Coulomb gluons and colour evolution René Ángeles-Martínez in collaboration with Jeff Forshaw Mike Seymour DPyC, BUAP JHEP 1512 (2015) 091 & arXiv:1602.00623 (accepted for publication) 2016
In this talk: Progress towards including the colour interference of soft gluons in partons showers. PDF Hard process ( Q ) Parton Shower: Approx: Collinear + Soft radiation Q � q µ � Λ QCD Underlying event Hadronization Hadron-Hadron collision, Soper (CTEQ School) … 2
Motivation • Why? Increase precision of theoretical predictions for the LHC • Is this necessary? Yes, for particular non-inclusive observables. • Are those relevant to search for new physics? Yes, these can tell us about the (absence of) colour of the production mechanism (couplings). 3
Outline • Coulomb gluons, collinear factorisation & colour interference. • Concrete effect: super-leading-logs. • Including colour interference in partons showers. (Also see JHEP 07, 119 (2015), arXiv:1312.2448 & 1412.3967) 4
One-loop in the soft approximation k µ ⌧ Q ij Hard subprocess is a vector � ↵ Colour matrices acting on � 2 colour + spin d d k Z − p i · p j ig 2 � ↵ � 2 s T i · T j (2 π ) d [ p j · k ± i 0][ − p i · k ± i 0][ k 2 + i 0] i : − i 0 i : − i 0 i : + i 0 j : + i 0 j : + i 0 j : − i 0 � � ↵
Introduction: one-loop soft gluon correction After contour integration: d 4 k (2 π ) δ ( k 2 ) θ ( k 0 ) (2 π ) 2 δ ( p i · k ) δ ( − p j · k ) � � Z + i ˜ g 2 s µ 2 ✏ T i · T j p i · p j ↵ � 2 δ ij (2 π ) 4 [ p j · k ][ p i · k ] 2[ k 2 ] (On-shell gluon: Purely real) (Coulomb gluon: Purely imaginary) 1 if i, j in , ˜ δ ij = 1 if i, j out , 0 otherwise . 6
Tree-level collinear factorisation For a general on-shell scattering: Colour + Spin operator. Depends only on non-collinear partons Depends only on collinear partons 7
Generalised factorisation beyond tree level Catani, De Florian & Rodrigo JHEP 1207 (2012) 026 This collinear factorisation generalises to all orders but Violation of strict (process- independent) factorisation! 8
Generalised factorisation: one loop The problem first seed at this order
Breakdown of color coherence Effectively, correction to Unavoidable colour correlation
Coulomb gluons and (the lack of) coherence Conclusion: coherence allows us to “unhook” on-shell gluons and recover process independent factorisation. But it fails for Coulomb gluons. instead of Can we make sense of these nested structure?
Concrete case: gaps-between-jets ( Forshaw, Kyrieleis & Seymour hep /0604094 ; /0808.1269 ) Y/ 2 , π − θ Y/ 2 , θ inside the gap q T < Q 0 ∼ Q ∼ Q Q ∼ Any q T Q � Q 0 Q ∼ outside the gap Soft corrections Super-leading logs (On-shell gluons) (On-shell + Coulomb gluons) Z |M ( q 1 , . . . , q m ) | 2 dPS σ m = ✓ Q 2 ◆ ✓ Q 2 ✓ Q 2 ◆ ◆ s ln 4 s ln 5 ∼ α 3 , α 4 s ln n ∼ α n Q 2 Q 2 Q 2 Q 0 ⌧ q i ⌧ Q 0 0 0 Origin: lack of coherence (strict factorisation). 12
Parton showers (Produce events from approximate x-sections) Typically: X-section approximated by - “ordering” real radiation Soft radiation included but no - colour interference. Virtual radiation included - indirectly via unitarity. Colour interference: - Ansatz (hep /0604094) : Order soft radiation, real & virtual, according to its “hardness”. - Is the specific ordering variable relevant?
The role of the ordering variable is crucial! (Banfi, Salam, Zanderighi JHEP06(2010)038) The coefficients of super-log varies for different ordering variables: - Angular ordering: zero. - Energy ordering: infinite. - Transverse momentum ordering: finite. - Virtuality ordering: 1/2. Our strategy to solve this problem: Brute force! Exact 3 & 4 -gluon vertex & ghosts Eikonal approximation Complete (1-loop) diagrammatic calculation assuming that all gluons are soft, but not relatively softness (RAM, Forshaw, Seymour: PhD thesis, JHEP 1512 (2015) 091 & arXiv:1602.00623)
Coulomb gluons and colour evolution Our fixed order calculations suggest that the one-loop amplitude of a general hard scattering with N soft-gluon emissions (ordered in softness q i λ ∼ q i +1 ) is p i − 1 N � E � n (1) J (0) ( q N ) · · · J (0) ( q m +1 ) I ij ( q ( ij ) m ) J (0) ( q m ) · · · J (0) ( q 1 ) | n (0) X X X m +1 , q ( ij ) = 0 i � N m =0 i =2 j =1 n + m − 1 n + m − 1 N J (0) ( q N ) · · · J (0) ( q m +1 ) I n + m,j ( q ( ij ) ) d jk ( q m ) J (0) ( q m − 1 ) · · · J (0) ( q 1 ) | n (0) X X X m +1 , q ( jk ) + 0 i , m m =1 j =1 k =1 where the virtual insertion operator: I ij ( a, c ) = I ij ( a, b ) + I ij ( b, c ) describes the non-emission evolution of partons i and j from a to c. q µ = α p µ j + ( q ( ij ) i + β p µ - Gauge invariant. ) µ T - Correct IR poles Interpretation: - ordered evolution! Key point: The … - Ordering variable is dipole kT 15
Sketch in the simplest case Decomposition in colour and spin i.e. non-trivial test of k_T ordering!
Diagrammatics of dipole: kT ordering 1.- Add N-emissions on external legs 2.- Add virtual exchanges Case a) Case b) and apply effective rules: Gauge invariance b) a)
Non-emission evolution operator Coulomb On-shell 2 3 √ 2 p + i /k ij ln I ij ( a, b ) = α s Z Z i πδ ij p i · p j d( k ( ij ) ) 2 ( k ( ij ) ) − 2 ✏ 6 7 d y 2 π T i · T j c Γ 2[ p j · k ][ p i · k ] − 6 7 ( k ( ij ) ) 2 4 5 √ − ln 2 p − j /k ij × θ ( a < k ( ij ) < b ) This is the same one-loop operator that appears at one-loop but kT ordered! k µ = α p µ i + β p µ j + ( k ( ij ) ) µ 18
Diagrammatics of dipole kT evolution � ↵ For a general scattering we need spheres � n The effective rules are the same: 19
Summary / Conclusion • Coulomb gluons introduce colour-interference & play an essential role in the evolution of hard processes: – super-leading logs – violations of coherence • Can be incorporated at amplitude level as an evolution in dipole transverse momentum, making sense of • Future: Monte Carlo Parton Shower for general observables.
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