1. Transverse Spin Physics (theory) Feng Yuan Lawrence Berkeley National Laboratory RBRC, Brookhaven National Laboratory

2. 2 Transverse Spin Theory Four Lectures Transverse Spin and Transversity  General introduction TMD and Semi-inclusive DIS  Global picture TMD in perturbative region and the relevant QCD dynamics Advanced topics: NLO calculations ad QCD resummation for SSA

3. 3 Exploring the nucleon: Of fundamental importance in science The nucleons (proton and neutron) are the most abundant particles around us!  Our human body is almost entirely made of nucleons  The sun ☼ and all other stars…   And all atomic nuclei… Nucleon as important tool for discovery  New Physics beyond Standard Model: Tevatron, LHC, Jefferson Lab, … Exploration of nucleon structure started century ago…

5. Deep Inelastic Scattering 6/12/20165 FriedmanKendallTaylor Bjorken Scaling: Q 2  Infinity Feynman Parton Model

6. 6 QCD and Strong-Interactions QCD: Non-Abelian gauge theory  Building blocks: quarks (spin½, m q, 3 colors; gluons: spin 1, massless, 3 2 -1 colors) Asymptotic freedom and confinement Clay Mathematics Institute Millennium Prize Problem Long distance:? Soft, non-perturbative Short distance hard, perturbative Nonperturbative scale  QCD ~1GeV ~1/Length

7. 7 The Proton in QCD Proton is made of  2 up quarks (e=2/3) + 1 down quark (e=-1/3)  valence  + any number of quark-antiquark pairs  sea  + any number of gluons Fundamental questions (from quarks to cosmos…)  Origin of mass? ~ 99% comes from the motion of quarks & gluons ~ l% from Higgs interactions (Tevatron, LHC)  How are Elements formed? the protons & neutrons interact to form atomic nuclei  How its constituents distributed in nucleon? momentum and space distributions  Proton spin budget: ½=?

8. Understanding the nucleon 6/12/20168 Solving QCD  Numerically simulation, like 4D stat. mech. systems  Feynman path integral  Wick rotation  Spacetime discretization  Monte Carlo simulation  Effective field theories (large N c, chiral physics,…) Experimental probes  Study the quark and gluon structure through low and high-energy scattering  Require clean reaction mechanism Photon, electron & perturbative QCD

9. 9 High energy scattering as a probe to the nucleon structure Many processes: Deep Inelastic Scattering, Deeply-virtual compton scattering, Drell-Yan lepton pair production, pp  jet+X  Momentum distribution: Parton Distribution  Spin density: polarized parton distribution  Wave function in infinite momentum frame: Generalized Parton Distributions DVCS Drell-Yan Hadronic reactions DIS (Q>>  QCD ) Q>>  QCD k Feynman Parton Momentum fraction

10. 10 Factorization is essential to study nucleon structure Universality of the parton distributions  Predictive power of QCD Collins-Soper-Sterman 85, Bodwin 85, … Ji-Ma-Yuan 04, Dominguez-Xiao-Yuan 10,…

11. Simple example: e+e-  hadrons Leading order  Eletron-positron annihilate into virtual photon, and decays into quark-antiquark pair, or muon pair  Quark-antiquark pair hadronize 6/12/201611 p1p1 p2p2 q k1k1 k2k2

12. Total cross section R ratio  Depends on the number of colors, electric charge of the quark 6/12/201612

13. High order corrections There are soft and collinear divergences in the real and virtual gluon radiation It can be shown that these divergences cancel out between real and virtual diagrams Total cross section is infrared safe 6/12/201613

14. Further remarks If we keep the light-cone k +, the cancellation is not complete  Left with the collinear divergence, factorized into the fragmentation function 6/12/201614 p1p1 p2p2 q k1k1 k2k2 phph

15. If we further keep the transverse momentum, there is large logarithms associated with the gluon radiation  TMD factorization and resummation 6/12/201615 p1p1 p2p2 q k1k1 k2k2 p h1 p h2

16. Back to nucleon structure 6/12/201616

17. 17 6/12/2016 qq GG LGLG LqLq Proton Spin Sum Rule Quark spin ~30% DIS, and pp coll. Gluon spin~ 0-70% RHIC, EIC, … Deeply Virtual Compton Scattering, Transverse spin physics, in DIS, pp coll.

