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Spin Physics and Transverse Structure Piet Mulders mulders@few.vu.nl 1 ABSTRACT Spin Physics and Transverse Structure Piet Mulders (Nikhef Theory Group/VU University Amsterdam) Spin is a welcome complication in the study of partonic structure that has led to new insights, even if experimentally not all dust has settled, in particular on quark flavor dependence and gluon spin. At the same time it opened new questions on angular momentum and effects of transverse structure, in particular the role of the transverse momenta of partons. This provides again many theoretical and experimental challenges and hurdles. But it may also provide new tools in high-energy scattering experiments linking polarization and final state angular dependence. 2 Parton distribution (PDF) and fragmentation (PFF) functions Parton densities (PDFs) and decay functions (PFFs) are natural way of dealing with quarks/gluons in high energy processes (several PDF databases) PDFs can be embedded in a field theoretical framework via Operator Product Expansion (OPE), connecting Mellin moments of PDFs with particular QCD matrix elements of operators (spin and twist expansion) Hard process introduces necessary directionality with light-like directions, P – M2 n and n = P’/P.P’ (satisfying P.n = 1) The lightcone momentum fraction x of the parton momentum p = x P is linked to xB = Q2/2P.q (DIS) or x1 = q.P2/P1.P2 and x2 = q.P1/P1.P2 (DY) Polarization, MS = SL P + MST - M2 SL n (longitudinal or transverse) is a welcome complication, experimentally challenging but new insights emerged Consideration of partonic transverse momenta, p = x P + pT + (p2 pT2) n opens new avenues to Transverse Momentum Dependent (TMD) PDFs, in short TMDs with experimental and theoretical challenges The measurement is mostly not for free, but requires dedicated final states (jets/flavor), polarized targets or polarimetry e.g. through decay orientation of final states (r or L) 3 Link to matrix elements hadron correlators Structure of PDFs and PFFs as correlators built from nonlocal field configurations, extending on OPE and useful for interpretation, positivity bounds, modelling, … ui ( p,s)u j ( p, s) Þ Fij ( p | p) ~ å P y j (0) | X >< X | yi (0) P d ( p - P + PX ) X yi (x ) = y j (0) dx i p.x ò 2p e P y j (0) yi (x ) P parametrized using symmetries (C, P, T) |X><X| gateway to models: e.g. diquarks WG6/12 Mao, Lu, Ma AUT predictions (diquark model) ui (k,s)u j (k,s) Þ Dij ( p | p) ~ å 0 yi (0) | K h X >< K h X | y j (0) 0 d (k - K h - K X ) X = ò dx i k .x e 0 yi (x ) ah+ ah y j (0) 0 2p no T-invariance! Collins & Soper, NP B 194 (1982) 445 4 Link to matrix elements hadron correlators Gluonic matrix elements e m ( p, l )e n * ( p, l ) Þ F g ab ( p | p) ~ å P Aa (0) | X >< X | Ab (x ) P d ( p - P + PX ) X = F g ab ( p) Also needed are multi-parton correlators a F A;ij ( p - p1 , p1 | p) = or better: ò ò dx i p.x e P F na (0)F nb (x ) P 2p FA(p-p1,p) d 4x d 4h i ( p- p1 ).x +ip1.h a e P y (0)A (h ) yi (x ) P j 8 (2p ) a F D;ij ( p - p1 , p1 | p) = a F F ;ij ( p - p1 , p1 | p) = (color gauge invariance) ò d 4x d 4h i ( p- p1 ).x +ip1.h a e P y (0)D (h ) yi (x ) P j 8 (2p ) ò d 4x d 4h i ( p- p1 ).x +ip1.h na e P y (0)F (h ) yi (x ) P j 8 (2p ) 5 (Un)integrated correlators F(x, pT , p.P) = F(x, pT ;n) = ò d 4x i p.x e P y (0) y (x ) P 4 (2p ) ò d(x .P)d 2xT (2p ) 3 e i p.x P y (0) y (x ) P unintegrated TMD (light-front) x .n=x + =0 p- = p.P integration makes time-ordering automatic. The soft part is simply sliced at a light-front instance F(x) = ò d(x .P) i p.x e P y (0) y (x ) P (2p ) x .n=xT =0 or x 2 =0 Is already equivalent to a point-like interaction F = P y (0) y (x ) P x =0 collinear (light-cone) local Local operators with calculable anomalous dimension 6 TMDs and color gauge invariance Gauge invariance in a non-local situation requires a gauge link U(0,x) y (0)y (x ) = å n 1 m1 m N x ...x y (0)¶m ... ¶m y (0) 1 N n! y (0)U (0,x ) y (x ) = å n x U (0, x ) P exp -ig ds A 0 1 m1 m N x ...x y (0) Dm ... Dm y (0) 1 N n! Introduces path dependence for F(x,pT) F[U ] (x, pT ) Þ F(x) x T x 0 x.P 7 Parametrization of F(x) collinear PDFs Quarks in nucleon: p=xP compare u( p,s)u( p,s) = p/ + m Fq (x) µ x f1q (x) P/ + .... = f1q (x) p/ + .... unpolarized quarks 8 Parametrization of F(x) collinear PDFs compare Quarks in nucleon: u( p,s)u( p,s) = p/ + m Fq (x) µ f1q (x) p/ + .... unpolarized quarks Operator connection through Mellin moments x N -1 F(x) = x = p+ = p.n ò = d(x .P) i p.x [n] e P y (0)(¶xn ) N -1U [0, y (x ) P x] (2p ) ò x .n=xT =0 d(x .P) i p.x [n] n N -1 e P y (0)U [0, (D ) y (x ) P x] x (2p ) x .n=xT =0 F( N ) = ò dx x N-1 F(x) = P y (0)(Dn ) N-1 y (0) P Anomalous dimensions can be Mellin transformed into the splitting functions that govern the QCD evolution. 9 Collinear PDFs without polarization Quarks in nucleon: p=xP compare u( p,s)u( p,s) = p/ + m Fq (x) µ x f1q (x) P/ + .... unpolarized quarks 10 Collinear PDFs with polarization æP ö + Mn ÷ + ST Quarks in polarized nucleon: S = S L ç èM ø 0 £ S L2 - ST2 £1 Fq (x) µ x f1q (x) P/ + S L xg1q (x) P/ g 5 + xh1q (x)S/ T P/ g 5 + ... unpolarized quarks chiral quarks in L-polarized N T-polarized quarks in T-polarized N compare u( p,s)u( p,s) = 12 ( p/ + m)(1+ g 5s/ ) spin spin DIS/SIDIS programme to obtain • Flavor PDFs: q(x) = f1q(x) • Polarized PDFs: Dq(x) = g1Lq(x) dq(x) = h1Tq(x) WG6/70 JAM global QCD analysis WG6/142 Drachenberg (STAR) Transversity in polarized pp WG6/150 Gunarathne (STAR) Longitudinal spin asymm. in W prod. 11 TMDs with polarization æP ö + Mn ÷ + ST Quarks in polarized nucleon: S = S L ç èM ø 0 £ S L2 - ST2 £1 q Fq (x, pT ) µ xf1q (x, pT2 ) P/ + S L xg1L (x, pT2 ) P/ g 5 + xh1Tq (x, pT2 )S/ T P/ g 5 + ... unpolarized quarks … but also chiral quarks in L-polarized N T-polarized quarks in T-polarized N compare u( p,s)u( p,s) = 12 ( p/ + m)(1+ g 5s/ ) ( pT × ST ) q F (x, pT ) µ ... + xg1T (x, pT2 ) P/ g 5 + ... M q spin spin chiral quarks in T-polarized N (worm-gear) 12 TMDs with polarization æP ö + Mn ÷ + ST Quarks in polarized nucleon: S = S L ç èM ø 0 £ S L2 - ST2 £1 q Fq (x, pT ) µ xf1q (x, pT2 ) P/ + S L xg1L (x, pT2 ) P/ g 5 + xh1Tq (x, pT2 )S/ T P/ g 5 + ... unpolarized quarks chiral quarks in L-polarized N T-polarized quarks in T-polarized N compare u( p,s)u( p,s) = 12 ( p/ + m)(1+ g 5s/ ) … but also ( pT × ST ) ^q 2 p F (x, pT ) µ ... + xh1T (x, pT ) / T P/ g 5 + ... M M q spin spin T-polarized quarks in T-polarized N (pretzelocity) WG6/103 Prokudin, Lefky Extraction pretzelosity distribution 13 TMDs with polarization … and T-odd functions p/ T ( pT ´ ST ) ^q F (x, pT ) µ ... + ih (x, p ) P/ + i xf1T (x, pT2 ) P/ + ... M M ^q 1 q T-polarized quarks in unpolarized N (Boer-Mulders) spin orbit 2 T unpolarized quarks in T-polarized N (Sivers) compare u( p,s)u( p,s) = 12 ( p/ + m)(1+ g 5s/ ) spin-orbit correlations are T-odd correlations. Because of T-conservation they show up in T-odd observables, such as single spin asymmetries, e.g. left-right asymmetry in p p­ ® p X 14 TMD structures for quark PDFs QUARKS U p/ f1T^ p/ g ag 5 h1^ f1 L T p/ g 5 g1L h1L^ g1T h1T , h1T^ 15 gluon TMD’s without polarization Also for gluons there are new features in TMD’s circularly polarized gluons in L-pol. N g F g mn (x, pT ) µ - gTmn xf1g (x, pT2 ) + iS LeTmn xg1L (x, pT2 ) m ö æ p m pn p mn ^g 2 T T T ç ÷ + g xh (x, p ) + ... unpolarized gluons T 1 T ç M2 2÷ 2M ø è in unpol. N quarks compare F g mn ( p) e m ( p, l )e n * ( p, l ) = -gTmn +... Collinear situation: Unpolarized gluon PDFs: g(x) = f1g(x) linearly polarized gluons in unpol. N (Rodrigues, M) WG6/75 Li (STAR) Jet and di-jet in pol. pp Polarized gluon PDF: Dg(x) = g1Lg(x) WG6/63 Guragain (PHENIX) ALL results, impact on Dg Some of the TMDs also show up in resummation. An example are the linearly polarized gluons (Catani, Grazzini, 2011) WG6/316 Pisano Linear polarization Higgs + jet prod 16 TMD structures for quark and gluon PDFs QUARKS U p/ GLUONS U g1L h1L^ f1T^ g1T h1T , h1T^ -gTab eTab pTab h1^g f1g L T p/ g ag 5 h1^ f1 L T p/ g 5 f1T^g g1Lg h1L^g g1Tg h1Tg , h1T^g 17 TMD structures for quark and gluon PFFs QUARKS U p/ GLUONS U G1L H1L^ D1T^ G1T H1T , H1T^ -gTab eTab pTab H1^g D1g L T p/ g ag 5 H1^ D1 L T p/ g 5 D1T^g G1Lg H1L^g G1T H1Tg , H1T^g 18 Requirements and possibilities to measure QT Parton collinear momentum (lightcone fractions) scaling variables Parton transverse momenta appear in noncollinear phenomena (azimuthal asymmetries) Jet fragments: k = Kh/z + kT or kT = k – Kh/z = kjet – Kh/z SIDIS: q + p = k but q + x PH ≠ Kh/z Hadrons: p1 + p2 = k1 + k2 but x1 P1 + x2 P2 ≠ k1 + k2 Two ‘separated’ hadrons involved in a hard interaction (with scale Q), e.g. SIDIS-like (g*H h X or g*H jet X), DY-like (H1H2 g* X or H1H2 dijet X or H1H2 Higgs X), annihilation (e+e- h1h2 X or e+e- dijet X) Number of pT’s in parametrization rank of TMDs azimuthal sine or cosine mf asymmetry Need for dedicated facilities at low and high energies 19 Rich phenomenology WG6/151 Seder (COMPASS) Charged kaon multiplicities in SIDIS WG6/75 Li (STAR) Jet and