Towards NLO Parton Shower MC S. JADACH M. Fabia nska, A. Gituliar, - PowerPoint PPT Presentation

Towards NLO Parton Shower MC S. JADACH M. Fabia´ nska, A. Gituliar, A. Kusina, W. Płaczek, M. Sapeta, A. Siódmok, M. Sławi´ nska and M. Skrzypek Institute of Nuclear Physics PAN, Kraków, Poland Partly supported by the grants of Narodowe Centrum Nauki DEC-2011/03/B/ST2/02632 and UMO-2012/04/M/ST2/00240 Presented at "HP2: High precision for hard process" , CGG Florence, Sept. 3-5th, 2014 S. Jadach (IFJ PAN, Krakow) NLO corrections in the parton shower Monte Carlo HP2 at CGG, Sept.2014 1 / 29

What is NLO parton shower? A litle bit of warm-up: What is the LO parton shower? ◮ The LO parton shower MC is built using LO class evolution kernels and/or LO PDFs for each incomming/ougoing shower/ladder. ◮ LO PS MC implements LO DGLAP evolution of the total cross section and of semi-inclusive distributions (structure functions). ◮ If hard process is corrected to the NLO level (N+LO), the all collinear/soft singularities of the LO PS MC are subtracted from the hard proces ME in the exclusive form. ◮ In N+LO schemes certain partons originally generated by the LO PS MC get promoted to the hard process, where their distributions get corrected to NLO level. S. Jadach (IFJ PAN, Krakow) NLO corrections in the parton shower Monte Carlo HP2 at CGG, Sept.2014 2 / 29

What is NLO parton shower? Now everything one order higher: ◮ The NLO parton shower MC is built using NLO class evolution kernels and/or NLO PDFs for each shower/ladder. ◮ NLO PS MC implements NLO DGLAP evolution of the total cross section and of semi-inclusive distributions (structure functions). ◮ If hard process is corrected to the N 2 LO level (N+NLO), collinear/soft singularities of the NLO PS MC are subtracted from the hard proces ME in the exclusive form. ◮ In N+NLO scheme certain partons originally generated by the NLO PS MC get incorporated into the hard process, where their distributions get corrected to N 2 LO level. S. Jadach (IFJ PAN, Krakow) NLO corrections in the parton shower Monte Carlo HP2 at CGG, Sept.2014 3 / 29

Problems and solutions ◮ NLO kernels have to be recalculated in the exclusive form. ◮ We have recalculatet all NLO kernels using Curci-Furmanski-Petronzio (CFP) scheme – explicit diagramatic calculation in axial gauge (also Ellis+Voghesang, Kunst+Heinrich). ◮ Technical improvements were proposed (Skrzypek+Gituliar) ◮ LO parton shower may miss some phase space regions which are present in NLO kernels/evolution, like q → qG ∗ , G ∗ → GG spliting ◮ One could add G ∗ → GG after LO PS generation is finished, ◮ Luckily, some modern LO PS MCs already populated this ph.sp. ◮ Introducing complete NLO real and virtual corrections into PS MC in the exclusive form, in accordance with the collinear factorization theorems (CFP), a formidable problem, theoreticaly and practicaly. ◮ Theoretical framework CFP-compatible formulated and tested, ◮ 3 methods of practical implementation of NLO corrections in the PS MC formulated and tested. One of them quite promissing. S. Jadach (IFJ PAN, Krakow) NLO corrections in the parton shower Monte Carlo HP2 at CGG, Sept.2014 4 / 29

Remarks on NLO kernel re-calculation ◮ Why CFP? Because there is nothing else in the literature. ◮ All inclusive MS kernels were reproduced, but we have listed/exploited all exclusive 2-real and 1real+1virtual distributions, before the phase space integration. ◮ CFP was modified in order to eliminate spurious 1 /ε 3 poles obscuring relation to MC at d = 4 dimensions. The so-called NPV prescription by Skrzypek and Gituliar, pulished recently. ◮ For subsets of diagrams in 2-real parton contributions, soft gluon limit was analyzed carefully. Expected gauge cancellations found. ◮ In CFP NLO kernel is extracted as coefficient of 1 /ε . An alternative method of taking derivative ∂/∂ ( ln µ 2 ) was tested. ◮ MS scheme produces technical artefact ∼ ε/ε 2 , which are source of the problems in the MC implementation of NLO corrections. These terms were clasified and their role was analyzed. S. Jadach (IFJ PAN, Krakow) NLO corrections in the parton shower Monte Carlo HP2 at CGG, Sept.2014 5 / 29

