Gauge-Invariant Gluon TMD from large- to small-x in the coordinate space I.O. Cherednikov 7th International Workshop on Multiple Partonic Interactions at the LHC ICTP, Trieste, 23 - 27 Nov 2015 I.O. Cherednikov Gauge-Invariant Gluon TMD from large- to small-x in the coordinate
Outline: ◮ Operator definition of TMD : gauge invariance, path dependence, universality, factorisation ◮ Path-dependence as the central issue: can one make any use of it? ◮ Stokes-Mandelstam gTMD: fully gauge-invariant, maximally path-dependent ◮ Evolution in the coordinate space: equations of motion in the loop space ◮ Abelian case : exponentiation, area derivative ◮ Outlook I.O. Cherednikov Gauge-Invariant Gluon TMD from large- to small-x in the coordinate
Definitions of TMD/uPDF ◮ TMD factorisation → operator definition ◮ uPDF via evolution/resummation : DGLAP, BFKL, CCFM ◮ High-energy/small-x regime: Balitsky, Kovchegov + extentions ◮ Looking for the an alternative approach I.O. Cherednikov Gauge-Invariant Gluon TMD from large- to small-x in the coordinate
Operator structure of TMD ‘Standard’ approach: factorisation in a convenient gauge (small- or large- x regime) → gauge-dependent pdf → gluon resummation → gauge-invariant pdf with Wilson lines, path-dependence as prescribed by the factorisation Alternative approach: operator structure related to a given pdf → generic gauge-invariant path-dependent object → evolution in the coordinate space to fit the factorisation scheme (small- or large- x regime) → gauge-invariant pdf with Wilson lines, path-dependence as prescribed by the factorisation I.O. Cherednikov Gauge-Invariant Gluon TMD from large- to small-x in the coordinate
Gluon TMD: from Small- x to Large- x @ [Mulders, Rodrigues (2001); Collins (2011); Dominguez, Marquet, Xiao, Yuan (2011); Balitsky, Tarasov (2014, 2015)] Small- x � � G ij dz − d 2 z ⊥ e ik ⊥ z ⊥ small − x ( x , k ⊥ ; P , S ) = � h | D i W LC ( z − , z ⊥ ) W † LC ( z − , z ⊥ ) D j W LC (0 − , 0 ⊥ ) W † LC (0 − , 0 ⊥ ) | h � � � dz − d 2 z ⊥ e ik ⊥ z ⊥ = � h | F il ( z − , z ⊥ ) W † LC ( z ) W LC (0) F lj (0 − , 0 ⊥ ) | h � � � dz − d 2 z ⊥ e ik ⊥ z ⊥ � h | F il ( z ) W γ LC ( z , 0) F lj (0) | h � = Rapidity cutoff: ln x ; single-logs α s ln x ; non-linear dynamics, BK Eq. I.O. Cherednikov Gauge-Invariant Gluon TMD from large- to small-x in the coordinate
Gluon TMD: from Small- x to Large- x @ [Mulders, Rodrigues (2001); Collins (2011); Dominguez, Marquet, Xiao, Yuan (2011); Balitsky, Tarasov (2014, 2015)] Large/Moderate- x � � d 2 z ⊥ e − ixp + z − + ik ⊥ z ⊥ G ij dz − small − x ( x , k ⊥ ; P , S ) = � h | F il ( z − , z ⊥ ) W † LC ( z ) W LC (0) F lj (0 − , 0 ⊥ ) | h � � � d 2 z ⊥ e − ixp + z − + ik ⊥ z ⊥ � h | F il ( z ) W γ LC ( z , 0) F lj (0) | h � dz − = Rapidity cutoff: η � = ln x ; double-logs are possible α s η ln x ; linear evolution. I.O. Cherednikov Gauge-Invariant Gluon TMD from large- to small-x in the coordinate
