Sivers Function in the Quasi-Classical Approximation

In this brief article we summarize the derivation of the Sivers function of a heavy nucleus, which we obtain by generalizing the quasi-classical McLerran-Venugopalan model to incorporate the role of spin and orbital angular momentum. In doing so we obtain a new channel which is capable of generating the Sivers function of the nucleus from the orbital motion of its nucleons. An essential role is played in this channel by the multiple rescatterings on spectator nucleons which screen the distribution function during the initial- or final-state interactions. The combination of orbital angular momentum together with multiple rescattering yields a new interpretation of the sign flip relation between the Sivers function measured in semi-inclusive deep inelastic scattering and the Drell-Yan process.


SIDIS and the Drell-Yan Process
• Lepton-hadron scattering producing a specific tagged particle.
• Leptonic part can be factorized out from the hadronic part, reducing to the scattering of a spacelike virtual photon:

Semi-Inclusive Deep Inelastic Scattering (SIDIS)
• A colorless initial state produces a colored final state: sensitive to final state interactions.

The Drell-Yan Process (DY)
• Hadron-hadron scattering producing a hard virtual photon, which decays into a tagged dilepton pair.
• The dileptons can be integrated out, reducing to the production of a timelike virtual photon: • A colored initial state produces a colorless final state: sensitive to initial state interactions.

TMD Factorization and Parton Distributions
• The parton distribution functions are now nonlocal in transverse coordinate space and require a nontrivial gauge link: vs • Transverse momentum permits many new types of parton distribution that couple spin and transverse momentum: • For these processes, transverse-momentum-dependent factorization has been proven when the hard scale is large: (Convolutions over longitudinal and transverse momenta)

The Sivers Function: A Lesson In Time Reversal
• Collins showed that, when one neglects the presence of the gauge link, timereversal invariance requires the Sivers function to vanish.
Nucl.Phys. B396 (1993) 161-182 • But the gauge link reflects more than just gauge invariance; it contains the physics associated with initial / final state interactions.
• The gauge link also transforms under time reversal: initial state interactions become final state interactions, so Collins' derivation instead predicts an exact sign reversal between SIDIS and DY: Phys. Lett. B536 (2002) 43-48 • One of these TMD's is the Sivers function, which measures the intrinsic STSA in a transverselypolarized hadron: • But the correlation is odd under time-reversal, and the hadron is an eigenstate of the T-Even QCD Hamiltonian....

The Importance of the Glue
• TMD's are not solely properties of the hadronic wave functions; they encapsulate aspects of the scattering dynamics as well.
• This "lensing" mechanism generates a preferred direction through the coherent attractive force the quark experiences as it escapes the hadron.

Regge Kinematics and Eikonal Scattering
• The high-energy Regge limit: kinematic simplifications occur when the scattering particles travel nearly along the light cone: • The interactions occur instantaneously and are naturally ordered along the light cone. This eikonal propagation of a projectile through the field of the target is re-summed into a Wilson line:

Quantum Evolution: the Small-x Gluon Cascade
• This increase in gluon bremsstrahlung with energy gives the BFKL equation, which drives up the gluon density at small-x.
• When the energy becomes so large that its logarithm can compete with the coupling, it reorders the perturbation series: • Emission of an extra longitudinally-soft (small-x) gluon is suppressed by the coupling, but contributes a factor of its phase space.
• These emissions must be re-summed to give the small-x gluon cascade which dominates the physics of high energies.

Unitarity and Gluon Saturation
• Nonlinear evolution yields a saturation scale that grows with energy, so eventually, it cuts off the IR while still in the perturbative domain.

BFKL:
Froissart: • BFKL gives a cross-section that grows too quickly with energy and would violate unitarity if it continued unabated.
• At high enough densities, nonlinear gluon fusion begins to compete with bremsstrahlung, saturating the gluon density to a parametrically large value.
• The proliferation of incoherent color sources generates a correlation length, described by the saturation scale, which cuts off the gluon distributions in the IR.

Heavy Nuclei: A Resummation Parameter
• The leading order effect in A 1/3 is a combinatoric enhancement that prefers each rescattering to occur on a different nucleon.
• Nature provides us with another way to approach the limit of dense color charges: using a heavy nucleus with a large number A of nucleons.
• The number of nucleons at a given transverse position, A 1/3 , provides a parameter to control the density of color charges and define the saturation scale: • DIS in the Regge Limit: the virtual photon has a long coherence length and fluctuates into a quark/antiquark pair, undergoing eikonal rescattering from the A 1/3 nucleons.
• Integrating over the momentum transfer k between nucleons puts the intermediate propagators on-shell and factorizes into a product of scattering on independent nucleons.
• The nucleus provides a well-defined regime to re-sum the high-density corrections, without the need for quantum evolution.

The Dense Limit is the Quasi-Classical Limit
• The 2-gluon/nucleon resummation parameter corresponds to interacting with the classical Weizsacker-Williams gluon field of the target.
• The gluon fields of the nucleus are characterized by high occupation numbers, reducing to their classical limit.
• Equivalently, one can solve the classical Yang-Mills equations for a heavy nucleus moving along the light-cone and recover the same formulas (McLerran -Venugopalan model).
• The high energy limit of QCD is the limit of high-density classical gluon fields.
• The resummation parameter embeds the perturbation series in a classical background field.

Spin in the Quasi-Classical
Bjorken Limit arXiv: 1310.5028

A Common Regime for Spin and Saturation
• There is a common limit that employs both the kinematics necessary for spin physics and the high densities necessary for saturation: Bjorken kinematics in a heavy nucleus • In SIDIS, the virtual photon has a short coherence length and interacts via a local "knockout" process due to the "large-x" Bjorken kinematics.
• But the struck quark has a long coherence length and can undergo eikonal finalstate rescattering on the spectator nucleons.
• Since , TMD factorization holds, and we can express the nuclear TMD's in terms of nucleons and Wilson lines.

Large-x and Large-A
• Since , we can do the calculations perturbatively, using the machinery of saturation physics.

The SIDIS Cross Section (1)
• We can relate the scattering amplitude on the nucleus to the light-cone wave functions of the nucleons and the rest of the amplitude (knockout + rescattering) • But, using the fact that W(p,b) varies with impact parameter only over macroscopic scales ~ A 1/3 , we can neglect the off-forwardness in the transverse momentum.
• After introducing the Wigner functions, the knockout + rescattering amplitudes are still off-forward.
• By Fourier transforming the momentum difference between the amplitude and C.C., we obtain the Wigner distribution of nucleons in the nucleus: