Feynman Rules for Piecewise Linear Wilson Lines

Transcription

1 Feynman Rules for Piecewise Linear Wilson Lines Evolution 2014, Santa Fe Frederik F. Van der Veken Universiteit Antwerpen May 15, 2014

2 Linear Wilson Lines Methodology Example Calculation Outline 1 Linear Wilson Lines 2 3 Piecewise Linear Wilson Lines: Methodology 4 Example Calculation

3 Linear Wilson Lines Methodology Example Calculation Path-Ordered Exponentials Wilson Line U[C] P exp ig dz µ A µ (z) P exp ig C b a dλ (z µ ) A µ (λ)

4 Linear Wilson Lines Methodology Example Calculation Path-Ordered Exponentials Path-Ordering for linear lines z µ r µ + ˆn µ λ 1 m! C C dz 1 dz m λ a... b b b a b dλ 1 dλ m λ 1 λ m 1 b λ m a a λ 2 a dλ m dλ 1

5 Linear Wilson Lines Methodology Example Calculation Wilson Line Bounded From Above l1 Feynman Rules ( U (r ; ) ( ig) m d 4 ) m k i 16π 4 ˆn A(k m ) ˆn A(k 1 ) m0 j m K(j) k l e i r K i ˆn K(j) + iη Feynman Diagram m 1 m k 1 k 1 + k k 2 j k j j1 j1 r µ k 1 k 2 k 3 k m 1 j1 k m

6 Linear Wilson Lines Methodology Example Calculation Feynman Rules Feynman Rules for Linear Wilson Lines 1) Wilson line propagator: 2) external point: k k i ˆn k + iη r k r µ e i 3) infinite point: + 1 (k 0) 4) Wilson vertex: j i µ, a ig ˆn µ (t a ) ij

7 Linear Wilson Lines Methodology Example Calculation Wilson Line Bounded From Below Feynman Rules ( U (+ ; r) ( ig) m d 4 ) m k i 16π 4 ˆn A(k m ) ˆn A(k 1 ) m0 j K(j) m k m l+1 e i r K i ˆn K(j) iη l1 j1 Feynman Diagram n k j j1 r µ k 1 n k j j2 k n 1 + k n k 2 k n k n 2 k n 1 k n +

8 Linear Wilson Lines Methodology Example Calculation Feynman Rules Reversals k i ˆn k + iη k i ˆn k iη

9 Linear Wilson Lines Methodology Example Calculation Feynman Rules Reversals k i ˆn k + iη k i ˆn k iη k i ˆn k iη k i ˆn k + iη j i ig ˆn µ (t a ) ij j i ig ˆn µ (t a ) ij µ, a µ, a

10 Linear Wilson Lines Methodology Example Calculation Finite Line More Types U (b ; a) ( ( ig) m d 4 ) m k i 16π 4 ˆn A(k m ) ˆn A(k 1 ) m l e ia K(l) i b K(m l) i m e ˆn K(j) n0 l0 j1 jl+1 i n K(j) Hermitian Conjugate U (r ; ) n0 ( (ig) m d 4 ) m k i 16π 4 ˆn A(k 1 ) ˆn A(k m ) m e i b K i ˆn K(j) + iη j1

11 Linear Wilson Lines Methodology Example Calculation More Types Finite Lines and Hermitian Conjugates a µ b µ ( ) ( )

12 Linear Wilson Lines Methodology Example Calculation Different Types Semi-Infinite Lines and Path Reversals r µ ˆn µ ( ig) m A m A 1 e i r K m m (ig) m i r K A m A 1 e m (ig) m i r K A m A 1 e m ( ig) m i r K A m A 1 e j j j i ˆn K(j) iη i ˆn K(j) iη j i ˆn K(j) + iη N A m (r, ˆn) N B m (r, ˆn) A m (r, ˆn) i ˆn K(j) + iη Bm (r, ˆn)

13 Linear Wilson Lines Methodology Example Calculation Relation Between A m and B m Relation Between A m and B m B m (r, ˆn) ( ) m A m (r, ˆn) k1 k n,...,k n k 1 ( )

