Combining fixed order QCD calculation with the parton shower Monte Carlo new PV prescription for IR singularities

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1 Combining fixed order QCD calculation with the parton shower Monte Carlo new PV prescription for IR singularities O. Gituliar, S. Jadach, A. Kusina, W. Płaczek, S. Sapeta, A. Siódmok, M. Skrzypek Partly supported by grants: DEC-0/03/B/ST/063, UMO-0/04/M/ST/0040, PITN-GA (LHCPhenoNet), DE-FG0-3ER4996 The first Symposium of the Division for Physics of Fundamental Interactions of the Polish Physical Society Various Faces of QCD Kielce, M. Skrzypek (IFJ PAN, Krakow) Kielce, / 4

2 Plan Introduction The KrkMC project MC-friendly NLO kernels New Principal Value prescription Results for NLO kernels The Axiloop package Summary M. Skrzypek (IFJ PAN, Krakow) Kielce, / 4

3 (Very) Long-term perspective The NNLO + NLO Parton Shower for LHC LO Hard process + LO Shower Pythia, Herwig (980-s) NLO Hard process + LO Shower MC@NLO, PowHEG (000-s) NLO Hard process + NLO Shower KrkMC (Jadach et.al., ongoing) NNLO Hard process + NLO Shower???????????????? Other developments: MINLO, Z. Nagy et.al., H. Tanaka et.al.... M. Skrzypek (IFJ PAN, Krakow) Kielce, / 4

4 The KrkMC project Based on collinear factorization. Requires: Reformulation of factorization in fully exclusive way Recalculation of the evolution kernels exclusive in four dimensions well defined relation to MS-bar Kinematical mappings Reweighting procedure (positive, convergent) Axial gauge instrumental allows for physical interpretation Here we discuss recalculation of the NLO evolution kernels real-real ones [JHEP 08 (0) 0] virtual-real ones HERE[PL B73 (04) 8-, < arxiv: ] M. Skrzypek (IFJ PAN, Krakow) Kielce, / 4

5 Collinear factorization and construction of the evolution kernel qq Z+nG Z LO cascade and construction of the ladder G q q qq Z Z Include NLO and group graphs in the ladder into "two-particle-ireducible" sets K q K0 K0 K0 K0 C 0 K 0 K 0 q K 0 K 0 Use "projection operators" P to split the ladder and extract kernels [ ˆn ] Γ qq = Tr 4nq K ˆp C K K K K P P P P q q M. Skrzypek (IFJ PAN, Krakow) Kielce, / 4

6 NLO-corrected Hard process γ * Born quark gluon W γ * q = g γ * + γ * quark gluon st order corrections Virt The KrkMC project Alternative to MC@NLO and POWHEG proton... WMC NLO = X n,m=0 ( n n m n X + j= n n j m mx + r= n n r m ) W NLO MC = + S+V + X j F β (ŝ, ˆp F, ˆp B ; a j, z Fj ) P(z Fj ) dσ B (ŝ, ˆθ)/dΩ + X j B β (ŝ, ˆp F, ˆp B ; a j, z Bj ) P(z Bj ) dσ B (ŝ, ˆθ)/dΩ, M. Skrzypek (IFJ PAN, Krakow) Kielce, / 4

7 gluon NLO-corrected middle-of-the-ladder kernel, CF The KrkMC project γ * quark gluon st order corrections quark Virt proton Born D [] B (x, Q) = e S ISR ( X n=0 x n n + nx p= n n W p + nx p X p= j= = + n p j ) = e S ISR ( δ x= + X ny Z ) nx nx p + d 3 () η i ρb (k i ) + β () 0 (zp) + X W ( k p, k j ) δ Q x= nj= xj. n= i= Q>a i >a p= p= j= i Prototype MC code based on the KrkMC scheme works! M. Skrzypek (IFJ PAN, Krakow) Kielce, / 4

8 Calculation of Monte Carlo friendly kernels Contributions to Non Singlet P qq kernel M. Skrzypek (IFJ PAN, Krakow) Kielce, / 4

