Transverse Spin Physics theory Feng Yuan Lawrence Berkeley
Transverse Spin Physics (theory) Feng Yuan Lawrence Berkeley National Laboratory RBRC, Brookhaven National Laboratory
Four Lectures n Transverse Spin and Transversity ¨ General n introduction TMD and Semi-inclusive DIS ¨ Global picture TMD in perturbative region and the relevant QCD dynamics n Advanced topics: NLO calculations ad QCD resummation for SSA n 2 Transverse Spin Theory
Exploring the nucleon: Of fundamental importance in science n n 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 ¨ n New Physics beyond Standard Model: Tevatron, LHC, Jefferson Lab, … Exploration of nucleon structure started century ago… 3
Deep Inelastic Scattering Friedman Kendall Taylor Bjorken Scaling: Q 2 Infinity Feynman Parton Model 2/26/2021 5
QCD and Strong-Interactions n QCD: Non-Abelian gauge theory ¨ Building blocks: quarks (spin½, mq, 3 colors; gluons: spin 1, massless, 32 -1 colors) n Asymptotic freedom and confinement ce n ta s i ive t d a rb ort u h t r S pe , Long distance: ? Soft, non-perturbative rd ha Clay Mathematics Institute Millennium Prize Problem ~1/Length Nonperturbative scale QCD~1 Ge. V 6
The Proton in QCD n 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 ¨ n 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=? ¨ 7
Understanding the nucleon n Solving QCD ¨ Numerically simulation, like 4 D stat. mech. systems Feynman path integral Wick rotation Spacetime discretization Monte Carlo simulation ¨ Effective field theories (large Nc, chiral physics, …) n Experimental probes ¨ Study the quark and gluon structure through low and high-energy scattering ¨ Require clean reaction mechanism n Photon, electron & perturbative QCD 2/26/2021 8
High energy scattering as a probe to the nucleon structure DIS Q>> QCD (Q>> QCD) k n Drell-Yan Feynman Parton Momentum fraction DVCS Hadronic reactions 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 9
Factorization is essential to study nucleon structure n Universality of the parton distributions ¨ Predictive power of QCD Collins-Soper-Sterman 85, Bodwin 85, … Ji-Ma-Yuan 04, Dominguez-Xiao-Yuan 10, … 10
Simple example: e+e- hadrons n Leading order k 2 k 1 q p 2 p 1 ¨ Eletron-positron annihilate into virtual photon, and decays into quark-antiquark pair, or muon pair ¨ Quark-antiquark pair hadronize 2/26/2021 11
n Total cross section n R ratio ¨ Depends on the number of colors, electric charge of the quark 2/26/2021 12
High order corrections There are soft and collinear divergences in the real and virtual gluon radiation n It can be shown that these divergences cancel out between real and virtual diagrams n Total cross section is infrared safe n 2/26/2021 13
Further remarks n If we keep the light-cone k+, the cancellation is not complete ¨ Left with the collinear divergence, factorized into the fragmentation function k 2 k 1 p 2 q p 1 ph 2/26/2021 14
n If we further keep the transverse momentum, there is large logarithms associated with the gluon radiation ¨ TMD factorization and resummation ph 2 k 2 p 2 q p 1 k 1 2/26/2021 ph 1 15
Back to nucleon structure 2/26/2021 16
Proton Spin Sum Rule Gluon spin~ 0 -70% RHIC, EIC, … Quark spin ~30% DIS, and pp coll. q Lq 2/26/2021 G LG Deeply Virtual Compton Scattering, Transverse spin physics, in DIS, pp coll. 17
Transverse spin - introduction n Pauli-Lubanski operator ¨ P – Momentum operator ¨ J – Angular momentum operator ¨ Dirac particle of momentum p, and spin s Polarization operator Transverse Spin Theory 18
n 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 sz/s. T ∞ when p ∞ Transverse Spin Theory 19
Helicity and chirality n Choose longitudinal polarization, s~p, Spin states are also helicity states n In the massless case, helicity states are also chirality states n Transverse Spin Theory 20
Transverse spin n n Transverse spin states are off-diagonal state in the helicity basis, e. g. , along the x-direction, Spin dependent observable, Transverse Spin Theory 21
Transverse polarized structure function n Polarized DIS structure contains two terms Longitudinal polarized, g 1 contributes n Transverse polarized, g 1+g 2 contribute; it is manifested as higher-twist effects, because ST suppressed by 1/P+ n Transverse Spin Theory 22
Factorize the parton distributions n In DIS, Quark distribution matri (P, k) Gauge links (next) Transverse Spin Theory 23
Leading order quark distribution n expands at leading order, Unpolarized helicity transversity n Although a leading-twist distribution, transversity is chiral-odd, and doesn’t contribute to the DIS structure function Transverse Spin Theory 24
Probability representation (helicity basis) + + + + q(x) q. L q. T (h 1, T q) + + - Transverse Spin Theory 25 + - + + - - +
Soffer bound n choose the coupling of nucleon and quark as ++ and +¨ q(x)=| ++|2+| +-|2 q. L(x)=| ++|2 -| +-|2 ¨ q. T(x)= ++ - -* ¨ Soffer bound: ¨ Transverse Spin Theory 26
Nucleon tensor charge n Tensor charge can be calculated from lattice QCD, Transverse Spin Theory 27
QCD evolution n No mixing with gluon + - + - + - This part is the kernel for unpolarized and longitudinal polarized quark distr. Transverse Spin Theory 28
Measuring transversity is difficult n 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) Transverse Spin Theory 29
Drell-Yan is an ideal place n Combining two transversity distributions in Drell-Yan lepton pair production q. T + + - Transverse Spin Theory 30
Angular distribution of the lepton pair n Double transverse spin asymmetry Transverse Spin Theory 31
Predictions at RHIC Vogelsang, et al, PRD, 1999 Transverse Spin Theory 32
Dedicated Drell-Yan study at RHIC II Matthias Grosse Perdekamp Transverse Spin Theory 33
Other processes Soffer, Stratmann, Vogelsang, 02 Transverse Spin Theory 34
Two-hadron interference frag. fun. n In DIS Collins, Heppelmann, Ladinsky, 94 Jaffe, Jin, Tang, 97 Bacchetta, Radici, 04, 06 Transverse Spin Theory Metz-Zhou, 2011 35
HERMES Results Transverse Spin Theory 36 ar. Xiv: 0803. 2367
Input from BELLE ar. Xiv: 1104. 2425 2/26/2021 37
2/26/2021 38
Semi-inclusive DIS n Collins fragmentation function is chiralodd (zk+p. T) (k, s. T) ~ p. TXs. T Combining with the quark transversity leads to single transverse-spin asymmetry in SIDIS n Open a whole window of SSAs in SIDIS (next lecture) n Transverse Spin Theory 39
e+e- collisions play important roles in this game n Reliable place to extract the Collins function and interference (two-hadron) fragmentation function Transverse Spin Theory 40 Belle Col. , PRL 06
Lecture II 3 D imaging of partons in nucleon/TMD physics n Transverse Momentum Dependent Factorization n 2/26/2021 41
Physics of SIDIS • Flavor decomposition for polarized or unpolarized quark distributions • Transverse momentum Dependent physics, transversity distribution, Sivers, Collins Transverse Spin Theory 42
Transverse-momentum-dependent (TMD) Parton distributions n Generalize Feynman parton distribution q(x) by including the transverse momentum. q(x, k. T) At small k. T, the transverse-momentum dependence is generated by soft nonperturbative physics. n At large k. T, the k-dependence can be calculated in perturbative QCD and falls like powers of 1/k. T 2 Transverse Spin Theory 43
Feynman Parton: one-dimension n Inclusive cross sections probe the momentum (longitudinal) distributions of partons inside nucleon Hadronic reactions 2/26/2021 44
Extension to transverse direction… n Semi-inclusive measurements ¨ Transverse momentum dependent (TMD) parton distributions n Deeply Virtual Compton Scattering and Exclusive processes ¨ Generalized parton distributions (GPD) 2/26/2021 45
Wigner function n Define as ¨ When Wigner 1933 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 distribution for the quark n The quark operator n Wigner distributions Ji: PRL 91, 062001(2003) After integrating over r, one gets TMD After integrating over k, one gets Fourier transform of GPDs 47
Wigner Distribution W(x, r, kt) dz kt d 3 r 2 d T. F. Generalized Parton Distr. H(x, ξ, t) dx d 2 kt Transverse Momentum Dependent PDF f(x, kt) GPD PDF f(x) Form Factors F 1(Q), F 2(Q) 2/26/2021 48