18. 18 Transverse Spin Theory Transverse spin - introduction Pauli-Lubanski operator  P  – Momentum operator  J  – Angular momentum operator  Dirac particle of momentum p, and spin s Polarization operator

19. 19 Transverse Spin Theory Spin vector s  is chosen such as, s 2 =-1, s.p=0,  In the rest frame, s  =(0,n)  Transverse part of s  is boost invariant, and s z /s T  ∞ when p  ∞

20. 20 Transverse Spin Theory Helicity and chirality Choose longitudinal polarization, s~p, Spin states are also helicity states In the massless case, helicity states are also chirality states

21. 21 Transverse Spin Theory Transverse spin Transverse spin states are off-diagonal state in the helicity basis, e.g., along the x-direction, Spin dependent observable,

22. 22 Transverse Spin Theory Transverse polarized structure function Polarized DIS structure contains two terms Longitudinal polarized, g 1 contributes Transverse polarized, g 1 +g 2 contribute; it is manifested as higher-twist effects, because S T suppressed by 1/P +

23. 23 Transverse Spin Theory Factorize the parton distributions In DIS, Quark distribution matrix  P,k) Gauge links (next)

24. 24 Transverse Spin Theory Leading order quark distribution   expands at leading order, Although a leading-twist distribution, transversity is chiral-odd, and doesn’t contribute to the DIS structure function Unpolarized helicity transversity

25. 25 Transverse Spin Theory Probability representation (helicity basis) qLqL  q T (h 1,  T q) q(x) + - + + - - - -- + + + + + + ++ ++ - - + + + + + -

26. 26 Transverse Spin Theory Soffer bound choose the coupling of nucleon and quark as ++ and +-  q(x)=| ++ | 2 +| +- | 2   q L (x)=| ++ | 2 -| +- | 2   q T (x)= ++ - - *  Soffer bound:

27. 27 Transverse Spin Theory Nucleon tensor charge Tensor charge can be calculated from lattice QCD,

28. 28 Transverse Spin Theory QCD evolution No mixing with gluon + -+ - + -+ - + -+ - This part is the kernel for unpolarized and longitudinal polarized quark distr.

29. 29 Transverse Spin Theory Measuring transversity is difficult Have to multiply another chiral-odd object (distribution or fragmentation)  Drell-Yan and other processes in hadronic collisions  Two-hadron production in DIS  Semi-inclusive single hadron production in DIS (connection to the next lecture)

30. 30 Transverse Spin Theory Drell-Yan is an ideal place Combining two transversity distributions in Drell-Yan lepton pair production + - - + qTqT qTqT

31. 31 Transverse Spin Theory Angular distribution of the lepton pair Double transverse spin asymmetry

32. 32 Transverse Spin Theory Predictions at RHIC Vogelsang, et al, PRD, 1999

33. 33 Transverse Spin Theory Dedicated Drell-Yan study at RHIC II Matthias Grosse Perdekamp

34. 34 Transverse Spin Theory Other processes Soffer, Stratmann, Vogelsang, 02

35. 35 Transverse Spin Theory Two-hadron interference frag. fun. In DIS Collins,Heppelmann,Ladinsky, 94 Jaffe,Jin,Tang, 97 Bacchetta,Radici, 04,06 Metz-Zhou, 2011

36. 36 Transverse Spin Theory HERMES Results arXiv: 0803.2367

37. Input from BELLE 6/12/201637 arXiv:1104.2425

38. 6/12/201638

39. 39 Transverse Spin Theory Semi-inclusive DIS Collins fragmentation function is chiral- odd Combining with the quark transversity leads to single transverse-spin asymmetry in SIDIS Open a whole window of SSAs in SIDIS (next lecture) (k,s T ) (zk+p T ) ~ p T Xs T

40. 40 Transverse Spin Theory e + e - collisions play important roles in this game Reliable place to extract the Collins function and interference (two-hadron) fragmentation function Belle Col., PRL 06

41. Lecture II 3D imaging of partons in nucleon/TMD physics Transverse Momentum Dependent Factorization 6/12/201641

42. 42 Transverse Spin Theory Physics of SIDIS Flavor decomposition for polarized or unpolarized quark distributions Transverse momentum Dependent physics, transversity distribution, Sivers, Collins

43. 43 Transverse Spin Theory Transverse-momentum-dependent (TMD) Parton distributions Generalize Feynman parton distribution q(x) by including the transverse momentum. Generalize Feynman parton distribution q(x) by including the transverse momentum. q(x,k T ) q(x,k T )  At small k T, the transverse-momentum dependence is generated by soft non- perturbative physics. At large k T, the k-dependence can be calculated in perturbative QCD and falls like powers of 1/k T 2 At large k T, the k-dependence can be calculated in perturbative QCD and falls like powers of 1/k T 2