9 PV prescription Axial gauge = physical interpretation as parton shower Axial gauge = spurious (unphysical) singularities ( gluon propagator: l g µν lµ n ν + n µ l ν ) nl Spurious singularities cancel in full set of diags, but need regularization Curci, Furmanski, Petronzio [80] Ellis, Vogelsang [96], Heinrich, Kunszt [97]: [ ] Principal Value: nl = nl PV (nl) + δ (pl) PV is more like "phenomenological rule" Rigorous prescription: Mandelstam [83], Leibbrandt [84]. Difficult in calculations: Bassetto, Heinrich, Kunszt, Vogelsang [97]. Side remark: Linear denominators used in NNLO calculations of rapidity distributions of EW bosons [Anastasiou, Dixon, Melnikov, Petriello, 004] d m kp k δ u d m kp k kp k(p up ) i0 c.c. M. Skrzypek (IFJ PAN, Krakow) Kielce, / 4

10 Problem with real emission graph Standard [Heinrich, Kunszt, 998]: [ N(ɛ, Q Pqq (x) P qq (x) ) ɛ 3 I 0 ɛ + p ( qq(x) I + 4I 0 + I 0 ln x I 0 ln( x)) ] ( ) + O ɛ ɛ /ɛ 3 bad for Parton Shower, needs Real-Virt. cancel. Parton Shower oriented [Jadach et.al. 0]: p qq (x) ɛ ( +I + 4I 0 + I 0 ln x I 0 ln( x) ) ( ) + O ɛ Good for PS, δ is a cut-off in 4-dimensions, easy to generate in MC: I 0 = 0 dx [x] PV δ dx x = ln δ, I = 0 dx ln x [x] PV ln δ, P qq(x) = p qq + ɛ( x), p qq = + x x M. Skrzypek (IFJ PAN, Krakow) Kielce, / 4

11 New use of PV prescription Standard: regularize with PV only the gluon propagator leave other singularities in (+)-component of integration momenta d m l l ɛ +, l + = nl np New proposal: regularize with PV all singularities of the integrand in (+)-component of integration momenta, real & virtual d m [ ] l ( l+ ɛ d m l + ɛ ln l + + ɛ ) l + PV ln l All (+)-singularities cancel in the final expression (kernel), so extension of "phenomenological PV rule" of Curci-Furmanski-Petronzio possible M. Skrzypek (IFJ PAN, Krakow) Kielce, / 4

12 Example: virtual three point integral Must perform (+)-integral as the last one, Ellis, Vogelsang [996]: d m l f (l + ) (π) m l (l q) (l p) = = i ( 4π ) ɛ Γ( ɛ) 6π q q ɛ + Γ ( + ɛ) ( x) ɛ Γ( + ɛ) x [ x dyf (l + ) zɛ ( z) ɛ ( + ɛ ln y ) y z 0 dyf (l + )( y) +ɛ ], x = q +/p +, y = l +/p +, z = y/x, p = (p q) = 0, m = 4 + ɛ, Singularities at y = 0 and y = x: only from gluon propagator. Singularity at y = : not from gluon propagator! Proposal: treat all (+)-singularities on equal footing Note: (+)-singularities lead to /ɛ 3 poles in kernel M. Skrzypek (IFJ PAN, Krakow) Kielce, / 4

13 Example: scalar non-axial integral kinematics: p = (p q) = 0 J3 F = d m l (π) m l (q l) (p l) The PV prescription: ( J3 F = C ) ɛ + π, C = i 6 Γ( ɛ) (4π) q ( ) 4π ɛ q New PV prescription: ( J3 F =C I ) 0 + ln( x) 4I + I 0 ln( x) + ln ( x), ɛ ɛ replaced by ln δ and ɛ ln δ M. Skrzypek (IFJ PAN, Krakow) Kielce, / 4

14 NLO kernels P qq and P gg in New PV scheme There are only four graphs with /ɛ 3 singularity: Γ (d R) qq (x, ɛ) Γ(d V ) qq (x, ɛ) Γ(d R ) gg (x, ɛ) Γ(d V ) gg (x, ɛ) related to P qq and P gg splitting functions in a standard way: Γ qq(gg) (x, ɛ) = δ x + ( αs ɛ π PLO qq(gg) (x) + ( αs ) ) P ( NLO ) qq(gg) π (x) O ɛ M. Skrzypek (IFJ PAN, Krakow) Kielce, / 4