TMD Parton Distributions n The definition contains explicitly the Collins-Soper 1981, gauge links Collins 2002, Belitsky-Ji-Yuan 2002 n The polarization and kt dependence provide rich structure in the quark and gluon distributions ¨ Mulders-Tangerman 95, Boer-Mulders 98 2/26/2021 49
Generalized Parton Distributions Mueller, et al. 1994; Ji, 1996, Radyushkin 1996 n n 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 2/26/2021 50
Impact parameter dependent parton distributions Soper 1977 Burkardt 2000, 2003 n Quark operator depends on bt n It is the F. T. of the GPD at ξ =0 2/26/2021 51
Transverse profile for the quark distribution: kt vs bt Quark distribution calculated from a saturation-inspired model A. Mueller 99, Mc. Lerran-Venugopalan 99 2/26/2021 GPD fit to the DVCS data from HERA, Kumerick-D. Mueller, 09, 10 52
Gluon distribution One of the TMD gluon distributions at small-x 2/26/2021 GPD fit to the DVCS data from HERA, Kumerick-Mueller, 09, 10 53
Deformation when nucleon is transversely polarized -0. 5 0. 0 0. 5 ky 0. 5 0. 0 -0. 5 kx Quark Sivers function fit to the SIDIS Data, Anselmino, et al. 20009 2/26/2021 Lattice Calculation of the IP density of Up quark, QCDSF/UKQCD Coll. , 2006 54
Access the GPDs n Deeply virtual Compton Scattering (DVCS) and deeply virtual exclusive meson production (DVEM) GPD In the Bjorken limit: Q 2>>(-t), ∧ 2 QCD, M 2 2/26/2021 55
Transverse momentum dependent parton distribution n Straightforward extension ¨ Spin average, helicity, and transversity distributions n Transverse momentum-spin correlations ¨ Nontrivial distributions, STXPT ¨ In quark model, depends on S- and P-wave interference 2/26/2021 56
Where can we learn TMDs Semi-inclusive hadron production in deep inelastic scattering (SIDIS) n Drell-Yan lepton pair, photon pair productions in pp scattering n Dijet correlation in DIS n Relevant e+e- annihilation processes n… n 2/26/2021 57
Semi-inclusive DIS n Novel Single Spin Asymmetries U: unpolarized beam T: transversely polarized target 58 2/26/2021
Two major contributions n Sivers effect in the distribution ST k. T P n Collins effect in the fragmentation (zk+p. T) (k, s. T) n ST (PXk. T) ~p. TXs. T Other contributions… 2/26/2021 59
Universality of the Collins Fragmentation ep--> e Pi X e+e---> Pi Pi X Metz 02, Collins-Metz 02, Yuan 07, Gamberg-Mukherjee-Mulders 08, 10 Meissner-Metz 0812. 3783 Yuan-Zhou, 0903. 4680 2/26/2021 pp--> jet(->Pi) X Exps: BELLE, HERMES, STAR at RHIC 60
Collins asymmetries in SIDIS Summarized in the EIC Write-up 2/26/2021 61
Collins effects in e+e- BELLE Coll. , 2008 Collins functions extracted from the Data, Anselmino, et al. , 2009 2/26/2021 62
Sivers effect is different It is the final state interaction providing the phase to a nonzero SSA n Non-universality in general n Only in special case, we have “Special Universality” n Brodsky, Hwang, Schmidt 02 Collins, 02; Ji, Yuan, 02; Belitsky, Ji, Yuan, 02
Sivers asymmetries in SIDIS Jlab Hall A 3 He data Non-zero Sivers effects Observed in SIDIS 2/26/2021 64
Quark Sivers function extracted from the data Leading order fit, simple Gaussian assumption for the Sivers function There are still theoretical uncertaintie In the fit: scale dependence, high order corrections, … Inner band is the impact from the planed EIC kiematics Alexei Prokudin, et al. 2/26/2021 65
DIS and Drell-Yan n Initial state vs. final state interactions * * Drell-Yan DIS HERMES n “Universality”: QCD prediction 66
RHIC predictions There have been proposals to Do this measurement at RHIC http: //spin. riken. bnl. gov/rsc/ 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 Kang, Qiu, 2008 2/26/2021 67
TMD gluon distributions It is not easy, because gluon does not couple to photon directly n Can be studied in two-particle processes n Di-photon In pp Dijet In DIS Vogelsang-Yuan, 2007 Dominguez-Xiao-Yuan, 2010 Boer-Brodsky-Mulders-Pisano, 2010 2/26/2021 Qiu-Schlegel-Vogelsang, 2011 68
Dijet-correlation at RHIC n Initial state and/or final state interactions Jet 2 Jet 1 Boer-Vogelsang 03 P, ST Standard (naïve) Factorization breaks! 69
