Effective Field Theory of Forward Scattering and Factorization - PowerPoint PPT Presentation

Effective Field Theory of Forward Scattering and Factorization Violation In collaboration with Iain Stewart SCET 2015 Santa Fe

Total Scattering Cross Sections in QFT Consider the total scattering cross section for neutral particles Z d σ σ ( s ) = dt dt some measure of traverse momentum t transfer at intermediate stage of calculation This integral is dominated by the region t ⌧ s

Power Counting Exercise Break up integral into regions with distinct power counting parameters SCET-I,II λ ≡ Λ / ( t, s ) 1: s ⇠ t � Λ SCETII-like λ ≡ t/s 2: t ⌧ s Consider region 1: There must exist some underlying hard event which must be integrated out generating some higher dimensional external operator Z d 4 x ¯ n ¯ ξ n 1 ξ n 2 ∼ λ 4 S ∼ C ( s, t ) ξ n ξ ¯

Region 2: No underlying hard interaction, at the scale t generate the interaction ( λ n , λ m , λ ) : ( m + n ) ≥ 3 The Glauber mode “Glauber” mode Z 1 d 4 x ¯ ¯ n ∼ λ − 2 λ 2 λ − 2 λ 2 ∼ 1 S ∼ ξ n ξ ¯ ξ n ξ ¯ n t “ V ” ∼ δ ( x + ) δ ( x − ) log( x 2 ⊥ ) Strong analogy with NRQCD, Coulomb kernel is dressed by soft gluons Note that while this operator is at the heart of region 2, it also exists at leading order in region 1, where it plants the seed of doubt on factorization proofs.

Note: no hard interactions to all orders in perturbation theory. The scale s = ( p 1 + p 2 ) 2 plays no dynamical role. Consider the dressing of the Glauber kernel by soft gluons M ∼ f ( s/t, t/m 2 ) t The scale s can only show up in logs: log( p + / ν ) + log( p − / ν ) + log( t/ ν 2 ) = log( s/t ) k + RG can not hope to capture the logs, need RRG n -coll. ¯ Q Regge behaviour F ( s/t ) ∼ ( s/t ) γ soft λ Q n -coll. λ 2 Q k − λ 2 Q λ Q Q

Matching onto the action 18 n n n n n n n n q q q q n n n n n n n n a) n n n n n n n n = = n n n n n n n n n n n n n n n n = = n n n n n n n n b) Operator 1 1 1 1 O qq O qC O gq O qC n = O qB O BC n = O gB O BC n , n , ns ¯ n s ns ¯ n s ¯ ¯ P 2 P 2 P 2 P 2 ⊥ ⊥ ⊥ ⊥ basis: 1 1 1 1 O qg O gC O gg O gC n = O qB O BC n = O gB O BC n , n . ns ¯ n s ¯ ns ¯ n s ¯ P 2 P 2 P 2 P 2 ⊥ ⊥ ⊥ ⊥ Must allow for soft ⊥ δ BC + .... O BC = 8 πα s P 2 s emission = χ n, ω T B ¯ n / = i n · ( P + P † ) B Dµ O qB O gB 2 f BCD B C 2 χ n, ω , n ? µ, � ω ¯ n ? , ω . n n

Matching Soft-collinear Operators 22 n n n n n n n n q q q q s s s s s s s s a) n n n n n n n n s s s s s s s s b) Matching is identical to the collinear-collinear case P ⊥ 1 1 1 1 O qq ns = O qB O q n B O qg ns = O qB O g n B O gq ns = O gB O q n B O gg ns = O gB O g n B , , , . n s n s n s n s P 2 P 2 P 2 P 2 ⊥ ⊥ ⊥ ⊥ S T B n / ⇣ ⌘ ¯ O q n B n 2 n = 8 ⇡↵ s , s S ⇣ i ⌘ S ⊥ µ n · ( P + P † ) B nDµ O g n B 2 f BCD B nC = 8 ⇡↵ s . s S ⊥

