Double parton scattering: factorisation, evolution and matching M. Diehl Deutsches Elektronen-Synchroton DESY REF (Resummation, Evolution, Factorization) Madrid, 13 to 16 Nov. 2017 DESY
Introduction DPS: Colour Evolution and cross section TMD matching Summary Hadron-hadron collisions ◮ standard description based on factorisation formulae cross sect = parton distributions × parton-level cross sect ◮ net transverse momentum p T of hard-scattering products: • p T integrated cross sect � collinear factorisation • p T ≪ hard scale of interaction � TMD factorisation ◮ particles resulting from interactions between spectator partons unobserved M. Diehl Double parton scattering: factorisation, evolution and matching 2
Introduction DPS: Colour Evolution and cross section TMD matching Summary Hadron-hadron collisions ◮ standard description based on factorisation formulae cross sect = parton distributions × parton-level cross sect ◮ net transverse momentum p T of hard-scattering products: • p T integrated cross sect � collinear factorisation • p T ≪ hard scale of interaction � TMD factorisation ◮ particles resulting from interactions between spectator partons unobserved ◮ spectator interactions can be soft � underlying event or hard � multiparton interactions ◮ here: double parton scattering with factorisation formula cross sect = double parton distributions × parton-level cross sections M. Diehl Double parton scattering: factorisation, evolution and matching 3
Introduction DPS: Colour Evolution and cross section TMD matching Summary Single vs. double parton scattering (SPS vs. DPS) ◮ example: prod’n of two gauge bosons, transverse momenta q 1 and q 2 q 1 q 1 q 2 q 2 double scattering: single scattering: both | q 1 | and | q 2 | ≪ Q | q 1 | and | q 2 | ∼ hard scale Q | q 1 + q 2 | ≪ Q ◮ for transv. momenta ∼ Λ ≪ Q : dσ SPS dσ DPS 1 ∼ ∼ Q 4 Λ 2 d 2 q 1 d 2 q 2 d 2 q 1 d 2 q 2 but single scattering populates larger phase space : ≫ σ DPS ∼ Λ 2 1 σ SPS ∼ Q 2 Q 4 M. Diehl Double parton scattering: factorisation, evolution and matching 4
Introduction DPS: Colour Evolution and cross section TMD matching Summary Single vs. double parton scattering (SPS vs. DPS) ◮ example: prod’n of two gauge bosons, transverse momenta q 1 and q 2 q 1 q 1 q 2 q 2 double scattering: single scattering: both | q 1 | and | q 2 | ≪ Q | q 1 | and | q 2 | ∼ hard scale Q | q 1 + q 2 | ≪ Q ◮ for small parton mom. fractions x double scattering enhanced by parton luminosity ◮ depending on process: enhancement or suppression from parton type (quarks vs. gluons), coupling constants, etc. M. Diehl Double parton scattering: factorisation, evolution and matching 5
Introduction DPS: Colour Evolution and cross section TMD matching Summary A numerical example integrated cross section gauge boson pair production W + W + single scattering: qq → qq + W + W + suppressed by α 2 s W + W + J Gaunt et al, arXiv:1003.3953 M. Diehl Double parton scattering: factorisation, evolution and matching 6
Introduction DPS: Colour Evolution and cross section TMD matching Summary Drell-Yan: factorisation for q T ≪ Q B B S H H ⇒ S H H A A ◮ fast-moving longitudinal gluons coupling to hard scattering • include in Wilson lines in parton density ◮ soft gluon exchange between left- and right-moving partons • include in soft factors = vevs of Wilson lines needs: eikonal approximation, Ward identities, Glauber cancellation • essential for establishing factorisation • permits resummation of Sudakov logarithms TMD factorisation Collins, Soper, Sterman 1980s; Collins 2011 M. Diehl Double parton scattering: factorisation, evolution and matching 7
Introduction DPS: Colour Evolution and cross section TMD matching Summary Drell-Yan: factorisation for q T ≪ Q B B S H H ⇒ S H H A A • absorb soft factor into parton densities √ √ σ = ˆ σBSA = ˆ σ ( B S )( SA ) = ˆ σf B f A • S requires a rapidity cutoff for the gluons: right-moving gluons � f A , left-moving ones � f B • separation at central rapidity Y (or equivalent variable) A e − Y ) 2 B e + Y ) 2 ¯ ζ ¯ ζ = 2( xp + xp − ζ = Q 4 ζ = 2(¯ • resum Sudakov logarithms log( q T /Q ) via evolution equations ζ f B (¯ d d d log ζ f A ( ζ ) and ζ ) d log ¯ M. Diehl Double parton scattering: factorisation, evolution and matching 8