44. Feynman Parton: one-dimension Inclusive cross sections probe the momentum (longitudinal) distributions of partons inside nucleon 6/12/201644 Hadronic reactions

45. Extension to transverse direction… Semi-inclusive measurements  Transverse momentum dependent (TMD) parton distributions Deeply Virtual Compton Scattering and Exclusive processes  Generalized parton distributions (GPD) 6/12/201645

46. 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 Wigner 1933

47. 47 Wigner distribution for the quark The quark operator Wigner distributions After integrating over r, one gets TMD After integrating over k, one gets Fourier transform of GPDs Ji: PRL91,062001(2003)

48. 6/12/201648 Wigner Distribution W(x,r,k t ) d3rd3r Transverse Momentum Dependent PDF f(x,k t ) Generalized Parton Distr. H(x,ξ,t) d 2 k t dz F.T. d2ktd2kt PDF f(x) dx Form Factors F 1 (Q),F 2 (Q) GPD

49. TMD Parton Distributions The definition contains explicitly the gauge links The polarization and kt dependence provide rich structure in the quark and gluon distributions  Mulders-Tangerman 95, Boer-Mulders 98 6/12/201649 Collins-Soper 1981, Collins 2002, Belitsky-Ji-Yuan 2002

50. Generalized Parton Distributions Off-diagonal matrix elements of the quark operator (along light-cone) It depends on quark momentum fraction x and skewness ξ, and nucleon momentum transfer t 6/12/201650 Mueller, et al. 1994; Ji, 1996, Radyushkin 1996

51. Quark operator depends on b t It is the F.T. of the GPD at ξ =0 Impact parameter dependent parton distributions 6/12/201651 Soper 1977 Burkardt 2000,2003

52. Transverse profile for the quark distribution: k t vs b t 6/12/201652 GPD fit to the DVCS data from HERA, Kumerick-D.Mueller, 09,10 Quark distribution calculated from a saturation-inspired model A.Mueller 99, McLerran-Venugopalan 99

53. Gluon distribution 6/12/201653 One of the TMD gluon distributions at small-x GPD fit to the DVCS data from HERA, Kumerick-Mueller, 09,10

54. Deformation when nucleon is transversely polarized 6/12/201654 Lattice Calculation of the IP density of Up quark, QCDSF/UKQCD Coll., 2006 Quark Sivers function fit to the SIDIS Data, Anselmino, et al. 20009 kyky kxkx -0.5 0.5 0.0 -0.5 0.0 0.5

55. Access the GPDs Deeply virtual Compton Scattering (DVCS) and deeply virtual exclusive meson production (DVEM) 6/12/201655 GPD In the Bjorken limit: Q 2 >>(-t), ∧ 2 QCD,M 2

56. Transverse momentum dependent parton distribution Straightforward extension  Spin average, helicity, and transversity distributions Transverse momentum-spin correlations  Nontrivial distributions, S T XP T  In quark model, depends on S- and P-wave interference 6/12/201656

57. Where can we learn TMDs Semi-inclusive hadron production in deep inelastic scattering (SIDIS) Drell-Yan lepton pair, photon pair productions in pp scattering Dijet correlation in DIS Relevant e+e- annihilation processes … 6/12/201657

58. 58 6/12/2016 Novel Single Spin Asymmetries U: unpolarized beam T: transversely polarized target Semi-inclusive DIS

59. Two major contributions Sivers effect in the distribution Collins effect in the fragmentation Other contributions… 6/12/201659 kTkT STST P (zk+p T ) ~pTXsT~pTXsT (k,s T ) S T (P X k T )

60. Universality of the Collins Fragmentation 6/12/201660 ep--> e Pi Xe + e - --> Pi Pi Xpp--> jet(->Pi) X Metz 02, Collins-Metz 02, Yuan 07, Gamberg-Mukherjee-Mulders 08,10 Meissner-Metz 0812.3783 Yuan-Zhou, 0903.4680 Exps: BELLE, HERMES, STAR at RHIC

61. Collins asymmetries in SIDIS 6/12/201661 Summarized in the EIC Write-up

62. Collins effects in e + e - 6/12/201662 BELLE Coll., 2008Collins functions extracted from the Data, Anselmino, et al., 2009

63. Sivers effect is different It is the final state interaction providing the phase to a nonzero SSA Non-universality in general Only in special case, we have “Special Universality” Brodsky,Hwang,Schmidt 02 Collins, 02; Ji,Yuan,02; Belitsky,Ji,Yuan,02