di-jet in pol. pp WG6/161 Vossen FFs at Belle WG6/136 Diefenthaler (SeaQuest) Polarized DY measurements WG6/135 Sbrizzai (COMPASS) Transverse spin asymmetries SIDIS WG6/153 Barish (PHENIX) Transverse Single-spin asymm. WG6/84 Anulli Collins Asymmetry with BaBar WG6/139 Allada Recent SIDIS results at JLab WG6/22 Lyu Collins Asymmetry at BESIII WG6/318 Liyanage Recent polarized DIS results at JLab 20 TMDs and factorization Collinear PDFs involve operators of a given twist x ‘measurable’ TMDs involve operators of all twist pT in a convolution (scale and rapidity cutoffs) ds ~ F(x1; m )F(x2 ; m )H(Q, m ) ds ~ ò dp 1T dp2T d ( p1T + p2T - qT )..... ..... F(x1 , p1T ; V 1 , m )F(x2 , p2T ; V 2 , m ) Impact parameter representation including also large bT ds ~ ò db T exp(iqT bT ).... ...F(x1 ,bT ; V 1 , m )F(x2 ,bT ; V 2 , m ) + ......... At small bT, large kT collinear physics (Collins-Soper-Sterman) WG6/33 Collins TMD factorization and evolution J.C. Collins, Foundations of Perturbative QCD, Cambridge Univ. WG6/312 Kang TMD evolution and global analysis 21 Non-universality because of process dependent gauge links F q[ C ] ij d (x .P)d 2xT i p.x [C ] ( x, pT ; n) e P (0) U j [0,x ] i (x ) P 3 (2 ) x .n 0 TMD path dependent gauge link Gauge links associated with dimension zero (not suppressed!) collinear An = A+ gluons, leading for TMD correlators to process-dependence: SIDIS DY … A+ … (resummation) … A+ … F[-] F[+] Time reversal Belitsky, Ji, Yuan, 2003; Boer, M, Pijlman, 2003 22 Non-universality because of process dependent gauge links [C ,C '] Fab (x, pT ;n) = g ò d(x .P)d 2xT (2p )3 [C '] nb ei p.x P U [[Cx ,0]] F na (0)U[0, F (x ) P x] x .n=0 The TMD gluon correlators contain two links, which can have different paths. Note that standard field displacement involves C = C’ F (x ) U[[C,x] ] F (x )U[[xC,] ] Basic (simplest) gauge links for gluon TMD correlators: Fg[+,+] Fg[-,-] Fg[+,-] Fg[-,+] gg H in gg QQ Bomhof, M, Pijlman, 2006; Dominguez, Xiao, Yuan, 2011 23 he t ransverse moment um of more[Ut han one hadron such as e.g. in t he pT · SisT involved, [U ] ] [U ] 2 2 for(x, of TMDs (x,be pT impossible ) = SL g1L pT ) −just a single g1T T(x, pT for ). (12) DY case above, Consequences itg1smay t oparametrizations have MD a given hadron M 5,6 because color get s ent angled . linkinclude dependence r quarks, t hese not only t helink functcan ions survive pT -intofegrat ion, T heGauge correlat ors including a gauge bet hat paramet rizedupon in t erms T MD q 7,8 2 h q (x) = δq(x), which are t he well-known collinear (x) = q(x), g (x) = ∆ q(x) and PDFs depending on x and pT , 1 1 quark TMDs quark and nucleon spin) but also moment um-spin denn-spinLeading densit ies (involving pT S T ⊥q 2 [U ion ] [U ] T [U ] es suchΦas f p ) (unpolarized a t ransversely [U ] t he Sivers funct 21T (x, ⊥ 2 (x, pT ; n) = f 1 (x, pT ) − f 1TT (x, pT ) +quarks g1s (x, in pT )γ 5 M larized nucleon) and spin-spin-moment um densit ies such as g1T (x, p2T ) (longit uγ5 p p P / / T nucleon). nally polarized quarks