Gluon TMD: from Small- x to Large- x Two definitions in two regimes—look so similar, but in fact very different: � � dz − d 2 z ⊥ e ik ⊥ z ⊥ � h | F il ( z ) W γ LC ( z , 0) F lj (0) | h � small = VS � � d 2 z ⊥ e − ixp + z − + ik ⊥ z ⊥ � h | F il ( z ) W γ LC ( z , 0) F lj (0) | h � dz − large = Factorization schemes are different, evolution is different: how to relate? Very complicated connection @ [Balitsky, Tarasov (2014, 2015)] However: the operator structure is the same. Let us start with it and forget (for a while) about the factorization issues. I.O. Cherednikov Gauge-Invariant Gluon TMD from large- to small-x in the coordinate
Quark and Gluon TMD: Generic Operator Definitions @ [Mulders, Rodrigues (2001); Collins (2011)] Highly gauge-dependent quark correlator for a hadron h with a momentum P and spin S � d 4 z e − ikz � h | ¯ Q g − d ( x , k ⊥ ; P , S ) = ψ ( z ) ψ (0) | h � Highly gauge-dependent gluon correlator for a hadron h with a momentum P and spin S � d 4 z e − ikz � h | A µ ( z ) A ν (0) | h � G µν g − d ( x , k ⊥ ; P , S ) = I.O. Cherednikov Gauge-Invariant Gluon TMD from large- to small-x in the coordinate
Gluon TMD: Gauge-Invariant Operator Definition @ [Mulders, Rodrigues (2001); Collins (2011)] � d 4 z e − ikz � h | F µν ( z ) W γ F ρσ (0) | h � G µν | ρσ ( x , k ⊥ ; P , S ) = Wilson line (system of lines) W γ in the adjoint representation F µν = F a µν T a Respects the desirable operator structure Knows nothing about any factorization scheme: maximally path-dependent, γ is entirely arbitrary Still difficult to evaluate I.O. Cherednikov Gauge-Invariant Gluon TMD from large- to small-x in the coordinate
Gluon TMD: Several Operator Definitions @ [Mulders, Rodrigues (2001); Collins (2011)] Gluon TMD from the generic gauge-invariant correlator G µν | ρσ ( k ; P , S ) = � d 4 z e − ikz � h | F µν ( z ) W γ F ρσ (0) | h � → � dk − G + i | + j ( k ; P , S ) = G ij ( x , k ⊥ ; P , S ) ∼ � dz − d 2 z ⊥ e − ikz � h | F + i ( z ) W γ F + j (0) | h � I.O. Cherednikov Gauge-Invariant Gluon TMD from large- to small-x in the coordinate
Equations of Motion in the Loop Space @ [Polyakov (1979); Makeenko, Migdal (1979, 1981); Kazakov, Kostov (1980); Brandt et al. (1981, 1982)] Wilson loops as the (fundamental) gauge-invariant degrees of freedom: � d ζ µ A µ ( ζ ) � �W γ � = �P γ exp γ or �W n γ 1 ,...γ n � = �T W γ 1 · · · W γ n � The Wilson functionals obey the Makeenko-Migdal loop equations: δ � dz µ δ (4) ( x − z ) �W 2 ∂ ν δσ µν ( x ) �W 1 γ � = N c g 2 γ xz γ zx � γ I.O. Cherednikov Gauge-Invariant Gluon TMD from large- to small-x in the coordinate