Theoretical challenges in the TMD n 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 n Relation to the Orbital angular momentum n Unified picture for GPDs and TMDs n… n 2/26/2021 70
2 d d 3 r Wigner Distribution W(x, r, kt) dz kt T. F. dz Transverse Momentum Dependent PDF f(x, kt) Generalized Parton Distr. H(x, ξ, t) W(x, kt, rt) PDF f(x) dx d 2 kt T-Wigner Distribution Form Factors F 1(Q), F 2(Q) 2/26/2021 71
Transverse Wigner Distributions n Integrate out z in the Wigner function ¨ Depends on x, kt, bt ¨ Also referred as GTMD in the literature n n See for example, Metz, et al. , 09; Pasquini, 10, 11 It has close connection to the small-x parton distributions in large nuclei e. g. , gluon number distr. Mueller, NPB 1999 2/26/2021 72
Further integrate out x AMO Exp. d-dbar rx kx Quark model calculation: Xiong, et al. Skovsen et al. (Denmark) PRL 91, 090604 2/26/2021 73
There is no known process can be used to measure the T-Wigner distribution n We have to either use a model (constrained by the GPD and TMDs) to calculate this function n Or parameterize them and fit to GPDs and TMDs simultaneously n 2/26/2021 74
Semi-Inclusive DIS at Low PT Transverse Momentum Dependent (TMD) Parton Distributions n Novel Single Spin Asymmetries n Interested kinematics: Q 2>>PT 2, Forward x. F>0. 1 Transverse Spin Theory 75
TMD Parton Distributions n The gauge invariant definition n In Feynman gauge Transverse Spin Theory Belitsky, Ji, Yuan (03) 76
Where does the gauge link come from? n Factorizable multiple gluon interactions Transverse Spin Theory 77
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 Transverse Spin Theory 78
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 Transverse Spin Theory 79
In light-cone gauge n Additional gauge link is needed to ensure the gauge invariance of the definition ¨ Which can also be derived from the previous diagrams Transverse Spin Theory 80
Why the gauge link is important for all these businesses n n Sivers proposed Sivers function in 1990 Collins claimed it vanishes because of timereversal 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 Transverse Spin Theory 81
Leading Order Quark Distributions Nucleon Quark Unpol. q(x, k┴) Long. Trans. Long. δq(x, k┴) Trans. q. T(x, k┴) Δq. L(x, k┴) Δq. T(x, k┴) δq. L(x, k┴) δq. T'(x, k┴) Boer, Mulders, Tangerman (96&98) 82 Transverse Spin Theory
Physical interpretation of some TMDs n kt-even: q(x, k. T), q. L(x, k. T), q. T(x, k. T) kt-odd: q. L, q. T’ T-odd: Sivers q. T, Boer-Mulders q Sivers function Transverse Spin Theory 83 Boer-Mulders function
TMD factorization for SIDIS At leading power of 1/Q Sivers Collins The structure functions depend on Q 2, x. B, zh, PT Transverse Spin Theory 84
Low PT SIDIS Factorization Ji, Ma, Yuan, 04 Collins, Soper, 81; Collins, Metz, 04 Transverse Spin Theory 85
Why Worry about Factorization? n Safely extract nonperturbative information ¨ Theoretically under control No breakdown by un-cancelled divergence n NLO correction calculable ¨ Estimate the high order corrections 86
What to Worry for Factorization? n n n 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 87
References on Factorization for back-back jet production in e+e- annihilation (in axial gauge) -- Collins-Soper, NPB, 1981 n Factorization for inclusive processes -- Collins, Soper, Sterman, NPB, 1985 -- Bodwin, PRD, 1985 -- Collins, Soper, Sterman, in Perturbative QCD, Mueller ed. , 1989 n 88
TMD: Naïve Factorization SIDIS Cross section Hadron tensor • Naïve factorization (unpolarized structure function) TMD distr. TMD frag. Mulders, Tangelman, Boer (96 & 98)
TMD Factorization • Leading order in pt/Q • Additional soft factor Collins-Soper, 81 Ji-Ma-Yuan, 04 Collins-Metz 04
One-loop Factorization Purpose: • Verify the factorization • Deduce the correct definition of TMD parton dis. • Estimate of one-loop correction to H Procedure: 1. 2. 3. 4. 5. Take an on-shell quark as target Calculate dis. and frag. to one-loop order Define and calculate the soft factor Full QCD calculation at one-loop order Extract the relevant hard part
TMD: the definition In Feynman Gauge, the gauge link v is not n to avoid l. c. singularity !!