polarized [U ] ⊥ [U ] ⊥ [U ] 2in a t/ransversely 2 /T + h1T (x, pT ) γ5 ST + h1s (x, pT ) + i h1 (x, pT ) , (11) M naming convent ion M in2 Ref. 9, is T he paramet rizat ion for gluons, following t he en by wit h t he spin vect or TMDs paramet rized as Sµ = SL P µ + STµ + M 2 SL n µ and short hand Leading gluon [U ] ⊥ [U ] pT S T not at ions for g1s and h1s , g[U ] ⊥ g[U ] µ ν µ ν 2x Γ µ ν [U ] (x,pT ) = − gT f 1 (x,p2T ) + gT T f 1T (x,p2T ) pT · ST M[U ] [U ] [U ] µ 2 2 ν )− 2 g (x, p ) = S g (x, p g (x, p ). (12) T L 1L pT pTT µ ν1s g[U ] µ ν MpT 1T ⊥ g[UT ] 2 + i T g1s (x,pT ) + − gT h1 (x,pT ) 2 2 M 2M For quarks, t hesepTinclude tνhat survive { µ ν } not only t he functpions } S T { µ ν }upon pT -int egrat ion, T {µ q q pT ⊥ g[U Swhich ] hq (x) = δq(x), g[U ] 2collinear T T T + Tare t p T well-known and he 1 (x) = q(x), −g1 (x) = ∆ q(x) 1 h (x,p ) − h (x,p (13) T T ). 1s 1T 2 2M 4M spin-spin densit ies (involving quark and nucleon spin) but also moment um-spin den⊥q sit ies such as t he Sivers funct ion f 1T (x, p2T ) (unpolarized quarks in a t ransversely polarized nucleon) and spin-spin-moment um densit ies such as g1T (x, p2T ) (longit u24 Transverse moments operator structure of TMD PDFs Operator analysis for TMD functions: in analogy to Mellin moments consider transverse moments involving gluonic operators pTa F[±] (x, pT ;n) = ò d(x .P)d 2xT (2p )3 ei p.x P y (0)U [0,±¥]iDTaU[±¥,x ]y (x ) P calculable T-even Fa¶ (x) = FaD (x) - FaA (x) FaD (x) = D FaG (x) = p FnFa (x,0 | x) T-odd T-even (gauge-invariant derivative) FaA (x) = PV ò ò dx Fa (x - x , x | x) 1 x .n=0 1 dx1 na F F (x - x1 , x1 | x) x1 1 T-odd (soft-gluon or gluonic pole, ETQS m.e.) Efremov, Teryaev; Qiu, Sterman; Brodsky, Hwang, Schmidt; Boer, Teryaev; Bomhof, Pijlman, M 25 TMD and collinear twist 3 There are many more links between TMD factorized results, pQCD resummation results and collinear twist 3 results: WG6/57 Pitonyak et al. Transverse SSA in pp /g X WG6/120 Koike Collinear twist 3 for spin asymm. WG6/309 Metz Twist-3 spin observables WG6/108 Dai Lingyun Sivers asymmetry in SIDIS WG6/104 Granados Left-right asymm. in chiral dynamics WG6/315 Burkardt Transverse Force on Quarks in DIS 26 Operator classification of TMDs according to rank factor TMD PDF RANK 0 1 F(x, pT2 ) [U ] CG,c [U ] CGG,c [U ] CGGG,c Buffing, Mukherjee, M 1 2 3 F¶ (x, pT2 ) F¶¶ (x, pT2 ) F¶¶¶ (x, pT2 ) FG,c (x, pT2 ) F{G¶},c (x, pT2 ) F{G¶¶},c (x, pT2 ) FGG,c (x, pT2 ) F{GG¶},c (x, pT2 ) FGGG,c (x, pT2 ) 27 Distribution versus fragmentation functions Operators: Operators: F[U ] ( p | p) ~ P | y (0)U[0,x ] y (x ) | P D(k | k) ~ å 0 | y (x ) | K h X out state K h X | y (0) | 0 X DaG (x) = p D nFa ( Z1 ,0 | Z1 ) = 0 T-even T-odd (gluonic pole) FaG (x) = p FnFa (x,0 | x) ¹ 0 Collins, Metz; Meissner, Metz; Gamberg, Mukherjee, M T-even operator combination, but still T-odd functions! 