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Wigner Functions; P T -Dependent Factorization in SIDIS

Wigner Functions; P T -Dependent Factorization in SIDIS. Xiangdong Ji University of Maryland. — COMPASS Workshop , Paris, March 1-3, 2004 —. Outline. Quantum phase-space (Wigner) distributions. GPD and phase-space picture of the nucleon.

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Wigner Functions; P T -Dependent Factorization in SIDIS

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  1. Wigner Functions; PT-Dependent Factorization in SIDIS Xiangdong Ji University of Maryland —COMPASS Workshop, Paris, March 1-3, 2004—

  2. Outline • Quantum phase-space (Wigner) distributions. • GPD and phase-space picture of the nucleon. • Transverse-momentum-dependent (TMD) parton distributions. • Factorization theorem in semi-inclusive DIS. • Single spin asymmetry: transversity and the connection between Collins, Sivers and Efremov-Teryaev-Sterman-Qiu.

  3. Motivation • Elastic form-factors provide static coordinate-space charge and current distributions (in the sense of Sachs, for example), but no information on the dynamical motion. • Feynman parton densities give momentum-space distributions of constituents, but no information of the spatial location of the partons. • But sometimes, we need to know the position and momentum of the constituents. • For example, one need to know r and p to calculate L=r×p !

  4. Phase-space Distribution? • The state of a classical particle is specified completely by its coordinate and momentum (x,p): phase-space • A state of classical identical particle system can be described by a phase-space distribution f(x,p). • In quantum mechanics, because of the uncertainty principle, the phase-space information is a “luxury”, but… • Wigner introduced the first phase-space distribution in quantum mechanics (1932) • Heavy-ion collisions, quantum molecular dynamics, signal analysis, quantum info, optics, image processing…

  5. Wigner function • Define as • When integrated over x (p), one gets the momentum (probability) density. • Not positive definite in general, but is in classical limit. • Any dynamical variable can be calculated as Short of measuring the wave function, the Wigner function contains the most complete (one-body) info about a quantum system.

  6. Simple Harmonic Oscillator N=5 N=0 Husimi distribution: positive definite!

  7. Measuring Wigner function of a quantum Light!

  8. Quarks in the Proton • Wigner operator • Wigner distribution: “density” for quarks having position r and 4-momentum k(off-shell) a la Saches Ji (2003) 7-dimensional distribtuion No known experiment can measure this!

  9. Custom-made for high-energy processes • In high-energy processes, one cannot measure k= (k0–kz) and therefore, one must integrate this out. • The reduced Wigner distribution is a function of 6 variables [r,k=(k+k)]. • After integrating over r, one gets transverse-momentum dependent (TDM) parton distributions. • Alternatively, after integrating over k, one gets a spatial distribution of quarks with fixed Feynman momentum k+=(k0+kz)=xM. f(r,x)

  10. Proton images at a fixed x • For every choice of x, one can use the Wigner distribution to picture the quarks; This is analogous to viewing the proton through the x (momentum) filters! • The distribution is related to Generalized parton distributions (GPD) through t= – q2  ~ qz

  11. A GPD or Wigner Function Model • A parametrization which satisfies the following Boundary Conditions: (A. Belitsky, X. Ji, and F. Yuan, hep-ph/0307383, to appear in PRD) • Reproduce measured Feynman distribution • Reproduce measured form factors • Polynomiality condition • Positivity • Refinement • Lattice QCD • Experimental data

  12. Up-Quark Charge Density at x=0.4 z y x

  13. Up-Quark Charge Denstiy at x=0.01

  14. Up-Quark Density At x=0.7

  15. Comments • If one puts the pictures at all x together, one gets a spherically round nucleon! (Wigner-Eckart theorem) • If one integrates over the distribution along the z direction, one gets the 2D-impact parameter space pictures of M. Burkardt (2000) and Soper.

  16. TMD Parton Distribution • Appear in the process in which hadron transverse-momentum is measured, often together with TMD fragmentation functions. • The leading-twist ones are classified by Boer, Mulders, and Tangerman (1996,1998) • There are 8 of them q(x, k┴), qT(x, k┴), ΔqL(x, k┴), ΔqT(x, k┴), δq(x, k┴),δLq(x, k┴), δTq(x, k┴), δT’q(x, k┴)

  17. UV Scale-dependence • The ultraviolet-scale dependence is very simple. It obeys an evolution equation depending on the anomalous dimension of the quark field in the v·A=0 gauge. • However, we know the integrated parton distributions have a complicated scale-dependence (DGLAP-evolution) • Additional UV divergences are generated through integration over transverse-momentum, which implies that ∫µ d2k┴ q(x, k┴)  q(x,µ)

  18. Consistency of UV Regularization • Feynman parton distributions are available in the scheme: dimensional regularization, minimal subtraction. • This cannot be implemented for TMD parton distributions because d=4 before the transverse-momentum is integrated. • On the other hand, it is difficult to implement a momentum cut-off scheme for gauge theories… ( I love to have one for many other reasons!) • Therefore, it is highly nontrivial that ∫d2k┴ q(x, k┴)  Fey. Dis. known from fits?

  19. Gauge Invariance? • Can be made gauge-independent by inserting a gauge link going out to infinity in some direction v (in non-singular gauges). • In singular gauges, the issue is more complicated • Ji & Yuan (2003) conjectured a link at infinity to reproduce the SSA in a model by Brodsky et. al. • Belitsky, Ji & Yuan (2003) derived the gauge link • Boer, Mulders, and Pijlman (2003): implications for real processes • If the link is not along the light-cone (used by Collins and Soper, and others…). The integration over k┴ does not recover the usual parton distribution.

