AA Based on work with Maarten Bu ffi ng and Markus Diehl MPI@LHC - - PowerPoint PPT Presentation

Matching and resummation in double parton scattering Tomas Kasemets Nikhef / VU √ AA Based on work with Maarten Bu ffi ng and Markus Diehl MPI@LHC - San Cristóbal de las Casas, November 29, 2016


 DPS di ff erential in transverse momenta q 1 q 1 q 2 q 2 Total cross section • σ DPS / σ SPS ∼ Λ 2 Q 2 DPS populates final state phase space in a di ff erent way than SPS 
 • 
 d σ SP S d σ DP S 1 | q 1 | , | q 2 | ∼ Λ << Q : ∼ ∼ d 2 q 1 d 2 q 2 d 2 q 1 d 2 q 2 Q 4 Λ 2 DPS same power as SPS Makes small transverse momentum region a very interesting region for • DPS Any factorization theorem for this region, must include both single and • double parton scattering MPI@LHC | San Cristóbal de las Casas 2016 | Tomas Kasemets 2


 
 Lessons from TMD factorization and pT resummation TMD Drell-Yan cross section (unpolarized) 
 • d 2 z d σ Z (2 π ) 2 e − i qz W q ¯ X q ( q 2 , µ 2 ) xd 2 q = ˆ q ( x, ¯ x, z ; µ ) σ q ¯ dxd ¯ q with 
 x, z ; µ, ¯ TMDs: 
 q = f q ( x, z ; µ, ζ ) f ¯ q (¯ ζ ) W q ¯ and born x hard-matching q ( q 2 , µ 2 ) = ˆ σ 0 q C H ( q 2 , µ 2 ) ˆ σ q ¯ q ¯ TMDs defined as combination of soft and collinear to cancel rapidity • divergencies Collins, 2011; Echevarria, Idilbi, Scimemi, 2011; 
 Echevarria, TK, Mulders, Pisano, 2015 Depends on two scales, UV and rapidity regularization. • ∂ ∂ ∂ log µ f a ( x, z ; µ, ζ ) = γ F,a ( µ, ζ ) f a ( x, z ; µ, ζ ) ∂ log µ K a ( z ; µ ) = − γ K,a ( µ ) ∂ log ζ f q ( x, z ; ζ , µ ) = 1 ∂ log ζ γ F,a ( µ, ζ ) = − 1 ∂ ∂ 2 K q ( z ) f q ( x, z ; ζ , µ ) . 2 γ K,a ( µ ) . γ K = Γ cusp MPI@LHC | San Cristóbal de las Casas 2016 | Tomas Kasemets 3

TMD factorization and pT resummation For perturbatively small , can match TMDs onto PDFs • z X C ab ( x 0 , z ; ζ , µ ) ⊗ x f b ( x 0 ; µ ) , f a ( x, z ; ζ , µ ) = b Solving the evolution equations gives the evolved TMDs • X C ab ( x, z ; µ 0 , µ 2 x f b ( x 0 ; µ 0 , ζ 0 ) f a ( x, z ; µ, ζ ) = 0 ) ⊗ b √ ζ √ ζ ⇢Z µ 1  � � dµ γ F,a ( µ, µ 2 ) − γ K,a ( µ ) log + 1 K a ( z , µ 0 ) log × exp µ µ µ 0 µ 01 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 Can be supplemented by non-perturbative 
 • λ f = λ h = λ Q = 0 λ f = λ h = λ Q = 0 . 5 1 1 1 1 1 1 1 1 transverse momentum dependence etc. NNLL NNLL NNLL NNLL NNLL NNLL NNLL NNLL b c = 1 . 5 GeV − 1 b c = 1 . 5 GeV − 1 b c = 1 . 5 GeV − 1 b c = 1 . 5 GeV − 1 b c = 1 . 5 GeV − 1 b c = 1 . 5 GeV − 1 b c = 1 . 5 GeV − 1 b c = 1 . 5 GeV − 1 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 Perturbative input alone gives pT resummed 
 • √ s = 13 TeV √ s = 13 TeV √ s = 13 TeV √ s = 13 TeV √ s = 13 TeV √ s = 13 TeV √ s = 13 TeV √ s = 13 TeV d σ /dq T d σ /dq T d σ /dq T d σ /dq T d σ /dq T d σ /dq T d σ /dq T d σ /dq T cross section 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 High scale processes, e.g. Higgs: 
 • 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 small dependence on non-pert. input 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 10 10 10 10 10 10 10 20 20 20 20 20 20 20 20 30 30 30 30 30 30 30 30 40 40 40 40 40 40 40 40 50 50 50 50 50 50 50 50 60 60 60 60 60 60 60 60 q T [GeV] q T [GeV] q T [GeV] q T [GeV] q T [GeV] q T [GeV] q T [GeV] q T [GeV] MPI@LHC | San Cristóbal de las Casas 2016 | Tomas Kasemets Echevarria, TK, Mulders, Pisano, 2015 4

