Factorization and Factorization Breaking with TMD PDFs

Factorization and Factorization Breaking with TMD PDFs Ted C. Rogers Vrije Universiteit Amsterdam trogers@few.vu.nl Standard PDFs, Gauge Links and TMD PDFs in DIS. TMD factorization breaking in pp hadrons. Next steps, new results. (TCR, Mulders (2010)) Seattle, Washington - September, 20 2010

PDFs in (SI)DIS Begin with tree-level. Start adding gluons. q P k Φ(k, P) 2

Ward Identity: Usual (Integrated) PDF Sum over graphs Eikonal factors, extra Feynman rules. i l + +iǫ Eikonal Propagator, igt a u µ J Vertex 3

Operator definition: Usual (Integrated) PDF f(x)=f.t. p ψ(0,w,0 t )V w (u J)γ + V 0 (u J )ψ(0) p (Example: DIS) u J =(0,1,0 t ) Wilson lines enforce gauge invariance. V w (n)=pexp ( igt a 0 ( V w (u J)V 0 (u J )=Pexp ) dλn A a (w+λn) igt a w 0 dλu J A a (λu J ) ) 4

Standard PDFs: Wilson Lines Paths of Wilson lines in coordinate space: _ w + Standard (Integrated) 6

Standard Factorization Separation of cross section into well-defined factors order-by-order in perturbation theory. Universal (process-independent) PDFs, FFs, etc. Same Wilson line / gauge link. 7

Transverse momentum dependent (TMD) Parton Distribution Functions Example: Drell-Yan (DY) q T distribution. Measured q t 2 q k k q dσ dq 2 T = dσ q q dq 2 T f q (x 1,k t ) f q (x 2,q t k t ) +highq 2 T (Much more complicated PDF definitions!) 8

TMD PDFs: Fields no longer evaluated along light-like separation. Φ(x,k F.T. p ψ(0,w,w t )V w(u J )I n;w,0 γ + t )=?? V 0 (u J )ψ(0) p Extend standard definition (first try): Link at infinity Φ [+] (x,k t )= F.T. p ψ(0,w,w t )V w(u J )I n;w,0 γ + V 0 (u J )ψ(0) p U[0,w] (Boer, Mulders, Pijlman (2003)) (Belitsky, Ji, Yuan (2003)) 9

Ward Identity: Usual (Integrated) PDF Sum over graphs Eikonal factors, extra Feynman rules. i l + +iǫ Eikonal Propagator, igt a u µ J Vertex 10

TMD PDFs: Wilson Lines Paths of Wilson lines in coordinate space: _ + w + w,w t Standard (Integrated) Unintegrated (First Try) 12

Rapidity Divergences in TMD - Factorization: Uncanceled light-cone divergences corresponding to gluons moving with infinite rapidity in minus direction. + Use non-light-like Wilson lines. Introduces new arbitrary parameter. Predictability recovered with new evolution equations (Collins- Soper equations). (Eg., Collins, Soper (1983)) 13

TMD PDFs: Wilson Lines Paths of Wilson lines in coordinate space: + + w w,w t Standard (Integrated) Unintegrated First Try _ + w,w t Still more complications! Unintegrated tilted Wilson lines 14

TMD PDFs: Other Divergences Precise definition of TMD PDFs still need some further modification. (Cherednikov, Stefanis (2008)) Extra divergences need to be cancel in definition. Full, correct definition is quite complicated, involves soft factors (see talk by J. Qiu). 15

Hadro-Production of Hadrons and TMD- Factorization Breaking Factorizes if integrated over parton transverse momentum. Cross sections with measured final state transverse momentum? 16

Hadro-Production of Hadrons and TMD- Factorization Breaking Problems with universality (Bomhof, Mulders, Pijlman (2004)) + Factorizes if integrated over parton transverse momentum. Cross sections with measured final state transverse momentum? 17

Breakdown of Universality ( Standard Use a simple model. Factorization Violated) (Collins, Qiu (2007)) Scalar quark / Dirac spectator. Massive Abelian gauge theory. Single spin asymmetry. 18