14 Linear Wilson Lines Methodology Example Calculation Relation Between A m and B m Relation Between A m and B m B m (r, ˆn) ( ) m A m (r, ˆn) k1 k n,...,k n k 1... F ( ) m ( ) m dki 16π 4 A m (r, ˆn)F µ 1 µ m a 1 a m (k m,..., k 1 ) (absorb gluon propagators in F )

15 Linear Wilson Lines Methodology Example Calculation Relation Between A m and B m Symmetrising the Blob symmetrise F simultaneously in k i, µ i and a i (identical to making all crossings) because all Lorentz indices are contracted with the same ˆn µ, F is automatically symmetric in µ i interchanging k i and k j is thus same as interchanging a i and a j sometimes F will have straightforward color symmetry

16 Linear Wilson Lines Methodology Example Calculation Relation Between A m and B m Symmetrising the Blob... F ( ) m... F {ai } Easy Blob Example: 3-Gluon Vertex

17 Linear Wilson Lines Methodology Example Calculation Outline 1 Linear Wilson Lines 2 3 Piecewise Linear Wilson Lines: Methodology 4 Example Calculation

18 Linear Wilson Lines Methodology Example Calculation Piecewise Path-Ordered Exponentials Wilson Line With M Segments U A (λ) λ a 1... a 2 U B (λ) λ a 2... a 3 U(λ). U M (λ) λ a M... a M+1 Result for Full Wilson Line U 1 U 2 M J1 U J 1 M U2 J + J1 M K 1 K2 J1 U K 1 U J 1

19 Linear Wilson Lines Methodology Example Calculation Piecewise Path-Ordered Exponentials Result for Full Wilson Line M M J 1 U 3 U3 J [ + U J 1 U2 K + U2 J U K ] M J 1 K U1 J U1 K U1 L J1 J2 K1 J3 K2 L1. i m J i j 1 1 All terms of the form U J j l U m j j1 i i1 j1 J j i j+1 J 0 1M such that l j m j1

20 Linear Wilson Lines Methodology Example Calculation Piecewise Path-Ordered Exponentials Illustration for U

21 Linear Wilson Lines Methodology Example Calculation Outline 1 Linear Wilson Lines 2 3 Piecewise Linear Wilson Lines: Methodology 4 Example Calculation

22 Linear Wilson Lines Methodology Example Calculation Basic Diagrams m 2 F F m 3 F F F

23 Linear Wilson Lines Methodology Example Calculation Basic Diagrams m 4 F F F F F

24 Linear Wilson Lines Methodology Example Calculation Blob Examples m 2 δ ab δ (4) (k 1 k 2 ) D µν (k 1 )

25 Linear Wilson Lines Methodology Example Calculation Blob Examples m 2 U 2 M U2 J + J1 M K 1 K2 J1 U K 1 U J 1 M J J φ J + { 0 1 M K 1 ( ) φ J +φ K K J K J

26 Linear Wilson Lines Methodology Example Calculation Blob Examples m 3 gf abc D k 1 µ 1 ν 1 D k 2 µ 2 ν 2 D k 3 µ 3 ν 3 g ν 1ν 2 (k 1 k 2 ) ν 3 +cross. etc.

27 Linear Wilson Lines Methodology Example Calculation Blob Examples m 3 U 3 M M U3 J + J1 J 1 J2 K1 [ U J 1 U K 2 + U J 2 U K 1 ] + M J 1 K 1 J3 K2 L1 U J 1 U K 1 U L 1 M J + M J J 1 J3 K2 L1 + M K 1 ( ) φ 2 (J K) J +φ K K J K 1 ( ) φ J +φ K +φ L J K L

28 Linear Wilson Lines Methodology Example Calculation Non-Trivial Color Structure Blob With Non-Trivial Color Structure F a 1 a m µ 1 µ m (k 1,..., k m ) perm C σ(a 1 a m) Fσ(µ1 µ m) (σ(k 1,..., k m )) Factorize Out Color... F σ C σ... F σ C σ t an t a 1 C σ(a 1 a m)