64. Sivers asymmetries in SIDIS 6/12/201664 Jlab Hall A 3 He data Non-zero Sivers effects Observed in SIDIS

65. Quark Sivers function extracted from the data 6/12/201665 Alexei Prokudin, et al. Leading order fit, simple Gaussian assumption for the Sivers function There are still theoretical uncertainties In the fit: scale dependence, high order corrections, … Inner band is the impact from the planed EIC kiematics

66. 66 DIS and Drell-Yan Initial state vs. final state interactions “Universality”: QCD prediction HERMES ** ** Drell-Yan DIS

67. RHIC predictions 6/12/201667 There have been proposals to Do this measurement at RHIC Collider or fixed target modes There is also a COMPASS Proposal in the near future It is very important to test the sign change of the quark Sivers function http://spin.riken.bnl.gov/rsc/ Kang, Qiu, 2008

68. TMD gluon distributions It is not easy, because gluon does not couple to photon directly Can be studied in two-particle processes 6/12/201668 Dijet In DIS Vogelsang-Yuan, 2007 Dominguez-Xiao-Yuan, 2010 Boer-Brodsky-Mulders-Pisano, 2010 Di-photon In pp Qiu-Schlegel-Vogelsang, 2011

69. 69 Initial state and/or final state interactions Dijet-correlation at RHIC Boer-Vogelsang 03 Jet 1 Jet 2 P,S T Standard (naïve) Factorization breaks!

70. Theoretical challenges in the TMD Q 2 dependence and soft gluon resummation, in particular, for the SSA  Kang-Xiao-Yuan, 2011, following lectures Global study at the Next-to-leading order Relation to the Orbital angular momentum Unified picture for GPDs and TMDs … 6/12/201670

71. 6/12/201671 Wigner Distribution W(x,r,k t ) d3rd3r d 2 k t dz F.T. Transverse Momentum Dependent PDF f(x,k t ) d2ktd2kt PDF f(x) Generalized Parton Distr. H(x,ξ,t) dx Form Factors F 1 (Q),F 2 (Q) T-Wigner Distribution W(x,k t,r t ) dz

72. Transverse Wigner Distributions Integrate out z in the Wigner function  Depends on x, k t, b t  Also referred as GTMD in the literature See for example, Metz, et al., 09; Pasquini, 10,11 It has close connection to the small-x parton distributions in large nuclei 6/12/201672 e.g., gluon number distr. Mueller, NPB 1999

73. Further integrate out x 6/12/201673 kxkx rxrx Skovsen et al. (Denmark) PRL91, 090604 Quark model calculation: Xiong, et al. d-dbar AMO Exp.

74. There is no known process can be used to measure the T-Wigner distribution We have to either use a model (constrained by the GPD and TMDs) to calculate this function Or parameterize them and fit to GPDs and TMDs simultaneously 6/12/201674

75. 75 Transverse Spin Theory Semi-Inclusive DIS at Low P T Transverse Momentum Dependent (TMD) Parton Distributions Novel Single Spin Asymmetries Interested kinematics: Q 2 >> P T 2, Forward x F >0.1

76. 76 Transverse Spin Theory TMD Parton Distributions The gauge invariant definition In Feynman gauge Belitsky, Ji, Yuan (03)

77. 77 Transverse Spin Theory Where does the gauge link come from? Factorizable multiple gluon interactions

78. 78 Transverse Spin Theory Example: FSI in DIS  This is just the leading order expansion of the exponential gauge link  Summing all final state gluon interactions will lead to the final gauge link in the parton distribution definition

79. 79 Transverse Spin Theory Initial state interaction in Drell-Yan  This leads to the gauge link in Drell-Yan process goes to -1, instead of +1 in DIS  Consequence is the Sivers functions change sign for these two processes

80. 80 Transverse Spin Theory In light-cone gauge Additional gauge link is needed to ensure the gauge invariance of the definition  Which can also be derived from the previous diagrams

81. 81 Transverse Spin Theory Why the gauge link is important for all these businesses Sivers proposed Sivers function in 1990 Collins claimed it vanishes because of time- reversal invariance, 1993 Brodsky-Hwang-Schmidt made a model calculation of SSA (Sivers type) in SIDIS, due to the Final State Interactions These final state interactions are actually built in the parton distribution definitions – the Gauge Link

