Transverse Momentum Dependent QCD Factorization for Semi-Inclusive DIS

1. Transverse Momentum DependentQCD Factorization for Semi-Inclusive DIS J.P. Ma, Institute of Theoretical Physics, Academia Sinica, Beijing The Sino-German Workshop 21.09.2006 DESY, Hamburg

2. Content • Physics of Semi-Inclusive DIS 2. Consistent Definitions of Transverse Momentum Dependent (TMD) Parton Distribution and Fragmentation 3.One-Loop Factorization in SIDIS 4. Factorization to all orders in Perturbation theory. 5. Outlook

3. k' q Ph X P 1. Physics of Semi-Inclusive DIS k • Photon momentum q is in the Bjorken limit. • Final state hadron h can be characterized by • fraction of parton momentum z and transverse • momentum Ph┴

4. A Brief History • European Muon Collaboration (CERN) • Measure the flavor dependence of the fragmentation functions (Duπ+(z), Duπ-(z)) • H1 and Zeus Collaboration (DESY) • Topology of the final state hadrons: Jet structure and energy flow. • Spin Muon Collaboration (CERN) and HERMES • Extracting polarized quark dis: Δq(x) • Single Spin Asymmetries Long history……..

6. A. Ph ┴~ Q : Ph┴generated from QCD hard scattering, factorization theorem exists. (Standard collinear factorization) B. Q >> Ph┴ >>ΛQCD : Still perturbative, but resummation is needed. It is important for many processes. C. Ph┴ ~ΛQCD Nonperturbative! Ph┴ is generated from partons inside of hadrons. Transverse momenta of partons: A transparent explanation for SSA It gives a possible way to learn 3-dimensional structure of hadrons!!!!! Three cases for measured Ph┴

7. A factorization theorem is needed for the case Ph┴ ~ΛQCD ! Single spin asymmetries observed in many experiments stimulated many theoretical works……… 1976: Nachtmann discussed SSA in parton fragmentation 1992: J. Collins suggested a factorization theorem, but without a proof and with some mistakes corrected in 2002. Many people use the theorem………. It was also realized: A consistent definition in QCD of TMD parton distribution was not there………. , and the factorization theorem?

8. 2. Consistent Definitions of TMD Parton Distribution and Fragmentation Light cone coordinate system: Two light cone vectors:

9. A hadron moves in the z-direction with Usual parton distribution: The parton distribution is the probability to find a quark with the momentum fraction x, defined as

10. A naïve generalization to include TMD would be: This is not consistent, because it has the light-cone singularity 1/(1-x) !!!!, and other drawbacks……….. The singularity is not an I.R. - or collinear singularity. If one integrates the transverse momentum, it is cancelled.

11. z QCD Definition v v n n t b v is not n to avoid l.c. singularity

12. Scale Evolution • Since the two quark fields are separated in both long. and trans. directions, the only UV divergences comes from the WF renormalization and the gauge links. • In v·A=0 gauge, the gauge link vanishes. Thus the TMD parton distribution evolve according to the anomalous dimension of the quark field in the axial gauge • Integrate over k┴ generates DGLAP evolution.

13. One-Loop Virtual Contribution Soft contribution Double logs : Energy of the hadron

14. One-Loop Real Contribution

15. The defined TMD distribution has 1. No light cone singularity. (good!!!) 2. double-logs ln2Q2/ΛQCD2 for every coupling constant. (can be resummed with Collins-Soper equation) 3. Beside collinear divergence, there are also infrared singularities, i.e., soft gluon contributions. (can be subtracted ……..)

16. For the double log’s: The TMD distributions depend on the energy of the hadron! (or Q in DIS) Introduce the impact parameter representation One can write down an evolution equation in ζ: (Collins and Soper, 1981 ) K and G obey an RG equation: μ independent!

17. Solve the RG equation: • Solving Collins-Soper equation: Double logs have been factorized!

18. Soft gluon contributions: • The soft gluon contribution can be factorized All soft gluon contributions are in the soft factor S:

19. We finally can give a consistent definition of TMD distribution: Similarly, one can perform the same procedure to define TMD fragmentation functions. It should be noted: Integration over the transverse-momentum does not usually yield Feynman distribution ∫d2k┴ q(x, k┴) = q(x,µ) !!