Wilson line problems in TMD-factorization SIDIS P q k l l What is the Wilson line? Abelian Model Theory 19

Wilson line problems in TMD-factorization SIDIS P q k l l What is the Wilson line? P 2 Drell- Yan k l l P 1 Sign flip!! Abelian Model Theory (Collins (2002)) 20

Wilson line problems in TMD-factorization SIDIS P q k l l gu µ J l + +iǫ igπu µ J δ(l+ ) P 2 Drell- Yan P 1 k l l gu µ J l + +iǫ igπu µ J δ(l+ ) Sign flip!! Abelian Model Theory (Collins (2002)) 21

Counter-example to Standard TMDfactorization Hadron 1 polarized. What is the Wilson line for hadron 1? P 2 P 2 k 4 l P 2 k 3 +l k 3 l + + P 1 P 1 P 1 Abelian Model Theory 22

Counter-example to Standard TMDfactorization Hadron 1 polarized. What is the Wilson line for hadron 1? P 2 P 2 k 4 l P 2 k 3 +l k 3 l + + P 1 P 1 P 1 Standard Abelian Model Theory 23

Counter-example to Standard TMDfactorization Hadron 1 polarized. What is the Wilson line for hadron 1? P 2 P 2 k 4 l P 2 k 3 +l k 3 l + + P 1 P 1 P 1 Standard Abelian Model Theory Violates Standard Factorization! (Non-universality!) Gluon attaches to wrong side of the graph. 24

Counter-example to Standard TMDfactorization Hadron 1 polarized. What is the Wilson line for hadron 1? P 2 P 2 k 4 l P 2 k 3 +l k 3 l + + P 1 P 1 P 1 Standard Violates Standard Factorization! ig 1 πn µ δ(l + ) + ig 2 πn µ δ(l + ) + ig 2 πn µ δ(l + ) Abelian Model Theory 25

Unpolarized Case: Unintegrated P 2 P 2 + + P 2 P 1 P 1 P 1 + Without integrating over final state hadron k T, no general cancellation between cuts! Factorization fails even for unpolarized case! (Collins (2007)) Abelian Model Theory 26

Generalized TMD Factorization Give up universality. Make gauge link structures process dependent in order to recover factorization. 27

Generalized TMD Factorization P 2 k 4 l + P 1 DIS Like _ + Wilson line Abelian Model Theory 28

Generalized TMD Factorization P 2 k 3 +l + P 1 Drell-Yan Like _ + Abelian Model Theory 29

Generalized TMD Factorization Together: Wilson loop from upper part of graph. Sum of all factorization anomaly contributions: _ + Wilson loop Abelian Model Theory 30

Generalized TMD Factorization Modification to standard TMD PDF: Φ [+( )] P 1 (x 1,k 1T ) F.T. P 1,s 1 φ 1 (0,w,w t )U[0,w]U ( ) φ 1 (0) P 1,s 1 Standard hook Wilson line (Bomhof, Mulders, Pijlman (2004)) Abelian Model Theory 31

Generalized TMD Factorization Modification to standard TMD PDF: Φ [+( )] P 1 (x 1,k 1T ) F.T. P 1,s 1 φ 1 (0,w,w t )U[0,w]U ( ) φ 1 (0) P 1,s 1 Standard hook Wilson line Extra Wilson loop insertion! Abelian Model Theory 32

Generalized TMD Factorization Generalized TMD-Factorization: Formal factorization formula exists, but parton distributions are non-universal! (!!?) dσ H Φ [+( )] P 1 (x 1,k 1T ) Φ [+( )] P 2 (x 2,k 2T ) (Bomhof, Mulders, Pijlman (2004)) 33

Possible Uses of TMD-factorization: q N 2 2 c [( ) + ] 2 [ + ] N % c 2 2 2 c c c + 1 5 Φ = Φ Φ Φ + π Φ N 1 N 1 N 1 G q 2 2 2 Nc [( ) + ] Nc [ + ] N % c 2 2 2 c c c 2 + 1 + 3 Φ = Φ Φ Φ π Φ N 1 N 1 N 1 G (Taken from P. Mulders) 34