82. 82 Transverse Spin Theory Leading Order Quark Distributions Quark Nucleon Unpol. Long. Trans. Unpol.Long.Trans. q(x, k ┴ ) q T (x, k ┴ ) Δq L (x, k ┴ ) Δq T (x, k ┴ ) δq(x, k ┴ ) δq L (x, k ┴ ) δq L (x, k ┴ ) δq T (x, k ┴ ) δq T (x, k ┴ ) δq T '(x, k ┴ ) Boer, Mulders, Tangerman (96&98)

83. 83 Transverse Spin Theory Physical interpretation of some TMDs k t -even: q(x,k T ),  q L (x,k T ),  q T (x,k T ) k t -odd:  q L,  q T,  q T ’ T-odd: Sivers q T, Boer-Mulders  q Sivers functionBoer-Mulders function

84. 84 Transverse Spin Theory TMD factorization for SIDIS At leading power of 1/Q The structure functions depend on Q 2,x B,z h,P T Sivers Collins

85. 85 Transverse Spin Theory Low P T SIDIS Factorization Ji, Ma, Yuan, 04 Collins, Soper, 81; Collins, Metz, 04

86. 86 Why Worry about Factorization? Safely extract nonperturbative information  Theoretically under control No breakdown by un-cancelled divergence NLO correction calculable  Estimate the high order corrections

87. 87 What to Worry for Factorization? Correct definition of TMD parton distributions Gauge Invariance? Soft divergence gets cancelled Hard Part can be calculated perturbatively The cross section can be separated into Parton Distribution, Fragmentation Function, Hard and Soft factors

88. 88 References on Factorization Factorization for back-back jet production in e + e - annihilation (in axial gauge) -- Collins-Soper, NPB, 1981 Factorization for inclusive processes -- Collins, Soper, Sterman, NPB, 1985 -- Bodwin, PRD, 1985 -- Collins, Soper, Sterman, in Perturbative QCD, Mueller ed., 1989

89. TMD: Naïve Factorization TMD distr.TMD frag. Mulders, Tangelman, Boer (96 & 98)  SIDIS Cross section Naïve factorization (unpolarized structure function) Naïve factorization (unpolarized structure function) Hadron tensor

90. TMD Factorization Collins-Soper, 81 Ji-Ma-Yuan, 04 Collins-Metz 04 Leading order in pt/Q Additional soft factor

91. One-loop Factorization Verify the factorizationVerify the factorization Deduce the correct definition of TMD parton dis.Deduce the correct definition of TMD parton dis. Estimate of one-loop correction to HEstimate of one-loop correction to H Purpose: Procedure: 1.Take an on-shell quark as target 2.Calculate dis. and frag. to one-loop order 3.Define and calculate the soft factor 4.Full QCD calculation at one-loop order 5.Extract the relevant hard part

92. TMD: the definition In Feynman Gauge, the gauge link v is not n to avoid l.c. singularity !!

93. TMDs are process dependent (Fragmentation is different)  Gauge link direction changes from DIS to Drell-Yan process  More complicated structure for dijet- correlation in pp collisions, standard factorization breaks Light-cone singularity beyond Born diagram  Transverse momentum resummation 6/12/201693

94. One-Loop Real Contribution energy dep.

95. Energy Dependence  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 One can write down an evolution equation in ζ  K and G obey an RG equation, Collins and Soper (1981) μ independent!

96. Subtract the soft factor in the Dis.  TMD distribution contains the soft contribution  Subtract the soft contribution Zero bin subtraction: Monahar, Stewart, 06; Lee, Sterman, 06; Idilbi, Mehen, 07;

97. TMD Fragmentation functions  Can be defined in a similar way as the parton distribution  Have similar properties as TMD dis.

98. One-loop Factorization (virtual gluon) Four possible regions for the 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) p p′ q k  Vertex corrections (single quark target)

99. One-Loop Factorization (real gluon)  Gluon Radiation (single quark target) Three possible regions for the gluon momentum k: 1) k is collinear to p (parton distribution) 2) k is collinear to p′ (fragmentation) 3) k is soft (Wilson line) p p′q k

100. At one-loop order, we verified the factorization The hard part at one-loop order,

101. All Orders in Perturbation Theory Consider an arbitrary Feynman diagram Find the singular contributions from different regions of the momentum integrations (reduced diagrams) Power counting to determine the leading regions Factorize the soft and collinear gluons contributions Factorization theorem.

102. Reduced (Cut) Diagrams Leading contribution to a cross section from a diagram. Can be pictured as real spacetime process (Coleman and Norton)

103. Leading Regions The most important reduced diagrams are determined from power counting. 1. No soft fermion lines 2. No soft gluon lines attached to the hard part 3. Soft gluon line attached to the jets must be longitudinally polarized 4. In each jet, one quark plus arbitrary number of collinear long.-pol. gluon lines attached to the hard part. 5. The number of 3-piont vertices must be larger or equal to the number of soft and long.-pol. gluon lines.