Theoretical framewework of PS MC: Collinear Factorization ◮ What is collinear factorization? F bare ( q h /µ, ε ) = σ Bare C ( ∞ ) � α, q h � � ⊗ Γ ( ∞ ) = ladder ( α, ε ) σ Born µ Ladders ⊗ in lightcone x and parton type, Γ inclusive, C can be kept unintegrated/exclusive. F ( 1 ) bare ( q h /µ, ε ) = [ ✶ + C [ 1 ] ( α, q h /µ )] ⊗ [ ✶ + Γ [ 1 ] ( α, ε )] Case LO : ◮ Physical distributions: Γ → PDF . LO example: C [ 1 ] ( q h /µ ) ≡ F [ 1 ] F Phys . = [ ✶ + C [ 1 ] ( α, q h /µ )] ⊗ PDF ( µ ) , bare ( q h /µ, ε ) − Γ [ 1 ] ( ε ) F Phys . factor. scheme independent; both C and PDFs are dependent: Γ [ 1 ] ( ε ) → Γ [ 1 ] + ∆Γ [ 1 ] , C [ 1 ] → C [ 1 ] − ∆Γ [ 1 ] , ∆ C [ 1 ] = − ∆Γ [ 1 ] . ◮ Evolution of F and/or PDFs and evolution kernels: ∂ P = α P [ 0 ] + α 2 P [ 1 ] + ... = Res 1 Γ( ε ) = ∂ ln ⊗ C ( q /µ ) ∂ ln µ 2 F ( µ ) = P ⊗ F ( µ ) , S. Jadach (IFJ PAN, Krakow) NLO corrections in the parton shower Monte Carlo HP2 at CGG, Sept.2014 ∂ ln µ 2 6 / 29

Theoretical framewework of PS MC: Collinear Factorization ◮ What is collinear factorization? F bare ( q h /µ, ε ) = σ Bare C ( ∞ ) � α, q h � � ⊗ Γ ( ∞ ) = ladder ( α, ε ) σ Born µ Ladders ⊗ in lightcone x and parton type, Γ inclusive, C can be kept unintegrated/exclusive. bare ( q h /µ, ε ) = [ ✶ + C [ 1 ] + C [ 2 ] ] ⊗ [ ✶ + Γ [ 1 ] + Γ [ 2 ] ] F ( 2 ) Case NLO : ◮ Physical distributions: Γ → PDF . LO example: C [ 1 ] ( q h /µ ) ≡ F [ 1 ] F Phys . = [ ✶ + C [ 1 ] ( α, q h /µ )] ⊗ PDF ( µ ) , bare ( q h /µ, ε ) − Γ [ 1 ] ( ε ) F Phys . factor. scheme independent; both C and PDFs are dependent: Γ [ 1 ] ( ε ) → Γ [ 1 ] + ∆Γ [ 1 ] , C [ 1 ] → C [ 1 ] − ∆Γ [ 1 ] , ∆ C [ 1 ] = − ∆Γ [ 1 ] . ◮ Evolution of F and/or PDFs and evolution kernels: ∂ P = α P [ 0 ] + α 2 P [ 1 ] + ... = Res 1 Γ( ε ) = ∂ ln ⊗ C ( q /µ ) ∂ ln µ 2 F ( µ ) = P ⊗ F ( µ ) , ∂ ln µ 2 S. Jadach (IFJ PAN, Krakow) NLO corrections in the parton shower Monte Carlo HP2 at CGG, Sept.2014 6 / 29