Loop space and differential operators Area derivative: �W γδγ x � − �W γ � δ δσ µν ( z ) �W γ � = lim | δσ µν ( z ) | | δσ µν ( z ) |→ 0 Path derivative: �W δ z − 1 µ γδ z µ � − �W γ � ∂ µ �W γ � = lim | δ z µ | | δ z µ |→ 0 Differential operators in the loop space → evolution of the Wilson loops in the coordinate representation = equations of motion in the loop space I.O. Cherednikov Gauge-Invariant Gluon TMD from large- to small-x in the coordinate
Stokes-Mandelstam Gluon TMD Non-Abelian Stokes’ theorem @ [Arefeva (1980) etc.] �� � �� � d σ ρρ ′ ( ζ ) F ρρ ′ ( ζ ) d ζ ρ A ρ ( ζ ) P γ exp = P γ P σ exp γ σ Mandelstam formula @ [Mandelstam (1968)] �� � �� � δ d ζ ρ A ρ ( ζ ) = P γ F µν ( x ) exp d ζ ρ A ρ ( ζ ) δσ µν ( x ) P γ exp γ γ I.O. Cherednikov Gauge-Invariant Gluon TMD from large- to small-x in the coordinate
Stokes-Mandelstam Gluon TMD δ δ G µν | µ ′ ν ′ ( z ; P , S ) = ˜ δσ µ ′ ν ′ (0) � h |W γ [ z , 0] | h � = δσ µν ( z ) δ δ � � h |W ′ γ [ z ] | X �� X |W ′ γ [0] | h � δσ µν ( z ) δσ µ ′ ν ′ (0) X Non-Abelian exponentiation �� a n W ( n ) � , W ( n ) = hadronic correlators � W γ [ z , 0] � = exp Gauge invariance, Path dependence, Universality I.O. Cherednikov Gauge-Invariant Gluon TMD from large- to small-x in the coordinate
Evolution in the Coordinate Space: Abelian Case Abelian exponentiation �� � d ζ ρ A ρ ( ζ ) �W γ � = � h |P γ exp | h � = γ − g 2 � � � � d ζ ′ ν D µν ( ζ − ζ ′ ) exp d ζ µ 2 γ γ Basic hadronic correlator D µν ( ζ − ζ ′ ) = � A µ ( ζ ) A ν ( ζ ′ ) � I.O. Cherednikov Gauge-Invariant Gluon TMD from large- to small-x in the coordinate
Evolution in the Coordinate Space: Abelian Case Parameterization D ρρ ′ ( z ) = g ρρ ′ D 1 ( z , P ) + ∂ ρ ∂ ρ ′ D 2 ( z , P ) + { P ρ ∂ ρ ′ } D 3 ( z , P ) + P ρ P ρ ′ D 4 ( z , P ) In general, the hadronic correlator contains all necessary information D ρρ ′ ( ζ − ζ ′ ) = � P , S | A ρ ( ζ ) A ρ ′ ( ζ ′ ) | P , S � I.O. Cherednikov Gauge-Invariant Gluon TMD from large- to small-x in the coordinate
Evolution in the Coordinate Space: Abelian Case Area derivative δ δσ µν ( z ) � h |W γ | h � = − g 2 � � δ � � d ζ ′ ρ ′ D ρρ ′ ( ζ − ζ ′ ) � h |W γ | h � d ζ ρ 2 δσ µν ( z ) γ γ Non-vanishing terms after taking the path-derivative ∂ ν – standard Makeenko-Migdal term � d ζ ν ∂ 2 D 1 ( z 2 , P 2 ) ∼ γ – hadron momentum-dependent term � d ζ ν ( P ∂ ) 2 D 4 ( z 2 , P 2 ) ∼ γ I.O. Cherednikov Gauge-Invariant Gluon TMD from large- to small-x in the coordinate
Evolution in the Coordinate Space: Abelian Case Shape evolution equation δ ∂ z δσ µν ( z ) � h |W γ | h � = µ − g 2 �� �� d ζ ν � ∂ 2 D 1 ( z , P ) + ( P ∂ ) 2 D 4 ( z , P ) � h |W γ | h � 2 γ Consistency check: Wilson loops in vacuum ∂ 2 D 1 ( z ) = − δ (4) ( z ) , D 4 = 0 δ � d ζ ν δ (4) ( z − ζ ) ∂ z δσ µν ( z ) � 0 |W γ | 0 � = g 2 µ γ = Makeenko-Migdal Eq. in the LO. I.O. Cherednikov Gauge-Invariant Gluon TMD from large- to small-x in the coordinate
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