104. Leading Region

105. Collinear And Soft Gluons The collinear gluons are longitudinally polarized 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. The soft part can be factorized from the jet using Grammer-Yennie approximation 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.

106. Factorization After soft and collinear factorizations, the reduced diagram become which corresponds to the factorization formula stated earlier.

107. Compared to the collinear factorization Simplification  Of the cross section in the region of pt<<Q, only keep the leading term Extension  To the small pt region, where the collinear factorization suffer large logarithms  Resummation can be done 6/12/2016

108. 108 Summary The TMD factorization has been shown for the semi-inclusive DIS process, and the hard factor been calculated for some observables Experiments should be able to test this factorization  Sign change between DIS and Drell-Yan for Sivers effects  Universality of the Fragmentation effects

109. Lecture III Unify the TMD and Collinear Factorization NLO example for SSA in Drell-Yan process (weighted asymmetry) 6/12/2016109

110. 110 Transition from Perturbative region to Nonperturbative region Compare different region of P T Nonperturbative TMDPerturbative region

111. Perturbative tail is calculable Transverse momentum dependence 6/12/2016111 Power counting, Brodsky-Farrar, 1973 Integrated Parton Distributions Twist-three functions

112. A unified picture (leading pt/Q) 6/12/2016112 Q Λ QCD PTPT Collinear/ longitudinal << Transverse momentum dependent Ji-Qiu-Vogelsang-Yuan,2006 Yuan-Zhou, 2009 PTPT

113. 113 Transverse Spin Theory Recall the TMD Factorization

114. 114 Transverse Spin Theory SIDIS: at Large P T When q T >>  QCD, the P t dependence of the TMD parton distribution and fragmentation functions can be calculated from pQCD, because of hard gluon radiation Single Spin Asymmetry at large P T is not suppressed by 1/Q, but by 1/P T

115. 115 Transverse Spin Theory Fragmentation function at p T >>  QCD See, e.g., Ji, Ma, Yuan, 04

116. 116 Transverse Spin Theory Sivers Function at large k T Quark-gluon Correlation Qiu, Sterman, 91,99

117. 117 Transverse Spin Theory Qiu-Sterman matrix element

118. 118 Transverse Spin Theory Sivers Function at Large k T 1/k T 4 follows a power counting Drell-Yan Sivers function has opposite sign Plugging this into the factorization formula, we indeed reproduce the polarized cross section calculated from twist-3 correlation

119. 119 Transverse Spin Theory SSA in the Twist-3 approach Twist-3 quark-gluon Correlation: T F (x 1,x 2 ) Fragmentation function: \hat q(x’) Collinear Factorization: Qiu,Sterman, 91

120. 120 Transverse Spin Theory Factorization guidelines Reduced diagrams for different regions of the gluon momentum: along P direction, P’, and soft Collins-Soper 81

121. 121 Final Results P T dependence Which is valid for all P T range  Resummation can be performed further Sivers function at low P T Qiu-Sterman Twist-three

122. Extend to other TMDs 6/12/2016122

123. 123 Polarized TMD Quark Distributions Quark Nucleon Unpol. Long. Trans. Unpol.Long.Trans. Boer, Mulders, Tangerman (96&98)

124. TMDs and Quark-gluon Correlations (twist-3) Kt-odd distribution 6/12/2016124 Boer-Mulders-Pijlman, 2003

125. Quark-gluon correlations (twist-three) Have long been studied, F-type and D-type are related to each other, Ellis-Furmanski-Petronzio 82, Eguchi-Koike-Tanaka 06 6/12/2016125

126. twist and collinear expansion 6/12/2016126 Twist-three matrix R.K. Ellis et al., 82; Qiu-Sterman, 90 Gauge invariant twist-3 Quark-gluon correlation functions: D- or F-type

127. Large kt TMDs 6/12/2016  Color factors, C F : a1-4,b1-4,c2,c4 1/2N c : c1,c3, d1-4 C A /2: e1-4  a1-4  b1-4,c1,c3,e1-4  b1-4,c1-4,d1-4,e1-4