No Generalized TMD-factorization! (TCR, Mulders (2010)) Counter-example: Again use scalar quark / spinor diquark model. Quarks and diquarks now carry color. Hard part: exchange of colorless vector boson. No color flowing through hard part! 35

Generalized TMD-factorization breakdown: Spectator Model: Extra Wilson loop for each PDF. Colorless Colored Non-Abelian Model 36

Generalized TMD-factorization breakdown: Spectator Model: Extra Wilson loop for each PDF. Colorless Colored dσ dq 2 T!? =H Φ [+( )] P 1 (x 1,k 1T ) Φ [+( )] P 2 (x 2,k 2T ). (Generalized factorization formula.) Non-Abelian Model Color-traced Wilson loops! 37

One gluon: One collinear gluon: Apply Eikonal approx. Lower hadron PDF: Non-Abelian Model (Standard Contribution) 38

One gluon: One collinear gluon: (Anomalous Attachment) Apply Eikonal approx. No factorization violating contribution with just one gluon. Wilson loop Non-Abelian Model 39

One gluon: One collinear gluon: (Anomalous Attachment) Apply Eikonal approx. No factorization violating contribution with just one gluon. Wilson loop Non-Abelian Model 40

Two gluons: + (Anomalous Attachments) Non-vanishing violation of standard TMD-factorization in both single spin dependence and unpolarized cross section. Still consistent with generalized TMD factorization. Non-Abelian Model (Vogelsang, Yuan (2006,2008)) 41

Summary so far: One extra gluon is consistent with standard factorization for both unpolarized case and single spin dependence. Two gluons from same hadron consistent with generalized TMD factorization for unpolarized case. What about extra gluons from both hadrons simultaneously? Non-Abelian Model 42

Generalized TMD factorization breaking: Graphs like: No SSA or unpolarized contribution. (Cancellation between cuts.) + What about double spin dependence? (No cancellation between cuts.) dσ dq 2 T How does it contribute to?:!? =H Φ [+( )] H 1 (x 1,k 1T ) Φ [+( )] H 2 (x 2,k 2T ). Non-Abelian Model 43

Generalized TMD factorization breaking: Graph like: Corresponds to contribution to generalized TMD-factorization formula: X X Non-Abelian Model 44

Generalized TMD factorization breaking: Gluons have color. Leads to contribution to generalized factorization formula: X X =0 Non-Abelian Model 45

Generalized TMD factorization breaking: Gluons have color. Actual color structure Color Entanglement 0! Leads to contribution to generalized factorization formula: X X =0 Non-Abelian Model M 46

Unpolarized / Higher Orders Generalized factorization violating graphs. 47

Summary: Universality of PDFs and Standard Factorization. Process independent definitions for gauge invariant correlation functions. Well-established for integrated case. Fails in hadro-production of hadrons with TMD PDFs. Non-universality and Generalized Factorization. Process still factorizes into hard part and separate correlation functions. Non-universal because Wilson lines are process dependent. 49

When is factorization still valid? Drell-Yan, SIDIS. Large q T / integrated PDFs. Back-to-back hadrons in P + e - H 1 + H 2 + X. It is not generally true that TMD-factorization fails for processes involving more than two hadrons! 52

Looking for Factorization Breaking A clear observation of generalized factorization breaking would be a discovery of exotic/interesting inter-nucleon correlations between partons. 53

Looking for Factorization Breaking Unpolarized back-to-back hadro-production: Need better/complete treatment of TMD PDFs, including evolution. Need parametrizations over large range of kinematics. (Ex: EIC kinematics) High luminosities allow for fully differential cross sections in x,z,q 2,P T 54