n TMDs are process dependent (Fragmentation is different) ¨ Gauge link direction changes from DIS to Drell-Yan process ¨ More complicated structure for dijetcorrelation in pp collisions, standard factorization breaks n Light-cone singularity beyond Born diagram ¨ Transverse momentum resummation 2/26/2021 93
One-Loop Real Contribution energy dep.
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 Collins and Soper (1981) K and G obey an RG equation, μ independent!
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;
TMD Fragmentation functions Can be defined in a similar way as the parton distribution Have similar properties as TMD dis.
One-loop Factorization (virtual gluon) Vertex corrections (single quark target) q p′ k p 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 (p. QCD correction)
One-Loop Factorization (real gluon) Gluon Radiation (single quark target) q p′ k p 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)
At one-loop order, we verified the factorization The hard part at one-loop order,
All Orders in Perturbation Theory n n n 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.
Reduced (Cut) Diagrams n n Leading contribution to a cross section from a diagram. Can be pictured as real spacetime process (Coleman and Norton)
Leading Regions n 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.
Leading Region
Collinear And Soft Gluons n n n 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.
Factorization n After soft and collinear factorizations, the reduced diagram become which corresponds to the factorization formula stated earlier.
Compared to the collinear factorization n Simplification ¨ Of the cross section in the region of pt<<Q, only keep the leading term n Extension ¨ To the small pt region, where the collinear factorization suffer large logarithms ¨ Resummation can be done 2/26/2021
Summary The TMD factorization has been shown for the semi-inclusive DIS process, and the hard factor been calculated for some observables n Experiments should be able to test this factorization n ¨ Sign change between DIS and Drell-Yan for Sivers effects ¨ Universality of the Fragmentation effects 108
Lecture III Unify the TMD and Collinear Factorization n NLO example for SSA in Drell-Yan process (weighted asymmetry) n 2/26/2021 109
Transition from Perturbative region to Nonperturbative region n Compare different region of PT Nonperturbative TMD Perturbative region 110
Perturbative tail is calculable n Transverse momentum dependence Power counting, Brodsky-Farrar, 1973 Integrated Parton Distributions Twist-three functions 2/26/2021 111
A unified picture (leading pt/Q) Transverse momentum dependent Collinear/ longitudinal ΛQCD PT << 2/26/2021 PT << Q Ji-Qiu-Vogelsang-Yuan, 2006 Yuan-Zhou, 2009 112
Recall the TMD Factorization Transverse Spin Theory 113
SIDIS: at Large PT When q. T>> QCD, the Pt dependence of the TMD parton distribution and fragmentation functions can be calculated from p. QCD, because of hard gluon radiation n Single Spin Asymmetry at large PT is not suppressed by 1/Q, but by 1/PT n Transverse Spin Theory 114
Fragmentation function at p. T>> QCD See, e. g. , Ji, Ma, Yuan, 04 Transverse Spin Theory 115
Sivers Function at large k. T Quark-gluon Correlation Transverse Spin Theory 116 Qiu, Sterman, 91, 99
Qiu-Sterman matrix element Transverse Spin Theory 117
Sivers Function at Large k. T n n n 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 Transverse Spin Theory 118
SSA in the Twist-3 approach Fragmentation function: hat q(x’) Twist-3 quark-gluon Correlation: TF(x 1, x 2) Collinear Factorization: Qiu, Sterman, 91 Transverse Spin Theory 119
Factorization guidelines Reduced diagrams for different regions of the gluon momentum: along P direction, P’, and soft Collins-Soper 81 Transverse Spin Theory 120
Final Results n PT dependence Sivers function at low PT n Qiu-Sterman Twist-three Which is valid for all PT range ¨ Resummation 121 can be performed further
Extend to other TMDs 2/26/2021 122
Polarized TMD Quark Distributions Nucleon Quark Unpol. Long. Trans. Unpol. Long. Trans. Boer, Mulders, Tangerman (96&98) 123