Final Glauber action L II(0) X X O ij X X O ij = n + ns ¯ ns G n, ¯ n i,j = q,g n i,j = q,g 1 1 1 X X O jC X X O iB O BC O iB O j n B n + . ≡ ¯ n s n s P 2 P 2 P 2 ⊥ ⊥ ⊥ n, ¯ n i,j = q,g n i,j = q,g Two rapidity Three rapidity sectors sectors

The form of the collinear operators are fixed but the soft can have a much more general form X O AB C i O AB O 1 = P µ ? S T = 8 πα s n S ¯ n P ? µ s i i O 2 = P µ ? S T n S n P ? µ ¯ O 3 = P ? · ( g e n )( g e B n S ? )( S T n ) + ( S T B ¯ n n S ¯ n S ¯ S ? ) ·P ? O 4 = P ? · ( g e n S n )( g e n S ? )( S T n S n ) + ( S T B n B ¯ S ? ) ·P ? ¯ ¯ nµ B nµ µ ( S T n )( g e B ¯ S ? ) + ( g e S ? )( S T O 5 = P ? n ) P ? n S ¯ n S ¯ µ B nµ nµ n S n )( g e S ? ) + ( g e B ¯ O 6 = P ? µ ( S T S ? )( S T n S n ) P ? ¯ ¯ µ O 7 = ( g B nµ S ? ) S T n n ( g B ¯ n S ¯ S ? µ ) O 8 = ( g B ¯ nµ S ? ) S T n S n ( g B n S ? µ ) ¯ n ν ( ig e O 9 = S T G µ ν n n µ ¯ s ) S ¯ n n ν ( ig e O 10 = S T G µ ν n n µ ¯ s ) S n ¯ Need to match up to 2 gluons to fix all of the coefficients

Not at all obvious that one collinear emission can be matched given that there are non-local TOP’s which contribute in the EFT 37 p p 3 n p p 3 n p p 3 n 2 n 2 n 2 n q G k q - q k n p n G - G p k k + μ, A 1 n n 4 n n p 1 n p 4 n p 1 n p 4 n q p 1 n p 4 n G k k n n a) μ, A μ, A n μ, A n n n μ, A n n n from n n n n Wilson line b) non-locality only eliminated after using k · A = 0 the on-shell condition similar matching works for gluon operators

Matching Soft Operator 39 s s n n n n n n ' q q q s n n n n q n n a) n n n n ' ' q q n n n n s s μ, C S n n q ' n n μ, C S = q n n n n b) Matching all polarizations w/o using on shell conditions at 1-gluon (simplifies 2 gluon matching)

First row is reproduced by TOP’s in EFT a) s s n n n n s s n n s n n n n n n n n s s s n n n n n n s s s n s n n n n n s s s n n n n n n s s s n n n n n n s n n s s s s s n n n n n n n n n n n n s s s s n n n n n n n n n n n n s s s s s s s s s s s s s s s s n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n s s s s s s s s s s s s TOP’s b) n n n n n n n n s s s s s s s s s s n n n n n n n n

C 2 = C 4 = C 5 = C 6 = C 8 = C 10 = 0 , C 9 = − 1 C 1 = − C 3 = − C 7 = +1 , 2 . ⇢ P µ B nµ B ¯ nµ B nµ µ g e n g e µ − g e n g e O BC ⊥ S T n P ⊥ µ − P ⊥ S ⊥ S T n − S T S ⊥ P ⊥ S ⊥ S T B ¯ n = 8 ⇡↵ s n S ¯ n S ¯ n S ¯ n S ¯ s S ⊥ µ � BC − n µ ¯ n ν n ig e S T G µ ν . s S ¯ n 2 At one gluon level this operator reproduces the Lipatov vertex and generalizes it to arbitrary number of gluons. The form is uniquely fixed to all loops as there are no hard corrections to the theory.