128. Generic results Kt-even TMDs 6/12/2016128 Splitting kernelLarge logs Zhou,Liang,Yuan,2010

129. Sivers and Boer-Mulders 6/12/2016129

130. g 1T and h 1L 6/12/2016130

131. Scale evolution for the quark- gluon correlation functions Non-singlet part, 6/12/2016131 Kang-Qiu, arXiv:0811.3101 Zhou-Liang-Yuan, arXiv:0812.4484 Vogelsang-Yuan, arXiv:0904.0410 Braun-Manashov-Pimay,arXiv:0909.3410

132. NLO corrections to SSA SSA in Drell-Yan as an example, Kt-moment of the asymmetry Collinear factorization 6/12/2016132 Vogelsang-Yuan, arXiv:0904.0410

133. Born diagram Hard coefficient 6/12/2016133 Boer-Mulders, 1998

134. Virtual diagrams 6/12/2016134 soft divergencecollinear divergence

135. Soft divergence from real diagrams Cancel out that from virtual diagrams 6/12/2016135

136. Collinear divergence--splitting Anti-quark splitting Sivers splitting 6/12/2016136

137. Finite terms +... 6/12/2016137

138. Threshold limits Large-z, The asymmetry should not change dramatically with energy in the forward region 6/12/2016138

139. Lecture IV QCD resummation for single spin asymmetries A N in pp  h+X 6/12/2016139

140. 140 Asymptotical Freedom and Factorization QCD is an asymptotical freedom theory (Gross, Politzer, Wilczek, 1973), where perturbation method becomes relevant at large scale. While, because of confinement, a typical hadronic process contains multiple scales, e.g., the nonperturbative scale  QCD, meaning that a QCD factorization must be proven in order to successfully separate different scales.

141. 141 One Large Scale Factorization If the physics only involves one large scale, the factorization is the simplest,  Inclusive DIS and Drell-Yan  Jet production  Inclusive particle production at hadron collider  Hard exclusive processes, Pi form factor, DVCS, …  (Q)=H(Q/  ) f 1 (  )…

142. 142 Additional Large Scale Introduces Large Double Logarithms For example, a differential cross section depends on Q 1, where Q 2 >>Q 1 2 >>  2 QCD We have to resum these large logs to make reliable predictions  Q T : Dokshitzer, Diakonov, Troian, 78; Parisi Petronzio, 79; Collins, Soper, Sterman, 85  Threshold: Sterman 87; Catani and Trentadue 89

143. 143 Why Resummation is Relevant Soft gluon radiation is very important for this kinematical limit Real and Virtual contributions are “imbalanced” IR cancellation leaves large logarithms (implicit)

144. 144 How Large of the Resummation effectsResum NLO Kulesza, Sterman, Vogelsang, 02

145. 145 General Structure of Large Logs LO1 NLO s L2s L2s L2s L2 s Ls Ls Ls L ssss NNLO s2 L4s2 L4s2 L4s2 L4 s2 L3s2 L3s2 L3s2 L3 s2 L2s2 L2s2 L2s2 L2+… N 3 LO s3 L6s3 L6s3 L6s3 L6 s3 L5s3 L5s3 L5s3 L5 s3 L4s3 L4s3 L4s3 L4+… N k LO  s k L 2k  s k L 2k-1  s k L 2k-2 +… LLNLLNNLL

146. 146 Two Large Scales Processes Include  DIS and Drell-Yan at small P T (Q T Resum)  DIS and Drell-Yan at large x (Threshold Resum)  Higgs production at small P T or large x  Semileptonic B Decays  Non-leptonic B Decays  Thrust distribution  Jet shape function  …

147. 147 Collins-Soper-Sterman Resummation Introduce a new concept, the Transverse Momentum Dependent PDF Prove the Factorization in terms of the TMDs  (P T,Q)=H(Q) f 1 (k 1T,Q) f 2 (k 2T, Q) S( T ) Large Logs are resummed by solving the energy evolution equation of the TMDs ( Collins-Soper 81, Collins-Soper-Sterman 85 )

148. 148 CSS Formalism (II) K and G obey the renormalization group eq. The large logs will be resummed into the exponential form factor  A,B,C functions are perturbative calculable. (Collins-Soper-Sterman 85)

149. 149 Factorization is Crucial here Because they make use of the TMD parton distributions, factorization and the gauge property of the TMDs are very crucial In Collins-Soper 81, Axial gauge was used to prove the factorization Ji-Ma-Yuan 2004, the gauge invariant TMD parton distributions were used

150. 150 Large Logarithms Resummation At low transverse momentum, P T <<Q, we must resum the large logarithms  s n ln 2n-1 (Q 2 /P T 2 ) -- Dokshitzer, Diakonov, Troian, 1978 -- Parisi, Petronzio, 1979 These large logarithms can be resummed by solving the energy evolution equation for the TMD parton dis. -- Collins-Soper 1981

151. QCD Factorization  Factorization for the structure function –q: TMD parton distribution –q hat: TMD fragmentation function –S: Soft factors –H: hard scattering. Impact parameter space

152. Large Logarithms Resummation Factorization form in b space, Large logs: Differential equation respect to The Solution No Large logs.