TMDs and Quark-gluon Correlations (twist-3) n Kt-odd distribution Boer-Mulders-Pijlman, 2003 2/26/2021 124
Quark-gluon correlations (twist-three) n Have long been studied, n F-type and D-type are related to each other, Ellis-Furmanski-Petronzio 82, Eguchi-Koike-Tanaka 06 2/26/2021 125
twist and collinear expansion R. K. Ellis et al. , 82; Qiu-Sterman, 90 Twist-three matrix Gauge invariant twist-3 Quark-gluon correlation functions: D- or F-type 2/26/2021 126
Large kt TMDs Color factors, CF: a 1 -4, b 1 -4, c 2, c 4 1/2 Nc: c 1, c 3, d 1 -4 CA/2: e 1 -4 2/26/2021 a 1 -4 b 1 -4, c 1, c 3, e 1 -4 b 1 -4, c 1 -4, d 1 -4, e 1 -4
Generic results n Kt-even TMDs Splitting kernel 2/26/2021 Zhou, Liang, Yuan, 2010 Large logs 128
n Sivers and Boer-Mulders 2/26/2021 129
n g 1 T and h 1 L 2/26/2021 130
Scale evolution for the quarkgluon correlation functions n Non-singlet part, 2/26/2021 Kang-Qiu, ar. Xiv: 0811. 3101 Zhou-Liang-Yuan, ar. Xiv: 0812. 4484 Vogelsang-Yuan, ar. Xiv: 0904. 0410 Braun-Manashov-Pimay, ar. Xiv: 0909. 3410 131
NLO corrections to SSA Vogelsang-Yuan, ar. Xiv: 0904. 0410 n SSA in Drell-Yan as an example, n Kt-moment of the asymmetry n Collinear factorization 2/26/2021 132
Born diagram n Hard coefficient Boer-Mulders, 1998 2/26/2021 133
Virtual diagrams soft divergence 2/26/2021 collinear divergence 134
Soft divergence from real diagrams n Cancel out that from virtual diagrams 2/26/2021 135
Collinear divergence--splitting n Anti-quark splitting n Sivers splitting 2/26/2021 136
Finite terms +. . . 2/26/2021 137
Threshold limits n n Large-z, The asymmetry should not change dramatically with energy in the forward region 2/26/2021 138
Lecture IV QCD resummation for single spin asymmetries n AN in pp h+X n 2/26/2021 139
Asymptotical Freedom and Factorization n n 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. 140
One Large Scale Factorization n 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( )… 141
Additional Large Scale Introduces Large Double Logarithms n n For example, a differential cross section depends on Q 1, where Q 2>>Q 12>> 2 QCD We have to resum these large logs to make reliable predictions ¨ QT: Dokshitzer, Diakonov, Troian, 78; Parisi Petronzio, 79; Collins, Soper, Sterman, 85 ¨ Threshold: Sterman 87; Catani and Trentadue 89 142
Why Resummation is Relevant n n Soft gluon radiation is very important for this kinematical limit Real and Virtual contributions are “imbalanced” IR cancellation leaves large logarithms (implicit) 143
How Large of the Resummation effects Resum NLO Kulesza, Sterman, Vogelsang, 02 144
General Structure of Large Logs LO 1 NLO s L 2 s L s NNLO s 2 L 4 s 2 L 3 s 2 L 2 +… N 3 LO s 3 L 6 s 3 L 5 s 3 L 4 +… Nk. LO sk L 2 k-1 sk L 2 k-2 +… LL NNLL 145
Two Large Scales Processes n Include and Drell-Yan at small PT (QT Resum) ü ¨ DIS and Drell-Yan at large x (Threshold ü Resum) ü ¨ Higgs production at small PT or large x ¨ Semileptonic B Decays ¨ Non-leptonic B Decays ¨ Thrust distribution ¨ Jet shape function ¨… ¨ DIS 146
Collins-Soper-Sterman Resummation Introduce a new concept, the Transverse Momentum Dependent PDF n Prove the Factorization in terms of the TMDs (PT, Q)=H(Q) f 1(k 1 T, Q) f 2(k 2 T, Q) S( T) n Large Logs are resummed by solving the energy evolution equation of the TMDs n (Collins-Soper 81, Collins-Soper-Sterman 85) 147
CSS Formalism (II) n n 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) 148
Factorization is Crucial here Because they make use of the TMD parton distributions, factorization and the gauge property of the TMDs are very crucial n In Collins-Soper 81, Axial gauge was used to prove the factorization n Ji-Ma-Yuan 2004, the gauge invariant TMD parton distributions were used n 149
Large Logarithms Resummation n n At low transverse momentum, PT<<Q, we must resum the large logarithms snln 2 n-1(Q 2/PT 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 150
QCD Factorization for the structure function – – q: TMD parton distribution q hat: TMD fragmentation function S: Soft factors H: hard scattering. Impact parameter space
Large Logarithms Resummation Factorization form in b space, Large logs: Differential equation respect to The Solution No Large logs.