Matching at one loop Glauber Loop Exegesis: Two insertions of O ns ¯ n 26 p k + p k + x  x  x  x    n n n n k q k + q k + k n n n n y  y  k - y  - k - y  p p   − k + d d − 2 k ⊥ d − − k − d Z ⌘ , I Gbox = ⇣ ⌘⇣ 2( ~ ⊥ )( ~ k + + p + 3 − ( ~ 4 − ( ~ q ⊥ / 2) 2 /p + q ⊥ / 2) 2 /p − − k − + p − k 2 q ⊥ ) 2 k ⊥ + ~ k ⊥ + ~ 2 + i 0 k ⊥ + ~ 1 + i 0 − k + d d − 2 k ⊥ d − − k − Z d ⌘ . I Gcbox = ⇣ ⌘⇣ 2( ~ ⊥ )( ~ k + + p + 3 − ( ~ 1 − ( ~ q ⊥ / 2) 2 /p + k 2 q ⊥ / 2) 2 /p − + k − + p − q ⊥ ) 2 k ⊥ + ~ k ⊥ + ~ 2 + i 0 k ⊥ + ~ 1 + i 0 (46) Neither integral is well defined. In the Abelian limit the sum of the two integrals is well defined. However to calculate the piece we need the individual diagrams to be well defined C A Introduce a rapidity regulator | 2 q 3 / ν | − η

− k 0 d − k z | k z | − 2 ⌘ ( ⌫ / 2) 2 ⌘ − d − 2 k ⊥ d d Z I Gcbox = ⇣ ⌘⇣ ⌘ ( ~ ⊥ )( ~ k 0 − k z + p + 3 − ( ~ k ⊥ + ~ q ⊥ 1 − ( ~ k ⊥ + ~ q ⊥ 2 ) 2 /p + 2 ) 2 /p − k 0 + k z + p − k 2 q ⊥ ) 2 k ⊥ + ~ 2 + i 0 1 + i 0 − k 0 d − k z | k z | − 2 ⌘ ( ⌫ / 2) 2 ⌘ − d − 2 k ⊥ d d Z I Gbox = ⇣ ⌘⇣ ⌘ ( ~ ⊥ )( ~ 3 − ( ~ k ⊥ + ~ q ⊥ 4 − ( ~ k ⊥ + ~ q ⊥ k 0 − k z + p + 2 ) 2 /p − − k 0 − k z + p − 2 ) 2 /p + k 2 q ⊥ ) 2 k ⊥ + ~ 2 + i 0 1 + i 0 Z − z | z | − 2 ⌘ h − d − 2 2 ⌘ Z ⊥ ⊥ ⊥ First term in build up of − d − 2 k ⊥ ⇣ − i d ⌘ Z h i = − i ⇡ + O ( ⌘ ) . ( ~ ⊥ )( ~ 4 ⇡ k 2 q ⊥ ) 2 Glauber Phase k ⊥ + ~ Is there a Non-Abelian contribution to the phase? Yes coming from soft loops The Abelian phase is universal as it does not care about the the spin of the collinear lines.

We can also consider soft-collinear forward scattering p p k + k +  n  n n n k q k + q k + k S S S S k - - k - p p   Just the boost of the previous case, yields the same result but now the the glaubers carry ( λ 2 , λ , λ 2 ) | k + − β k − | η Can tweak regulator to insure homogenous scaling Gives same result as collinear-collinear

Other source of rapidity divergences are the Wilson lines which need to be regulated ⇢ − g ⇢ − g  w | 2 P z | − η / 2  w | 2 P z | − η / 2 �� �� X X S n = exp n · A s , S ¯ n = exp n · A s ¯ , ⌫ − η / 2 ⌫ − η / 2 n · P n · P ¯ perms perms (56) ⇢ − g ⇢ − g n · P| − η  w 2 | n · P| − η � w 2 | ¯ �� �� X X W n = exp n · A n ¯ , W ¯ n = exp n · A ¯ . n n · P ¯ ⌫ − η n · P ⌫ − η perms perms Note we regulate every gluon to be consistent with the Glaubers (important for zero bin cancellation)

Zero Bin Subtractions Soft, Collinear and Glauber all overlap S = S � S ( G ) C n = C n � C ( S ) � C ( G ) + C ( GS ) , n n n S ( G ) = G n n S G This is why we don’t see the Glauber in hard matching n n Using the rapidity regulator many of the zero- bin diagrams vanish but crucially not all as in this example.