28 Operator classification of TMDs factor TMD PDF RANK 0 1 F(x, pT2 ) [U ] CG,c 1 2 F¶ (x, pT2 ) F¶¶ (x, pT2 ) F¶¶¶ (x, pT2 ) FG,c (x, pT2 ) F{G¶},c (x, pT2 ) F{G¶¶},c (x, pT2 ) FGG,c (x, pT2 ) F{GG¶},c (x, pT2 ) [U ] CGG,c FGGG,c (x, pT2 ) [U ] CGGG,c factor TMD PFF RANK 0 1 3 D(z -1,kT2 ) Buffing, Mukherjee, M 1 D¶ (z -1 ,kT2 ) 2 D¶¶ (z -1 ,kT2 ) 3 D¶¶¶ (z -1,kT2 ) 29 Operator classification of quark TMDs (unpolarized nucleon) factor QUARK TMD PDF RANK UNPOLARIZED HADRON 0 1 2 3 f1 1 CG[U ] h1^ [U ] CGG,c Example: quarks in an unpolarized target are described by just 2 TMD structures; in general rank is limited to to 2(Shadron+sparton) T-even Gauge link dependence: (coupled to process!) T-odd [B-M function] h1^[U ] (x, pT2 ) = CG[u]h1^ (x, pT2 ) 30 Operator classification of quark TMDs (polarized nucleon) factor QUARK TMD PDFs RANK SPIN ½ HADRON 0 1 f1 , g1L , h1T CG[U ] [U ] CGG,c 1 2 g1T , h1L^ h1T^( A) h1^ , f1T^ h1T^( B1) , h1T^( B2) Three pretzelocities: Process dependence also for (T-even) pretzelocity, [U ] [U ] h1T^[U ] = h1T^(1)( A) +CGG,1 h1T^( B1) +CGG,2 h1T^(1)( B2) Buffing, Mukherjee, M 3 A : y ¶¶y = Trc éë¶¶yy ùû B1: Trc éëGGyy ùû B2 : Trc éëGG ùû Trc éëyy ùû 31 Operator classification of TMDs (including trace terms) factor GLUON TMD PDF RANK UNPOLARIZED HADRON 0 1 f1g 1 2 3 h1g^( A) h1g^( Bc) [U ] CGG,c Note also process dependence of (T-even) linearly polarized gluons [U ] ^( Bc) h1g^[U ] = h1^( A) + åCGG,c h1 c Buffing, Mukherjee, M 32 Operator classification of TMDs (including trace terms) factor TMD PDF RANK 0 1 F(x, pT2 ) [U ] CG,c [U ] CGG,c 1 2 3 F¶ (x, pT2 ) F¶¶ (x, pT2 ) F¶¶¶ (x, pT2 ) FG,c (x, pT2 ) F{G¶},c (x, pT2 ) F{G¶¶},c (x, pT2 ) FGG,c (x, pT2 ) F{GG¶},c (x, pT2 ) FG.G,c (x, pT2 ) [U ] CGGG,c FGGG,c (x, pT2 ) Trace terms affect pT width (note FG.G(x) = 0) Boer, Buffing, M, arXiv:1503.03760 33 Operator classification of TMDs (including trace terms) factor GLUON TMD PDF RANK UNPOLARIZED HADRON 0 1 2 1 f1g h1g^( A) [U ] CGG,c d f1g ( Bc) h1g^( Bc) 3 Note process dependence, not only of linearly polarized gluons but also affecting the pT width of unpolarized gluons [U ] ^( Bc) h1g^[U ] = h1^( A) +CGG,c h1 [U ] f1g[U ] = f1g +CGG,c d f1g ( Bc) Boer, Buffing, M, M arXiv:1503.03760 with df1g(Bc)(x) = 0 34 Operator classification of TMDs (including trace terms) factor QUARK TMD PDFs RANK SPIN ½ HADRON 0 1 CG[U ] [U ] CGG,c f1 , g1L , h1T 1 2 g1T , h1L^ 3 h1T^( A) h1^ , f1T^ d f1, d g1, dh1 h1T^( B1) , h1T^( B2) Process dependence in pT width of TMDs is due to gluonic pole operator (e.g. Wilson loops) [U ] f1[U ] = f1 +CGG,c d f1( Bc) with df1(Bc)(x) = 0 Boer, Buffing, M, arXiv:1503.03760 35 Entanglement in processes with two initial state hadrons Resummation of collinear gluons coupling onto external lines contribute to gauge links …. leading to entangled situation (Rogers, M), breaking universality y (x 2 ) y (02 ) [ - ,0 1 ] [x 1 , - ] [ ,x 1 ][ ,x 2 ] Gauge knots (Buffing, M) [0 1 , ][02 , ] [x 2 , - ] [ - ,02 ] y (x1 ) y (01 ) 36 t ransverse separat ions involving collinear and t ransverse into account gauge links, which are also leading gluons. In hadrons t he t he color remains ent angled as or ilCorrelators