Looking for Factorization Breaking Compare unpolarized, single spin asymmetries and double spin asymmetries in hadron production of jets. Generalized factorization breakdown means we cannot write: dσ H P 1 S 1 WL 1 P 1 S 1 P 2 S 2 WL 2 P 2 S 2 If generalized factorization is true, then unpolarized cross section + single Sivers completely determines double Sivers effect, even if universality is violated. See Lu, Ma, Schmidt (2007) for DY. Try to extend this to jet production?? 55

Implementing Evolution (Preliminary) Collins-Soper Evolution: RG: ln F(x,b T,µ,ζ) ln ζ = K(b T ;µ) d K dlnµ = γ K(g(µ)) dln F(x,b T ;µ,ζ) dlnµ K(b T ;µ)= 1 2 = γ F (g(µ);ζ/µ 2 ) y n ln S(b T ;y n, ) S(b T ;+,y n ) Collab. with M. Aybat 56

Implementing Evolution (Preliminary) After solving evolution (and taking maximal input from known results): F f/h (x,b T,µ,ζ)= j exp { ln 1 ζ µ dµ K(b ;µ b )+ µ b µ b µ exp x dˆx ˆx C f/j (x/ˆx,b ;µ b,g(µ b ))f j/h (x,µ b ) [ γ F (g(µ );1) ln { g j/h (b T )+g K (b T )ln ζ xm p ]} ζ µ γ K(g(µ )) } Collab. with M. Aybat 57

Implementing Evolution (Preliminary) After solving evolution (and taking maximal input from known results): Perturbatively calculable coefficient functions Standard (int) PDF F f/h (x,b T,µ,ζ)= j 1 x dˆx ˆx C f/j (x/ˆx,b ;µ b,g(µ b ))f j/h (x,µ b ) b (b T ) b T 1+b 2 T /b 2 max µ b (b T ) 1/b Collab. with M. Aybat 58

Implementing Evolution (Preliminary) After solving evolution (and taking maximal input from known results): F f/h (x,b T,µ,ζ)= j exp { ln 1 ζ µ dµ K(b ;µ b )+ µ b µ b µ x Perturbatively calculable dˆx ˆx C f/j (x/ˆx,b ;µ b,g(µ b ))f j/h (x,µ b ) [ γ F (g(µ );1) ln ]} ζ µ γ K(g(µ )) b (b T ) b T 1+b 2 T /b 2 max µ b (b T ) 1/b Collab. with M. Aybat 59

Implementing Evolution (Preliminary) After solving evolution (and taking maximal input from known results): F f/h (x,b T,µ,ζ)= j exp { ln 1 ζ µ dµ K(b ;µ b )+ µ b µ b µ exp x dˆx ˆx C f/j (x/ˆx,b ;µ b,g(µ b ))f j/h (x,µ b ) [ γ F (g(µ );1) ln { g j/h (x,b T )+g K (b T )ln ]} ζ µ γ K(g(µ )) } ζ xm p Non-Pertubative b T dependence Collab. with M. Aybat Universal (same for PDFs, FFs etc ) 60

Momentum Space Implementation (Preliminary) F(x,k T ;µ;ζ)= 1 (2π) 2 d 2 b T e ik T b T F(x,bT ;µ;ζ) Collab. with M. Aybat 61

Momentum Space Implementation (Preliminary) Q= ζ=4gev U-quark TMD PDF (GeV -2 ) MSTW coll. dists. (arxiv:0901.0002) BLNY non-pert b T -dependence: (Landry et al., (2003)) Q= ζ=22gev Gaussian Model (Schweitzer,Teckentrup,Metz (2010)) Collab. with M. Aybat k T (GeV) 62

TMD-Factorization (For Factorizable Processes) Foundations of Perturbative QCD, John Collins http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9780521855334 Consistent TMD PDF definition. Requirements for factorization, universality, cancellation of spurious divergences etc. leads to unique TMD PDF definition. Soft factor universality. σ SIDIS H F(x,b T ;µ;ζ) D(z,b T ;µ;ζ) (No explicit soft factor in final factorization formula!) See also TMD 2010 Trento Workshop Slides: http://www.pv.infn.it/~bacchett/tmdprogram.htm 63

Thanks 64