Goal of project: Set up the theoretical (DTMD) framework, within QCD • As few assumptions as possible • As much perturbative input as possible, to enhance predictive power • Provide the basis, correctly including and treating the di ff erent e ff ects. • Once set up in place, can introduce modeling and approximations to • connect with experiments Additional di ffi culties compared to TMDs for SPS • Di ff erent regions which require di ff erent matchings • Color (and polarization) structure • talk by Markus Diehl etc. • Compared to the pocket formula, it represents the other end of DPS • research MPI@LHC | San Cristóbal de las Casas 2016 | Tomas Kasemets 5


 
 
 
 
 Soft and collinear functions DPS cross section proportional to • F T us ,gg ( Y R ) s T − 1 ( Y R − Y C ) s − 1 ( Y C − Y L ) F us ,gg ( Y L ) = F T gg ( Y C ) F gg ( Y C ) We define rapidity divergency free DTMDs as • Y L →−∞ s − 1 ( Y C − Y L ) F us ,gg ( Y L ) , F gg ( Y C ) = lim √ AA Collinear matrix element 
 • Z dz − dz − 2 ) p + 1 2 π dy − e − i ( x 1 z − 2 1 + x 2 z − F us ,gg ( x 1 , x 2 , z 1 , z 2 , y ) ⇠ 2 π ⇥ h p | O g (0 , z 2 ) O g ( y, z 1 ) | p i , Diehl, Schäfer, Ostermeier, 2011 operators dressed by Wilson lines (adjoint rep.) O g i ( y, z i ) = g T µ ν W † G + ν W G + µ � i = y + =0 , � � z + Soft function, matrix in color space 
 • � W W † W W † W W † W W † � talk by Markus Diehl � 0 � ⌦ ↵ S ∼ 0 perturbative calculation at NNLO Vladimirov, 2016 MPI@LHC | San Cristóbal de las Casas 2016 | Tomas Kasemets 6


 
 
 
 
 DTMD cross section For color singlet production (photon, z, Higgs etc.) at 
 • | q 1 , 2 | ⇠ q T ⌧ Q = 1 d σ DPS X σ a 1 b 1 ( q 2 1 , µ 2 σ a 2 b 2 ( q 2 2 , µ 2 ˆ 1 ) ˆ 2 ) x 2 d 2 q 1 d 2 q 2 dx 1 dx 2 d ¯ x 1 d ¯ C a 1 ,a 2 ,b 1 ,b 2 d 2 z 1 d 2 z 2 Z (2 π ) 2 d 2 y e − i q 1 z 1 − i q 2 z 2 W a 1 a 2 b 1 b 2 (¯ x i , x i , z i , y ; µ i , ν ) × (2 π ) 2 with: x i , z i , y ; µ i , ¯ X R F b 1 b 2 (¯ ζ ) R F a 1 a 2 ( x i , z i , y ; µ i , ζ ) W = Φ ( ν y + ) Φ ( ν y − ) R y ± = y ± 1 removes UV region . Choose . 
 2 ( z 1 − z 2 ) y ± ⌧ 1 / ν Φ ( ν y ± ) • ν ∼ Q dependence cancelled by subtraction Φ talk by Jo Gaunt Double TMDs (DTMDs) depend on: 
 R F a 1 a 2 ( x i , z i , y ; µ i , ζ ) • color label, parton and polarization label 
 a 1 , 2 , b 1 , 2 = R = 1 , 8 , ... momentum fractions, transverse distances 
 x 1 , 2 = y , z 1 , 2 = UV renormalization scales, rapidity regularization scale, ζ = ζ = Q 2 1 Q 2 ζ ¯ µ 1 , 2 = 2 MPI@LHC | San Cristóbal de las Casas 2016 | Tomas Kasemets 7