description of process hard (e.g. process (e.g. DY) Twoininitial state DY) cont rat ed tin 1 and given lust ratribut ed inions, Fig. is 2. illust Bypassing heFig. det ails of get t ingby gauge links in t he ﬁrst place, we not e t hat at measured qT t he ∗ dσDY ∼ cont Tr c ribut Φ(xe1 ,tpo1Tt he )Γ gauge Φ(x 2 , p 2T )Γappear in ingredient s t hat links different part s of t he 1 diagram and∗ cannot be t rivially ab = Φ(xof1 , tphe Φ(xcorrelat 1T )Γ 2 , p2T )Γ, sorbed in t he deﬁnit ion T MD ors, nor can Nc t hey be incorporat ed by a simple redeﬁnit ion of t he corre Complications transverse momentum ofher twot han lat or. T herefore, tifhethe name gauge connect ion rat initial state hadrons is involved, resulting for DY at gauge link is used at t his point . T he result is measured QT in dσDY = Tr c U−† [p2 ]Φ(x 1 , p1T )U− [p2 ]Γ ∗ × U−† [p1 ]Φ(x 2 , p2T )U− [p1 ]Γ (2) † 1 [− ] ∗ [− ] = Φ (x 1 , p1T )Γ Φ (x 2 , p2T )Γ, Nc This leads color just ascross for twist-3 suppressing allto part s offactors t he (part ial) sect ion t hat are squared in collinear DY not of direct import ance for our purpose, e.g. t he phase 1 (xors. , x ,qT ) =argument f (x , sp of) Ä f1 (xWilson , p2T ) lines we have spacesfact t he DY 1 2 As 2 N c 1 1 1T used a not at ion wit h t he moment a p1 and p2 in square 1 ^ from which ^ bracket s, merely- t 1o indicat e orj )t he h (x , p ) Ä h (x2 correlat , p2T )cos(2 1 1 1T 1 2 N c -1 cont ribut ions in t he form o gauge connect ionsNreceive c gluon emissions. Also, in Eq. 2 t he dagger indicat es t he Buffing, M, PRL 112 (2014), 092002 37 Attempts to provide a useful database have started TMDlib project http:/ / tmdlib.hepforge.org 4 38 Conclusions and outlook TMDs extend collinear PDFs (including polarization 3 for quarks and 2 for gluons) to novel TMD PDF and PFF functions Although operator structure includes ultimately the same operators as collinear approach, it is a physically relevant combination/resummation of higher twist operators that governs transverse structure (definite rank linked to azimuthal structure) Knowledge of operators for transverse structure is important for interpretation, relations to twist 3 and lattice calculations. Transverse structure for PDFs (in contrast to PFFs) requires careful study of process dependence linked to color flow in hard process Like spin, the transverse structure does offer valuable tools, but you need to know how to use them! 39 Some other (related) contributions WG6/21 Lowdon (ArXiv:1408.3233) Spin operator decompositions WG6/58 Courtoy, Bacchetta, Radici Collinear Dihadron FFs WG6/102 Prokudin et al. Tensor charge from Collins asymm. WG2/87 Yuan TMDs at small x WG6/304 Kasemets Polarization in DPS WG6/315 Burkardt Transverse Force on Quarks in DIS WG2/112 Tarasov Rapidity evolution of gluon TMD WG2/87 Yuan TMDs at small-x WG2/326 Stasto Matching collinear and small-x fact. WG4/88 Yuan Soft gluon resumm for dijets WG7/297 Zhihong SoLID-SIDIS: future of T-spin, TMD WG1/325 Kasemets TMD (un)polarized gluons in H prod 40