Double parton sca/ering for perturba3ve transverse momenta Maarten - PowerPoint PPT Presentation

Double parton sca/ering for perturba3ve transverse momenta Maarten Buffing In collaboration with Markus Diehl and Tomas Kasemets QCD evolution workshop 2016 May 31, 2016

Content - outline • Brief motivation/introduction • Soft factors • Color for DPDFs/DTMDs • Evolution equations – Writing them down for DTMDs – Solving them • Matching: cross section contributions for large y • Conclusions 2

Motivation • DPDs: double parton distribution functions • Factorization: stick to singlets in final states – Double Drell-Yan – Higgs + W/Z • For perturbative q T → significant predictive results • Motivation and goals – Formulate description to handle soft factors – Write down evolution equations – Solve evolution equations – Matching equations for DPDFs/DTMDs 3 Diehl, Ostermeier, Schäfer, JHEP 1203 (2012) 089

Short-distance expansion • Differences compared to TMDs – Two hard processes involved – Two coefficient functions per DTMD – Positions z 1 and z 2 (compare with b T for the TMD case) – Additional distance y Figure: modified from Diehl, Ostermeier, • Consider the limit Schäfer, JHEP03 (2012) 089 – | z 1 |, | z 2 | much smaller than 1/ Λ – | z 1 |, | z 2 | ≪ y , with y fixed • Gives separate matching factors F us ( x i , z i , y ) = C f ( x 0 x 1 C f ( x 0 x 2 F us ( x 0 1 , z 1 ) ⊗ 2 , z 2 ) ⊗ i , y ) with Z 1 ⇣ x dx 0 ⌘ C ( x 0 ) ⊗ x F ( x 0 ) = x 0 C ( x 0 ) F x 0 x 4

Soft factors • Wilson line structure from factorization formula. • Nontrivial color complications. Collinear and soft factors carry color indices. • Wilson line self-interactions drop out in cross section. Figure: Diehl, Gaunt, Ostermeier, Plößl, Schäfer, JHEP01 (2016) 076 Collins, Foundations of perturbative QCD , (2011); Aybat, Rogers, PRD 83 (2011) 114042; Diehl, Gaunt, Ostermeier, Plößl, Schäfer, JHEP01 (2016) 076 5

Soft factors • TMDs • Soft functions for the single TMD related to K through  ∂ � K ( z ; µ ) = 1 ∂ log S ( z ; y A , −∞ ) − log S ( z ; + ∞ , y B ) 2 ∂ y A ∂ y B • Soft function not matrix valued • Square root construction for TMD (see Collins’ book) • DTMDs • For DPDs: matrix valued functions (working hypothesis) h i S ( z 1 , z 2 , y , y A , y B ) = exp ( y A − y B ) K ( z 1 , z 2 , y ) • Soft function matrix valued • Square root construction extended to matrix expressions 6 Collins, Foundations of perturbative QCD , (2011); Aybat, Rogers, PRD 83 (2011) 114042

Soft factors (technical details) • Subtracted DPD distributions are defined as L → 0 S − 1 F qq ( v c ) = lim qq ( v L , v C ) F us,qq ( v L ) v 2 with F us vector in color space and S a matrix. • Matrix equivalent of square root construction S − 1 ( v L , v C ) = S 1 / 2 ( − v C , v R ) S − 1 / 2 ( v L , v R ) S − 1 / 2 ( v L , v C ) using composition law S ( v A , − v B ) S ( v B , v C ) = S ( v A , v C ) and a similar expression for left moving particles. • Wilson line self-interactions drop out in F . 7 Collins, Foundations of perturbative QCD , (2011); Aybat, Rogers, PRD 83 (2011) 114042

Soft factors • Wilson line structure for double Drell-Yan with Wilson lines � z + = z − =0 Z 0  − igt a d λ vA a ( z + λ v ) W ij ( z , v ) = P exp ij −∞ and similarly for the adjoint representation. • We will need uncontracted color indices in the middle. 8