29 Linear Wilson Lines Methodology Example Calculation Non-Trivial Color Structure Blob With Non-Trivial Color Structure F a 1 a m µ 1 µ m (k 1,..., k m ) perm C σ(a 1 a m) Fσ(µ1 µ m) (σ(k 1,..., k m )) Factorize Out Color... F σ C σ... F σ C σ t a1 t an C σ(a 1 a m)

30 Linear Wilson Lines Methodology Example Calculation m 4 Blob Example With Non-Trivial Color Structure σ f a 1a 2 x f xa 3a 4 Fµ1 µ 4 (k 1,..., k 4 )

31 Linear Wilson Lines Methodology Example Calculation m 4 Blob Example With Non-Trivial Color Structure σ f a 1a 2 x f xa 3a 4 Fµ1 µ 4 (k 1,..., k 4 )

32 Linear Wilson Lines Methodology Example Calculation m 4 Blob Example With Non-Trivial Color Structure σ f a 1a 2 x f xa 3a 4 Fµ1 µ 4 (k 1,..., k 4 ) C 1 F C 2 F C 3 F1423

33 Linear Wilson Lines Methodology Example Calculation m 4 Blob Example With Non-Trivial Color Structure σ f a 1a 2 x f xa 3a 4 Fµ1 µ 4 (k 1,..., k 4 ) C 1 F C 2 F C 3 F1423 C 3 F1234 C 2 F1324 C 1 F1423

34 Linear Wilson Lines Methodology Example Calculation Outline 1 Linear Wilson Lines 2 3 Piecewise Linear Wilson Lines: Methodology 4 Example Calculation

35 Linear Wilson Lines Methodology Example Calculation Quadrilateral Wilson Loop Quadrilateral Wilson Loop ˆn a µ 1 b µ ˆn 4 ˆn 2 d µ ˆn 3 c µ

36 Linear Wilson Lines Methodology Example Calculation Quadrilateral Wilson Loop First Order F (r J, ˆn J ) G(r J, r K, ˆn J, ˆn K )

37 Linear Wilson Lines Methodology Example Calculation First Order U 2 M U2 J + J1 M K 1 K2 J1 M F (r J, ˆn J ) + Result for Light-Like Loop J U K 1 U J 1 K2 J1 Quadrilateral Wilson Loop M K 1 ( ) J+K G(r J, r K, ˆn J, ˆn K ) U 2 α sc F π ( 2πµ2 ) ɛ Γ(1 ɛ) [ 1 ( (b d) 2 ɛ 2 iη ) ɛ 1 ( + (c a) 2 ɛ 2 iη ) ] ɛ

38 Linear Wilson Lines Methodology Example Calculation Conjecture Geometric Evolution of Light-Like Quadrilateral ( µ µ + β(g) ) δ ln W γ Γ cusp g δ ln Σ cusps Gamma cusp at NLO: Γ cusp (g) α s π C F + ( αs ) ( ( ) ) 2 67 CF C A π 36 π2 5 N f 12 18

39 Linear Wilson Lines Methodology Example Calculation Conjecture l 2 l 2 l 2 l 1 l 3 l 1 l 3 l 1 l 3 l 4 l 4 l 4

40 Linear Wilson Lines Methodology Example Calculation Reusability Example ig 2 C F ˆn 2 ( 4πµ 2 ) ɛ Γ(ɛ)X(ˆn 2, ɛ) 16π η ( g 2 C F ˆn 1 ˆn 2 2πiµ 16π ˆn 1 ñ2 ˆn 2 1 ) ɛ R Y ɛ (iη RN 2 ) 2 η N 2

41 Linear Wilson Lines Methodology Example Calculation Conclusions Conclusions & Outlook framework to minimize number of diagrams for piecewise linear Wilson lines lesser diagrams in exchange for more general (and thus more complicated) integrals only interesting for M > 2 calculate higher orders include final-state cut try framework for TMD Wilson line structure