Theoretical framewework of PS MC: Collinear Factorization ◮ What is collinear factorization? F bare ( q h /µ, ε ) = σ Bare C ( ∞ ) � α, q h � � ⊗ Γ ( ∞ ) = ladder ( α, ε ) σ Born µ Ladders ⊗ in lightcone x and parton type, Γ inclusive, C can be kept unintegrated/exclusive. ◮ Physical distributions: Γ → PDF . LO example: C [ 1 ] ( q h /µ ) ≡ F [ 1 ] F Phys . = [ ✶ + C [ 1 ] ( α, q h /µ )] ⊗ PDF ( µ ) , bare ( q h /µ, ε ) − Γ [ 1 ] ( ε ) F Phys . factor. scheme independent; both C and PDFs are dependent: Γ [ 1 ] ( ε ) → Γ [ 1 ] + ∆Γ [ 1 ] , C [ 1 ] → C [ 1 ] − ∆Γ [ 1 ] , ∆ C [ 1 ] = − ∆Γ [ 1 ] . ◮ Evolution of F and/or PDFs and evolution kernels: ∂ P = α P [ 0 ] + α 2 P [ 1 ] + ... = Res 1 Γ( ε ) = ∂ ln ⊗ C ( q /µ ) ∂ ln µ 2 F ( µ ) = P ⊗ F ( µ ) , ∂ ln µ 2 S. Jadach (IFJ PAN, Krakow) NLO corrections in the parton shower Monte Carlo HP2 at CGG, Sept.2014 6 / 29

Theoretical framewework of PS MC: Collinear Factorization ◮ What is collinear factorization? F bare ( q h /µ, ε ) = σ Bare C ( ∞ ) � α, q h � � ⊗ Γ ( ∞ ) = ladder ( α, ε ) σ Born µ Ladders ⊗ in lightcone x and parton type, Γ inclusive, C can be kept unintegrated/exclusive. ◮ Physical distributions: Γ → PDF . LO example: C [ 1 ] ( q h /µ ) ≡ F [ 1 ] F Phys . = [ ✶ + C [ 1 ] ( α, q h /µ )] ⊗ PDF ( µ ) , bare ( q h /µ, ε ) − Γ [ 1 ] ( ε ) F Phys . factor. scheme independent; both C and PDFs are dependent: Γ [ 1 ] ( ε ) → Γ [ 1 ] + ∆Γ [ 1 ] , C [ 1 ] → C [ 1 ] − ∆Γ [ 1 ] , ∆ C [ 1 ] = − ∆Γ [ 1 ] . ◮ Evolution of F and/or PDFs and evolution kernels: ∂ P = α P [ 0 ] + α 2 P [ 1 ] + ... = Res 1 Γ( ε ) = ∂ ln ⊗ C ( q /µ ) ∂ ln µ 2 F ( µ ) = P ⊗ F ( µ ) , ∂ ln µ 2 S. Jadach (IFJ PAN, Krakow) NLO corrections in the parton shower Monte Carlo HP2 at CGG, Sept.2014 6 / 29

Collinear Factorization – Fixed order calculations ◮ Fixed order calculation (like MCFM): C [ 1 ] ≡ [ F [ 1 ] F ( q h ) = [ ✶ + C [ 1 ] ] J ⊗ PDF ( µ ) , bare ( q h /µ, ε ) − Γ [ 1 ] ( ε )] q h = µ [ ... ] J means experimental acceptance J ( x , y ) kept in integrand. ◮ Typical example: ISR gluonstrahlung part of DIS, def. y = q / q h ∈ ( 1 , 0 ) : � ¯ C [ 1 ] ( z , y ) = δ z = 1 δ y = 0 ( 1 +∆ SV )+ C F α � 1 � P ( z ) � + + β ( z , y )+ δ y = 0 Σ( z ) π y 1 − z + � ¯ ¯ ( 1 − z ) 2 � β ( z , y ) = | ME exact | 2 − C F α 1 P ( z ) Σ( z ) = C F α P ( z ) 1 − z , π y π 1 − z z + where ¯ P ( z ) = ( 1 − z ) P qq ( z ) = ( 1 + z 2 ) / 2. ◮ Soft-collinear counterterm technique (eg. Catani-Seymour) often used to facilitate computing codes (MCFM): � ¯ C SC ( z , y ) = C F α 1 P ( z ) � C [ 1 ] = [ F [ 1 ] bare − C SC ] d = 4 +[ C SC − Γ [ 1 ] ] d � = 4 , y 1 − 2 ε π 1 − z + S. Jadach (IFJ PAN, Krakow) NLO corrections in the parton shower Monte Carlo HP2 at CGG, Sept.2014 7 / 29