 
 
 
 
 Scale evolution UV and rapidity scale 
 • ∂ R F a 1 a 2 ( x i , z i , y ; µ i , ζ ) = γ F,a 1 ( µ 1 , x 1 ζ /x 2 ) R F a 1 a 2 ∂ log µ 1 R F a 1 a 2 ( x i , z i , y ; µ i , ζ ) = 1 ∂ RR 0 K a 1 a 2 ( z 1 , z 2 , y ) R 0 F a 1 a 2 ∂ log ζ 2 Complicated functions (3 transverse vectors!), little predictive power • When : 
 • Λ ⌧ q T ⌧ Q | q 1 | ∼ | q 2 | ∼ | q 1 ± q 2 | ∼ q T d 2 z 1 d 2 z 2 Z (2 π ) 2 d 2 y e − i q 1 z 1 − i q 2 z 2 W a 1 a 2 b 1 b 2 (¯ x i , x i , z i , y ; µ 1 , µ 2 , ν ) (2 π ) 2 then region of perturbative dominates result | z i | ∼ 1 /q T But what about the size of 
 • y — can be either small or large | y | ∼ 1 /q T | y | ∼ 1 / Λ MPI@LHC | San Cristóbal de las Casas 2016 | Tomas Kasemets 8

Region of large y | z i | ∼ 1 , | y | ∼ 1 Scalings • Λ ⌧ q T ⌧ Q Λ q T Match DTMDs onto the DPDFs • X R F a 1 a 2 ( x i , z i , y ) = R C f,a 1 b 1 ( x 0 R C f,a 2 b 2 ( x 0 R F b 1 b 2 ( x 0 1 , z 1 ) ⊗ 2 , z 2 ) ⊗ i , y ) x 1 x 2 b 1 b 2 Mixing between quark and gluon distributions • RR S qq R F us ,a 1 b 1 Combine and into subtracted DTMD possible since 
 • (independent of parton type) RR S qq ( y ) = RR S gg ( y ) We calculate soft function and matching coe ffi cients at one-loop order 
 • (all parton types, polarizations and color representations, CSS and SCET) Coe ffi cients equal to TMDs — PDFs matching coe ff s. appart from: 
 • 1) Color factors for non-singlet 
 2) Di ff erent vector dependence, since DTMDs and DPDs are parametrized in terms of same distance between partons 
 3) additional polarizations possible MPI@LHC | San Cristóbal de las Casas 2016 | Tomas Kasemets 9

Region of large y Rapidity evolution kernel simplifies considerably 
 • RR 0 K a 1 a 2 ( z i , y ; µ i ) = δ RR 0 ⇥ R K a 1 ( z 1 ; µ 1 ) + R K a 2 ( z 2 ; µ 2 ) + R J ( y ; µ i ) ⇤ Diagonal in color, distance dependence separated 
 • 1 K a 1 ( z 1 ; µ 1 ) usual Collins-Soper kernel R J ( y ; µ i ) remains for DPDFs (rapidity scale evolution for collinear func.) • Solution to evolution equations: • R F a 1 a 2 ( x i , z i , y ; µ i , ζ ) X R ⇥ ⇤ x 2 F b 1 b 2 ( x 0 = C a 1 b 1 ( z 1 ) ⊗ x 1 C a 2 b 2 ( z 2 ) ⊗ i , y ; µ 0 i , ζ 0 ) b 1 b 2 ⇢Z µ 1 p p  x 1 ζ /x 2 � x 1 ζ /x 2 dµ + R K a 1 ( z 1 ) log × exp γ F,a 1 − γ K,a 1 log µ µ µ 01 µ 01 Z µ 2 p p dµ  x 2 ζ /x 1 � x 2 ζ /x 1 + R K a 2 ( z 2 ) log + γ F,a 2 − γ K,a 2 log µ µ µ 02 µ 02 √ ζ � + R J ( y ) log √ ζ 0 MPI@LHC | San Cristóbal de las Casas 2016 | Tomas Kasemets 10