Further factorization for the TMD distribution at large 1/b (Collins&Soper 81) Integrated dis. CSS resummation at large 1/b (CSS’ 85) C functions Our one-loop results for the TMD dis. and frag. can reproduce the C functions,
n n After resummation, large logarithms associated with Q 2 can be factorized into the Sudakov form factors, e. g. And the Sudakov form factor 154
Double Logarithmic (DL) Approx. n n n 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 PT distribution at higher Q 2 from that of lower Q 2 The PT spectrum of the polarization asymmetry will be the same for different Q 2 at fixed x. B and z 155
n Phenomenogical applications of the QCD resummation to the PT spectrum of EW bosons production have been very successful Yuan, Nadolsky, Ladinsky, Landry, Qiu, Zhang, Berger, Li, Laenen, Sterman, Vogelsang, Kulesza, Bozzi, Catani, de. Florian, Kulesza, Stirling, and many others, … working even at NNLL level for some 156
Drell-Yan at Fixed Target QT spectrum from E 288, PRD 23, 604(81) 157
At very large Q 2 (e. g. , Z 0 and W boson), DL Approx. breaks down 158
Qiu-Zhang, 2000
SSAs: DY as an example n PT dependence Sivers function at low PT n Qiu-Sterman Twist-three Which is valid for all PT range 160
CSS Resummation n Separate the singular and regular parts n TMD factorization in b-space 2/26/2021 161
Leading order n Small-b expansion, 1/b>>intrinsic kt 2/26/2021 162
Virtual diagrams soft divergence 2/26/2021 collinear divergence 163
Soft divergence from real diagrams 2/26/2021 164
Collinear divergence--splitting n Sivers function 2/26/2021 165
Hard factor at one-loop order n Same as the spin-average case 2/26/2021
Final resum form n Sudakov the same 2/26/2021
Coefficients at one-loop order n It will be important to apply this resummation formalism to study the energy dependence of the SSAs ¨ Work in progress… 2/26/2021 168
Back to AN in pp h+x n Large transverse momentum hadron production ¨ Single n scale: collinear factorization Higher-twist effects ¨ Qiu-Sterman, 91, 98 2/26/2021 169
Twist-3 factorization for Pion SSA in hadronic collisions n Collinear factorization Qiu-Sterman matrix element TF(x, x) Hard factor 170 Unpolarized parton distr. Fragmentation function
Initial and final state phases 171
Final results Kouvaris, Qiu, Vogelsang, Yuan, 06 Leading order only, to demonstrate the factorization, one need to go beyond this order n Additional contributions: soft-fermion pole, tri-gluon corrections were carried out recently (Koike, et al. ; Kang, et al. ) n 172
Twist-3 Fit to data RHIC STAR E 704 BRAMHS Kouvaris, Qiu, Vogelsang, Yuan, 06 173
Compare to 2006 data from RHIC J. H. Lee, SPIN 174 2006
175
challenge from STAR data (2006) Talks by Ogawa and Nogach in SPIN 2006 2/26/2021 Users Meeting, BNL 176
Some comments n It’s difficult to explain this pattern in the current twist-3 theoretical approaches ¨ Fragmentation (Collins effect) contributions? 177
A kt-dependent Model Kang, Yuan, 2011 H Quark distribution From the projectile Dense medium Spin-average case Dumitru-Jalilian-Marian, 02 Dumitru-Hayashigaki-Jalilian-Marian, 06 2/26/2021 178
SSA at low PT n Cross section dominated by low transverse momentum UGD 2/26/2021 179
SSA at high PT n n Cross section is dominated by large pt UGD Ratios 2/26/2021 180
Compare to data 2/26/2021 181
Summary n n n 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 Transverse Spin Theory 182
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