Soft factors • Uncontracted indices in the middle • Soft factor for DTMDs factorizes in small-distance expansion as S ( z 1 , z 2 , y ) = C s ( z 1 ) C s ( z 2 ) S ( y ) • Wilson lines in S( y ) pairwise at the same transverse position. • We require a simplification of the color indices. 9

Color structure • Recall full Wilson line structure • Hard scattering couples four parton lines, insert color projectors • Examples of color projectors – Quarks: = 1 p j 1 j 0 1 k 1 k 0 δ j 1 j 0 1 δ k 1 k 0 1 1 N c 1 p j 1 j 0 1 k 1 k 0 = 2 t a 1 t a 1 j 1 j 0 k 1 k 0 8 1 – For gluons: more possibilities – Mixed quark-gluon projectors also exist • Highly nontrivial whether color structure can be factorized. 10

Color structure • Recall full Wilson line structure • Hard scattering couples four parton lines, insert color projectors Color trick (in collinear situation: WW † = 1 ) • 11 For proof: use color Fierz identity

Color structure Color trick (in collinear situation: WW † = 1 ) • • For proof: use color Fierz identity: jj 0 = δ ij 0 δ i 0 j − 1 2 t a ii 0 t a δ ii 0 δ jj 0 N c • Trick also works for adjoint Wilson lines. Use color Fierz identity and   t a Wt b W †  W ab = 2 Tr  12

Color structure Color trick (in collinear situation: WW † = 1 ) • • Dynamical and not just some color algebra • With same trick show that S( y ) is color diagonal. 13

Implications for soft factor • Color projection of fields at infinity rather than ξ + = ξ - = 0 . Related ⇐ ⇒ • Allows for relating most general soft function with open indices in the middle with soft function with contracted indices in the middle. • For collinear factorization case only! 14

Renormalization and rapidity evolution • Short-distance expansion – The two hard processes are separated • Evolution equations for DTMDs – Two renormalization scales: µ 1 and µ 2 • Soft factor recap – Working hypothesis h i S ( z 1 , z 2 , y , y A , y B ) = exp ( y A − y B ) K ( z 1 , z 2 , y ) – Soft factor becomes RR 0 S ( z 1 , z 2 , y ) = R C s ( z 1 ) R C s ( z 2 ) RR S ( y ) δ RR 0 • For phenomenology: only four independent collinear soft functions 15

Renormalization and rapidity evolution • TMDs ∂ ∂ log µF ( x, z ; µ, ζ ) = γ F ( µ, ζ ) F ( x, z ; µ, ζ ) ∂ log ζ F ( x, z , µ, ζ ) = 1 ∂ 2 K ( z ; µ ) F ( x, z , µ, ζ ) • DTMDs ∂ R F ( x i , z i , y ; µ i , ζ ) = γ F ( µ 1 , x 1 ζ /x 2 ) R F ( x i , z i , y ; µ i , ζ ) ∂ log µ 1 ∂ R F ( x i , z i , y ; µ i , ζ ) = γ F ( µ 2 , x 2 ζ /x 1 ) R F ( x i , z i , y ; µ i , ζ ) ∂ log µ 2 R F ( x i , z i , y , µ i , ζ ) = 1 ∂ RR 0 K ( z i , y ; µ i ) R 0 F ( x i , y , µ i , ζ ) X ∂ log ζ 2 R 0 • DTMD renormalizations are independent, since they are separated. 16 Collins, Foundations of perturbative QCD , (2011); Aybat, Rogers, PRD 83 (2011) 114042

Evolution: TMDs vs DTMDs PDF/TMDs DPDF/DTMDs • Soft function not matrix valued • Soft function matrix valued • Just the position of one parton • Positions of two partons and the distance y • Renormalization scale µ • Renormalization scales µ 1 , µ 2 • Rapidity evolution scale ζ • Rapidity evolution scale ζ – ζ dependence also for collinear distri- bution if R ≠ 1. • Two coefficient functions per • One coefficient function per DTMD TMD 17