20. How many TMD’s at leading twist? In general, in Semi-DIS or other processes, if factorization can be proven, one can access the quark density matrix in experiment: It provides all information about the quark inside of the hadron with an arbitrary spin s, it is characterized with some scalar distributions. : certain gauge links……

21. Nucleon H = proton: (uncompleted list) Unpol. Long. Trans. Quark Unpol. q(x, k┴) qT(x, k┴) Long. ΔqL(x, k┴) ΔqT(x, k┴) δqT(x, k┴) δqT'(x, k┴) Trans. δq(x, k┴) δqL(x, k┴) Boer, Mulders, Tangerman et al.

22. 3. One-Loop Factorization in SIDIS Cross section Hadronic Tensor: At tree-level:

23. One-loop Factorization (virtual gluon) • Vertex corrections (single quark target) q p′ k p Four possible regions of gluon momentum k: 1) k is collinear to p (parton distribution) 2) k is collinear to p′ (fragmentation) 3) k is soft (Wilson line) 4) k is hard (pQCD correction)

24. One-Loop Factorization (real gluon) • Gluon Radiation (single quark target) q p′ k p The dominating topology is the quark carrying most of the energy and momentum 1) k is collinear to p (parton distribution) 2) k is collinear to p′ (fragmentation) 3) k is soft (Wilson line)

25. Factorization Theorem: • Factorization for the structure function: with the corrections suppressed by (P┴, ΛQCD/ Q)2 Impact parameter space

26. 4. Factorization to all orders in Perturbation theory Main steps for all-order factorization: • Consider an arbitrary Feynman diagram • Find contributions singular contribution from the different regions of the momentum integrations (Landau equation, reduced diagrams) • Power counting to determine the leading regions • Factorize the soft and collinear gluons contributions • Factorization theorem.

27. Reduced (Cut) Diagrams • A Feynamn diagram, if it contains collinear- and infrared singularities, will give the leading contribution • These singularities can be analyzed with Landau equation, represented by reduced diagram. For our case, the reduced diagram looks: Physical picture Coleman & Norton

28. The most important reduced diagrams are determined from power counting.(Leading region) The leading region is determined by: • No soft fermion lines • No soft gluon lines attached to the hard part • Soft gluon line attached to the jets must be longitudinally polarized • In each jet, one quark plus arbitrary number of collinear long.-pol. gluon lines attached to the hard part. • The number of 3-piont vertices must be larger or equal to the number of soft and long.-pol. gluon lines.

29. Leading Region

30. Factorizing the Collinear Gluons • The collinear gluons are longitudinally polarized • One can use the Ward identity to factorize it off the hard part. The result is that all collinear gluons from the initial nucleon only see the direction and charge of the current jet. The effect can be reproduced by a Wilson line along the jet (or v) direction.

31. Factorizing the Soft Part • The soft part can be factorized from the jet using Grammer-Yennie approximation • Neglect soft momentum in the numerators. • Neglect k2 in the propagator denominators • Potential complication in the Glauber region • Use the ward identity. • The result of the soft factorization is a soft factor in the cross section, in which the target current jets appear as the eikonal lines.

32. Factorization • After soft and collinear factorizations, the reduced diagram becomes: which corresponds to the factorization formula stated earlier.

33. An interesting feature of our factorization theorem for P┴ ~ΛQCD : when P┴ becomes large so that P┴ >>ΛQCD , the famous Collins-Soper-Sterman resummation formula can be reproduced from our factorization theorem. The topics discussed here can be found in X.D. Ji, J.P. Ma and F. Yuan: Phys.Rev.D71:034005,2005

34. 5 . Summary and outlook In general there are 3 classes of distributions to characterize the quark density matrix in a nucleon: ▪Novel distributions that vanish without final state interactions: (Siver’s function, SSA) ▪ the ordinary parton distributions: ▪ New effects with the transverse momentum:

35. They delivery information about 3-dimentional structure, like orbital angular momenta, etc……… What we have done: We establish a factorization theorem of semi-DIS for the first classes of distributions, JMY: hep-ph/0404183, Phys.Rev.D71:034005, 2005 extend the theorem of Drell-Yan process, JMY: hep-ph/0405085 , Phys.Lett.B597:299, 2004 and also extend the theorem with TMD gluon distributions, JMY: hep-ph/0503015 , JHEP 0507:020,2005 Outlook: To establish factorization theorem for other two class distributions, and applications………

36. Thank you !