  20. Evolution In Gluon Rapidity • The transverse momentum of the quarks can be generated by soft gluon radiation. As the energy of the nucleon becomes large, more gluon radiation (larger gluon rapidity) contributes to generate a fixed transverse-momentum. • The evolution equation in energy or gluon rapidity has been derived by Collins and Soper (1981), but is non-perturbative if k┴, is small.

  21. Factorization for SIDIS with P┴ • For traditional high-energy process with one hard scale, inclusive DIS, Drell-Yan, jet production,…soft divergences typically cancel,except at the edges of phase-space. • At present, we have two scales, Q and P┴(could be soft). Therefore, besides the collinear divergences which can be factorized into TMD parton distributions (not entirely as shown by the energy-dependence), there are also soft divergences which can be taken into account by the soft factor. X. Ji, F. Yuan, and J. P. Ma (to be published)

  22. Example I • Vertex corrections q p′ k p Four possible regions of gluon momentum k: 1) k is collinear to p (parton dis) 2) k is collinear to p′ (fragmentation) 3) k is soft (wilson line) 4) k is hard (pQCD correction)

  23. Example II • Gluon Radiation q p′ k p The dominating topology is the quark carrying most of the energy and momentum 1) k is collinear to p (parton dis) 2) k is collinear to p′ (fragmentation) 3) k is soft (Wilson line) The best-way to handle all these is the soft-collinear effective field theory… (Bauer, Fleming, Steward,…)

  24. A general leading region in non-singular gauges Ph Ph J H H s J P P

  25. Factorization theorem • For semi-inclusive DIS with small pT ~ • Hadron transverse-momentum is generated from • multiple sources. • The soft factor is universal matrix elements of Wilson • lines and spin-independent. • One-loop corrections to the hard-factor has been • calculated

  26. Sudakov double logs and soft radiation • Soft-radiation generates the so-called Sudakov double logarithms ln2Q2/p2T and makes the hadrons with small-pT exponentially suppressed. • Soft-radiation tends to wash out the (transverse) spin effects at very high-energy, de-coupling the correlation between spin and transverse-momentum. • Soft-radiation is calculable at large pT

  27. x z y What is a Single Spin Asymmetry (SSA)? • Consider scattering of a transversely-polarized spin-1/2 hadron (S, p) with another hadron (or photon), observing a particle of momentum k k p p’ S The cross section can have a term depending on the azimuthal angle of k which produce an asymmetry AN when S flips: SSA

  28. Why Does SSA Exist? • Single Spin Asymmetry is proportional to Im (FN * FF) where FN is the normal helicity amplitude and FF is a spin flip amplitude • Helicity flip: one must have a reaction mechanism for the hadron to change its helicity (in a cut diagram). • Final State Interactions (FSI): to general a phase difference between two amplitudes. The phase difference is needed because the structure S·(p× k) formally violate time-reversal invariance.

  29. 1/2 1/2−1 −1/2 1/2 Parton Orbital Angular Momentum and Gluon Spin • The hadron helicity flip can be generated by other mechanism in QCD • Quark orbital angular momentum (OAM): the quarks have transverse momentum in hadrons. Therefore, the hadron helicity flip can occur without requiring the quark helicity flip. Beyond the naïve parton model in which quarks are collinear

  30. Novel Way to Generate Phase Coulomb gluon Some propagators in the tree diagrams go on-shell No loop is needed to generate the phase! Efremov & Teryaev: 1982 & 1984 Qiu & Sterman: 1991 & 1999

  31. k k’ X P Single Target-Spin Asymmetry in SIDIS • Observed in HERMES exp. • At low-Pt, this can be generated from Siver’s distribution function and Collins fragmentation function (twist-2). • At large-Pt, this can be generated from Efremov-Taryaev-Qiu-Sterman (ETQS) effect (twist-3). • Boer, Mulders, and Pijlman (2003) observed that the moments of Siver’s function is related to the twist-3 matrix elements of ETQS.

  32. Low P┴ Factorization • If P┴ is on the order of the intrinsic transverse-momentum of the quarks in the nucleon. Then the factorization theorem involved un-integrated transversity distribution, • One can measure the un-integrated transversity • To get integrated one, one can integrate out P┴ • with p┴ weighted. (soft factor disappears…)

  33. When P┴ is large… • Soft factor produces most of the transverse-momentum, and it can be lumped to hard contribution. • The transverse-momentum in the parton distribution can be integrated over, yielding the transversity distribution. Or when the momentum is large, it can be factorized in terms of the transversity distribution. • The transverse-momentum in the Collins fragmentation function can also be integrated out, yielding the ETSQ twist-three fragmentation matrix elements. Or when the momentum is large, it can be factorized in terms of the transversity distribution. • Likewise for Siver’s effect. • The SSA is then a twist-three observable.

  34. Physics of a Sivers Function • Hadron helicity flip • This can be accomplished through non-perturbative mechanics (chiral symmetric breaking) in hadron structure. • The quarks can be in both s and p waves in relativistic quark models (MIT bag). • FSI (phase) • The hadron structure has no FSI phase, therefore Sivers function vanish by time-reversal (Collins, 1993) • FSI can arise from the scattering of jet with background gluon field in the nucleon (collins, 2002) • The resulting gauge link is part of the parton dis.

  35. Conclusion • GPDs are quantum phase-space distributions, and can be used to visualize 3D quark distributions at fixed Feynman momentum • There is now a factorization theorem for semi-inclusive hadron production at low pt, which involves soft gluon effects, allowing study pQCD corrections systematically. • According to the theorem, what one learns from SSA at low pt is unintegrated transversity distribution. • At large pt, SSA is a twist-three effects, the factorization theorem reduces to the result of Efremov-Teryaev-Qiu-Sterman.

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