15 Virtual contribution to NLO P qq kernel in NPV The virtual graph contributes: Γ (d V ) qq (x, ɛ) = ɛ P qq( + ɛ ln( x) ) ZdV [ ɛ p ( ) qq I 0 ln x + ln( x) 6I Li ( x) + ln x ] π + + x ɛ x, Z dv = 4I 0 + ln( x) + ln x 3, Inclusive sum of Real and Virtual graphs identical as in standard PV scheme Γ (d) qq M. Skrzypek (IFJ PAN, Krakow) Kielce, / 4

16 Contributions to inclusive kernel P qq in NPV prescr. Columns "SUM" agree with standard PV scheme M. Skrzypek (IFJ PAN, Krakow) Kielce, / 4

17 Exclusive contributions to P qq Part C F : { ( ( k αs Γ( ɛ) ) ɛ C F (4π) ɛ k ɛ 4 ln x µ ) P qq R ( ) ( k ) ɛ } + p qq 4 Li( x) ( x) + ( + x) µ R Parts C F C A, C F T F : α S C F Γ( ɛ) { (4π) ɛ k ɛ [ C A 3 T 4 F 3 4 C ( ) ( k ) ɛ ] A ln( x) + I0 + 4 C A [ p qq ( Li() Li( x) + I 0 ln( x) I ) x Monte Carlo friendly µ R ] ( k ) ɛ } µ R P qq M. Skrzypek (IFJ PAN, Krakow) Kielce, / 4

18 Real contribution to NLO P gg kernel in NPV Only ɛ poles the calculation can be done in 4-dimensions, much simpler than in standard PV The real graph in New PV prescription: ( gg (x, ɛ) = C (d R) ( ) S [P gg 4I + 4I 0 ln( x) ln(x) ɛ Γ (d R) + ln ( x) + ln (x) 4 ln(x) ln( x) 8 ln( x) + ) ( ln(x) + π ln(x) 3 x x ) 3x 3 x x x ]. M. Skrzypek (IFJ PAN, Krakow) Kielce, / 4

19 Virtual contribution to NLO P gg kernel in NPV The virtual graph in New PV prescription: [ gg,npv (x, ɛ) = C(d V ) ( S P gg + ɛ ln( x) V ) Z ɛ GS + ( 4I 0 ln( x) + 8I 0 ln(x) 6I + 4 ln (x) ɛ + π 6 34 )] C (d V ) S 9 3ɛ x Z GS V =I ln( x) + 4 ln(x) 3, Γ (d V ) Inclusive sum of Real and Virtual graphs identical as in standard PV scheme Γ (d) gg Both schemes, PV and New PV, give the same P qq and P gg kernels M. Skrzypek (IFJ PAN, Krakow) Kielce, / 4

20 Axiloop Fully automated package for symbolic calculation of NLO kernels in axial (light-cone) gauge Written in Mathematica Can be installed from here M. Skrzypek (IFJ PAN, Krakow) Kielce, / 4

21 The Axiloop package Library of integrals, in PV and NPV prescriptions One-loop integration and renormalisation (keeping track of the UV and IR poles) M. Skrzypek (IFJ PAN, Krakow) Kielce, / 4

22 The Axiloop package One-particle FS integration M. Skrzypek (IFJ PAN, Krakow) Kielce, / 4

23 Axiloop - example Input Exclusive kernel (bare) M. Skrzypek (IFJ PAN, Krakow) Kielce, / 4

24 Summary The NLO parton shower project KrkMC advances: Factorization theorem modified Kinematical mappings formulated Weights defined New, MC-friendly NLO kernels calculated (P qq ): New PV prescription proposed and tested /ɛ 3 poles replaced by (/ɛ) ln δ etc. Inclusive real+virtual components agree with standard PV results Prototype MC code based on KrkMC scheme already works M. Skrzypek (IFJ PAN, Krakow) Kielce, / 4

arxiv:1008.4792v2 [hep-ph] 20 Jun 2013

arxiv:1008.4792v2 [hep-ph] 20 Jun 2013 A Note on the IR Finiteness of Fermion Loop Diagrams Ambresh Shivaji Harish-Chandra Research Initute, Chhatnag Road, Junsi, Allahabad-09, India arxiv:008.479v hep-ph] 0 Jun 03 Abract We show that the mo