Introduction DPS: Colour Evolution and cross section TMD matching Summary Drell-Yan: factorisation for q T ≪ Q B � 0 � − igt a dλ vA a ( λv + z ) W ( z ) = P exp � −∞ S H H W † W L ( − z/ 2) L ( z/ 2) W R ( z/ 2) W † R ( − z/ 2) A ◮ transverse variables • z Fourier conjugate to q : � x, z ; ¯ dσ/d 2 q ∝ d 2 z e i zq f A ( x, z ; ζ ) f B (¯ ζ ) � � � � � tr W † 2 ) W † 1 L ( z 2 ) W R ( z R ( − z 2 ) W L ( − z � 0 • soft factor S = 0 2 ) N c � d 2 q ( dσ/d 2 q ) have z = 0 • collinear factorisation: in ⇒ S = 1 � soft gluon exchanges cancel in sum over all graphs � no Sudakov logarithms M. Diehl Double parton scattering: factorisation, evolution and matching 9
Introduction DPS: Colour Evolution and cross section TMD matching Summary Double parton scattering ◮ can generalise previous treatment from single to double Drell-Yan and other DPS processes M Buffing, T Kasemets, MD 2017 B B H 1 H 1 S H 1 H 1 S ⇒ H 2 H 2 H 2 H 2 A A ◮ basic steps can be repeated: • collinear gluons � Wilson lines in DPDs • soft gluons � soft factor MD, D Ostermeier, A Sch¨ afer 2011; MD, J Gaunt, P Pl¨ oßl, A Sch¨ afer 2015 M. Diehl Double parton scattering: factorisation, evolution and matching 10
Introduction DPS: Colour Evolution and cross section TMD matching Summary Double parton scattering: colour complications ◮ DPDs have several colour combinations of partons • colour projection operators k ′ j ′ j k • singlet: P jj ′ ,kk ′ = δ jj ′ δ kk ′ / 3 z 1 / 2 + y z 2 / 2 − z 2 / 2 − z 1 / 2 + y 1 as in usual PDFs x 1 x 2 x 2 x 1 • octet: P jj ′ ,kk ′ = 2 t jj ′ a t kk ′ a 8 • for gluons: 8 A , 8 S , 10 , 10 , 27 ◮ corresponding combinations in soft factor z 1 / 2 + y z 2 / 2 − z 2 / 2 − z 1 / 2 + y • soft factor → matrix in colour space • for colour octet (and other non-singlets): W R t a W † R � = 1 when at same position ⇒ S � = 1 t a � Sudakov factors even in collinear factoris’n t a M Mekhfi 1988; A Manohar, W Waalewijn 2012 M. Diehl Double parton scattering: factorisation, evolution and matching 11
Introduction DPS: Colour Evolution and cross section TMD matching Summary Coloured final states ◮ processes with coloured final states (jets etc) collinear factorisation only with measured small q T no TMD factorisation even for single scattering P Mulders, T C Rogers 2010 B • soft factor with more open colour indices H 1 H 1 S • to be contracted with hard scattering H 2 H 2 • for large distance y non-perturbative A ◮ looks grim for phenomenology . . . M. Diehl Double parton scattering: factorisation, evolution and matching 12
Introduction DPS: Colour Evolution and cross section TMD matching Summary Simplification for collinear factorisation P ii ′ ,j ′ j R j j ′ i i ′ ◮ projector identity for Wilson lines at same position W † ( z ) = W † ( z ) W ( z ) W ( z ) ◮ includes all interactions ◮ also for adjoint Wilson lines j ′ j k k ′ (gluons) and mixed case P jj ′ ,k ′ k R ◮ use this to show • S for jet production etc. same as for Drell-Yan: = • S ( y ) is diagonal in colour: RR ′ S ( y ) ∝ δ RR ′ with R = 1 , 8 , . . . and octet 88 S ( y ) is same for quarks and gluons M. Diehl Double parton scattering: factorisation, evolution and matching 13
Introduction DPS: Colour Evolution and cross section TMD matching Summary Collinear factorisation ◮ in collinear factorisation simple colour structure � σ DPS ∼ � R ˆ R ˆ d 2 y R F B ( y ) R F A ( y ) σ 1 σ 2 R √ with R F A = RR S R A and R F B likewise ◮ evolution of R F ( x 1 , x 2 , y ; µ 1 , µ 2 , ζ ) with Collins-Soper type equation: R F = R J ( y ; µ 1 , µ 2 ) R F R J = − R γ J ( µ 1 ) 2 ∂ ∂ ∂ log ζ ∂ log µ 1 • can choose separate factorisation scales µ 1 , µ 2 for hard scatters • for colour singlet have 1 J = 0 • for colour octet: 8 J ( y ) = kernel for rapidity evolution of single gluon TMD A Vladimirov 2016 ◮ solution has form R F ( x 1 , x 2 , y ; µ 1 , µ 2 , ζ ) = e − R E ( x 1 ,x 2 , y ; µ 1 ,µ 2 ,ζ ) R � F ( x 1 , x 2 , y ; µ 1 , µ 2 ) where R � F follows DGLAP equations in µ 1 and µ 2 with kernels R P ( µ ) M. Diehl Double parton scattering: factorisation, evolution and matching 14
Introduction DPS: Colour Evolution and cross section TMD matching Summary TMD factorisation RR ′ S = S ( z 1 , z 2 , y ; Y ) nontrivial matrix in colour space ◮ ◮ rapidity evolution of S understood at perturbative two-loop level A Vladimirov 2016 ◮ assume that general structure valid beyond two loops: ∂Y S ( Y ) = � ∂ K S ( Y ) for Y ≫ 1 work towards an all-order proof: A Vladimirov 2017 √ ◮ define F A = s A ( s = matrix equivalent of S ) � R F B R F A ◮ cross section σ ∝ ˆ σ 1 ˆ σ 2 R B H 1 H 1 S H 2 H 2 A M. Diehl Double parton scattering: factorisation, evolution and matching 15
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