DTMD evolution • The evolution of DTMDs is in the short-distance matching given by R F ( x i , z i , y ; µ 1 , µ 2 , ζ ) ⇢Z µ 1 p p dµ  x 1 ζ /x 2 � x 1 ζ /x 2 + R K ( z 1 , µ 01 ) log γ F ( µ, µ 2 ) − γ K ( µ ) log = exp µ µ µ 01 µ 01 Z µ 2 p p  � dµ x 2 ζ /x 1 x 2 ζ /x 1 γ F ( µ, µ 2 ) − γ K ( µ ) log + R K ( z 2 , µ 02 ) log + µ µ µ 02 µ 02 √ ζ � + R J ( y , µ 01 , µ 02 ) log √ ζ 0 × R C ( x 0 R C ( x 0 R F ( x 0 1 , z 1 ; µ 01 , µ 2 2 , z 2 ; µ 02 , µ 2 01 ) ⊗ 02 ) ⊗ i , y ; µ 01 , µ 02 , ζ 0 ) x 1 x 2 • From additive structure of the Collins-Soper evolution kernel we have the sum for the two contributions for the µ 1 and µ 2 dependences. • K ( z 1 , z 2 , y )-kernel splits in three separate contributions: K ( z 1 , µ 01 ), K ( z 2 , µ 02 ) and J ( y , µ 01 , µ 02 ) when collinear soft function becomes diagonal. 18

Cross section contribution • Cross section contribution given by ⇢Z µ 1 γ F ( µ, µ 2 ) − γ K ( µ ) log Q 2 + R K ( z 1 , µ 01 ) log Q 2  � dµ X 1 1 W large y = exp µ 2 µ 2 µ µ 01 01 R Z µ 2 γ F ( µ, µ 2 ) − γ K ( µ ) log Q 2 + R K ( z 2 , µ 02 ) log Q 2  � � dµ 2 2 + µ 2 µ 2 µ µ 02 02 × R C ( x 0 1 , z 1 ; µ 01 , µ 2 R C ( x 0 2 , z 2 ; µ 02 , µ 2 01 ) ⊗ 02 ) ⊗ x 1 x 2 × R C ( x 0 1 , z 1 ; µ 01 , µ 2 R C ( x 0 2 , z 2 ; µ 02 , µ 2 01 ) ⊗ 02 ) ⊗ x 1 x 2 p Q 2 1 Q 2  � i 2 h R J ( y , µ 0 i ) log 2 R F ( x i , y ; µ 0 i , ζ 0 ) R F ( x i , y ; µ 0 i , ζ 0 ) Φ ( ν y ) exp × ζ 0 • The z 1 , z 2 and y contributions nicely factorize. 19

Cross section contribution • Cross section contribution given by ⇢Z µ 1 γ F ( µ, µ 2 ) − γ K ( µ ) log Q 2 + R K ( z 1 , µ 01 ) log Q 2  � dµ X 1 1 W large y = exp µ 2 µ 2 µ µ 01 01 R Z µ 2 γ F ( µ, µ 2 ) − γ K ( µ ) log Q 2 + R K ( z 2 , µ 02 ) log Q 2  � � dµ 2 2 + µ 2 µ 2 µ µ 02 02 × R C ( x 0 1 , z 1 ; µ 01 , µ 2 R C ( x 0 2 , z 2 ; µ 02 , µ 2 01 ) ⊗ 02 ) ⊗ x 1 x 2 × R C ( x 0 1 , z 1 ; µ 01 , µ 2 R C ( x 0 2 , z 2 ; µ 02 , µ 2 01 ) ⊗ 02 ) ⊗ x 1 x 2 p Q 2 1 Q 2  � i 2 h R J ( y , µ 0 i ) log 2 R F ( x i , y ; µ 0 i , ζ 0 ) R F ( x i , y ; µ 0 i , ζ 0 ) Φ ( ν y ) exp × ζ 0 • There is ζ – dependence for color non-singlet DPDFs. 20