule for the proton and related generalized parton distributions, arxiv

  • Transverse spin sum rule of the proton, arxiv

  • iro Kodaira and Kazuhiro Tanaka, Twist - 3 polarized structure functions, arxiv

  • Operator relations for gravitational form factors of a spin-0 hadron, arxiv

  • Twist–three relations of gluonic correlators for the transversely polarized nucleon, arxiv

  • Contribution of Twist-3 Multi-Gluon Correlation Functions to Single Spin Asymmetryin Semi-Inclusive Deep Inelastic Scattering, arxiv

  • Xiangdong Ji, Transverse Polarization of the Nucleon in Parton Picture, arxiv

  • Operator Constraints for Twist-3 Functions and Lorentz Invariance Properties of Twist-3 Observables, arxiv


  • QCD evolution of the orbital angular momentum of quarks and gluons: Genuine twist-three part, arxiv

  • QCD factorization for twist-three axial-vector parton quasidistributions, arxiv

  • TMDs of Spin-one Targets: Formalism and Covariant Calculations, arxiv

  • Matching of transverse momentum dependent distributions at twist-3, arxiv

higher dim

  • Experimental constraint on quark electric dipole moments, arxiv

  • Parton distribution function for the gluon condensate, arxiv

  • Relating hadronic CP-violation to higher-twist distributions, arxiv

multipole (angular momentum expansion) expansion of bilinear current

  • Momentum-Current Gravitational Multipoles of Hadrons, arxiv

  • Spherical-harmonic tensors, arxiv

  • The tensor spherical harmonics, course

bilinear current

The bilinear current is non-local. In position space, we can do a multipole expansion for the field.

Heavy quark effective theory

For quark current, one could separate heavy mode and perform expansion on v/M.

The case of nonrelativistic QCD (NRQCD) for quarkonium states, in which there are also three distinct scale M, M beta, M beta^2, beta is the typical velocity of a quark inside a quarkonium.

Heavy quark field: small off-shellness, HQET is formulated in a frame independent manner

boosted Heavy Quark Effective Theory (bHQET) arxiv 1606.07737, describe a heavy hadron carrying high momentum, highly boosted heavy quarks.

Collinear soft effective theory

One field could have different modes. The mode is “frame dependent”, in every direction it have different scaling.

  • The frame dependence is shown in arxiv. Deep inelastic scattering (DIS) could be analyzed in SCET in either the target rest frame, where only ultrasoft and n-collinear modes were required, or the Breit frame, where n-collinear, n ̄-collinear and ultrasoft modes were required – the degrees of freedom may differ in different reference frames, so the theory is not manifestly frame independent.

  • Therefore, SCET depends on frame, and need a phase space separation. fast particle, collinear.

  • Collinear effective theory at subleading order and its application to heavy-light currents, arxiv

Let us consider a reference frame in which a light quark carries a large energy E. We can construct a different type of an effective theory by taking the energy E of the energetic light quark to infinity. In this limit, nonperturbative effects can also be systematically obtained. In fact, this effective theory is more complicated than the HQET and the naive power counting in 1/E should be modified since the system involves several energy scales.

If a light quark moves with a large energy, the momentum has three distinct scales. It is similar to the case of nonrelativistic QCD (NRQCD) for quarkonium states.

The collinear quark can emit either a soft gluon or a collinear gluon to the large momentum direction and can still be on its mass shell. Thus in the collinear effective theory, the power counting in 1/E is troublesome, but the expansion in the small parameter λ offers a consistent power counting and there is no mixing of operators with different powers of λ.

The following two stages:

  • Between E and Eλ, we have collinear modes and soft modes for the light quark. The effective theory at this stage is called the collinear-soft effective theory. Including the effects of collinear gluons obtains the IR behavior.

  • Below the scale Eλ and above Eλ^2, we integrate out all the collinear modes, and there remain only soft modes in the final soft effective theory. This actually corresponds to the large-energy effective theory suggested by Dugan and Grinstein, in which there are only soft modes. Furthermore, the large-energy effective theory is still troublesome since it does not include the effects of collinear gluons properly.

The bilinear current is non-local. Position space vs momentum space:

  • In position space, we can do a multipole expansion for the field. This position space multipole expanded result ensures that ultrasoft momenta are added to collinear momenta in momentum space interactions. For the collinear particle, the only nontrivial position dependence that can appear in the interactions involving the ultrasoft fields. This appears as an apparent non-conservation of momentum when Fourier transforming the interaction.

  • Momentum Space: the Label Formalism. Here one implements the multipole expansion by applying power counting to the momenta that flow through the Feynman diagrams directly by separating the “large” component of momentum from the “small” component. Emitting ultrasoft modes from (anti-)collinear modes leaves the (anti-)collinear labels unchanged.

This is described in the Cohen’s TASI lecture arxiv.

Large momentum effective theory

Formulated in Euclidean space (no time dependence), instead of Minkowski space. Also consider Light-front in Lagrangian formalism, instead of Hamiltonian formalism, so it is an effective theory in light-front space-time. In the past, Hamiltonian formalism is not a field theory, just a modeling.

Why there is no need for collinear and soft mode separation in this case? Is it equivalent to scet theory?