153. Further factorization for the TMD distribution at large 1/b (Collins&Soper 81) Integrated dis. CSS resummation at large 1/b (CSS’85) Our one-loop results for the TMD dis. and frag. can reproduce the C functions, C functions

154. 154 After resummation, large logarithms associated with Q 2 can be factorized into the Sudakov form factors, e.g. And the Sudakov form factor

155. If Q 2 is not too large, DL approx. applies. The Sudakov suppression form factor then only depends on Q 2 The Q 2 dependence of the structure functions can be factorized out We can predict the P T distribution at higher Q 2 from that of lower Q 2 The P T spectrum of the polarization asymmetry will be the same for different Q 2 at fixed x B and z 155 Double Logarithmic (DL) Approx.

156. 156 Phenomenogical applications of the QCD resummation to the P T spectrum of EW bosons production have been very successful Yuan, Nadolsky, Ladinsky, Landry, Qiu, Zhang, Berger, Li, Laenen, Sterman, Vogelsang, Kulesza, Bozzi, Catani, deFlorian, Kulesza, Stirling, and many others, … working even at NNLL level for some

157. 157 Drell-Yan at Fixed Target Q T spectrum from E288, PRD23,604(81)

158. 158 At very large Q 2 (e.g., Z 0 and W boson), DL Approx. breaks down

159. Qiu-Zhang, 2000

160. 160 SSAs: DY as an example P T dependence Which is valid for all P T range Sivers function at low P T Qiu-Sterman Twist-three

161. CSS Resummation Separate the singular and regular parts TMD factorization in b-space 6/12/2016161

162. Leading order Small-b expansion, 1/b>>intrinsic kt 6/12/2016162

163. Virtual diagrams 6/12/2016163 soft divergencecollinear divergence

164. Soft divergence from real diagrams 6/12/2016164

165. Collinear divergence--splitting Sivers function 6/12/2016165

166. Hard factor at one-loop order Same as the spin-average case 6/12/2016

167. Final resum form Sudakov the same 6/12/2016

168. Coefficients at one-loop order It will be important to apply this resummation formalism to study the energy dependence of the SSAs  Work in progress… 6/12/2016168

169. Back to A N in pp  h+x Large transverse momentum hadron production  Single scale: collinear factorization Higher-twist effects  Qiu-Sterman, 91, 98 6/12/2016169

170. 170 Twist-3 factorization for Pion SSA in hadronic collisions Collinear factorization Qiu-Sterman matrix element T F (x,x)Unpolarized parton distr. Hard factor Fragmentation function

171. 171 Initial and final state phases

172. 172 Final results Leading order only, to demonstrate the factorization, one need to go beyond this order Additional contributions: soft-fermion pole, tri-gluon corrections were carried out recently (Koike, et al.; Kang, et al.) Kouvaris,Qiu,Vogelsang,Yuan, 06

173. 173 Twist-3 Fit to data E704 STAR Kouvaris,Qiu,Vogelsang,Yuan, 06 RHIC BRAMHS

174. 174 Compare to 2006 data from RHIC J.H. Lee, SPIN 2006

175. 175

176. 176 6/12/2016 Users Meeting, BNL challenge from STAR data (2006) Talks by Ogawa and Nogach in SPIN2006

177. 177 Some comments It’s difficult to explain this pattern in the current twist-3 theoretical approaches  Fragmentation (Collins effect) contributions?

178. A kt-dependent Model 6/12/2016178 H Quark distribution From the projectile Dense medium Dumitru-Jalilian-Marian, 02 Dumitru-Hayashigaki-Jalilian-Marian, 06 Kang, Yuan, 2011 Spin-average case

179. SSA at low P T Cross section dominated by low transverse momentum UGD 6/12/2016179

180. SSA at high P T Cross section is dominated by large pt UGD Ratios 6/12/2016180

181. Compare to data 6/12/2016181

182. 182 Transverse Spin Theory Summary We are at a very exciting era of transverse spin physics studies Existing data from current experiment and future ones from the planed experiments will provide a detailed understanding of the spin degrees of freedom, especially for the quark orbital motion We will learn more about nucleon structure from these studies and the strong interaction dynamics